Form Finding of Grid Shells -
a Parametric Approach using Dynamic Relaxation Henrik Green & Daniel Lauri
PREFACE
This folder presentation is a summary of the Master thesis:
Form Finding of Grid Shells - a Parametric Approach using Dynamic Relaxation. The master thesis was written at the Department of Mechanics at the Royal Institue of Technology as part of the masters program Civil and Architectural Engineering in collaboration with structural engineering consultancy Tyréns. Examiner/Supervisor: Prof. Anders Eriksson Mechanics Department, KTH Co-supervisor: Karl Graah-Hagelbäck Head of Sructural Engineering Department, Tyréns AB. Authors: Henrik Green, henrik.green@tyrens.se Daniel Lauri, daniel.lauri@tyrens.se
CONTENTS Preface 2 Case Study 6 Reference Project 8 Structural Analysis 10 Form 12 Conclusions 15
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BACKGROUND
The rapid development of computational capacity in recent years has expanded the possibilities of digital modelling in architectural design. Parametric design has emerged from these possibilities with a capacity to generate complex geometries which call for advanced structural systems. Especially for form found structures, where the geometry is determined by structural mechanics, collaboration between architects and structural engineers is crucial in early design.
Rules
P
Constraints
D
A parametric approach to form finding of grid shells has been developed. The algorithm is based on dynamic relaxation with kinetic damping coupled with a structural evaluation by the finite element method. The algorithm is applied on steel and glass grid shells in general and in greater detail on two case studies.
D*
Parametric Design
D*
Computational Design
The relation of parametric and computational design adopted in this study is shown above. In the parametric part, the parameter set, P produces the design space, D by the rules formulated in the parametric model. The design space, D contains all the possible design outcomes from different parameter combinations. To arrive at a narrower design space, constraints are formulated according to structural mechanics and the design space D* is established. This subspace of D is ment to contain all the possible designs which fulfills the constraints. The constraints are:
Grasshopper plot of grid shell
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•
Static equilibrium
•
High membrane action, i.e. no bending
•
Sufficient member buckling resistance
METHOD
1 -1
1. Grasshopper
2. Initial geometry
3. Branch node data
4. Script
5. Form found grid shell
6. FEM verification
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CASE STUDY
To form find the grid shells, the method of dynamic relaxation is applied. The method mimics a physical haning chain model numerically where the structural shape is a result of the loads applied. The initial geometry is a 2D planar grid pattern. This grid pattern is subjected to a uniform load giving a 3D cupola shape. The form found shapes are evaluated based on structural performance to generate structurally efficient forms.
Process of form finding a grid shell with single boundary
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The process is similar in the case of an interior boundary. Once again the initial geometry is the planar grid, but dynamic relaxation is applied in two steps. First, a cone-like form is found under self stressed conditions. Then, the load is applied and the final shape is form found. With the interior boundary placed assymetrically, the result is a flowing shape where the structure height varies with the span. The largest height occurs at the longest span and it is evident that the structural shape is a result of form follows force.
Process of form finding grid shell with interior boundary
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REFERENCE PROJECT
Section of the winning proposal for Uppsala new town hall by Henning Larsen Architects (henninglarsen.com, 2017)
In 2016, Henning Larsen Architects and TyrĂŠns won the competition to design the new town hall in Uppsala, Sweden. The project is used as a reference project in the case study
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The winning proposal consists of an L-shaped building added to an existing L-shaped building. Together the two volumes form a closed quadratic courtyard. In the yard, an additional, inner, buidling volume is proposed. In the proposal the yard is covered by a glass roof supported by beams and tension cables below.
Section with design proposal of grid shell with interior boundary
In the design proposal with interior boundary resulting from the master thesis, the roof is supported on the inner buidling. The roof is self supporting and rests on the main building as well as on the one story lower inner building.
Perspective view of grid shell on building volumes
By supporting the roof on the inner building, an efficient structure can be achieved without increasing the height excessively compared to the winning proposal.
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STRUCTURAL ANALYSIS
Vertical displacements for single boundary grid shell, maximum 23 mm
Vertical displacements for grid shell with interior boundary, maximum 9 mm
The case study forms are verified in the commercial FEM software, Comsol Multiphysics. The figures show analysis results under ideal conditions used in form finding. These conditions are: •
Uniformly distributed load
•
All boundary nodes pinned
The low displacements and even distributions indicate effective membrane action. To the right, a plot of material stress shows a symmetric stress distribution where the maximum stress is well below the yield stress of steel. Von Mises stress for single boundary grid shell, maximum 107 MPa
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First buckling mode for single boundary grid shell, safety factor 1.15
Plot of critical load factor against number of pattern divisions, a load factor of 1
The structures are further analysed with regards to global stability and asymmetric loading. The top left picture illustrates the first buckling mode of the single boundary grid shell, given by linearized buckling. The plot at top right shows how the critical load factor goes down with increased pattern density. This reveals a shortcoming of the constraints in the algorithm and highlights the importance of studying stability at the stage of form finding. With a critical load factor lower than one, the structure collapses before the design load is reached. This is shown in the figure to the right. Collapse of grid shell with 40 pattern divisions for 74 percent of the design load
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FORM
Wa/Wtot
0.9976
Wa/Wtot
0.9963
Wa/Wtot
0.9964
Dmax [mm]
4.60
Dmax [mm]
18.74
Dmax [mm]
16.40
Wa/Wtot
0.9973
Wa/Wtot
0.9963
Wa/Wtot
0.9993
Dmax [mm]
16.84
Dmax [mm]
14.12
Dmax [mm]
6.75
Series of single boundary grid shells form found by the algorithm along with linear membrane action and vertical displacement
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Wa/Wtot
0.9985
Wa/Wtot
0.9990
Wa/Wtot
0.9992
Dmax [mm]
7.22
Dmax [mm]
4.97
Dmax [mm]
2.66
Wa/Wtot
0.9991
Wa/Wtot
0.9995
Wa/Wtot
0.9987
Dmax [mm]
2.72
Dmax [mm]
5.90
Dmax [mm]
4.36
Series of grid shells with interior boundary form found by the algorithm along with linear membrane action and vertical displacement
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CONCLUSIONS
Structurally meaningful geometry can be achieved by a parametric modelling approach where constraints limits the solution space even if multiple parameters are included. The direct link between computational procedures and software commonly used in architectural design enables successful collaboration between architect and structural engineer. The developed algorithm is shown to generate early design solutions of grid shells based on structural performance. In design of form found structures structural engineering is especially crucial and has the capacity to push the boundaries of form and expression.
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