Technical Studies- Scherk's Minimal Surface

| ANDREEA-LAURA NICA | DS10 | May 2016

TECHNICAL STUDIES IN PRACTICE | Scherk’s

Explorations | - Continuous minimal surfaces in the design of shell structures -

4ARC651

| TECHNICAL STUDIES IN PRACTICE | William McLean & Peter Silver

University of Westminster 4ARC651

| TECHNICAL STUDIES IN PRACTICE | William McLean & Peter Silver

w1560320 May 2016

| ANDREEA-LAURA NICA | DS10

CONTENTS

I

PROJECT DESCRIPTION

7

Brief (DS10) Future House Context Design Proposal Technical Proposal II

III

SCHERK’s MINIMAL SURFACE (small scale visual & experimental system diary)

10

Minimal Surfaces Frei Otto

11

Geometry generation Methods of Fabrication & Tests

12

DESIGN DEVELOPMENT (GPS Ghettos & Drone Houses) 26

A. Tower Construction Strategy Structural Strategy Structural Analysis Material Study: Ferro-cement Construction Options Precedent Study: Taichung Opera B. House Prototype

38

Structural Strategy Sketches / Precedents Material Study: Ultra-light materials Precedent Study: Spatium Gelattum IV

ENVIRONMENTAL STRATEGY

46

Climate Sun Path Analysis Wind Analysis Ventilation M&E Systems V

BIBLIOGRAPHY

53

Technical Proposal

Studio DS10 is passionated about natural and structural systems using physical modelling and parametric experiments tested with digital tools, for analysis, formal generation and fabrication.

The technical report provides the opportunity to consider how the building is designed, constructed and delivered. Reflecting upon the relationship to technology, the environmental and the profession, the following strategies have been developed:

Future Housing Context The project is placed in a dystopian world, where contamination land levels have reached their peak. New Flying house typologies form clusters of drone capsules made of custom novel building materials. The new Towers, ‘Google Ghettos’districts, represent landing platforms for the flying capsules.

Developed through a study in Minimal Surfaces, the the project offers mobile accommodation for the future dwelles.

Design Proposal The future relies on flying houses. Drones will replace the need of streets, elevators and cable cars as they will be a reliant tecnology for movement.

The future house relies on adaptability. The new proposed capsules will adapt to the users activities, need and budget by constantly changing functions and space organisation. Integrated smart technology will monitor internal systems, help harvest fog, grow its own garden and improve day light and ventilation. The new house proposal will represent a Plug-in modules that may be transported into the nearsest Tower to change, as well as allowing the users to chose the Tower which will serve their purpose (market/plaza/ office etc.).

MULTIPLE FACILITY SPACES (Office, market, school, garden)

The design of the Towers is driven by the structural and geometrical properties of the Scherk’s Minimal Surfaces.

‘Minimal Surfaces’ are the smallest surfaces and the minimal energy form within defined boundaries, and the surface tension is equal and uniform at any point. Therefore, these geometries have highly structural efficency, material distribution and overall area minimization.

This research investigates the application of Doubly periodic continuous minimal surfaces in the design of shell structures. It presents different formal outcomes derived from the implementation of a computational algorithm which generates Minimal surfaces having a Quadri-rectangular Tetrahedron as a kaleidoscopic cell, as well as derived from the inclusion of those preliminary results into a parametric system.

The research explored several ways of fabrication, both at small scale and larger scale. Using new methods, materials and tools of fabrication, new strategies can be developed to represent these geometries.

Material Strategy - Light-weight skins The design of the House is driven by the structural and properties of light weight materials. Each Living capsule is in-closed within an ETFE membrane.

The material strategy of the housing proposal looks at new light-weight materials such as Aerogel, aluminium foam mesh and latex syntehic skins.

HOUSING

This year DS10 expands on the HOUSING. Living in one of the most rapidly challenging periods of human history, the Anthropocene era, where humanity’s actions for the first time are actually affecting the finite resource of spaceshp earth. The main issues that the next generation will face , rising population, climate change, mass urbanisation, labour market shifts, soaring housing prices and worldwide immigration crisis are just a few examples.

Structural Strategy - Continious Shells

The dystopian view of the future is that conventional natural resources will slowly disminish and cellulose will become the base component for most of the man matter. For the built environment, cellulose mixed with water and salt will produce a structural material, having higher strenght properties than steel and being lighter.

Aerogel, a synthetic porous ultralight material will replace conventional transparent, translucent and infill wall components. It is based on silica which can be found in sand, the second most abundant mineral on the globe.

Environmental Strategy- Building performance Due to its geometry, the Tower will proof to have enhanced thermal and ventilation properties that guide the upper level wind flows. The environmental strategy will follow the climate range, solar analysis and wind prevailling.

ELECTRICITY / DATA RECEIVER

On project work, DS10 crafts solutions that interpret design visions which respond intelligently to its enironment and sit within the wider cultural and environmental context. The process finds its roots in the understaning of manufacturing and digital fabrication. This allows to select materials and processes effectively so innovative and intelligent design solutions emerge by using unexpected uncommon or adaptive techniques and smarter design methods.

COORDINATION SYSTEMS

Studio Brief (DS10)

Laura Nica | Technical Studies in Practice

7

SCHERK’S MINIMAL SURFACE SMALL SCALE EXPLORATION - Visual Diary, Research Fabrication and Experiments -

RESEARCH INTRODUCTION

MINIMAL SURFACES

FREI OTTO’S EXPERIMENTS

In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having a mean curvature of zero. ‘Minimal Surfaces’ are the smallest surfaces and the minimal energy form within defined boundaries, and the surface tension is equal and uniform at any point. Therefore, these geometries have highly structural efficency, material distribution and overall area minimization.

Frei Otto is a revolutionary architect and structural engineer. He is renowned for his development and use of ultra-modern and super-light tent-like structures, and for his innovative use of new materials.

The term ‘minimal surface’ is used because these surfaces originally arose as surfaces that minimized total surface area subject to some constraint. Physical models of area-minimizing minimal surfaces can be made by dipping a wire frame into a soap solution, forming a soap film, which is a minimal surface whose boundary is the wire frame.

However the term is used for more general surfaces that may self-intersect or do not have constraints.

Minimal surfaces can be defined in several equivalent ways in R3. The fact that they are equivalent serves to demonstrate how minimal surface theory lies at the crossroads of several mathematical disciplines, especially differential geometry, calculus of variations, potential theory, complex analysis and mathematical physics.

Scherk’s Minimal Surface- soap experiment

For Frei Otto, experimentation with models and maquettes was a fundamental part of his work. In 1961, he began to conduct a series of experiments with soap bubbles. His experiments centered on suspending soap film and dropping a looped string into it to form a perfect circle. By then trying to pull the string out a minimal surface was created.

Another experiment were the famous ‘Wool Experiments’ that started at the beginning of the 1990s. The experiments were influenced by Gaudi’s catenary chain models used to create the Sagrada Familia. These experiments were meant to create strategies for calculating two-dimensional city infrstructure, as well as three-dimensional canellous bone structures. These are considered analog computational and form finding.

Frei Otto with his model of an arctic city under an air conditioned container structure

Numerical Analysis in the Design Process of Lightweight Structures

10

APPROXIMATION OF MINIMAL SURFACES IN NATURE

BUILT MINIMAL SURFACES IN ARCHITECTURE

AA school-EmTech minimal surface through bending wood

Vlad Tenu - Minimal Surface Installations

Living creatures

Playground - Prefabricated Triply periodic surfaces

Degenerated skin cells

Microscopic Bone Structure

Frei Otto- Munich Olympic Stadium

Toyo Ito - Taichung Opera House (China)*

Playground Gyroid- San Francisco Exploratorium * Main Case study further explored in the report; Laura Nica | Technical Studies in Practice

11

SCHERK’S MINIMAL SURFACE

MATHEMATICAL FORMULA & PROPERTIES y

W²(∂)

W¹(∂)

2 A²∂

A¹∂

1 b∂ b∂ cos ∂ -3

-2

∂

-1

1

3

2

x

-1

A²∂

A³∂ -2 W³(∂)

W³(∂)

intersection with z=0 intersection with z=� asymptote planes A ∂i

-1

i

∞

0

Diagrams showing the transformation from singly periodic to doubly periodic.

-i

1

In mathematics, a Scherk surface (named after Heinrich Scherk in 1834) is an example of a minimal surface. A minimal surface is a surface that locally minimizes its area (or having a mean curvature of zero). The classical minimal surfaces of H.F. Scherk were initially an attempt to solve Gergonne’s problem, a boundary value problem in the cube. The term ‘minimal surface’ is used because these surfaces originally arose as surfaces that minimized total surface area subject to some constraint. Physical models of area-minimizing minimal surfaces can be made by dipping a wire frame into a soap solution, forming a soap film, which is a minimal surface whose boundary is the wire frame. However the term is used for more general surfaces that may self-intersect or do not have constraints. For a given constraint there may also exist several minimal surfaces with different areas (for example, minimal surface of revolution, Saddle Towers etc.).

Scherk’s minimal surface arises from the solution to a differential equation that describes a minimal monge patch (a patch that maps [u, v] to [u, v, f(u, v)]). The full surface is obtained by putting a large number the small units next to each other in a chessboard pattern. The plots were made by plotting the implicit definition of the surface.

An implicit formula for the Scherk tower is:

Scherk’s second surface can be written parametrically as:

sin(x) · sin(z) = sin(y),

x = ln((1+r²+2rcosθ)/(1+r²-2rcosθ)) | y = ((1+r²-2rsinθ)/(1+r²+2rsinθ)) | z = 2tan-1[(2r²sin(2θ))/(r-1)]

where x, y and z denote the usual coordinates of R3.

for θ in [0,2), and r in (0,1).

* http://mathworld.wolfram.com/ScherksMinimalSurfaces.html; * https://en.wikipedia.org/wiki/Scherk_surface;

12

MATHEMATICAL TRANSLATION & REPRESENTATION

t=1

t = 1/2

t=0

t=1

t = 1/2

t=0

Diagrams of concentric circles under f and corresponding minimal surfaces for various values of t

Translation surface generated by two helices

Scherk Surface is one of minimal translation surfaces

Translation surface generated by two helices

The theory of translation surfaces is always one of interesting topics in Euclidean space. Translation surfaces have been onvestigated from various viewpoints by many differential geometers: L. Verstraelen, J.Walrave and S.Yaprak have investigated minimal translation surfaces in n-dimensional Euclidean spaces.

A surface that can be generated from two space curves by translating either one of them parallel to itself in such a way that each of its points describes a curve that is a translation of the other curve.

Gauss curvature of a translation surface generating by space curves in zero if and only if at least one of generator curves is an asiymptotic line of surface.

In general, Scherk Minimal surfaces can be represented through: a. Weiserstrass Representation (Every regular minimal surface in RÂł has a local isothermal parametric representation) b. Planar harmonic mappings

Laura Nica | Technical Studies in Practice

13

DIGITAL REPRESENTATION & MATHEMATICAL TRANSLATION Grasshopper definition of Scherk’s Minimal Surface - second surface

1. Define 2 range of domains with resolution set as nr. of steps. Evaluate with function of Scherk Surface.

5. Complete Scherk Surface

2. Build a mesh plane and deconstruct the mesh.

6. Copy, mirror and flip.

* definition taken from the Grasshopper Forum: http://www.grasshopper3d.com/forum/topics/singly-periodic-scherk-surface

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3. Reconstruct the mesh (points from flat place and function points)

7. Create a box and trip the edges

4. 3D proximity and join curves.

8. Create a box and trip the edges

EXPERIMENTS: ITERATIONS

Scherk's Minimal Surface Iterations (surpassing limits)

r: o to pi*2 rez.: 3 r2: 0 to 0.9

r: o to pi*2 rez.: 5 r2: 0 to 0.9

r: o to pi*2 rez.: 7 r2: 0 to 0.9

r: o to pi*1 rez.: 80 r2: 0 to 0.9

r: o to pi*2 rez.: 80 r2: 0 to 0.5

r: o to pi*2 rez.: 9 r2: 0 to 0.9

r: o to pi*5 rez.: 80 r2: 0 to 0.9

r: o to pi*2 rez.: 80 r2: 0 to 0.8

r: o to pi*15 rez.: 80 r2: 0 to 0.9

r: o to pi*9 rez.: 80 r2: 0 to 0.9

r: o to pi*2 rez.: 80 r2: 0 to 1

r: o to pi*2 rez.: 30 r2: 0 to 0.9

r: o to pi*2 rez.: 15 r2: 0 to 0.9

r: o to pi*2 rez.: 80 r2: 0 to 5

r: o to pi*2 rez.: 80 r2: 0 to 9

r: o to pi*2 rez.: 80 r2: 0 to 0.9

r: o to pi*30 rez.: 80 r2: 0 to 0.9

r: o to pi*2 rez.: 80 r2: 0 to 15

After building the digital representation of the model, several parameters have been changed to test the geometry, by surpassing the limits of the function.

The Vertices will proove to be very useful in building and fabrication of the future prototypes, representing the minimum substructure necessary for the shape to resist.

Laura Nica | Technical Studies in Practice

15

SCHERK'S MINIMAL SURFACE- MAIN ITERATION Transformation Diagrams of the possible geometrical transformations possible

1.

number of saddle branches - 2 number of holes/ stories - 1 strech x - 1 strech y - 1 strech z - 1

2.

number of saddle branches - 2 number of holes/ stories - 3 strech x - 1 strech y - 1 strech z - 1

3.

number of saddle branches - 2 number of holes/ stories - 3 turn around axis- 180ยบ strech x - 1 strech y - 1 strech z - 1

4.

number of saddle branches - 2 number of holes/ stories - 3 flange- 2.6 turn around axis- 180ยบ bend towards axis - 360ยบ strech x - 1 strech y - 1 strech z - 1

1. The main module of Scherk Second Surface, results from the basic parametric function.

16

2. By copying, mirror it and flipping it, the surface becomes a Saddle Tower.

3. By rotating the saddle branches around its own axis, the surface becomes a twisted Saddle Tower.

4. And finally if the ends of the Saddle

branches are connected by twisting 360ยบ, the desired surface closes.

DESIGN ITERATIONS

Scherk's Minimal Surface Multiple Shape and Iteration Options following the principles on the previous page

31 36 32 41

37

33

1 34

38

42 2

35 3

43 39

6 4 7

11

44

40

5

12

45

8

13

9 14

10 15

16 21 17

#- number of saddle branches n- number of holes/stories t- overall axial twist a- turn around axis b- bend towards axis x- stretch x y- stretch y z- stretch z

1

6

#- 2 n- 1 x- 1 y- 1 z- 1

2

#- 4 n- 2 x- 1 y- 1 z- 1

7

11 #- 2

n- 3 t- 15º x- 1 y- 1 z- 1

#- 2 n- 2 x- 1 y- 1 z- 1

3

#- 5 n- 2 x- 1 y- 1 z- 1

8

12 #- 2

n- 3 t- 30º x- 1 y- 1 z- 1

22

#- 2 n- 3 x- 1 y- 1 z- 1

4

#- 6 n- 2 x- 1 y- 1 z- 1

9

13 #- 2

n- 3 t- 45º x- 1 y- 1 z- 1

#- 2 n- 4 x- 1 y- 1 z- 1

5

#- 8 n- 3 x- 1 y- 1 z- 1

10 #- 2

14 #- 2

15 #- 2

n- 3 t- 90º x- 1 y- 1 z- 1

#- 3 n- 2 x- 1 y- 1 z- 1

n- 6 x- 1 y- 1 z- 1

n- 3 t- 180º x- 1 y- 1 z- 1

16

21

26

#- 2 n- 3 b- 30º x- 1 y- 1 z- 1

17

#- 2 n- 2 height- 1.5 flange- 2.9 x- 1.2 y- 1.3 z- 0.2

22

#- 6 n- 13 height- 1.5 flange- 2.6 t- 60º a- 30º b- 360º x- 1.1 y- 1.1 z- 0.9

27

#- 2 n- 3 b- 60º x- 1 y- 1 z- 1

18

#- 3 n- 2 height- 1.5 flange- 2.9 x- 1.2 y- 1.3 z- 0.2

23

#- 3 n- 13 height- 1.5 flange- 2.6 t- 360º a- 190º b- 360º x- 1.1 y- 1.1 z- 0.9

28

#- 2 n- 3 b- 90º x- 1 y- 1 z- 1

19 #- 2

#- 7 n- 5 height- 1.5 flange- 0.5 x- 1.2 y- 1.3 z- 0.2

24 #- 7

#- 2 n- 13 height- 1.5 flange- 2.6 t- 360º a- 190º b- 360º x- 1 y- 1 z- 1

29 #- 4

n- 3 b 180º x- 1 y- 1 z- 1 n- 5 t- 60º height- 1.5 flange- 2.9 x- 1.2 y- 0.9 z- 0.4 n- 5 height- 1.5 flange- 2.6 t- 360º a- 0º b- 360º x- 1.2 y- 4.1 z- 1

20

25

30

#- 2 n- 3 b- 360º x- 1 y- 1 z- 1 #- 4 n- 5 t- 60º height- 1.5 flange- 2.9 x- 1.2 y- 0.8 z- 0.4 #- 4 n- 16 height- 1.5 flange- 1.3 t- 90º a- 0º b- 360º x- 1.6 y- 3.9 z- 1.6

18 19

26 23 27

20 28 24

29

25 30

Scherk’s Minimal Surface Iteration Catalogue with main param changed - brief 1 Laura Nica | Technical Studies in Practice

17

METHODS OF FABRICATION & TESTS

FABRICATION 1- TENSILE MODEL (abandoned proccess*) Transformation Diagrams of the possible geometrical transformations possible

The second experiments with fabric were inteded to try and understand the construction of the Scherk Saddle Tower.

The initial experiments were inteded to facilitate the understanding of minimal surfaces and help decide on a type that would be further developed.

The concept and research on the burnt skin cells was continued by looking at surfaces that would imitate their behaviour.

The Schwarz minimal surfaces are periodic minimal surfaces originally described by Hermann Schwarz.

Schoen named this surface ‘diamond’ because it has two intertwined congruent labyrinths, each having the shape of an inflated tubular version of the diamond bond structure.

2 Edge simulations

By decomposing the geometry in grasshopper, the form was reduced to initial directory lines that could be inscribed into a paralelogram.

The main lines were represented by drawing with thread and then fabric was streched inbetween the main points to form the base. The model was not completed, but showed another fabrication method.

3 Edge simulations

The following tensile model is not a Scherk Surface, but a D-Schwrz minimal surface.

4 Edge simulations

* The process was eventually abandoned due to the amount of time that required to tight all the tensile fabric onto the right vertices of the geometry.

18

Basic directory lines inscribed into a paralelogram.

Main lines of the vertices that form the geometry.

FABRICATION 2- KERF CUTS ON WOODEN MATERIAL (abandoned proccess*) Scherk's Minimal Surface Multiple Shape and Iteration Options following the principles on the previous page

The second experiment was inteded to try and understand the construction of a minimal surface testing with wood bending, through kerf cutting plywood. Kerf Cutting Timber is the process by which kerf cut lines are used to program bending, stretching, and warping in hardwood timber.

By decomposing the geometry in Grasshopper, one of the surfaces of the Scherk geometry was unrolled and its form was reduced to initial directory lines that could mark the places where the cuts could be made. The spacing between the cuts was parametriclly programmed.

Unfortunatelly , this method of fabrication was abandoned due to the inability to discover and solve a precise rule of cutting the gaps in the material. Nevertheless, this represents a future challenge to be explored and tested.

110ยบ

app. 0.5 cm app. 1cm

app. 0.25 cm app. 0.5 cm

Digital Model Logic

* The process was eventually abandoned due to the lack of precise known cut in the flat geometry.

Laura Nica | Technical Studies in Practice

19

FABRICATION 3- UNROLLING SURFACES Methods of unrolling and laser cutting a Scherk Minimal Surface Scherk’s Surface can be adapted to several design possibilities, with multiple ways of fabrication.

The first fabrication exploration included a paper model. In order to produce and test the first paper model of the Scherk’s Surface, the ‘UnrollSrf’ command was used to flatten the Saddle Tower. By identifying the two longest edges and isolate the isocurves from the isocurves from the surface, I manage to create developable surfaces. The command works only for surfaces or polysurfaces with curvature in one direction to a planar surface. After lofting it into grasshopper, I managed to smash the surfaces and obtain a 2D file proper for laser cutting and fabrication.

By applying the same principle to any flat sheets of material (metal, polypropylene, plywood), we are able to transform the surface and its properties into something three dimensional as well as flexible.

4

In order to produce and test the first model of my Scherk Surface, I used the UnrollSrf command which flattens the Saddle Tower. By identifying the two longes edges and isolate the isocurves from the surface, I manage to create developble surfaces. The command works only for surfaces or polysurfaces with curvature in one direction to a planar surface. After lofting it into Grashopper, I managed to smash the surfaces and obtain a 2D file proper for laser cutting.

20

FABRICATION 4- CNC MACHINING MOULDS

FABRICATION 5 - INTERLOCKED SLICES A way of defining the mesh is to divide the model into a number of slicing counts and interlocks between them. Despite being a pretty quick formula, to solve the model, the process takes long time to fit in right order all the slices. The denser the slicing is, the higher resolution the geometry will have. x

1. Basic Scherk Surface

2. CNC fabrication

3. Framework fabrication

- Creating a Scherk Surface using the function in Grasshopper; - Combine and clean the mesh; - Weld the vertices; - Obtain the naked vertices; - Obtain the points; - Connect the points;

First method of fabrication would be CNC milling fabrication of the negative shape of the surface.

By extruding the lines created by the connections of the naked vertices, the result is a extruded framing with the main guides of the geometry.

Possible rib assembly of the Scherk Surface, with very dense interlocking several customised laser-cut layers.

2. CNC Milling Machine Fabrication Foam Model for a basic Scherk Surface.

Assembly rib 1

3. Possible framing of the Scherk Surface that could be used for mouldind and acsting with several material options

3. Cardboard model, main frame for a basic Scherk Surface/

Assembly rib 7

Assembly rib 32

Assembly rib 40

Possible rib assembly of the Scherk Surface, interlocking several customised laser-cut layers to form the general frame of the geometry.

Laura Nica | Technical Studies in Practice

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FABRICATION 6 - STACKED SLICED LAYERS Contouring layers of the Geometry, staked on top of each other

The Sliding Cave Exploded Axonometry of the Stacked Construction Sequence layers that form a Scherk’s Minimal - brief 2a surface

Initial mesh from the Scherk’s Iteration

Sliced Mesh

Wooden Stacked Layered Structure

Final Layers

Stacked layers of wood to for half of the geometry of a Scherk’s Minimal surface

Layers 14- 27

Layers 5-14

Layers 1-3

22

Exploded Axonometric View showing the construction sequence and the layer orders

Steps in Assembly of the Stacked layers after CNC fabrication

m

m

1m

0.63m

1m

CNC Machining

1m

1m

1m

CNC Machining CNC Machining 18 mm MDF sheet 18 mm MDF sheet 18 mm MDF sheet 2440 x 1220 mm bed size bed size mm bed size 2440 x 1220 £18 x 11 sheets = 2440 £198 x 1220 mm = x£198 £18 11 sheets = £198 app.: 8 h 30 min £18 x 11 sheets app.: 8 h 30 min app.: 8 h 30 min

1m

0.63m

0.75m

0.63m

0.75m

0.41m

0.70m 0.75m

0.70m

CNC Machining

0.70m

CNC Machining CNC Machining 9 mm MDF sheet 9 mm MDF sheet 9 mm MDF sheet 2440 x 1220 mm bed size 2440 x 1220 mm bed size mm bed size 2440 x 1220 £12 x 3 sheets = £36 £12 x 3 sheets£12 = £36 x 3 sheets = £36 app.: 2 h app.: 2 h app.: 2 h

0.41m

0.47m

0.41m

0.47m

0.49m 0.47m

0.49m

0.49m

Laser Cutting

Laser CuttingLaser Cutting 6 mm MDF sheet mm MDF sheet 6 mm MDF sheet 600 x 400 mm bed6 size 600 x 400 mm600 bedx size 400 mm bed size £1 x 33 sheets = £33 = x£33 33 sheets = £33 app.: 16 h 30 min£1 x 33 sheets£1 app.: 16 h 30 min app.: 16 h 30 min

Final Prototype Built (app. 0.7 x 0.7 x 0.4 m)

Several Options in reducing the material waste and cost

CNC fabrication

Final Prototype Built with its structural properties

Laura Nica | Technical Studies in Practice

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DESIGN DEVELOPMENT LARGE SCALE EXPLORATION

- Structural Strategy, Site Preparation, Bespoke components, Assembly -

STRUCTURAL & CONSTRUCTION STRATEGY

MAIN STRUCTURE - ‘Scherk’ Tower 1

EXCAVATION & PILE FOUNDATION

1. soil dug out and replaced with slurry; 2. reinforcing steel cage inserted into the trench; 3. concrete poured into trench (slurry pumped out); 4. completed foundation pile;

2

3

IN SITU RC SLAB

a. Existing Site (Enclosing the site & Demolition)

b. Excavation & Pile Foundation

c. In situ reinforced concrete (ground floor slab)

d. Main steel branches of the secondary structure

e. Main RC cylindrical Core (pre-cast concrete)

f. Temporary scaffolding for each module asembly

g. House steel structure connected to the secondary structure

h. Layers of wooden planks (part of the module), supported by the temporary work

CYLINDRICAL CORE

- interlocking pre-cast concrete parts -

Butt connection includes bottom segments of lower and upper columns vertically fixed on reinforced concrete framing axis with height fracture, and coverage placed between end faces of columns. Screw-threaded longitudinal reinforcement escapes pass through coverage.

4

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SCAFFOLD SUPPORTING MODULE

HOUSE ‘Scherk’ MODULES

Temporary scaffold location

SCREWED WOODEN PLANKS

HOUSE MODULE STRUCTURE

CENTRAL CORE

ASSEMBLY PROCESS Stacked Layers of Pre-cut Wooden Planks

The supporting system for the House prototype is a Scherk’s Minimal Surface geometry. This will be constructured using temporary scaffold to hold the assemby of the stacked wooden planks that form the main geometry. If residents desire, the exterior can be rendered using spray-concrete system, after extracting the vertices to form an intense mesh.

POSSIBLE MODULE CONNECTIONS

STEEL COLUMNS CONNECTION

The main structure of the Tower is composed by a main RC core which support transversal steel beams that hold the secondary structure. (House)

FERRO-CEMENT HOUSE MODULE

FOUNDATION

WOODEN HOUSE MODULE

STRUCTURAL STRATEGY

Columns fixed in RC curved mesh

PRE-STRESS TENSION ANCHOR (BRANCHING STRUCTURAL SYSTEM OF THE MESH)

Laura Nica | Technical Studies in Practice

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MAIN GEOMETRY ANALYSIS

- Scherk Minimal Surfaces typologies analysis using Scan & Solve Rhino plug-in* -

SCHERK’S MINIMAL SURFACE

SCHERK’S SADDLE TOWERS -TYPOLOGY 1-

SCHERK’S EXPANDED BRANCHES -TYPOLOGY 2-

LINEAR STRESS ANALYSIS

MAIN GEOMETRY / PRIMITIVE

-UNIT MODULE-

MAX- Fragile (dangerous for construction) MIN- Resistent (optimal under stress)

POSSIBLE DEFORMATIONS / FAILURES

Applied Load + Restrictions

* http://www.scan-and-solve.com/;

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Normal Gravity Load

SCHERK’S BRANCHES -TYPOLOGY 3-

FINAL SCHERK - TYPOLOGY 4-

FINAL TOWER- STRUCTURE ANALYSIS

- Scherk Minimal Surfaces analysis using Scan & Solve Rhino plug-in* -

MAIN GEOMETRY

LINEAR STRESS ANALYSIS

POSSIBLE DEFORMATIONS

Several loads were applied to the modules that form the final Scherk Tower. The restrictions and limits were formed by the ends of the geometry, and gravity forces were applied, As noticed from the diagrams above, the geometry becomes more stiff when bent towards the centre.

When stacked on top of each other, the modules become very rigid in the centre, leaving small vulnerabilities on the exterior rings.

MAIN GEOMETRY WITH FORCES AND LOADS APPLIED

1. Restrictions were implemented in the Reinforced Concrete; 2. Main loads were simulated (as in a house just landed)

NORMAL GRAVITY LOAD + BOUNDARY RESTRICTIONS

3. Resistant bent core modules; 4. Vulnerabilities on the lower part of the building;

GRAVITY + EXTRA LOAD (HOUSES) + BOUNDARY RESTRICTIONS

5. Main deformations are visible form the original geometry. When high load forces are applied, the central axis of the tower gets displaced, as well as small twists and bends appear in the exterior cantelivered rings.

Laura Nica | Technical Studies in Practice

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MATERIAL RESEARCH Fine Aggregate

FERRO - CEMENT

Reinforcing Mesh

(main material for Scherk Towers)

Skeletal Steel

Ferrocement or ferro-cement (also called thin-shell concrete or ferro-concrete) is a system of reinforced mortar or plaster (lime or cement, sand and water) applied over layer of metal mesh, woven expandedmetal or metal-fibers and closely spaced thin steel rods such as rebar, metal commonly used is iron or some type of steel. It is used to construct relatively thin, hard, strong surfaces and structures in many shapes such as hulls for boats, shell roofs, and water tanks. Ferrocement originated in the 1840s in France and is the origin of reinforced concrete. It has a wide range of other uses including sculpture and prefabricated building components. The term “ferrocement” has been applied by extension to other composite materials, including some containing no cement and no ferrous material.

Ferro-cement is a relatively new construction material consists of wire meshes and cement mortar. It was developed by P.L.Nervi, an Italian architect in 1940. Ferro cement is widely used due to the low self weight, lack of skilled workers, no need of framework etc. Quality of ferro-cement works are assured because the components are manufactured on machinery set up and execution time at work site is less.

Maintenance cost of ferro-cement is low. Ferro-cement construction has come into widespread use only in the last two decades.

What is Ferro-cement?

Typical cross section of ferro-cement structure.

• Highly versatile form of reinforced concrete.

• Its a type of thin reinforced concrete construction, in which large amount of small diameter wire meshes uniformly through out the cross section.

• Mesh may be metal or suitable material.

• Instead of concrete Portland cement mortar is used.

• Strength depends on two factors quality of sand/cement mortar mix and quantity of reinforcing materials used.

Advantages of Ferro-Cement:

Disadvantages of Ferro-Cement:

• Basic raw materials are readily available in most countries.

• Fabricated into any desired shape.

• Low labour skill required.

• Ease of construction, low weight and long lifetime.

• Low construction material cost.

• Better resistance against earthquake.

• Structures made of it can be punctured by collision with pointed objects.

• Corrosion of the reinforcing materials due to the incomplete coverage of metal by mortar.

• It is difficult to fasten to Ferro-cement with bolts, screws, welding and nail etc.

• Large no of labors required.

• Cost of semi-skilled and unskilled labors is high.

• Tying rods and mesh together is especially tedious and time consuming.

Constituent Materials: Cement Fine Aggregate Water Admixture Mortar Mix Reinforcing mesh Skeletal Steel Coating

* http://www.fao.org/docrep/003/v9468e/v9468e09.htm; (Construction of Ferro-cement Hull); * https://en.wikipedia.org/wiki/Ferrocement (Properties of Ferro-Cement);

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Cement

METHODS OF FABRICATION - MAIN TOWER TYPOLOGIES TYPOLOGY 1 - A, B, C, D, E -

TYPOLOGY 2 - F1, F2 -

TYPOLOGY 3 - G, H, I , J, K -

1. The geometry is generated as a developable surface.

1. The geometry obtained is a mesh.

2. Using the ‘UnrollSrf’ (fabrication method 3 from small scale tests) the Saddle Tower can be flatten into fabrication sheets.

2. Using digital computation, vertices can be extracted and transformed into steel trussts, which support the entire structure.

F2

A

2. Main steel frame can be designed, fabricated and assabled on site. Then fabric is woven onto the frame and concrete is then sprayed to create the shape.

3. Reinforced mortar or plaster (lime or cement, sand and water) is then applied over layer of metal mesh, to construct relatively thin, hard, strong surface, and concrete-render.

3. By applying the same principle to any flat sheets of material (metal, polypropylene, plywood), we are able to transform the surface and its properties into something three dimensional as well as flexible.

F2

1. The geometry obtained is a relaxed mesh. (simulation of a stretched fabric placed on a metal frame).

F1

F1

K

B

H

J

G

B C D E

F1 K F2 G H

I J

G ‘STRETCHED’ Laura Nica | Technical Studies in Practice

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FABRICATION OF TYPOLOGIES 1 & 3 (A, B, C, D, E & G, H, I, J , K) - Scherk Minimal Surfaces analysis reinforcing mesh and concrete sprayed -

1. The geometry is generated as a developable surface. any sheet material

2.

Using the ‘UnrollSrf’ (fabrication method 3 from small scale tests) the Saddle Tower can be flatten into fabrication sheets.

The majority of the main metal bars that form the shape are twisted in all 3 directions. A proposed way to fabricate these would be the use of a robotic arm and a programmed tortion machine.

3. By applying the same principle to any flat sheets of material (metal, polypropylene, plywood), we are able to transform the surface and its properties into something three dimensional as well as flexible.

main metal bars

1. The geometry obtained is a relaxed mesh. (simulation of a stretched fabric placed on a metal frame).

2. Main steel frame can be designed, fabricated and assabled on site. Then fabric is woven onto the frame and concrete is then sprayed to create the shape.

Main relaxed mesh on metal frame

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Main metal frame

Using spray-concrete speeds up the process of construction and it can be used both for rigidity formation and structural repair. Installed properly by experienced applicators, sprayed concrete provides designers with a cost effective and adaptable method to create concrete structures.

MODULES FOR TYPOLOGIES 3 ( G, H, I, J , K)

METAL FRAME

- Scherk Minimal Surfaces analysis reinforcing mesh and concrete sprayed -

RELAXED FABRIC

Perspective View - cubic polyhedron

Perspective View - relaxed mesh Monkey-Saddle M

Left-handed screw S

Right-handed screw Z

Scherk minimal surface (singly- periodic)

Scherk minimal surface two-fold rotational symmetry

Perspective View - relaxed mesh module development The edges of the mesh can be fixed upon a metal frame that can be modular and easy to assemble. In order to control the tension in the mesh, the vertices and points need to be extracted from the generated geometry and applied in real-life in the shape of fabric holes and stretching needles.

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GENERATION OF VERTICES AND TRUSS WALLS - Transformed simulation from Scherk’s geometry mesh to buildable walls -

1. The curved web plate is cut out from the steel plate with the method of laser-cutting.

2. Another plate acts as a flange is weld on the web plate, following the curve of web plate.

3. Each frame will be set up by following the curvature of building geometry.

4. The frames are connected with each other by the horizontal frame and the basic for is created.

5. The shotcrete is applied on the expand metal mesh installed in the both sides. This makes it possible to realize free-form surfaces without an installation of the complicated frameworks, Concrete-sprayed surface 2.

1.

SCHERK’S TOWER MODULE GEOMETRY (MESH)

Extracted vertices from the mesh geometry

CONSTRUCTION PROCESS The structural system is developed together with the construction method to realize the freeform geometry in rational and efficient manner. The freeform concrete surfaces are shotcrete (spray concrete). It can be shot horizontally or vertically. Rather than in constructing doubly curved formwork that is expensive and time consuming on site, the temporary structure in the void created faceted surfaces that best-fit the finished surface.

Between the temporary steel work, the expanded metal mesh spans between the temporary steel work to act as faceted framework. 150 mm thick concrete can be shot at one time. The surface layer of 25 mm is shot separately without large aggregate to achive smooth surface finish. Concrete thickness varies between 200 mm and 350 mm.

* http://www.designboom.com/architecture/toyo-ito-taichung-metropolitan-opera-house-taiwan-21-08-2014/ (Design & Construction Drawings for Catenoid Construction)

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3.

4.

5.

GENERATION OF VOIDS & INVERTED VOLUMES (typology 3) - Transformed simulation from Scherk’s geometry mesh to CNC Mould Fabrication -

8.

7.

9.

6.

6. The desired geometry is obtained after relaxing the mesh on the metal frame.

10.

7. A standard module is extracted and translated into a cube.

8. Each curve that forms the surface is then translated into x, y, z axis, for future fabrication of frames.

9. The surface is thickened equally.

10. The thickened surface modules and be assembled on site as pre-cast concrete pieces that followed CNC moulds (see image below).

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PRECEDENT

Taichung Metropolitan Opera House (Taiwan) - Toyo Ito The National Taichung Theater is an opera house in the 7th Metropolitan area of Taichung, Taiwan.

The estimated area of the structure is 57,685 square metres (620,920 sq ft). It was designed by Japanese architect Toyo Ito. It was contracted on 11 November 2009 with construction planned for 45 months.

The porous three-dimensional sponge-like design is an ecosystem in itself. advanced building systems are implemented to recycle rainwater, passively control temperature and light, and even influence the sustainability of the surrounding sites. the most challenging aspect of the project by far is the construction of the organic interior.

Construction Process A mesh of steel beams lays the foundation for the three-dimensional curving walls, shaped by a smaller metal mesh and finally solidified in ‘shotcrete’, a spray-able concrete used in tunnel work. it can be shot horizontally or vertically. rather than constructing doubly curved formwork that is expensive and time consuming on site, the temporary structure in the void creates faceted surfaces that best-fit the finished surface. between the temporary steel work,expanded metal mesh is expanded metal mesh spans between the temporary steel work to act as faceted formwork. 150 mm thick concrete can be shot at one time.

It is applied in two layers, the bulk of the structure is sprayed on with a pre-mixed batch to get the massing as accurate as possible. afterwards, workers plaster over with a slightly different concrete mix by hand to get the exact size and textures desired. it is without a doubt the first structure in the world to implement such methods at this scale and will eventually encase three theaters of 200, 800, and 2000 seats in a transformed block of Taichung.

Sprayed concrete construction method Typically, an expanded metal mesh is used as a permanent back shutter to which the reinforcing mesh is affixed. the concrete is sprayed onto the expanded metal and the reinforcement is fully enclosed. the concrete is typically sprayed using one of two methods. with the dry process, the dry constituents of the concrete are mixed in a portable batching plant and the water is added to the mix at the nozzle. with the wet process, the water is added to the batching plant and premixed with the dry constituents and the wet concrete is sprayed from the nozzle. the benefits of the wet process are that there is greater control over the concrete mix as the concrete is often mixed off site by ready-mix contractors and delivered in lorries. it is common practice to apply the concrete in two layers. the first thick layer is usually applied using the wet process. once sufficiently cured, a second, thin finishing layer is then applied using the dry process.

It is essential that the finished product is cured appropriately to mitigate shrinkage and to ensure that design strength is achieved. spraying concrete is a messy process. some concrete will rebound and some will pass through the expanded metal back shutter. it may be necessary to install temporary protection to avoid polluting the surrounding area.

Project info site area: 57,685 sqm floor area: 43,264 sqm 2009 seat grand theatre 4,450 sqm 800 seat playhouse 1,770 sqm 200 seat black box 520 sqm backstage space 2,630 sqm other space 16,090 sqm management space 1,850 sqm arts plaza 6,000 sqm arts/creative workshop 3,100 sqm total 36,410 sqm 36

* http://www.designboom.com/architecture/toyo-ito-taichung-metropolitan-opera/(Drawings and process of Opera); * https://en.wikipedia.org/wiki/Ferrocement (Properties of Ferro-Cement);

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STRUCTURAL & CONSTRUCTION STRATEGY

MAIN HOUSE PROTOTYPE STRUCTURE

DRONE-HOUSE CAPSULE AXONOMETRIC

CANTILEVERED PLATFORM DETAIL

SLICED AXONOMETRIC VIEW OF LIVING UNIT 38

INFLATABLE SYSTEM PIPE DETAIL

Sketches of Design Development & Principles SKETCHES

CAPSULE GEOMETRY EXPLORATION & SPACE ANALYSIS

Drone system attached

Air-ship Inflatable Deck Aerodynamic shape to suit flying mode

Elliptic Pre-fabricated Capsule

Soft, gellatine gel plug-in system

Fisrt Drone prototype (flow lines for aerodynamics)

Current Proposal Laura Nica | Technical Studies in Practice

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CAPSULE CONCEPT

- Pre-fabricated cage with retractable doors and inflatable rooms-

Sketched Sectioned Plan View

Sketched Long Section

Sketched Short Elevation

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CAPSULE PRINCIPLES & TECHNICAL PRECEDENTS - Drone system, Inflatables and Pre-fabricated CapsulesDRONE SYSTEM

INFLATABLES

PRE-FABRICATED CAPSULES Capsule Portable House

Blob Mobile

Possible Inflatable Section (furniture-wall)

The Embryological House- Greg Lynn Bartlett - Deployable Wall system

Living Pod- David Greene

*https://www.mentor.com/products/mechanical/industries/aerospace-defense (Mechanical Analysis )

AA EmTECH School - inflatable tests

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MATERIAL RESEARCH

LIGHT-WEIGHT MATERIALS (main material for Drone House)

Lightweighting is a concept in the auto industry about building cars and trucks that are less heavy as a way to achieve better fuel efficiency and handling.[1][2] Carmakers make parts from carbon fiber, windshields from plastic, and bumpers out of aluminum foam, as ways to lessen vehicle load.[3] Replacing car parts with lighter materials does not lessen overall safety for drivers, according to one view, since many plastics have a high strength-to-weight ratio.

AEROGEL

METALIC FOAM (ALUMINIUM)

METAL MICROLATTICE (ALUMINIUM)

Aerogel is a synthetic porous ultralight material derived from a gel, in which the liquid component of the gel has been replaced with a gas.

The result is a solid with extremely low density and low thermal conductivity. Nicknames include frozen smoke, solid smoke, solid air, or blue smoke owing to its translucent nature and the way light scatters in the material. It feels like fragile expanded polystyrene to the touch. Aerogels can be made from a variety of chemical compounds. Aerogels are produced by extracting the liquid component of a gel through supercritical drying. The first aerogels were produced from silica gels. Kistler’s later work involved aerogels based on alumina, chromia and tin dioxide.

A metal foam is a cellular structure consisting of a solid metal, frequently aluminium, as well as a large volume fraction of gas-filled pores. The pores can be sealed (closed-cell foam), or they can form an interconnected network (opencell foam). The defining characteristic of metal foams is a very high porosity: typically 75–95% of the volume consists of void spaces making these ultralight materials.

Ultralight Metallic Microlattice owes its extreme lightness to a lattice of interconnected hollow tubes, with a wall thickness 1,000 times thinner than a human hair, made from 99.99% air, and 0.01% solids.

GELATIN COMPOSITE

BALLISTICS GEL

(a) A cartoon schematic of a possible monolayer reinforcement architecture produced in (b) an alumina–gelatin composite. (c) Swelling of such a system leads to shape change in correspondence with the local reinforcement architecture. The gelatin matrix swells 50% more in the direction perpendicular to the local reinforcement orientation. (d) A cartoon schematic of a possible bilayer structure produced in (e) an alumina–gelatin bilayer composite. (f) Swelling of the bilayer system leads to local regions that curl or twist depending upon the previously programmed reinforcement architecture. Scale bar, 1 cm.

The basic mechanism of gelation of gelatin is a coil-to-helix transition during which the helices that are created are similar to the collagen structure (Djabourov, Lechaire, & Gaill, 1993). This transition occurs at temperatures below 30 °C, even at low gelatin concentrations. It was reported that at low gelatin concentration, the formation of crosslinks is very likely to occur between two segments of the same molecule or molecules already joined (Ferry & Eldridge, 1949 ). The present knowledge of the gelation mechanisms is the gelatin gel, tested on different experimental approaches (polarimetry, electron microscopy, rheology).

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PRECEDENT SPATIUM GELATUM (Poland) - Zbigniew Oksiuta Spatium Gelatum technology studies the rules for forming liquid and gelling objects from biological polymers. Polymers in a liquid state were used in the reasearch. The technology uses the ball bearing principle to rotate the liquid mass in flexible, sphere-shaped forms (plastic spheres and PVC balls). The sphere-like shape of the forms enables rotation within the liquid. The goal is to study biological forms of habitation.

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ENVIRONMENTAL STRATEGY - Climate, Temperatures, Sun Diagrams, Wind Simulation, Site Conditions, Ventilation and Integrated Systems -

CLIMATE

- General climate analysis of San Francisco Location: San Francisco, California, USA (37.6ยบ- 122.4ยบ); Date: January -December 2016; Time: 00:00 - 24:00;

Average Temperature

Maximum Temperature

Minimum Temperature

Relative Humidity

Direct Solar Radiation (W/ m2))

Average Cloud Cover (%)

Diffuse Solar Radiation

Average Daily Rainfall

Average Wind Speed

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SOLAR ANALYSIS

- Proposed scheme sunpath analysis -

Sun Path

Summer Solstice (21st June)

Spring Equinox (21st March)

Shadow range (11 o’clock)

Autumn Equinox (21st September)

Winter Solstice (21st December)

Shadow Range during year 48

PREVAILING WIND ANALYSIS - Wind frequency (Hrs) -

Location: San Francisco, California, USA (37.6ยบ- 122.4ยบ); Date: January -December 2016; Time: 00:00 - 24:00;

January

February

March

April

May

June

July

August

September

October

November

December Laura Nica | Technical Studies in Practice

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Wind simulations *

Monthly diurnal averages

Average Wind speed (km/h)

APRIL - Average Wind speed

JANUARY- Average Wind speed

Exterior Ventilation

Interior Ventilation

Stack- Ventilation The difference between the air inside the building and outdoors drives the flow. The colder air is heavier.

When it enters the building it gets warmer and lighter and escapes through openings situated on the top of the building.

Average Wind speed (km/h) 50

SYSTEMS ANALYSIS

- Energy Supply, Distribution and Usage, Water Circulation -

BATTERY SYMILARITIES

POSSIBLE HEAT STRATEGY

MAIN PIPES FOR ELECTRICITY HOUSE SUPPLY Laura Nica | Technical Studies in Practice

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References

I

Books Arturo Tedeschi- AAD Algorithms- Aided Design. Parametric Strategies using Grasshopper; Gani, M. S. J. Cement and Concrete. London: Chapman & Hall, 1997. 8. Print.

II

Digital Resources The MIT Introduction to Algorithms: http://courses.csail.edu/6.006/fall11/notes.shtml Mathworld- Scherk Surface gemeration:

http://mathworld.wolfram.com/ScherksMinimalSurfaces.html; M. Elgensatz- Case Studies in Cost-Optimized Paneling of Architectural Freeform Surfaces

http://vecg.cs.ucl.ac.uk/Projects/SmartGeometry/paneling_aag/paper_docs/paneling_aag10.pdf Construction of Ferro-cement Hulls http://www.fao.org/docrep/003/v9468e/v9468e09.htm; Properties of Ferro-Cement https://en.wikipedia.org/wiki/Ferrocement Taichung Opera House- Catenoid Construction

http://www.designboom.com/architecture/toyo-ito-taichung-metropolitan-opera-house-taiwan-21-08-2014/ Natural Ventilation - Stack Ventilation http://www.architecture.com/RIBA/Aboutus/SustainabilityHUb/Designstrategies/Air Foundations & Footing http://collections.infocollections.org/ukedu/en/d/Jg09c2e/5.3.html

III

Drawings Taichung Opera House- Catenoid Construction

http://www.designboom.com/architecture/toyo-ito-taichung-metropolitan-opera-house-taiwan-21-08-2014/

University of Westminster 4ARC651

| TECHNICAL STUDIES IN PRACTICE | William McLean & Peter Silver

w1560320 May 2016

| ANDREEA-LAURA NICA | DS10