Master’s thesis in Master Program Structural Engineering and Building Technology by Sara Eidenvall

Conceptual design and development of preliminary calculation methods for wood-wood connections Through design of two visible connections

Master’s thesis in Master Program Structural Engineering and Building Technology Sara Eidenvall & Erica Samuelsson

Department of Architecture and Civil Engineering Chalmers University of Technology Gothenburg, Sweden 2021 www.chalmers.se

Master’s thesis ACEX30

Conceptual design and development of preliminary calculation methods for wood-wood connections Through design of two visible connections

Sara Eidenvall Erica Samuelsson

Department of Architecture and Civil Engineering Division of Structural Engineering Wood structure Chalmers University of Technology Gothenburg, Sweden 2021

Conceptual design and development of preliminary calculation methods for wood-wood connections Through design of two visible connections

Master’s Thesis in the Master’s Programme Structural Engineering and Building Technology © SARA EIDENVALL, 2021 © ERICA SAMUELSSON, 2021 Supervisor: Robert Jockwer (CTH), Elzbieta Lukaszewska (Sweco AB) and Eric Rundqvist (Sweco AB) Examiner: Robert Jockwer Department of Architecture and Civil Engineering Division of Timber Engineering Research Group for Light-Weight Structures Chalmers University of Technology SE-412 96 Göteborg Sweden Telephone: + 46 (0)31-772 1000

Name of the publisher or Department of Architecture and Civil Engineering Gothenburg, Sweden, 2021

Cover. Illustration of one analysed failure mode and a resulting connection.

Conceptual design and development of preliminary calculation methods for wood-wood connections Through design of two visible connections

Master’s thesis in the Master’s Programme Structural Engineering and Building Technology SARA EIDENVALL ERICA SAMUELSSON Department of Architecture and Civil Engineering Division of Structural Engineering Research Group for Light-Weight Structures Chalmers University of Technology

Erica Samuelsson

Sara Eidenvall

Chalmers University of Technology August 2019 - January 2021 Msc, Structural Engineering and Building Technology

Buro Happold Engineering January 2018 - July 2018 Industrial placement

DTU - Technical University of Denmark January 2020 - July 2020 Msc, Structural Engineering and Building Technology

Chalmers University of Technology January 2017 - January 2019 Msc, Architecture and Urban Design

Kungsbacka Municipality Plan and Construction May 2019 - September 2019 Internship

École Nationale Supérieur d’Architecture de Lyon August 2016 - January 2017 Msc, Architecture and Urban Design

Sweco Architects June 2017 - October 2017 Internship

Chalmers University of Technology August 2013 - June 2016 Bsc in Architecture and Engineering

Chalmers University of Technology August 2015 - April 2019 Bsc in Architecture and Engineering

ABSTRACT The method is designed by a literature review of current methods for carpentry joints. Different designs are compared against each other and evaluated to specify the method for the two specific connections. By using the construction of a timber building in Borlänge the method and thesis is narrowed down to a Swedish context. The method itself is, due to this, adapted for modern construction even more by utilizing the timber engineered materials glulam and CLT, which are the structural materials used in the building.

Timber structures get more and more common in the Swedish context of construction. With an increasing interest in new ways of building sustainably, and with structures getting a more relevant role in the visual architecture of a building, the connection of structural members hold an important role. By constructing these connections of only wood instead of more commonly with metal screws, nails and plates, a more sustainable solution can be investigated. This also gives way for new solutions with a different architectural appearance, similar to the more traditional carpentry joints.

The calculation method in the end is transformed to illustrate a more general approach. Where it can be used for more connections than the two utilized in this study. The method is a good starting point for increasing the interest in using carpentry connections as simple and tangible results can be calculated. Although for an holistic approach, and as a future step, more elements of construction need to be regarded for the connections to be fully plausible in design.

Despite many advantages of using these types of connections, both due to the technological progress of manufacturing and the interest due to sustainability, amongst others, they are very rarely used in a building context today. One of the main problems that cause this is the lack of information as well as the lack of calculation methods and guidelines. Carpentry connections are today referred to as an historical connection. By designing two carpentry joints the aim is to investigate a method of design and open up the possibility of using these types of connections in a modern context. Key words: Wooden joinery Timber Carpentry joints Connections EC5 Preliminary calculation method Glulam CLT (Cross-laminated timber)

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INDEX Abstract i Index ii List of figures

List of tables

vii

Preface viii Abbreviations ix 1. Introduction

Thesis background �� 2 Thesis question �� 3 Aim and Objectives �� 3 Delimitations �� 3 Method �� 4 Reading instructions �� 6

2. Project background

Project background �� 8 Context �� 8 Focus �� 8 Connection one, C1 �� 9 Geometry ��10 Loads ��11 Material ��11 Connection two, C2 ��12 Geometry ��12 Loads ��13 Material ��13

3. Research

Wood as a material ��15 Building with wood ��15 Engineered timber products ��15 Modern manufacturing ��15 Wood anatomy ��16 Moisture, fire and acoustics ��16 Design of connections ��17

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Types of joints ��17 Traditional wooden joinery and library of connections ��18 C1: Glulam - Glulam ��19 C2: CLT - CLT ��20 Reference projects ��20 Vidy theatre ��20 Tamedia office building ��20 SWG Production hall ��21 X-Fix ��21 Traditional carpentry joints in Sweden ��22 Chapter conclusion ��22

4. Architectural design

First iteration ��24 C1 ��25 C2 ��25 Second Iteration ��27 C1 ��27 C2 ��27 Evaluation ��27 Criteria ��29 C1 ��30 C2 ��31 Chapter conclusion ��31

5. Analysis

FE-Modelling ��34 Input C1 ��34 Input C2 ��34 Analysis ��38 Preliminary calculations ��46 Method investigation ��46 Additional geometrical effect ��51 Chapter conclusion ��52

6. Preliminary calculation method

7. Preliminary Proposal

Design verifications ��91 C1 ��91

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C2 ��92 Prototypes ��93 Design Selection ��95 C1 ��95 C2 ��96

8. Discussion and conclusion

Appendix A

APPENDIX A1 �� II APPENDIX A2 �� VIII APPENDIX A3 �� X APPENDIX A4 �� XIII

Appendix B

APPENDIX B1 �� XXI APPENDIX B2 �� XXX APPENDIX B3 �� XXXIV APPENDIX B4 ��XLV

Appendix C

LVIII

APPENDIX C1 �� LIX APPENDIX C2 �� LXIV

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IMAGE LIST OFLIBRARY FIGURES Figure 1.1. Illustrative scheme of project method. �� 5 Figure 2.1. Situation plan, scale 1:500 with building index (Drawing credit: AIX Architecture). �� 8 Figure 2.2. Floor plan, Level 1, Building B and C, Scale 1:400 (Drawing credit: AIX Architecture). �� 9 Figure 2.3. Illustration of window geometry for connection, C1. �� 10 Figure 2.4. Zoomed in illustration of location for C1. �� 11 Figure 2.5. Illustration of main wind load direction and load-bearing walls (Drawing credit: AIX Architecture). �� 11 Figure 2.6. Illustration of stair geometry. �� 12 Figure 2.7. Zoomed in illustration of location for C2. �� 13 Figure 3.1. Illustration of trees a. deciduous, b. coniferous. �� 16 Figure 3.2. Phenomenon of the charring process and fire penetration zones of wood. �� 17 Figure 3.3. Wedged dovetail joint with a half-shoulder. �� 18 Figure 3.4. Pinned tenon and mortise joint. �� 18 Figure 3.5. Reverse dovetail through, pinned tenon and mortise joint. �� 19 Figure 3.6. Notched joint. �� 19 Figure 3.7. Standard connections for glulam. �� 19 Figure 3.8a-d. Standard connections for CLT members. �� 20 Figure 3.9a. Structure details of Vidy Theater (EPFL, 2020) b. Assembly the roof to the building (EPFL, 2020). �� 20 Figure 3.10. Structure of the Tamedia Office Building (Shigeru Ban Architects, 2014) �� 21 Figure 3.11. Illustrations of the "Puzzle Connection" and carpentry joint in the framework system. �� 21 Figure 3.12. Two dovetail-shaped and tapered coupling strips, X-fix L (X-fix, retrieved 2021). �� 22 Figure 3.13. Connections Pelarne Church in Småland. �� 22 Figure 4.1. Process illustration for the Architectural design. �� 24 Figure 4.2a. Location for C1. b. Location for C2. �� 24 Figure 4.3. Examples of excluded connections for C1 after the first iteration. �� 25 Figure 4.4. Examples of some continued interesting alternatives for C1 after the first iteration. �� 26 Figure 4.5. Examples of excluded alternatives for C2 after the first iteration. �� 26 Figure 4.6. Examples of some continued interesting alternatives for C2 after the first iteration. �� 27 Figure 4.7. Extract from Appendix - A3. Example of centred tenon connections for C1 with small differences. �� 28 Figure 4.8. Extract from Appendix - A3. Example of a part cross-section insertion connections for C2 with small dif-

ferences. �� 28 Figure 4.9. An extract from Table 1 in Appendix A4 of the grading for C1. �� 30 Figure 4.10. An extract from Table 2 in Appendix A4 of the grading for C2. �� 30 Figure 4.11a. Plane dovetail full depth, PD1. b. Plane dovetail half depth, PD2. �� 31 Figure 4.12a. Tenon and mortise with thick peg, P1. b. Tenon and mortise with thin peg, P2. �� 31 Figure 4.13a. Tenon and mortise, TM1. b. Top notch, N1. c. Bottom notch, N2. �� 32

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Figure 4.14. Rounded dovetails 1-5. a. RD1. b. RD2. c. RD3. d. RD4. e. RD5. �� 32 Figure 5.1. Boundary conditions and mesh definition, Abaqus, C1. �� 35 Figure 5.2. Boundary conditions and mesh definition, Abaqus, C2. �� 37 Figure 5.3. Default load case for C2. �� 38 Figure 5.4. Average bending stress for dovetail joints, RD1, RD2 and RD3 for tf and bf. �� 39 Figure 5.5. Geometry comparison for RD1-RD3. �� 39 Figure 5.6. Illustration of paths to consider bending for RD1. An average value are calculated from these paths. �� 39 Figure 5.7. Geometry comparison for TM1, N1 and N2. �� 40 Figure 5.8. Average bending stress for tenon and mortise, TM1, notches, N1 and N2 for tf and bf. �� 40 Figure 5.9. Average shear stress in the tenon/notch for a. TM1, b. N1, c. N2. �� 41 Figure 5.10. Average axial stress in the tenon/notch for a. TM1, b. N1, c. N2 along the z-axis. �� 41 Figure 5.11. Illustration of paths to consider shear for TM1 where an average value are calculated from these paths. 42 Figure 5.12. Stress-strain-curve of clear wood, exposed to tensile and compression stresses (perp. to the grain - dashed

line) and (parallel to the grain - solid line). �� 42 Figure 5.13. Geometry comparison for RD1 and N1. �� 43 Figure 5.14. Average shear stress for N1. �� 43 Figure 5.15. Average shear stress for RD1. �� 43 Figure 5.16. Average shear, tension and compression stress for RD1 and N1. �� 43 Figure 5.17. Geometry comparison for C2, RD3-5. �� 44 Figure 5.18. Shear stress at the same location in DT1 and DT2 for the geometry of RD4. �� 44 Figure 5.19. Geometry comparison for C1, PD2 and C2, N2. �� 45 Figure 5.20. Average shear stress for PD2. �� 45 Figure 5.21. Average shear stress for N2. �� 45 Figure 5.22. Relevant geometry of Notched joint b. top notch, c. bottom notch and a. Tenon joist. �� 46 Figure 5.23. Comparing the two functions that could be used for a bottom notch. �� 47 Figure 5.24. Geometry for comparing the different positions of a tenon. �� 47 Figure 5.25. Shear capacity for different positions of a tenon and a notch. �� 48 Figure 5.26. Relevant geometry of Dovetail joint. �� 48 Figure 5.27. Shear capacity at the notch for a dovetail with 5 different methods. �� 50 Figure 5.28. Shear capacity for a dovetail header comparing two different methods. �� 50 Figure 5.29. Reduction factor, kv for different ratio of lB/hB depending on alpha. �� 51 Figure 5.30. Shear stress mortise depending on the width of the header bH. �� 51 Figure 5.31. Relevant geometry definitions of the header. �� 52 Figure 7.1. Model photography of connection PD1 for C1. �� 96 Figure 7.2. Model photography of connection P1 for C1. �� 96 Figure 7.3a-b. Model photographies of connection RD1 for C2 in two different views. �� 96 Figure 7.4a-b. Model photographies of connection N2 for C2 in two different views. �� 97

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Figure 7.5. Illustration of preliminary connection PD1. �� 98 Figure 7.6. Illustration of preliminary connection P1. �� 98 Figure 7.7. Illustration of preliminary connection RD1. �� 99 Figure 7.8. Illustration of preliminary connection N2. �� 99 Figure 7.9. Assembly of all four selected connections. a. PD1, C1, b. P1, C1, c. RD1, C2, d. N2, C2. �� 100

List of Tables LIST OF TABLES Table 2.1. Geometry C1. �� 11 Table 2.2. Loads C1. �� 11 Table 2.3. Geometry C2. �� 13 Table 2.4. Loads C2. �� 13

Table 3.1. Change in the properties with a one percent change in wood moisture. The benchmark figure is the properties at a moisture content of 12 % (Blaß & Sandhaas, 2017). �� 17 Table 5.1. Glulam engineering constants. �� 34 Table 5.2. CLT engineering constants. �� 36 Table 5.3. Investigative load cases for C2 �� 38 Table 5.4. Equations for verifying a dovetail joist in shear and tension perpendicular to the grain. �� 49 Table 7.1. A summary of the utilization rate for the different connections for C1. �� 94 Table 7.2. A summary of the utilization rate for the different connections for C2. �� 95 Table 7.3. A summary of the strengths and weaknesses of each prototype. �� 97

vii

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PREFACE In this study two carpentry connections have been designed with the intent of developing a general calculation method for carpentry joints in modern construction. The study was performed from September 2020 to January 2021.

University of Technology and Elzbieta Lukaszewska and Eric Rundqvist at Sweco AB. Your feedback and expertise within the subject of timber construction has been greatly appreciated and this thesis would not have been possible without your guidance.

The work has been carried out at both the Division of Structural Engineering, Department of Architecture and Civil Engineering at Chalmers University of Technology, Sweden and at Sweco AB, Structures division in Gothenburg, Sweden.

We would also like to thank AIX Architecture who together with Sweco AB let us use their project in Bältartäppan 7 as a reference project to carry out the study on. Finally we would like to thank our fellow students who has helped us with reading through drafts giving us feedback during our process, with special thanks to Alexander Angrén and Ola Sjöberg.

We would like to thank our three supervisors in the project. Robert Jockwer, Assistant Professor at the Division of Structural Engineering at Chalmers

Sara Eidenvall och Erica Samuelsson Gothenburg, January 2021

viii

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ABBREVIATIONS avg.

Average

P1, 2

TM with a peg joint, 1, 2

b.c

Boundary condition

PD1, 2

Plate dovetail joints, 1, 2

b.f.

Bottom fibre

perp.

Perpendicular

b.n.

Bottom notch

RD1-5

Rounded dovetail joint, 1-5

b.t.

Bottom tenon

s-s

Side-side

Connection one

Connection two

S-ZZ

The stress on the z-face in the z-direction

Centred

t.f.

Top fibre

c-c

Center-centered

t.n.

Top notch

c.f.

Centred fibres

Tenon and Mortise joint

c.t.

Central tenon

CLT

Cross Laminated Timber

CNC

Computer Numerical Control

Dovetail joint

EC5

Eurocode 5

eq.

Equation

LC1

Load Combination 1

LC2

Load Combination 2

l.t.

Lower tenon

N1, 2

Notched joint, 1, 2

min.

Minimum

max.

Maximum

m.f.

Middle fibre

n.c.

Notch corner

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1. INTRODUCTION

”A structure is a constructed assembly of joints separated by members” - McLain, 1998

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

By trying to reduce this steel one can decrease the CO2 emissions substantially since the amount of energy it takes to produce a ton of steel is 24 times as high as for a ton of wood (Mayo, 2015).

THESIS BACKGROUND Timber is an increasingly used material for the construction of buildings, both large and small. Despite its decline over the last centuries with the immersion of new materials, such as steel and concrete, and its relationship to fire, it has now taken an upswing due to its many environmental advantages and aesthetics. Architects and engineers are becoming more and more conscious of the structure of a building as something to show rather than hide which opens up the discussion of joints as both structural elements and architectural features.

When looking into the historic examples of carpentry joining, many of the ancient timber structures using these still stand today which is a perfect example of the durability of this material when cared for (Gustafsson et al., 2019). There are also a few modern examples where the amount of steel in joints have been reduced to a minimum, such as the Tamedia office building in Zurich. The technological development of manufacturing is also a resource which introduces many possibilities of using new materials, such as CLT and glulam. Creating geometrical connections using for example CNC machinery is one of these advancements. For many cases though, the guidelines for connections that uses very little or no steel are very few and does not promote the use of these connections.

For many years fire has been the main issue when using timber as a construction material and it was not until the mid 90's that laws for functional requirements in fire regulations were instituted to help increase the use of wood once more. Today, few engineers are not aware of the advantages of building with wood, as it is the only main construction material that reduces the amount of CO2 in the air during its service life. When it comes to the question of sustainability though a main issue is that of how it affects deforestation. For many countries this is an issue that needs to be evaluated, but by looking into the Swedish local context the forest industry is continuously increasing with a yearly net growth of forest mass. Therefore, by sourcing timber locally, wood as a construction material is not likely to harm the forest sustainability in Sweden (Gustafsson et al., 2019).

Today’s technology and market for timber, especially in Sweden, brings a lot of possibilities and so far is not fully utilized. By embracing the old ways of carpentry jointing in a modern way and clarifying preliminary calculation methods, these resources can be used in a way to promote building in wood even further.

The amount of built timber structures is increasing in Sweden as the demand and resources exists (Trähusbarometern, 2020). Although, when it comes to the joining of structural members a lot of steel is still used in timber construction. 2

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THESIS QUESTION

How can a modern carpentry design approach be developed? How can a preliminary calculation method be designed for carpentry joints? How can this be adapted for glulam and CLT? AIM AND OBJECTIVES

DELIMITATIONS

The aim of this master thesis is to present a possible preliminary calculation method for timber joints only made of wood. This is made to elevate the living and renewable material of wood and hopefully increase the use of these types of carpentry connections within construction.

How to structurally solve two visible woodwood connections for the ongoing project in Borlänge, is the main focus in this project. The structural aspects are limited for the load cases and initial stresses for the chosen design proposals. The calculation method will be created for the designed connections and similar connections for future work. The live load is specifically designed for the building's purpose. Wind loads are based on the location, Borlänge.

Some of the objectives to be able to fulfil this aim are:

-- Compiling a resource library of possible carpentry joint techniques.

The characteristics of moisture, acoustic and fire are certainly important and must be further described and examined in a next research step to meet all demands and requirements. These factors will be briefly mentioned, how they affect the connection design in general, and to some extent the designed connections, but will not particularly be part of this study nor calculated.

-- Compile design proposals for two wood-wood connections.

-- Investigate the important failure modes --

of the calculation method through the use of simple FE-models. Propose a preliminary calculation method for these designed connections by investigating existing calculation methods. Discussing future features of the design of carpentry connections apart from the calculations, to fulfil a holistic view of the design.

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METHOD

DESIGN PHASE

GENERAL

Preliminary design of the connections are developed by using references and by sketches of the design. The iterations of the designs are preliminary made by evaluating how reasonable they are and their grading with regards to certain factors. From this grading two types of designs from each connection is brought into the next phase of developing a preliminary calculation method.

The main method of the thesis is to design two wood-wood connections for an office building in Borlänge. This will be done first with a research step of historical ways of solving these types of connections. This will also include modern ways of solving these as well as research into methods of manufacturing and calculations. The second part of the thesis is the design phase and the development of a preliminary calculation method that will go hand-inhand as the design will work as a basis for the calculation method and the method will work as a process step in iterating the design solutions. The preliminary design will be evaluated in simple FE-models to distinguish important failure modes of the design before finally designing the general calculation method.

ANALYSIS In this phase the designed connections are analysed in simple FE-models in Abaqus in order to identify possible failure modes. In this analysis minor modifications are made to the designs (to create several options for one connection type) in order to investigate the influence of geometry as well. PRELIMINARY CALCULATION METHOD

RESEARCH

From the literary review and the analysis the general calculation method is developed. The method is then used in order to analyse the chosen connections in a final step.

The research phase include mainly the research of historical carpentry connections as references for the design. It also includes some references of contemporary projects where carpentry joints have been used as a support for the design as well as the background of the thesis question. Other relevant parts of the research is the information on modern timber connections that normally use steel reinforcement or connectors as well as research into how the built structure is influenced by the wood-wood connections with regards to moisture, fire and acoustics.

CHOICE OF PRELIMINARY DESIGN As a final design step prototypes are made of some selected connections to identify possible weaknesses of the connections beside the structural verifications. Together with these and results from calculations two different solutions are selected for the preliminary design of each connection.

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Research

Design iterations

FE-analysis

Preliminary calculations research

Development of calculation method

Geometrical investigations

Development of models

Development of calculation method

Calculation on connections

Finalizing report

Final discussion Figure 1.1. Illustrative scheme of project method.

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READING INSTRUCTIONS With regard to references and chapter numbering Chapter 6 which is the preliminary calculation method has its own index and numbering to be able to stand on its own outside of this report. For all references outside of Chapter 6 that refers to a chapter or equation stated within Chapter 6 will be explicitly put. The same works for referencing inside Chapter 6 to other parts of the report. The reference will then mention the report. If nothing explicit is mentioned the reference is to the same part as one is in. For example:

Regarding page numbering, Chapter 6 also includes an internal page numbering in order to be able to stand on its own outside of the report. The internal page numbering is illustrated in black and the corresponding report page numbering on the same page in round brackets and grey, as illustrated below. (Report nr.)

Internal page nr.

Ex. (58)

Reading in Chapter 5: - The text refers to "eq. (5)" → eq. is found in the report. - The text refers to "eq. (5) in the general calculation method" → eq. is found in Chapter 6. Reading in Chapter 6: - The text refers to "Chapter 5.2" → Chapter is found in Chapter 6. - The text refers to "Chapter R.5.2" → Chapter is found in Chapter 5 in the report.

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2. PROJECT BACKGROUND

In this chapter the project utilized in the thesis is presented briefly together with the chosen connections. The connections are presented with their location in the building, their loads, materials and data that is needed for the preliminary calculation method.

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PROJECT BACKGROUND

FOCUS

CONTEXT

The challenge is to focus on two visible connections in the open and public areas. This is also made with a goal that the design solution of the connection should display that the connection is only made out of wood. This was decided with the intent to market the possibility of these type of connections as an architectural element as well as functional.

Sweco AB are currently planning and designing the structural system for an ongoing project based in Borlänge. They are working together with the architectural office AIX. The project is called Bältartäppan 7 (BT7) and consists of residences and offices divided into three volumes A, B and C as seen in the situation layout plan in Figure 2.1.

Figure 2.1. Situation plan, scale 1:500 with building index (Drawing credit: AIX Architecture).

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lD Figure 2.2. Floor plan, Level 1, Building B and C, Scale 1:400 (Drawing credit: AIX Architecture).

The location of the two selected connection locations can mainly be seen in building complex B and C in the entrance halls, as marked red in Figure 2.2.

CONNECTION ONE, C1 The first connection, C1, investigated is a glulam – glulam connection between a beam attached sideways to a pillar. These are the load-bearing slats inside the glass panes of the entrance halls of building B and C as seen in Figure 2.2. The horizontal slats are inserted at different heights in every other pillar opening. This makes the assembly easier as the joists can be attached

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to the pillar from the outside.

GEOMETRY The base cross-section for the glulam can be seen in Table 2.1.

An illustration of how the beams are connected to the glulam pillars and in relation to the rest of the building can be seen in Figure 2.3 and a zoomed in illustration of the connection can be seen in Figure 2.4.

The geometry used for calculating the load is the approximate geometry of building B, these values can be seen in Table 2.1 and their referred locations can be seen in Figure 2.2.

Figure 2.3. Illustration of window geometry for connection, C1.

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

Figure 2.4. Zoomed in illustration of location for C1. Figure 2.5. Illustration of main wind load direction and load-bearing walls (Drawing credit: AIX Architecture).

Table 2.1. Geometry C1.

Variable (relevance)

Value

Height (wind load)

11.5 m

Width Volume B

13.7 m

Length Volume B

21.4 m

Width, beam and pillar

90 mm

Depth, beam and pillar

225 mm

ccdistance, pillars

1000 mm

ccdistance, beams

1780 mm

lD (wind load)

≈ 2.76 m

lE (wind load)

≈ 2.23 m

Some of the loads are received from Sweco AB. Both the calculated loads and given loads can be seen in Table 2.2. The full load calculations for connection 1, C1, and connection 2, C2, can be seen in Appendix B. Table 2.2. Loads C1.

Load

Value

Self-weight glulam

4.3 kN/m3

In-plane wind load

2.44 kN/m

Out-of-plane wind load

-1.09 kN/m2

MATERIAL

LOADS

The material used in this connection is glulam and the material class is chosen as GL30c with material values according to Gustafsson et al. (2019).

The window slats used for the calculation of the loads are the slats on the short side of the facade. As this is a load bearing wall it is assumed to have the most critical loads. It is assumed that the full load on the outer walls are divided 50% - 50% between the two load bearing walls of the entrance. The load division and load bearing walls are marked in Figure 2.5. 11

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CONNECTION TWO, C2

GEOMETRY The base cross-section of the CLT is assumed to be a 3-layer CLT plate and the thickness and layer thickness is decided on based on the selection from Martinsons (2020). The thickness's and other geometrical data used can be seen in Table 2.3.

The second connection is the joint between the step of the stair and the railing. Both members are made from CLT. This type of stair can be seen in building B and C as seen in Figure 2.2, but also in building A. An illustration of the connection area can be seen in Figure 2.6 and a zoomed in illustration in Figure 2.7.

Figure 2.6. Illustration of stair geometry.

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MATERIAL The material used in this connection is cross laminated timber with internal strength classes given from Martinsons (2020) for the chosen thickness.

Figure 2.7. Zoomed in illustration of location for C2. Table 2.3. Geometry C2.

Variable

Value

Thickness Step

120 mm (40+40+40)

Thickness Railing

100 mm (30+40+30)

Length, step

1000 mm

Depth, step

300 mm

LOADS Any loads that might act on the railing are excluded from the calculations. The railing is assumed stiff enough and the connection is through this put more in focus in the calculations. The loads acting on the step is self-weight of the CLT and imposed loads taken from Swedish Standards Institute [SIS] (2008). These can be seen in Table 2.4. Table 2.4. Loads C2.

Load

Value

Imposed Point load Imposed horizontal load Self weight, CLT

2 kN 0.5 kN/m 4.5 kN/m3

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3. RESEARCH

Timber has been used in construction for a very long time and whilst newer materials, such as concrete and steel, has many structural advantages it is important to remember that timber does to. This will be presented shortly in this chapter as an introduction to the thesis question and background for the design of connections.

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WOOD AS A MATERIAL

Glulam

BUILDING WITH WOOD

Glulam is a material built up of board lamellae with the grain oriented in the same direction for all boards. The main structural qualities of this is the freedom in dimensions and that using smaller timber pieces reduces the likeliness of defects such as knots.

Timber as a construction material has been given an upswing in the past decade due to mainly its environmental impact. It is our belief that this increased the positive view on using timber as an architectural element as well. Timber is the only green building material so far with the advantage of storing CO2 during its service life.

Cross laminated timber CLT is, as glulam, built up of boards but where every other layer of boards is oriented across from the one before, hence "cross-lamination". This greatly minimizes the effect on moisture movements which is a difficulty in timber construction.

A great advantage of timber products is its weight-to-strength ratio which is very low in comparison to other materials. With further development of engineered timber products it is possible to incorporate timber in larger and larger projects where it is not yet possible today.

MODERN MANUFACTURING A main advantage of today's fast technological advancement is the increasing number of production methods when it comes to building materials.

When it comes to timber in building most failures are caused in the connections which is therefore one of the most important aspects to consider in timber design. There are many advantages in doing these connections of solely wood when it comes to for example moisture, fire and acoustics which is described more further on.

With regard to carpentry joints the robotic technology increases the possibility of using these type of designs, as the manufacturing no longer need to be done by hand. By the use of a five-axis CNC (Computer Numerical Control) machine, for example, both the angle and direction of cutting and drilling as well as the cutting tool itself can be controlled.

Still, the European design standard rules gives very little guidance to design of carpentry joints and only about 20 % of the current Eurocode 5, EC5, is about connections (Sandhaas, 2018). This is why the main focus of this thesis, to develop a calculation method, has arisen.

The interaction between the design, which is today mainly done in a computer, and the manufacturing increases the accuracy of production and reduces production errors. When also taking into account the possibility of assembling a structure automatically also this step of the production can eliminate errors by fast assembly and precise positioning in the future (Dedijer, 2016).

ENGINEERED TIMBER PRODUCTS Many attempts are ongoing today with trying to minimize the orthotropic qualities that makes timber a difficult material to handle construction wise. The second goal of these attempts are to be able to create timber products not limited in size by the log itself. 15

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WOOD ANATOMY

Moisture

When it comes to the structure of wood there are two vital distinctions of types. This is softwood and hardwood. Softwood generally comes from conifer trees like spruce or pine. These are trees that grow faster than hardwood trees and therefore tend to be less dense. Hardwood is derived from deciduous trees such as ash, beech, birch and oak (Middleton & Middleton, 2020). Figure 3.1 illustrates the different wood types.

Wood’s moisture content regulates quite a lot of the other properties of the material. This becomes especially clear when the material goes below its fibre saturation point. With an increasing moisture content the stiffness and strength decrease (Blaß & Sandhaas, 2017). Table 3.1 show the impact on the mechanical properties with a changing moisture content level. With regards to designing connections in solely wood this might give an advantage in that the whole connection deforms with a changing moisture content instead of just the main timber members when for example steel is used within the connection. This hypothesis would require more investigation though, as it is very dependant on the grain direction of the concerning members as is visible in Table 3.1.

Softwood is generally used as a main structural material due to its faster growth and its lighter weight. Hardwood is a good material to use in minor connections because it is a more homogeneous material and has a higher compression capacity both parallel and perpendicular to the grain. MOISTURE, FIRE AND ACOUSTICS These three properties are vital for the design of timber structures. They will be taken into consideration mainly in discussion in this thesis. a)

Fire Historically fire has always been a main problem for timber constructions and has discouraged many from using wood in this purpose.

Whether a fire develops or subsides depends on the heat generation during a current fire situation. If a fire is set on a visible wooden surface the combustion will proceed in constant speed inwards the material. Due to a covered layer of carbon on the surface the penetration is slow usually about 0.6 – 1.1 mm/minute. Through this the carbon layer wood protects itself and its inner parts and works as a heating-insulation. It counteracts the heat flow to the pyrolysis zone and is a favourable property of the wood. The fire penetration zones can be seen in Figure 3.2.

Figure 3.1. Illustration of trees a. deciduous, b. coniferous.

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Table 3.1. Change in the properties with a one percent change in wood moisture. The benchmark figure is the properties at a moisture content of 12 % (Blaß & Sandhaas, 2017).

Acoustics At an early stage, sound and sound insulation should be considered in the construction process. Sound will move through any cavities and connections are therefore especially important with regard to acoustics. By using timber connections that are properly manufactured difficulties in connections can be minimized as the timber geometries are closely assembled with no gaps. With regard to carpentry connections these might face a problem as they generally remove and cut more holes in the involved members, than a normal steel nail connection for example, therefore this is an important aspect to look into in further studies.

Change (%)

Property Compression || to the grain

Compression ⊥ to the grain

Bending strength

Tension || to the grain Tension to ⊥ the grain

2.5 2

Shear

2.5

E-modulus || to the grain

1.5

Metallic fasteners such as nails, screws and steel dowels increase the burning and make connections the most critical part when considering fire for buildings. An efficient way to obtain fire protection in the connections when steel embraces the timber elements, is to cover it with either wood dowels, wood wedges or glulam or plywood board. Therefore, excluding steel components in the connections entirely with the use of carpentry connections would be of great advantage to the fire protection of a construction (Crocetti et al., 2016).

DESIGN OF CONNECTIONS TYPES OF JOINTS Timber joints affect the constructive behaviour of the elements by, for example, contributing moment resisting joints or hinged attachments and have to be joined together to function as a system. Two main types of joints can be distinguished depending on their working principle, either by transmitting the forces through a contact area or by transmitting the forces by a mechanical connector. Joints can be divided into glued joints, carpentry joints and joints using various metal fasteners. In contemporary timber structures the most common connections are produced with steel plates and laterally loaded metal dowel-type fasteners. The ease of design, production, and assembly, as well as their high load-carrying capacity and ductility are advantages of these connections (Palma et al. 2016). The following chapter will go through the most common traditional contact joints and their advantages in design.

1. 2. 3. 1. Charred wood, the surface cracks and erodes. 2. Pyrolysis zone 3. Unaffected wood Figure 3.2. Phenomenon of the charring process and fire penetration zones of wood.

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A few examples of such joints can be seen in Figure 3.3-3.5. A structural advantage is that they can generally take forces in many different directions due to the locking mechanisms. These could preferably be made of hardwood.

TRADITIONAL WOODEN JOINERY AND LIBRARY OF CONNECTIONS Traditional wooden joinery has a lot of interesting features that, with modern dowel type and metal reinforced joints, has disappeared from the profession today (Branco, 2015). With modern manufacturing techniques, the possibility of bringing back these traditional joinery methods has increased, since the manufacturing is no longer a handmade issue. Important to try and incorporate in this thesis is the idea of these traditional joints but with modern manufacturing techniques and contemporary timber materials. Traditional carpentry joints can be categorized into four different categories. These are “Tenon and mortise joints”, “Lap joints”, “Notched joints” and “Scarf joints” (Branco, 2015).

Figure 3.3. Wedged dovetail joint with a half-shoulder.

The next paragraphs show some of the most common traditional wooden joints related to this thesis and a short description of their main features. Below, a short summary of the main features of the category of tenon and mortise joints and notched joints are shown as well as images of some examples of such joints. More examples for each joint type can be seen in Appendix A1. Tenon and mortise joints

Figure 3.4. Pinned tenon and mortise joint.

The tenon and mortise joints normally connect members to form an “L” or “T” configuration. The two components consist of the tenon tongue and the mortise hole (Branco, 2015). They can be locked by wedges or pinned into place and the connection can include shoulders to stabilize as well as different shapes of the tenon, such as rectangular or dovetail. 18

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The following two sections describe some common solutions for the two connections treated in this thesis. The goal is to create carpentry connections that are able to transfer the forces as these common connections show. C1: GLULAM - GLULAM To attach glulam frames of horizontal and vertical elements in front of a glass facade the same solution as for connecting beams and columns can be used. The connections in Figure 3.7 are designed so the compression of the beam is transmitted to the column by contact pressure between the end of the beam and the column. This type of joint is dimensioned on the basis that the beam's transverse force and normal force are transmitted by the wood screws, which are subjected to shear and tension.

Figure 3.5. Reverse dovetail through, pinned tenon and mortise joint.

Notched joints In order for this type of joint to work they need appropriate joinery at multiple locations for example tenons or pegs and pins. A notch is a V shaped groove most commonly cut perpendicular from the beam (Branco, 2015) as seen in Figure 3.6. The joints shown below can more commonly be rounded or notched out with a diagonal to reduce the number of sharp corners and the stress extremes in these points. Structurally a notch, similar to a tenon, can help hold two members in place whilst they are locked together with a peg or wedge. It prevents both rotation and translation.

Figure 3.7. Standard connections for glulam.

Figure 3.6. Notched joint.

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C2: CLT - CLT

VIDY THEATRE

Long self-drilling wood-screws are today the most standard solution for connecting CLT members. This is seen in Figure 3.8a. This is done by placing the horizontal CLT plate on top of the vertical one. This helps transfer the vertical forces and reduces the shear forces that occurs in the connection as seen in Figure 3.8d. Figure 3.8b. shows how the same components can be joined using angle brackets or nail plate angles. Generally larger transverse forces can be taken, compared to merely screwing. Another, alternative is slotted fittings in Figure 3.8c. These reduce the visibility of the connecting parts.

Vidy Theatre is a timber pavilion in Lausanne, Switzerland. The structural support is completely made of doublelayered timber folded plates. This holds its mechanical performance from the stiffness of its joints in a span of 20 m. The connections are double through tenon joints. Some metal connectors are used but have been greatly reduced by these connections. The tenon connections and structure can be seen in Figure 3.9a-b.

b) Figure 3.9a. Structure details of Vidy Theater (EPFL, 2020) b. Assembly the roof to the building (EPFL, 2020).

TAMEDIA OFFICE BUILDING c)

Tamedia office building is one of the most well known contemporary structures that uses carpentry joints. It was the first 7-storey timber construction in Switzerland so it is both known for its daring size and techniques. The free form connections hold a special hardwood filling that is precisely CNC milled (BlumerLehmann, 2021). The beams and pillars are produced of glulam elements and can be seen in Figure 3.10.

Figure 3.8a-d. Standard connections for CLT members.

REFERENCE PROJECTS The following section includes a couple of reference projects to support the discourse of this thesis. These projects use both carpentry connections, computer controlled manufacturing and engineered timber materials. 20

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Figure 3.10. Structure of the Tamedia Office Building (Shigeru Ban Architects, 2014)

SWG PRODUCTION HALL The SWG production hall is another example of a timber construction that, to some extent, uses carpentry connections. The hall's most outstanding part is the highly stressed central joint. It involves notches which absorb and transmit large forces down to the ground from the 40 meters long roof trusses on both sides. This connection is illustrated in Figure 3.11. An advancement in this construction is that it mainly uses hardwood in its construction which made it possible to design very slender cross-sections.

Figure 3.11. Illustrations of the "Puzzle Connection" and carpentry joint in the framework system.

X-FIX X-fix is a produced timber-to-timber coupling system for joining CLT elements. This system is patented and proof that construction, at least CLT constructions, with only carpentry connections is feasible. This system contains of two pieces of LVL with a double dovetail as can be seen in Figure 3.12. It is simple to assemble where the only on-site tool needed is a hammer (X-fix, 2021).

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Figure 3.12. Two dovetail-shaped and tapered coupling strips, X-fix L (X-fix, retrieved 2021).

TRADITIONAL CARPENTRY JOINTS IN SWEDEN The pictures in Figure 3.12 show some simple traditional carpentry joints from Pelarne Church in Småland. This church which is believed to have been built in the 13th century still stands today showing both the durability of the carpentry joints as well as the tradition of constructing in timber in Sweden. The church's construction consist of a combination of tenon and mortise, notched and peg joints.

CHAPTER CONCLUSION What is brought into the next phase of design is the library of historical connections. This will be a relevant starting point in the design. The implementations of carpentry joints in the reference projects are relevant to bring into a discussion in a later stage as a description of how the designed connections could be applied on larger construction parts and in more complex ways. The research about manufacturing and moisture, fire and acoustics is also a relevant discussion subject and are used in a later stage to grade the designed connections based on a few criteria. This is described more in Chapter 4.

Figure 3.13. Connections Pelarne Church in Småland.

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4. ARCHITECTURAL DESIGN

Throughout this chapter the architectural design process from an early stage to the investigation phase (where the types will be studied as FE-models) will be shown for the two visible wood-wood connections.

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Figure 4.1 shows a more detailed illustration of the general development of the connections and of what is described in this chapter. Figure 4.2 illustrates the location for the two connections C1 and C2. For more detail into this, see Chapter 2.

1st Iteration Analysis degrees of freedom Simplification

FIRST ITERATION

2nd Iteration

A library was gathered of traditional carpentry connections in the research and listed into four main joint categories, Tenon and Mortise, Lap, Notched and Scarfed joints. This library introduce essential possibilities of what wood-wood connections can look like. By combining the historical connections from this library with ideas for the specific two connections in this project, different suggestions were implemented in schematic tables. These connection types were divided into degrees of freedom. In this phase both applicable and less applicable varieties are considered to keep the design open and to keep an easy comparison with the different degrees of freedom. The tables including these are shown in Appendix A2.

Connection trees

Grading

Final selection Minor alterations of chosen geometries

Design for further investigation Figure 4.1. Process illustration for the Architectural design. a)

Figure 4.2a. Location for C1. b. Location for C2.

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The solid arrows explain the possible translations and the dashed grey arrows together with a solid red the possible rotations for the next coming figures.

C1 Figure 4.3 show design examples that failed for their complex design and lack of functionality. The first can rotate thanks to a circular geometry and the circular hole in the plate inside of the pillar. A dowel connects the beam to the column. The second is characterized by its puzzlecomponents, where the first component consist of a dovetail inserted in the column, then a cut-out component connects the two elements and lastly two wedges which keep the beam in place. This connection is set as rigid, hence no arrows. The third consists of a cube with a dovetail, inserted from the side into the pillar which is not a reasonable geometry due to the high possibility of movement as indicated. Alternatives which managed to solve the connection demands and at the same time represent simple and smart solutions continued on as interesting alternatives. Three examples of these are shown in Figure 4.4. C2 All dynamic solutions were removed as this did not seem to be a likely connection solution for a staircase. Nor the solution with a single pin insertion which can rotate was reasonable as a stair step. Some examples of these solutions can be seen in Figure 4.5. A slight modification to this which could work instead would be to have two pins instead of one. These were all designed in this iteration but removed in the second iteration two. After this first iteration the step height was also increased which rendered a few of the previous solutions unnecessary and to intricate for the next step.

Figure 4.3. Examples of excluded connections for C1 after the first iteration.

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Figure 4.4. Examples of some continued interesting

Figure 4.5. Examples of excluded alternatives for C2

alternatives for C1 after the first iteration.

after the first iteration.

For full tables of design proposals after the first iteration see Appendix A2. Some of the remaining proposals for C2 can be seen in Figure 4.6. These seemed reasonable enough to be moved into the next iteration step.

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made in a group from simple to more sophisticated design. This is found in Appendix A3. This was done as a first step to categorize the connections and to not get too many extremely different solutions. This is also in order to be able to compare them more easily in the next step. C1 A division of groups were named "Centred Insertion" with subclasses of rectangular and dovetail cross-sections. "Side insertion" is the other main group where the inserted part is visibly connected from the side of the pillar. An example of this is presented in Figure 4.7, where a rectangular cube is developed by adding cuts, amount of elements and insertion depth. C2

ternatives for C2 after the first iteration.

The connections for C2 were divided into three different groups. The first one, “Full cross-section”, where the full cross-section of the step is inserted in the supporting CLT railing. The second, “Part cross-section”, where more tenon and mortise like connections are used. The last is “Dovetail tenon”, similar to a tenon and mortise joint but where the tenon includes a slanting of its edges to create a dovetail. In Figure 4.8 the tree structure of the part cross-section is shown as an example.

SECOND ITERATION

EVALUATION

In the second iteration the connections are simplified as much as possible to, in stepby-step, add new more advancing features of the connections. Different examples were grouped into categories due to the insertion method, the joints geometry and position. Small advancements and differences of the connections were then

The last step before finalising the chosen designs is an evaluation of the connections based on several criteria. These are designed to be as easily measurable as possible for a clear comparison of the qualities of each connection design.

Figure 4.6. Examples of some continued interesting al-

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Full cross-section Full cross-section Shape of the Shape of the insert geometry insert geometry

CENTERED INSERTION Part cross-section Rectangular cross-section

Dowels to reinforce Dowels reinforce and locktothe step and lock the step

The beam is slanted Slanted tenon Add a wooden into the column into the column wedge above

Increase the depth of Increase depth of insertionthe in the railing insertion in the railing Outside reinforcement Outside reinforcement of the step of the step Dowels added to the inserted part

Extra and locking parts

Figure 4.7. Extract from Appendix - A3. Example of centred tenon connections for C1 with small differences.

Dovetail tenon Dovetail cross-section

One wedge block added

Two wedges as lockers

Increasing the Increasing amount of the dowels amount of dowels

Part cross-section Part cross-section

Rotated dovetails in 90°

Amount of tenon Amount of tenon Full or half shoulders Full or half shoulders

Shape of tenon Shape of tenon

Figure 4.8. Extract from Appendix - A3. Example of a part cross-section insertion connections for C2 with small differences.

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mentioned in Chapter 2. Therefore this criteria is taken into consideration. The questions to answer are how many sides are shown? As well as if these connecting pieces are separate (for example dowels) and can be made out of another type of wood and can make them more visible?

CRITERIA Manufacturability In all projects, manufacturability is an important aspect to consider since this often goes hand in hand with economy and how easy it is to assemble. For this category the connections are looked at based on how many cuts it would take to create the geometry. Dowels and the drilling holes are also counted as one cut.

Structure A measure of how well the connection is optimized for what the construction demands. If the connection takes forces in a lot more directions and moments than is actually needed the design could be too intricate.

Assembly For future restorations, demolitions, reusability as well as erection time and economy, assembly is an aspect to consider. This is measured for the connections by how many movements are needed for joining the connection.

Applicability Applicability has to do with future intent of this thesis and the involving connections. At this point it is not an easily measurable category but for future reference it is important to think of how easily the connections could be applied to other situations.

Fire & Acoustics resistance Influencing this criteria is depth of penetration for the connected members, the risk of gaps and cracks forming between the pieces and if the joining geometry is mostly covered (how many sides are exposed).

This criteria also takes in the “essence” of some of the other criteria such as assembly, manufacturing, fire and acoustics. How it performs within these will in some part determine how easily it could be applied to other scenarios as well. This criteria is an important part of the discussion in Chapter 8.

Regarding this one question that is considered is how likely is it that small gaps are created in the joining? Several angled sides (dovetails) increase the risk of gaps more than a block.

Grading tables are found in Appendix A4 for full overview of the evaluation. A grading for all connection types is set, with respect to the mentioned parameters, between one to three dots. They are ranked in comparison to each other where one dot stands for good, two dots for better and three for the best concept concerning the different categories for the connections.

With regard to C1 and C2 the acoustical qualities are not considered in this step as these connections are within one room and will not open up for vibrations to move between rooms. Architectonic qualities For C1 and C2 this feature matters since these are two visible connections. One of the goals for these connections is that it is visible and the connection is shown as 29

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TYPE OF CONNECTION

ARCHI. QUALITIES

MANU FACTURING

ASSEMBLY

FIRE AND STRUCTURE ACOUSTICS

MANU FACTURING

ASSEMBLY

twoAND selected options from ARCHI.the evaluation FIRE APPLICA STRUCTURE ACOUSTICS BILITY are shown with theirQUALITIES grading result. One

From the evaluation phase, two major different concepts per connection were selected for further investigation and calculation comparisons. In Figure 4.9 the

APPLICA BILITY

geometry is a plane dovetail tenon inserted from the side and the other is a rectangular tenon with a locking peg.

TYPE OF CONNECTION

MANU FACTURING

ASSEMBLY

FIRE AND STRUCTURE ACOUSTICS

ARCHI. QUALITIES

APPLICA BILITY

TYPE OF CONNECTION

MANU FACTURING

ASSEMBLY

FIRE AND STRUCTURE ACOUSTICS

ARCHI. QUALITIES

APPLICA BILITY

TYPE OF CONNECTION

MANU FACTURING

ASSEMBLY

FIRE AND STRUCTURE ACOUSTICS

ARCHI. QUALITIES

APPLICA BILITY

ARCHI. QUALITIES

APPLICA BILITY

Figure 4.9. An extract from Table 1 in Appendix A4 of the grading for C1.

TYPE OF CONNECTION

MANU FACTURING

ASSEMBLY

FIRE AND STRUCTURE ACOUSTICS

Figure 4.10. An extract from Table 2 in Appendix A4 of the grading for C2.

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The different geometries are analysed in FE-models in the next Chapter as a further iteration before bringing a last selection into the hand calculation phase.

C2 The two major concepts brought forward from the evaluation phase was a notched geometry protruding the railing as well as a tenon and mortise geometry. A dovetail tenon is also chosen as a version of a notched geometry. Both kept alternatives for further studies are found in Figure 4.10.

CHAPTER CONCLUSION From this chapter chosen options for C1 and C2 that will be further investigated, as a help to develop the preliminary calculation method, can be seen in Figure 4.11 - 4.14.

For C1 a side inserted plane dovetail and a centred tenon with a locked peg are analysed with two options each. These are variations of depth of the dovetail and a variation in peg size for the tenon and mortise as seen in Figure 4.11 and 4.12 with their respective shortened names. They are named "Plane dovetail 1", PD1, "Plane dovetail 2", PD2, "Peg 1", P1, and "Peg 2", P2.

b) Figure 4.11a. Plane dovetail full depth, PD1. b. Plane dovetail half depth, PD2.

For C2 the investigated geometries are three types of tenon and mortise joints where the position of the tenon differs and two different geometries are used. These different geometries are named "Tenon and mortise", TM1, "Upper notch", N1, and "Bottom notch", N2, as seen in Figure 4.13. For the dovetail joints five different models are investigated where the shape of the dove is taken into account as well as the number of doves in the connections. These are classified as "Round dovetail 1-5", RD1 - RD5. All varieties of the dovetail can be seen in Figure 4.14.

b) Figure 4.12a. Tenon and mortise with thick peg, P1. b. Tenon and mortise with thin peg, P2.

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b) b)

c) c) Figure 4.13a. Tenon and mortise, TM1. b. Top notch, N1. c. Bottom notch, N2.

e) Figure 4.14. Rounded dovetails 1-5. a. RD1. b. RD2. c. RD3. d. RD4. e. RD5.

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5. ANALYSIS

The following chapter includes an analysis from the FEM-program, Abaqus, together with analytical calculations comparisons. In Appendix C more specific results are find from Abaqus.

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The interaction added in the contact surfaces in the connection are simple normal force contact. No additional friction is added. This is to simulate that it is merely a contact connection and that no additional glue or joining mechanisms are used. This is the same for C2.

FE-MODELLING The FE-models in Abaqus are made to locate the most critical stress points and by this try to locate important failure modes. It is also used as a method to investigate the differences between the connections in both stress locations and magnitudes. Through a discussion on the differences between the connections a further iteration and choice can be made of which connections to perform hand calculations on. The first step of the FE-modelling was also utilized to identify the most critical load locations for the C2 connection.

Material The material values can be seen in Table 5.1 for glulam (Atashipour ,2020). Table 5.1. Glulam engineering constants.

Variable

Value

E1 (x)

12 600 MPa

Loads

E2 (y)

390 MPa

The loads are applied with two different load cases for the pinned connections but a conclusion was made that load case one is the most critical one. Therefore, the focus on the analysis are stress diagrams related to this load case. The loads are calculated and can be seen in Appendix B1.

E3 (z)

700 MPa

υ12 (xy)

0.03

(-)

υ13 (xz)

0.04

(-)

υ23 (yz)

0.35

(-)

G12 (xy)

350 MPa

G13 (xz)

720 MPa

G23 (yz)

30 MPa

INPUT C1

Boundary conditions and Geometry Regarding the geometry modelled for C1 and C2 it is important that enough of the model is considered to portray a reasonable stiffness with the chosen boundary conditions. Any other locking addition of nodes is merely made in order to prevent rigid body motion of the model. The mesh and boundary conditions for C1 can be seen in Figure 5.1. The end of the pillars are locked in translation in all directions. An additional boundary condition is used for the pinned connection where the connection is locked in one end in y-direction. This is to prevent rigid body motion which otherwise occurred.

INPUT C2 Loads The loads applied in the FE-models are the ones seen in Table 2.2 and 2.4 in Chapter 2 with minor differences in order to be applied in Abaqus. These differences are described below and calculated in Appendix B1 and B2. In order to eliminate as many singularities as possible and apply the loads upon the structures all line loads are divided by the area and applied as a positive or negative pressure.

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Locked translation axis Y

X Figure 5.1. Boundary conditions and mesh definition, Abaqus, C1.

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should have been. A test was made using the proper values for some of the analysis and the conclusion was made that this only affect the magnitude of the stresses and not the location of the stress peaks. Therefore the discussion is not related to the size of the stresses which also is not the primary intention.

The point load acting on C2 is applied over an area of 100x100 mm instead of a single node. The values of the loads applied are calculated in Appendix B2. Boundary conditions and Geometry The boundary conditions, B.C., applied to C2 can be seen in Figure 5.2. The B.Cs added on the railing are fully rigid with no translation or rotation. The B.C. added to the end of the step geometry where the connection toward the wall would have been has been kept unspecified and is therefore added as a symmetry condition and therefore locked in translation in x-direction and locked in rotation in yand z-direction. Material The material values can be seen in Table 5.2 for CLT (Al-douri & Hamodi, 2009)*. Table 5.2. CLT engineering constants.

Variable

Value

E1 (x)

12 000 MPa

E2 (y)

600 MPa

E3 (z)

900 MPa

υ12 (xy)

0.015

(-)

υ13 (xz)

0.035

(-)

υ23 (yz)

0.558

(-)

G12 (xy)

600 MPa

G13 (xz)

600 MPa

G23 (yz)

50 MPa

* For C2 there was a mix-up in some of the material values in the modelling. This results in slightly different values than what

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Locked rotation axis Locked translation axis

Z X

Figure 5.2. Boundary conditions and mesh definition, Abaqus, C2.

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Table 5.3. Investigative load cases for C2

ANALYSIS In this section the results of the FEManalysis are presented. The stress results have been modified slightly to not account for stress singularities. This is done by applying a maximum, max., and minimum, min., magnitude of the stresses for each diagram. The analysed diagrams are taken from several locations surrounding the connections for shear, tension and compression stresses. Or on several locations along the step for the bending stress case . The analysed diagrams are an average of these stress lines.

Load

Critical load case C2

P.L. on edge-side

Illustration

Point load on centre-centre

P.L. on centre-side

P.L. on edge-centre

The first iteration of the FEM-analysis was made on the most simple tenon and mortise joint for C2. The different load cases investigated for this can be seen in Table 5.3. For all of the cases a body load to simulate the self-weight was also added. The most critical location for the point load and the horizontal loads were then put together to give the most critical load case as seen in Figure 5.3. This load case is applied to the other connections as well as used in hand calculations.

Horizontal length Pressure Horizontal length Traction

Side length - Pressure

Side length - Traction

Figure 5.3. Default load case for C2.

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Comparing RD1 with RD2 it is clear that a wider dovetail decrease the stresses. The highest stresses also seem to occur around the notch which is similar to the other geometries as well and indicates that this area is important to investigate for failure modes.

C2 - Comparisons RD1, RD2 and RD3 This comparison focuses on the influence of the dovetail size on the bending stresses. The bending stresses, as an average along the length of the step, can be seen in Figure 5.4 for the top and the bottom fibres, t.f., and b.f. In the graphs positive values represents tension and negative compression stresses. The graphs are taken along the x-direction of the step as illustrated in Figure 5.6. The geometries compared are seen in Figure 5.5. The bending stresses are positive in the top of the connection, fibres in tension, and the b.f. are in compression around the connection. This has to do with the constraint of the railing and contact surfaces. This will differ in the hand calculations for the geometries as the step in C2 and beam in C1 are then assumed to be simply supported for the sake of the load calculations. Because of this, the tension in the t.f. around the connection will not be investigated in the calculation method. Although, verifying the compression stresses in the b.f. might be a relevant failure mode.

RD1

RD2

Bending stress, RD1, RD2 & RD3

Stress

[MPa]

3 2

RD3

Figure 5.5. Geometry comparison for RD1-RD3.

250

500

750

1000

-1

-2

-3

Length [mm] RD1 tf RD2 tf RD3 tf

RD1 bf RD2 bf RD3 bf

Figure 5.4. Average bending stress for dovetail joints, RD1, RD2 and RD3 for tf and bf.

Figure 5.6. Illustration of paths to consider bending for RD1. An average value are calculated from these paths.

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C2 - Comparisons TM1, N1 and N2

Bending stress, TM1, N1 & N2

In general the bending graph for the three geometries shown in Figure 5.7 illustrates the same type of failure modes as for RD1RD3.

[MPa]

Stress

The bending graphs for the t.f. shown in Figure 5.8 for TM1, N1 and N2 show that the stresses are less critical in tension for N2 around the notch than for the other two. This correlates to the theory on notches. When it comes to stresses around the notch, a notch at the same side as the support (N1) is generally a more critical solution.

250

500

750

1000

-1

-2

Length [mm] TM 1 tf N1 tf N2 tf

TM 1 bf N1 bf N2 bf

Figure 5.8. Average bending stress for tenon and mortise, TM1, notches, N1 and N2 for tf and bf.

The graphs in Figure 5.9 show the shear stresses along the height of the step around the connections. Figure 5.11 illustrate the area from which the stresses are taken for TM1. This is similarly done for N1 and N2. The shear stresses are generated when bending occur by loads perpendicular to the x-axis. It is clear in the graphs in Figure 5.9 that the stresses are influenced by the layer properties of the CLT. This is therefore important to reflect on in the calculation method. For the two shear planes the layers with grain parallel to the plane take the highest stresses which reflects to the materials highest shear strength in panel shear as well.

TM1

In comparison to solid timber and glulam, where the grain runs parallel with the length of the beam for the full crosssection, it could be relevant to investigate the rolling shear strength for a CLT crosssection, since this is the weakest material property of timber.

N2 Figure 5.7. Geometry comparison for TM1, N1 and N2.

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Stress

-1

[MPa]

[mm]

-3

Length

120

100

[mm]

Avg. yz-pla ne

100

Length

-0,5

Length

Avg. x z-plane

-0,4

[mm]

100

[MPa]

120

-0,3

-0,2

Stress

120

Shear stress, avg. TMv.1

-0,1

Tens. & Comp. stress, avg. TMv.1 Tension & Compression stress, TM1

Shear stress, TM1

Tens. & Comp. stress, avg. TMv.1

-5

0 0

Length-0,2[mm]-0,3 -0,1

-0,5

-1

-3

-5

120

100

Avg. yz-pla ne

Length Stress [mm] [MPa] Avg. S 33

120

100

[MPa]

Stress Avg. x z-plane

-0,4

[MPa]

-0,4

-0,5

20 0

-3

-0,4

Stress Avg xz-plane

-0,3

Avg xz-plane

[mm]

[mm] Length

Length

-0,2

Avg yz-plane

120

-0,1

100

-1

[MPa]

-0,3

[MPa]

[mm]

Stress -5

-0,2

Stress

3 -3

-0,5 -0,1

-0,4 0

5 -1

120

100

Length

100

[mm]

[MPa]

Length

Avg. S 33

-0,3

[MPa]

100

Tension & Compression stress, N1

120

Stress

120

Shear stress, avg, TMv.2

-0,2

Stress

Avg yz-plane

Shearstress, stress,avg. N1 TMv.1 Shear

-1

-3

Stress [MPa] Avg. S 33

-5

60 40

100

[mm]

Tens. & comp. stress, avg. TMv.3

80 60

Avg. xz-plane

Length

[mm]

100

120

Length

Shear stress, avg. TMv.3

120

Tens. & &comp. stress, avg. TMv.2 Tension Compression stress, N2

40 20

0 0

-0,1

-0,2

Stress Avg. x z-plane

-0,3

[MPa]

-0,4

-0,5

Avg. yz-pla ne

Figure 5.9. Average shear stress in the tenon/notch for a. TM1, b. N1, c. N2.

-1

Stress [MPa] Avg. S 33

-3

-5

Figure 5.10. Average axial stress in the tenon/notch for a. TM1, b. N1, c. N2 along the z-axis.

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Tens. & comp. stress, avg. TMv.3

Shear stress, N2 Shear Shear stress, stress, avg. avg, TMv.3 TMv.2

Avg. yz-pla ne

-5

-0,5

Tens. & comp. stress, avg. TMv.2

-0,1

σ y

ft,0

Figure 5.11. Illustration of paths to consider shear for TM1 where an average value are calculated from these paths.

ft,90

With regard to critical stress points, it is visible in Figure 43b-c that high stresses occur in the notches and especially in the notch corner for N1.

ε fc,90

fc,0

Axial stresses in the tenon (z-direction, see Figure 5.11) are presented in Figure 5.10 for the same area from where the shear stresses are taken. Positive values define tension and negative compression. For this axis all stresses act perpendicular to the grain in all layers which is more critical than stresses parallel to the grain. Most critical of these are in tension which is illustrated in Figure 5.12 with a stressstrain relationship curve for timber.

Figure 5.12. Stress-strain-curve of clear wood, exposed to tensile and compression stresses (perp. to the grain dashed line) and (parallel to the grain - solid line).

C2 - Comparisons RD1 and N1 In many cases a dovetail can be approximated as a notch on the same side as the support, as both the geometries include the critical point of a notch corner at the bottom of the cross-section seen in Figure 5.13. When comparing the shear stresses for the two, as seen in Figure 5.14 and 5.15, as well as the axial stresses, as seen in Figure 5.16, it can be concluded that the highest stresses occur at similar points of the geometries. This would suggest that approximating the dovetail as a notch is not unreasonable for cases where verifications does not yet exist for dovetail geometries but for notches.

Cracks primarily appear at the notch and tenon tip when having a combination of shear and tensile stresses. Such situations could rapidly happen without any markable deformation or visible signs (Blaß & Sandhaas, 2017) and is therefore a relevant failure mode. The axial stresses around the notch for N1 show tensile stresses under the notch. This could portray a likeliness of a combination with high shear stresses and a relevant failure mode especially for this type of notch. The graphs in Figure 5.10 also show that a relevant failure mode might be compression perpendicular to the grain at the bottom surface of the notches and tenons as already mentioned from the bending stresses discussion as well.

162

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Length

[mm] 120

100

0 -0,2

Stress

Length 80 60

Avg xz-plane

[mm]

120

100

Stress

Length

[mm]

100 0

-0,3 0 -0,4-0,5 -0,5 -1

[MPa]

Avg yz-plane

-0,1

Shear stress, N1 120

40 20

-1,5

[MPa]

0 0

-0,1

-0,2

-2

Stress Avg xz-plane

-0,3

-0,4

[MPa]

-0,5

Avg yz-plane

-2,5

Figure 5.14. Average shear stress for N1.

120

Shear stress, avg. DTv.1

Shear Shearstress, stress,avg. RD1DTv.1 Shear stress, avg, TMv.2

Length

[mm]

100 80 60 40 20 0

-0,5

-1

Stress Avg. x z-plane

-1,5

-2

[MPa]

-2,5

Avg. yz-pla ne

Figure 5.15. Average shear stress for RD1.

Tension &Bending Compression stress, stress, TM1 RD1 & N1 1202

Stress Length [MPa] [mm]

100

RD1 Figure 5.13. Geometry comparison for RD1 and N1.

1 80 600 40

101

151

201

-1 20 0 -2

Length [mm] 1 -1 -3 -5 Stress [MPa] RD1 avg. S33 N1 avg. S 33 3

Figure 5.16. Average shear, tension and compression stress for RD1 and N1.

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Shear stress, avg, TMv.2

The shear stresses for RD1 are quite a lot larger than for N1. This is most likely mainly due to singularities in the model and since the width of RD1 is a lot smaller than for N1. This could also be due to the extra restraint in movement that the angled geometry of the dovetail entails. A main difference between the two geometries is the high rolling shear stresses in the bottom layer of the dovetail, which could relate to a failure mode only valid for this type of geometry. This might be due to the angled geometry of the dovetail and therefore the short fibres that are disconnected from the rest of the step which is a critical design point also in manufaturing.

C2 - Comparisons RD3, RD4 and RD5

Shear stress at same point for DT1 and DT2 on RD4

Length [mm]

A comparison of affecting the number of dovetails (which could correlate to number of notches or tenons as well based on the previous discussion) is also made where the geometries are seen in Figure 5.17. When comparing the shear stresses along the same line for the two dovetails in RD4, only minor differences occur, see Figure 5.18. It is therefore not a reasonable assumption that the load is divided equally between the used number of dovetails.

120 110Shear stress at same point for DT1 100 and DT2 on RD2 90 80 70 60 50 40 30 20 10 0 0

-1

Stress

DT1, xz-plane DT1, yz-plane

-2

[MPa]

-3

DT2, xz-plane DT2, yz-plane

Figure 5.18. Shear stress at the same location in DT1 and DT2 for the geometry of RD4.

Comparisons C1, PD2 and C2, N2 A comparison between PD2 and N2, whose geometries can be seen in Figure 5.19, also means a comparison between the material glulam and CLT.

RD3

The main differences between the graphs shown in Figure 5.20 and 5.21 is that the glulam only take shear forces in one direction while the layer orientation of the CLT makes it possible for this to take the shear forces in both xz- and yz-plane. Due to the orientations of the grains the failure mode of rolling shear will not be valid for the connections using glulam.

RD4

RD5 Figure 5.17. Geometry comparison for C2, RD3-5.

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-0,1 -0,2

Stress

-0,3

[MPa]

-0,4 -0,5

Shear stress, N2 Shear stress, avg. TMv.3

120

Length [mm]

100 80 60 40 20

PD2

0 0

-0,1

-0,2

Stress Avg. x z-plane

-0,3

[MPa]

-0,4

Avg. yz-pla ne

Figure 5.21. Average shear stress for N2.

Length

[mm] 90

Shear stress, PD2

-0,15

[MPa]

[mm]

Avg. x z-plane

-0,25

Length

30 20 10

Shear stress, avg, v.2

Stress

Avg. yz-pla ne

-0,05

Figure 5.19. Geometry comparison for C1, PD2 and C2, N2.

0 -0,05

Stress Avg. x z-plane

-0,15

-0,25

[MPa]

-0,5

Avg. yz-pla ne

Figure 5.20. Average shear stress for PD2.

Shear stress, avg, v.2

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The functions for notches and tenons are quite similar and verifies the same type FUNCTIONS of failure mode. Different sources state similar functions but with small changes.

PRELIMINARY CALCULATIONS The analysis of the preliminary calculation method include two parts. The first part investigates different methods used for the same failure mode. The second part illustrate some geometrical effects that the chosen connections might have.

FUNCTIONS

Notch:

Figure 5.22 show the relevant geometrical data for 3this comparison where the width Vd kv joint fv,d types. ofdthe= beam, b, is same for all 2 h bef FUNCTIONS The verification given in SIS (2009) for notches can be rewritten to a force capacity hN verification in thejoint equations below. for notch Notch:h as seen = he Vd FUNCTIONS 3 for tenon joint kv fv,d d = h 2 h bef

METHOD INVESTIGATION Shear and tension perpendicular to the grain with a notch at the opposite side of the support

FUNCTIONS

Notch: FUNCTIONS 2

Fv, Rd h= 3 h bef kv fv, d = ... FUNCTIONS N (1) Vfor 2 3h d notch joint kv fv,d d == 2 b h k f Notch: ... = 3 ef Nh bv v,d he ef tenon joint Vfor 3h d Tenon and Mortise: Different α's are used depending = kv fv,d on if a d h2N h b notch function for is used or joint a tenon function. efnotch h 2 Fv,= Rd = 3 h bef kv fv, d = ... h 2h Ne 2 for tenon joint befk notch h k joint k f for bottom tenon ... = 3 hh bef 3h Nfor v efv,dz v v, d Fv,= Rd = h 4 hN e 2 bef tenon h k kv fv, d for central tenon for 9 Mortise: Tenon Fv, Rd =hand h beef kzv joint fv, d = ... h 3 2 ... = 3 bef kv fv,d 2 eth N al. In (2016) different Fv, Müller = h kv fv, two = ... Rd 3 2 usedbeffor formulas are thisd verification bef h e kz kv fv, d for bottom tenon 2 3 the depending position of the tenon in Tenon befonhMortise: F...v, =Rd3=and N kv fv,d h 4 the cross-section. hN b h k k f for central tenon 9 ef e z v v, d

ho he

b) hN

Tenon and2 Mortise:

for(2)bottom tenon 3 bef h e kz kv fv, d Fv, Rd = 4 2 b h k k f for(3)central tenon 9 bef h e kz kv fv, d for bottom tenon 3 ef e z v v, d Fv, Rd = 4 kz lower kv fv, tenons, forl.t.,central tenon Equation (2)9 is bused ef h e for d which also can be seen as a bottom notch, b.n., when he = h. Equation (3) is used for central tenons, c.t..

Figure 5.22. Relevant geometry of Notched joint b. top z0,Ta. Tenon joist. z0,T notch, c. bottom notch and

z0,B

46 tr

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The first comparison seen in Figure 5.23 is made to determine which function should be used for a bottom notch, b.n., (1) or (2). In equation (1) for b.n. kv = 1.0 according to EC5. In equation (2) when using an α = 0.90 (α is really 1 but then the kv function fails as this means a division by 0) kv still equals to 1.0 so this coefficient is the same for the two equations. The differences then comes down to the different heights hN and he and the additional correctional coefficient kz.

z0,T he

Fv,Rd [kN]

Figure 5.24. Geometry for comparing the different positions of a tenon.

The equations according to Müller et al. (2016), eq. (2) and (3), are the functions to use for the tenon where z0,T > z0,B. Hence an underlying tenon as these functions give a requirement that ho ≥ hu, as seen in Figure 5.22. For a central tenon, c.t. eq. (3) is used (Müller et al., 2016) but in other sources eq. (2) is used for both a l.t. and a c.t (Blaß & Sandhaas, 2017). With the same reasoning as for a b.n. eq. (3) is used for a lower tenon as this gives the most conservative result.

When testing the different functions for a tenon where z0,T < z0,B, hence a higher placed tenon, this should most likely give a lower value for Fv,Rd than a central tenon and a higher value than a top notch, t.n., with the same height of the notch. This results in the use of eq. (3) as seen in Figure 5.25.

85 78 72 65 58 51 44 37 30 24 17

hN /h [%] Fv,Rd,BN

z0,B

Fv,Rd for different ratios hN/h, comparing eq. (1) & (2)

z0,T

Fv,Rd,BT

Figure 5.23. Comparing the two functions that could be used for a bottom notch.

The equations give a lower capacity the smaller the tenon or notch height is, which is reasonable. Comparing the two results, eq. (1) is more conservative and gives a lower capacity for a b.n. than eq. (2). Therefore, eq. (1) is used for all notched geometries.

For most part the results in Figure 5.25 show that the capacity of the joint decrease with the height of the tenon. The only anomaly is that the capacity for a lower tenon is higher than that of a bottom notch since this equation uses he instead of hN. This function also uses kz but for this position of the tenon and geometry kz< 1.0 and will therefore not increase the capacity any further. Given that this function for a lower positioned tenon is mentioned in several sources it is still considered to be a safe equation for the verification.

The next comparison is made to see which equation should be used if the tenon centre of gravity doesn't correlate with the beam centre of gravity as seen in Figure 5.24.

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b h k k f for bottom tenon 2 3 ef e z v v, d F == 4 h b k f Fv, = ... v, Rd Rd 3 b h ef k v k v, df for central tenon 9 ef e z v v, d 2 ... = 3 bef h N kv fv,d Dovetail: Tenon and Mortise: 2 Fv, Rd = 3 bB1 h B1 r kv fv, d 2 b h kz kto fv, d for The three equations Fv,Rdbottom tenon v calculate 3 ef eused h f 2 Fv, be = below. This is equations N v, d (4), (5) can seen Fv, Rd = 4 b k h Rd and (6). 3 bef crh e Nkz khv fv, dkcal for central tenon 9

Fv,Rd for different ratios hN/h, comparing all different positions for a tenon and notch 6

Fv,Rd [kN]

Fv,Rd,IBT (2) Fv,Rd,ITT (3)

Figure 5.25. Shear capacity for different positions of a tenon and a notch.

Shear and tension perpendicular to the grain for a dovetail For the typically used dovetail geometry seen in Figure 5.26 many different methods have been found to verify the cross-section for the same failure mode as for the top notch’s bottom corner. The five methods used for comparison are summarized in Table 5.4. b bN

bB1

2 Fv, Rd = 3

bB1

h B1

fv, d

1.0 kv fv, d 6.5 fv, d lZ 0.4 kcal h

r 2

hN-r

Figure 5.26. Relevant geometry of Dovetail joint.

kv b

hH,u

-r

Architecture and Civil Engineering, Master’s Thesis ACEX30 bCHALMERS B1 1

cal

hB1

2 To note on b. Inb these bB1 h N = isr a limitation Fv, Rd b hN r k fv,B1d 3 as h calculations. The limit is that AB= = bB1 2 tan 2 h B1bNv ≥ 0.8·b 2 B1 2 h mentioned in the sources. Hence: 1.0 2 hbB1 N 6.5 bk = 0.8 kvv = minh lZ 1 2 2 This limitation,has well as other geometrical 0.4 h V 3 given Ed in the sources, is 4most conditions 1.0 Fv, Rd = 3 bH h H, u fv, d Ed = 4 b h likely based Hon H, theu load-bearing capacity 0.2 3600 kv =also but onr common dimensions of these hmin N AB the width of the2 joist = For our joints. case 18 h H, u h hN isF wider= than this in6.5 relation to the widthl h 0.8 f 2 v, Rd z H t, 90, d h r h of the dove. By using the "real" H value of h B1 N 2 the joist the capacity in eq. (4) and (6) gets kv = h higher which is not reasonably the case if the geometry of the dove stays the same. For this reason 1.0 the width of the joist is therefore limited in this 0.2 calculation to get 3600 = min akconservative validation of the connection. v AB The comparison l of the l1 shear capacity for the different methods with different heights of the dovetail can be seen in Figure 5.27. hH,u

hN /h [%]

Fv,Rd,CT (3) Fv,Rd,TN (1)

b hN

hN kv = min 2 Fv, Rd = 3 bh kcr h N h

85 78 72 65 58 51 44 37 30 24 17

Fv,Rd,BN (1)

2 Fv, Rd = 3 Dovetail:

bB1 8

notch efjoint hh e 2 for notch joint h 2 for bef tenon h k joint k f for bottom tenon for 3 h h = he h b h ek zk vf v, d for bottom tenon ef tenon e z joint v v, d Fv, Rd = e 34Nfor joint Fv, Rd =h 24 for hfork notch k f joint for central tenon h h b tenon fv, dv, =d ...for central tenon Fv, Rd == 399 befefhh ebeefkz kzvkv fvv, d 2 h e Fv, Rd 2= 32 h beffor kv tenon fv, d =joint ... =

h b k f

...v, Rd = 23 b3 h k eff v v, d ef N v v,d Dovetail: Dovetail: ... = 32 bef h N kv fv,d

= ...

b h 2k f

... =

Tenon 3andef 22Mortise: = N b v v,d hh bef kvr fv, dk = ... Fv, RdFv, Tenon and F ==Rd3Mortise: kv v fv, fdv, d B1 h B1B1 r 3 b3B1 v, Rd

Tenon and2Mortise: ... = 3 bef h N kv fv,d 2 h Nh N fv,fdv, d 22 for l bottom l1 tenon = 233 bbbbkefkh hhehk kzk kvf hfH,u d Fv, = tenon v, Rd Rd cre NzN hvperpendicular h v, dv, 3 crtension kcalkcal for bottom Table 5.4. Equations for verifying a dovetail joist in shear and to the grain. ef 3 Tenon and Mortise: bbN Fv, Rd = 24 Fv, Rd = 4 b h k k f for bottom tenon h for central r 39 befef hee kzz kvv fv, d N l1 tenon bB1 hB1 h b h k k f v, d for l central tenon b Fv, Rd = 2924 ef e z v v, d k Fv,Rd Area bN F == 3 bb2h N r kv k fv,v fd Fv, v, Rd v v, dl 3 b hhbNk h kr khfH,u b 1 tenontenon for lbottom lZRd 9 ef3 e ef z e v z v,kvd fv, d for central bγN Dovetail: l l Dovetail: Fv, Rd = 4 1 r h 1.01.0 h hN bN bB1 hB1 bef h e kz H,u k6.5 fv, d for central tenon 9 v 2 Dovetail: 2 6.5 F == bb h h r hrH,ukv kv fv, k =Rd min l Zfdv, d 1 r min 33 B1B1 B1B1 2 lZ bB1 k hB1 kvvv,v,lZ=Rd h hN γ 1 2 2 2 0.4 2 h v h h H,u k0.4hf F = b h r I eq. (4) h hN Dovetail: h Nh fv,fd v v,hd r 23 B1 B1 2 bB1 hB1 v,l Rd N v, d F Fv,ZRd == 33 bbkcrk h Nh h γ v, Rd cr N h kcalkcal h r 2 N hN r bB1 hB1 h h N fv, d 2= l F b b h r kv fv, d γ kv F =Z h Nv,h=Rdr b 3k h B1 B1 hH,u v,= Rd cr N h k 23 lZ =h 2 b h F r r kv k cal fv, fd γ Fv, b h Rd = 3 N h Nv v, fv,dd v, Rd h 32 2 N kkv Fv,B1Rd22= 3 b kcr h N h k v k = h b h=B1 Fv k1.0 f cal 3 b hN r v 1.0v, d hH,u kv k v,=Rd v h 6.5 6.5 2 kv k = min b Fv, Rd = 1.0 b h 2 r 1.0k l Z f 1 v 2 bB1 hH,u kv kv = min h 31.0 0.2 N 2 0.4 vh l Zv, d 1 2 3600 6.5 kv = min b h 0.4 h II eq. (5) 0.2 hH,u AB 3600 k = min lZ 1.0 1 2 2 b kv kvv = min hA 0.4 h h r hH,u kv 6.5 N B bB1 = h h r kv kN = min l 1 = vh 2 Z 2 h 0.4 h k h r

hN-r

hN-rhN-r hN-r

hN-r

bbef ˙ b B1

hN hN hN

IV eq. (5)

hB1-rhhB1N-rhB1-r hB1-rhB1-rhB1h-rB1h-rB1-r hB1-r

III eg. (5)

bB1 bB1 bB1

hB1-r

eq. (6)

bB1 bB1 bB1 bB1 bB1 bB1 bB1 bB1 bB1 bef ˙ b bef ˙ b bef ˙ b bef ˙ b

*** (Blaß & Sandhaas, 2017)

kv kv kv

kv v red.kv

kv==

kv =

red.

1.0

B 3600 h 1.0 AB 3600 0.2 AB 1.0

kv = min

3600 AB

kv = min

AB = bB1 b

0.2 3600 1.0 h A 2 0.2 B1

= kv = kmin v

kv red.

red.

hhhB1 2 h hN r h=B1 2 h

kv = min

red.

N h

2 tan 2

0.2

h B1

bB1 2

bB1

h B1

bB1

0.8

red. can beV approximated to a reduction constant 3 4 Ed similar av,ratio the notch height *** = of Ed = 4 to kv as it is F Rd 3 bH h H, u fv, d bH h H, u and the joist height in eq. (6). Fv, Rd =

6.5

18 h

H, u hH 2

l z hH

0.8

ft, 90, d

*** (CEN, 2019) *** (CEN, 2017)

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Shear and tension perpendicular to the grain for the header of a dovetail

Fv,Rd for different ratios hN/h 6

For a dovetail header two different equations can be used for this verification. These can be seen as eq. (7) and (8). The comparison between these can be seen in Figure 5.28.

Fv,Rd [kN]

4 3 2

Fv,Rd for the header

1 50

85 78 72 65 58 51 44 37 30 24 17

Fv,Rd,I

Fv,Rd,II

Fv,Rd,III

Fv,Rd,IV

Fv,Rd,V

Fv,Rd,N

Ft,90,Rd [kN]

hN /h [%]

Figure 5.27. Shear capacity at the notch for a dovetail with 5 different methods.

Fv,Rd,H,M ü

0.8 3 = 4

Fv, Rd =

VEd

Fv,Rd,H,FC

Figure 5.28. Shear capacity for a dovetail header comparing two different methods.

Eq. (7) is represented by the red curve from CEN (2017) and eq. (8) is taken from Müller et al. (2016) and includes more detailed geometrical variables. Both the equations give similar capacities for larger dovetails, hence where the height 2 under theb crack is smaller for a set total bB1 B1 header height. Since eq. (7) shows quite a h 8 the header where lot B1 smaller2capacity for the values of the two equations start and then differ from each other, this equation is used for the most conservative solution.

4 Fv, Rd = 3 bH h H, u fv, d

bH h H, u hN

85 78 72 65 58 51 44 37 30 24 17

hN /hH

Through this comparison the most conservative methods are either method II or III, depending on the ratio of the height of the dovetail to the height of the joist. Method III is more similar to the other curves and is the most conservative for the most critical ratios (smaller ratios, hence smaller dovetails). Therefore, this is chosen for the calculation of the capacity in the b general calculation method for dovetail B1 AB = bB1 2 tan 2 h B1 2 joints for this particular failure mode.

20 10

The dotted line in this graph show the shear force of a regular notch in the top of the cross-section with a similar geometry of the dovetail.

6.5

18 h H, u2

l z hH

hH 2

0.8

ft, 90, d

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(7)

(8)

ADDITIONAL GEOMETRICAL EFFECT

kv for different ratios of lZ/hN

This section investigates how the geometries and different calculation factors and formulas affect the loadbearing capacity. Some geometrical effects can be seen in the previous analysis as well, for example the effect of the tenon position and notch height on the load-bearing capacity.

1,00 0,90 0,80 0,70

kv [-]

0,60 0,50 0,40 0,30 0,20 0,10

Reduction factor, kv (-)

0,00

The risk of crack growth in a notched and tenon member can be taken into consideration through a reduction factor kv, which is found in EC5 (SIS, 2009). A further definition for this reduction factor is found in Chapter 6.

1 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0

⍺ [-]

0,83

0,50

0,25

Figure 5.29. Reduction factor, kv for different ratio of lB/ hB depending on alpha.

Geometry of the header

Figure 5.29 shows how kv changes depending on alpha (which is a geometry ratio), as defined below. The three geometries (one per graph) in this evaluation have different ratios of the depth of the tenon, lz , divided by the height of the beam, hB. In this test all curves has same height of the beam, set to 120 mm. Therefore it is the depth that changes between them. A low ratio between lz and hB gives a higher value of kv. Hence, a smaller lz gives a stronger notch.

The height under the cut geometry of a header for a notch, tenon and mortise or dovetail header influences the strength in shear and tension perpendicular to the grain. This is shown in Figure 5.30. The geometry illustration and related failure mode can be seen in Figure 5.31.

1,50

Once alpha reach approximately 0.5 the blue curve (0.25) turns up and gives a higher value. Having a short length of the tenon or notch and at the same time a very small height of the tenon seems to give a stronger connection which could be misleading. Although taking into consideration the purpose of the connections it is highly unlikely that such a small notch or tenon is possible for the loads required in a construction. At least from a production point of view as well.

𝜏𝜏Ed for different hu,H [mm]

𝜏𝜏 Ed [MPa]

1,00

0,50

0,00

200

160

120

bH [mm] TM , h.u,H = 100

DT, h.u,H = 100

TM , h.u,H = 200

DT, h.u,H = 200

Figure 5.30. Shear stress mortise depending on the width of the header bH.

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hH,u A

hH,u

B-B

CHAPTER CONCLUSION

hH,u

As a conclusion of this chapter what seem to be important failure modes based on the FE-analysis are: - Compression perpendicular to the grain at the bottom of the connection (notch). lz - Shear stresses at the notch. - TensionbH stresses at the notch (in combination with shear stresses).

It is also important to investigate the influence of layer orientation for the CLT connections. This would include rolling shear failure of the connection geometry.

Figure 5.31. Relevant geometry definitions of the header.

Another important part that might be needed to consider in the calculation method are approximations of a dovetail geometry to a similar notched geometry.

The results are quite reasonable. If the height of the header underneath the crack is smaller the larger the stresses are. What can be seen is also that even though this height is doubled the difference in stress size is very small.

Important to note is that the analysis in this chapter is focused on the joist geometry therefore research of similar failure modes for the header of the connection is needed for the development of the calculation method.

No clear correlation between height under the crack and width of the header can be deduced for when the geometry would be critical. Both the different heights show approximately the same width for when the stress increase dramatically towards a singularity, at 30 - 50 mm. The equation used in this comparison from CEN (2017) includes a restriction on the width for a dovetail geometry. This is seen as the dashed lines in Figure 5.30. The influence of this is minor and this type of geometrical restriction is therefore not taken into consideration in the general calculation method.

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6. PRELIMINARY CALCULATION METHOD

This chapter includes the full general calculation method for the carpentry connections investigated for C1 and C2. This chapter has its own chapter index and numbering separate from the report.

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INDEX 1. General info

1.1. Loads �� 1 1.2. Verification of the partial coefficient method �� 1 1.2.1. Design values for material properties �� 1 1.2.2. Partial coefficient �� 2

2. Limitations

3. Material properties

3.1. Modification factors �� 3 3.1.1. Factor, kmod �� 4 3.1.2. Factor, kcr �� 4 3.1.3. Factor, kn �� 4 3.1.4. Factor, kv �� 5 3.1.5. Factor, kc,90 �� 5

4. Verification of notched joint

4.1. Notched Member �� 6 4.1.1. Geometry �� 6 4.1.2. Shear and tension failure perpendicular to the grain at the notch corner �� 6 4.1.3. Compression failure perpendicular to the grain at the bottom of the notch ��11 4.1.4. Rolling shear failure in the notch (only for CLT) ��13 4.2. Header member ��15 4.2.1. Geometry ��15 4.2.2. Shear and tension failure perpendicular to the grain at the notch corner ��15 4.2.3. Compression failure perpendicular to the grain at the bottom of the mortise ��18 4.2.4. Compression failure perpendicular to the grain due to an axial force in the joist ��19

5. Verification of tenon and mortise joints

5.1. Tenon member ��21 5.1.1. Geometry ��21 5.1.2. Shear and tension failure perpendicular to the grain at the lower tenon corner ��21 5.1.3. Compression failure perpendicular to the grain at the bottom of the tenon ��23 5.1.4. Rolling shear failure in the tenon (only for CLT) ��24 5.2. Mortise member ��24

6. Verification of dovetail joints

6.1. Dovetail member ��24 6.1.1. Geometry ��24

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

6.1.2. Shear and tension failure perpendicular to the grain at the lower dovetail corner ��24 6.1.3. Compression failure perpendicular to the grain at the bottom of the dovetail ��28 6.1.4. Rolling shear failure in the tenon (only for CLT) ��28 6.2. Header member ��29 6.2.1. Geometry ��29 6.2.2. Shear and tension failure perpendicular to the grain at the notch corner ��30 6.2.3. Tension failure perpendicular to the grain due to an axial force in the joist ��32 6.2.4. Compression failure perpendicular to the grain at the bottom of the mortise ��32 6.2.5. Compression failure perpendicular to the grain due to an axial force in the joist ��32

7. Verification of peg joints

7.1. Peg ��33 7.1.1. Geometry ��33 7.1.2. Shear failure of a laterally loaded peg joint ��33 7.2. Supporting member ��34 7.2.1. Geometry ��34 7.2.2. Tension failure perpendicular to the grain of the supporting member ��34

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1. GENERAL INFO The basic calculations regards toDIMENSIONING material, loads and dimensioning of a cross-section the 1.with GENERAL RULES 1. DIMENSIONING RULES 1. GENERAL GENERAL DIMENSIONING RULES method in SIS (2009) should be followed. The calculation method in this report only include basicfrom calculations with regards to additional material, loads and dimensioning the parts that mightThe differ this standard or where verifications are made of a cross-section The regards to loads and dimensioning of The basic basic calculations calculations with regards to material, material, loadsbe and dimensioning of aa cross-section cross-section the with method in SIS (2009) should followed. The calculation method in this report only except from in this. the SIS be calculation in this report only the method method in SIS (2009) (2009) should be followed. followed. The calculation method inor this report only includeshould the parts that mightThe differ from thismethod standard where additional verifications are include the parts that might differ from this standard or where additional verifications include the parts thatmade might differ fromthis. this standard or where additional verifications are are except from When it comes tothis. the preliminary calculation method for glulam and CLT the verifications made made except except from from this. made for solid timber are itthe sametoasthe forpreliminary glulam butcalculation with givenmethod coefficients and changed When comes for glulam and CLT the verifications When comes preliminary method and CLT the verifications When it it data. comes to to the the preliminary calculation method for glulam and but CLT thegiven verifications material made for solid calculation timber are the samefor as glulam for glulam with coefficients and material made for timber are the same but given coefficients and material madeCLT for solid solid timber arethe theorientation same as as for forofglulam glulam but with with coefficients materialand data changed. For depending on the board somegiven geometrical data and is changed data changed. data changed. For CLT depending on the orientation of the board some geometrical verifications added due to theorientation different grain directions of the material.data is changed and data is changed and For on of board some For CLT CLT depending dependingverifications on the the orientation of the the board some geometrical geometrical data of is the changed and added due to the different grain directions material. verifications added due to the different grain directions of the material. verifications added due to the different grain directions of the material. 1.1. LOADS 1.1. LOADS 1.1. 1.1. LOADS LOADS Loads used for dimensioning are calculated from respective part of SIS (2009) for self-weight, Loads used for dimensioning are calculated from respective part of SIS (2009) for selfimposed load, load andimposed wind load. Loads for dimensioning are from part of Loads used used forsnow dimensioning are calculated calculated from respective respective partload. of SIS SIS (2009) (2009) for for selfselfweight, load, snowload and wind weight, imposed load, snowload and wind load. weight, imposed load, snowload and wind load. The impact of load duration and of service class is also SIS (2009) The impact load duration andstated serviceinclass is also .stated in SIS (2009) . The The impact impact of of load load duration duration and and service service class class is is also also stated stated in in SIS SIS (2009) (2009) .. 1.2. VERIFICATION OF PARTIAL COEFFICIENT METHOD 1.2.THE VERIFICATION OF THE PARTIALCOEFFICIENT METHOD 1.2. 1.2. VERIFICATION VERIFICATION OF OF THE THE PARTIALCOEFFICIENT PARTIALCOEFFICIENT METHOD METHOD 1.2.1. Design values 1.2.1. for material properties Design values for material properties 1.2.1. Design values for material properties 1.2.1. Design values for material properties The design values of the strength properties, Xd , used in this method is calculated according The design values of the strength properties, Xd , used in this method is calculated The design values of the strength properties, X ,, used in to equation (1) from SIS (2009). The design values ofaccording the strength properties, X used in this this method method is is calculated calculated d to equation (1) from d SIS (2009). according to equation (1) from SIS (2009). according to equation (1) from SIS (2009). Xk X Xkk X = k (1) (1) d mod X (1) Xdd == kkmod (1) M mod Where: Where: Where: X Xkk

M M

kkmod mod

Where:

M M

Xk is the characteristic value of the load-bearing properties. is value of is the value ofproperties. the load-bearing properties. is the the characteristic characteristic valuecharacteristic of the the load-bearing load-bearing properties. partial coefficient materialproperties properties M is is thethe partial coefficient forfor thethe material is for properties is the the partial partial coefficient coefficient for the the material material properties (Recommended values in Table 1). (Recommended in (Recommended (Recommended values values in Table Table 1). 1).values in Table 1). kmod isisthe thecorrectional correctionsfactor factorwith withregard regardtotoload loadduration durationand and moisture conten is with regard to load duration and moisture conten is the the corrections corrections factor factor with regard to load duration and moisture conten (Recommended values in Table values 3). in Table 3). moisture content (Recommended (Recommended (Recommended values values in in Table Table 3). 3). Factor, Factor, M M Solid timber Solid timber Glulam Glulam CLT CLT

Factor,

Source Source Source SS EN 1995 1 1 : 2004 Sv 1.3 SS SS EN EN 1995 1995 11 11 :: 2004 2004 Sv Sv SS EN 1995 1 1 : 2004 Sv 1.25 SS EN 1995 1 1 : 2004 SS EN 1995 1 1 KL-Trähandboken : 2004 Sv Sv 1.25 KL-Trähandboken KL-Trähandboken

Solid timber 1.3 1.3 Glulam 1.25 1.25 CLT 1.25 1.25

Table 1: Partial coefficients. Table Table 1: 1: Partial Partial coefficients. coefficients.

2.3.2. Design values for load-bearing capacities 2.3.2. 2.3.2. Design Design values values for for load-bearing load-bearing capacities capacities The design value, Rd , is calculated in a similar way as the strenght properties in equation The design value, R , is calculated in a similar way The design value, R(1) is calculated similar way as as the the strenght strenght properties properties in in equation equation 1 (56)1995-1-1:2004. dd , and accordingintoaS-EN (1) and according to S-EN 1995-1-1:2004. (1) and according to S-EN 1995-1-1:2004. Rk R Rkk R = k d mod R Rdd == kkmod M mod where: where: R Rkk

where:

M M

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Rk is the characteristic values of the load-bearing capacity. is is the the characteristic characteristic values values of of the the load-bearing load-bearing capacity. capacity.

1.2.2. Partial coefficient Table 1. Partial coefficients.

Material

Factor, γM

Source

Solid timber

1.3

(SIS, 2009)

Glulam

1.25

(SIS, 2009)

CLT

1.25

(Gustafsson et al., 2019)

2. LIMITATIONS This calculation method is limited to CLT, glulam and solid timber although might be relevant and applicable, to some extent, for other timber engineered materials as well. The calculation material is limited to verifications in ultimate limit state (ULS) and no consideration is taken for the connections in a service limit state (SLS). The calculation material is based on moment free connections. The equilibrium equations are therefore limited to simply supported members or connections with moment free supports. A geometrical constraint of this calculation method is that the geometry of the header follows the geometry of the joint (notch, tenon or dovetail). The header is the "inverted" shape and therefore the geometries of the two members are restricted by one another.

(57)

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3. MATERIAL PROPERTIES 3.1. MODIFICATION FACTORS Table 2 below includes all correctional and modification factors with small explanations of their purpose. It also states the equation number of the equation or equations including the factor. Table 2. Modification factors.

Factor

Source

Purpose

Equations

kmod

Correction factor with regard to load duration and moisture content. Found in Table 3.

(1), (25) & (26)

(SIS, 2009)

kcr

Factor taking into account the risk of cracks.

(2), (6), (14), (18) & (20)

(SIS, 2009)

Proportionality constant established by testing

Reduction factor that takes into consideration the risk of crack growth in notched members.

(3) & (7)

(Blaß & Sandhaas, 2017)

(6), (7), (20), (23) & (24)

(Blaß & Sandhaas, 2017)

Correction coefficient to take into account the non-uniformly distributed tensile stresses.

(11)

(Blaß & Sandhaas, 2017)

kc,90

Factor that take into consideration the fact that compression perpendicular to the grain is distributed over areas larger than the directly loaded area, here the support area of the connection.

(4), (5), (13) & (19)

(SIS, 2009)

Correction factor based on test results that takes into consideration influences of the tenon geometry.

(20) & (21)

(Blaß & Sandhaas, 2017)

(58)

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3.1.1. Factor, kmod

Table 3. Modification factor, kmod .

Material

Solid timber

Glulam

CLT

Load duration class

Climate class

Permanent

Long

Average

Short

Instantaneous

0.60

0.70

0.80

0.90

1.10

0.60

0.70

0.80

0.90

1.10

0.50

0.55

0.65

0.70

0.90

0.60

0.70

0.80

0.90

1.10

0.60

0.70

0.80

0.90

1.10

0.50

0.55

0.65

0.70

0.90

0.60

0.70

0.80

0.90

1.10

0.60

0.70

0.80

0.90

1.10

0.50

0.55

0.65

0.70

0.90

** *

** (SIS, 2009) ** (Gustafsson et al., 2019)

3.1.2. Factor, kcr

4.1.2. Solid timber 2.0 Solidtimber timber 4.1.2. Solid timberSolid f v, k

kcr =

2.0 2.5 fv, k

Solid timber Glulam Glulam

v, k

(2)

2.5 CLT CLT* kcr = 0.67 Glulam fv, k * For general CLT boards the cross lamination of the perpndicular layers mean that kcr * For general CLT boards the cross lamination of the perpendicular layers mean that kcr does CLT doesn't 0.67 need to be taken into consideration and is therefore equal to 1.0, but for not need to be taken into consideration andwith is therefore 1.0 (Blaß 2017). constructions a small equal boardtowidth such & as Sandhaas, a connection point kcr = 0.67 is used (KLFor general CLT boards cross lamination perpndicular mean that kcr But for constructions with a *small board width, such asthe a connection point, kof =the 0.67 is used layers cr trähandboken, p.57 and Timber Engineering). according to Gustafsson et al. (2019). doesn't need to be taken into consideration and is therefore equal to 1.0, but for constructions small timber board width such as a connection point kcr = 0.67 is used (KL5.0 with a Solid 3.1.3. Factor, kn trähandboken, p.57 and Timber Engineering). Glulam k = 6.5 n

6.5 CLT Solidtimber timber 5.0 Solid Glulam Glulam kknn =is a 6.5 proportionality constant established by testing which (3) gave a range of constants for 6.5 CLT * solid timber, glulam CLT and LVL and simplified to the seen above to be used in the factor kv .

knThe is afactor proportionality constant established by testing for which gaveisa applied range ofasconstants for isvalue not specified soas the glulam an * The factor kn is not specified* for CLT so kthe for glulamfor is CLT applied anvalue approximation. n solid timber, glulam and LVL and simplified to the seen above to be used in the factor kv . approximation. *k The kn (59) is factor not specified forinto CLT so the valuethe forrisk glulam is applied is afactor reduction that takes consideration of4crack growthasinannotched v approximation. members. k must not exceed 1.0 to ensure no shear failure occur in the remaining crossv

section. For beams withthat notched theconsideration opposite sidethe of the = 1.0 in cannotched be used. kv is a reduction factor takes at into risksupport of crackkgrowth v members. kv must not exceed 1.0 to ensure no shear failure occur in the remaining cross* kv for notch and TM section. For beams with notched at the opposite side of the support kv = 1.0 can be used. * kv for and dovetail CHALMERS Architecture Civil Engineering, Master’s Thesis ACEX30 * kv for notch and TM

4.1.3. Glulam

kn is a proportionality constant established by testing which gave a range of constants for solid timber, glulam and LVL and simplified to the seen above to be used in the factor kv (Blaß & Sandhaas, 2017). 3.1.4. Factor, kv

kv is a reduction factor that takes into consideration the risk of crack growth in notched members. kv must not exceed 1.0 to ensure no shear failure occur in the remaining crosssection. For beams with a notch at the opposite side of the support kv = 1.0 can be used (Blaß & Sandhaas, 2017). Different functions are used for kv depending on the geometry of the connection. The discussion of this can be seen in Chapter 5 in the Report. If different equations are used in the equations this will be stated. direction one or more layer would have grain parallel to the compression and this failure mode would sieze to be relevant. 3.1.5. Factor, kc,90 is a factorthe thatfact take intocompression consideration the fact that to compression This factor take into consideration that perpendicular the grain isperpendicular to the grain is distributed over areas larger than the directly loaded area, here the support area. Therefore, distributed over areas larger than the directly loaded area. This is the support area for the joists this factor increases the compression strength perpendicular to the grain. This factor is given with either a notch, tenon or this method thefor loaded area from joist(s) for the distance between fordovetail discrete in supports (EC5)orand softwood. The the values for when the headers. Therefore, thiscompression factor increases the compression strength perpendicular to the zones is long enough to not interact with each other (hence, l 1 2 h N ) for all grain. This factor is given for supports for for softwood 2009). of discrete the notches. Theand values CLT is(SIS, taken from KL-trähandboken. For a joist the compression For zones will notifinteract withone each other the geometries for these this case more than notch is as acting on the header the compression zones might does not interact. Althoughinteract for a header joists this amight be value the case. Figure 1 and l 1with2multiple h N , which gives smaller for the coefficient, kc, 90 (EC5). This is and equation (5) display theillustrated conditions for such an interaction. in Figure X below. h = h N

kc, 90 = b

H, u

1.0

for all materials,for l1< all 2∙hmaterials, l1 H,u

eq. (5)

l1 ≥ 2∙hH,u

2 h H, u

2 h H, u (4)

Figure X. Header with interacting tenons in compression. l l1

5. VERIFICATION OF NOTCHED JOINT h

hH,u 5.1 GEOMETRY

bB1

hB1 lZ

hN-r

The geometry of a notched beam is illustrated with the relevant geometry constants in Figure 1.

Figure 1. Geometrical conditions for kc,90 .

kv b

hH,u kv

(60)

A discussion for what equations to use when validating the geometry where the notch is on the opposite side of the support can be seen in Chapter X.

bB1 hB1-r

Figure 1.

5.2 NOTCHED MEMBER kv

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

5.2.1. Geometry

5.2.1 Failure in shear and tension at the notch corner

approximation. kv is a reduction factor that takes into consideration the risk of crack growth in notched members. kv must not exceed 1.0 to ensure no shear failure occur in the remaining crosssection. For beams with notched at the opposite side of the support kv = 1.0 can be used.

kc, 90 =

1.5

Solid timber for solid timber

1.5

CLT *

1.75

for glulam, Glulam, l ≤ 400 mm for CLT

400 mm (5)

* This coefficient is only used for CLT orientation where compression perpendicular to the * This coefficient is only used forisCLT orientation whereofcompression perpendicular to theoriented in a different grain acting upon all layers the geometry. For CLT boards grain is acting upon all layers of the geometry (Gustafsson et al., 2019). For CLT boards oriented in a different direction one or more layer would have grain parallel to the compression and this failure mode would seize to be relevant.

4. VERIFICATION OF NOTCHED JOINT 4.1. NOTCHED MEMBER 4.1.1. Geometry All geometrical data for the verifications can be seen in Figure 2 for this member of the connection. The notch can be either on the opposite side as the support (referred to in this thesis as a bottom notch [b.n.]) or on the same side as the support (called a top notch [t.n.]). Figure 2.a. shows a top notch with two different cross-section possibilities. One where the width of the beam is larger than the width of the notch (this is also a possibility for the b.n. even though this is not visualized) and one where they are the same. For both of them it is the width of the notch that is used in the calculations. Figure 2.b shows a bottom notch. The discussion on which equations to use with regards to a b.n. as seen in Figure 2.b. can be read in Chapter 5 in the report. 4.1.2. Shear and tension failure perpendicular to the grain at the notch corner For notched beams one failure occur where a crack appear in the tip of the notch and propagate parallel to the grain, see Figure 3. These cracks are generally generated by a combination of Figureand 2. Failurestresses due to shear and tension perpendicular to the grain. shear perpendicular to the Figure 2. tensile Failure dueFigure to shear tension perpendicular to the grain. 2. and Failure due to grain. shear and tension perpendicular to the grain.

Since infinite stresses can not occur fracture mechanics are used to estimate the following Fracture mechanics used tooccur estimate thecan following formula for verifyingthe the shear and Since infinite stressesisSince can not fracture mechanics are used to estimate following infinite stresses notof occur fracture mechanics are used formula for verifying the shear and tensile capacity a notched beam. The formula isto estimate the following formulacapacity for verifying the shear and tensile capacity of a notched beam. The formula is tensile of a notched beam. The formula is reduced to shear verification but this formulabut forthis verifying theverifies shear and capacity of a notched beam. toThe formula is reduced to shear verification formula thetensile capacity for tension perpendicular reduced to shear verification but this formula verifies to reducedfor to tension shear verification but the this formula verifies theperpendicular capacity perpendicular to formula capacity perpendicular to capacity the fibresfor as tension well (SIS, 2009). for tension the fibersverifies as wellthe (EC5). the fibers as well (EC5). the fibers as well (EC5). VEd 3 VEd 3 kv 3 fv, d VEd d = 2 = kv fv, d d = kv fv, d h b d (6) 2 2 h bef h bef ef Where: Where: Where: VEd VEd

(4) (4)

bef = kcr bN bef = kcr bN bN bN

kv = min kv = min

(4)

Where: is the dimensioning shear force. is dimensioning shear force. is theVthe dimensioning shear force. is the dimensioning shear force. Ed

is theiswidth of theofnotch reduced to account for thefornegative influence of the width the notch reduced to account the negative influence is the bwidth ofbthe notch reduced to account forreduced the negative influence of negative influence of = k is the width of the notch to account for the cr (k N can be found in Chapter 3.1). ofef(k cracks cracks can be cr found in Chapter 4.1). cr cracks (kcr can be found in Chapter cracks (kcr can 4.1). be found in Chapter 4.1).

is the width of the notch. May be modified for CLT see Chapter 5.2.1.1. is thebwidth of the notch. be of modified for May CLTbe seemodified Chapter for 5.2.1.1. is theMay width the notch. CLT see Chapter 5.2.1.1. N

h h

1.0 1.0 (61) 1.1 i 1.5 1.1 i 1.5 kn 1 kn 1 kn 1 h h kv = min x 1 2 1 2 2 0.8 xh 0.8h h

1.0

1.1 i

2 2

1.5

x 0.8 h

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

is a reduction factor that takes into consideration the risk of crack growth in notched

(5) (5)

(5)

2 v, d Where: (4) hh bbefef kvv fv, ef d = 2 (4) d to the fact that the stresses are not uniformly distributed Where: The factor 1.3his bapproximated due ef (4) VEd is the shear force.in peak stresses and an approximate increase at the notch corner, see dimensioning Figure X, which results Where: (4) Where: (4) VEd is the dimensioning shear force. of the tension force component by 30 %. Approximated through FE simulations by Henrici (4) Where: V isthe thewidth dimensioning shear force. to account for the negative influence of Where: (Timber engineering). V the dimensioning shear force. befEd isis of the notch reduced Where: Ed= k b bVef = kcrcr bN is the width of the notch reduced to account for the negative influence of Where: is the dimensioning shear N cracks (k canthe be bond found in force. Chapter 4.1). V isthe thewidth dimensioning shear force. For reinforcement layers line stresstoasaccount well as thethe tensile stressinfluence is calculated Ed=these cr V is the dimensioning shear force. bbefEd b is of the notch reduced Ed= k k b is the width of the notch reduced to account for the negative negative influence of of cracks (k can be found in Chapter 4.1). for cr N cr N and cr VefEd verified. is the dimensioning shear force. cracks (kcr can can be found found inMay Chapter 4.1). for CLT == kkcr bbN is the width of the notch reduced to account the negative influence of cracks (k be in Chapter 4.1). bbef is the width of the notch. be modified see Chapter 5.2.1.1. is the the width width of the the notch notch reduced reduced to to account account for for the the negative negative influence influence of of cr of bbefN kcrcr bNN bond is ef = effective is the width of the notch. May be modified for CLT see Chapter 5.2.1.1. The line area taken into account is the area below the critical area of the cracks (k can be found in Chapter 4.1). N= k b b is the width of the notch reduced to account for the negative influence of cracks (kcrcr can can be found inMay Chapter 4.1). for CLT see Chapter 5.2.1.1. failure area the notch corner). cr(hence N theis is the below width of be the notch. be modified modified cracks (k found in Chapter 4.1). bbefN the width notch. May be for CLT see Chapter 5.2.1.1. cr of the 1.0found N cracks (k can be in Chapter 4.1). is the width of the notch. May be modified for CLT see Chapter cr 1.0notch. b is the width of the May be modified for CLT see Chapter 5.2.1.1. Fis ist,90,d the4.1.2.1. width of of the the notch. May be be modified modified for for CLT CLT see see Chapter Chapter 5.2.1.1. 5.2.1.1. 1.1 i 1.5May bbNN the width notch. N 1.0 1.5 1.0 k 1 1.1 i May be modified for CLT see Chapter 5.2.1.1. fthe ed, d = bN is the width of notch. n Rv, d h 1.5 kv = min nb h h N knl r 1 1.0 1.1 ii 1.5 1.1 1.0 kv = min kknn 11 1.0 hx1.5 1 2 1.0 2 1.1 i 1.5 (7) min 1.1 kkvv == min h 0.8 ihhixh1.5 1 1.1 kkn 211 2 1.5 h k 12 0.8 ihxxh 1.1 k = tensile min The in theknnreinforcement hlayers is11 then22acting parallel to the fibers and verified (7) min stress 12 h 0.8 hh kkvvv == min h 0.8 n as: h x 1 h (5) 2 2 kv = min 2 xx 11 hh 0.8 22 2 (5) 0.8 xhh h 0.8 1 the2 risk of crack growth in notched Ft,90,d ft, d takes 2 into is a reduction factor that consideration (5) h (5) h 0.8consideration is that takes into the risk of crack growth in notched = (See factor t,a dreduction h members Chapter kX). n t l (5) k takes members (Seer factor Chapter r is reduction factor that takes into into factor, consideration the risk risk of crack crack growth growth in in notched notched (5) is aa reduction that consideration the is X). a reduction see Chapter 3.1.of (5) *members For beams with notch at the opposite side of the support k = 1.0 can be used. members (See (See Chapter Chapter X). X). (5) v is a reduction factor that takes into consideration the risk of crack growth in notched (8) * For beams with notch the into opposite side of the support = 1.0growth can bein is reduction factor that at takes into consideration the risk of ofkcrack crack growth inused. notched is aa reduction factor that takes consideration the risk v notched members (See Chapter X). For beams with notch at the into opposite side ofsee the support = 1.0 1.0 can be beinused. used. members (Seewith Chapter X). **k For beams the opposite side of the support kkcrack can is a reduction factor takes consideration the risk of growth notched is notch a that proportionality constant, Chapter 4.1. is X). a at proportionality constant, see Chapter members (See Chapter v =3.1. v Where: n k is a proportionality constant, see Chapter 4.1. members (See Chapter X). * nFor beams with notch at the opposite side of the support k = 1.0 can be used. For beams beams with with notch at the the opposite opposite side of ofsee theChapter support4.1. 1.0 can can be be used. used. is proportionality constant, **kki nFor at side the support kkvvv == 1.0 is aathe proportionality constant, see Chapter isnotch notch inclination (see Figure 1). 4.1. 2 n if only internal layers act as reinforcement n *i For beams withisnotch at the opposite side of the support k = 1.0 can be used. is the notch inclination, see Figure 2. notch inclination (see Figure 1). 4.1. v athe proportionality constant, see Chapter nk = 2 n 1is is proportionality constant, see Chapter Chapter 4.1. if external as reinforcement is height of thelayers beamare inincluded mm. is aathe the notch inclination (see Figure 1). khkiinnnb proportionality constant, see 4.1. is the notch inclination (see Figure 1). is athe height of the constant, beam in mm. khn is proportionality see Chapter 4.1. ihi theh number ofis is the notch inclination (see Figure isheight the joist height, see Figure 2.1). the height of the beam in mm. is bond-lines. is the notch inclination (see Figure 1). h is the of the beam in mm. N i h is the notch inclination (see Figure 1). = is the ratio between the height of the notch and the height of the beam. May ih hN is the notch inclination (see Figure 1). is the height of the beam in mm. h = is the ratio between the height of the and the the height of theMay beam. n is the number of reinforcement layers. h is the height of the beam in mm. h is the ratio between the notch height joist height. be May h modified the see height of the5.2.1.1. beam in mm. notch and N hN be for is CLT Chapter = is the ratio between the height of the notch and the height of the beam. May h is the height of the beam in mm. = is the ratio between the height of the notch and the height of the beam. May be forthe CLT see of Chapter 5.2.1.1. hhhhN modified CLT, seeFigure Chapter h modified is height thefor beam (See X).4.1.2.1. h N x is the distance from the support reaction to the corner of the notch. be==modified modified foris CLT see Chapter Chapter 5.2.1.1. the ratio between the height of the notch and the height of the beam. May N be for see 5.2.1.1. isCLT the ratio between the height of the the notch notchtoand and the height of the the beam. May May x = hhhhN is isthe the ratio distance between from the the height support of reaction thethe corner height of the of notch. beam. is the height of the notch, see Figure 2. h is the height of the notch (See Figure X). be modified for CLT see Chapter 5.2.1.1. N = is the ratio between the height of the notch and the height of the beam. May f is the design shear strength of the material. is distance from the support support reaction reaction to to the the corner corner of of the the notch. notch. be modified for CLT CLT Chapter 5.2.1.1. xx v,modified is the see distance from the h be for Chapter fv, dd is the see design shear5.2.1.1. strength of the material. be modified for CLT see Chapter 5.2.1.1. lx is the effective width ofstrength reinforcement (See is the distance from the support reaction to the corner of the notch. isdesign the distance from theof reaction theX). corner the notch, is the shear ofsupport the material. is the distance from the support support reaction to Figure thetocorner corner of the theofnotch. notch. is shear strength the material. 5.2.1.1. Modifications for CLT xxrffv, is the design distance from the reaction to the of v, dd see Figure 2. for CLT x5.2.1.1. is the distance fromstrength the support reaction to the corner of the notch. fv, d Modifications is the design shear of the material. is the design rolling shear strength of material. the reinforcement layer(s). is the the design design shear strength of the the material. 5.2.1.1. Modifications for CLT ffv, is shear strength of Rv, v, dd d Modifications 5.2.1.1. for CLT fv, d is theisdesign shearshear strength of theofmaterial. the design strength the material. 5.2.1.1. Modifications for CLT t5.2.1.1. is the thickness of one reinforcement layer(s) (See Figure X). 5.2.1.1. Modifications for CLT Modifications for CLT r 5.2.1.1. Modifications for CLT 4.1.2.1. Modifications for CLT kk = 2.0 is a correction coefficient to take into account the non-uniformly distributed For cross laminated timber the verification in eq. (6) may need to be modified in order to account for the grain direction in the different layers. For a notched geometry of CLT where the load is applied perpendicular to the grain direction of all layers the height of the cross section in the calculation may need to be altered. This is done if the notch corner occurs in a layer where the grain is perpendicular to the crack propagation as seen in Figure 3. This type of connection can be seen in Figure 4. This change will affect α and make it smaller than for a notched beam where all layers act parallel to the crack. This in turn will give a lower capacity through the calculations as Fv,Rd decreases for a decreasing height of the notch.

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CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

A-A

A I

bN a. B-B a. B-B

x 1

VEd

VEd 1

lz B

b h

lz B

Figure 2. a. Geometry of a top notch. b. Geometry of a bottom notch.

VEd x

lz VEd

Figure 3. Shear and tension failure perpendicular to the grain at the notch corner for the joist.

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CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

For cross laminated timber the verification in equation (4) may need to be modified in order For cross laminated timber the verification in equation (4) mayinneed to be modified to account for the grain direction the different layers. in Fororder a notched geometry of CLT to account for the grain direction in the different layers. For a notched geometry of CLT where the load is applied perpendiculat to the grain direction of all layers the height of the where the load is applied perpendiculat to in thethe grain directionmay of all layers heightThis of theis only if the notch corner cross section calculation need to bethealtered. cross section in the calculation mayinneed to be altered. if the notch corner occurs a layer where the This grain is is only perpendicular to the crack propagation as seen in Figure occurs in a layer where the grain is perpendicular to the crack as seen X. This type of connection can bepropagation seen in Figure X. in Figure X. This type of connection can be seen in Figure X.

h Figure X.

Figure X.

This change will affect and make it smaller than for a notched beam where all layers act This change will affect and make to it smaller than forin a notched layers act parallel the crack. This turn willbeam give where a lowerallcapacity through the calculations as parallel to the crack. This inFturn will give a lower capacity through the calculations as decreases for a decreasing height of the notch. h v, Rd Fv, Rd decreases for a decreasing height of the notch. hN Also depending on the lamination direction, one or more lamination layers may act as Also depending on the lamination direction, or more lamination layers for maya possible act as tension failure perpendicular to reinforcement of one the cross section. This could reinforcement of the cross section. This could for a possible tension failure perpendicular to the grain be proven sufficient to strengthen the cross-section against this type of failure, see the grain be provenof sufficient toifstrengthen cross-section againstofthis of failure, see of the layer needs to be Figure 4. Modification height for CLT notch a athe Figure widthtype for CLT with not all Figure X.atFor layer to act5.asModification reinforcement the grain direction l z,ef Figure X. For ato layer topropagation. act as reinforcement direction thepropagation. layer needs to be layer perpendicular the crack layers parallel to theofcrack perpendicular to the the grain tension failure. perpendicular to the tension failure. When one or more layers act as reinforcement the shear stress calculated for verification of Depending the layer one or more layers may act as reinforcement When one oronmore layersorientation, actthe as reinforcement the lamination shear stress calculated of of the cross-section, but only crack propagation does not occur accrossfor theverification whole width of the cross section. This for a possible tension failure perpendicular to the grain beillustrated in Figure X. the crack propagation doescould not occur accross the whole width of the cross-section, but only in the layers with a grain direction parallel to the crack. This is in the layers withto a grain direction to theagainst crack. This is illustrated proven sufficient strengthen the parallel cross-section thishNtype of failure,inseeFigure FigureX.5. For a layer to act as reinforcement the grain direction of the layer needs to be perpendicular hto the tension failure. When one or more layers act as reinforcement the shear stress calculated for verification of bN lz the crack propagation does not occur across the whole width of the cross-section, but only in the layers with a grain direction parallel to the crack. This is illustratedlz,efin Figure 5. Figure X. Figure X. For verifying the cross-section for shear and tension failure perpendicular to the grain the For verifying to cross-section for shear and tension failure perpendicular to the grain the sameverifying methodtoascross-section in Chapter 4.1.2. is used but with a reduced width of thetocross-section. For for shear tension perpendicular the grain same as in and 5.2.1 is usedfailure but with a reduced width of thethe cross-section. same as in 5.2.1 is used but with a reduced width of the cross-section. h bN = t (8) i, bN = t i,

Where: Where:

Where:

t is the thickness of each layer with grain direction parallel to the crack. i, each layer with grain direction parallel to the crack. is the thickness of is the thickness of each layer with grain the crack. i, z0,Ndirection parallelzto 0,N z By decreasing the width of the cross-section for shear the shear stresses increase zT,i the zT,iverification By decreasing the width of the shear verification increase A shear stresses andcross-section might result for in failure for the cross-section. In this case itA is therefore relevant to also y By decreasing theinwidth the cross-section for In shear verification the shearrelevant stressestoincrease and might result failureoffor the cross-section. this case it is therefore also verify the reinforcement layer. B relevant to zalso verify the reinforcement layer. and might result in failure of the cross-section. In this case it is therefore B,i verify B Anisapproximation for this type to of the verification done by usingtothe formulas for externally the reinforcement layer which the layer or layers parallel load andisperpendicular zT,ithe fibers in a similar notched x of verification An approximation for this type is doneplates by using formulas for externally on reinforcement withthe fibers perpendicular to the crack propagation. platesglued glued on reinforcement with fibers perpendicular to the fibers in a similar notched cross section. The geometrical data for this type of verification can be seen in Figure X. cross section. The geometrical data for this type of verification can be seen in Figure X. An approximation for this type of verification is donebby using the formulas for externally bi i glued on reinforcement plates on a similarly notched cross-section. The geometrical data for Vverification b. this type of can be seen in Figure 6. a. Ed t

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CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

formula for verifying the shear and tensile capacity of a notched beam. The formula is reduced to shear verification but this formula verifies the capacity for tension perpendicular to the fibers as well (EC5). hN

3 = 2

h VEd

kv fv, d

h bef

(4) Where: z0,T he

VEdz0,T bef = kcr bN

z0,B

is the dimensioning shear force. is the width of the notch reduced to account for the negative influence of cracks (kcr can be found in Chapter 4.1). h is the width of the notch. May be modified for CLT see Chapter 5.2.1.1. tr kn

1.0

1.1 i

1.5

h kv = min Figure 6. Geometry verifying one orperpendicular several reinforcement layers for CLT. Figure 2. Failure due to shearforand tension to the grain. x 1 2 2 h 0.8fracture Since infinite stresses can not occur mechanics are used to estimate the following h This reinforcement layer verification is generally only needed for notches at the same side as formula for verifying the shear and tensile capacity of a notched beam. The formula is This reinforcement layer verification is the onlycapacity needed for for tension notches perpendicular at the same side the support and tenons. reduced to shear verification but this formula verifies to as the (5) support. the fibers as well (EC5). The generatedfactor force that component tension perpendicular to thegrowth grain in cannotched be calculated is a reduction takes intoforconsideration the risk of crack The generated force component for tension perpendiculat to the grain can be calculated as according the V eq. (9) (Blaß members3 (See Chapter X). & Sandhaas, 2017). Ed (Timber engineering). kv fv, d d = 2 h befnotch at the * For beams with opposite side of 2the support kv =31.0 can be used. (9) Ft, 90, d = 1.3 VEd 3 1 2 1 (4) k is a proportionality constant, see Chapter 4.1. n

(6) This reinforcement layer verification is only needed for notches at the same side as the is the notch inclination (see Figure 1). support. The factor 1.3 is approximated due to the fact that the stresses are not uniformly distributed is the dimensioning shear force. isheight the dimensioning shear at is the notch corner, Figure X,force. which results in peak stresses and an approximate increase of thesee beam in mm. The the generated force component for tension perpendiculat to the grain can be calculated as of the tension force component by 30 %. Approximated through FE simulations by Henrici (Timber engineering). engineering). bef =hkcr bN (Timber is theiswidth of the notch reduced toheight accountand forthe thejoist negative influence the ratio between the notch height. May beof N = h is cracks the modified ratio between the height of the notch and the height of the beam. May 2 3 see can be CLT found Chapter 4.1). Ft, 90, =(k 1.3 Vfor 3 layers 1 in Chapter 24.1.2.1. 1 stress as well as the tensile stress is calculated For these reinforcement the bond line cr d Ed be modified for CLT see Chapter 5.2.1.1. and verified. b is the width ofdue the to notch. Maythat be modified forare CLT Chapterdistributed 5.2.1.1. The factor 1.3 is approximated the fact the stresses not see uniformly (6) xN is the distance from the support reaction to the corner ofbelow the notch. The effective bond line area taken into account is the area the critical area of the at the notch corner, which results in peak stresses and an approximate increase of the tension failure (hence the area below thedue notch the corner). The factor isshear approximated fact that the stresses are not uniformly distributed 1.0Sandhaas, force 30design %1.3 (Blaß & 2017). fv, d component isbythe strength of theto material. at the notch corner, see Figure X, which results in peak stresses and an approximate increase Ft,90,d 1.1 i 1.5 by 30 %. Approximated through FE simulations by Henrici of the tension force component For theseModifications reinforcement layers kfor 1 the bond line fstress as well as the tensile stress is calculated 5.2.1.1. CLT ed, d = engineering). (Timber Rv, d nnb h h N hl r kv =verified. min and For these reinforcement layers the line stress as well as the tensile stress is calculated x 1 bond 2 (7) The effective bond line area taken into is the2area below the critical area of the failure h verified. 0.8 account and h (the area below the notch corner). The bond line is verified by calculating the stresses over the The stress inline thearea reinforcement is isthen parallel the fibers verified The tensile effective taken into layers account theacting area below thetocritical area and of the surface and verifying againstbond the rolling shear strength of the material. as: (5) failure (hence the area below the notch corner). Where: Where: i VEd h

is a reduction factor thatFtakes into fconsideration the risk of crack growth in notched t,90,dFt,90,d t, d members (See Chapter X). = t, d fRv, d ed, d = n t l nb r hr h Nkk l r * For beams with notch at the opposite side of the support kv = 1.0 can be used. kn

(10) (7) (8)

is a proportionality constant, see Chapter 4.1. Where: The tensile stress in the reinforcement layers is then acting parallel to the fibers and verified as: i is the notch inclination (see Figure 1). 2 n if only internal layers act as reinforcement n = F f h in mm.layers are included as reinforcement bis the2height nt,90,dof 1 the beam t, dif external 10 t, d = (65) k n tr l r k hN is the number of bond-lines. = h is the ratio between the height of the notch and the height of the beam. May (8) n is the number of reinforcement layers. be modified for CLT see Chapter 5.2.1.1. Where: h is the distance is thefrom height the beam (See Figure X). of the notch. theof support reaction to the corner 2Architecture n the height if only internal layersFigure act as Thesis reinforcement hN is the notch (See CHALMERS and of Civil Master’sX). ACEX30 fv, d is the design shear strength ofEngineering, the material. nb = 2 n 1 if external layers are included as reinforcement lr the effective width of the reinforcement (See Figure X). 5.2.1.1. Modifications forisCLT x

Ft, 3 1 2 1 and verified. ddd = 1.3 VEd t, 90, 90, Edthese and verified. t, 90, ForEd reinforcement layers the bond line stress as well as the tensile stress is calculated and verified. and verified. The effective bond line area taken into account is the area below the critical area of the (6) The effective bond line area into is The effective bond linebelow area taken taken into account account is the the area area below below the the critical critical area area of of the the (6) failure (hence the area the notch corner). failure (hence (hence the the area belowbond the notch notch corner). Thearea effective line area taken into account is the area below the critical area of the failure below the corner). The is due to the that the stresses The factor factor 1.3 1.3failure is approximated approximated due tobelow the fact fact thecorner). stresses are are not not uniformly uniformly distributed distributed (hence the area the that notch F at the notch corner, see Figure X, which results in peak stresses and an at the notch corner, see Figure X, which results in peak stresses and an approximate approximate increase increase Ft,90,d F t,90,d of the force component 30 %. t,90,d == ffby dd tension ofed, the tension force component by 30 %. Approximated Approximated through through FE FE simulations simulations by by Henrici Henrici F ed, Rv, d n h h l = f ed, d t,90,d Rv, (Timber n l Rv, dd nbb hhed, d hh N (Timber engineering). engineering). N = l rrr fRv, d b N nb h h N l r For these reinforcement layers the bond line stress (7) For these reinforcement layers the bond line stress as as well well as as the the tensile tensile stress stress is is calculated calculated (7) (7) and verified. and verified. (7) The tensile stress in the reinforcement layers is then acting parallel to the fibers and verified The tensile stress in the reinforcement layers is is then then acting parallel parallel to to the the fibers and verified The tensile stress in the reinforcement layers acting fibres and The tensile stress in the reinforcement then fibers verified as: The line taken is area the critical area The effective bond bond line area area takenininto into account is the the layers area below below theacting critical area of oftothe the as: effective The tensile stress the account reinforcement is then parallel the fibers and verified as: failure (hence the area below the notch corner). failure (hence the area below the notch corner). as: F fft, d F Ft,90,d ft, d t,90,d = F t,90,d t, dF t, d t,90,d ft, d t, ttr llFr t,90,d = n t,90,d k t, dd = kk t,90,d f n ed, d = n t l ed, d (11) fRv, ed, d = n rr hrrt, d h= kkk l Rv, d d nbbb h h N N nl rrrtr l r Rv, d kk N (8) (8) (8) (7) (7) (8) Where: Where: Where: Where: The in The tensile tensile stress stress in the the reinforcement reinforcement layers layers is is then then acting acting parallel parallel to to the the fibers fibers and and verified verified Where: if only internal layers act as reinforcement 2 n if only internal layers act as reinforcement as: as: 22 n if only internal layers act as reinforcement n if only internal layers act as reinforcement n = n n layers if internal layers act as reinforcement n 11 if layers are included reinforcement if external areonly included as as reinforcement nbbb == 222 F n if2 external external layers are included as reinforcement f Fnt,90,d 1 = ft, t,90,d t, dddif external layers are included as reinforcement n t,90,d t, t, b 2k n 1 if external layers are included as reinforcement t, ddd = =number t, n llof isis bond-lines. the bond-lines. kkkk n ttrrr of is the the number number of bond-lines. rrr bond-lines. is the number of is thenumber number ofreinforcement bond-lines. layers. n is is the of number of reinforcement (8) n is the the number number of reinforcement layers. layers. (8) n is the of reinforcement layers. n is the number of reinforcement layers. hh is the beam Where: is the the height height ofjoist the height, beam (See (See Figure X). is theof see Figure Figure X). 2. Where: h is the height of the beam (See Figure X). h is the height of the beam (See Figure X). hh N is the of the notch (See Figure X). tensile stresses. is the height height ofheight theinternal notch (See Figure X). 22 is n if only layers act as reinforcement hN the height of the notch (See Figure X). is the of the notch, see Figure 2. n if only internal layers act as reinforcement N hN is the height of the notch (See Figure X). n nl bbb = = 22 n 1 if external layers are included as reinforcement iswidth the design tensile strength of the reinforcement layer. of the reinforcement Figure X). n is 1ft, deffective if external layers as(See reinforcement is the the effective width ofwidth the are reinforcement (See Figure X). is the effective ofincluded the reinforcement, seeX). Figure 6. ll rr is the effective width of the reinforcement (See Figure r lr is the effective width of the reinforcement (See Figure X). is number of bond-lines. isff the the number of bond-lines. The effective width, l r , strength of theisreinforcement to: is the design shear of layer(s). The effective width, lr , of rolling the reinforcement limited to: is limited is the design rolling shear strength of the the reinforcement reinforcement layer(s). d fRv, is the design rolling shear strength of the reinforcement layer(s). Rv, Rv, dd fRv,number isof design rolling shear strength of the reinforcement layer(s). n is reinforcement layers. nt isis the the number ofthe reinforcement layers. d h the layer(s) h Nof lr 0.50 (12) is 0.25 the thickness thickness of one one reinforcement reinforcement layer(s)h (See (Seeh NFigure Figure X). X). ttrr is the thickness of one reinforcement layer(s) (See Figure X). hhr is the height of the beam (See Figure X). is tthe height ofisthe beam (See Figure X). the thickness of one reinforcement layer(s) (See Figure X). r kk == 2.0 is coefficient to into distributed (9) 2.0the minimum is aaa correction correction coefficient to take take into account accountinthe the non-uniformly distributed Where value prevent crack development thenon-uniformly area accentuated in Figure = 2.0 is correction coefficient to take into account the non-uniformly distributed hhkkkkN is the height of the notch (See Figure X). is the height of the notch (See Figure X). N kk = 2.0 is a correction coefficient to development take into account non-uniformly distributed 6 Nand the maximum value ensures that area of the reinforcement into account isin Where the minimum valuethe prevent crack in taken thethearea accentuated Figure X limited by theis notch area subjected tension perpendicular tothe thereinforcement grain. maximum ensures that the (See area of taken into account is ll rr the effective width of reinforcement Figure X). isand the the effective widthvalue oftothe the reinforcement (See Figure X). r limited by the notch area subjected to tension purpendicular to the grain. is therolling designshear rolling shear strength of the material. ffRv, is strength of layer(s). is the the design design rolling shear strength of the the reinforcement reinforcement layer(s). Rv, Rv, ddd 5.2.2 Compression failure at the bottom of the notch ttrr is the thickness of one layer(s) Figure is the of the the reinforcement layer(s), see X). Figure 6. grain needs to be verified isFor the tenon thickness of one reinforcement reinforcement layer(s) (See (See Figure X). andthickness mortise joints compression perpendicular to the r in the bottom of the tenon and as a notch can be compared to a tenon a similar verification kkkk == 2.0 is aa correction coefficient to into distributed can be is made for this geometry. modethe cannon-uniformly be seen in Figure X. Formula (9) is given a correction coefficient, see account Chapter 3.1. 2.0 is correction coefficient to take take Failure into account the non-uniformly distributed tensile stresses. k in the French contribution. is the design tensile strength parallel to the grain of the reinforcement ft, d is the design tensile strength of the reinforcement layer. layer.

The effective width, l r , of the reinforcement is limited to: 4.1.3. Compression failure perpendicular to the grain at the bottom of the notch

For to the grain needs to be verified 0.25tenon h and h Nmortise joints l r the compression 0.50 h perpendicular hN in the bottom of the tenon and as a notch can be compared to a tenon a similar verification (9) can be made for this geometry (CEN, 2017). The failure mode can be seen in Figure 7. Where the minimum value prevent crack development in the area accentuated in Figure X VEd that the area of the reinforcement taken into account is and the maximum value ensures = f purpendicular to the grain. limited by the notch area subjected to ktension (13) c, 90, d c, 90 c, 90, d bN l z, ef

5.2.2 Compression failure at the bottom of the notch

For tenon and mortise joints the compression perpendicular to the grain needs to be verified in the bottom Where: of the tenon and as a notch can be compared to a tenon a similar verification 11 can be made for this geometry. Failure mode(66) can be seen in Figure X. Formula (9) is given Vd is the dimensioning shear force. in the French contribution. bN

is the width of the notch (See Figure X).

l 30 mm z l = min CHALMERS Architecture z, ef 2 l and Civil Engineering, Master’s Thesis ACEX30 z

is the effective length of the transverse compression (mm) (See Figure X). In theory the

(10)

Since infinite stresses can not occur fracture mechanics are used to estimate the following formula for verifying the shear and tensile capacity of a notched beam. The formula is (10) VEd 3to shear reduced verification k butf this formula verifies the capacity for tension perpendicular to d = 2 the fibers as well Where: h b(EC5). v v, d ef

VEd the dimensioning shear force. 3 is VEd k fv, d d = 2 kc, 90v fc, 90, c, 90, d = bef width d bNishl z, b the of the notch (See Figure X). Where: ef N VEd

(4) (4) (10)

is the dimensioning shear force. lz 30 mm z

Where: Where: l = min Where: bz,ef ef= kcr bN is 2thel width of the notch reduced to account for the negative influence of z V is dimensioning the dimensioning shear force. is the dimensioning shear force.4.1). cracks (k can beshear found in Chapter VdEd is the force. cr is the effective length of the transverse compression (mm) (See Figure X). In theory the effective the compression zone extended depending on CLT grain direction and5.2.1.1. onof both bN of is of reduced account for the negative influence bef = kcr area is the the width of the the notch. May beto modified see Chapter is the width the notch Figure X). iswidth theofwidth ofnotch theis(See notch. N of the sides support. l takes into consideration the distribution beyond the tenon length. cracksz,(k ef can be found in Chapter 4.1). cr The evaluated lgeometries thesis limit this zone to only be extended in one direction 30 mmin this1.0 z due to the interaction of the joist and its1.5 header. is the width of the notch. May be modified for CLT see Chapter 5.2.1.1. 1.1 i lbz,N ef = min 2 lz k 1 n kv = min 1.0 h hN is the effective length compression (mm) (See(See Figure X). In the the is effective lengthofofthe thetransverse transverse compression [mm] Figure 8).theory In theory 1 2 2 1.1 iisx1.5 hextended effective area area of the compression zone extended depending on grain direction and on 0.8 effective ofhthe compression zone is depending on grain direction and on both both kn 1 h sides of the support. l takes into consideration the distribution beyond the tenon length. h sides of the support. lz,ef z, eftakes into consideration the distribution beyond the tenon length. kv = min The evaluated geometries geometries in limit this this zone zone to to only only be The evaluated in 2this this thesis thesisxlimit be extended extended in in one onedirection direction (5) 1 2 due to the interaction of the joist and its header. h 0.8 its due to the interaction of the joist and header. h is a reduction factor that takes into consideration the risk of crack growth in notched members (SeeisChapter kc, 90 a correctional factor, see 3.1. is X). a correctional factor, see Chapter 3.1. hChapter (5) h * For beams at the opposite side of the support k = 1.0 can be used. N with notch is the design strength to the grain of the v is factor the design strenght ofconsideration the material in compression perpendicular to the isfc,a90,reduction that takes intocompression the riskperpendicular of crack growth in notched d material. members (See Chapter X). grain. kn is a proportionality constant, see Chapter 4.1. * For beams with notch at the opposite side of the support kv = 1.0 can be used. lz,ef i is the notch inclination (see Figure 1). k is a proportionality constant, see Chapter 4.1. the height of the beam in mm.3.1. khc,n 90 is aiscorrectional factor, see Chapter i h is the notch inclination (see Figure 1). fc, 90, N is the design strenght of the material in compression perpendicular to the hN the height of the beam. May = hd is the ratio between the height of the notch and h is the height of the beam in mm. grain. h be modified for CLT see Chapter 5.2.1.1. h x= N theratio distance fromthe theheight support thethe corner of the notch. isisthe between of reaction the notchtoand height of the beam. May h bef modified for is CLT see Chapter the design shear5.2.1.1. strength of the material. b v, d

lz x is the distance from the support reaction to the corner of the notch. lz,ef 5.2.1.1. Modifications for CLT fv, d 7. Compressionisperpendicular the designtoshear strength material. Figure the grain at the of theFigure 8. Geometry of the effective compression length, N

bottom of the notch.

lz,ef.

5.2.1.1. Modifications for CLT

4.1.3.1. Modifications for CLT

h This failure mode is only possible when all layers are perpendicular to the load and therefore the compression surface. lz The effective area is only extended in the direction of the fibres (Gustafsson et al., 2019). Therefore, if the first layer closest to the compression surface has fibres oriented perpendicular z0,N z to the grain as seen in Figure 8 then z lz,ef = lz. This is illustrated0,Nzin Figure 9. zT,i T,i A A y zB,i

x VEd

zT,i bi

(67) a.

bi b.

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

hN hN hN

hN hN

h h h h h

lz lz,ef lz,ef lz,ef h lz

hN hN

Figure 9. Geometry of the effective compression length, lz,ef if surface in compression has grain perpendicular to the main grain direction of the beam. z0,N z0,N

VEd

h h

z bNz lz z A T,i 5.2.2.1. Modifications forinCLT A 4.1.4. Rolling shear failure the notchT,i(only for CLT) y b lz 5.2.2.1. Modifications for CLT N lz,ef If the compression the bottom of themember CLT is applied in a layer perpendicular to the beam When a shear force at is acting on a CLT where all layers are perpendicular force z B B,i B lz,efin If theX. compression bottom ofthe thebending CLTl isor applied astay layer perpendicular to the beam length, see Figure should the latZ, the be reduced to only should it because the one or more layers will act in rolling shear due to of the member. This is verified ef Z length, see Figure X. should the l Z, ef be reduced to only l Z or should it stay because the z x T,i factor kc, 90 is lower than that of stress glulam? See KL-trähandboken. by controlling the rolling shear specifically for the layers acting in rolling shear and factor is lower that ofAn glulam? SeeofKL-trähandboken. verifying against the kstrength of thethan material. example what layers are concerned for a c, 90 Yes, extension onlyisoccurs parallel to the 10 grain so perpendicular is the one with h bi in Figure biaifbending notched geometry illustrated with aroundlayer the y-axis. Yes, extension compression the l.Z,ef = l.Z.only occurs parallel to the grain so if perpendicular layer is the h one with compressiona.the l.Z,ef = l.Z. b. 5.2.3 Rolling shear failure of the notch (only for CLT) l 5.2.3 Rolling shear failure of the notch (only for zCLT) lz When a shear force is acting on a CLT member where all layers are perpendicular to the When a shear acting shear. on a CLT where all layers are failure perpendicular to the force one or more layers willforce act inisrolling Panelmember shear for the most critical z the most critical failure z5.2.1. forcemember one or more layers will act in rolling shear. 0,N Panel shear for 0,N point of a notched is already controlled in Chapter z point of a notched member is already controlled in Chapter 5.2.1. z0,NzT,i z0,NzT,i A A z y zT,i A zT,i A zB,i y B B zB,i B zT,i B x zT,i x

b bi Figure 2. Failure due to shear and tension perpendiculari to the grain. bi bi VEdstresses can not occur fracture mechanics a. are used to estimate b. Since infinite the following VEd formula for verifying the shear and tensile capacity of a. a notched beam.b.The formula is reduced to shear verification but this formula verifies capacity for tension perpendicular to For a 10. force acting theofmember as seen X.the The equations for control of rolling Figure Rolling shearon failure the layers with grainin Figure Figure 11. Illustration of different cross-sections for the the fibers as well (EC5). For a force acting on the member ascalculation seen inofFigure X. Theof equations for control of rolling shear are below. parallel to thestated bending axis. the first moment inertia. shear are stated below. 3V SVEd Ed R, x d = VfEd kSv R,fxv, d Rd = 2 bef Rv, d I x, nethRd bef (14) = fRv, d I x, net bef (4) (11) (11) Where: Where: Where: is the dimensioning shear force. is the dimensioning shear force. VEd is the dimensioning shear force. Ed VEd is the dimensioning shear force. the first moment of inertia the rolling shear layers, see around Chapter =k b theis width ofofthe notchfor reduced toforaccount for the negative influence ofthe SbR, isis the moment inertia the rolling shear layers with bending ef x cr N 4.1.4.1. is the moment of inertia for the rolling shear layers with bending around the SR,cracks (ktocr the canforce be found in Chapter 4.1).section 5.2.2.1. x plane axis perpendicular (See Figure X). See plane axis perpendicular to the force (See Figure X). See section 5.2.2.1. Ibx, net isisthe of inertia the full cross-section the notch. See thesecond width moment of the notch. May for be modified for CLT seeofChapter 5.2.1.1. N I is the second moment of inertia for the full cross-section of the 13 notch. See (68) section 5.2.2.2.x, net section 5.2.2.2. 1.0 bef = kcr bN is the width of the cross-section reduced for cracks. bef = kcr bN is the1.1 width i 1.5of the cross-section reduced for cracks. kn 1 f d is the design rolling shear h strength of the material. kvRv, = min fRv, d is the design rolling shear strength of the material. x of inertia 1 for 2 moment 2 rolling 5.2.2.1 Calculation of the first shearThesis ACEX30 CHALMERS Architecture and Civil Engineering, Master’s h 5.2.2.1 Calculation0.8 of the first moment of inertia for rolling shear h For a symmetrical cross-section such as that seen in Figure X.a where the centre of gravity a symmetrical cross-section suchatasthe that seenpoint in Figure where the centre of gravity of the notchedFor geometry as a solid member occurs same as theX.a centre of gravity

Rv, d I x, verifying b formula for net ef the shear and tensile capacity of a notched beam. The formula is reduced to shear verification but this formula verifies the capacity for tension perpendicular to Where: the fibers as well (EC5). (11)

if only internal layers act as reinforcement Where: 3 2 nVEd nb = = 2 n 1 kv fv, dlayers are included as reinforcement if external d 2 VEd is shear force. h the bef dimensioning is the number of bond-lines. SR, x is the moment of inertia for the rolling shear layers with bending around the (4) n is the number of reinforcement layers. plane axis perpendicular to the force (See Figure X). See section 5.2.2.1. Where: h is the height the beam (See Figure X). theofsecond moment of inertia full cross-section the notch, I x, net is the is second moment of inertia for thefor fullthecross-section of theofnotch. See VEd is the shear force. seedimensioning Chapter 4.1.4.2. hN is the height of the notch (See Figure X). section 5.2.2.2. the width the notch reduced to account the negative influence bef = kcr bN is theiswidth of theofnotch reduced to account for thefornegative influence of b = k b is the width of the cross-section reduced for cracks. l ref cr N is the effective width (See 3.1). Figure X). of cracks (kcr of canthe bereinforcement found in Chapter cracks (kcr can be found in Chapter 4.1). the design rolling shear strength the material. is design theisdesign rolling strength of reinforcement theofmaterial. ffRv, dd is the rolling shearshear strength of the layer(s). bNRv, is the width of the notch. May be modified for CLT see Chapter 5.2.1.1. 5.2.2.1 Calculation of the first moment of inertia layer(s) for rolling shear tr the thickness of one 4.1.4.1. Firstismoment of inertia for reinforcement the rolling shear layers(See Figure X). 1.0 For aa symmetrical symmetrical cross-section cross-section such such as as that that seen seen in in Figure Figure 11.a. X.a where ofofgravity For wherethe thecentre centre gravity kk the = 2.0 is geometry a correction into account the non-uniformly distributed ito1.5take of notched as coefficient a solid1.1 member occurs at the same point as the centre of gravity knas a1 solid member occurs at the same point as the centre of gravity of the notched geometry for the same member in CLT. The moment of inertia is the same for both part A and B and kv =the min for same in CLT. The firsthmoment of inertia is the same for both part A and B only one needmember to be calculated. and only one need to be calculated. x 1 2 2 h 0.8 h For a non-symmetrical cross-sections as seen in Figure 11.b. the first moment of inertia should be calculated for both the top and bottom half and the larger one should be used to (5) Change this image to only notch! verify the largest rolling shear stresses. is a reduction factor that takes into consideration the risk of crack growth in notched members (See Chapter X). SR, x, A *SFor beams with (15) = max S notch at the opposite (12)side of the support kv = 1.0 can be used. R, a x non-symmetrical For R, x, B cross-section as seen in Figure X.b. the moment of inertia should be S calculated for both the top and bottom half and the larger one should be used to verfiy the SSR, kn is x, R, x,a A Aproportionality constant, see Chapter 4.1. z R, x, A S largest rolling shear stresses. SR, S == max (12) T, i SR, max and (12) S R,Sx, x, A A are calculated xx = S max (12) accordingEto eq. (16).z 2 Where S E S zB, i2 R, R, x, B R, x R,z,A R,z,B S SSiR, x == max (12) x, R, i x, B x,i is the notch inclination (see Figure 1). T, i R, x, B max (12) S SR, = b z dz = (13) R, xx ESx,R, E x,ref bi 2 2 R, x, x, B Bi z h isrefthe height of the beam in mm. zzzT, T, ii zzB, E E zzT, i222 zzB, i222 T, ii E E ii E x, E x,i zT, zB, T, ii x, x,i i 2 T, x, i x,i T, i B, ii22 E E zzB, S bbi zz dz == bbi zzT,2 i 2 (13) h==N S dz (13) E E (16) x, i x,i R, x E E 22 ii SWhere: = b z dz = b (13) R, x i i x, i x,i T, i B, E E 2 R, x i i x,isref refthebratio between x,ref E x, E 2 2 SSR,= x == z dz = b (13) the height of the notch and the height of the beam. May x,ref b z dz = b (13) z x, ref x,ref E i E i 2 2 h z R, x Ex, i zB, i Ex,ref i 2 2 ref ref index ii each rolling zzB, i modified for isx,the shearx,ref layer in the notched cross section on either B,of be CLT see Chapter 5.2.1.1. B, i B, i side of the centre of gravity. Where: Where: Where: x is the distance from the support reaction to the corner of the notch. Where: Where: Where: is the elasticity modulus forshear the rolling shear layers, which is the materials iiiE x, i is the index of each rolling layer in cross on is index of each rolling in the notched cross section on either is the the index ofindex each rolling shear layer in the the notched notched cross section section on either eitheron isdesign the ofstrength eachshear rolling shear layer in the notched cross-section fv, of isofthe shear oflayer the material. iside is the index of each rolling shear layer in the notched cross section on side the centre gravity. d E . gravity. i 90 of is theof of each rolling shear layer in the notched cross section on either either side of the the centre centre ofindex gravity. either side of the centre of gravity. side side of of the the centre centre of of gravity. gravity. E is the elasticity modulus for the rolling shear layers, which is the materials 5.2.1.1. Modifications for CLT E is elasticity modulus for the rolling layers, is the materials i a chosen value modulus, generally EEx, isisthe the elasticity modulus forfor thethe rolling shear layers, which iswhich thechosen materials is thereference elasticity modulus forelasticity theshear rolling shearwhich layers, is theas the x, x, x,iiref E is the elasticity modulus for the rolling shear layers, which is the materials is the elasticity modulus for the rolling shear layers, which is the materials E x, . i E . x, i. . materials 90 strongest elasticity modulus ofEall E 90 90 the internal layers. E .. E90 90 90 is a reference chosen forX). the elasticity in chosen the x-direction, Ebx, chosen reference value(See forvalue the elasticity modulus,modulus generally as the the is width of eachreference layer Figure E aaa chosen value elasticity modulus, E x, is the chosen reference value for for the the elasticity modulus, generally generally chosen chosen as as the i ref x, ref ref E is a chosen reference value for the elasticity modulus, generally chosen as the generally chosen as the strongest E-modulus of all the internal layers. E is a chosen reference value for the elasticity modulus, generally chosen as the strongest elasticity modulus modulus of of all all the the internal internal layers. layers. x, strongest elasticity x, ref ref strongest elasticity modulus of all the internal layers. z is the distance from z to the point of the layer furthest away from z strongest elasticity modulus of all the internal layers. strongest elasticity modulus of all the T, i 0, Ninternal layers. 0, N is the the width width of width each layer layer (Seelayer, Figure X). is the of each seeX). Figure 11. bbb(See is of each (See Figure i Figure X). is the width of each layer (See Figure X). i bbii is is the the width width of of each each layer layer (See (See Figure Figure X). X). i is the distance from z to the point of the layer furthest away from is the distance from z to the point of the theof layer closest to z0, (See zzzzT, is the distance from z to the point of layer furthest away zzz0, is the distance from z to the point each layer that isfrom furthest away T, 0,0,N NN to the 0,Figure N 0,N point of the layer furthest away is the distance from z0, from B,iii N N i N to N zzT, is the distance from zz0, the point of the layer furthest away from zz0, is the distance from to the point of the layer furthest away from (See Figure X). from z , see Figure 11. T, i 0, N 0, N X). the centre in the layer zB, i = 0 (See Figure X.a.). (See Figure X). 0,N T, i If 0, N 0, N (See Figure X). of gravity occurs (See (See Figure Figure X). X). is the distance from zthe to the of point of each layer to that is closest to z0,N, zzzB, i is from to the layer (See 0,N point is the distance from the layer closest Figure is the the distance distance from zzz0, to the the point offor therolling layer closest closest to zzz0, (See Figure Figure 0, N to 0, N N (See 5.2.2.2 Calculation of the second moment ofpoint inertiaof shear to B, ii N B, 0, N 0, N z is the distance from z to the point of the layer closest to z (See Figure see Figure 11. If the centre of gravity occurs in the layer z = 0. zB, is the distance from zin to the zzpoint== 00of(See the layer closest B,i(See Figure X). If the centre of gravity occurs the layer Figure X.a.).to z0, 0, N X). the of occurs in Figure B, iiIf 0, the N layer 0, N N ii = 0 (See X). Ifsecond the centre centre of gravity gravity occurs in the layerfor zB, (See Figure X.a.). X.a.). B, B,the i = net The moment of inertia is calculated cross-section of the notch with a X). If the centre of gravity occurs in the layer z 0 (See Figure X.a.). X). If the centre of gravity occurs in the layer zB, = 0 (See Figure X.a.). B, ii layers included. 5.2.2.2 Calculation of the second moment of inertia for rolling shear 5.2.2.2 Calculation Calculation of of the the second second moment moment of of inertia inertia for for rolling rolling shear shear 5.2.2.2 5.2.2.2 of the second moment of inertia for rolling shear 5.2.2.2 Calculation Calculation of the second moment of inertia for rolling shear 3 E x, j bof tinertia E x, j The is for (69)the The second second moment moment of inertia is calculated calculated for the net net cross-section cross-section of of the the notch notch with with aaa 14 j inertia j The second moment of is calculated for the net cross-section of the notch with 2 The second moment of inertia is calculated for the net cross-section of the notch with a layers included. I = b t a (14) layers included. The second moment of calculated x, net E 12inertia is E j jforj the net cross-section of the notch with a layers included. x,ref layers layers included. included.x,ref E b t 33 E E E Ex, Ex, x, jj bbjj ttjj33 x, jj 2 x, j j j x, j b E b t E 3 IIIWhere: = (14) = b tt aa 22 (14) E b t x, j j j x, = (14) x, net net E x,ref 12j EEx,ref x, E 12 E x, j j x, jj bbjjj ttjjj aajjj22 net = E 12 E IIx, (14) x,ref x,ref = b t a (14) jx, net is the index of each layer in the cross-section of the notch. x,ref x,ref x, E 12 E j j j net Ex,ref 12 Ex,ref j j j x,ref x,ref Where: Where: CHALMERS Architecture Master’s Thesis ACEX30 Where: E x, j is the elasticity modulusand forCivil each Engineering, layer in x-direction. For layers with grain Where: Where: is the the index of each layer in the the cross-section cross-section of the the notch. notch.y this is E . direction along x this is of E each andlayer for layers grain direction jjj is index in of is the index of each layer in thewith cross-section of thealong notch. 90 j is the index of 0each layer in the cross-section of the notch.

x, ref E . E90 isstrongest a chosenelasticity reference value foroftheallelasticity modulus, generally chosen as the 90 the internal layers.generally x, ref E x, ref is a chosen referencemodulus value for the elasticity modulus, chosen as the x, ref E x, is a chosen reference value for the elasticity modulus, generally chosen as the strongest elasticity modulus of all the internal layers. E x, ref is a chosen reference value for the elasticity modulus, generally chosen as the strongest elasticity modulus of all the internal layers. bi is the width of each layer (See Figure X). ref strongest elasticity modulus of all the internal layers. b is the width of each layer (See Figure X). strongest elasticity modulus of all the internal layers. bii is the width of each layer (See Figure X). the layer distance z0,X). to the point of the layer furthest away from z0, N bii is zthe ofiseach (Seefrom Figure T, i width N is the width of from each layer (See Figure X). zbiT, i is(See the Figure distance z to the point of the layer furthest away from z 0, N to the point of the layer furthest away from z 0, N zT, i is the distanceX). from z0, T, i Figure X). 0, N N to the point of the layer furthest away from z0, 0, N N zT, is the distance from z0, (See N zT, ii Figure X). iszthe distanceisfrom z0, N to from the point oftothe layer furthest away from z0, (See the distance z the point of the layer closest to z0, N (See Figure N 0, N 0, N (See Figure X). B, i zB, i Figure X). isX). theIfdistance from zgravity tooccurs the point of layer the layer closest toFigure z0, N (See Figure (See the centre of in the z = 0 (See X.a.). 0, N zB, i is the distance from z0, N to the point of the layer B, iclosest to z0, N (See Figure B, iIf the centre 0, the N to N (See Figure z4.1.4.2. is the from zin the the layer closest X). ofdistance gravityofoccurs layer zpoint = of 0 (See Figure X.a.).to z 0, N Second moment inertia for the notch B, i 0, N B, i z is the distance from z to the point of the layer closest (See Figure X). If the centreS of gravity occurs in0,the layer zB, i = 0 (See Figure X.a.).to z0, B, i N 0, N 5.2.2.2 Calculation of the second moment of inertia for rolling shear B, i X). If the centre R, ofx,gravity occurs in the layer zB, i = 0 (See Figure X.a.). A The second moment inertia is in calculated the cross-section X). the centre of gravity occurs the layer zinertia = 0net (See Figure X.a.).of the notch with all 5.2.2.2 ofofthe second moment offor for rolling shear SR, If = Calculation max (12) B, i SR, x,second x included. 5.2.2.2 Calculation of the second moment of inertia for rolling shear layers The moment of inertia is calculated for the net B the second moment of inertia for rolling shearcross-section of the notch with a 5.2.2.2 Calculation of The second moment inertia is calculated forinertia the netforcross-section layers included. 5.2.2.2 Calculation ofofthe second moment of rolling shearof the notch with a The second moment of inertia is calculated for the net cross-section of the notch with a z layers included. The second moment of inertia for the net cross-section of the notch with a T, i is calculated layers included. The second moment for the net cross-section E x, i of inertia E x,i zT, i2 zB,ofi2 the notch with a E x, jis calculated bj tj3 E layers included. x, j (17) 3 layers E x, SR, x =included. =j bi b t 2a 2 (13) I x,j netbb=ji tj33 E z dz E x,12 (14) E x, E 2 E j j j E b t E 2 I x, net = (14) x, jjref bjj12 x, jj b t ax,refx,ref E x, tjj3 x,ref E E x, E I x, = (14) x, j j tj 3zB, i x, j b j t j a j222 E b E x,ref x,ref net E 12 E j j j x, net = x, j j j x, j b j t j a j I x, (14) Where: x,ref x,ref j j j 2 E x,ref 12 E x,ref I x, net = (14) Where: x,ref bj tj a j net Ex,ref 12 E Where: x,ref x,ref Where: j is the index of each layer in the cross-section of the notch. Where: is the of each layer in theincross-section of thesection notch. on either j isis the of each in the cross-section the notch. iWhere: the index index of index each layer rolling shear layer the of notched cross j of the centre is E theofindex ofiseach layer in themodulus cross-section of layer the notch. the elasticity for each in x-direction. For layers with grain side gravity. x, jindex of each layer in the cross-section of the notch. j is the Ej x, j is direction the elasticity modulus for each layer in x-direction. For layers with grain index of each layer in the cross-section of the notch. is the elasticity modulus for each layer in x-direction. For layers along x this isforE 0each andlayer for layers with grainFor direction alonggrain y with this is E 90 . E is the elasticity modulus in x-direction. layers with E is the elasticity modulus for the rolling shear layers, which is the materials jj x, E x, is the elasticity modulus for each layer in x-direction. For layers with grain direction along x this is E and for layers with grain direction along y this is E . and for layers with grain direction grain direction along x this is E x, ij x, 0 90 0 E x, j is the elasticity for each indirection x-direction. Fory layers direction along x this is E 0 modulus and for layers withlayer grain along this iswith E grain . E 90 . athis chosen reference the elasticity generally chosen as the 0yand 90. is E . with value direction alongExx,this is Eis for layers grain for direction along ymodulus, this is E 90 90 ref along 0 and for layers 90 . direction along x this is E with grain direction along y this is E E x, ref is strongest a chosen reference value for the elasticity modulus, generally chosen 0 90 as the elasticity modulus of all the internal layers. is reference a reference chosen value reference value for the modulus, elasticity in the x-direction, E isisa achosen for generally chosen asasthe E chosen value forthe theelasticity elasticity modulus,modulus generally chosen the x, ref ref E x, is a chosen reference for the elasticity modulus, generally chosen as the strongest elasticity modulus of allvalue the internal layers. x, ref x, ref generally chosen as the strongest E-modulus of all the internal layers. E is a chosen reference value for the elasticity modulus, generally chosen as the strongest elasticity modulus of all the internal layers. b is the width of each layer (See Figure X). x, ref strongest elasticity j modulus strongest elasticity modulus of of all all the the internal internal layers. layers. bj is the width of each layer Figure X). strongest elasticity modulus of all the(See internal layers. bbj isisthe width of each layer (See Figure X). is the width of each layer, see Figure the width of each layer (See Figure X). t is the thickness of each layer (See11. Figure X). j bij is thej width of each layer (See Figure X). bt j isis the width of each layer (See Figure X). the thickness of each layer (See Figure X). j isisthe thickness of each (See Figure zttjjT, i distance from z0,layer tofrom thelayer point ofX). the athe is thickness theeach distance the centre of layer gravity of theaway layerfrom to thez0, centre of gravity for is the of layer (See Figure X). is the see Figure 11.furthest Nof each N j thickness tajj is the thickness of each layer (See Figure X). is the distance from the centre of gravity of the layer to the centre of gravity for (See thedistance notch cross-section (SeeofFigure a jj FigureisX). the from the centre gravityX). of the layer to the centre of gravity for a jj notch cross-section is the distance from the centre of gravity of layeroftoeach the centretoofthe gravity forof the (See Figure X). is the distance from the centre of the gravity centre a is the distance from the centre of gravity to the layer centre gravity for the cross-section (Seefrom Figure zB, is the distance z0,X). to the point of the layer closest to z0,of (See Figure j notch inotch cross-section N N the (See Figure X). gravity for the net cross-section. the notch (Seeoccurs Figure X). If the cross-section centre of gravity in X). the layer zB, i = 0 (See Figure X.a.). 5.3 HEADER MEMBER 5.3 HEADER MEMBER 4.2. 5.2.2.2 Calculation of the second moment of inertia for rolling shear 5.3 HEADER MEMBER 5.3.1 Geometry 5.3 HEADER MEMBER 4.2.1. Geometry 5.3 HEADER MEMBER 5.3.1 Geometry The second moment of inertia is calculated for the net cross-section of the notch with a 5.3.1 Geometry layers included. Two different geometries of the header is illustrated in Figure 12 and Figure 13 with the 5.3.1 Geometry 5.3.1 Geometry relevant geometrical notations for all verifications. Both of these geometries have the same E j bj tj3 E x, j failure modes.x,The force arrows display the load2 from the joist on the header. I x, net = (14) E x,ref 12 E x,ref bj tj a j 4.2.2. Shear and tension failure perpendicular to the grain at the notch corner Where: For headers with a grain direction as shown in Figure 12 and 13 the header is loaded

perpendicular grain therefore subjected toofshear stresses and tension stresses j is to thethe index ofand eachislayer in thealso cross-section the notch. perpendicular to the grain at the notch corner similarly to the joist. This type of failure mode E is the elasticity modulus for each layer in x-direction. For layers with grain can x, jbe seen in Figure 14. In a similar way to eq. (6) the verification is reduced to control shear direction along this is E and for layers with grain y this is E 90 . but through thisx verifies the tension perpendicular to direction the grain along as well. 0 Note: This type of failure mode value is onlyforrelevant when modulus, the headergenerally is not supported E is a chosen reference the elasticity chosen as directly the x, ref under the notch or if the height under the notch of the header is relatively large. strongest elasticity modulus of all the internal layers. The is verified according to eq. (18) X). (CEN, 2017). bj failure ismode the width of each layer (See Figure tj

is the thickness of each layer (See Figure X).

aj is the distance from the centre of gravity of the layer to the centre of gravity for the notch cross-section (See Figure X). 15 (70) 5.3 HEADER MEMBER

5.3.1 Geometry CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

A-A A-A A-A

bH bH bH

A A

B B

A A

B B

hH,u hH,u hH,u

bN bN bN

B-B B-B B-B

hH hH hH

A-A A-A

Figure 12. Geometry of a header where the notch extends through the full width.

A-A

bH bH bH

bN bN bN

A A

B B

A A

B B

hH,u hH,u hH,u

B-B B-B B-B

hH hH hH

lz lz

bH bH lz

Figure 13. Geometry of a header where the notch is embedded in the width of the header.

B B

hH,u hH,u hH,u

hH hH hH

lz lz

bH bH lz

Figure 14. Shear and tension failure perpendicular to the grain at the notch cornerbfor the header. H

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CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

1.0 Figure 2. Failure due to shear and1.1 tension i 1.5 perpendicular to the grain. kn 1 Since infinite stresses can not occur fracture mechanics are used to estimate the following h kv = min formula for verifying the shear and tensile capacity of a notched beam. The formula is x 1verifies2 the capacity for tension perpendicular to 2 reduced to shear h verification but this 0.8 formula h the fibers as well (EC5). Figure X. VEd (5) 3 Figure X. VEd = k f d 3 Figure X. 2 v v, d hd bef= that takes Figure X. is a reduction factor into consideration growth in notched (18) fv,d the risk of crack (15) V 4 b h H, u 3 members (See Chapter X). Ed H, ef V fv,d (15) d = 3 34 b VEd (4) Ed h H, u = f (15) d H, ef (15) * dFor=beams at thefv,d opposite side of the support kv = 1.0 can be used. Where: 44 bwith hhnotch v,d b H, ef H, u Where: H, ef H, u Where: Where: Vis is the dimensioning shear force. kn Ed a proportionality constant, see Chapter 4.1. Where: Where: VEd is the dimensioning shear force. is the dimensioning shear force. VEd is the dimensioning shear force. b = k b is the width of the V is the dimensioning shear force. iVEd is efthe notch (seeshear Figure 1). header reduced to account for cracks. is dimensioning force. H, cr the Hinclination befEd= kcr bN is theiswidth of theofnotch reduced to account for the negative influence of the width the header reduced negative b =k b is the width of the header reduced to to account account for for the cracks. hbH, ef = kcr bHb iscracks theinfluence height of the beam in mm. is the width of the header (See Figure X). is the width of the header reduced to account for cracks. of cracks (k can be found in Chapter 3.1). can be found incr Chapter 4.1).to account for cracks. bH, is (k the reduced ef = kcr bH H cr width of the header H, ef cr H bH h is the width of the header (See Figure X). h H, isthe the height from the 12 crack to13. thesee bottom of the header (See Figure bbH N is width of header (See Figure X). iswidth width ofnotch. the header, see Figure and of the May modified for CLT Chapter 5.2.1.1. utheratio isthethe the width of the header (See Figure X). = h isisthe between the height ofbe the notch and the height of the beam. May N H X). h H, u is the height from the crack to the bottom of the header (See Figure be for CLT see Chapter 5.2.1.1. hh H,modified is the crack the bottom of header (See Figure 1.0from is the the height height from the the notch crack to to bottom of the the header u X). of the thestrength header, of see Figure 12 (See and Figure 13. H, u fv, d is the height under is the design shear the material. X). x is the distance from support reaction to the corner of the notch. X). 1.1the i 1.5 fv, d isk the1design shear strength of the material. nthe design shear 5.3.2.1 Modifications for CLT fv, is strength of the material. is the design shearshear strength of theofmaterial. h strength is the is thedesign design shear strength ofthe thematerial. material. d kfvv, = min v, dd 5.3.2.1 Modifications for CLT In the case for2CLT where x not all 1 layers 2 run parallel to the crack some modifications may be 5.3.2.1 for 5.2.1.1. Modifications forCLT CLT 5.3.2.1 Modifications Modifications for CLT h 0.8of failure needed. For this type the ortogonal grain patterns that may occur can be seen in 4.2.2.1. Modifications for CLT h run parallel to the crack In the case for CLT where not all layers some modifications may be Figure X. not all layers run parallel to the crack some modifications may be In the case for CLT where needed. Forfor this type ofisfailure thethe ortogonal grain that may occur can berun seen in be In CLT where not as all layers run parallel to the crack some modifications may the case where CLT used material of thepatterns header, where not all layers parallel needed. For this type of failure the ortogonal grain patterns that may occur can be seen in Figure X. (5) needed. For this type of failure the ortogonal grain patterns that may occur can be seen in to the crack some modifications may be needed. For this type of failure the orthogonal grain Figure X. Figure X. patterns that may occur beinto seenconsideration in Figure 15.the Thisrisk illustration shows the minor changes is a reduction factor thatcan takes of crack growth in notched of the failure as well. members (Seemode Chapter X). * For beams with notch at the opposite side of the support kv = 1.0 can be used. kn

is a proportionality constant, see Chapter 4.1.

is the notch inclination (see Figure 1).

ha.

is the height of the beam hH,u in mm. c.

ti,||

= h is the ratio between the height of the notch and the height of the beam. May be modified for CLT see Chapter 5.2.1.1. x

is the distance from the support reaction to the corner of the notch. hH,u is the design shear strength of the material. d.

f b.v, d

Figure 15. Changes of the failurefor mode due to orientation of the CLT material of the header. 5.2.1.1. Modifications CLT

For the orientation as shown in Figure 15.a and 15.b. The crack may occur in a layer with the grain perpendicular to the crack propagation which is a weaker grain bdirection. This results N in a smaller height below the crack of the header similarly to the modification for the joist as seen in Figure 4. This result as mentioned in the reduced height of hH,u as seen in Figure 14.a. hH,u

hH,u (72)

bN,ef

AN Architecture and Civil Engineering, Master’s Thesis ACEX30 CHALMERS

For theX.layer orientation as seen in Figure 15.b. no changes need to be made for this Figure verification. For the X. pattern in a) The crack may occur in a layer with the grain perpendicular to the crack Figure ure X. For the X. orientation as athat seengrain in Figure 15.c. Similar not all oftothe grain parallel to the propagation which is weaker direction. thelayers crack have pattern shown in Figure Figure X.b.where the crack moves downwards to the closest layer with grain direction parallel to the crack propagation and one or more layers will act as reinforcement similarly to that explained Foroccur the pattern in a) Thethe crack may occur in a layer with the grain perpendicular to the crack the pattern in a) The crack may in a layer with grain perpendicular to the crack For the pattern in a) The crack may occur in a layer with the grain perpendicular to the crack crack. This will reduce the height to the distance between the bottom of the header to the propagation which is ato4.1.2.1. weaker grain direction. to the crack pattern shown in Figure for the joist in Chapter pagation which is a weaker grain direction. Similar the crack pattern shownSimilar in Figure propagation which a weaker grain direction. Similar tothe the crack pattern in seen Figure top of to thethe layer withislayer grain parallel todirection thetocrack closestlayer to the bottom of theshown notch as X.b.where the crack moves downwards the closest with grain direction parallel to in the where the crack moves downwards closest with grain parallel to X.b.where the crack moves downwards to the closest layer with grain direction parallel tothe the Figure X.a. crack. will between reduce the height to of thethe distance thelayers bottom the header to to the orientation as that seen in Figure 15.c. notbetween allthe of the haveofgrain parallel k. This will reduce the heightFor to the theThis distance the bottom header to crack. This will reduce the height to the distance between the bottom of the header to the top the layerclosest with parallel to of the to notch as seen in of the layer with grain parallelcrack to of the crack toone theor bottom ofthe thecrack notch as seentointhe bottom propagation andgrain more layers actclosest similarly explained For the layer orientation as parallel seen in to Figure X.b. the reinforcement width of the header bH notch is that reduced to in top of the layer with grain the will crack closest to the bottom of the as seen Figure X.a. ure X.a. for the joist in Chapter 4.1.2.1. For this the width is similarly reduced to what is seen in eq. Figure X.a. only include layers with grain parallel to the crack as seen in Figure X.b. (8). Other than thiswidth the header and theb reinforcement verified according eq. (18)toand For the layer orientation as in Figure X.b. the width the header b istoreduced the layer orientation as seen in Figure X.b. the of seen the header is reduced toare of H X.b. the width of the header bH is reduced to For(9), the(10) layerand orientation as seen in Figure eq. (11). only layersaswith parallel to the crack as seen in Figure X.b. H bH =include b crack include layers with grain parallel to the seengrain in Figure X.b. i, layers with grain parallel to the crack as seen in Figure X.b. only include For eq. (9) some changes are made where the dimensioning shear force is divided in 2 since bH =perpendicular b The also verified as reinforcement by the same calculations as = b the onlayers both could sides of thebenotch in the header and therefore reinforcement layers i, i, bH crack = bappear VEd i, act on both sides. Only one side is verified as the geometry is symmetrical around the notch. in Chapter 5.2.1.1 but bycould usingby Vthe =same asasthe forceasis divided between two sidesasof The layers also verified reinforcement by the same the calculations perpendicular layers could also beperpendicular verified as reinforcement Edbe 2 calculations The perpendicular layers could also be verified as reinforcement by the same calculations as VEdone side needs to be verified for the reinforcement V symmetrically and therefore only the notch Ed V in Chapter 5.2.1.1 but by = 2Edtheastwo the sides force is Chapter 5.2.1.1 but by using Vcapacity = 2of the as the force is using dividedV between of divided between the two sides of layer(s). inEdChapter 5.2.1.1 but by using VEd = 2 as the force is divided between the two sides of Ed the notch symmetrically therefore onereinforcement side needs to be verified for the reinforcement notch symmetrically and therefore only one side needs and to be verifiedonly for the Where: the notchofsymmetrically and: capacity the layer(s). and therefore only one side needs to be verified for the reinforcement city of the layer(s). capacity of the layer(s). h H, u Where: ere: the header. Where: = h is is thethe αα forratio the of header (See Figure X). H h h H, u H, u = hhhH,(See the α for the header (See Figure X). u is the α for thehheader Figureis X). hH For==eq. (10) the geometrical variables changed are: X). is the α for the header (See Figure hHH H the height of the joist is changed to the height of the header, see h N==hh H, u = hH h = hH Figure 12 and 13. H h is the height height of under the mortise, see Figure h H, u = h H, u the the notch is changed to the X. height under the notch of N u= h H, N = h H, u the header, see Figure 12 and 13. h H the mortise, see the height fullX. height themortise, header (See X). isis the underofthe see Figure X. is the height under Figure u u h H, is the height under the mortise, see Figure X. H, u The layer orientationis of Figure 15.d. is possible for aa member the The layer Figure X.c. is only possible forFigure member with aa support support under under the h H of the full X). height ofonly the header (See X). with is the full height the orientation header (Seeof Figure h is the full height of the header (See Figure X). notch and this type of failure mode in the header will therefore not be possible. notch H and this type of failure mode in the header will therefore not be possible.

Theislayer orientation X.c. is only possible forthe a member with a support under the layer orientation of Figure X.c. only possible failure forofaFigure member with a support under 4.2.3. Compression perpendicular to the grain at the bottom of theamortise The orientation offailure Figure X.c. ispossible. only possible a member support under the 5.3.3 Failure in compression perpendicular tofor the grain notch and this type oftherefore mode the header will therefore not with be possible. h and this type of failure mode inlayer the header will not bein notch and this type of failure mode in the header will therefore not be possible. Depending on the grain direction of the header compression can occur in the header in Depending on the grain direction of the secondary member compression can occur in the 5.3.3 Failure compression perpendicular the grain same asinfor the geometry in Chapter 4.1.3. This typethis of failure mode can be .3 Failure in compressionthe perpendicular to theasnotch grain header in way the same way for the notch geometry into 5.2.2, is for the geometry 5.3.3 Failure in compression perpendicular toChapter the grain seen in Figure 16. For this case if more than one joists are interacting with the header in secondary Figure Depending on X.a. themember grain direction of thecan secondary compression can occur in thethe ending on the grain directionseen of the compression occur inmember the Depending on the grain direction of the secondary member compression in the interact as notch explained Chapter 3.1.5. header in thezones same way as for the in Chapter 5.2.2, this iscan foroccur the geometry der in the same way as for thecompression notch geometry in can Chapter 5.2.2, this isgeometry forinthe geometry header in the same way as for the notch geometry in Chapter 5.2.2, this is for the geometry Other thisX.a. the same verification as for the joist in Chapter 4.1.3. is used with only lz,ef seen inthan Figure in Figure X.a. seen in Figure modified as seenX.a. in Figure 17.

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CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

H,u

ti,||

hH,u b.

bN For this case if more than one notch is acting on the header the compression zones might For this case if more than one notch is acting on the header the compression zones might interact 2 hN , which gives isa smaller value the coefficient, kc, 90 zones (EC5). This is For this and casell 1if more than one notch acting on the for header the compression interact and 2 hN , which gives a smaller value for the coefficient, kc, 90 (EC5).might This is 1 interact andinl Figure 2 hX below. , whichhgives illustrated = ha smaller value for the coefficient, kc, 90 (EC5). hH,uThis is N = h H, u illustrated in 1Figure XN below. hhH,u u illustrated in Figure X below. h N = h H, N H, u 1.0 for all materials, l 1 2 h N 1.0 for all materials, l 1 2 h NbN,ef 1.0 for all materials, l 1 2 h N kc, 90 = Table X l 1 2 h H, u ? kc, 90 16. = Compression Figure failure perpendicular to the Table X l 2 h Figure ? 17. Geometrical extension of the compression kc, 90 = Table X l 1 2 h H, u ? grain at the bottom of the mortise of the header.

H,area u bN .

For theX. header thewith geometrical variables for this method are: Figure Header interacting tenons inchanged compression. Figure X. Header with interacting tenons in compression. A Figure X. Header with interacting tenons in compression. the N width of the joist is changed to the depth of the compression area, bN = l z bN = l z A h see Figure 13. bN = l z B l z, ef = bN, ef the effective depth of the joist is changed to the effective width of the l z, ef = bN, ef Acompression hH,uarea , see Figure 17. B,ef l z, ef = bN, ef bN 60 mm b 60 mm bN, ef = min bN b60 mm bN, ef = min N 2 bN 2 bN bN, ef = min 2 bbN N Except for this small modification the verification is exactly the same. Except for this small modification the verification is exactly the same. for this small is exactly the same. bExcept is extended in themodification direction of the the verification grain and this will therefore be on both sides of the N,ef 5.3.4 Failure in compression perpendicular to the grain due to an axial force notch the header, hence extension of 60 mm. to the grain due to an axial force 5.3.4 in Failure in compression perpendicular on theFailure joist in compression perpendicular to the grain due to an axial force 5.3.4 on the joist 4.2.3.1. Modifications on the joistpressure onfor For an axial theCLT header due to an axial force in the primary member the For an axial pressure on thetoheader due to anbe axial force infor thea primary member thearea where compression perpendicular the grain calculated decreased pressure This failure mode is only possible when allan layers are perpendicular to the load and therefore For an axial pressure on thetoheader due may to axial force infor thea primary member thearea compression perpendicular the grain may be calculated decreased pressure where the "hole" of the notch is removed as seen in Figure X. compression perpendicular to the grain may be calculated for a decreased pressure area where the compression the "hole" of the surface. notch is removed as seen in Figure X. the "hole" of the notch is removed as seen in Figure X. The effective area is only extended in the direction of the fibres (Gustafsson et al., 2019). Therefore, Figure X. if the first layer closest to the compression surface has fibres oriented perpendicular Figure X. as seen in Figure 17 then b = b and l is extended to l instead in the same to the grain Figure X. N,ef N z z,ef Fc, d in Chapter 4.1.3. The only constraint to the extension of l is that it cannot way as explained Fc, d z,ef kc, 90 fc, 90, d (16) c, 90, d = A Fc, dmodification exceed is illustrated similarly for the joist in Figure 9. k f (16) c, 90, dbH=. This B, ef kc,c, 90 fc, 90, d (16) c, 90, d = A AB, ef 90 c, 90, d B, ef Where:Compression failure perpendicular to the grain due to an axial force in the joist 4.2.4. Where: Where: For the axial header due to anforce. axial force in the joist the compression Fc, d an axial pressure on is the compression F is the axial compression force. perpendicular to the grain be calculated for aforce. decreased pressure area where the "hole" of Fc,c, dd is may the axial compression the is removed as seen inarea Figure 18.header subjected to the pressure (See Figure X). AB,notch = A A is the of the AB, ef = ABB AN is the area of the header subjected to the pressure (See Figure X). AB, ef = AB AN is the area of the header subjected to the pressure (See Figure X). ef N geometry of the joist the pressure area might extend in width outside of Depending on the AB is the area of the beam (See Figure X). AB notch. is the area of the beam (See Figure X). the AB is the area of the beam (See Figure X). AN is the area of the notch (See Figure X). AN is the area of the notch (See Figure X). AN is the area of the notch (See Figure X). 1.0 for all materials, l 1 2 h N 1.0 for all materials, l 1 2 h N 1.0 for all materials, l 1 2 h N kc, 90 = Table X l 1 (74) 2 hN kc, 90 = Table X 19 l 2 h kc, 90 = Table X l 1 2 hN 1

5.3.4.1. Modifications for CLT 5.3.4.1. Modifications for CLT 5.3.4.1. Modifications for CLT For a header with a grain orientation as seen in Figure X.b. the compression perpendicular For a header with a be grain orientation as seen in Figure X.b. the perpendicular to the grainCHALMERS should verified for CLT headers in the same waycompression asThesis seenACEX30 in Chapter 5.3.4. Architecture and Civilin Engineering, Master’s For a header with a be grain orientation as seen Figure X.b. the compression perpendicular to the grain should verified for CLT headers in the same way as seen in Chapter 5.3.4. For other orientations of the CLT part of this area will have grain direction parallel to the to the grain should be of verified for CLT headers in the same way asdirection seen in Chapter 5.3.4. For other orientations the CLT part of this area will have grain parallel to the axialother forceorientations and this typeofofthe verification isofnot needed. For CLT part this area will have grain direction parallel to the axial force and this type of verification is not needed.

c, 90 Tablethe X compression l 1 2 lto h1H,the ? h N needs to be verified 1.0mortise for all materials, For and joints perpendicular u 2 grain kc, 90tenon = Table 1 X l 1 for 2 hallH,materials, ? 1.0 l 2 hN kinc, 90 = u the=bottom to1 a tenon Na similar verification TableofXthe tenon and as a notch l 1 can 2 hbeH,compared ? kc, 90 u Table h H, be ? kc, 90be= made can for X this geometry. Failurell 1mode22 can u ? seen in Figure X. Formula (9) is given Table X h 1 H, u in the French contribution. Figure X. Header with interacting tenons in compression. Figure X. Header with interacting in compression. hH,utenons hH,u Figure X. Header tenons in compression. bN with = l zinteracting Figure X. Header with interacting tenons in compression. bN = X. l Header with interacting tenons in compression. Figure bN = l zz bN,ef l z, ef = bN, ef bN = l z bl N = =l zb ef = bN, ef l z, ef = bN, ef l z, bN 60 mm z, ef l z, ef = bN, ef N, ef bb 60 mm = min N ef 60 mm bN, 2 bN bN, ef = min bN 60 mm bN, ef = min bN 2 60 bN mm bN, ef = min V N 2 bN Ed 2 A bfor bN, ef = min Except this fsmall modification the verification is exactly the same. Nk c, 90, d = 2 bN bN lsmall N c, 90 c, 90, dthe verification is exactly the same. Except for this modification h z, ef modification the verification is exactly the same. Except forAthis small B 5.3.4 in compression perpendicular to the grain due to an axial force Except for this smallFailure modification the verification is exactly the same. ExceptFailure for this in small modificationperpendicular the verification is the same. 5.3.4 compression to exactly the grain due to an axial force(10) joist AB,ef 5.3.4 Failureon in the compression perpendicular to the grain due to an axial force hH,u on theFailure joist in compression perpendicular to the grain due to an axial force 5.3.4 on theFailure joist For Where: axial pressureperpendicular on the header due axial force member the 5.3.4 in an compression to to theangrain due in to the an primary axial force on joistpressure For the an axial on theperpendicular header due totoanthe axial force in be thecalculated primary member the compression grain may for a decreased pressure area where on the joist For the headershear due to an axial force in the primary member the bon Vd an axial pressure isthe the"hole" dimensioning force. compression perpendicular the grain be calculated for a primary decreased pressure area where of to the notch is may removed asforce seeninin Figure X. member N the For an axial pressure on header due to an axial the the compression perpendicular toheader the grain may be calculated for decreased pressure where For an axialofpressure onisthe dueseen to an axial force theaa primary member thearea the "hole" the notch removed as inbe Figure X. infor compression perpendicular to the grain may calculated decreased pressure area where b the "hole" of the notch is removed as seen in Figure X. compression thenotch grain mayinFigure beFigure calculated bN "hole" of perpendicular is the width oftothe (See X). the the notch is removed as seen X. for a decreased pressure area where the "hole" of the notch removed to astheseen Figure X. force in the joist. Figure 18. Compression failureisperpendicular grainindue to an axial Figure X. l 30 mm Figure X. z Figure X. F lFigure = min z, ef X. 2 l z = c, d Figure X. Fc, d c, 90, (19) kc, 90 fc, 90, d (16) d Fc, d A ef kc, 90 fB,c, 90, (16) c, 90, d = A F d k 90 fc, 90, d (16) c, def 90, d = A isc,c,the the transverse compression (mm) = AFB, fc, 90, (16) (See Figure X). In theory the c, deflengthkc,of 90, deffective B, Where: c, 90 d = k f (16) c, 90, d area effective of the compression zone is extended depending on grain direction and on both B, ef AB, ef c, 90 c, 90, d Where: Where: sides of the support. l z, ef takes into consideration the distribution beyond the tenon length. Where: Fc, d is the axial compression force. Where: The this thesis limit zone to only be extended in one direction Fc, d evaluated geometries is in the axial compression force. Where: is the dimensioning axialthis force. Fc, d to the interaction ofisthe thejoist axialand compression force. due its header. Fc, d compression AB, ef = AisB theAaxial is the area force. of the header subjected to the pressure (See Figure X). N F is compression force. Ac,B,def = AB AN is the the axial area of the header subjected to the pressure (See Figure X). AB, ef = AB AN is the is the of the header subjected to the (See Figure X). 18. areaarea of the header that is subjected topressure the pressure, see Figure is the areasubjected of the beam (Seepressure Figure (See X). Figure X). AB, ef = AB AABN is the area of the header to the A = AB AN is subjected the pressure (See Figure X). AB, is the the area area of of the the header beam (See FiguretoX). ABB ef is the area of the beam (See Figure X). AN is the areaFigure of the notch AB is the of the beam (See Figure X).(See Figure X). is the areaarea of the joist, see 18. A is the the area area of of the the notch beam (See (See Figure Figure X). X). ABN is AN is the area of the notch (See Figure X). AN is the of the notch (See Figure 1.0 area forX). all materials, l 1 2 h N AN is the of the notch (See Figure is the areaarea of the notch, See Figure 18.X). 1.0 for all materials, l 2 h kc, 90 = Table X 1.0 for all materials, l 11 h 2 h N l 2 1.0 for all materials, l 1 N 2 hN 1 kc, 90 = Table X l 2 h 1.0 for all materials, l 2 hN kkc, 90 = Table is aXcorrectional factor, see 3.1. 1 see N N lChapter 2 hChapter is a correctional factor, 3.1. 1 kc,c, 90 90 = 1 N X l 1 2 hN kc, 90 = Table Table X l 1 2 hN fc, 90, d Modifications is the design material in compression perpendicular to the 4.2.4.1. for strenght CLT of the 5.3.4.1. Modifications for CLT grain. 5.3.4.1. Modifications CLTorientation as seen in Figure 15.c. the verification should be For a CLT header with for a grain 5.3.4.1. Modifications for CLTa grain orientation as seen in Figure X.b. the compression perpendicular For a header with 5.3.4.1. Modifications for CLT performed in the same way as seen inasChapter For a header with a grain seen for in4.2.4. Figure X.b. theincompression perpendicular to the grain should CLT headers the same way as seen in Chapter 5.3.4. 5.3.4.1. Modifications fororientation CLT be verified For a header with a grain orientation as seen in Figure X.b. thegrain compression perpendicular For other orientations of the CLT part of this area will have direction parallel to the to the grain should be verified for CLT headers inpart the way as seen in grain Chapter 5.3.4. For aother orientations of the CLT ofsame this areacompression will have direction parallel to the For a header with grain orientation as seen in Figure X.b. the perpendicular to the grainand should be verified for CLT headers in the same waycompression as seen in Chapter 5.3.4. For a header with a grain orientation as seen in Figure X.b. the perpendicular axial force this type of verification is not needed. other orientations of the CLT part of this area will have grain direction parallel to the axial force and this type ofheaders verification is same not needed. to the grain should be verified for CLT in the way as seen in Chapter 5.3.4. For other orientations the CLT part of this area willsame have grainasdirection parallel 5.3.4. to the to the grain should be of verified for CLT headers in the seen in Chapter axial force and this type ofthe verification not For other orientations of CLT partis of thisneeded. area will have way grain direction parallel to the axial force and this type of verification is not needed. For CLT partisofnot thisneeded. area will have grain direction parallel to the axialother forceorientations and this typeofofthe verification axial force and this type of verification is not needed.

6. VERIFICATION OF TENON AND MORTISE JOINT 6. VERIFICATION OF AND 6. VERIFICATION OF TENON TENON AND MORTISE MORTISE JOINT JOINT 6.1 TENON MEMBER 6. VERIFICATION OF TENON AND MORTISE JOINT 6. VERIFICATION 6.1 TENON MEMBEROF TENON AND MORTISE JOINT 6.1 TENON MEMBER 6.1 TENON MEMBER 6.1 TENON MEMBER

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CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

5. VERIFICATION OF TENON AND MORTISE JOINTS 5.1. TENON MEMBER 5.1.1. Geometry The tenon can be seen as a combination of a top notch and a bottom notch when it comes to the failure modes. Therefore, even though the tenon has two notch corners, it is the lower notch corner that is the most critical and the tenon is therefore, in most verifications, approximated as a top notch, see Figure 19. A-A

hu bN

VEd

Figure Geometry of a tenon joist. 6.1.119.Geometry

z0,T z0,T The tenon can be seen as a combination of a top notch and a bottom notch when it comes 5.1.2. tension failure perpendicular to the grain the notch lower corners, tenon corner to the Shear failureand modes. Therefore, even though the tenon hasattwo it is the lower notch corner that is the most critical and the tenoon is therefore, in most verifications, h N Similar to that of a notched member crack propagation in the grain direction at the heighth approximated as a top notch. mode is illustrated in Figure hu is the major failure mode for the tenon member. The failurez0,B h 20. u 6.1.2 Failure at lower notch corner V This is verified in aa notched similar way wherecrack the equation verifies thegrain tension perpendicular to the Similar toEdthat of member propagation in the direction at the height grain been reduced tofor shear verification. h u is but the has major failure mode the tenon member. lz For the tenon joist different functions are used depending on the placement of the tenon. This is verified a similarfor waya notch where (eq. the equation verifies the tension the Compared to theinequation (6)) the correctional factor kperpendicular is also addedto(Blaß z grain but has been reduced to shear verification. & Sandhaas, 2017). The equations for this verification depend on the position of the tenon as Chapter in the report and this can be seen in Figure where zof is the 0,B the Forexplained the tenoninand mortise5 different functions are used depending on the 21 placement geometrical centre of gravity for the tenon and z is the centre of gravity for the tenon. tenon. Compared to the function for a notch the 0,T correctional factor k is also added. (Source) z

2 3 bef h N kz kv f

Fv, Rd = 4 VEd b h k k 9 ef N z v lz

VEd VEd

v, d

for z0, T

z0, B

for z0, T

z0, B

(20)

(16)

For discussion regarding the use of this function for different tenon possitions see Chapter XX. Where: (76) to account for the negative influence of 21 is the width of the tenon reduced cracks (kcr can be found in Chapter 4.1).

bef = kcr bN hN Figure X). hT

kz = h N

is the height from the bottom of the tenon to the top of the beam (See CHALMERS Architecture and Civil h 2 h Engineering, Master’s Thesis ACEX30

o approximated as a top notch. hmost notch corner that is the critical and theperpendiculat tenoon is therefore, in most verifications, N The generated force component for tension to the grain x can be calculated as approximated II I as a top h notch.hT (Timber engineering). 6.1.2 Failure atdue lower notch Figure 2. Failure to shear andcorner tension perpendicular to the grain.

2 crack propagation 3 hu corner 6.1.2 Failure atalower notch Similar to= that of in the grain direction at the height Ft, 90,infinite 1.3stresses VEd notched 3can 1notmember 2 1 mechanics are Since occur fracture used to estimate the following d major hFigure is the failure mode for the tenon member. 2. Failure due to shear and tension perpendicular grain. u Similar to of a notched member crackcapacity propagation intothethegrain direction at the height formula forthat verifying the shear and tensile of aAnotched beam. The formula is h u is thetomajor failure modebut for this the tenon member. reduced shear verification formula verifies the capacity for tension perpendicular(6) to b b N stresses Nthe equation This isinfinite verified in(EC5). a similar way where verifies the tension perpendicular to the Since can not occur fracture mechanics to estimate the following the fibers as well lzare used grain butfor has been reduced to shear verification. formula verifying the shear and tensile capacity a stresses notchedare beam. The formula is The factor 1.3 is approximated to the that of the not uniformly distributed This is verified a similar waydue where thefact equation verifies the tension perpendicular to the b inverification reduced to shear but this formula verifies the capacity for tension perpendicular at the notch corner, see Figure X, which results in peak stresses and an approximate increaseto V VEd grain but3tenon has been reduced different to shear functions verification. Ed mortise For the and are used depending on the placement of the the fibers as well (EC5). of force component by 30 %. Approximated through FE simulations by Henrici = tension kv fv, d the 2 d tenon. Compared to the function for a notch the correctional factor k is also added. (Source) h b due (Timber engineering). Figure Failure to shear and tension perpendicular to the grain. For the 2. tenon and mortise different functions are used depending on zthe placement of the z z0,T VEd efdue to shear and tension perpendicular to the grain. FigureCompared 2. 3 Failure tenon. to the function for a notch the correctional factor 0,T kz is also added. (Source) =infinite k f For reinforcement layers the bond line stress as well as the tensile stress isfollowing calculated(4) d these Since stresses can not occur fracture mechanics are used to estimate the 2 2 v v, d h b Since infinite stresses can not mechanics are to the following and verified. kzshear kv foccur Vfracture z0,used z0,estimate formula for verifying and tensile capacity offor a notched beam. The formula is 32 befef h Nthe Ed T B formula for verifying the shear and tensile capacity of a notched beam. The formula is reduced to shear verification but this formula verifies the capacity for tension perpendicular to h Where: Nfor z bverification h N karea k but f this Vformula zthe Fv, Rd =to shear The effective line into is the areacapacity below of the h(4) reduced the for tensionarea perpendicular to 3bond ef(EC5). z vtaken Edaccountverifies 0, T 0, Bcritical the fibers as4well z0,B failure (hence the area below the notch corner). the fibers as well (EC5). b h k k V for z z V is the dimensioning shear force. F = 94 ef due N htoz shear v v, and d tension Ed perpendicular to 0, the T grain. 0, B Ed Figure v, Rd 2. Failure Where: VEd u 3 b h k k V for z z F V z vk of v, the Ed reduced to account T B 3b 9 stresses fdv, dnotch bVefd = =kinfinite iseft,90,d theN width for the negative influence of(16) Since can not occur fracture mechanics are0,used to0,estimate the following 2VEd hEd shear force. kvv fRv, bisef the dimensioning ed, dEdd= cr= 2 N d v, d formula for the shear and tensile capacity of a notched beam. The formula is nbverifying h h l h cracks bef N (k r can be found in Chapter 4.1). (16)to cr For discussion thebut usethis of this function for different tenon see Chapter reduced to shear verification formula verifies the capacity forpossitions tension perpendicular lz regarding b = k b is the width of the notch reduced to account for the negative influence of (4) XX. the ef fibers cr as N well (EC5). (7) (4) For the of usethe ofnotch. this function differentfortenon see5.2.1.1. Chapter bN discussion regarding is the width May befor modified CLTpossitions see Chapter cracks (kshear canand be found Chapter 4.1). Figure and tension failure perpendicular to inperpendicular Figure 21.toGeometrical data for the positioning of the Figure20.2.Shear Failure due to tension the grain. cr XX. Where: Where: VEd the grain at3the notch corner for the joist. tenon. The tensile stress in the reinforcement layers is then acting parallel to the fibers and verified Where: = k f 1.0 Since stresses can not occur fracture mechanics are used to estimate the following as: bVdN infinite is the width of the notch. May be modified for CLT see Chapter 5.2.1.1. Where: is the dimensioning shear force. vof the v, d tenon reduced to account for befEd= kcr 2for bN verifying width the negative influence hisbisefthe formula the shear and1.1 tensile capacity of a notched beam. The formula is of V the dimensioning shear force. 1.5 i Where: Ed ft,k(k reduced verification butbe this formula verifies the capacity fornegative tensioninfluence perpendicular to cracks found in Chapter 4.1). 1of bbef == kkcr to bFshear the tenon reduced to for of t,90,dis dn cr can 1.0 is (EC5). theiswidth width of the the notch reduced to account account for the thefor negative influence of (4) =cr bN the as well the width of the tenon reduced to account the negative influence h N kbveft,==dfibers min kk(k can kcr bnN t l iscracks the width of the notch1.5in reduced to account for the negative influence of be found 4.1). ef r r cracks (k found in Chapter Chapter 4.1). cr canfrom 1.1 can found in the Chapter ofheight cracks (kbe hN is the theibe bottom of tenon3.1). to the top of the beam (See cr cr x 1 Where: 3 21 be found in Chapter2 4.1). VEdcracks k(kncr can h is the 0.8 Figure X). (8) hhbottom hbkN height from the of tenon the topmodified of the beam (See see iswidth the kwidth ofnotch. the notch, see Figure 19.to May be for 5.2.1.1. CLT fthe dv = =min is the of May bethe modified for CLT see Chapter 2 v v, d V is the dimensioning shear force. N h b bNEd X). is eftheChapter width of5.1.2.1. the notch. for CLT see Chapter 5.2.1.1. Figure x h May 1 be modified 2 Where: h TVEd h T2 2 (5) h 0.8 h N kbz == khh b 1 is2 the1 width 2 1.0notch of 2the reduced to account for the negative influence of (4) h h h h ef crN N l 1.0 T 2 zn T internal layers N act as reinforcement N if only isk a=reduction that consideration the 4.1). risk of crack growth in notched (21) 1 factor 2cracks 1 (ktakes 2 i 1.5in caninto be1.1 found Chapter nzb = h N2 (See 1.5h crh N1 (5) Where: k 1.1 i members Chapter X). n 1 ifn external layers are included as reinforcement (17) kn 1 h kv = min isisnotch the ofopposite the notch. May bethe modified CLT seebeChapter 5.2.1.1. hshear a=reduction factor that takes into consideration risk ofkfor crack growth inused. notched the force. kbVFor min *is beams with atcorrection the side ofChapter the support = 1.0 can iswidth adimensioning factor, see N is number of bond-lines. is athe correction factor based on test results consideration influences of the (17) vEd v x that 1takes 2into3.1. members (See Chapter X). 2 tenon geometry.h This is based on 0.8 a limited geometry discussed further in Indesign. x 1 2 test 2 andconsideration h reduced is based on results that takes influences of theof kcr bN is factor is the widthof ofreinforcement the notch to into account for the negative influence 1.0 layers. hthe 0.8 knbnefa=correction anumber proportionality constant, see Chapter 4.1. h side * For beams with notch at the opposite of the support kv = 1.0 further can be used. tenon based onthe a limited geometry and discussed in Indesign. isisheight the tenon height, see Figure 19. h T geometry. This is the of tenon. cracks (kcr can be1.1 found1.5in Chapter 4.1). is the beam i(See (5) kn of 1the ihh isis height the notch inclination (seeFigure FigureX). 1). the height of the tenon. (5) knT is a proportionality constant, see Chapter 4.1. h the isto the height from bottom of thetotenon to thesee top of the beam, see hhkb = min is the full height of the joist. Figure 2. Failure shear and tension perpendicular is due the width ofinto the notch. May bethe modified for grain. CLT 5.2.1.1. the height of the notch (See Figure X). is that takes consideration risk ofthe crack growth Chapter in notched v Na reductionisfactor h is the height of the beam in mm. Figure 20.intoofconsideration is that risk of crack growth in notched 1 the21). members (See factor Chapter X).takes hi a reduction is full height thex joist. 2inclination is the the notch (seemechanics Figure Since infinite stresses can not occur fracture are used to estimateh N the following h 0.8 members (See Chapter X). lformula is the effective width of the reinforcement (See Figure X). The h h capacity 1.0tensile k*rvFor Nbeams is a reduction factor according to equation (5). With =behhformula . for verifying the shear and of a notched beam. is is the joist height, see Figure 19. with notch at the opposite side of the support k = 1.0 can used. h= isthe theratio height of thethe beam in mm. is between height of the notch and height of beam. Mayto v =the Nthe 1.5 reduced to shear verification but this formula verifies the capacity for tension perpendicular *k For beams with notch at the opposite side of the support k 1.0 can be used. h 1.1 according i is a reduction factor to equation (5). With = . v f is the design rolling shear strength of the reinforcement layer(s). v modified h (5) k Chapter 1 shear5.2.1.1. the fibers wellCLT is(EC5). design strength ofsee theChapter cross-section. be Rv, hdN as for kfnv, is athe proportionality constant, 4.1.and Chapter 3.1. with issee a nreduction factor to eq. (7) h according kv==d min k is a proportionality constant, see Chapter 4.1. isisthe ratio between the heightofofthe the notch and the growth height of beam. May the design strength cross-section. is that into the risktoof crack in the notched α = htakes /h .shear h 3 isVfactor trfnv,a dreduction the thickness one consideration reinforcement layer(s) (See Figure xV isis the distance the support reaction the corner X). of the notch. N offrom Ed the dimensioning shear force. x 1 2 2 ibedEdmodified is the notch inclination (see Figure 1). members (See Chapter X). 5.2.1.1. = 2 forh CLT see Chapter k fv, d0.8 if isbthe thedesign notchvinclination (see 1). h strength hisis shear strength ofFigure theofmaterial. V the dimensioning shear force. is the design shear the material. ef k = 2.0 is a correction coefficient toside take into account the =non-uniformly distributed v, d beams Ed is notch the distance height the beam in mm. *hxkFor with at CLT theofopposite of the support 1.0 can be used. 6.1.2.1 Modifications for from the support reaction to kthe v corner of the notch. h is the height of the beam in mm. (5) (4) 5.2.1.1. Modifications forCLT CLT 6.1.2.1 Modifications for The use of CLT for of this of failure mode are the same as seen in is shear strength thetype material. kfnv, dhhmodifications is the awith proportionality constant, see Chapter 4.1. 3.1. N isdesign athe proportionality constant, see Chapter Chapter 5.2.1.1. withtakes the modifications andofand width =a reduction is But thethat ratio between the heightofofheight the theapplied height to of the beam.(16) May is factor consideration thenotch risk crack growth in equation notched Where: hN The modifications with thebetween useinto of CLT for this type of failure mode are the same as seen = (17) is the ratio the height of the notch and the height of the beam. Mayin and instead of eq. (4). h 5.2.1.1. 5.2.1.1. Modifications forChapter CLT members (SeeforChapter X). be CLT see 5.2.1.1. Chapter the modifications of height i modified isBut thewith notch inclination (see Figure 1).and width applied to equation (16) VEdmodified is eq. the dimensioning shear force. issee the dimensioning shear force. be for CLT Chapter 5.2.1.1. and (17) instead of (4). *xh For beams with at theofopposite side of of the support = 1.0 can be used. 6.1.3 Compression failure at the bottom the tenon is notch the distance from the support reaction to kthe height the beam in mm. v corner of the notch. xb = k b is the the width distance from the support reaction to the corner of the influence notch. of is of the notch reduced to account for the negative 6.1.3 failure at the bottom oftenon the tenon For mortise a possible failure ofofthe is also compression perpendicular to ef ah tenon crCompression N and is f the design shear strength the material. knv, d N is a proportionality constant, see Chapter 4.1. cracks (k can be found in Chapter 4.1). f is the design shear strength of the material. the ratioacrpossible between the height of the notch andcompression the height ofperpendicular the beam. May For=v, adhtenon andismortise failure of the tenon is also to 5.2.1.1. Modifications for CLT i modified for CLT is the see notch inclination (see Figure 1). be Chapter 5.2.1.1. bN is the width of the notch. May be modified for CLT see Chapter 5.2.1.1. 5.2.1.1. Modifications for CLT is the the distance height offrom the beam in mm.reaction to the corner of the notch. xh is the support 1.0 22 (77) fv, dh N is the design shear strength 1.5 of the material. 1.1 i = h is the ratio the height of the notch and the height of the beam. May kn between 1 5.2.1.1. Modifications for CLT h be k modified = min for CLT see Chapter 5.2.1.1. v

fv, d

x 1 2 2 h is the distance from 0.8theh support reaction to the corner of the notch. is the design shear strength of the material.

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

5.2.1.1. Modifications for CLT is a reduction factor that takes into consideration the risk of crack growth in notched

(5)

hT limited by the notch area subjected to toIIthe grain. I tension h purpendicular

5.2.2 Compression failure at the bottom of thehunotch tensile stresses.

For tenon and mortise joints the compression perpendicular to the grain needs A to be verified is the design tensile strength of the reinforcement layer. inft,the d bottom of the tenon and as abnotch can be compared toba tenon a similar verification N N can be made for this geometry. Failure mode can be seen in Figure X. Formulalz (9) is given The width, l r , of the reinforcement is limited to: in theeffective French contribution. b VEd 0.25 h h N lr 0.50 h h N

z0,T (9) The modifications with the use of CLT for this type of failure mode are the same as seen Where the minimum value prevent crack development in thetoarea in of Figure X in in Chapter 4.1.2.1. But with the modifications of applied eq.accentuated (20) instead eq. (6) hNaccount and the maximum value ensures that the area of the reinforcement taken into is Chapter 4.1.2. z0,B limited by the notch area subjected to tension purpendicular to the grain. h u 5.1.3. Compression failure perpendicular to the grain at the bottom of the tenon 5.2.2 Compression failure at the bottom of the notch VEd VEdfailure of the tenon is also compression perpendicular to For a tenon and mortise a possible kc, 90 fthe c, 90, d = and mortise joints For tenon compression perpendicular to the grain needs to be verified c, 90, d bNthe l z,tenon. the grains in This verification is similar to that of the notch under the same failure ef tenon and as a notch in the bottom of the can be compared to a tenon a similar verification lz mode, Chapter and can be seen in Figure can be see made for this4.1.3. geometry. Failure mode can be22. seen in Figure X. Formula (9) is given (10) in the French contribution. Figure 2. Failure due to shear and tension perpendicular to the grain. Where: Since infinite stresses can not occur fracture mechanics are used to estimate the following Vd is the dimensioning force. formula for verifying the shear andshear tensile capacity of a notched beam. The formula is reduced to shear verification but this formula verifies the capacity for tension perpendicular to the bN fibers as well is the(EC5). width of the notch (See Figure X). 5.1.2.1. Modifications for CLT

z0,T

VEd 3 l 30 mm k f = 2 z v v, d h b l z, ef = min VEd 2ef l z VEd (4) fc, 90, d lz compression (mm) (See Figure X). In theory the 90, effective d = is c,the ofkc,the 90 transverse bN llength z, efgrains the in the tenon. verification is bottom similar totenon. that of the notch Where: area of effective the compression zone This is extended depending onof grain direction and onunder both the same Figure 22. Compression perpendicular to the grain at the the failure (Chapter sides of the support. mode l z, ef takes into5.2.2.). consideration the distribution beyond the tenon length. (10) VEd is the dimensioning shear force. The evaluated geometries in this thesis limit this zone to only be extended in one direction Fv, Rd = 1.7 bN l z, ef fc, 90, d V (22) Where: due to the interaction of the joist and its header. Ed b = k b is the width of the notch reduced to account for the negative influence of Figure Failure due to shear and tension perpendicular to the grain. ef cr2. N (18) Vd is the dimensioning cracks (k can beshear foundforce. in Chapter 4.1). Where: Since infinite stresses cancrnot occur fracture mechanics are used to estimate the following This formula is derived from tests and the factor 1.7 is derived from this. This factor formula for verifying the shear and tensile of a notched beam. TheChapter formula5.2.1.1. is bbN is the width of the notch (Seecapacity Figure X). is the of May be, Figure modified CLT see corresponds thethe coefficient kc, 90 as seen 19. forfor notches, that take into consideration the fact iswidth the to width ofnotch. the notch, see N reduced to shear verification but this formula verifies the capacity for tension perpendicular to that(EC5). compression perpendicular to the grain is distributed over areas larger than the directly the fibers as well l 30area, mm here the 1.0support area. Therefore, this factor increases the compression strength loaded z Vperpendicular l z, ef = min to the grain with 70 %. 3 1.1 i 1.5 Ed 2 l z k k1 f d = 2 v, d width of the notch (See Figure X). n vis the hbNbef h kv = min k is a correctional factor, seecompression Chapter 3.1.[mm] is the effective length of the transverse (mm) (See is c,the the (See Figure Figure X). 8). In theory the 90 effective length 1 2see Chapter 2 effective area ofl the compression zone isx extended depending on grain direction and on both 5.2.2.strength perpendicular to the grain hthe 0.8 isl thestrenght designinto compression of length. the (4) z, efdesign fc, 90, of is of the material in compression perpendicular to the sides the support. takes consideration the distribution beyond the tenon h d z, ef material. Where: grain. The evaluated geometries inisthis limit thiscompression zone to onlystrength be extended in one direction fc, 90, d thethesis dimensioning perpendicular to the grain. (5) due to the interaction of the joist and its header. VEd is the dimensioning shear force. is the dimensioning shear force. VEd is the dimensioning shear force. is a reduction factor that takes into consideration the risk of crack growth in notched members Chapter X). of the notch reduced to account for the negative influence of bef = formula kcr b(See isderived the width This from tests and the factor 1.7 is derived from this. This factor N is 6.1.3.1. Modifications for CLT cracks (k can be found in Chapter 4.1). , as seen for that takes into consideration the fact corresponds to the coefficient k * For beams with notch atcr the opposite side ofnotches, the support kv = 1.0 can be used. c,90 No modifications need to be made for this failure mode for a CLT member that compression perpendicular to the grain is distributed over areas larger than the directlybut it is only relevant for members where the load, and therefore the compression, is applied perpendicular isthe width ofarea. the Therefore, notch. May beChapter modified for CLT Chapter 5.2.1.1. kbnN athe proportionality constant, see 4.1. loaded area, hereis support this factor increases the see compression strength to all layers of the CLT. perpendicular to the grain with 70 %. 1.0 i is the notch inclination (see Figure 1). 6.1.4 Rolling shear failure of the notch (only for CLT) 1.5 1.1beam i Chapter khc, 90 is aiscorrectional factor, see 3.1. height in mm. knmethod 1of the The the same as in Chapter 5.2.2 is used but modified with the geometrical data of the tenon as seen in Figure hX. kv =hmin fc, 90, N is the design strenght of the material in compression perpendicular to the d 23 1of the2notch and the height of the beam. May (78) = h is the ratio between the xheight 2 grain. 6.2 h MORTISE/HEADER 0.8 MEMBER h be modified for CLT see Chapter 5.2.1.1. The geometry and verifications of the header for a tenon and mortise joint are the same as for theissecondary member of the notchreaction and calculation method cannotch. be followed (5) in Chapter 5.3. x the distance from the support to the corner of the d

is the design shear strength of thethe material. isfv, a dreduction factor that takes into consideration risk of crack growth in notched Add Images for all Failure modes. members (See Chapter X). CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30 5.2.1.1. Modifications for CLT * For beams with notch at the opposite side of the support kv = 1.0 can be used.

7. VERIFICATION OF DOVETAIL JOINT

5.1.3.1. Modifications for CLT See Chapter 4.1.3.1.

5.1.4. Rolling shear failure in the tenon (only for CLT) The same method as in Chapter 4.1.4 is used but modified with the geometrical data of the tenon as seen in Figure 19. 5.2. MORTISE MEMBER The geometry and verifications of the header for a tenon and mortise joint are the same as for the header of the notch and calculation method can be followed in Chapter 4.2.

6. VERIFICATION OF DOVETAIL JOINTS In general a dovetail joint is very similar to a notched joint and is the reason why many of the failure modes and calculations are similar for the two but with modified geometrical data. The difference of the dovetail is that this joint can also handle axial tension between the dovetail and the header. This creates an additional failure mode. 6.1. DOVETAIL MEMBER 6.1.1. Geometry This thesis include two different dovetail geometries with minor geometrical modifications between these. Figure 23 illustrate these geometries with their variables. The only difference between a. and b., and c. and d. is the orientation of the dovetail geometry. This is differentiated since the dovetail geometries are not symmetrical with regard to these two directions and this entail differences in the calculation methods for the two cases. Figure 23.a2. and 23.b2. illustrates the geometry for when the dovetail is widened to more than the smallest width due to the radius as is shown in Figure 23.a1. and b1. 6.1.2. Shear and tension failure perpendicular to the grain at the lower dovetail corner The point of failure is very similar to that of the notched member but the crack appears in the transition zone between the straight tenon and the fillet instead of at the notch corner for the geometry in Figure 23.a. (Blaß & Sandhaas, 2017). For the geometry in Figure 23.b and d. the failure occurs in the notch corner. For the geometry in Figure X.c. the base of the dovetail is a straight notch and the failure occurs for this in the notch corner as well. All four failure modes can be seen in Figure 24.

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CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

hB1

hN h hN

bB1

a2. Type I

hB1

bB1

VEd

bB1

b2. Type II

hN h hN

b lZ

c. Type III

VEd

d. Type IV

VEd

Figure 23. a. and b. Geometry of a rounded dovetail for two different load cases. c. and d. Geometry of a straight dovetail for two different load cases.

6.1.2.1. Dovetail type I

Dovetail type I is the most common geometry when it comes to calculations and could be found in several sources. The tension perpendicular to grain and shear stress verification for this is very similar as for a notch. A comparison with motivation of the choice of equation for verification of this failure mode can be seen in Chapter 5 in the report. The resulting equation is eq. (23) (CEN, 2019).

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CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

hB1

lZ lZ The point of failure is very similar to that of the notched member but the crack appears in The point of failure very similar to that the of the notched member butfillet crack appears in notch corner for the transition zone between straight tenon and the instead of at the Vthe Vis Ed crack appears in The point of failure isEdvery similar to that of the notched member but the the transition zone between the straight tenon and the fillet instead of at the notch corner the geometry in Figure (Timber Engineering). a. Type c. Typemember III The pointI of failure is very similar to X.a that of the crack appears infor the transition zone between the straight tenon andnotched the fillet insteadbut of the at the notch corner for the geometry in Figure X.a (Timber Engineering). the zone between the straight tenon and the fillet instead of at the notch corner for the transition geometry in Figure X.a (Timber Engineering). Geometry for dovetail with a very wide base! the geometry Figure with X.a (Timber Engineering). Geometry forindovetail a very wide base! Geometry for dovetail with a very wide base! For the geometry in Figure X.c and d. the failure occurs in the notch corner. Geometry for dovetail with aX.c veryand wide base! For the geometry in Figure d. the failure occurs in the notch corner. β For the geometry X.c in andFigure d. theX.b. failure in is theapproximated notch βcorner.as a dovetail like the one in For in theFigure geometry theoccurs dovetail For in the failure occurs in theasnotch corner. For the the geometry geometry in Figure Figure X.b.and the d. dovetail is approximated a dovetail like the one in Figure X.d. X.c For the geometry in Figure X.b. the dovetail is approximated as a dovetail like the one in Figure X.d. For the Figure X.b. the dovetail is approximated bgeometry Figure b X.d. N Allinfour failure modes can be seen in Figure b bN X.as a dovetail like the one in Figure X.d. All four failure modes can be seen in Figure X. All four failure modes can be seen in Figure X. All four failure modes can be seen in Figure X.

Figure X. Failure modes for the 4 types of dovetails. Figure X. Failure lZ modes for the 4 types of dovetails. lZ Figure X. Failure modes for the type 4 types 7.1.2.1. Dovetail I of dovetails. Figure X. Failure modes for the 4 types of dovetails. 7.1.2.1. Dovetail type I VEdI 7.1.2.1. Dovetail type Ed Dovetail type I is the most common geometry when it V comes to calculations and could be 7.1.2.1. Dovetail type I d.perpendicular Type IV to calculations b. Type II Dovetail type Ifound is theinmost common geometry when it comes could be verification for several sources. The tension to grain andand shear stress Dovetail type I issources. the most common geometry whentoit grain comesand to shear calculations and could be found24. inShear several The tension perpendicular stress verification forchoice of this is very similar to that of a notch. A comparison with motivation of the Figure and tension failure perpendicular to the grain at the notch corner for the four different geometries shown Dovetail type I issources. the most common geometry whento it comes to calculations and could be found in several The tension perpendicular grain and shearofstress verification for this is very similar to that of a notch. A comparison with motivation the choice of equation forThe verification of this failuretomode can seenstress in Chapter X of the in Figure 23.several found in sources. tension perpendicular grain andbeshear verification for report. (CENthis is very similar to that of a notch. A comparison with motivation of the choice of equation forsimilar verification failureAmode can be seen Chapter Xofofthe thechoice report.of(CENTC250) this is very to thatofofthis a notch. comparison within motivation equation TC250) for verification of this failure mode can be seen in Chapter X of the report. (CENequation TC250) for verification of2 this failure mode can be seen in Chapter X of the report. (CENTC250) 2 F (23) = b h r kv fv, d VEd (19) Fv, Rd = 23 bv,B1Rd h B1 3 r B1 kv fB1 V (19) v, d Ed Fv, Rd = 23 bB1 h B1 r kv fv, d VEd (19) Where: Fv, = b h r k f V (19) v, Rd 3 B1 B1 v v, d Ed Rd B1 B1 v v, d Ed Where: Where: Where: b is the smallest width of the dovetail (See Figure X). Where: B1 bB1 isis the dovetail (See Figure23.a. X). the smallest smallest width of the dovetail, see Figure bB1 is the smallest width of the dovetail (See Figure X). h B1 is the smallest iswidth the largest the Figure dovetailX). (See Figure X). bB1 of the height dovetailof(See hB1 is the largest height of the dovetail (See Figure X). B1 h B1 is Figure 23.a. X). is the largest height of the dovetail dovetail,(See see Figure r is theof radius of the bottom fillet X). of the dovetail (See Figure X). h B1 is the largest height the dovetail (See Figure rB1 is the radius of the bottom fillet of the dovetail (See Figure X). r is the2 radius of the bottom fillet of the dovetail (See Figure X). his the r the radius the bottom fillet of ofthe the dovetail, dovetail see (SeeFigure Figure23.a. X). B1 radius of of the bottom fillet h B1 22 k = is a reduction factor that takes into consideration the risk of crack growth h v h kv = h B1 is a reduction factor that takes into consideration the risk of crack growth h 22 kv = B1 isisa the reduction that takes into consideration of crack growth design factor compression strength perpendicularthe to risk the grain of the B1 h kv = is a reduction factor that takes into consideration the risk of crack growth material. v h and size-effect of the member. and size-effect of the member. and size-effect of member. f the is the design shear strength. and d member. fv, dsize-effect ofv,the is the design shear strength. fv, d is the design shear strength. VEd is the design shear is thestrength. dimensioning shear force. fv, dd Vv, is the dimensioning shear 26 (81)force. VEd is the dimensioning shear force. Ed VEd is the dimensioning shear force. Ed

7.1.2.2. Dovetail type II 7.1.2.2. Dovetail type II 7.1.2.2. Dovetail typedovetail II For type the as in Figure X.b. the shear capacity is calculated according to 7.1.2.2. Dovetail II as in oriented For the dovetail oriented Figure the shear that capacity is calculated Z-9.1-649. Where it is X.b. approximated the reaction force according is appliedto at the centre of the For the dovetail oriented as in Figure X.b. the reaction shear capacity is applied calculated according tothe Z-9.1-649. Where it is approximated that the force is at the centre of width of the dovetail height. Even though the dovetail is not symmetrical over it's width this For the dovetail oriented as in Figure X.b. the shear capacity is calculated according to CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30 Z-9.1-649. Where it height. is approximated that the the dovetail reaction is force issymmetrical applied at the centre of thethis width of theWhere dovetail Even though notis over it's width is an itapproximation. Z-9.1-649. is approximated that the reaction force applied at the centre of the width of the dovetail height. Even though the dovetail is not symmetrical over it's width this is an approximation. width of the dovetail height. Even though the dovetail is not symmetrical over it's width this is an approximation. 2

h B1 iswidth the(see largest height the dovetail (See Figure X). ihbb is the notch inclination Figure 1). of iscan the not largest height ofof (See Figure X). Since infinite stresses occur fracture mechanics are used to estimate is the smallest the dovetail (See Figure X). is the smallest width ofthe thedovetail dovetail (See Figure X). the following B1 B1 B1 formula for verifying the shear and tensile capacity of a notched beam. The formula is hr theisheight of the in mm. r isverification isbeam thebottom radius of thethe bottom fillet oftension theFigure dovetail reduced to shear but this formula verifies capacity forX). perpendicular to X). the largest radius of the fillet of the dovetail (See X).(See Figure hh B1 is height of dovetail (See Figure is the the largest height of the the dovetail (See Figure X). B1fibers as well (EC5). the h h B1 2 2 rr = Nh B1 Vkis =the ratio is radius bottom fillet of dovetail (See Figure X). height offactor the and the height the beam. is the thebetween radius isof ofthe the bottom filletnotch of the the dovetail (See of Figure X).the May athe reduction that takes into consideration risk of crack growth kv = h h3 v Ed is ah reduction factor that takes into consideration the risk of crack growth = k f bed modified see Chapter hh B1 2 22forh CLT v v, d5.2.1.1. bef B1 kkv == is a reduction factor that takes into the risk of is adistance reduction factor takesreaction into consideration consideration of crack crack growth growth xv is the from the that support to the cornerthe of risk the notch. hh and size-effect and size-effect of the member.of the member. (4) fv, d shearshear strength of the isdesign the design design strength ofmaterial. thestrength. material. fisv, the is the design shear f is the shear strength. d and v, dsize-effect Where: and size-effect of of the the member. member. 5.2.1.1. Modifications for CLT is the dimensioning shear force. V VffEd is dimensioning shear force. is the design shear strength. Edis the dimensioning force. the dimensioning shear force. isisthe the design shearshear strength. v, d Ed v, d

V is the force. befEd= kcr Dovetail bN istype the of the notch shear reduced to account for the negative influence of V is width the dimensioning dimensioning shear force. 6.1.2.2. II Ed 7.1.2.2. Dovetail type II cracks (kcr can be found in Chapter 4.1). 7.1.2.2. Dovetail typethe II For the dovetail with load as in Figure 23.b. the shear capacity is calculated according to

For the dovetail asthat inthe Figure X.b. the shear capacity is to DIBT Where isasapproximated the force applied thecalculated centre the For dovetail oriented inoforiented Figure X.b. shear capacity is iscalculated according toof according bN the(2018). is the itwidth the notch. May bereaction modified for CLT seeat Chapter 5.2.1.1. 7.1.2.2. Dovetail type II 7.1.2.2. Dovetail type II Z-9.1-649. Where it is approximated that the reaction force is applied at the centre of the dovetail height. Even the dovetail is not symmetrical it's at height, hN , this is an Z-9.1-649. Where it isthough approximated that the reaction force is over applied the centre of the width of the dovetail height. Even though the dovetail is not symmetrical over it's width this width of the dovetail height. Even though the dovetail is not symmetrical over it's width this approximation. For the as For the dovetail dovetail oriented as in in Figure Figure X.b. the the shear shear capacity capacity is is calculated calculated according according to to 1.0 X.b. is oriented an approximation. is an approximation. Z-9.1-649. the reaction force is applied at the centre of the Z-9.1-649. Where Where it it is is approximated approximated that that 1.5 the reaction force is applied at the centre of the 1.1 i width of of the the dovetail dovetail height. height. Even though the dovetail is is not not symmetrical symmetrical over over it's it's width width this this width Even though the dovetail 2k 1 2 n b isF an an approximation. approximation. F = h k f V (20) (24) is = 3 bN, efv, Rd h N k3v fv,N,d ef N VvhEdv, d (20) Ed k v,=Rdmin v

22 2 Fv, Rd == 3 bbN,Where: Where: h hh N kkv ffv, d F Where: v, Rd 3 N, ef ef N v v, d Where: = bN Where: bN, ef = bN bN,h Nef tan 2

x 0.8VEd h Ed

(20) (20)

h N tanis 2the effectiveiswidth the effective width of(See theFigure dovetail of the dovetail X).(See (5)Figure X).

is a reduction that takes into consideration thewidth ofof crack growth in notched bNh tan isrisk theof inner width of(See the dovetail (See Figure X). = bbN factor isisthe the effective thedovetail dovetail (See FigureX). X). the inner Figure is the2 approximated width of width the dovetail. bbN, hN is effective width ofthe the dovetail (See Figure X). N, ef = (See N tanX).2 N ef members N Chapter hN is the inner height of theFigure dovetail isof the inner height of the dovetail (See Figure X).(See Figure X). is the inner width dovetail (See X). is the inner widthis theof dovetail, seeof Figure 23.b. *bbhNN For beams with notch at the opposite side thewidth support kthe =dovetail 1.0 can (See be used. the inner Figure X). vthe dovetail (See Figure X). is the vertical angleofofthe N is the the vertical vertical angle angle of of the the dovetail dovetail (See (See Figure Figure X). X). is is vertical angle strength. of the the dovetail (See Figure Figure X). X). is the the inner height of the dovetail (See Figure X). is a proportionality constant, see dovetail, Chapter 4.1. hhknfN is the inner height of dovetail (See is the smallest height of the see Figure 23.b. is the design shear Nv, d is the design design shear strength. strength. fffv,v,dd is is the the design shear shear strength. iV v, d is theis notch inclination (see Figure 1). is dimensioning shear the vertical angle of dovetail, seeof Figure 23.b. (See Figure X). is the thethe vertical angle the force. dovetail Ed VEd is the the dimensioning dimensioning shear shear force. force. VV is is the shear force. hfEd the height of the beam in dimensioning mm. Ed calculated is is design shear strength. accordingto toequation equation(7) (5)the where: according where: kkvvv,isisdcalculated is calculated according according to equation equation (5) where: where: kkkvv is ishcalculated calculated according to to equation (5) (5) where: N v V is the dimensioning shear force. is the inclination of the notch corner. i =Ed the inclination the notch corner. =0 is theisratio between theofheight of the notch and the height of the beam. May ii == 00h is the inclination of the notch corner. is of the corner. ih ==modified 0 isis the the inclination ofbeam, the notch notch corner. be for CLT see inclination Chapter 5.2.1.1. of limited. kv isb calculated according to equation where: isthe thewidth width ofthe the(5) joist, this is limited to ≤ 0.8 ∙ bN as explained in h = b is the width of the beam, limited. hhx == bb is the width of the beam, limited. isChapter the width ofthe the beam, limited. 5from in report. is the distance the support reaction to the corner of the notch. i = 0l z is the inclination of the notch corner. xf = lll2z is design the length from the reaction force to the notch corner. is the shear strength of the material. v, d zz isisthe the width of thethe beam, limited. thelength lengthfrom from thereaction reaction forcetotothe thenotch notchcorner. corner. is the length from the reaction force the xxxh === b22 is force is the length from the reaction force to to the notch notch corner. corner. 2 0.5 bN bN,for 5.2.1.1. Modifications CLT ef 0.5 bbN bbN, ef and l z = 0.5 x = = 0.5 bNN b isbN, the N, ef eflength from the reaction force to the notch corner. and and and 2 == bbb 7.1.2.3. Dovetail III 0.5 bN typebN, ef 7.1.2.3. Dovetail type III 7.1.2.3. Dovetail type III 7.1.2.3. Dovetail typeand III and type =Dovetail 6.1.2.3. type This of dovetail b III orientation is the same as a notched joint and the same Thistype type of of dovetail dovetail and orientation orientation is the same as as aa notched notched joint and the same verifications are performed accordingin torelation Chapter This type and is the same joint and same and load is the same a notched This typeof ofdovetail dovetail andorientation orientation is the sameto5.2. asthe a notched joint andasthe the same joint and verifications are performed according to Chapter 5.2. verifications are according to 7.1.2.3. Dovetail type IIIperformed verifications are performed performed to Chapter Chapterto5.2. 5.2. the same Dovetail verifications Chapter 4.1.2. 7.1.2.4. typeare IV according according 7.1.2.4. Dovetail type IV IVorientation is the same as a notched joint and the same 7.1.2.4. type This typeDovetail of dovetail 7.1.2.4. Dovetail typeand IV For the type of dovetail in Figure X.d. the same verification is made as in Chapter 7.1.2.2. verifications are performed according to Chapter 5.2. For the type of dovetail in Figure X.d. the same verification is made as in Chapter Chapter 7.1.2.2. For the type of X.d. verification as where only difference that there nosame vertical angle, ,is the dovetail and the7.1.2.2. vertical For thethe type of dovetail dovetail in inisFigure Figure X.d.isthe the same verification isofmade made as in in Chapter 7.1.2.2. 7.1.2.4. Dovetail type IV where the only difference is that there is no vertical angle, , of the dovetail and the vertical where the only difference is that there is no vertical angle, , of the dovetail and the vertical reduction, , isdifference not relevant either. Hence: where the only is that there is no vertical angle, , of the dovetail and the vertical reduction, , is not relevant either. Hence: reduction, , is not either. Hence: For Figure X.d. the same verification is made as X). in Chapter 7.1.2.2. reduction, not relevant relevant either. Hence: bN, efthe=type bN ,ofisdovetail isinthe effective width of the dovetail (See Figure 27 where the only difference is that there is no vertical angle, , of the dovetail and the vertical b = b is the effective width of the dovetail (See Figure X). (82) bbN, is N, efef == bbN N is the the effective effective width width of of the the dovetail dovetail (See (See Figure Figure X). X). N, ef N hN is theeither. full height of the notch (See Figure X). reduction, , is not relevant Hence: is the the full full height height of of the notch notch (See Figure Figure X). hhhNN is is the full height of the the notch (See (See Figure X). X). N b = b is the effective width of the dovetail (See Figure X). 7.1.2.1. Modifications for CLT N, ef N 7.1.2.1. Modifications Modifications for CLT CLT 7.1.2.1. 7.1.2.1. Modifications for for CLT The for top notches tenons according to chapter 5.2.2.1. h N same modificationsisare themade full as height of the notchor(See Figure X). samemodifications modifications made are made made as for for top notchesofor orthe tenons according to chapter chapter 5.2.2.1. The only are that of the height rolling shear in to the The same modifications are as top notches tenons according to The same modifications are made as and for Civil top notches or tenons according tocomparison chapter 5.2.2.1. 5.2.2.1. CHALMERS Architecture Engineering, Master’s Thesis ACEX30 The only modifications made are that of the height of the rolling shear in comparison to the height of the crack as seen in Figure X. The only modifications are 7.1.2.1. for CLT The onlyModifications modifications made made are that that of of the the height height of of the the rolling rolling shear shear in in comparison comparison to to the the height of the crack as seen in Figure X. height of the crack as seen in Figure X. height of the crack as seen in Figure X. Figure X. modifications Crack in the rolling shear moving to or higher layer for dovetail. The same are made as layer for top notches tenons according to chapter 5.2.2.1.

hN is the full height of the notch (See Figure X). 0.5 bN bN, ef 7.1.2.2. Dovetail type II and = b 7.1.2.1. Modifications for CLT For the dovetail oriented as in Figure X.b. the shear capacity is calculated according to Z-9.1-649. Wheretype it isIII approximated that thenotches reaction is according applied at to thechapter centre 5.2.2.1. of the The sameDovetail modifications are made as for top orforce tenons 7.1.2.3. widthonly of the dovetail height. though theheight dovetail is not symmetrical over it's width this The modifications madeEven are that of the of the rolling shear in comparison to the is an type approximation. height of the crack as and seenorientation in Figure X. This of dovetail is the same as a notched joint and the same verifications are performed according to Chapter 5.2. Figure X.2 Crack in the rolling shear layer moving to higher layer for dovetail. Fv, Rd = Dovetail h N kIV f VEd (20) 7.1.2.4. 3 bN, ef type v v, d Discuss critical placement of the dovetail in the report for CLT mention some demands in 6.1.2.4. Dovetail type VIin this report. As quite critically reduce theverification cross-section and in to be 7.1.2.2. For the type of dovetails dovetail Figure X.d. the same is made as order in Chapter Where: structurally stable the area of the dovatail with grain direction in lengthwise orientation For thethe type ofdifference dovetail inis Figure 23.d. same verification is the made as in Chapter 6.1.2.2. where only that there is the no vertical angle, , of dovetail and the vertical should be maybe a certain ispercentage. where the only difference that there are no angles, γ and β, of the dovetail, hence: reduction, bN, ef = bN, is not h N relevant tan 2 either. Hence: is the effective width of the dovetail (See Figure X). bN, ef = bN bN hN hN

effectivewidth widthofofthe thedovetail. dovetail (See Figure X). is isthethe effective is the inner width of the dovetail (See Figure X).

is the full height of the notch (See Figure X). the innersee height of 23.d. the dovetail (See Figure X). is the height of theis dovetail, Figure

7.1.2.1. Modifications for CLT 6.1.2.1. Modifications for CLT The same modifications are made as for top notches or tenons according to chapter 5.2.2.1. The same modifications made are made as for orthe tenons according chapter 4.1.2.1. The only modifications are that oftop thenotches height of rolling shear intocomparison to the height of the crack as seen in Figure X. 6.1.3. Compression failure perpendicular to the grain at the bottom of the dovetail 7.1.3 thelayer bottom of the notchlayer for dovetail. Figure Compression X. Crack in thefailure rolling at shear moving to higher This is also a similar failure mode as that for a notched beam therefore this verification is This a similar as thatin for a notched beam therefore verfication madeisbyalso following thefailure method in dovetail Chapter 4.1.3. but where: Discuss critical placement of mode the the report for CLT mentionthis some demandsis in made by following the method in Chapter 5.2.2. but where: this report. As dovetails quite critically reduce the cross-section and in order to be structurally stable the area of the dovatail with grain direction in lengthwise orientation Dovetail type I and III should bebNmaybe a certain percentage. bN = h Dovetail type II and IV N 6.1.4. Rolling shear failure in the tenon (only for CLT) This failure is the same as that shown in Chapter 4.1.4. but with minor modifications to better approximate the geometry of the dovetail. Depending on orientation of the dovetail and the load the geometry is approximated differently. See Figure 25. For all of the dovetail types the rolling shear verification is simplified as much as possible by approximating each layer to a rectangular cross-section within the dovetail. 7.1.3 Compression failure at the bottom of the notch 6.1.4.1. Dovetail type failure I This is also a similar mode as that for a notched beam therefore this verfication is made by following the method in Chapter 5.2.2. bbut= where: Very similar to the notched geometry but where b is used, which is the smallest width N

of the base dovetail. is III done to get the most conservative solution as can be seen bN of the Dovetail typeThis I and in Figure 25.a. bN = h Dovetail type II and IV N 6.1.4.2. Dovetail type II

For the rounded dovetail as seen in Figure 25.b. the cross-sections of each layer is limited by the smallest width bB1 and the smallest dovetail height hN. 6.1.4.3. Dovetail type III and IV

For the straight dovetail in Figure 25.c. and d. the rolling shear layers are limited by the smallest width bN and the height hN.

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CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

bB1

a. Type I

b. Type II

tj ti tj

ti tj

hN d. Type IV

c. Type III

Figure 25. Example of rolling shear layer approximation for the four dovetail types and load cases.

6.2. HEADER MEMBER The failure modes for the header of the dovetail are the same as for the header of the notched geometry seen in Chapter 4.2 with hH one additional failure mode for when an axial hHtension force is applied hH,uon the connection. hH,u 6.2.1. Geometry The general geometry of the header corresponding to all four types of dovetails in Chapter a. Type b. Type II 6.1 can beI seen in Figure 26.

hH hH,u

hH,u c. Type III

d. Type IV

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CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

B bN

A-A

B-B hN-r

hH,u

b. Type II

B bN

A-A

B-B hH

hH,u

a. Type I

A-A

B-B hH

A-A

B-B hH

hH,u

c. Type III

hH,u

d. Type IV

Figure 26. Geometry of a header for the four different types of dovetail geometries.

6.2.2. Shear and tension failure perpendicular to the grain at the notch corner 6.2.2.1. Dovetail type I

This failure is similar to that shown in Chapter 4.2.2. for a notch but where the crack propagation occur at the height of the transition between the straight tenon and the lower fillet similarly as for the joist with a dovetail (Blaß & Sandhaas, 2017). This can be seen in Figure 27.

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CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

bB1 tj 6.2.2.2. Dovetail type II hN For these of h type of geometries of thetiheader dovetail IIγ is tapproximated to the geometry bB1 i dovetail IV by ignoring the slight angle, γ , of the dovetail and assume that the crack occurs at the height where the width of the dovetail is largest. This will give a hH,u as seen in Figure 27 and the verification can be performed by the method in Chapter 4.2.2. hN γ 6.2.2.3. Dovetail type III h a. Type I b. Type of II the crack or the verifications as The geometry of this dovetail does not affect the location the bottom surface of the dovetail is plane and horizontal the same as a regular straight notch b and the same verifications as Nin Chapter 4.2.2. can be followed. 6.2.2.4. Dovetail type IV

tj tj For geometry type IV the crack is assumed to appear at the smallest width of the dovetail h hN ti b ti which gives the geometrical values as seen in Figure 27. By this the same verificationsN as seen tj in Chapter 4.2.2. can be followed. tj 6.2.2.5. Modifications for CLT

hN The same modifications as in Chapter 4.1.2.1. are made for all the different geometries of the h dovetail but with the already made modifications of the height of the crack made in Chapter d. Type IV c. Type III 6.2.2.1 - 6.2.2.4.

hH,u

hH hH,u

a. Type I

b. Type II

hH hH,u

hH,u c. Type III

d. Type IV

Figure 27. Shear and tension failure perpendicular to the grain in the header for the four different geometries shown in Figure 26.

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CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

F90, Rd = 14 bS F90, Rd = 14 bS

dS dS dS dS 1 hS 1 hS

kmod kmod

M AMHD

FEd FEd

(24) (24)

AHD

Where:Tension failure perpendicular to the grain due to an axial force in the joist 6.2.3. Where: This similar failure inelement shear and tension to grain at the bS failure mode is theiswidth of to thethe supporting (See Figureperpendicular X). bS notchediscorner the width ofheader. the supporting (See Figure X). load acts perpendicular inner of the For that element failure mode the vertical to When theofgeometry ofthe thisend failure mode in a similar way dS the crack propagation. is the distance fromtreating the centre the peg to of the supporting element d is the distance from the centre of the peg to the end of the supporting element the tension forceofact to theX). crack propagation and the failure can be verified in inS the direction theperpendicular force (See Figure the direction the forcein(See Figure X). ainsimilar way as of described Chapter 4.2.2. h S failure mode is theisfull of for the asupporting element (See Figure X). header, hence where This not depth relevant dovetail that goes through the full hS is the full depth of the supporting element (See Figure X). a. Type & IIof the dovetail as seen in Figure 28. b. Type III & IV lz = bH , as the crack appears at theI end kmod is the modification factor for duration of load and moisture content according to kmod is the modification factor for duration of load and moisture content according to Table X. Table X. M M

FEd FEd

is is is is

the partial factor for the material according to Table X. the partial factor for the material according to Table X. the dimensioning lateral force (See Figure X). the dimensioning lateral force (See Figure X).

7.2.3 Failure in tension perpendicular to the grain due to an axial force on the 7.2.3 Failure in tension perpendicular to the grain due to an axial force on the joist (Second method) joist (Second method) This failure mode is similar to theperpendicular failure in to shear andintension perpendicular to grain at the Figureis28. Tension to failure the grain the headerperpendicular for an axial This similar the For failure infailure shear and tension to grain inner failure notchedmode corner of the header. that mode the vertical reaction force actsat the tension force on the joist. inner notched to corner of acting thepropagation. header. For that failure mode vertical of reaction force mode acts in a perpendicular the crack When treating thethe geometry this failure perpendicular totension the crack propagation. When treating the geometry of this modecan in be a similar way the force act perpendicular to the grain propagation andfailure the failure The only differences between the two verifications are minor geometrical differences where: similar way the tension force act perpendicular to the grain propagation and the failure can be verified in a similar way as discribed in Chapter 5.3.2. but where: verified in a similar way as discribed in Chapter 5.3.2. but where: FEd bH, ef = h H, ef using the h height insteadisofused bH .insteaddof the width of the header. bH, ef = h H, ef using h H instead of b . H H FEd h H, u = l z h H, u = l z relates to the depth A of the crack. A hS

6.2.3.1. Modifications for CLT

FEd For a header with a grain A-A orientation as seen in Figure to the dS 15.c. the tension perpendicular grain should be verified for CLT headers in the same way as seen in Chapter 6.2.3. For other bS/2 parallel to the axial force orientations of the CLT part of this area will have grain direction FEdis not needed. and this type of verification bS/2 6.2.4. Compression failure perpendicular to the grain at the bottom of the mortise Similar to a notch the compression perpendicular to the grain can be verified for the header of a dovetail joint. This is done by following the verifications in Chapter 4.2.3. 6.2.4.1. Modifications for CLT

The same modifications are made as in Chapter 4.2.4. but where AB,ef = AD,ef as illustrated in Chapter 6.2.3.

6.2.5. Compression failure perpendicular to the grain due to an axial force in the joist Similar to a notch the compression perpendicular to the grain can be verified for the header of a dovetail joint. This is done by following the verifications in Chapter 4.2.4 but where the area AN of the dovetail can be approximated for simplicity. This should be done so that the outcome is a larger area than the true area for a conservative solution. (87)

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

Similar to a notch the compression perpendicular to the grain can be verified for the header of a dovetail joint. This is done by following the verifications in Chapter 5.3.3.

7.2.5 Failure in compression perpendicular to the grain due to an axial force on the joist The same verifications are made as in Chapter 5.3.4. but where AB, ef = AD, ef as described in Chapter 7.2.3. AHD AD AHD AD 7.2.5.1. Modifications for CLT 6.2.5.1. Modifications fororientation CLT For a header with a grain as seen in Figure X.b. the tension perpendicular to the For header with a grainfor orientation as seen in Figure X.b.asthe tension perpendicular to the AHD AD headers grainasame should beThe verified CLT inmade the same way seen in Chapter 7.2.3. For samefor modifications as inway Chapter 5.3.4.1. Aseen Aillustrated = AD,ef as in The modifications are made as in are Chapter 4.2.4. butaswhere Ain grain should be verified CLT headers in the same 7.2.3. For HD D B,efChapter other orientations of the CLT part of this area will have grain direction parallel to the axial Chapter 6.2.3. of the CLT part of this area will have grain direction parallel to the axial other orientations force and this type of verification is not needed. force and this type of verification is not needed. ForVERIFICATION the orientation inOF Figure X.a and c. no verification is needed since one or more layers 7. PEG JOINTS 8. VERIFICATION PEG For thereinforcement orientation inwhere Figure andOF c. no verification sincetension one or force. more layers act as theX.a grain direction isJOINT parallelistoneeded the acting act as reinforcement where the grain direction is parallel to the acting tension force. The laterally loaded peg andloaded it's supporting element is verified according to CEN (2017). The laterally peg and it's supporting element is verified according to the French Figure X. CLT orientations with failure mode in tension contribution. Figure X. CLT orientations with failure mode in tension 7.1. PEG & IV a. TypeFailure I & II in compression perpendicularb.toType 7.2.4 the III grain 8.1 PEG 7.2.4 Failure in compression perpendicular to the grain 7.1.1. Geometry Similar notch the compression perpendicular to the grain can be verified for the header b.thesis Type III & IVbe a. Type Ito&aa II Similar to notch the compression perpendicular the grain can verified for the The geometry of the peg evaluated in this can be in Figure 29 header with its 8.1.1. Geometry of a dovetail joint. This is joint done by following the to verifications in seen Chapter 5.3.3. of a dovetail joint. This is done by following the verifications in Chapter 5.3.3. relevant variables. Shear failure of a laterally peg joint 7.2.5 Failure8.1.2. in compression perpendicular to loaded the grain due to an axial force 7.2.5 Failure in compression perpendicular 7.1.2. Shear failure of a laterally loaded peg joint to the grain due to an axial force on the joist This failure mode of a timber peg can be seen in Figure X. on the joistmode 1. GENERAL DIMENSIONING RULES This failure of the peg is illustrated in Figure 30. The same verifications areShear madefailure as in of Chapter 5.3.4. but where AB, ef = AD, ef as described Figure X. peg. The same verifications are made as in Chapter 5.3.4. AB, ef = AofD,a efcross-section as described The basic calculations with regards to material, loadsbut andwhere dimensioning in Chapter 7.2.3. 2 in theChapter method7.2.3. in SIS (2009)9.5 should The calculation method in this report only n d bekfollowed. mod include parts might differ from this standard or where additional verifications are(25) 7.2.5.1.the Modifications for CLT Fv,that = F (23) Rd Ed 7.2.5.1. Modifications for CLTM made except from this. The same modifications are made as in Chapter 5.3.4.1. The same modifications are made calculation as in Chapter 5.3.4.1. When it comes to the preliminary method for glulam and CLT the verifications Where: made for solid timber are the same as for glulam but with given coefficients and material is the number of pegs in the joint (See Figure X). data changed. n For CLT depending on the orientation of the board some geometrical data is changed and F 8. VERIFICATION OF PEG JOINT d the d dueOF is diameter the peg Ed (See Figure X). 8. VERIFICATION PEG JOINT verifications added to the different grainofdirections of the material. The laterally loaded peg and it's supporting element is verified according to the French FEd laterally kmod peg and isdit's the supporting modification factorFis for duration of load and moisture The element verified according to the Frenchcontent according to Ed 1.1. LOADS loaded contribution. contribution. TableAX. A F Loads used for dimensioning are calculated from Figure respective partfailure of SIS for selfEd PEG 30. Shear of a (2009) laterally loaded peg. 8.1 hS and wind load. 8.1 PEGimposed load, snowload weight, is the partial A factor for the material according to Table X. M A 8.1.1. Geometry 8.1.1. Geometry Edforcein hand the service dimensioning (See The of Fload duration class is lateral alsoFstated SISFigure (2009)X). . dS is A-Aimpact S Ed 8.1.2. Shear failure of a laterally loaded peg joint 8.1.2. Shear failure of aPARTIALCOEFFICIENT laterallyb loaded peg joint FEd /2 1.2. VERIFICATION OF THE METHOD S 8.2 SUPPORTING ELEMENT dS A-A This failure mode of a timber peg can be seen in Figure X. FEd failure mode of a timber peg can be seen in Figure X. This 1.2.1. Design8.2.1. values for material properties /2 bbSS/2 Figure X. Shear failureGeometry of peg. Figure X. Shear failure of peg. F Ed design values of the strength properties, X , used in this method is calculated The d failure perpendicular the failure grainperpendicular to the grain in 9.5 8.2.2. n d 2 kTension bSthe /2 Figure 29. Geometry joint illustrating Figure 31.to Tension modfrom according to SIS both (2009). 9.5equation n ofda 2pegk(1) modrelevantFvariables. Fv,and = (23) peg supporting member with the supporting element. This failure mode can be seen in Figure X. Rd = Ed Fv, Rd FEd (23) M Xk M Figure failure of supporting element. Xd =X. kTension (1) Where: Where: mod Where: M is the number in the joint. n is the number of pegs of in pegs the joint (See Figure X). n is the number of pegs in the joint (See Figure X). Where: d is the diameter of the peg (Seepeg, Figure X). 29. is the diameter of the the see Figure dX is the diameter ofvalue the peg (See Figure X). properties. is the characteristic of load-bearing k kmod is the modification factor for duration of load and moisture content according to kmod is the modification factor for duration load andproperties moisture (Recommended content according to is the partial coefficient for theofmaterial Table X. is the partial coefficient for the material properties M Table X. valuesvalues in Table 1). 1). (Recommended in Table is the partial factor for the material according to Table X. M is the partial factor the regard material to Table X. kM is the corrections factorfor with to according load duration and moisture conten mod FEd is the dimensioning lateral force (See Figure X). (Recommended values in Table 3). 33 (88) F is the dimensioning lateral force (See Figure X). Ed

8.2 SUPPORTING ELEMENT 8.2 SUPPORTING ELEMENT Factor, M 8.2.1. Geometry 8.2.1. GeometrySolid timber

Source

SS EN 1995 1 1 : 2004 Sv 1.3 8.2.2. Tension failure perpendicular to the grain SS EN 1995 1 1 : 2004 Sv Glulam perpendicular to1.25 8.2.2. Tension failure the grain Master’s CHALMERS Architecture and Civil Engineering, Thesis ACEX30 This failure modeCLT can be seen in Figure X. 1.25 KL-Trähandboken This failure mode can be seen in Figure X. Table 1: Partial coefficients. element. Figure X. Tension failure of supporting

X act reinforcement where the grain is parallel to the acting tension force. Fv,as = F k direction (23) Rd XdM = kmod Ed (1) Figure X. CLT orientations with M failure mode in tension Where: Where: 7.2.4 Failure in compression perpendicular to the grain n is the number of pegs in the joint (See Figure X). Xk the characteristic value perpendicular of the load-bearing Similar to is a notch the compression to the properties. grain can be verified for the header d is the diameter of the peg (See Figure X). of a dovetail joint. This is done by following the verifications in Chapter 5.3.3. is the partial coefficient for the material properties kM is the modification factor for duration of load and moisture content according to 7.2.5 Failure in compression perpendicular to the grain due to an axial force (Recommended values in Table 1). 1.modGENERAL DIMENSIONING RULES Table on theX.joist is the correctional factor with regard to load duration and moisture kmodbasiciscalculations the corrections factor with regard toloads load and duration and moisture The with regards to material, dimensioning of a conten cross-section content (Recommended values inbut Table 3). The same verifications areshould made as infollowed. Chapter 5.3.4. where AB, efX.=inAthis as described the(2009) partial factor for the material according to Table the inisSIS be The calculation method M method(Recommended D, ef report only values in Table 3). include the 7.2.3. parts that might differ from this standard or where additional verifications are in Chapter FEd except from is thethis. dimensioning lateral force Figure X). 29. is the dimensioning lateral(See force, see Figure made 7.2.5.1. Modifications for CLT Source and CLT the verifications When it comes toFactor, the preliminary calculation method for glulam M 8.2 SUPPORTING SUPPORTING ELEMENT 7.2. MEMBER The same modifications made for solid timber areare themade sameas as in forChapter glulam 5.3.4.1. but with given coefficients and material SS EN 1995 1 1 : 2004 Sv Solid timber 1.3 data changed. 7.2.1. Geometry 8.2.1. Geometry EN 1995data 1 is1 :changed 2004 Sv For CLT depending on the orientation of the board some SS geometrical and Glulam 1.25 The geometry of thedue supporting member of the peg is illustrated in Figure 29 with its relevant verifications added to the different grain directions of the material. 8.2.2. Tension failure perpendicular to the grain KL-Trähandboken CLT 8. VERIFICATION OF PEG JOINT 1.25 variables. 1.1. ThisLOADS failure mode can1:be seencoefficients. in Figure X. Table Partial The laterally and it's supporting element verified according 7.2.2. Tensionloaded failure peg perpendicular to the grain of theissupporting member to the French Loads for dimensioning are calculated from respective part of SIS (2009) for selfcontribution. Figureused X. Tension failure of supporting element. This failure mode of thesnowload supporting can be seen in Figure 31. weight, imposed load, andmember wind load. 2.3.2. Design values for load-bearing capacities 8.1 PEG The , is calculated in aclass similar way stated as the in strenght properties in equation The design impact value, of loadRduration and service is kalso SIS (2009) . d dS mod 8.1.1. Geometry (1) and according to S-EN 1995-1-1:2004. dS b kmod F90, Rd = 14 FEd (24) dS PARTIALCOEFFICIENT kmod d S 1.2. VERIFICATION OF THE (26) M F = 14 b FSEd METHOD (24) R 90, Rd S d Shear loaded pegk joint(24) F8.1.2. = 14 bSfailure of adSlaterally F M1 90, Rd h SEd = k 1 forhmaterial S MRproperties 1.2.1. Design values This failure mode of a1 timber peg can dbe seenmod in Figure X. M h SS Where: The design values of the strength properties, Xd , used in this method is calculated Figure X. Shear failure of peg. Where: Where: where: according to equation (1) from SIS (2009). Where: bS is the width of the supporting element (See Figure X). 2 k 9.5 n d bRS is the width the supporting element member, (See Figure ismod theofwidth of the supporting see X). Figure 29. is the characteristic values of the load-bearing Xsupporting bFS k is the width of theF element (See capacity. Figure X). kdistance from(23) = d is the the centre of the peg to the(1) end of the supporting element v, Rd Ed SX M = k d distance mod dS isinthe from the centre of the peg to the end of the supporting element is the distance from the centre of the peg to the end of the supporting the direction ofMthe (See Figure is is thethe partialcoefficient forforce the material properties dSM distance from the centre of the peg X). to the end of the supporting element in Figure the direction inWhere: the direction of theelement force (See X). of the force, see Figure 29. in the directionhof the force is (See X). of the supporting element (See Figure X). theFigure full depth Where: S hnS is the full depth of the supporting element (SeeX). Figure X). is the number of pegs in the joint (See Figure hXS is the full depth the of supporting elementmember, (See Figure is the fullof depth ofthe theload-bearing supporting see X). Figure 29. is the characteristic value properties. kmod is the modification factor for duration of load and moisture content according to k isthethe diameter of factor the peg FigureofX). kdmod is Table modification for(See duration load and moisture content according to X. kmod is the modification factor for duration ofmaterial load andproperties moisture content according to is the partial coefficient for the (Recommended thethe partial coefficient for the material properties M X. is is Table kmod modification factor for duration of load and moisture content according to valuesvalues in Table 1). 1). Table X. (Recommended in Table is the partial factor for the material according to Table X. M Table X. is the partial for the factor material according toload Table X. is thefactor correctional with regard to duration and moisture kM is the corrections factor with load duration and moisture isFthe partial factor for the regard materialtolateral according to(See Table X. X). conten M is the dimensioning force Figure mod content (Recommended values in Table 3). Ed the dimensioning partial factor the material to Table X. M FEd isis the lateral force Figure X). (Recommended values for in Table 3). (Seeaccording FEd is the dimensioning lateral force (See Figure X). FEd is the dimensioning lateral force (See Figure X). is the dimensioning lateral force, see Figure 7.2.3 Failure in tension perpendicular to the29. grain due to an axial force on the 7.2.3 Failurejoist in tension perpendicular to the grain due to an axial force on the (Second method) 7.2.3 Failure in tension perpendicular to the grain Source due to an axial force on the Factor, joist (Second method) M 8.2 SUPPORTING ELEMENT joist (SecondThis method) failure mode is similar to the failure in shear and tension perpendicular to grain at the SS EN 1995the1 vertical 1to: 2004 timbercorner This failure mode isnotched similar to the of failure in 1.3 shearFor andthat tension perpendicular grain atSv the force acts innerSolid the header. failure mode reaction This failure mode is similar to the failure in shear and tension perpendicular to grain at the 8.2.1. Geometry inner notched corner of the header. For that failure mode the vertical reaction force acts perpendicular to the crack propagation. When treating the geometry of this SS EN 1995 1 1 : 2004 Sv Glulam 1.25 mode the vertical reaction force acts failure mode in a inner notched to corner of thepropagation. header. For that failure perpendicular the crack When the geometry of thispropagation failure mode a failure can be similar way the tension force acttreating perpendicular to the grain andinthe perpendicular totension the crack propagation. When treating the geometry of this failure modecan in be a KL-Trähandboken 8.2.2. way Tension failure to the grain CLT 1.25 similar the act perpendicular to the propagation the failure verified inforce a perpendicular similar way as discribed ingrain Chapter 5.3.2. butand where: similar way the tension force act perpendicular to the grain propagation and the failure can be verified in a similar way asseen discribed in Chapter 5.3.2. but where: This failure mode can in Figure X. Table 1:be Partial coefficients. verified in a similar in Chapter 5.3.2. of butb where: bH, ef way = hasH,discribed using h instead . ef H H bFigure = h using h instead of b . failure supporting ef =X.hTension H, ef bH, usingofh H instead ofelement. bH . H, ef Design H, efh values H H = l 2.3.2. for load-bearing capacities H, u z h H, u = l z h H, udesign = l z value, R , is calculated in a similar way as the strenght properties in equation The d (1) and according to S-EN 1995-1-1:2004. mod

Rd = (89) kmod

Rk M

where: Rk M

is the characteristic values of the load-bearing capacity. Architecture Engineering, Master’s Thesis ACEX30 isCHALMERS the partialcoefficient forand theCivil material properties

7. PRELIMINARY PROPOSAL

The preliminary selection of the joints for C1 and C2 is based on the results of hand calculations but also to some extent on the criteria from Chapter 4. A couple of chosen connections are made as prototypes in models which also better illustrate some strengths and weaknesses of each connection. This chapter presents the results of hand calculations, prototypes and the final selection of two possible connections for each joint.

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

The failure modes not calculated, as shown in the tables, are not needed based on either the material used in the connection, grain orientation or due to the geometry of the connection.

DESIGN VERIFICATIONS The full design verification is presented in Appendix B3 and B4 for the connections analysed for C1 and C2. The results are presented as utilization rates where either the load capacity is compared to the applied load or a stress is compared to the strength of the material.

C1 For the design verification see Table 7.1.

Table 7.1. A summary of the utilization rate for the different connections for C1.

Failure Mode

Ratio

Load PD1

PD2

Vertical

0.4 %

Horizontal

11 %

10 %

8.0 %

Vertical

0.2 %

0.1 %

0.2 %

Horizontal

3.8 %

2.4 %

1.6 %

2.3 %

3.0 %

3.2 %

1.6 %

1.3 %

6. Failure in compression perpendicular to the grain due to an axial compression force on the Axial joist.

10 %

19 %

7. Failure in tension perpendicular to the Axial grain due to an axial force on the joist.

18 %

8. Failure in shear of a laterally loaded peg Axial joint.

40 %

160 %

9. Failure in tension perpendicular to the Axial grain.

28 %

Joist 1. Failure in shear and tension perpendicular to the grain. 2. Failure in compression perpendicular to the grain.

3. Rolling shear failure of the notch (only for Vertical CLT). Header 4. Failure in shear and tension perpendicular to the grain. 5. Failure in compression perpendicular to the grain.

Vertical Horizontal Vertical Horizontal

Peg

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

Based on the utilization ratio the only connection that fails in one regard is connection P2 with the thinnest peg.

C2 For the design verification see Table 7.2.

Table 7.2. A summary of the utilization rate for the different connections for C2.

Failure Mode

Ratio

Load RD1

TM1

Vertical

88 %

15 %

Horizontal

7.5 %

2.6 %

Vertical

11 %

4.0 %

2.0 %

1.7 %

0.2 %

1.1 %

7.3 %

8.6 %

Horizontal

Vertical

Horizontal

6. Failure in compression perpendicular to the grain due to an axial compression force on the Axial joist.

0.2 %

0.4 %

7. Failure in tension perpendicular to the Axial grain due to an axial force on the joist.

8. Failure in shear of a laterally loaded peg Axial joint.

9. Failure in tension perpendicular to the Axial grain.

Joist 1. Failure in shear and tension perpendicular to the grain. 2. Failure in compression perpendicular to the grain.

Horizontal

3. Rolling shear failure of the notch (only for Vertical CLT). Header 4. Failure in shear and tension perpendicular to the grain. 5. Failure in compression perpendicular to the grain.

Vertical

Peg

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

PROTOTYPES

Model prototypes were designed in order to identify possible strengths and weaknesses with regard to manufacturing, architectural design and assembly. Four of the connections were selected for these prototypes, PD1 and P1 for C1, and RD1 and N2 for C2. The models are made in scale 1:2.5 and by using hand tools as well as a three-axis CNC-milling machine. The materials used are made to scale with lamellae size from Martinsons (2020). Photos of the prototypes can be seen in Figure 7.1 - 7.4.

Figure 7.1. Model photography of connection PD1 for C1.

Figure 7.3a-b. Model photographies of connection RD1 for C2 in two different views.

Figure 7.2. Model photography of connection P1 for C1.

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

Figure 7.4a-b. Model photographies of connection N2 for C2 in two different views.

The manufacturing process itself is difficult to asses since the models are made using a three-axis CNC-machine. These would most likely be made in a five-axis machine with interchangeable tools in reality. This would therefore diminish the problems of manufacturing that occurred for these prototypes. What could be concluded from this are some geometrical difficulties, especially using CLT in the dovetail. It can be difficult to control that not too small pieces of single lamella occur, depending on where the lamellae and cuts end up.

The architectural design was mainly assessed by how clearly visible the connections are. The connections PD1, PD2 and TM1 look to be less visible since the prototype of PD1 only consist of two members with the same materiality. With regard to assembly, connection P1 is the most difficult as the pillars need to be "moved" in order to assemble the system which would not be preferable. The strengths and weaknesses for each prototype can be seen in Table 7.3.

Table 7.3. A summary of the strengths and weaknesses of each prototype. Strengths PD1 P1 RD1 N2

Weaknesses Difficult geometry to manufacture

Easy on-site assembly

Not very visible

Not locked for loads in all directions

Mixes different types of wood which increases visibility Locks in all load directions

Difficult assembly Difficult geometry to manufacture Difficult geometry to manufacture

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

DESIGN SELECTION Based on the calculations and conclusions from the prototypes four of the connections were selected, two for C1 and two for C2. Minor modifications are made to account for weaknesses. A general difference between the geometries used for the hand calculations and the prototypes and selected designs are some small adjustments needed for manufacturing and assembly. This is for example the rounded corners that were made by the drill of the CNC-machine and the hardwood pegs and blocks for connection C2. The rounded corners will theoretically also improve the design by reducing stress increments in the corners. The pegs and blocks were added both to improve the visual impact and to permit the assembly of the members.

Figure 7.5. Illustration of preliminary connection PD1.

The assembly process for all selected connections is illustrated in Figure 7.9. C1 PD1 Modifications for PD1 were made in order to lock the connections for all load directions and to make it more visual. In order to achieve this pegs are inserted on the outside of the header (pillar), see Figure 7.5. This will not affect the on-site assembly process substantially. P1 Connection P1 is selected in comparison to P2 since this solution fails for the peg. Despite the slightly more difficult assembly process this is still regarded as a possible solution. An illustration of the connection can be seen in Figure 7.6. Figure 7.6. Illustration of preliminary connection P1.

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

C2 RD1

No failures occur for connection RD1. As for many of the connections, a lot of the utilization ratios are well under their limit. The geometry of the connections are kept in order to simplify manufacturing and in order to look structurally sound for the users. The preliminary outcome of connections RD1 can be seen in Figure 7.7.

Connection N2 shown in Figure 7.8 is selected mainly due to its strong structural capacity, simplicity to manufacture and visual impact. The added element of pegs on the outer side of the railing are verified in shear failure.

Figure 7.7. Illustration of preliminary connection RD1.

Figure 7.8. Illustration of preliminary connection N2.

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

Figure 7.9. Assembly of all four selected connections. a. PD1, C1, b. P1, C1, c. RD1, C2, d. N2, C2.

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

8. DISCUSSION AND CONCLUSION

This chapter includes the final discussion of the results and analysis of the FEmodelling, calculation method and connection design. Discussed is also possible future studies relating to the thesis.

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

The credibility of the method can be discussed based on the limitations of geometry that are not taken into account for many of the verifications. This was a choice in order to be able to adapt the method for larger connections since many of the limitations result in smaller proportions. It is likely that many of the limitations are established by the development of the verifications through testing. The geometrical limits are set based on the test geometries even though it is likely that the verifications hold for other sizes as well. Through this some constants in the equations could be needed to be modified in order to include other geometrical constraints. This could also be an interesting future investigation regarding this thesis.

General calculation method The primary result of this thesis is the general calculation method in Chapter 6. The main difference between this method and other standards published are the modifications for CLT and the comprehensive collection of different geometries and their respective failure modes. From the compiled research it can be concluded that most of the validated calculation methods that already exist rarely include many different types of carpentry joints and the correlation between these. Few of the sources utilized include as many failure modes for each connection type as the method in Chapter 6. For future reference it would therefore be interesting to investigate if all the possible failure modes within the calculation method actually need to be verified. It is our conclusion that the verifications analysed are needed in order to validate the connections for all different types of load application. An interesting future study with regard to the calculation method would be to include a moment in the connections.

One possible failure mode not included in the method or any of the found sources is the slipping of the dovetail and failure of the angled joist geometry in shear. For future iterations this would be relevant to investigate and add to the method. The validity of the results of the thesis are not assumed to be unreasonable, based on that the geometries do not differ extensively from the limits set in the sourced equations. Further improvements for the calculation method, might be the possibility of slight modifications to include even more geometries than what is already included. Similarly to the many small modifications made to the dovetail in order to approximate it as a notch. Another prospect of the method is to use it for other materials such as LVL and Plywood that are similar to the analysed materials but have slightly different grain structures and material properties.

With regard to the adaptation of the calculation method for different materials, glulam do not vary from the case of solid timber, except for the specific material constants. CLT, on the other hand, includes many modifications that would be interesting to validate with further studies and live testing. This is a material that is quite new within construction and very little material on the subject of connections in CLT exist today. This is for example the equation for validating one or several layers of CLT as reinforcement. This equation is adapted from the verification of external reinforcement of glued-on wooden plates. This might differ for CLT as this is both internal and might be adhered to the other layers differently. 99

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

Assumptions and modifications of the design

Implementation The aim of this thesis is to design a calculation method for some general carpentry joints. The objectives within this is to investigate the possibility of using carpentry joints in larger constructions.

With regard to the preliminary design some assumptions and simplifications made within the verifications may result in non-conservative solutions. This is for example the simplification to disregarding the grain orientation of the railing in connection C2. This could be another iteration to implement in the calculation method, to consider grain orientations that differ from orthogonal solutions.

This is made by an iterative process of designing several possible solutions for two different connections, one in CLT and one in glulam. The connections chosen in this thesis are smaller connections excluded from the main load-bearing system of the building and therefore not exposed to larger loads. What can be concluded from the results of the hand calculations for the connections is that most of them are designed with capacities well above the applied load. They will therefore, most likely, be able to handle larger loads as in, for example, a connection between a timber roof member and outer wall member.

Another simplification made is the assumption of calculating the reaction forces for a simply supported beam for both connections. With analysing the FEmodels it can be seen that this might not be the closest approximation to the real load case that can be made. Another load case would result in a moment in the connection which is why the simplification was made. It is rare to assume that a timber connection can take moments and it is concluded that this is a reasonable assumption to make for the hand calculations.

For further studies it would be relevant to implement the method in larger construction joints with higher applied loads in order to determine this explicitly. With regard to many coefficients used in the equations it would also be relevant to live test larger connections. To see if the results are proportional to the size and load or if the size would affect the equations themselves.

With this mentioned it could be relevant for future development of carpentry connections to use FE-modelling as a part of the design. In this thesis the FEmodelling was mainly used as a starting point of designing the hand calculation method. Within the deign process FEmodelling is an important tool to use. In reality it would be preferable to use hand calculations as a preliminary design step and then finalize the design with the help of this type of analysing tool. This might then be able to validate the connection more thoroughly and together with the hand calculations give a strong motivation for the design.

100

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

Summary As a final review of this thesis the primary results with the general calculation method do include many interesting features of carpentry joints. The resulting designs from the research made and the verification based on the calculation method does suggest that carpentry joints are reasonable to use in construction. This was the main goal and hypothesis of this thesis. Further studies would be needed in order to validate this type of method in proper practice. Although, what can be concluded from this study together with the reference projects, that are already using carpentry connections to some extent, is that using these type of joints in larger constructions are not an unreasonable idea and has many arguments promoting its case.

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CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

BIBLIOGRAPHY AIX Architecture. (2020).Drawings for BT7. Drawing credit: architect Karin Löfgren. Al-douri, H. and Hamodi, K. (2009). Compression Strength Perpendicular to Grain in Cross-laminated Timber (CLT). (Master’s thesisSchool of Technology and Design, Växjö University, Växjö, Sweden). Retrieved 2020 from https://www.diva-portal.org/smash/ get/diva2:223627/FULLTEXT01.pdf Atashipour, R. (2020). Timber Engineering - FE Lab [PDF]. Retrieved from Chalmers University of Technology, Timber Engineering, Canvas. Blaß, H. J., & Sandhaas, C. (2017). Timber Engineering - Principles for Design. Karlsruhe: Karlsruher Institut für Technologie Scientific Publishing. Almere: Centrum Hout. Blumer-Lehmann. Tamedia office building: First seven-storey wood structure. (n.d.). Retrieved January 08, 2021, from https://www.lehmann-gruppe.ch/en/holzbau/referenz/ tamedia.html Branco, J. (2015). Analysis and strengthening of carpentry joints. Construction and Building Materials Vol. 97, No. 30 October 2015, pp. 34-47 Comité Européen de Normalisation. (2017). French contribution for traditional carpentry joints (Standards No. CEN/TC 250/SC 5 - N 672). Comité Européen de Normalisation. (2019). Background Carpentry connections, Dove Tail (Standards No. CEN/TC 250/SC 5 - N 1086).

Crocetti, R. (2016). Limträhandbok Del 2. Stockholm, Sweden. Skogsindustrierna Svenskt Trä. Dedijer, M. (2016) Shear Resistance and Failure Modes of edgewise multiple tab-and-slot joint (MTSJ) Connection with dovetail design for thin LVL Spruce Plywood KERTO-Q Panels. WCTE2016 World Conference on Timber Engineering. Deutsches Institut für Bautechnik. (2018). Allgemeine bauaufsichliche Zulassung/Allgemeine Bauartgenehmigung: Schwalbenschwanz-Verbindung in Bauteilen [General building approval / general type approval: Dovetail connection in components] (Standards No. Z-9.1-649). Retrieved 2020 from https://www.lohn-abbund.de/wp-content/uploads/ ABZ-Z-9_1-649-Juni-2018-high.pdf EPFL. Timber Pavilion of the Vidy-Lausanne Theatre. Photo Credit: Ilka Kramer. Retrieved 2020 from https://www.epfl.ch/labs/ibois/projects/completed-projects/vidy-pavilion/ Gustafsson, A.( 2019). KL-trähandbok. Stockholm, Sweden. Skogsindustrierna Svenskt Trä. 102

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

McLain, T. E. (1998). Connectors and fasteners: Research needs and goals. Wood Engineering in the 21st Century: Research Needs and Goals, K. J. Fridley, ed, ASCE, Reston, Virginia. Martinsons. (2020). Materialguide för Martinsons KL-trä [Brochure]. https://martinsons.se/byggnader-i-tra/limtra-och-kl-tra-for-byggnadsobjekt/kl-tra/ Mayo, J. (2015). Solid wood: Case studies in mass timber architecture, technology and design. London: Routledge. Middleton, B., & Middleton, B. (2020, March 09). Hardwood vs Softwood: What's the Difference?. Retrieved October 09, 2020, from https://www.laver.co.uk/blog/hardwood-vs-softwood-whats-the-difference/ Müller, A., Vogel, M., Lang, S., Sauser, F. (2016). Historische Holzverbindungen: Untersuchung des Trag- und Lastverformungsverhaltens von histori-schen Vollholzverbindungen und Erstellung eines Leitfadens für die Baupraxis. [Historical Timber Connections: Investigation of the load-bearing and load deformation behavior of historical solid wood connections and creation of a guideline for building practice]. Bern Institute of Applied Sciences. Palma, P. (2016). Dowelled Timber Connections with Internal Members of Densified Veneer Wood and Fibre-Reinforced Polymer Dowels. WCTE2016 World Conference on Timber Engineering. Sandhaas, C. (2018). Design of Connections in Timber Structures. Germany. A state-ofthe-art report by COST. Shigeru Ban Architects. (2014). Tamedia Office Building Published 24 Feb 2014. ArchDaily. Retrieved January 08 2021. a. Swedish Standards Institute. (2009). Eurocode 5: Design of timber structures – Part 1-1: General – Common rules and rules for buildings (Standards No. SS-EN 1995-11:2004). b. Swedish Standards Institute. (2009). Eurocode 1: Actions on Structures – Part 1-1: General actions – Densities, self-weight, imposed loads for buildings (Standards No. SSEN 1991-1-3:2005). Swedish Standards Institute. (2008). Eurocode 1: Actions on Structures – Part 1-4: General actions – Wind actions (Standards No. SS-EN 1991-1-4:2005). Trähusbarometern. (n.d.). Retrieved October 09, 2020, from https://www.tmf.se/statistik/ statistiska-publikationer/trahusbarometern/ X-fix. Fix Brettsperrholz BSP Verbinder / CLT Connectors. (n.d.). Retrieved 2020 from http://www.x-fix.at/ 103

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

APPENDIX A

Appendix A consists of material for the architectural design process.

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

APPENDIX A1 LIBRARY OF CARPENTRY JOINTS Tenon and mortise joints The tenon and mortise joints normally connect members to form an “L” or “T” configuration. The two components consist of the tenon tongue and the mortise hole. This joint is normally used when the adjoining pieces are at an angle of 45 to 90 degrees (Branco, 2015). They can be locked or pinned into place and the connection can include shoulders to stabilize as well as different shapes of the tenon, such as rectangular or dovetail.

Figure 2. Reverse dovetail through, pinned tenon and mortise joint.

Figure 1. Wedged dovetail joints with a half-shoulder.

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

Figure 3. Pinned tenon and mortise joints.

Figure 4. Wedged dovetail joint with a cut-out angeled top notch.

III

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

Lap joints In the full lap joint no material is removed to create the joint. In a half-lap joint material is removed so that the resulting thickness of the joint is that of one of the members. The half-lap can be made with additional cogs. In the full lap joint no material is removed to create the joint. In a half-lap joint material is removed so that the resulting thickness of the joint is that of one of the members (Branco, 2015). The half-lap can be made with additional cogs. The two last connections shown can be seen as both lap joints or tenon and mortise joints and the last is also a form of notched joint. The dovetail lap joint is a way to reinforce the joints tensile strength.

Figure 6. Dovetail lap joint.

Figure 5. Pinned full lap joint.

Figure 7. Half lap joint.

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

Figure 9. Birdsmouth joint.

Figure 8. Full lap joint with a vertical locking.

Notched joints This type of joint is connected to the development of king post-like frames. In order for these joints to work they need appropriate joinery at multiple locations for example tenons or pegs and pins. A notch is a V shaped groove most commonly cut perpendicular from the beam (Branco, 2015). A notch like this is usually called a birdsmouth joint. The joints shown below can more commonly be rounded or notched out with a diagonal to reduce the number of sharp corners and the stress extremes in these points.

Figure 10. Notched joint.

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

Scarf joints Scarf joints allow connection of two members at the end to, for example, extend the length of a post or beam. The scarf joints are more efficient when used with a “key” which is a wedge, or several wedges used to lock the joint that is usually made of hardwood (Branco, 2015).

Figure XX. Wedged scarf joint.

Figure XX. Pinned scarf joint.

Figure 11. Wedged scarf joints with steps.

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

Figure 12. Illustrations of different types of wedged or pinned scarf joints.

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CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

APPENDIX A2 CONNECTION TABLES C1 Table 1. Connections divided into degrees of freedom

Rotation - Locked

Rotation - 1-way

Rotation - 2-way

Translation - 2-way

Depending on the horizontal loading, rotation is allowed in two directions

Depending on the horizontal loads, rotation is allowed in one direction

Translation - 1-way

Translation - Locked

2-way rotation is not really possible due to the geometry

Depending on the depth of the dove, the slight tilting, rotation in all directions are locked

X VIII

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

C2 Table 2. Connections divided into degrees of freedom

Translation - 2-way

Rotation - 2-way Rotation - 1-way Rotation - Locked

Translation - 1-way

2-way rotation is not really possible due to the geometry.

The translation in this case would make the connection fail

Translation - Locked

X 2-way rotation is not really possible due to the geometry.

The translation in this case would make the connection fail

X IX

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

APPENDIX A3 CONNECTION TREES C1 Table 1. Small advancements of the connections, staggered from simple to more sophisticated design

CENTERED INSERTION Part cross-section Rectangular cross-section

The beam is slanted Slanted tenon Add a wooden into the column into the column wedge above

Dowels added to the inserted part

Extra and locking parts

Dovetail tenon Dovetail cross-section

One wedge block added

Two wedges as lockers

Rotated dovetails in 90°

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

SIDE INSERTION Full cross-section cross-section Part cross-section Full From a rectangular to a plate dovetail

Change of the Shape of of the the Shape depth of insertion insert geometry insert geometry

Change of geometry and depth of insertion of the beam

Increase the the depth depth of of Increase insertion in the railing insertion in the railing

Full cross-section

Dowels to to reinforce reinforce Dowels and lock the step and lock the step

Outside reinforcement reinforcement Outside of the step of the step From a rectangular to a plate dovetail along the whole width of beam

Increasing the Increasing the amount of dowels amount of dowels

Table 2. Small advancements of the connections, staggered from simple to more sophisticated design.

Part cross-section cross-section Part

Amount of of tenon tenon Amount Full or or half half shoulders shoulders Full

Shape of of tenon tenon Shape XI

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

Full Full cross-section cross-section Shape of the Shape of the insert geometry insert geometry

Dowels to reinforce Dowels to reinforce and lock the step and lock the step

Increase the depth of Increase the depth of insertion in the railing insertion in the railing Outside reinforcement Outside reinforcement of the step of the step

Increasing the Increasing the amount of dowels amount of dowels

Dovetail tenon Dovetail tenon Part Part cross-section cross-section One-sided dovetail with lock One-sided dovetail with lock

Full dovetail Full dovetail

Amount of tenon Amount of tenon

Rotate direction of dovetail and with Rotate direction of size. different shape and dovetail and with different shape and size.

Full or half shoulders Full or half shoulders

Shape of tenon Shape of tenon

XII

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

APPENDIX A4 GRADING TABLES C1 Table 1. Gradign C1 TYPE OF CONNECTION

MANU FACTURING

ASSEMBLY

FIRE AND STRUCTURE ACOUSTICS

ARCHI. QUALITIES

XIII

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

APPLICA BILITY

TYPE OF CONNECTION

MANU FACTURING

ASSEMBLY

FIRE AND STRUCTURE ACOUSTICS

ARCHI. QUALITIES

XIV

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

APPLICA BILITY

TYPE OF CONNECTION

MANU FACTURING

ASSEMBLY

FIRE AND STRUCTURE ACOUSTICS

ARCHI. QUALITIES

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

APPLICA BILITY

TYPE OF CONNECTION

MANU FACTURING

ASSEMBLY

FIRE AND STRUCTURE ACOUSTICS

ARCHI. QUALITIES

APPLICA BILITY

MANU FACTURING

ASSEMBLY

FIRE AND STRUCTURE ACOUSTICS

ARCHI. QUALITIES

APPLICA BILITY

C2 Table 2. Grading C2 TYPE OF CONNECTION

XVI

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

TYPE OF CONNECTION

MANU FACTURING

ASSEMBLY

FIRE AND STRUCTURE ACOUSTICS

ARCHI. QUALITIES

XVII

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

APPLICA BILITY

TYPE OF CONNECTION

MANU FACTURING

ASSEMBLY

FIRE AND STRUCTURE ACOUSTICS

ARCHI. QUALITIES

XVIII

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

APPLICA BILITY

TYPE OF CONNECTION

MANU FACTURING

ASSEMBLY

FIRE AND STRUCTURE ACOUSTICS

ARCHI. QUALITIES

XIX

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

APPLICA BILITY

APPENDIX B

Appendix B includes all hand calculations and verifications for the design process.

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

APPENDIX B1 Indata

LOAD CALCULATIONS C1 Self-weight Indata Given from Sweco Structures Self-weight kN from gGiven 1.23Sweco : Structures roof m2 pillar. kN groof 1.23 2 : m kN pillar. gouter, wall 0.90 2 : m kN kN gouter, wall 0.90 : m gglulam 4.3 :2 m3 kN gglulam 4.3 : m3 Snow load

Calculation sheets for the building. Calculation sheets the building. This load is usedfor to calculate to Euler cracking of the This load is used to calculate to Euler cracking of the This load is used to calculate to Euler cracking of the pillar. This load is used to calculate to Euler cracking of the pillar. Self wieght of glulam (Martinsons) Self wieght of glulam (Martinsons)

Roof angle is 30 degrees. This load is also given from weco Structures Calculation sheets Snow for the load building. Roof angle is 30 kNdegrees. This load is also given from weco Structures Calculation sheets for the building. sk, max 2.64 2 : This load is used to calculate to Euler cracking of the pillar. m kN sk, max 2.64 2 : This load is used to calculate to Euler cracking of the pillar. m Wind load Borlänge, terrain type II (Given in the Sweco Calculation Sheet) Wind load m Borlänge, vb 22 sterrain : type II (Given in the Sweco Calculation Sheet) m vb 22 s kN : qp 0.68 2 : m kN q p pressure 0.68 coefficients : The for the relevant parts of Volume B is calculated for each load case. m2

Assumptions The pressure coefficients for the relevant parts of Volume B is calculated for each load case. The beam is for hand calculations simplified as a simply supported beam with two moment Assumptions free supports. The beam is for hand calculations simplified as a simply supported beam with two moment free Addsupports. sketch of beam

Load case 1 of (LC1) Add sketch beam

The loads are calculated for the wind on Volume B, for a special verification of the window Load case 1 (LC1)C a special load case need to be checked but for the design verification of raster in Volume C1 it is enough to check the load case for Volume B. The loads are calculated for the wind on Volume B, for a special verification of the window raster Volume 0Cdegrees a special load case need to be checked but for the design verification of Windin direction: C1 it is enough to check the load case for Volume B. Volume B Geometry Wind direction: 0 degrees Geometric data is taken from given drawingsXXI form Volume B. Volume B Geometry hB 11.5 m : Geometric data is taken from given drawings form Volume B. bhB1 B

21.4m m: : 11.5

db B1 B1

13.7CHALMERS m :: 21.4 m (b is the cross wind dimension) Architecture and Civil Engineering, Master’s Thesis ACEX30

ed min bB1m , 2: h B 13.7 B1

(b is the cross wind dimension)

= 21.4 m

Add sketch of beam

Load case 1 (LC1) The loads are calculated for the wind on Volume B, for a special verification of the window raster in Volume C a special load case need to be checked but for the design verification of C1 it is enough to check the load case for Volume B. Wind direction: 0 degrees Volume B Geometry Geometric data is taken from given drawings form Volume B. hB

11.5 m :

bB1

21.4 m :

dB1

13.7 m :

(b is the cross wind dimension)

min bB1, 2 h B

= 21.4 m

B A lD

The redimage area marks the window raster area used to calculate the forces out of the two sides. Insert of geometry Geometry Glulam dglulam

225mm :

wglulam

90mm :

cpillar

1000mm :

cbeam

1790 mm :

Wind load

B A

Internal pressure coefficents: cpip

0.2 :

cpin

0.3 :

For each load case the internal pressure coefficents are applied to find the most critical load case. wi, p wi, n

q p cpip q p cpin

= 0.136 m2 =

kN 0.204 m2

XXII

External pressure coefficents: ze

hB bB1 Architecture and Civil Engineering, Master’s Thesis ACEX30 if mCHALMERS m , h B, 0 = 11.5 m

Wind beam load Internal pressure coefficents: Wind load Internal pressure coefficents: cpip 0.2 : cpin

0.3 :

cpip each0.2 : case thecpin : For load internal0.3 pressure coefficents are applied to find the most critical load case. For each load case the internal pressure coefficents are applied to find the most critical load kN case. wi, p q p cpip = 0.136 m2 kN wi, p q p cpip = 0.136 kN m2 wi, n q p cpin = 0.204 m2 kN wi, n qpressure c = 0.204 External p pin coefficents: m2 External pressure hB bcoefficents: B1 ze if m , h , 0 = 11.5 m m hB bB1 B ze if , h B, 0 = 11.5 m The wind m load onm the wall perpendicular to the load bearing wall is divided 50 % between the two walls so that half the load on this wall area is applied as an axial Thewindow wind load on the wall perpendicular to the load bearing wall is divided 50 % the raster. between the two walls so that half the load on this wall area is applied as an axial Insert picture of load bearing walls and area indexes the window raster. h B picture of load bearing walls and area indexes Insert dhB1 0.839 B 0.839 d cpe, 1.4 : cpe, 1, B 1.1 : c B11, A

pe,1,D

cpe, 1, A wind 1.4load: : In-plane

cpe, 1, B

1.1 :

pe,1,D

- 50 % force on - 50 % force on

1.0 : 1.0 :

In-plane wind load: lD 2755.9mm = 2.76 m

lD 2755.9mm = 2.76 m This is the 50 % length of the wind load that goes to the window raster from wall D. This is the 50 % length of the wind load that goeskN to the window raster from wall D. q wind, D, pos cpe, 1, D cpip q p l D = 1.499 m kN q wind, D, pos cpe, 1, D cpip q p l D = 1.499 kN m q c c q l = 2.436 m wind, D, neg pe,1,D pin p D kN q c c q l = 2.436 m wind, D, neg pe,1,D pin p D Abaqus load: this is simulated as a pressure on the side of the modelled pillar with the q wind, D, neg Abaqus load: this is simulated as a=pressure on the side of the modelled pillar with the value: q wind, 0.011 MPa IP, A dglulam q wind, D, neg value: q wind, IP, A = 0.011 MPa d Hand calculation load is glulam applied on the connection as an axial point load on the beam with the value:

q wind, IP, H

q wind, D, neg cbeam

= 4.361 kN

Out-of-plane wind load: Since the outer wind pressure creates suction on these walls the worst case is when the internal pressure is positive. To calculate the maximum load on the beams the wind pressure for wall A is used as this becomes the dimensioning value. q wind, A

cpe, 1, A

cpip

= 1.088 m2

Assuming an even distribution of the horizontal loads between beams and pillars. It is more likely that the pillars will take more load since the load is transfered directly XXIII from these to the supports but for simplicity 50 % - 50 % is used and this also gives a worse case for loads on the beams. For the applied load in Abaqus the load is applied is applied as a pressure over the surface over the beam and pillars, the highest value of q.w for the beam and pillar is used.

q w, A, beam

cbeam

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30 q wind, A

wglulam

0.011 MPa

the value:

qqwind, c q wind,=D,4.361 c kN wind,IP, D,H neg beam neg beam

qthe value: wind, IP, H

= 4.361 kN

Out-of-plane wind Out-of-plane load: wind load: Since the outerSince windthe pressure outer wind createspressure suctioncreates on these suction wallson thethese worst walls casethe is when worstthe case is when the internal pressure internal is positive. pressure Toiscalculate positive.the Tomaximum calculate the loadmaximum on the beams load on the the wind beams the wind pressure for wall pressure A is used for wall as this A isbecomes used as the thisdimensioning becomes the dimensioning value. value. kN 1.088 cpip q p = m2

cpe, q wind, c c q = 1, A A pip pe, p1, A

q wind, A

1.088

kN m2

Assuming an Assuming even distribution an evenofdistribution the horizontal of the loads horizontal betweenloads beamsbetween and pillars. beams and pillars. It is more likely It isthat more the likely pillarsthat willthe take pillars morewill loadtake since more the load load since is transfered the loaddirectly is transfered directly from these to from the supports these tobut thefor supports simplicity but for 50 simplicity % - 50 % is 50used % - and 50 %this is used also gives and this a also gives a worse case forworse loads case on the forbeams. loads on the beams. For the applied For load theinapplied Abaqusload thein load Abaqus is applied the load is applied is applied as aispressure appliedover as a the pressure surface over the surface over the beam over and the pillars, beamtheand highest pillars, value the of highest q.w for value the of beam q.w and for the pillar beam is used. and pillar is used.

q w, A, beam

cbeam cbeam q wind, A q 2 wind, A 2 q w, A,wbeam w0.011 MPa 0.011 MPa

q w, A, pillar

q wind, A q w, A, pillar w

q wind, min OP, q w,A,beam min , q w,A,pillar q w,A,beam=, q w,A,pillar 0.012 MPa = 0.012 MPa A

glulam

cpillar 2 q2wind, A

glulam

wind, OP, A

glulam

cpillar 2 2

= 0.012 MPa w= 0.012 MPa glulam

For the hand calculations For the handthis calculations load is applied this load as aishorizontal applied asline a horizontal load withline the load magnitude: with the magnitude: kN q wind, OP, H q wind, q wind,OP, wqglulam 1.088 wglulam OP, HA wind, OP, A m

kN 1.088 m

Self weight Self weight q glulam

kN kN gglulam q glulam wglulamgglulam dglulamwglulam = 0.0871 dglulamm= 0.0871 m

Summary of loads Summary of loads LOAD

LOAD

Self weight, q glulam Self weight, : q glulam :

0.0871

Horizontal axial Horizontal load, qwind,IP,H axial load, : qwind,IP,H : Horizontal lineHorizontal load, q wind,OP,H line load, : q wind,OP,H :

kN kN : 0.0871 : m m

4.36 kN :

kN kN 1.088 m : 1.088 m :

Table 1: Summary Table of1: loads Summary - Loadofcase loads 1 - Load case 1 Reaction forces Reaction forces Vertical loads Vertical loads RC1, v

q glulam cpillar q glulam cpillar RC1, = 0.044 2 v 2 kN

= 0.044 kN XXIV

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

Mv x

RC1,v x

Vv x

RC1,v

Mv x

RC1,v x

Vv x RC1,v Horizontal loads

q glulam x2

2 x2 : q glulam : 2

q glulam x : q glulam x :

The axial horizontal load does not entail any further reaction force calculations since it Horizontal loads simply reacts to the axial load applied. The axial horizontal load does not entail any further reaction force calculations since it simply reactsqto the axialc load applied. RC1, h RC1, h Mh x Mh x Vh x

wind,OP,H pillar

= 0.544 kN 2 c q wind,OP,H pillar = 0.544 kN 2 q x2 wind,OP,H RC1, h x : 2 q wind,OP,H x2 RC1, h x : 2 RC1, h q wind,OP,H x :

RC1, h q wind,OP,H x : Vh x Load case 2 (LC2)

For 2 the wind load is applied in a 90 degree angle from load case 1. Loadload casecase 2 (LC2) For load case 2 the load is applied in a 90 degree angle from load case 1. Wind direction: 90 wind degrees Wind direction: 90 degrees Volume B Geometry Volume B Geometry Geometric data is taken from given drawings form Volume B. Geometric hB 11.5data m : is taken from given drawings form Volume B. hB bB2

bB2 dB2

11.5 m : 13.7 m :

b is the cross wind dimension

13.7 m : 21.4 m :

b is the cross wind dimension

dB2 21.4 m : e min bB2, 2 h B

= 13.7 m

e min bB2,of 2 geometry h B = 13.7 m Insert image Insert image of geometry Geometry Glulam Geometry Same as in Glulam LC1. Same asload in LC1. Wind Wind load Internal pressure coefficents: Internal pressure coefficents: The internal pressure coefficents are the same as in LC1. The internal pressure coefficents are the same as in LC1. External pressure coefficents: External pressure coefficents: bB2 hB ze if m , h B, 0 = 11.5 m hB bm B2 XXV ze if m m , h B, 0 = 11.5 m The in-plane wind load on the wall perpendicular to the wind is divided 50 % - 50 % between the two walls so that half the load on this wall area is applied as an axial force on The in-planeraster. wind load on the wall perpendicular to the wind is divided 50 % - 50 % the window between the two walls so that half the load on this wall area is applied as an axial force on the window raster. Insert picture of load bearing walls and area indexes CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

Insert picture of load bearing walls and area indexes

For load case 2 the wind load is applied in a 90 degree angle from load case 1. D B Wind direction: 90 degrees Volume B Geometry A

Geometric data is taken from given drawings form Volume B. lD hB

11.5 m :

bB2

13.7 m :

dB2

21.4 m :

b is the cross wind dimension

min bB2, 2 h B

= 13.7 m

Insert image of geometry Geometry Glulam

Same as in LC1.

Wind load

Internal pressure coefficents:

lD The internal pressure coefficents are the same as in LC1. External pressure coefficents: ze

hB if m

bB2

= 11.5 m

m , h B, 0

The in-plane wind load on the wall perpendicular to the wind is divided 50 % - 50 % between the two walls so that half the load on this wall area is applied as an axial force on the window raster. h B picture of load bearing walls and area indexes Insert dB2 = 0.537 cpe, 1, A

1.4 :

1.0 :

pe,1,D

In-plane wind load: lA

= 2.76 m

2755.9mm

This is the 50 % length of the wind load that goes to the window raster from wall A. q wind, A, pos

cpe, 1, A

wind, A, neg

pe, 1, A

cpip

qp lA

q l

= 2.998 m A

= 2.061 m

Abaqus load: this is simulated as a pressure on the side of the modelled pillar with the q wind, A, pos value: q wind, IP, A = 0.013 MPa d glulam

XXVI as an axial point load on the beam with Hand calculation load is applied on the connection the value: q wind, IP, H q wind, A, pos cbeam = 5.367 kN

Out-of-plane wind load: Since the outer wind creates pressure on these walls the worst case is when the internal pressure is negative. CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30 q

= 0.884

This is the 50 % length of the wind load that goes to the window raster from wall A. q wind, A, pos

cpe, 1, A

wind, A, neg

cpip

qp lA

q l

pe, 1, A

= 2.998 m A

= 2.061 m

Abaqus load: this is simulated as a pressure on the side of the modelled pillar with the q wind, A, pos value: q wind, IP, A = 0.013 MPa d glulam

Hand calculation load is applied on the connection as an axial point load on the beam with the value: q wind, IP, H q wind, A, pos cbeam = 5.367 kN Out-of-plane wind load: Since the outer wind creates pressure on these walls the worst case is when the internal pressure is negative. q wind, D

cpe, 1, D

cpin q p

= 0.884 m2

Assuming an even distribution of the horizontal loads between beams and pillars. It is more likely that the pillars will take more load since the load is transfered directly from these to the supports but for simplicity 50 % - 50 % is used and this also gives a worse case for loads on the beams. For the applied load in Abaqus the load is applied is applied as a pressure over the surface over the beam and pillars, the highest value of q.w for the beam and pillar is used. q wind, D

q w, D, beam

cbeam

wglulam

q wind, D

q w, D, pillar

cpillar 2

wglulam

q wind, OP, A

= 0.0088 MPa 2

= 0.0098 MPa

max q w, D, beam, q w, D, pillar = 0.0098 MPa

For the hand calculations this load is applied as a horizontal line load with the magnitude: q wind, OP, H

q wind, OP, A wglulam

= 0.884 m

XXVII

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

Summary of loads Summary of loads LOAD

LOAD

Self weight, q glulam Self weight, : q glulam :

0.0871

Horizontal axial Horizontal load, qwind,IP,H axial load, : qwind,IP,H :

kN kN : 0.0871 : m m

5.37 kN :

kN kN Horizontal lineHorizontal load, q wind,OP,H line load, : q wind,OP,H : 0.884 : 0.884 m m : Table 2: Summary Tableof2:loads Summary - Loadofcase loads 2 - Load case 2 Reaction forces Reaction forces Vertical loads:Vertical loads: RC2, v

q glulam cpillar q glulam cpillar RC2, = 0.0435 kN= 0.0435 kN 2v 2

Mv x

RC2, Mvv x

Vv x

RC2,Vvv x q glulam RC2,xv:

q glulam x2 q glulam x2 RC2,2 v x : : 2 q glulam x :

Horizontal loads: Horizontal loads:

The axial horizontal The axial load horizontal does notload entaildoes anynot further entail reaction any further force reaction calculations forcesince calculations it since it simply reacts simply to the axial reactsload to the applied. axial load applied. RC1, h

q wind,OP,H cpillar q wind,OP,H cpillar RC1, 2h = 0.442 kN = 0.442 kN 2 q wind,OP,H x2q wind,OP,H x2 RC1, h2 x : : 2

Mh x

RC1, Mhh x

Vh x

RC1,Vhh x q wind,OP,H RC1, h x q: wind,OP,H x :

Comparison reaction Comparison forcesreaction forces

L1 L2

L1/L2 L2

L1/L2

0.044 0.044 kN kN

0.044 100 kN %

100 %

4.36 kN

4.36 -5.37 kN kN

-5.37 81 % kN

81 %

Horizontal maximum Horizontal shear maximum force, Vshear : V h0.544 x : kN h x force,

0.544 0.442 kN kN

0.442 123 kN %

123 %

FORCE

Vertical maximum Vertical shear maximum force, Vshear : V v x0.044 : kN v x force, Horizontal axial Horizontal force, QAxial axial:force, QAxial :

Table 3: ForceTable comparisson 3: Force comparisson

XXVIII

Bibliography Bibliography

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

wind,OP,H pillar

RC1, h Mh x Vh x

RC1, h c q wind,OP,H pillar 2

Mh x RC1, h x

= 0.442 kN 2 = 0.442 kN q wind,OP,H x2

x x2 qR C1, h wind,OP,H 2

Vh x RC1, h q wind,OP,H x : RC1, h q wind,OP,H x :

Comparison reaction forces Comparison reaction forces

FORCE

L2 L1/L2 0.044 kN 100 % -5.37 kN 81 % 0.442 kN 123 %

L2 Vertical maximum shear force, V v x : L1 0.044 kN Vertical maximum shear force, V v x : 0.044 kN 0.044 kN Horizontal axial force, QAxial : 4.36 kN Horizontal axial force, QAxial : kN Horizontal maximum shear force, V h 4.36 x : kN 0.544-5.37 kN Horizontal maximum shear force, V h x : 0.544 kN 0.442 kN Table 3: Force comparisson Table 3: Force comparisson Bibliography Bibliography Bibliography Swedish Standards Institute. (2008). Eurocode 1: Actions on Structures – Part 1-4: FORCE

General actions – Wind actions (Standards No. SS-EN 1991-1-4:2005).

XXIX

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

L1/L2 100 % 81 % 123 %

APPENDIX B2 LOAD CALCULATIONS C2

Indata Indata

Stair geometry

wstep Stair geometry wstep l step Astep t1,step tstep Vstep

300 mm :

300l step mm : 1.0 m : Indata wstep l step : 1.0 Amstep: Stair geometry t1,step 40 mm : wstep l step : 300 wstep mm :

t2,step 40tl step mm : t1.0 1,step m : t2,step step

t2,step

40 mm :

t3,step

t40 : : mm 3,step

t3,step

40 mm :

t2,step

40 mm :

t3,step

40 mm :

Vstep t Astep ttstep : : t1,step Astep 2,step wstep l3,step : step

Image of step geometry Astep t tstep : 40 mm : 1,step

Image of step tstepgeometry t1,step t2,step t3,step : (mm) GEOMETRY Vstep Astep tstep : 120 THICKNESS wstep (mm) GEOMETRY 300 WIDTH Image of step geometry 120 THICKNESS 1000 LENGTH 300 WIDTH Table 1: Geometry - Stair Step 1000 (mm) LENGTH GEOMETRY THICKNESS Self-weight Table 1: Geometry - Stair Step 120

40 mm :

tstep R1

lstep

300 Gravity WIDTH R2 Self-weight m 1000 LENGTH g 9.82 2 : s Gravity Table 1: Geometry - Stair Step m C24-C14-C24 g 9.82 Type: : s2 Self-weight kg Type: C24-C14-C24 54 2 : (Martinsons, 2020) Gravity m m kg 9.82 : kg 2020) 54 2 : g (Martinsons, s 2 54 m kg m2 Self-weight: = 450 Type: C24-C14-C24 tstep kg m3 54 2 kg kgm Self-weight:54 t : = 450 kg 3 (Martinsons, 2020) m 2 450 Qstepmstep V g = 159.08 N 3 step m kg kg Since theVstair54 is 1 meter Qstep 450 gstep = 159.08 N long. The distributed load per m is the same: 2 3 kg step m m Self-weight:159.0840000 = 450 N N XXX tstep m3 Since the stair step is 1 meter long. The distributed load per m is the same: Qstep, m = 159.08 m 1m 159.0840000 kg N N In Abaqus the self-weight Qstep 450 V 159.08 g is= applied 159.08 asNa body load of magnitude: Qstep, m = 3 step 1m m m kg kN Since the stair is g1 =meter The distributed load per m is the same: In Abaqus the self-weight is applied as a long. body load of magnitude: Qstep, 450step 4.42 3 A m m3 and Civil Engineering, Master’s Thesis ACEX30 CHALMERS Architecture 159.0840000 N kg kN N Imposed Qstep, m3 loads = 159.08 m Qstep, A 450 g = 4.421m 3 m m

Self-weight Gravity g

9.82

m : s2

Type: C24-C14-C24 54

kg : m2

(Martinsons, 2020)

Self-weight: Qstep

450

kg m2

= 450

tstep kg m3

step

kg m3

= 159.08 N

Since the stair step is 1 meter long. The distributed load per m is the same: 159.0840000 N 1m

Qstep, m

= 159.08 m

In Abaqus the self-weight is applied as a body load of magnitude: Qstep, A

450

kg kN g = 4.42 m3 m3

Imposed loads P

step

= Q : from (SIS, 2009).

Pstep

2000 N :

There is also an imposed load that is in the horizontal direction of the step. This is a line load. q k, h

kN 0.5 m :

In Abaqus the imposed point load is applied in 4 different points to first and foremost check which location of the point load that gives the most critical load case and to check the influence of the location of the pointload depthwise of the step. The four points will distribute the point load on different areas depending if it's in the centre, on the side or at the corner. These pressure loads are calculated below. Pstep

Pstep, A, CC

= 0.20 MPa

100mm 100mm

Pstep, A, CS, SC

Pstep

50mm 100mm Pstep

Pstep, A, CC

50mm 50mm

= 0.40 MPa

= 0.80 MPa

The horizontal line load is applied in abaqus as a pressure or suction over an area, this line load is therefore divided by the height of the step. q k, h, A

q k, h tstep

= 0.0042 MPa

XXXI

Assumptions For this case hinged supports are assumed for the railing and the wall. R1 : = reaction force fromArchitecture wall support : = reaction force from support (along CHALMERS andand CivilREngineering, Master’s Thesisrailing ACEX30 2 l step : , see image below).

step 100mm 50mm = 0.80 MPa 50mm 50mm

A, CS, SC Pstep, step, A, CC

P The line loadstepis applied=in0.80 abaqus as a pressure or suction over an area, this line Pstep,horizontal MPa CC 50mm load isA,therefore divided 50mm by the height of the step. The horizontal q k, h line load is applied in abaqus as a pressure or suction over an area, this line load is therefore divided by the height of the step. q k, h, A = 0.0042 MPa t step

q k, h qAssumptions k, h, A t step

= 0.0042 MPa

For this case hinged supports are assumed for the railing and the wall.

Assumptions R1 : = reaction force from wall support and R2 : = reaction force from railing support (along l stepthis : , see image below). For case hinged supports are assumed for the railing and the wall. R : = reaction from wall support and R2 : = reaction force from railing support (along Image of loads force and reactions 1 l step : , see image below). The depth of the step and the position of the point load is not taken into consideration as simple FEM models have been analysed which showed the most critical load case where the Imageload of loads and reactions point is centered on the step in depth. In hand calculations this is also a simplification so that the supports will hinged. With thetaken pointinto loadconsideration applied with as an The depth of the step andbethemoment positionfree of and the point load is not exentricity from the support a console case would be created and the connection would have simple FEM models have been analysed which showed the most critical load case where the neededload to be forstep a moment. point is dimensioned centered on the in depth. In hand calculations this is also a simplification so that the supports will be moment free and hinged. With the point load applied with an exentricity from the- Point support console Load case 1 (LC1) loada on railingcase would be created and the connection would have needed to be dimensioned for a moment. Equilibrium for moment around railing support (positive clockwise): l step : = at support 2.

Load case 1 (LC1) - Point load on railing R2 : Global equilibrium gives the following equations: Equilibrium for moment around railing support (positive clockwise): l step : = at support 2. Qstep, m l step l step M l = 0 : equations: R1 R2 l:step GlobalRequilibrium gives2the following 1 step

Qstep, m l step l step

Vertical force balance: Qstep, m l step l step M l step R1 l step =0: 2 R1 Pstep Qstep,m l step R2 = 0 :

Qstep, m l step l step

2 l step

= 79.54 N = 79.54 N

Vertical force balance: Reaction force for the railing support, R2 : R Pstep Qstep,m l step R2 = 0 : R2 1 R1 Pstep Qstep,m l step = 2.08 kN Reaction force for the railing support, R2 :

Load case 2 (LC2) - Point load on the centre of the step Equilibrium for moment around (positive clockwise): l step : = at support 2. M l step

R1 l step

Qstep, m l step l step 2

Pstep l step

l step

Pstep

=0:

l step 2

= 1.08 kN

Vertical force balance: R1

Pstep

Qstep,m l step

R2 = 0 :

Reaction force for the railing support, C2: R2

Pstep

XXXII

Qstep,m l step = 1.08 kN

Horizontal load case The horizontal load case also gives reaction forces in the supports in the horizontal direction. Because the supports are counted as hinged. CHALMERS Architecture andin-plane Civil Engineering, Thesis ACEX30 The horizontal load is applied in both directions Master’s which will result in one axial force at the connection as well. Q q w = 0.15 kN

step 2. Equilibrium for moment around (positive clockwise): l step : = at support

M l step

Qstep, m l step l step Pstep l step Qstep, l step l step Pstep l step M l step R1m l step =0: 2 R1 l step =0: 2 2 2

Qstep, m l step l step Qstep, m l step l step l step 2 P step 2 R1 2 l step l step

Pstep

l step 2

= 1.08 kN

Vertical force balance: Vertical force balance: R P Q l R1 Pstep Q1 step,mstep l step step,m R2 = 0 :step

= 1.08 kN

R2 = 0 :

Reaction force for the railing support, C2: Reaction force for the railing support, C2: R2

R2 Pstep

R1 Pstep Qstep,m l step = 1.08 kN Qstep,m l step = 1.08 kN

Horizontal load case Horizontal load case The horizontal load case also gives reaction forces in the supports in the horizontal The horizontaldirection. load caseBecause also gives in the supports the reaction supportsforces are counted as hinged.in the horizontal direction. Because the supports areiscounted The horizontal load applied as in hinged. both in-plane directions which will result in one axial The horizontalforce loadatisthe applied in both connection as in-plane well. directions which will result in one axial force at the connection as well. Q Axial q k, h wstep = 0.15 kN Qh, Axial q k, hh, w = 0.15 kN step q k, h l step q k, h l step Rh = 0.25 kN Rh = 0.252 kN 2 Summary of reaction forces Summary of reaction forces

RESULT LC1, R2 : LC2, R2 :

(kN) 2.08 1.08

Horizontal load case, QAxial : 0.15 Horizontal load case, QAxial : Horizontal load case, Rh : 0.25 Horizontal load case, Rh :

(kN) 2.08 1.08 0.15 0.25

Table 2: Reaction forces at the railing Table 2: Reaction forces at the railing

XXXIII

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

APPENDIX B3 VERIFICATIONS, CONNECTION C1 1.Loads Loads 1. 1. Loads 0.044 kN kN:: VVEd, 0.044 Ed,VV VEd, V 0.044 kN : 0.544 kN kN:: VVEd, 0.544 Ed,HH VEd, H 0.544 kN : 4.36kN QQEd 4.36kN :: Ed QEd 4.36kN :

2.Material Materialproperties properties 2. 2. Material Thechosen chosenproperties material for forthis this connection connectionisis Glulam GlulamGL30c GL30cwith withthe thevalues values according according to to The material Gustafsson al. (2019). (2019). Gustafsson al. The chosenetet material for this connection is Glulam GL30c with the values according to Gustafsson et al. (2019). 2.1.Correction Correctionand andmodification modificationfactors factors 2.1. 2.1. Correction and modification factors 1.25:: M 1.25 M 1.25 : M 0.90:: (for (forclimate climateclass class 11and andshort shortduration durationload loadasas dimensioning) dimensioning) kkmod 0.90 mod kmod 0.90 : (for climate class 1 and short duration load as dimensioning) 2.5MPa 2.5MPa = 0.78 (according to toeq. eq. (2)) (2)) kkcrcr (according 3.2MPa 2.5MPa = 0.78 3.2MPa kcr = 0.78 (according to eq. (2)) 3.2MPa k 6.5 : (according to eq. (3)) knn 6.5 : (according to eq. (3)) kn 6.5 : (according to eq. (3)) 1.75 : (according toeq. eq. (5)) (5)) kkc,c,90 1.75 : (according to 90 kc, 90 1.75 : (according to eq. (5)) 2.2.Design Designvalues valuesfor formaterial materialproperties properties 2.2. 2.2. Design values for material properties Characteristicstrength strength Design Designstrength strength Characteristic strength strength : : ffc,c,90, ffc,c,90, Compressivestrength strength⊥⊥ to tothe thegrain grain Characteristic 2.5MPa MPa Design 1.80MPa MPa Compressive 2.5 1.80 90,kk: 90,dd: f : f : Compressive strength ⊥ to the grain 2.5 MPa fc, 90, d: 1.80 MPa k: fft,c,t,0,90, Tensionstrength strength to tothe thegrain grain 19.5MPa MPa ft,t,0,0,dd: 14.04MPa MPa Tension 19.5 14.04 0,kk: f : f : Tension strength to the grain 19.5 MPa ft, 0, d : 14.04 MPa 0, k : : fft,t, Tensionstrength strength tothe thegrain grain 0.5MPa MPa ft,t,90, 0.36MPa MPa Tension to 0.5 0.36 t,90, 90,kk 90,dd: ft, 90, k : ft, 90, d : Tension strength to the grain 0.5 MPa 0.36 MPa ffv,v,kk:: Panel shear shear 3.2MPa MPa ffv,v,dd:: 2.30MPa MPa Panel 3.2 2.30 fv, k : fv, d : Panel shear 3.2 MPa 2.30 MPa : : ffRv, Rolling shear shear 1.2MPa MPa ffRv, 0.86MPa MPa Rolling 1.2 0.86 Rv,kk: Rv,dd: f : f : Rolling shear 1.2 MPa Rv, d 0.86 MPa Rv, k

MaterialProperties Properties--glulam, glulam,GL30c GL30c Material Material Properties - glulam, GL30c

2.5MPa 2.5MPa = 1.80 MPa ffc,c,90, kkmod 90,dd mod 2.5MPa = 1.80 MPa M M fc, 90, d kmod = 1.80 MPa M 19.5MPa 19.5MPa = 14.04 MPa fft,t,0,0,dd kkmod mod 19.5MPa = 14.04 MPa M M ft, 0, d kmod = 14.04 MPa M 0.5MPa 0.5MPa = 0.36 MPa fft,t,90, kkmod 90,dd mod 0.5MPa = 0.36 MPa XXXIV M M ft, 90, d kmod = 0.36 MPa M 3.2MPa 3.2MPa = 2.30 MPa ffv,v,dd kkmod mod 3.2MPa = 2.30 MPa M M fv, d kmod = 2.30 MPa M 1.2MPa 1.2MPa = 0.86 MPa ffRv, kkmod Rv,dd mod 1.2MPa = 0.86 MPa CHALMERS M Architecture and Civil Engineering, Master’s Thesis ACEX30 M fRv, d kmod = 0.86 MPa

3.Geometry Geometry 3.

Rv, k

Rv, d Material Properties - glulam, GL30c

fc, 90, d

2.5MPa

kmod

ft, 0, d

kmod

ft, 90, d

19.5MPa 0.5MPa

fRv, d

kmod

= 0.36 MPa

3.2MPa

kmod

= 14.04 MPa

kmod

fv, d

= 1.80 MPa

1.2MPa M

= 2.30 MPa = 0.86 MPa

3. Geometry Four different geometries are evaluated with the relevant indata below. The geometries of two of these can be seen in Figure 1 and 2. The other two geometries are similar, compared comparedto toFigure Figure11the thesecond seconddovetail dovetail portrudes portrudes the thefull full width widthof ofthe thepillar pillarand and compared comparedto toFigure Figure22the thefourth fourthgeometry geometryhas has aaslightly slightlysmaller smallerpeg. peg. The Thegeometrical geometrical data data isis stated statedin inthe thetable tablebelow. below. PD1 PD1

PD2 PD2

P1 P1

P2 P2

Tenon Tenon bbN, 1 N, 1

60mm 60mm:: bbN,1 N,1

bb1 1 hh1 1

bbN,2 N,2

: 0.8 0.8 :

bb2 2

225mm 225mm::

hh2 2

hhN, 1 N, 1

112.5mm 112.5mm::

60mm 60mm:: bbN,2 N,2

bbN, 3 :: N, 3

==P1 P1

225mm 225mm::

hh3 3

225mm 225mm::

==P1 P1

hhN,2 N,2

00::

llz,2

112.5mm 112.5mm::

90mm 90mm::

z,2

==P1 P1

bb3 3

45mm 45mm::

z,1

90mm 90mm::

: 0.8 0.8 :

-llz,1

bbN,3 N,3

hhN, 3 N, 3

142.5mm 142.5mm::

==P1 P1

hhT, 3 T, 3

60mm 60mm::

==P1 P1

llz,3

90mm 90mm::

z,3

00::

==P1 P1 ==P1 P1

00::

Header Header bbH, 1 H, 1

90mm 90mm::

bbH, 2 H, 2

90mm 90mm::

bbH, 3 H, 3

90mm 90mm::

==P1 P1

hhH, 1 H, 1

225mm 225mm::

hhH, 2 H, 2

225mm 225mm::

hhH, 3 H, 3

225mm 225mm::

==P1 P1

hhH, u, 1 H, u, 1

112.5mm 112.5mm:: hhH, H,u,u,22

hhH, u, 1 :: H, u, 1

hhH, u, 3 H, u, 3

82.5mm P1 82.5mm:: ==P1

Peg Peg dd3 3

4. 4.Verifications Verifications 4.1 4.1Joist Joistmember member

40mm 40mm::

dd4 4

20mm 20mm::

XXXV

4.1.1 4.1.1 Failure Failurein inshear shearand andtension tensionat atthe thenotch notchcorner corner 4.1.1.1 4.1.1.1Vertical Vertical loads loads Verified Verifiedfor forPD1 PD1and andPD2 PD2according according to toeq. eq. (24) (24)for foraadovetail dovetail type typeIV. IV.

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

V, V,11

bbN, 1 N, 1 bb

== 0.80 0.80

V, V,22

bbN, 2 N, 2 bb

== 0.80 0.80

compared to Figure 1 the second dovetail portrudes the full width of the pillar and compared to Figure 2 the fourth geometry has a slightly smaller peg. The geometrical data is stated in the table below. PD1

PD2

Tenon bN, 1

60mm :

bN,2

bN,1

b1 h1

0.8 :

225mm :

h N, 1

112.5mm :

60mm : bN,2

bN, 3 :

= P1

225mm :

= P1

h N,2

112.5mm : -

45mm :

l z,2

Header bH, 1

90mm :

h H, 1

225mm :

h H, u, 1

h N, 3

142.5mm :

= P1

h T, 3

60mm :

= P1

l z,3lz

90mm :

h H, u, 2

h H, u, 1 :

= P1

90mm : 0 : Ed VEd,V

bNi

V bH, 2 Ed,H90mm : hN h H, 2 225mmh :

112.5mm :

= P1

90mm :

0.8 :

l z,1

bN,3

= P1

hH,u

bH, 3

90mm :

= P1

h H, 3

225mm :

= P1

h H, u, 3

= P1

82.5mm :

Peg d3

40mm :

20mm :

Figure 1. Geometry for PD1. PD2 is the same but with the dovetail protruding fully through the pillar.

4. Verifications 4.1 Joist member a) b) 4.1.1 Failure in shear and tension at the notch corner

4.1.1.1 Vertical loads Verified for PD1 and PD2 according to eq. (24) for a dovetail type IV. bN, 1

V, 1

l z, 1 2

= 0.80

V, 2

= 22.50 mm

l z,2

x2 kn

kv, V, 1

bN, 2

1.1 i

V,1

1.5

min 1.0, b1

= 45 mm

1 2

V,1

= 0.80

mm x1

0.8 b 1

1 V,1

= 1.0

V,1

XXXVI

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

Ed VEd,V

ho hT hu

hH,u

VEd,H hN

Figure 2. Geometry for P1. P2 is the same but with a smaller peg.

a) kv, V, 2

k b)n

min 1.0, b2

V,2

2 3 bN,1 h N,1 kv,V,1 fv,d

Fv,Rd,V,1

1.1 i

1.5

V,2

mm x2

0.8 b 2

c) 1 V,2

= 0.97

V,2

= 10.37 kN

2 3 bN, 2 h N, 2 kv, V, 2 fv,d = 10.04 kN

Fv, Rd, V, 2

PD1

PD2

Fv, Rd, V

10.37 kN

10.04 kN

VEd, V

0.044 kN

0.42 %

0.44 %

Utilization ratio

OK! Failure mode 1, Vertical loads

OK!

XXXVII

4.1.1.2 Horizontal loads PD1 and PD2 are verified as dovetail type III and the other two as tenon and mortise joints. Verified according to eq. (6) CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

H, 1

h N, 1 h

= 0.50

H, 2

h N, 2 h

= 0.50

V,2

2 bN,1 h N,1 k2v,V,1 f = 10.37 kN 3 Fv,Rd,V,1 3 bN,1v,dh N,1 kv,V,1 fv,d

Fv,Rd,V,1

x2 2 0.8 b V,2 V,2 2

0.8 b 2 V,2 2

b2 V,2

V,2

= 10.37 kN

2 2k h f h = 10.04 kN F k f = 10.04 kN 3 v, bRd, N, 2 N, 2 v, b V, 2 3 V,N,2 2 v,d N, 2 v, V, 2 v,d

Fv, Rd, V, 2

PD1

PD2

PD1

PD2

Fv, Rd, V

10.37 kN

10.04 kN

VEd, V

0.044 kN

0.42 %

0.44 %

Utilization ratio Utilization ratio

OK!

Failure mode 1, Vertical loads Failure mode 1, Vertical loads

OK!

4.1.1.2 Horizontal loads 4.1.1.2 Horizontal loads

PD1 and PD2 PD1 are verified as dovetail typeasIII and thetype other and PD2 are verified dovetail IIItwo andasthetenon otherand twomortise as tenon and mortise joints. Verifiedjoints. according to eq.according (6) Verified to eq. (6) H, 1

ef, 1

h N, 1 h1

h N, 1

=H, 10.50

= 0.50

kcr bN,b1 = 46.9 kcr mm bN, 1 ef, 1

mink 1.0, v, H, 1

min 1.0,

H,1

kn kv, H, 2

mink 1.0, v, H, 2

d, H, 2

3 2

d, H, 1

3 2

min 1.0,

VEd, H

d, H, 2h H, 2 2

3 bef, 22

VEd, H

d, H, 1h H, 1 1

3 bef,1 2

H, 2

=H, 20.50

h N, 2 h2

= 0.50

bef, 2mm kcr bN,b2 = 46.9 = 46.9 kcr mm bN, 2 = 46.9 mm ef, 2 kn

kv, H, 1

h N, 2

H,2

1.1 i 1.5 hkn 1

h 1 H,1

1.1 i 1.5 mm h1 2

0.8 h H,1 1

H,1

1.1 i 1.5 hkn 1

h 2 H,2

0.8 h H,2 2

H,2

V1Ed, H 1 kv, H, 2 = 0.25k MPa H, 2 h 2 bef, 2 v, H, 2 1V 1 Ed, H kv,H,1 = 0.22 kMPa H, 1 h 1 bef,1 v,H,1

1 x1 2 0.8 h H,1 H,1 1

1.1 i 1.5 mm h2 2

mm 1

H,1

1 x2 2 0.8 h H,2 H,2 2

= 0.72

H,1

= 0.61

H,2

= 0.72

= 0.61

H,2

= 0.25 MPa

= 0.22 MPa

P1 and P2 has P1 theand same for the tenon and need to be one verified. P2geometry has the same geometry for therefore the tenononly and one therefore only need to be verified. Due to symmetry and placement of the tenon z = z : and therefore the strength is the strength is Due to symmetry and placement 0,ofTthe 0tenon z0, T = z0 : and therefore calculated according to eq.according (20b) to eq. (20b) calculated

XXXVIII

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

H, 3 H, 3

h N, 3 h hN, 3 3

l z, 3

70.3 mm h 3 70.3 mm 70.3 mm kv, H, 3 kv, H, 3

N, 3 z, 3 =H,0.63 x = 45 x3 mm 2 =bef, 453 mm kcr bN,b3ef,=3 3 h 3 = 30.63 l z,2 3 = 0.63 x3 bef, 3 kcr bN, 3 = 2 = 45 mm

kn kn

1 1

1.1 1.1

min k1.0, min 1.0, v, H, 3 2 min 1.0, h H, 3 h H, 3 3 3 2 h3 H, 3 H, 3

h T, 3 h T, 3 kz, 3 1 2 1 k h h N, h N, 3 T, 33 z, 3 kz, 3 1 2 1 h N, 3 4 Fv, Rd, H, 3 h 94Fv, bRd, ef, 3H, 3N, 3 Fv, Rd, H, 3 9 bef, 3 h N, 3 PD1 PD1 0.25 MPa d, H / VEd, H d, H / V Ed, H 0.25 MPa d, H / VEd, H fv, d / Fv,Rd,H f MPa / F2.30 v, d v,Rd,H fv, d / Fv,Rd,H 2.30 MPa Utilization ratioUtilization10.9 ratio% Utilization ratio 10.9 % OK! Failure mode 1, Horizontal loads OK!

i 1.5 k1.5 1 hi n 3

1.1 i 1.5 mm mm h 3

x3 2 0.8 xh 3H, 3 H, 3 0.8 h 3 3

x3 2 1 H, 3 0.8 1 2 H, 3 h 3 H, 3

H, 3

= 0.66 1 = 0.66 H, 3

h T, 3 2 h N, 3 2 h T, h N, 3 3 2 12 = 0.96 1 2 2 = 0.96 h h h N, h N,hN,33 3 h3 T, 3 3 2 = 0.96 h N, 3 h3 4 kz, 3b kv, H,h 3 fv,kd = 6.52 kN 9 ef, 3 N, 3 z, 3 kv, H, 3 fv, d = 6.52 kN kz, 3 kv, H, 3 fv, d = 6.52 kN

PD2 PD1 PD2 0.22 MPa 0.25 MPa 0.22 MPa 2.30 MPa 2.30 MPa 2.30 MPa 10.99.6 %% 9.6 % OK! OK! OK!

Failure mode 1, Horizontal loads

P1 PD2 P1 0.54 kN 0.22 MPa 0.54 kN 6.52 kN 2.30 MPa 6.52 kN 9.68.3 %% 8.3 % OK! OK! OK!

P1P2 P2 0.54kN kN 0.54 0.54 kN 6.52kN kN 6.52 6.52 kN 8.3% % 8.3 8.3 % OK! OK! OK!

P2 0.54 kN 6.52 kN 8.3 % OK!

Failure mode 1, Horizontal loads

4.1.2 Compression failure at the bottom of the notch 4.1.2 Compression failure at the bottom of the notch 4.1.2 Compression failure at the bottom of the notch Verified according to eq.according (13). Verified to eq. (13). Verified according to eq. (13). 4.1.2.1 Vertical4.1.2.1 loads Vertical loads 4.1.2.1 Vertical loads PD1 and PD2 PD1 and PD2 PD1 and PD2 l z, ef, 1 min ll z, 1 30mm, = 75 mm min l2z, l1z, 1 30mm, 2 l z, 1 = 75 mm z, ef, 1 l z, ef, 1 min l z, 1 30mm, 2 l z, 1 = 75 mm l z, ef, 2 min ll z, 2 30mm, = 120 2mm min l2z, l2z, 2 30mm, l z, 2 = 120 mm z, ef, 2 l z, ef, 2 min l z, 2 30mm, 2 l z, 2 = 120 mm VEd, V 1 VEd, V 1 = 0.0030 MPa= 0.0030 MPa c, 90, d, V, 1 V c, 90, d, V, 1 h N, 1Ed, l z,Vef, 1 khc,190 l kc, 90 N, 1 =z,0.0030 ef, 1 MPa c, 90, d, V, 1 h N, 1 l z, ef, 1 kc, 90 VEd,V 1 VEd,V 1 = 0.0019 MPa= 0.0019 MPa c, 90, d, V, 2 V d,l V, 2 khc,90 1 hc,N,90,2Ed,V l ef, 2 kc,90 z, ef, 2 N, 2 = z,0.0019 MPa c, 90, d, V, 2 h N, 2 l z, ef, 2 kc,90 P1 and P2 P1 and P2 P1 and P2 l z, ef, 3 min ll z, 3 30mm, = 120 2mm min l2z, l3z, 3 30mm, l z, 3 = 120 mm z, ef, 3 l z, ef, 3 min l z, 3 30mm, 2 l z, 3 = XXXIX 120 mm VEd,V 1 VEd,V 1 = 0.0035 MPa= 0.0035 MPa c, 90, d, V, 3 V c, 90, d, V, 3 h T, 3 Ed,V l z, ef, 3 kc,90 h1T, 3 l z, ef, 3 kc,90 = 0.0035 MPa c, 90, d, V, 3 k h l T, 3

PD1 PD1 0.003 MPa

z, ef, 3

c,90

PD2 P1 and Civil Engineering, PD1 PD2 P1P2 P2 ACEX30 CHALMERS Architecture Master’s Thesis

PD2 0.002 MPa 0.003 MPa

kcr bN, 3

P1 0.003 MPa 0.002 MPa

P2 0.003 MPa 0.003 MPa

0.003 MPa

H, 3

= 0.66

l z, ef, 2

min l z, 2

30mm, l2 l z, 2 z, ef, 2

VEd, V

c, 90, d, V, 1

h N, 1 l z, ef, 1 VEd,V

c, 90, d, V, 2

h N, 2 l z, ef, 2

P1 and P2 l z, ef, 3

min l z, 3

h N, 2 l z, ef, 2

c,90

z, ef, 3

h T, 3 l z, ef, 3

PD2 c, 90, d, V

fc, 90, d 1.80 MPa fc, 90, d

= 0.0030 MPa

= 0.0019 MPa

kc,90

=min 120l mm 30mm, 2 l = 120 mm z, 3 z, 3 V

Ed,V 0.0035 MPa c, 90,= d, V, 3 kc,90 h T, 3 l z, ef, 3

PD1 P1

PD2 P2

1.80 MPa 1.80 MPa 1.80 MPa 1.80 MPa 1.80 MPa 1.80 1.80 MPaMPa

Utilization ratio 0.2 %Utilization ratio 0.1 % OK!

OK!

Failure mode 2, Vertical loads Failure mode mode 2, 2, Vertical Vertical loads loads Failure

ailure ailure mode mode 2, 2, Vertical Vertical loads loads

0.2 % 0.2 %0.2 % OK! OK! OK!

4.1.2.2 Horizontal loads PD1 and PD2 VEd, H

c, 90, d, H, 1

bN, 1 l z, ef, 1 VEd, H

c, 90, d, H, 2

bN, 2 l z, ef, 2

P1 and P2 VEd, H

c, 90, d, H, 3

1 kc,90

= 0.0035 MPa

bN, 3 l z, ef, 3

0.1 % 0.1 %0.2 % OK! OK!OK!

4.1.2.2 Horizontal loads 4.1.2.2 Horizontal loads PD1 and PD2 PD1 and PD2 VEd, H VEd, H c,190, d, H, 1 bN, 1 l z, ef, 1 = 0.069 MPa c, 90, d, H, 1 kc, 90 b l N, 1

z, ef, 1

fc, 90, d 1.80 MPa fc, 90, d

Utilization ratio

1.80 MPa 1.80 MPa

0.2 % 0.2 % OK! OK!

1 1 kc, 90 kc, 90

= 0.069 MPa = 0.069 MPa

VEd, H 1 VEd, H 1 c,190, d, H, 2 bN, 2 l z, ef, 2 kc,90 = 0.043 MPa c, 90, d, H, 2 kc,90 bN, 2 l z, ef, 2 kc,90

= 0.043 MPa = 0.043 MPa

VEd, H 1 VEd, H 1 k bN, 3 l z, ef, 3 c,90 MPa bN, 3 l z, ef, 3 kc,90

= 0.029 MPa = 0.029 MPa

P1 and P2 P1 and P2

c,190, d, H, 3 0.029 c, 90, = d, H, 3 kc,90

ailure mode 2, Horizontal loads

PD2

1.80 MPa 1.80 MPa 1.80 MPa 1.80 MPa 1.80 1.80 MPa MPa 1.80 1.80 MPaMPa 1.80 MPa

Utilization ratio 3.8 %Utilization ratio 2.4 % OK!

OK!

Failure mode 2, Horizontal loads Failure mode 2, Horizontal loads

0.003 MPa

1.80 MPa 1.80 MPa

PD1 PD2 P1 PD1 P1 PD2 P2 P1 0.069 MPa 0.043 MPa 0.029 MPa c, 90, d, H 0.043 MPa 0.0690.029 0.069 MPa MPa MPa 0.0430.029 MPa MPa 0.029 MPa c, 90, d, H PD1

fc, 90, d

0.0019 MPa d, V, 2 k c, 90, =

1 kc, 90

0.003 MPa MPa MPa 0.0020.003 MPaMPa 0.003 MPa c, 90, d, V 0.002 MPa 0.0030.003

Utilization ratio

c, 90, d, H

VEd,V

30mm, l2 l z, 3 VEd,V

PD1

fc, 90, d

1 Ed, V =V,0.0030 MPa c, 90, d, 1 kc, 90 h N, 1 l z, ef, 1

P1 and P2

c, 90, d, V, 3

c, 90, d, V

=min 120l mm 30mm, 2 l = 120 mm z, 2 z, 2

3.8 % 3.8 %1.6 % OK! OK! OK!

2.4 % 2.4 %1.6 % OK! OK! OK!

P2 P2 0.029 MPa 0.029 MPa 1.80 MPa 1.80 MPa

1.6 % 1.6 % OK! OK!

4.2 Header member 4.2 Header member XLnot to be verified for the vertical loads as the grain is then parallel to the The header need The header need not to be verified the vertical the grain parallel to as thethe grain is then parallel to the The for header need notloads to beasverified for isthethen vertical loads reaction forces. reaction forces. reaction forces. 4.2.1 Failure in shear and tension perpendicular to the grain 4.2.1 Failure in shear and tension perpendicular to the and graintension perpendicular to the grain 4.2.1 Failure in shear PD1 and PD2 are verified as dovetail type III according to eq. (18) PD1 and PD2 are verified as dovetail type IIIare according (18) type III according to eq. (18) PD1 and PD2 verified to as eq. dovetail 4.2 Header member

bH, ef, 1

bH, ef, 1and Civil kcr Engineering, bH, 1 = 70.31 mm Thesis ACEX30 bH, ef, 2 CHALMERS Architecture Master’s kcr bH, 1

= 70.31 bmm H, ef, 1

bkH, ef, : mm b 2 1 =bH, 70.31 ef, 1 cr H,

VEd, H V

bH, ef, 2

bH, ef, 1 : bH, ef, 1 :

c, 90, d, H

0.069 MPa

0.043 MPa

0.029 MPa

fc, 90, d

1.80 MPa

3.8 %

2.4 %

1.6 %

OK!

Utilization ratio

ailure mode 2, Horizontal loads

4.2 Header member The header need not to be verified for the vertical loads as the grain is then parallel to the reaction forces. 4.2.1 Failure in shear and tension perpendicular to the grain PD1 and PD2 are verified as dovetail type III according to eq. (18) bH, ef, 1 d, H, 1

kcr bH, 1 3 4

= 70.31 mm

VEd, H

bH, ef, 1 h H, u, 1

bH, ef, 2

bH, ef, 1 :

= 0.052 MPa

d, H, 2

d, H, 1 :

P1 and P2 are verified similarly. bH, ef, 3 d, H, 3

kcr bH, 3 3 4

PD1 d, H fv, d

d, H 2.30 MPa f

ailure mode 4, Horizontal loads

VEd,H

bH, ef, 3 h H, u, 3 PD2

0.052 MPa

Utilization ratio

= 70.31 mm

v, d

= 0.070 MPa

PD1 P1

PD2 P2

0.052 MPa 0.0520.070 MPaMPa 0.0520.070 MPaMPa 0.070 MPa

0.070 MPa

2.30 MPa 2.30 2.30 MPaMPa

2.30 MPa

2.3 %Utilization ratio 2.3 % OK!

2.30 2.30 MPaMPa 2.30 MPa

2.3 %3.0 %

OK!

2.3 %3.0 %

OK! OK!

3.0 %

OK!

Failure mode 4, Horizontal loads

4.2.2 Failure in compression perpendicular to the grain 4.2.2 Failure in compression perpendicular to the grain No interaction in compressionNo zones occur as in l 1 compression 1790mm zones : interaction occur as l 1

1790mm :

Calculated similarly as for the Calculated joist. similarly as for the joist. z, ef, 1

min bN, 1

60mm,l 2 bN, 1 min = 120 bN,mm 60mm, 2 bN, 1 z, ef, 1 1

= 120 mm

l z, ef, 2

min bN, 2

60mm,l 2 bN, 2 min = 120 bN,mm 60mm, 2 bN, 2 z, ef, 2 2

= 120 mm

l z, ef, 3

min bN, 3

60mm,l 2 bN, 3 min = 150 bN,mm 60mm, 2 bN, 3 z, ef, 3 3

= 150 mm

bN, 1

l z, 1 :

c, 90, d, H, H, 1

c, 90, d, H, H, 2

c, 90, d, H, H, 3

bN, 2 b l z, 2 : l : N, 1 z, 1

VEd, H

bN, 1 l z, ef, 1 VEd, H

XLI

bN, 3 l z, ef, 3

bN, 1 l z, ef, 1 V

= 0.029 MPa Ed, H

d, H, H, 2 bN, 2 l z, ef, 2 c,k90, c,90

VEd, H

= 0.058 MPaEd, H

c,k90, d, H, H, 1 c, 90

bN, 3 b l z, 3 : l : N, 2 z, 2

bN, 2 l z, ef, 2 V

= 0.023 MPa Ed, H

c,k90, d, H, H, 3 c,90

kc, 90 1 kc,90 1

kc,90

bN, 3

= 0.058 MPa = 0.029 MPa = 0.023 MPa

N, 3 z, Master’s ef, 3 CHALMERS Architecture and Civil Engineering, Thesis ACEX30

l z, 3 :

l z, ef, 1

min bN, 1

min b 1 60mm, 2 bN, 1 60mm,l z, 2 ef,bN, 1 1 = 120N,mm

= 120 mm

l z, ef, 2

min bN, 2

min b 2 60mm, 2 bN, 2 60mm,l z, 2 ef,bN, 2 2 = 120N,mm

= 120 mm

l z, ef, 3

min bN, 3

min b 3 60mm, 2 bN, 3 60mm,l z, 2 ef,bN, 3 3 = 150N,mm

= 150 mm

bN, 1

bN, 2 bN,l z,1 2 : l z, 1 :

l z, 1 :

c, 90, d, H, H, 1

c, 90, d, H, H, 2

c, 90, d, H, H, 3

PD1

VEd, H

c, 90, d, H, 1 = H,0.058 bN, 1 l z, ef, 1 kc, 90

VEd, H MPa bN, 1 l z, ef, 1

VEd, H c, 90, d, H, 2 = H, 0.029 MPa b l k

VEd, H

z, ef, 2

kc,90

VEd, H MPa bN, 3 l z, ef, 3

1 kc,90

N, 2

c,90

VEd, H

c, 90, d, H, 3 = H, 0.023 bN, 3 l z, ef, 3 kc,90

PD1 P1

1 kc, 90 1

bN, 2 l z, ef, 2

PD2

bN, l z,23 : l z, 2 :

bN, 3

PD2 P2

bN, 3

l z, 3 :

= 0.058 MPa = 0.029 MPa = 0.023 MPa

c, 90, d, H, H

MPaMPa 0.0290.023 MPaMPa 0.023 MPa 0.058 MPa MPa 0.0580.023 c, 90, d, H,0.029 H

0.023 MPa

fc, 90, d

fc, 90, d 1.80 MPa

1.80 MPa

Utilization ratio

ailure mode 5, Horizontal loads

MPaMPa 1.80 MPa 1.80 1.80

1.80 1.80 MPaMPa 1.80 MPa

3.2 %Utilization ratio 1.6 %

3.2 %1.3 %

1.6 %1.3 %

1.3 %

OK!

OK! OK!

OK!OK!

OK!

Failure mode 5, Horizontal loads

4.2.3 Failure in compression to the grain to an axial compression force on the 4.2.3 Failure in compression perpendicular to the grain due toperpendicular an axial compression forcedue on the joist joist This This is verified according to eq. (19)is verified according to eq. (19) AB

225mm 90mm

2 AB mm225mm 90mm = 20250 mm 2 = 20250

AN, 1

bN, 1 h N, 1 :

AN, 1 A bN, 1 h N, AB1 : AN, 1 B, ef, 1

2 A AB, ef,mm = 13500.0 1 B

AN, 1

= 13500.0 mm 2

AN, 2

bN, 2 h N, 2 :

AN, 2 A bN, 2 h N, AB2 : AN, 2 B, ef, 2

2 A AB, ef,mm = 13500.0 2 B

AN, 2

= 13500.0 mm 2

AN, 3

bN, 3 h N, 3 :

c, 90, d, Q, 1

c, 90, d, Q, 2

c, 90, d, Q, 3

AB, ef, 3

QEd

1 kc, 90

= 0.18 MPa

QEd

AB, ef, 2

1 kc, 90

= 0.18 MPa

QEd

AB, ef, 1

AB, ef, 3

kc, 90

AN, 3

= 0.34 MPa

PD1

PD2

c, 90, d, Q

0.18 MPa

0.34 MPa

XLII 0.34 MPa

fc, 90, d

1.80 MPa

10 %

19%

19 %

OK!

Utilization ratio

ailure mode 6, Axial load

OK!

= 7425.0 mm 2

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

c, 90, d, Q, 2 c, 90, d, Q, 2 c, 90, d, Q, 3 c, 90, d, Q, 3

c, 90, d, Q c, 90, d, Q

fc, 90, d fc, 90, d Utilization ratio Utilization ratio

ailure mode 6, Axial load ailure mode 6, Axial load

AB, ef, 1 kc, 90 QEd 1 QEd AB, ef, 2 kc, 190 AB, ef, 2 kc, 90 QEd 1 Q 1 AB, Ed k ef, 3 c, 90 AB, ef, 3 kc, 90

PD1 PD2 PD1 PD2 0.18 MPa 0.18 MPa c, 90, d, Q 0.18 MPa 0.18 MPa c, 90, d, Q 1.80 MPa fc, 90, d 1.80 MPa 1.80 MPa fc, 90, d 1.80 MPa 10 %Utilization ratio 10 % 10 %Utilization ratio 10 % OK! OK! OK! OK! Failure mode 6, Axial load Failure mode 6, Axial load

B, ef, 1

c, 90, d, Q, 1

= 0.18 MPa

c, 90, d, Q, 2 c,=90,0.18 d, Q, MPa 2

= 0.34 MPa

c, 90, d, Q, 3 c,=90,0.34 d, Q, MPa 3

c, 90

AB, ef, 1 kc, 90 QEd 1 QEd AB, ef, 2 kc,190 AB, ef, 2 kc, 90 QEd 1 QEd AB, ef, 3 kc,190 AB, ef, 3 kc, 90

PD1 P1 PD2 P2 PD1 P1 PD2 P2 0.34 MPa 0.18 MPa 0.18 0.34 MPaMPa 0.18 0.34 MPaMPa 0.18 0.34 MPaMPa 1.80 MPa 1.80 1.80 MPa 1.80 MPaMPa 1.80 1.80 MPaMPa 1.80 1.80 MPaMPa 19% 10 % 10 %19 % 19% 10 % 10 %19 % OK! OK! OK!OK! OK! OK! OK! OK!

= 0.18 MPa = 0.18 MPa = 0.18 MPa = 0.34 MPa = 0.34 MPa

P1 P1 0.34 MPa 0.34 MPa 1.80 MPa 1.80 MPa 19% 19% OK! OK!

P2 P2 0.34 MPa 0.34 MPa 1.80 MPa 1.80 MPa 19 % 19 % OK! OK!

4.2.4 Failure in tension perpendicular to the grain due to perpendicular an axial force to onthe thegrain joist due to an axial force on the joist 4.2.4 Failure in tension 4.2.4 Failure in tension perpendicular to the grain due toperpendicular an axial force to onthe thegrain joist due to an axial force on the joist 4.2.4 Failure in tension This failure mode is only validThis for the dovetail andvalid in this only PD1 sinceand PD2 failure modejoints is only forcase the dovetail joints in this case only PD1 since PD2 This failure is onlyand valid for the dovetail and intherefore this case onlyfailure PD1 since PD2 protrudes themode full header therefore this cannot occur. This failure mode ismode only valid for thethis dovetail joints and in thisoccur. case only PD1 since PD2 protrudes thefailure fulljoints header and mode cannot protrudes the full header and therefore modeand cannot occur.this failure mode cannot occur. protrudesthis thefailure full header therefore h H, ef, 1 kcr h H, 1 = 175.78h mm kcr h H, 1 = 175.78 mm ef, 1 h H, ef, 1 kcr h H, 1 = 175.78hH, mm k h = 175.78 mm H, ef, 1 cr H, 1 QEd 3 Q 3 Ed QEd = 0.41 MPa 3 d, Q, 1 QEd = 0.41 MPa d, Q, 1 43 h l 4 h l z, 1 = 0.41 MPa = 0.41 MPa H,ef,1 z, 1 d, Q, 1 H,ef,1 d, Q, 1 4 h l 4 h l H,ef,1 z, 1 H,ef,1 z, 1

d,Q,1 d,Q,1

fv, d fv, d Utilization ratio Utilization ratio

ailure mode 7, Axial load ailure mode 7, Axial load

PD1 PD1 0.41 MPa 0.41 MPa 2.30 MPa 2.30 MPa 18 % 18 % OK! OK!

d,Q,1 d,Q,1

fv, d fv, d Utilization ratio Utilization ratio Failure mode 7, Axial load Failure mode 7, Axial load

PD1 PD1 0.41 MPa 0.41 MPa 2.30 MPa 2.30 MPa 18 % 18 % OK! OK!

4.3 Peg 4.3 Peg 4.3 Peg 4.3 Peg 4.3.1. Shear failure of a laterally4.3.1. loaded peg joint Shear failure of a laterally loaded peg joint 4.3.1. Shear failure of a laterally loaded peg joint 4.3.1. Shear failure of a laterally loaded peg joint This failure mode is verified according to eq. (25).is verified according to eq. (25). This failure mode This failure mode is verified according to eq. (25). This failure mode is verified according to eq. (25). Where: n 1: Where: n 1: Where: n 1: Where: n 1: 9.5MPa n d3 2 kmod 9.5MPa n d3 2 kmod 9.5MPa n d3F2 kmod = 10.94 Fv, Rd, P, 3 kN 9.5MPa n d3 2 kmod = 10.94 kN M Fv, Rd, P, 3= 10.94 kN Fv, Rd, P, 3 M = 10.94 kN v, Rd, P, 3 M

Fv, Rd, P, 4

9.5MPa n d4 2 kmod M

= 2.74 kN

XLIII

Fv, Rd, P

10.94 kN

2.74 kN

4.36 kN

40 %

160 %

OK!

NOT OK!

Utilization ratio

ailure mode 8, Axial load

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

9.5MPa nn dd4 22 kkmod 9.5MPa n d4 2 kmod 9.5MPa 4 mod Fv, Rd, P,=4 2.74 kN F v, Rd, P, 4

Fv, Rd, P, 4

M M

= 2.74 2.74 kN kN =

P1 P2 P1

P2 P2

Fv, Rd, P

Fv, Rd, P 10.94 kN F v, Rd, P

10.94 kN 2.74 kN 10.94 kN

2.74 kN kN 2.74

Q Q

4.36 kN kN 4.36 4.36 kN

4.36 kN kN 4.36

40 % % 160 % 40

160 % % 160

OK! NOT OK! OK!

NOT OK! OK! NOT

Utilization ratio

ailure mode 8, Axial load

4.36 kN

Ed Ed

Utilization ratio ratio 40 % Utilization OK! Failure mode mode 8, 8, Axial Axial load load Failure

4.3.2. Tension failure perpendicular perpendicular to to the the grain grain 4.3.2. Tension failure perpendicular to Tension the grainfailure 4.3.2. Verified according according to to eq. eq. (26) (26) where: where: Verified according to eq. (26) where: Verified

= 165 mmbbS

h H, 3

h T, 3

bH, 3

= 90 mm

2 = 45 mm

F90, Rd

14MPa bS

= 165 165 mm mm =

hhH, H, 33

hhT, T, 33

hhSS

bbH, H, 33

= 90 90 mm mm =

ddSS

hhSS 22

dS F90, Rd F dS 90, Rd 1 h

= 45 45 mm mm = kmod mm 14MPa bbS 14MPa mm S M

evalf F90, Rd

ddSS = 4.99 d 10.00 dSS 11 hh S S

evalf F F90, Rd = = 15.78 15.78 kN kN = 15.78 kN evalf 90, Rd P1 and P2

P1 and and P2 P2 P1

F90, Rd

F90, Rd F 90, Rd

15.78 kN

15.78 kN kN 15.78

QEd

QEd Q Ed

4.36 kN

4.36 kN kN 4.36

Utilization ratio

Utilization ratio ratio Utilization

28 %

28 % % 28

OK!

OK! OK!

ailure mode 9, Axial load

mm mm kN mm mm

Failure mode mode 9, 9, Axial Axial load load Failure

XLIV

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

kkmod mod M M

= 4.99 4.99 10.00 10.00 kN kN =

APPENDIX B4 VERIFICATIONS, CONNECTION C2

1. Loads VEd, V

2.08 kN :

VEd, H

0.25 kN :

QEd

0.15kN :

2. Material properties The chosen material for this connection is CLT from the products of Martinsons with the values according to Gustafsson et al. (2019). 2.1. Correction and modification factors M

1.25 :

kmod

0.80 :

(for climate class 1 and average duration load as dimensioning)

kcr

0.67 :

(according to eq. (2))

6.5 :

(according to eq. (3))

kc, 90

1.5 :

(according to eq. (5))

2.2. Design values for material properties C24-C14-C24

Characteristic strength

Design strength

Compressive strength ⊥ to the grain

fc, 90, k :

2.7 MPa

fc, 90, d :

1.73 MPa

Tension strength

ft, 90, k :

0.4 MPa ft, 90, d :

0.26 MPa

Panel shear

fv, k :

4.0 MPa

fv, d :

2.56 MPa

Rolling shear

fRv, k :

3.0 MPa

fRv, d :

1.92 MPa

to the grain

Material Properties - CLT, 3-layer panel

fc, 90, d

kmod

ft, 90, d

kmod

2.7MPa M

0.4MPa M

= 1.73 MPa

fv, d

= 0.26 MPa

fRv, d

4.0MPa

kmod

3.0MPa M

Characteristic strength C14 XLV C24

(Values in MPa) Compressive strength ⊥ to the grain, fc, 90 Tension strength Tension strength

to the grain, ft, 0

to the grain, ft, 90

2.5

2.0

14.5 0.4

= 2.56 MPa = 1.92 MPa

Design strength C24 C14 1.6

1.28

7.2

9.28

4.61

0.4

0.26

Panel shear, fv 4.0 Master’s3.0 2.56 CHALMERS Architecture and Civil Engineering, Thesis ACEX30 Elasticity module, E 0

11000

7000

1.92

Material Properties - CLT, 3-layer panel

fc, 90, d

kmod

ft, 90, d

kmod

2.7MPa M

0.4MPa M

= 1.73 MPa

fv, d

= 0.26 MPa

fRv, d

to the grain, ft, 0

Tension strength Panel shear, fv

Elasticity module, E 0

Elasticity module, E 90 E 0, C24

11000MPa :

E 90, C14

230MPa :

kmod

3.0MPa M

2.5

2.0

14.5

to the grain, ft, 90

= 2.56 MPa

Characteristic strength C24 C14

(Values in MPa) Compressive strength ⊥ to the grain, fc, 90 Tension strength

4.0MPa

kmod

= 1.92 MPa

Design strength C24 C14 1.6

1.28

7.2

9.28

4.61

0.4

0.26

4.0

3.0

2.56

1.92

11000

7000

370

230 Material Properties - CLT, Internal layers

3. Geometry Three different geometries are evaluated with the relevant indata below. The geometries can be seen in Figure 1, 2 and 3. The geometrical data is stated in the table below. RD1

Tenon

bN, RD

55.6mm : bN, RD

bRD tB1

40mm :

tB3

tB1 :

h N, RD

tB2

tB3

20mm :

tH1

30mm :

tH3

30mm :

tH2

40mm :

100mm :

h H, u, RD

tB1 :

tB1 : 300mm :

225mm :

150mm :

tH,i h/ t hB1 B,i

tB3

tB2

h N, TM

2 = RD 360 for Figure 1. Geometry RD.

40mm :

tB1

h TM

: V100mm Ed,H h

i 0: Header

H,u 93.2mm :

h RD :

N2 bN, N

300mm :

tB1 :

h RD tB1 120 mm

h B1

bN,TM

0.8

tB2

TM1

1 15

l z, TM

100 mm :

tH1

40mm :

tB3

tB1 :

tB2

tB1 :

h RD :

h N, N

60mm : -

l z, N

100mm : Q Ed

XLVI

30mm :

tH2

40mm :

100mm :

tH3

tB1

VEd,V

30mm :

tH1

30mm :

tH3

30mm :

tH2

40mm :

100mm :

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

220mm :

h H, u, TM

180mm :

h H, u, N

180mm :

E 0, C24

11000MPa :

E 90, C14

230MPa :

3. Geometry Three different geometries are evaluated with the relevant indata below. The geometries can be seen in Figure 1, 2 and 3. The geometrical data is stated in the table below. RD1

Tenon

bN, RD

55.6mm : bN, RD

bRD tB1

40mm :

tB3

tB1 :

h N, RD

100mm :

20mm :

12 360 2

tH1

30mm :

tH3

30mm :

tH2

40mm :

100mm :

h H, u, RD

93.2mm :

i 0: Header

h RD :

tB1

40mm :

tB3

tB1 :

tB2

120 mm

h RD :

N2 bN, N

300mm :

tB1 :

h B1

bN,TM

0.8

tB2

TM1

tB1 :

h TM

300mm :

h N, TM hT

225mm :

150mm : -

l z, TM 1

= 15

220mm :

100 mm :

tB1

40mm :

tB3

tB1 :

tB2

tB1 :

h RD :

h N, N

l z, N

100mm :

tH1

30mm :

tH3

30mm :

tH2

40mm :

100mm :

h H, u, TM

60mm :

180mm :

tH1

30mm :

tH3

30mm :

tH2

40mm :

100mm :

h H, u, N

180mm :

4. Verifications 4.1 Joist member 4.1.1 Failure in shear and tension at the notch corner 4.1.1.1 Vertical loads Verified for RD1 and N2. The TM1 joist for the vertical load is a simple rectangular beam with a reduced cross-section. XLVII

The dovetail is verified according to eq. (23) and the notch according to eq. (6). RD1 As the crack seem to appear in a perpendicular layer to the crack propagation the height in CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

N, RD

bRD tB1

40mm :

tB3

tB1 :

tB2

tB1 :

h N, RD

tB2

100mm :

20mm :

tB3

tB1 :

300mm :

h N, TM hT

225mm :

150mm : -

100 mm :

= 15

i bH

tH1

30mm :

tH3

30mm : h H,u

tH2

40mm :

100mm :

220mm :

4. Verifications

tB1

40mm :

tB3

tB1 :

tB2

tB1 :

h RD :

h N, N

60mm : -

l z, N

100mm :

tH1

30mm :

tH3

30mm :

tH2

40mm :

100mm :

h H, u, TM

tH,i

VEd,H

tB1 :

l z, TM

12 360 2

h H, u, RD

40mm :

h TM

i 0: Header

tB3

93.2mm :

h RD :

tB1 tB2

h RD tB1 120 mm

h B1

0.8

180mm :

tH1

30mm :

tH3

30mm :

tH2

40mm :

100mm :

h H, u, N

tB,i

4.1 Joist member hT

Q Ed

VEd,V

180mm :

4.1.1 Failure in shear and tension at the notch corner

Figure 2. Geometry for TM.

4.1.1.1 Vertical loads

Verified for RD1 and N2. The TM1 joist for the vertical load is a simple rectangular beam with a reduced cross-section. The dovetail is verified according to eq. (23) and the notch according to eq. (6). RD1 As the crack seem to appear in a perpendicular layer to the crack propagation the height in this formula is reduced. The height h B1 is reduced by the distance from the crack to the nearest layer with grain parallel to the crack according to the modifications for CLT for this geometry. 2 Fv, Rd = 3

bB1

evalf cos

red

33.2mm :

h B1, red

h B1

red

2 Fv,Rd = 3 bB1 tB1 kv fv,d

kv fv, d

= 39.78 mm

= 86.8 mm

h RD

Fv, Rd, V, RD

r r 2

h B1, red

kv, V, RD

h B1

2 3 bB1

XLVIII

= 0.52 h B1, red

kv, V, RD fv,d = 2.37 kN

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

For the notch the crack appear in a perpendicular layer as well. Since the notch is on the opposite side as the support k 1:

bH lz hH,u The height h is reduced by the distance from the crack to the this formula is reduced. B1 nearest layer with grain parallel to the crack according to the modifications for CLT for this geometry.

2 Fv, Rd = 3 VEd,H

bB1

evalf cosb

red

33.2mm :

h B1 hN

Figure 2. Geometry for N.

h B1, red

h B1

red h RD

2 3 bB1

Fv, Rd, V, RD

tB,i

r 2

2 Fv,Rd = 3 bB1 tB1 kv fv,d

tH,i

kv fv, d

Q Ed

= V39.78 mm Ed,V

= 86.8 mm

h B1, red

kv, V, RD

= 0.52 h B1, red

kv, V, RD fv,d = 2.37 kN

N2 For the notch the crack appear in a perpendicular layer as well. Since the notch is on the opposite side as the support kv, V, N 1: V, N

d, V, N

tB3 hN

= 0.33 3 2

bef, V, N VEd, V

V, N

h N bef, V, N

RD1

kcr bN, N

= 201.00 mm

= 0.39 MPa

XLIX

Fv, Rd, V / fv, d

2.37 kN

2.56 MPa

VEd, V /

2.08 kN

0.39 MPa

d,V

Utilization ratio

CHALMERS Architecture 88 % and Civil Engineering, Master’s 15 % Thesis ACEX30

OK!

tB3

V, N

= 0.33 V, N

3 2

d, V, N

V, N d, V, N

tB3

v, V, N

bef, V, N

kcr bN, N

= 0.33 VhEd, N V

= 201.00 mm

bef, V, N

= 0.39 MPa h N b3ef, V, N VEd, V 2 V, N h b N

ef, V, N

kcr bN, N

= 0.39 MPa

RD1

Fv, Rd, V / fv, d

2.37 kN

VEd, V /

2.08 kN

d,V

Fv, Rd, V / fv, d

V / Utilization ratio Ed, V

d,V

RD1

2.56 MPa

0.39 MPa

2.08 kN

0.39 MPa

15 %

88 %

OK!

2.56 MPa

2.37 kN

88 %

Utilization ratio

= 201.00 mm

15 %

OK!

Failure mode 1, Vertical loads

OK!

Failure mode 1, Vertical loads

4.1.1.2 Horizontal loads

Horizontal RD1 is verified4.1.1.2 as dovetail type II loads and then TM1 is verified. Verification according to eq. (24) and eq. (20b). Both of these are modified for CLT because the centre layer act as RD1 isthis verified reinforcement against crack.as dovetail type II and then TM1 is verified. Verification according to eq. (24) and eq. (20b). Both of these are modified for CLT because the centre layer act as reinforcement against this crack. RD1 RD1 evalf bN, RD

= 45.80 mm 2 RD bkN,1.0, evalf bN, RD h N, RD tan 2 = 45.80 mm kv, H, RD 0.5 min min 1.0, ef, RD b b l z, RD v, N, H, RD RD N, ef, RD = 0.73 xRD = 50 mm H, RD bRD0.5 b 2 b l z, RD N, RD N, ef, RD 1.5 = 0.73 x = 50 mm H, RD 1.1 i bRD RD 2 1.1 i 1.5 kn 1 mm k 1 mm bRDn bRD kv, H, RD min 1.0, xRD 1 x 2 21 2 RD 2 bRD H, RDb H, RDH, RD 0.8 H, H, RD 0.8 RD H, RD b RD RD H, RDbRD H, RD bN, ef, RD

= 0.80

= 0.80 k 1 n

Fv, Rd, H, RD TM1

bRD

= 0.80

h N, RD tan

1.1 i

1.5

mm bRD 2 2 bN, ef, RD h N, RD tB2 khv, H, RD ftv,d = k3.34 kN f F3v, Rd, xRD N, RD 1 B2 v, H, RD2 v,d = 3.34 kN H, RD 32 bN, ef, RD 0.8 H, RD H, RD H, RD bRD TM1 H, RD

bbN, TM tB2 mm kcr = bN,53.60 tB2 = 53.60 mm ef, TM TM 2 Fv, Rd, H, RDh bN, ef, RD h N, RD fv,d h T t2B2 kv, H,hhRD 2 = 3.34 hkN hT T 3 TN,TM 1 2 1 2 =2 1.02 N,TM kz,H 1 2 1 h N,TM kz,H h N,TM h TM h h h TM TM1 N,TM N,TM bef, TM

kcr

h TM l z, TM l z, TM kcr N,bN, tB2h N,=TM 53.60 mm TM= 0.75 x = 50 mm = 0.75 x h TMH, TM TM 2 h TM TM 2 2 hT hT h N,TM 2 = 1.02 h N,TM 1 2 1 h N,TM h TM

bef, TM

H, TM

kz,H k

hevalf k min 1.0, evalf min 1.0, N, TM v, H, TM xTM h TM = 0.75

v, H, TM

H, TM

k k

v, H, TM

evalf min 1.0, h

= 0.68

1.1 i h

l z, TM 2

1.5

2 0.8 H,TM h H,TM H,TM TM

L 1.1 i 1 mm h

h H,TM TM

= 1.02

= 50 mm

= 50 mm 1.5

1 x 2 TM H,TM 0.8 h H,TM TM

1 H,TM

H,TM

= 0.68 1 1.1 iand Civil Engineering, mm CHALMERSk Architecture Master’s Thesis ACEX30

Fv, Rd, H, TM

1.5

h 4 4 TM bef, TM h N, TM bkz, Hx kv,hH, TM fkv, d = 9.53 kN F9v, Rd, H, TM 9 ef, TM N, TM 1z, H kv, H, TM fv, d n

= 9.53 kN

z, TM = 0.75 N, TM = 0.75 xTM = 50 mm 50 mm l z, 2TM xTM H, TM t h = h53.60 kcrh hN,TM bTM mm 2 l z, = N, TM TM N, TM= 0.75 B2 TM x = 50 mm = 0.75 x = 50 mm H, TM h TM 2 h TM TM TM 2 2 hT hT h N,TM 1 2 1 2 = 1.02 h h h

H, TM

bef, TM

H, TM

kz,H k k

v, H, TM

N,TM evalf min 1.0,

v, H, TM

H, TM

h TM

= 0.75 k k

= 0.68 TM = 0.68

H,TM

1.1 i

xTM

1.5

= 50 mm

1.1 i 1.5 k 1 mm 1.5 mm 1.1 hi 1.5 n h 1.1 i TM 1 k mm 1 TM mm h x n 1 hxTM TM TM 2 2 1 TM 2 0.8 x H,TM H,TM 0.8 hH,TM x 1 H,TM h H,TM TM TM 2 H,TM 1 2 2 TM 0.8 h 0.8 H,TM H,TM H,TM H,TM h h TM 1

evalf min 1.0, H,TMh

v, H, TM TM

N,TM

k evalf min 1.0, v, H, TM hevalf evalf min 1.0, l z, TM N, TMmink 1.0, v, H, TM

= 0.68 4 = k0.681

1.1 i

H,TM

1.5

H,TM

2 H,TM

mm 4 k n Fv, Rd, H, TM F 9 bef, TM h N, TM f = 9.53 kN hb z, H kv, h f = 9.53 kN H, TM k v, d k TM v,4 Rd, H, TM 9 ef,4 TM N, TM z, H v, H, TM v, d Fv, Rd, H, TM b h k k f = 9.53 kN F b h k k f x 9 ef, v,TM H, TM 1v, d z, H v, H, TM v, d = 9.53 kN v, TM Rd, H, N, TMTM z, 9 H ef, N, TM TM h

H,TM

= 0.68

0.8

RD1

H,TM

TM1

RD1

RD1 RD1 4 Fv, Rd, H 3.34 kN 3.34 kN f FF b h k k Rd,H,HTM v,v,Rd, 9 ef, TM N, TM z, H v, H, TM v, d Fv, Rd, H 3.34 kN Fv, Rd, H 3.34 kN VEd, H 0.25 kN VEd, H 0.25 kN VEd, H 0.25 kN V 0.25 kN Ed, H Utilization ratioUtilization 7.5 % ratio 7.5 % RD1 Utilization ratio 7.5 % Utilization ratio 7.5 % OK! OK! Fv, Rd,mode 3.34 kN OK! Failure loads OK! H 1, Horizontal Failure mode 1, Horizontal loads

TM1 9.53 kN = 9.53 kN 9.53 kN 0.25 kN 0.25 kN 2.6 % TM1 2.6 % OK!

9.53 kN OK!

TM1 TM1 9.53 kN 9.53 kN 0.25 kN 0.25 kN 2.6 % 2.6 % OK! OK!

Failure mode 1, Horizontal loads mode 1, Horizontal loads Failure VEd, H 0.25 kNgives a capacity that is 0.25 kN the As the reduced As cross-section well above the reducedstill cross-section still gives a capacity that is

dimensioning well above the load dimensioning load no verification of the reinforcement layer is needed. no verification of the reinforcement layer is needed. As the reduced cross-section still a capacity still that gives is well above thethat dimensioning load As the reduced cross-section a capacity is well above the dimensioning load Utilization ratio 7.5 % gives 2.6 % no verification of the reinforcement layer isreinforcement needed. no verification of thethe layer is needed. 4.1.2 Compression failure at the bottom of notch 4.1.2 Compression failure at the bottom of the notch OK! OK! 4.1.2 Compression the bottom offailure the notch Verified accordingfailure to4.1.2 eq. at (13). Compression at the bottom of the notch

Failure mode 1, Horizontal loads

4.1.2.1 Vertical loads

As the reduced cross-section still gives a capacity that is well above the dimensioning load l z,verification minofl z,the 30mm, 2 l z, RD = 130 mm (The same for all three no is needed. ef, V RDreinforcement layer connections) 4.1.2 Compression failure at the bottom of the notch VEd, V 1 = 0.1918 MPa c, 90, d, V, RD k bN, RD l z, ef, V c, 90 c, 90, d, V, TM

c, 90, d, V, N

VEd,V

h T l z, ef, V

kc,90

VEd,V

1 kc,90

bN, N l z, ef, V

= 0.0711 MPa = 0.0356 MPa

RD1

TM1

c, 90, d, V

0.19 MPa

0.07 MPa

0.04 MPa

fc, 90, d

1.73 MPa

11 %

OK!

Utilization ratio

1.73 MPa

2% OK!

Failure mode 2, Vertical loads

4.1.2.2 Horizontal loadsArchitecture and Civil Engineering, Master’s Thesis ACEX30 CHALMERS This failure mode is not relevant in this direction as not all layers have fibres perendicular

bN, RD l z, ef, V

N, RD z, ef, V

VEd,V

c, 90, d, V, TM

VEd,V

Utilization ratio

Failure mode 2, Vertical loads

TM1

kc,90

Ed,V = 0.0356 MPa

c,kc,90 90, d, V, N

bN, N l z, ef, V

RD1

fc, 90, d

V = 0.0711 MPaEd,V kc,c,90 90, d, V, TM h T l z, ef, V

h T l z, ef, V

c, 90, d, V, N

c, 90, d, V

bN, N l z, ef, V N2

RD1

c, 90

= 0.0711 MPa

1 kc,90

= 0.0356 MPa

TM1

0.19 MPa

0.07 MPa 0.19 MPa

0.04 MPa 0.07 MPa

0.04 MPa

1.73 MPa fc, 90, d

1.73 MPa 1.73 MPa

1.73 MPa

c, 90, d, V

11 % Utilization ratio

OK!

11 %

OK!

Failure mode 2, Vertical loads

4.1.2.2 Horizontal loads

4.1.2.2 Horizontal loads This failure mode is not relevant in this direction as not all layers have fibres perendicular This failure mode is not relevant in this direction as not all layers have fibres perendicular to the load. to the load. 4.1.3 Rolling shear failure of the notch (only for CLT) 4.1.3 Rolling shear failure of the notch (only for CLT) Verified according to eq. (14) for the vertical force case. For the horizontal force the Verified according to eq. (14) for the vertical force case. For the horizontal force the orientation of the layers doesn't give a rolling shear case. orientation of the layers doesn't give a rolling shear case. The dovetail is approximated as shown in Chapter 6.1.4.1. in the general calculation The dovetail is approximated as shown in Chapter 6.1.4.1. in the general calculation method. method. 4.1.3.1 Calculation of the first moment of inertia for rolling shear 4.1.3.1 Calculation of the first moment of inertia for rolling shear RD1 RD1 tR, 1, RD 13.2 mm

tB1 : tR, 1, RD

a S, 1, RD

tR, 2, RD t : t h t tB2 = tR, 1, RDB2 tB1 : R, 3, RD tR, 2, RDN, RDtB2 : B1 tR, h N, RD tB1 3, RD 13.2 mm tR, 2, RD tR, 1, RD tR, 2, RD = 20 mm a S, t 2, RD = R, 1,mm RD 2 a = 60 mm a S, 1, RD 20 t 2 S, 2, RD R, 1, RD 2 tR, 3, RD tR, 3, RD tR, 2, RD = 86.60 mm a S, 3, RD 2 tR, 1, RD tR, 2, RD = 86.60 mm 2

a S, 3, RD

tR, 1, RD

bR, 3, RD

bB1, RD

bR, 2, RD

evalf bR, 3, RD

bR, 1, RD

evalf bR, 2, RD

z0, RD t t

= 39.78 bmm R, 3, RD

R, 1, RD R, 1, RD

R, 3, RD

S, 3, RD

RD tR, 3, RD 2 = 42.56 mm evalf bR, 3, RD tan 2

tan bR, 2, RD2 tan

tR, 2, RD 2

90, C14

S, 1, RD

0, C24

b E

0,C24

tR, 3, RD 2

= = 50.96 mm t

R, 2, RD R, 2, RD

90,C14

R, 1, RD R, 1, RD

R, 3, RD

= 39.78 mm

bB1, RD

S, 2, RD

R, 2, RD R, 2, RD

R, 3, RD

LII

= 34.00 mm

The rolling shear layer is only to one side of the centre of gravity. zB, 2, RD zT, 2, RD

= 6.00 mm

tR, 1, RD

z0, RD

zB, 2, RD

tR, 2, RD

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

= 46.00 mm

= 42.56 mm

tB2

= 60 mm

bR, 1, RD z0, RD t t

evalf bR, 2, RD b

R, 1, RD R, 1, RD

R, 3, RD

tan

90, C14

S, 1, RD

0, C24

S, 3, RD

tR, 2, RD 2

b E

R, 1, RD R, 1, RD

R, 2, RD R, 2, RD

90,C14

= = 50.96 mm

0,C24

S, 2, RD

R, 3, RD

R, 2, RD R, 2, RD

R, 3, RD

= 34.00 mm The rolling shear layer is only to one side of the centre of gravity.

= 6.00 mm

zB, 2, RD

tR, 1, RD

z0, RD

zT, 2, RD

zB, 2, RD

tR, 2, RD

E 90,C14

SR, x, RD

= 46.00 mm zT, 2, RD2

bR, 2, RD

E 0,C24

zB, 2, RD2

= 925.31 mm 3

TM1 Symmetrical cross-section where all layers have the same width and height hence: bN, TM

z0, TM

= 60 mm

E 90,C14

SR, x, TM

E 0,C24

zB, TM

zT, TM 2 2

0mm :

zB, TM 2

zT, TM

tB2 2

= 20 mm

= 627.27 mm 3

N2 All layers have the same width. t2, N a S, 2, N

20mm : t2, N 2

t3, N = 10 mm

E 90, C14 z0, N

E 0, C24

z0, N

zT, N

z0, N

SR, x, N

a S, 3, N

bN, N t2, N a S, 2, N E 90,C14 E 0,C24

zB, N

tB3 :

t2, N

bN,N t2,N

t2, N

t3, N 2

= 40 mm

bN, N t3, N a S, 3, N bN,N t3,N

= 39.69 mm

= 19.69 mm

= = 39.69 mm

E 90,C14 E 0,C24

bN, N

zT, N 2 2

zB, N 2 2

LIII

= 3724.70 mm 3

4.1.3.2 Calculation of the second moment of inertia for rolling shear RD1 a I, 1, RD

z0, RD

a S, 1, RD

= 14.00 mm

a I, 2, RD

z0, RD

a S, 2, RD

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

bN, N t2, N a S, 2, N

E 0, C24

z0, N

E 90,C14

bN,N t2,N

E 0,C24

zB, N

z0, N

zT, N

z0, N

bN,N t3,N

= 39.69 mm

= 19.69 mm

t2, N

= = 39.69 mm

E 90,C14

SR, x, N

bN, N t3, N a S, 3, N

zT, N 2

bN, N

E 0,C24

zB, N 2

= 3724.70 mm 3

4.1.3.2 Calculation of the second moment of inertia for rolling shear RD1 a I, 1, RD

z0, RD

= 14.00 mm

a S, 1, RD

a I, 2, RD

z0, RD

a S, 2, RD

26.00 mm a I, 3, RD I x, net, RD

z0, RD

= 52.60 mm

a S, 3, RD

3 t R, 1, RD R, 1, RD

3 t R, 2, RD R, 2, RD

90,C14

E b

R, 1, RD R, 1, RD

0,C24

3 t R, 3, RD R, 3, RD

R, 3, RD R, 3, RD

= 1548100.10 mm 4

2 I, 1, RD

R, 2, RD R, 2, RD

2 I, 2, RD

2 I, 3, RD

TM1 a I, 1, TM

40mm :

a I, 2, TM

a I, 3, TM

a I, 1, TM

= 40.00 mm

h t T

I x, net, TM h t T

0mm :

h t

I, 1, TM

90,C14

0,C24

h t

a I, 2, N

z0, N

a S, 2, N

= 29.69 mm

a I, 3, N

z0,N

a S, 3, N

= 0.31 mm

= 20816727.27 mm 4

I, 3, TM

h t T

h t

I, 2, TM

90,C14

I x, net, N a

0,C24

I, 3, N

N, N 2, N

I, 2, N

N, N 3, N

= 1715922.71 mm 4 4.1.3.3 Calculation of the rolling shear stress LIV

RD1 bef, V, RD Rd, V, RD

TM1

kcr bB1, RD I

= 26.65 mm

VEd, V SR, x, RD b

= 0.033 MPa

N, N 3, N

x, net, RD ef, V, RD and Civil Engineering, Master’s Thesis ACEX30 CHALMERS Architecture

a I, 3, N

z0,N

a S, 3, N

90,C14

I x, net, N

0,C24

I, 3, N

N, N 2, N

= 0.31 mm t

N, N 2, N

I, 2, N

N, N 3, N

= 1715922.71 mm 4 4.1.3.3 Calculation of the rolling shear stress RD1 bef, V, RD

kcr bB1, RD

= 26.65 mm

VEd, V SR, x, RD

Rd, V, RD

I x, net, RD bef, V, RD

= 0.033 MPa

TM1 bef, V, TM

kcr h T

= 100.50 mm

VEd, V SR, x, TM

Rd, V, TM

I x, net, TM bef, V, TM

= 0.004 MPa

N2 kcr bN, N

bef, V, N

= 201.00 mm

VEd,V SR, x, N

Rd,V,TM

I x, net, N bef, V, N

f Rv, d

Utilization ratio

ailure mode 3, Vertical loads

= 0.022 MPa I x, net, N bef, V, N = 0.022 MPa

Rd, V MPa 0.033

RD1 TM1 TM1 N2 0.033 MPa 0.004 MPa 0.004 MPa 0.022 MPa

fRv, d 1.92 MPa

1.92 MPa 1.92 MPa

RD1 Rd, V

VEd,V SR, x, N

Rd,V,TM

Utilization ratio 1.7 %

0.2 %

OK!

Failure mode 3, Vertical loads

4.2 Header member

OK!

N2 0.022 MPa

1.92 MPa 1.92 MPa

1.7 %

1.1 %

OK!

1.92 MPa

0.2 %

1.1 %

OK!

4.2 Header member

For the header the grain direction is, for simplification, assumed to be orthogonal to the For the header the grain direction is,with for simplification, assumed to be orthogonalgrain to the loads the two outer layers having a horizontal direction and the centre layer a loads with the two outer layersvertical havinggrain a horizontal grain direction and the centre layer a direction. vertical grain direction. 4.2.1 Failure in shear and tension perpendicular to the grain 4.2.1 Failure in shear and tension perpendicular to the grain 4.2.1.1 Vertical loads 4.2.1.1 Vertical loads The width is modifies to only include the layers with grain parallel to the crack The width is modifies to onlypropagation include the according layers withtograin parallel to theofcrack Chapter 4.2.2.1. the general calculation method. propagation according to Chapter 4.2.2.1. of the general calculation method. Verified according to eq. (18). Verified according to eq. (18). bH, ef d, V, RD

d, V, TM

k cr t H1

t H3

H, ef mm cr = 40.20

tH1 LV

tH3

= 40.20 mm

VEd, V 3 V Ed, V 3 d, V, RD 4 b h = 0.18 MPa H, ef H, u, RD 4 b h H, ef H, u, RD

= 0.18 MPa

VEd, V 3 VEd, V 3 = 0.22 MPa d, V, TM 4 b h Master’s Thesis ACEX30 CHALMERS Architecture and Civil MPa Engineering, = 0.22 H, ef H, u, TM 4 b h H, ef

H, u, TM

VEd, V

kcr

tH1

d, V, RD

3 4

d, V, TM

3 4 3 4

d, V, N

fv, d

Utilization ratio

ailure mode 4, Vertical loads

tH1

= 40.20 mm

tH3

VEd, V

d, V, RD

VEd, V 3 4 b h 0.18 MPa H, ef H, u, RD

= 0.18 MPa

VEd, V

d, V, TM

VEd, V 3 4 b h 0.22 MPa H, ef H, u, TM

= 0.22 MPa

bH, ef h H, u, RD

bH, ef h H, u, TM VEd, V

bH, ef h H, u, N

VEd, V 3 4 b h 0.22 MPa H, ef H, u, N

d, V, N

= 0.22 MPa

Rd, VMPa 0.18

RD1 TM1 0.18 MPa 0.22 MPa

0.22 MPa 0.22 MPa

0.22 MPa

fv, d 2.56 MPa

2.56 MPa 2.56 MPa

2.56 MPa

RD1 Rd, V

H, ef mm cr = 40.20

tH3

Utilization ratio 7.3 %

8.6 %

OK!

7.3 %

Failure mode 4, Vertical loads

8.6 %

OK!

TM1

8.6 %

OK!

As the reduced cross-section still gives stresses much lower than what the material can As the reduced cross-section still gives stresses much lower thanthe what the materiallayer. can handle it is not necessary to verify reinforcement handle it is not necessary to verify the reinforcement layer. 4.2.1.1 Horizontal loads 4.2.1.1 Horizontal loads With regards to the horizontal loads the width from the edge is so far and the reinforcement layers are more than the layers for the crack propagation that this failure is not deemed necessary. 4.2.2 Failure in compression perpendicular to the grain Since not all layers have grain perpendicular to the load in either the case of the vertical or horizontal load this is not a relevant failure mode for the railing. 4.2.3 Failure in compression perpendicular to the grain due to an axial compression force on the joist This is verified according to eq. (19) AB

300mm 120mm

= 36000 mm 2

AN, RD

bN, RD h N, RD :

AB, ef, RD

AN, RD

= 30818.08 mm 2

AN, TM

bN, TM h T :

AB, ef, TM

AN, TM

= 18000 mm 2

AN, N

LVI

bN, N h N, N :

c, 90, d, Q, RD

AB, ef, N

QEd

AB, ef, RD kc, 90

AN, N

= 18000 mm 2

= 0.003 MPa

CHALMERS Q Architecture and Civil Engineering, Master’s Thesis ACEX30

c, 90, d, Q, TM

AB, ef, TM

1 kc, 90

= 0.006 MPa

4.2.3 Failure to the grain 4.2.3 Failure in compression perpendicular to in thecompression grain due toperpendicular an axial compression forcedue on to thean axial compression force on the joist joist This is verified according to eq. (19) This is verified according to eq. (19) AB

AN, RD

bN, RD h N, RD :

AN, RD

h A: AbN, RDRD N, RD B, ef, B

AN, RD

A2 = A30818.08 B, ef, RD mmB

AN, RD

= 30818.08 mm 2

AN, TM

bN, TM h T :

AN, TM

h :A AbB, N,ef, TM TM T B

AN, TM

= A18000 mm 2 AB B, ef, TM

AN, TM

= 18000 mm 2

AN, N

c, 90, d, Q, TM

c, 90, d, Q, N

QEd

AB, ef, N

Utilization ratio

ailure mode 6, Axial load

B, ef, N

AN, N

QEd

2 A AB, ef,mm = 18000 N B

1 MPa AB, ef, RD kc, 90

= 0.003 MPa

QEd MPa AB, ef, TM

= 0.006 MPa

c, 90, d, Q, TM kc, 90 = 0.006

AB, ef, TM QEd

bN,A N h N, N : A

c, 90, d, Q, RD kc, 90 = 0.003

AB, ef, RD

c, 90, d, Q, N kc, 90 = 0.006

QEd MPaA B, ef, N

TM1 RD1

RD1

fc, 90, d

AN, N

bN, N h N, N :

c, 90, d, Q, RD

c, 90, d, Q

2 AB mm 300mm 120mm = 36000 mm 2 = 36000

300mm 120mm

kc, 90

1 kc, 90

TM1

0.003 MPa 0.006 MPa 0.006 MPa 0.006 MPa

0.006 MPa

fc, 90,MPa 1.73 d

1.73 MPa 1.73 MPa

1.73 MPa

0.4 % 0.2 %

0.4 % 0.4 %

0.4 %

OK!

= 18000 mm 2

= 0.006 MPa

0.003 c, 90, MPa d, Q

Utilization 0.2 % ratio

AN, N

Failure mode 6, Axial load

4.2.4 Failure in tension 4.2.4 Failure in tension perpendicular to the grain due toperpendicular an axial forcetoonthe thegrain joist due to an axial force on the joist failure modejoint is only thecase dovetail but not in this case since the dovetail This failure mode is only validThis for the dovetail but valid not inforthis sincejoint the dovetail the full cross-section of the header. protrudes the full cross-sectionprotrudes of the header.

LVII

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

APPENDIX C

Appendix C consists of stress graphs for all FE models from Abaqus.

LVIII

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

APPENDIX C1 BENDING STRESS FOR C1 - PD1, PD2, P1 & P2, LC1 z

Bending stress, PD1

[MPa]

Stress

3 1 -1 0

250

500

750

1000

-3 -5 -7

Length [mm] Tf avg. Figure 1. Paths to consider bending stress for PD1.

Bf avg.

Figure 2. Bending stress, tf and bf for PD1.

Bending stress, PD2

Stress

[MPa]

3 1 -1 0

250

500

750

1000

-3 -5 -7

Length [mm] Tf avg. Figure 3. Paths to consider bending stress for PD2.

Bf avg.

Figure 4. Bending stress, tf and bf for PD2.

Bending stress, P1

Stress

[MPa]

3 1 -1 0

250

500

750

1000

-3 -5 -7

Length [mm] Tf avg. Figure 5. Paths to consider bending stress for P1.

Figure 6. Bending stress, tf and bf for P1.

LIX

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

Bf avg.

Stress

[MPa]

Bending stress, P2

1 -1 0

250

500

750

1000

-3 -5 -7

Length [mm] Tf avg. Figure 7. Paths to consider bending stress for P2.

Figure 8. Bending stress, tf and bf for P2.

BENDING STRESS FOR C1 - P1 & P2, LC2 7

Bf avg.

Bending stress, P1, LC2

[MPa]

Stress

3 1 -1 0

250

500

750

1000

-3 -5 -7

Length [mm] Tf avg.

Figure 10. Bending stress, tf and bf for P1, LC2.

Figure 9. Paths to consider bending stress for P1, LC2.

Bf avg.

Bending stress, P2, LC2

Stress

[MPa]

3 1 -1 0

250

500

750

1000

-3 -5 -7

Length [mm] Tf avg.

Bf avg.

Figure 12. Bending stress, tf and bf for P2, LC2.

Figure 11. Paths to consider bending stress for P2, LC2.

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

Length

[mm] 90

0 1 0,5

Stress 0

[MPa]

-0,5

SHEAR, COMPRESSION AND TENSION STRESS FOR C1 - PD1, PD2, P1, P2 & LC1 -1

Tension & Comp. stress, v.1

y x

x [mm] 90 0

-0,05

-0,15

Stress Avg. x z-plane

-0,25

[MPa]

Avg. yz-pla ne

70 60 40

Avg. S 33

[mm]

Length

Shear stress, avg, v.2 Tension & Compression stress, PD2 90 Tension & Comp. stress, avg. v.2

Length

C beam C int

Tension & Comp. stress, avg v.1

-1

Tension & Compression stress, PD1 Tension & Comp. stress, avg v.1 90

50 30 20 10

0 1

0,5

Stress

[MPa]

-0,5

-1

Avg. S 33 -ZZ

0,5

Stress

[MPa]

-0,5

-1

33 Avg. S -ZZ

Figure 14. Average shear, tension and compression stress for PD1.

Figure 16. Average shear, tension and compression stress for PD2.

LXI

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

Tension & Comp. stress, avg. v.2

-1

Shear stress, avg, v.1

Shear stress, avg, v.2

Avg. yz-pla ne Avg. x z-plane

[mm] Length

[mm]

Length

Avg. yz-pla ne

[MPa]

-0,5

Stress Avg. x z-plane

-0,25

[MPa]

-0,5

[MPa]

Stress

Shear stress, PD2

Length 20

-0,25

0,5

Stress

-0,15

0 1

-0,05

90 70

-0,15

[mm]

-0,25 0,5

0 1

Length

[MPa]

-0,15

[MPa]

Stress

-0,05

Figure 15. Paths to consider shear, tension and compression stress for PD2.

Shear stress, PD1

Figure 13. Paths to consider shear, tension and compression stress for PD1. CN C int C beam NC

[mm]

Avg. yz-pla ne

Length

Avg. x z-plane

Length

Shear stress, avg, v.1

Length

[mm]

-0,05

Stress

Figure 17 Paths to consider shear, tension and compression stress for P1.

Shear stress, avg v.4, LC1

[mm]

Avg. x z-plane

20 10

0,5

Stress

-0,15

-0,2

[MPa]

Tens. & Comp. stress, avg. v.4 LC1

[mm] Length

0,25

Stress

0 -0,25 0,5 -0,5 0,25

[MPa] Avg. S -ZZ 33

Stress

-0,4

0,5 -0,5

Shear stress, avg v.4, LC2

Stress

0,25

Stress

0,25

LXII

Avg. yz-pla ne

Avg. yz-pla ne Avg. x z-plane

Avg. S 33

[mm]

Length

Tens. & Comp. stress, avg v.3 LC1

Length

[mm] Length

Shear stress, avg, v.3 LC1

[mm] Length

[mm]

[mm] 50

Length

-0,3

0 0,5

Shear stress, avg, v.3 LC2

Figure 20. Average shear, tension and compression stress for P2. Length [mm]

Avg. yz-pla ne

-0,05

Avg. yz-pla ne

[MPa]

Figure 18. Average shear, tension and compression stress Length [mm] for P1. Avg. x z-plane

[MPa]

-0,2

-0,25

Stress [MPa] 33 Avg. S -ZZ

-0,5

-0,25

Length 20

0,25

Stress

0,5

-0,25

10 -0,1

-0,15

[MPa]

Tension Compression stress, P2 Tens. Shear &&Comp. stress, stress, avgavg. v.4, v.4 LC1LC1

80 60

Stress

& Comp. stress, avg v.3 LC1

-0,05

60Tens.

-0,5

-0,15

Stress Avg. x z-plane

80 60

0 -0,05

Avg. yz-pla ne

[mm]

[MPa]

-0,25

[MPa]

-0,25

Tension & Compression stress, P1 Length

-0,5

Stress Avg. x z-plane

-0,15

-0,05

90 70

-0,25

Stress

[MPa]

0,25

0-0,25

-0,150,25

0,5

[mm]

[MPa] Stress

Length

0,5

Shear stress, avg, v.3 LC1

Length [mm] P2 Shear stress,

Stress

80 60

-0,05

-0,25

Shear stress, P1 90 70

-0,15

[MPa]

[mm] tension and Figure 19. Paths to Length consider shear, compression stress for P2.

[MPa]

-0,25

[MPa]

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

-0,25

[MPa]

-0,5

0 -0,25 0,5

Stress

0,25

-0,15

[MPa]

Length

[mm]

0,5

[MPa]

Avg. x z-plane

Avg. yz-pla ne

-0,25

Stress

0,25

Shear stress, avg v.4, LC1 -0,5

SHEAR, COMPRESSION TENSION Shear stress, avg,AND v.3 LC1 STRESS FOR C1 - P1 & P2 LC2 0 -0,25

[MPa]

Tens. & Comp. stress, avg. v.4 LC1 Length

-0,2

Stress

Avg. yz-pla ne

[mm]

-0,1

Avg. x z-plane

x Tens. & Comp. stress, avg v.3 LC1

-0,5

-0,3

[MPa]

Stress

-0,25

[MPa]

-0,5

0,5

0,25

-0,25

Stress [MPa] Avg. S -ZZ 33

-0,2

-0,3

-0,4

-0,5

[MPa] Avg. yz-pla ne

Tension & Compression stress, P2 LC2 90 70 60 40

Tens. & Comp. stress, avg v.4 LC2 Avg. S 33

50 30 20 10 0,25

Stress

-0,25

[MPa]

-0,5

-ZZ Avg. S 33

Figure 22. Average shear, tension and compression stress for P1, LC2.

Figure 24. Average shear, tension and compression stress for P2, LC2.

LXIII

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

Tens. & Comp. stress, avg v.4 LC2

Stress Avg. x z-plane

0,5

-0,5

Shear stress, avg v.4, LC2

Avg. yz-pla ne

Length [mm] Avg. x z-plane

Avg. x z-plane

Shear stress, avg, v.3 LC2

-0,1

[mm]

Tens. & Comp. stress, avg. v.3 LC2

Avg. S 33

[mm] Length

Length

[mm] Length

Shear stress, avg v.4, LC1

Avg. yz-pla ne

Avg. yz-pla ne Avg. x z-plane Avg. S 33

Tens. & Comp. stress, avg. v.4 LC1

[mm]

Tens. & Comp. stress, avg. v.3 LC2

[mm]

-0,5

Avg. yz-pla ne

Tension & Compression stress, P1 LC2 Shear stress, avg, v.3 LC2 90

[MPa]

-0,25

Stress Avg. x z-plane

[MPa]

-0,15

Length

-0,25-0,05

Avg. S 33

Shear stress, P2, LC2

Shear stress, avg v.4, LC2

Stress

0,25

-0,15

90 80 70

0,5

Length

-0,5

-0,4

Figure 23. Paths to consider shear, tension and compression stress for P2.

-0,25

Stress

[mm]

[MPa]

-0,05

Stress

Length

-0,25

-0,150

0,25

Avg. S 33 Shear stress, P1, LC2

[MPa]0,5

-0,05

Stress

Figure 21. PathsLength to consider [mm] shear, tension and compression stress for P1.

APPENDIX C2 BENDING STRESS FOR C2 - TM1, N1 & N2 z

Bending stress, TM1 3 2

[MPa]

y x

Stress

250

500

750

1000

-1 -2 -3

Length [mm] Tf avg. Figure 25. Paths to consider bending stress for TM1.

Bf avg.

Figure 26. Bending stress for tenon and mortise, top and bottom fibres, TM1.

Bending stress, N1 3

Stress

[MPa]

1 0

250

500

750

1000

-1 -2 -3

Length [mm] Tf edge avg. Bf edge avg. Figure 27. Paths to consider bending stress for N1.

Bending stress, N2 3

[MPa]

Figure 28. Bending stress for a top notch, top and bottom fibres, N1.

Stress

250

500

750

1000

-1 -2 -3

Length [mm] Tf edge avg. Figure 29. Paths to consider bending stress for N2.

Bf edge avg.

Figure 30. Bending stress for bottom notch, top and bottom fibres, N2.

LXIV

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

BENDING STRESS FOR C2 - RD1, RD2, RD3, RD4 & RD5 Bending stress, RD1 y

[MPa]

Stress

1 0

250

500

750

1000

-1 -2 -3

Length [mm] Tf avg.

Figure 32. Bending stress for dovetail joints, top and bottom fibres, RD1.

Figure 31. Paths to consider bending stress for RD1.

Bending stress, RD2

[MPa]

Bf avg.

Stress

1 0

250

500

750

1000

-1 -2 -3

Length [mm] Tf avg. Figure 33. Paths to consider bending for RD2.

Figure 34. Bending stress, tf and bf for RD2.

Bending stress, RD3

[MPa]

Bf avg.

Stress

1 0

250

500

750

1000

-1 -2 -3

Length [mm] Tf avg. Bf avg. Figure 35. Paths to consider bending for RD3.

Figure 36. Bending stress, tf and bf for RD3.

LXV

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

Bending stress, RD4

[MPa]

Stress

2 1 0

250

500

750

1000

-1 -2 -3

Length [mm] Tf avg. Figure 37. Paths to consider bending for RD4.

Figure 38. Bending stress, tf and bf for RD4.

Bending stress, RD5

[MPa]

Bf avg.

Stress

1 0

250

500

750

1000

-1 -2 -3

Length [mm] Tf avg.

Figure 39. Paths to consider bending for, RD5.

Bf avg.

Figure 40. Bending stress, tf and bf for RD5.

LXVI

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

SHEAR, COMPRESSION AND TENSION STRESS FOR C2 - TM1, N1, N2 & RD1

Avg yz-plane

20 0

120

100

-3

Avg. S 33

80 60 40

-5

100

-1

[MPa]

Shear stress, avg, TMv.2

120

Stress

Tension & Compression stress, N1

100

Tens. & Comp. stress, avg. TMv.1 Tension & Compression stress, TM1 Shear stress, avg. TMv.1 120

Avg. yz-pla ne

-0,1 -0,2 -0,4 -0,5 Length [mm] -0,3 Stress [MPa] Avg xz-plane Avg yz-plane

20 0

Stress

-1

[MPa]

-3

-5

-ZZ Avg. S 33

Stress

-1

[MPa]

-3

-5

Avg. S -ZZ 33

Figure 42. Average shear, tension and compression stress for TM1.

Figure 44. Average shear, tension and compression stress for N1. Tens. & comp. stress, avg. TMv.2

LXVII

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

Tens. & comp. stress, avg. TMv.2

-0,5

[MPa]

-0,4

-5

Stress Avg. x z-plane

-0,3

Avg. S 33

-0,2

[mm]

-0,1

Length

-3

Shear stress, avg, TMv.2

Tens. & Comp. stress, avg. TMv.1

-1

[MPa]

-0,5

1-0,5

Length

Avg xz-plane

[mm]

Avg. x z-plane

[mm]

100

-0,4

Stress

3-0,4

[mm]

120

-0,3

[MPa]

Length

-0,2

Stress

120

100

5-0,3

-0,2

Stress

Shear stress, N1 Shear stress, avg. TMv.1

-0,1

-0,1 100

Avg. yz-pla ne

Length Shear[mm] stress, TM1 120

Length

120

100

[mm] shear, tension and Figure 43. PathsLength to consider compression stress for N1.

120

100

Length [mm] Figure 41. Path planes to consider shear, tension and compression stress for TM1.

Length

-0,5

-0,1

Stress

-1 -1,5

[MPa]

-0,2

Stress

-0,3

[MPa]

-2

Figure 47. Paths to consider shear, tension and compression stress forLength RD1. [mm]

Avg. yz-pla ne

-5 20 0Tens.

Stress Avg. x z-plane

-0,3

[MPa]

-0,4

-0,5

Avg. yz-pla ne

Tens. & comp. stress, avg. TMv.3

-5

Tension & Compression stress, N2 120

Tens. & comp. stress, avg. TMv.3

[mm]

100 80 60 40 20 0

& Comp. stress, avg. DTv.1 -0,5

-1

Stress Avg. x z-plane

-1,5

-2

[MPa]

-2,5

Avg. yz-pla ne

Tension & Compression stress, RD1 120 100

Avg. S 33

-0,2

[mm]

-0,1

Length

-3

Shear stress, avg. DTv.1

Avg. x z-plane

[mm]

-3

Length

80 60 40 20 0

Stress

-1

[MPa]

-3

-5

33 Avg. S -ZZ

Stress

-1

[MPa]

-3

-5

-ZZ Avg. S 33

Figure 46. Average shear, tension and compression stress for N2.

Figure 48. Average shear, tension and compression stress for RD1.

LXVIII

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

Tens. & Comp. stress, avg. DTv.1

Shear stress, avg. TMv.3

-1

[MPa]

[mm]

1 100

-1

[MPa]

120

100

120

Stress

Shear stress, avg. DTv.1 Shear stress, RD1

120

100

0 3

-0,5 80

100

-2,55

-0,4

Shear stress, N2 Shear[mm] stress, avg. TMv.3 Length

120

Length

120

100

Figure 45. Paths to consider shear, tension and compression stress for N2.

Length

[mm]

120

100

[mm]

Length

SHEAR STRESS FOR C2 - RD1, RD2, RD3, RD4 & RD5 Length

[mm]

120 110 100 90 80 70 60 50 40 30 20 10 0

[mm]

-1

-0,2

-1,5

[MPa]

Stress

Figure 51. Paths to consider shear stress for RD3.

20 0 0

-0,1

-0,2

Stress Avg. x z-plane

-0,3

[MPa]

-0,4

[mm]

Shear stress, avg. DTv.3

-0,5

Avg. yz-pla ne

Figure 50. Average shear stress for RD2.

Shear stress, avg. DTv.3

Shear stress, RD3

120 110 100 90 80 70 60 50 40 30 20 10 0

Avg. x z-plane

Length

-0,5

Shear stress, avg. DTv.2

-2,5

-0,4 100 80

Avg. yz-pla ne

-2

-0,3

[MPa]

Shear stress, RD2

120

[mm]

Stress

-0,1

Figure 49. Paths to consider shear stress for RD2.

Length

-0,5

120

100

0 0

Length

-0,5

-1

Stress Avg. x z-plane

-1,5

-2

[MPa]

-2,5

Avg. yz-pla ne

Figure 52. Average shear stress for RD3.

LXIX

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

Length

[mm] 120

100

0 0 -0,5

Length

[mm] 120

100

-0,5

-1,5

[MPa]

-1

Stress

-1

Stress

-2

Figure 55. Paths to consider shear stress for RD5.

-2,5

-1,5

[MPa]

Figure 53. Paths to consider shear stress for RD4.

60 40 20 0

Shear stress, avg. DTv.5

100 80 60 40 20 0

-0,5

-1

Stress Avg. x z-plane

-1,5

-2

-2,5

[MPa] Avg. yz-pla ne

-0,5

-1

Stress

Avg. xz-plane

Figure 54. Average shear stress for RD4.

-1,5

[MPa]

-2

-2,5

Avg. yz-pla ne

Figure 56. Average shear stress for RD5.

LXX

CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30

Shear stress, avg. DTv.5

120

Avg. x z-plane

Shear stress, avg. DTv.4 Length [mm]

[mm] Length

Shear stress, RD5

-2,5

100

Avg. yz-pla ne

-2

Shear stress, RD4 Shear stress, avg. DTv.4

120