Form Finding of Arches in Tensile Structures

161 Pages·2015·19.8 MB·English

Form Finding of Arches in Tensile Structures
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Giorgio Piscitelli
Student number: 4317572

First mentor: Ir. Andrew Borgart, Structural Mechanics
Second mentor: Arch. Michela Turrin, Design Informatics
Third mentor: Dr. Craig Lee Martin, Climate Design and Sustainability
External examiner: Ir. Ype Cuperus, Architectural Engineer and Technology

Sustainable Design Graduation Studio
Master Graduation in Building Technology
Structural Mechanics, Form-Finding

Faculty of Architecture and the Built Environment
TUDelft, 05/05/2015
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00 CONTENT

01 INTRODUCTION .................................................................................................................................. 8
Background ......................................................................................................................................... 8
Context of the research project ...................................................................................................... 8
Basic problem analysis .................................................................................................................... 8
Problem statement ............................................................................................................................. 8
Main problem ................................................................................................................................. 8
Possible sub-problems .................................................................................................................... 8
Objectives............................................................................................................................................ 9
General objectives .......................................................................................................................... 9
Sub-objectives ................................................................................................................................. 9
Final products .................................................................................................................................. 9
About the direction of the solution ................................................................................................ 9
Boundary conditions ....................................................................................................................... 9
Research questions ........................................................................................................................... 10
Main research question ................................................................................................................ 10
Sub-questions ............................................................................................................................... 10
Background questions ................................................................................................................... 10
Approach and methodology ............................................................................................................. 10
Research questions and objectives definition .............................................................................. 10
Design tool developments ............................................................................................................ 10
Design tool application ................................................................................................................. 10
Structural verification ................................................................................................................... 10
Methodology scheme ................................................................................................................... 11
Relevance .......................................................................................................................................... 12
Societal relevance ......................................................................................................................... 12
Scientific relevance ....................................................................................................................... 12
02 BACKGROUND .................................................................................................................................. 14
Context .............................................................................................................................................. 14
Existing Situation ........................................................................................................................... 14
Existing Project .............................................................................................................................. 14
Concept Design ................................................................................................................................. 15
Functional concept ........................................................................................................................ 15
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Functional proposal ...................................................................................................................... 16
Structural concept ......................................................................................................................... 18
Structural design proposal ............................................................................................................ 19
03 THEORY ............................................................................................................................................ 20
Form Finding in Architecture ............................................................................................................ 20
Early strategies .............................................................................................................................. 20
Form-finding nowadays ................................................................................................................ 21
Force Density Method FDM .............................................................................................................. 22
The mathematics beyond the “Stuttgart Direct Approach” ......................................................... 22
Force Density and equivalent cable nets for membranes ............................................................ 26
Extending FDM to compressive elements .................................................................................... 28
Minimal Bending Energy Arch ........................................................................................................... 30
2D Thrust Line of a vertically loaded Arch .................................................................................... 30
2D Thrust Line of any generic loaded Arch ................................................................................... 33
Minimal Complementary Energy and Arches ............................................................................... 36
From 2D to 3D Arches ................................................................................................................... 40
From Generic 3D Arches to Planar 3D Arches .............................................................................. 41
Remarks for the Analysis ................................................................................................................... 44
Equivalent cable net properties .................................................................................................... 44
Effective Young Modulus of ties ................................................................................................... 46
04 COMPONENTS .................................................................................................................................. 48
From theory to design tools .............................................................................................................. 48
Grasshopper as a working environment ....................................................................................... 48
Default Components, VB Components, Clusters and Groups ....................................................... 48
FDM Components ............................................................................................................................. 49
The FDM group ............................................................................................................................. 49
About the Default Matrix Components ........................................................................................ 50
FDM Engine Component ............................................................................................................... 51
C and Cf Component ..................................................................................................................... 52
Diagonal Q Component ................................................................................................................. 53
Strut Force Component ................................................................................................................. 54
Fixed Pt Component ..................................................................................................................... 55
Arch Components ............................................................................................................................. 56
Arch form-finding Component ...................................................................................................... 56
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Energy calculation Component ..................................................................................................... 57
Planar arch Component ................................................................................................................ 59
3D Arch Form Finding Component ............................................................................................... 60
3D Planar Arch Form Finding Component .................................................................................... 62
Analysis Components ........................................................................................................................ 64
Equivalent Cable Properties Component ...................................................................................... 64
Effective Young Modulus Component .......................................................................................... 65
Loop Component .............................................................................................................................. 66
Write to Text Component ............................................................................................................. 66
Point Loop Group .......................................................................................................................... 67
05 DESIGN APPLICATION ....................................................................................................................... 68
Designing Membrane Structures ...................................................................................................... 68
Stiffness of Membrane structures ................................................................................................ 68
Typologies of Membrane Structures ............................................................................................ 72
Parameterization of the Design ........................................................................................................ 74
Design Background ....................................................................................................................... 74
Overall Structure of the Design Algorithm .................................................................................... 75
Geometry Definition ..................................................................................................................... 76
FDM Form-Finding ........................................................................................................................ 78
Model Output ............................................................................................................................... 83
Arch Form-Finding ......................................................................................................................... 85
Loop and Convergence ................................................................................................................. 90
Exporting to GSA ........................................................................................................................... 92
Overall Algorithm Scheme ............................................................................................................ 93
Design Variants ................................................................................................................................. 94
Design inputs and geometries ...................................................................................................... 94
Curvature Analysis ........................................................................................................................ 98
FEM Model Properties ................................................................................................................ 100
FEM Analysis Results ................................................................................................................... 107
FEM Results Verification and Interpretation .............................................................................. 124
Visualizations .............................................................................................................................. 126
06 CONCLUSIONS ................................................................................................................................ 134
Considerations on the product ....................................................................................................... 134
About the developed algorithm .................................................................................................. 134
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About components ..................................................................................................................... 134
About structural performances of light weight structures ......................................................... 135
About Form-Finding in the overall design process ..................................................................... 135
Conclusions ..................................................................................................................................... 136
Learning from the process .......................................................................................................... 136
Acknowledgements ......................................................................................................................... 137
07 REFERENCES ................................................................................................................................... 138
Bibliography .................................................................................................................................... 138
Software .......................................................................................................................................... 139
08 APPENDIX ....................................................................................................................................... 140
VB Scripts ........................................................................................................................................ 140
FDM Engine ................................................................................................................................. 140
Cs ................................................................................................................................................. 143
C and Cf ....................................................................................................................................... 147
Q values ....................................................................................................................................... 148
Diagonal Q ................................................................................................................................... 151
Px ................................................................................................................................................. 152
Py ................................................................................................................................................. 152
Pz ................................................................................................................................................. 152
Arch form-finding ........................................................................................................................ 153
Energy calculation ....................................................................................................................... 156
Planar arch .................................................................................................................................. 159
Write to Text ............................................................................................................................... 159
Cluster Structures ........................................................................................................................... 160
3D Arch Form Finding ................................................................................................................. 160
3D Planar Arch Form Finding ...................................................................................................... 160
Strut Force ................................................................................................................................... 161
Equivalent Cable Properties ........................................................................................................ 161
Effective Young Modulus ............................................................................................................ 161

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01 INTRODUCTION

Background
Context of the research project
The main context of the research project is the Milan Expo 2015. A structure is designed and
calculated in order to stimulate the research and apply the gained knowledge. According to the
existing context the design concept is elaborated and several form-finding tools are developed and
combined together to achieve the set goals.

Basic problem analysis
Dealing with digital form-finding techniques, there are two main issues that should be taken into
account since the beginning. The first point regards the boundary conditions of the process, the fixed
parameters determining the final shape. The second aspect concerns the process itself, the
algorithm generating the final result according to the set boundary conditions. Those kind of
processes can be direct or iterative, linear or non linear, and the level of accuracy of the final result
is strictly linked to these aspects.

Problem statement
Main problem
The main problem is how to develop a form-finding algorithm for tensile membranes supported by
arches. The final equilibrium shape should have the rigid supports working in pure compression
according to their self-weight and the pre-stress forces of the membrane and any other tensioned
element like cables and ties. In order to do that an iterative combination of different form-finding
methods might be required.

Possible sub-problems
The first sub-problem is the selection of the existing form finding methods by evaluating their limits
and potentials.

The second sub-problem regards the way of combining the different form finding methods in an
iterative way.

The third sub-problem is dealing with the convergence of the combined methods into a unique
result. This aspect is strictly linked with both the selected form-finding methods and their
combination. However this issue will appear just in the last step of the algorithm development. In
order to avoid any inconvenience, the convergence aspect should be taken into account since the
beginning of the tool development.
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Objectives
General objectives
The general objective is to create a tool able to form-find structures composed by pure compressed
arches, a membrane and tensioned ties. An algorithm will be drawn in the Grasshopper environment
and several design tools will be developed for Rhinoceros users.

Sub-objectives
The first sub-objective is to select or elaborate form-finding direct methods. Since they do not
require nonlinear iterations, they can be easily scripted and combined together.

The second sub-objective is to combine the selected direct methods in an iterative process able to
preserve the fixed boundary conditions.

The third sub-objective is to guarantee the convergence of the iterative process to a unique solution.

A last sub-objective is to demonstrate the structural efficacy of the form-finding tool developed.

Final products
The main product will be the form-finding tools themselves and their design application.

The secondary product is the report explaining the tools background, their development, their final
structure, a generic design application and their verification.

The third product is the elaboration of the supporting material for the oral presentation.

The fourth product is a DVD containing the all digital materials developed during the research.

About the direction of the solution
The main direction of the solution is combining direct form-finding methods. Iterations should just
be used to link different form-finding techniques into a hybrid form-finding algorithm.

Boundary conditions
A pavilion typology is developed in order to set the design boundary conditions. A clever structural
concept is needed before the application of the developed tool. The parameterization of the pavilion
according to its boundary conditions is the starting point to compare different form-found
structures. The only shaping forces taken into account are the pre-stresses of the tensile elements
and the self weight of the compressed ones. External loads, like wind, are not involved in the form-
finding process but just used to test the efficacy of the designed structure by FEM analysis.

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Research questions
Main research question
The main research question is how to design optimal arches in membranes and cable net structures.
In order to do that a hybrid form-finding tool will be developed.

Sub-questions
The first sub-question is what the most suitable form-finding methods are.

The second sub-question is how the form-finding methods should be combined together

The third sub-question is how and if a unique solution exists according to the set boundaries

The forth sub-question is if the developed tool applications are performing as desired and if not
what are the lacks of the developed method.

Background questions
The main background question is how a form-finding tool should be used in the design process.

Background sub-questions are about the stage in which these tools should be involved in the design
process and their efficacy in improving the structural performances of the proposed design.

Approach and methodology
Research questions and objectives definition

- Research of study material
- Selection of study material theories
Design tool developments

- Mathematics, mechanics and scripting studies needed to achieve the set goals
- Translation of the theory into design tools for the design algorithm
- Structural concept elaboration for a generic pavilion
- Translation of the concept into a parametric model with its BC (boundary conditions)
Design tool application

- Combination of the form-finding tools with the structural concept of the pavilion
Structural verification

- FEM analysis of the design pavilion
- Reflections and considerations
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