graphic statics in arches and curved beams

96 Pages·2014·3.94 MB·English

TU DELFT - FACULTY OF ARCHITECTURE
GRAPHIC STATICS IN
ARCHES AND CURVED
BEAMS
FINDING FORCE EQUILIBRIUM THROUGH TOTAL MINIMUM
COMPLEMENTARY ENERGY

Niels van Dijk

28-1-2014

GRAPHIC STATICS IN ARCHES AND CURVED BEAMS

Date 28-01-2014

Name Niels van Dijk
Student number 1361201
Address Juniusstraat 161
Postal code 2625 WZ
Place of residence Delft
Phone number [phone]
e-mail [email protected]

University Technical University of Delft
Faculty Architecture
Department Architectural Engineering + Technology
Graduation track Building technology
Graduation lab Computation and technology

Main mentor Ir. A. Borgart
Chair of structural mechanics
e-mail [email protected]
Second mentor Ir. T.R. Welman
Chair of design informatics
e-mail [email protected]
External examiner Ir. E.J. van der Zaag
e-mail [email protected]
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PART I INTRODUCTION ............................................................................................................................... 1
1 BACKGROUND ................................................................................................................................................ 3
2 RESEARCH QUESTION AND OBJECTIVES ................................................................................................................ 5
3 THESIS OUTLINE .............................................................................................................................................. 6
PART II THEORY ........................................................................................................................................... 7
4 GRAPHIC STATICS ............................................................................................................................................ 9
5 STATIC-GEOMETRIC ANALOGY .......................................................................................................................... 15
6 MINIMUM COMPLEMENTARY TOTAL ENERGY ...................................................................................................... 18
PART III METHOD DEVELOPMENT EXCEL..................................................................................................... 23
7 PROBLEM AND HYPOTHESES ............................................................................................................................ 25
8 LINE OF THRUST INSIDE THE MATERIAL ............................................................................................................... 26
9 LINE OF THRUST OUTSIDE THE MATERIAL ............................................................................................................ 28
10 OWN WEIGHT .............................................................................................................................................. 36
PART IV METHOD DEVELOPMENT GRASSHOPPER ...................................................................................... 39
11 EXCEL TO GRASSHOPPER ................................................................................................................................ 41
12 GRASSHOPPER EXPLORATORY METHODS ............................................................................................................ 43
13 EXPLORATION FOR OTHER SOLUTIONS ............................................................................................................... 48
14 FINAL GRASSHOPPER METHOD ......................................................................................................................... 63
15 DEFINITION DESCRIPTIONS GRASSHOPPER .......................................................................................................... 68
PART V RESULTS......................................................................................................................................... 75
16 CONCLUSIONS .............................................................................................................................................. 77
17 LIST OF FIGURES ............................................................................................................................................ 80
18 LITERATURE LIST ........................................................................................................................................... 82
19 APPENDIX .................................................................................................................................................... 84

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GRAPHIC STATICS IN ARCHES AND CURVED BEAMS

PART I INTRODUCTION ............................................................................................................................... 1
1 BACKGROUND ................................................................................................................................................ 3
2 RESEARCH QUESTION AND OBJECTIVES ................................................................................................................ 5
3 THESIS OUTLINE .............................................................................................................................................. 6
PART II THEORY ........................................................................................................................................... 7
4 GRAPHIC STATICS ............................................................................................................................................ 9
4.1 Head-tails method ............................................................................................................................. 9
4.2 Cable analysis .................................................................................................................................... 9
4.3 Arch analysis .................................................................................................................................... 10
4.4 Practice ............................................................................................................................................ 11
4.5 Thrust network analysis .................................................................................................................. 13
5 STATIC-GEOMETRIC ANALOGY .......................................................................................................................... 15
5.1 Plate calculations ............................................................................................................................ 15
5.2 Shell calculations ............................................................................................................................. 15
6 MINIMUM COMPLEMENTARY TOTAL ENERGY ...................................................................................................... 18
6.1 Complementary energy ................................................................................................................... 18
6.2 Normal forces .................................................................................................................................. 19
6.3 Bending moments ........................................................................................................................... 19
6.4 Rewriting ......................................................................................................................................... 20
6.5 Application of minimum complementary energy ............................................................................ 20
PART III METHOD DEVELOPMENT EXCEL..................................................................................................... 23
7 PROBLEM AND HYPOTHESES ............................................................................................................................ 25
7.1 Problem ........................................................................................................................................... 25
7.2 Hypothesis ....................................................................................................................................... 25
8 LINE OF THRUST INSIDE THE MATERIAL ............................................................................................................... 26
9 LINE OF THRUST OUTSIDE THE MATERIAL ............................................................................................................ 28
9.1 Test compressive complementary energy ....................................................................................... 28
9.2 Test compressive and bending complementary energy .................................................................. 29
9.2.1 Relating forces ............................................................................................................................................. 29
9.2.2 Forces construction ..................................................................................................................................... 30
9.3 Unknown horizontal support ........................................................................................................... 33
9.4 Review ............................................................................................................................................. 35
10 OWN WEIGHT .............................................................................................................................................. 36
10.1 Formula catenary ............................................................................................................................ 36
10.2 Discretization ................................................................................................................................... 37
10.3 Conclusions ...................................................................................................................................... 37
PART IV METHOD DEVELOPMENT GRASSHOPPER ...................................................................................... 39
11 EXCEL TO GRASSHOPPER ................................................................................................................................ 41
11.1 Grasshopper over Excel ................................................................................................................... 41
11.2 Differences Excel to Grasshopper .................................................................................................... 41
11.2.1 Line of thrust .......................................................................................................................................... 41
11.2.2 Calculation of supports ........................................................................................................................... 41
11.3 Goal of the script ............................................................................................................................. 41
12 GRASSHOPPER EXPLORATORY METHODS ............................................................................................................ 43
12.1 Validation ........................................................................................................................................ 45
12.2 Limitations ....................................................................................................................................... 47
13 EXPLORATION FOR OTHER SOLUTIONS ............................................................................................................... 48
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13.1 Mathematical approach .................................................................................................................. 49
13.1.1 ............................................................................................................................................................. 49
13.1.2 .............................................................................................................................................................. 50
𝑵
13.1.3 Combining calculations ........................................................................................................................... 50
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13.2 Graphical relations between construction, force polygon and line of thrust .................................. 50
13.2.1 Relation force polygon – line of thrust ................................................................................................... 51
13.2.2 Relation total area line of thrust – force polygon ................................................................................... 53
13.3 Changing parameters of force polygon ........................................................................................... 56
13.3.1 Changing vertical support ....................................................................................................................... 56
13.3.2 Changing horizontal support .................................................................................................................. 56
13.3.3 Changing divisions .................................................................................................................................. 57
13.3.4 Application ............................................................................................................................................. 58
13.4 Proof of equal areas ........................................................................................................................ 59
13.4.1 Area construction ................................................................................................................................... 59
13.4.2 Area line of thrust ................................................................................................................................... 60
13.4.3 Minimum complementary energy .......................................................................................................... 61
13.4.4 Conclusions ............................................................................................................................................. 62
14 FINAL GRASSHOPPER METHOD ......................................................................................................................... 63
14.1 Closing line ...................................................................................................................................... 63
14.1.1 Finding the closing line ........................................................................................................................... 64
14.1.1 Application closing line ........................................................................................................................... 65
14.1.2 Validation ............................................................................................................................................... 65
14.2 Using the script ................................................................................................................................ 66
14.3 Stresses ............................................................................................................................................ 66
14.4 Conclusions ...................................................................................................................................... 67
15 DEFINITION DESCRIPTIONS GRASSHOPPER .......................................................................................................... 68
15.1 Calculation description .................................................................................................................... 68
15.2 Graphical representation................................................................................................................. 70
15.2.1 Visual check ............................................................................................................................................ 70
15.2.2 Display component ................................................................................................................................. 70
15.3 Differences different scripts ............................................................................................................ 73
PART V RESULTS......................................................................................................................................... 75
16 CONCLUSIONS .............................................................................................................................................. 77
16.1 Process............................................................................................................................................. 77
16.2 Calculation method and component ............................................................................................... 77
16.3 Recommendations ........................................................................................................................... 78
17 LIST OF FIGURES ............................................................................................................................................ 80
18 LITERATURE LIST ........................................................................................................................................... 82
19 APPENDIX .................................................................................................................................................... 84

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GRAPHIC STATICS IN ARCHES AND CURVED BEAMS

Part I Introduction

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GRAPHIC STATICS IN ARCHES AND CURVED BEAMS

Background
Throughout history people created and built the most fascinating structures. Mankind developed a
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good sense for how to construct certain structures, first by trial and error, later by learning how to
analyse whether structures would collapse or not. With the development of computer technology
more complex building shapes could be modelled, not only in design but also in structural analysis.
Simple constructions, like beams and columns, are easily calculated analytically by hand. More
complex to calculate are plates and beam networks, but it is still possible to do this by hand. For shell
structures, it is nearly impossible to make analytical hand calculations. This is only possible for a small
basic set of shell shapes. For more complicated structures, Finite Element Methods are needed to
gain insight in the stresses and displacements caused by certain load cases. These FEM calculations
are numerical calculations for complicated mechanical problems.
The most efficient way of transferring loads is trough axial (stretching and compression) forces. A
structurally good shell uses this efficiency and acts mostly in compression. When architects like Gaudi
started designing structures with mainly compression, an analogy with chains was used. Chains only
transfer loads trough stretching forces, inverting the shape of the hanging chain, gives a structure
with only compression (Huerta 2005, Block, DeJong et al. 2006). When it is not possible to build the
exact inverse shape of a hanging chain, bending moments are introduced. With bending in a
material, large deformations can be expected, and bending becomes the most important factor
dimensioning your structure.

Figure 1 Left; hanging chain models by Gaudi (Daniel 2007) Right; Sagrada Familia, Barcelona, by Gaudi (Mak 2013)
Due to the minimum presence of bending forces, compression structures can be designed with little
structural height, compared to ordinary structures. This slenderness is part of the elegance, and
probably the allure, of shell structures. Famous shell builders are architect Felix Candela and
engineer Heinz Isler. Candela designed his shells trough mathematical models, using only basic
shapes like hyperbolic paraboloids, making sure he could perform hand calculations on his designs.
Isler was more of a mystery man, never revealing his real way of designing. What is known is that he
used a lot of physical models to test different shells and shapes to create his final design.
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Figure 2 Left; Deitingen service station, by Isler (BauNetz 2009) Right; the Manantiales Restaurant, Xochilmico, by
Candela (Lukas 2013)
Until present days, there is still no way to get analytical information about freeform shells. If a
designer wants to see the structural performance of a shell design, the Finite Element Method is
essential. The way of calculating in this software is numerical, so the relation between geometry and
structural performance is lost. In other words, the direct relation between geometrical en structural
properties of a shell is unknown. This in itself is not a problem, since FEM is mostly used in the final
stages of a design, when the shape is already determined. Here it is used to check material
thicknesses and strengths, to see if the construction will fail or not.
With the development of computer technology, a great new range of software became available.
Tools like Grasshopper are used by designers to make parametric relations in 3D modelling software,
giving more flexibility in designing complex shapes. The strength of this parametric way of modelling
is that a designer can create a large set of different designs in a matter of seconds. The downside
here is that the designer has to choose one of these designs.
Designers can use different goals in a design and assess if a certain option, generated by a parametric
model, fulfils these goals. If these goals are related to the field of structural mechanics, or other
properties that can be expressed in numbers for that matter, these goals can be optimized for
chosen values. This means that a certain parameter, or set of parameters, can be varied until the
chosen output reaches a chosen value. This value, for example deflections in a structure, then has to
be calculated by a structural mechanics component. If a FEM calculation is used, the designer can see
if the design gives the desired results in numbers, but if this is not the case, it is not directly clear how
changing the design can improve the performance.
In this thesis the power of this parametric software is used to find ways to analyse geometrical
properties of shells and give insight in the relations between geometry and structural properties,
with the goal that designers whom are using parametric models, can implement this insight in their
models. So when a designer does not retrieve the desired results, the parametric software can give
graphical and geometrical output to give new leads for the designer to improve his or her design.
This direct feedback is lacking in FEM calculations.

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