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graphic statics in arches and curved beams PDF

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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 +316 24 151 272 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] i | P age 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 ii | P age 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 iii | P age 13.1 Mathematical approach .................................................................................................................. 49 13.1.1 ............................................................................................................................................................. 49 13.1.2 .............................................................................................................................................................. 50 𝑵 13.1.3 Combining calculations ........................................................................................................................... 50 𝒍 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 iv | P age GRAPHIC STATICS IN ARCHES AND CURVED BEAMS Part I Introduction 1 | P age 2 | P age GRAPHIC STATICS IN ARCHES AND CURVED BEAMS Background Throughout history people created and built the most fascinating structures. Mankind developed a 1 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. 3 | P age 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. 4 | P age

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e-mail [email protected]. University. Technical University of Delft. Faculty. Architecture. Department. Architectural Engineering + Technology. Graduation GRAPHIC STATICS . Figure 61 Output generated by Grasshopper, in orange the stresses higher than the yield strength of the material, the.
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