Structural analysis and deployable development of cable strut systems

STRUCTURAL ANALYSIS AND DEPLOYABLE DEVELOPMENT OF CABLE STRUT SYSTEMS SONG JIANHONG NATIONAL UNIVERSITY OF SINGAPORE 2007 STRUCTURAL ANALYSIS AND DEPLOYABLE DEVELOPMENT OF CABLE STRUT SYSTEMS SONG JIANHONG (B.Eng. Xi’an Jiao Tong University) A THESIS SUBMITTED FOR THE DEGREE OF DOCTER OF PHILOSOPHY DEPARTMENT OF CIVIL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2007 ACKNOWLEDGEMENT Firstly, I would like to thank my supervisors, Professor Koh Chan Ghee and Associate Professor Liew Jet Yue, Richard for their invaluable advice, continuous guidance and generous support throughout my graduate study at the department of civil engineering, National University of Singapore. Secondly, I would thank all my classmates, colleagues and friends who have helped me in any way and to any extent since the beginning of my graduate study in January 2003. Thirdly, I owe my thanks to my parents, wife and parents-in-law for their continuous love, trust and support. They have been looking after my daughter since her birth in mid 2003, and thus I can concentrate my attention on the study. I would also say thanks to my lovely daughter who gives me much courage and happiness. Finally, scholarship and other financial assistances from the National University of Singapore are gratefully acknowledged. ii TABLE OF CONTENTS TITLE PAGE . i ACKNOWLEDGEMENT ii TABLE OF CONTENTS . iii SUMMARY . vi NOMENCLATURE viii LIST OF FIGURES . xiv LIST OF TABLES xxii CHAPTER INTRODUCTION 1.1 Introduction . 1.2 Cable strut systems . 1.2.1 Structural types . 1.2.2 Tension cable strut systems 1.2.3 Free standing cable strut systems 1.2.3.1 Non-deployable structures 1.2.3.2 Deployable structures 1.3 Structural analyses of the two focused systems 12 1.3.1 Analysis types and methods 12 1.3.2 Simplified analysis of cable truss . 16 1.3.3 Simplified analysis of cable-strut truss . 19 1.4 Objectives and scope . 23 1.5 Organization of thesis 24 CHAPTER STATIC ANALYSIS OF RADIALLY ARRANGED CABLE TRUSS . 27 2.1 Introduction . 27 2.2 Initial configurations . 27 2.3 Simplified solutions 30 2.3.1 Irvine’s solution 30 2.3.2 Improved solution . 31 2.3.3 Solutions for strut force 35 2.4 Numerical verification 38 2.4.1 Finite element theory 38 2.4.2 Numerical verification 39 2.4.3 Nonlinear effect 48 2.4.4 Validity on other structural types 52 2.5 Effect of different parameters on structural behavior . 54 iii 2.6 Summary . 57 CHAPTER FREE VIBRATION ANALYSIS OF RADIALLY ARRANGED CABLE TRUSS 58 3.1. Introductions . 58 3.2. Analytical free vibration solution . 59 3.2.1. Irvine’s solution for cable truss with a cubic shape 59 3.2.2. Solution for cable truss with a parabolic shape 60 3.2.2.1. Single layer circular shallow membrane . 60 3.2.2.2. Double layer shallow membrane 63 3.2.2.3. Solution for radially arranged cable truss . 65 3.3 Numerical free vibration analysis . 68 3.3.1 Finite element theory for free vibration analysis 68 3.3.2 Numerical verification 70 3.4 Summary . 73 CHAPTER EARTHQUAKE ANALYSIS OF RADIALLY ARRANGED CABLE TRUSS 89 4.1 Introduction . 89 4.2 Structural behavior study based on finite element analysis 90 4.2.1 Finite element analysis methods . 90 4.2.2 Input parameters for numerical model 93 4.2.3 Structural behavior under earthquake . 97 4.3 Proposed simplified procedure . 104 4.3.1 Formula for estimation of response using response spectra . 104 4.3.2 Evaluation of the proposed simplified procedure . 106 4.4 Summary . 113 CHAPTER NOVEL DEPLOYABLE CABLE STRUT SYSTEM . 114 5.1 Introduction . 114 5.2 Proposed cubic truss system . 114 5.3 Structural behavior studies 130 5.3.1 Evaluation method 130 5.3.2 Comparison of structural behavior between different systems . 133 5.3.3 Optimal study on novel cubic truss system . 138 5.4 Summary . 139 CHAPTER ENHANCED DEPLOYABLE CUBIC TRUSS SYSTEM 140 6.1 Introduction . 140 6.2 Enhanced cubic truss system 140 6.2.1. Type-A enhanced cubic truss system 140 6.2.2. Type-B enhanced cubic truss system 143 iv 6.2.3. Type-C enhanced cubic truss system 152 6.3. Proposed deployable shelter . 184 6.4 Summary . 188 CHAPTER STATIC AND DYNAMIC ANALYSIS OF THE NOVEL CUBIC TRUSS SYSTEM . 189 7.1 Introduction . 189 7.2 Static analysis 190 7.2.1 Plate Analogies . 190 7.2.1.1 Thin plate analogy . 190 7.2.1.2 Thick plate analogy . 193 7.2.1.3 Numerical verification 194 7.2.2 Novel method based on 2-D planar truss 195 7.2.2.1 Introduction . 195 7.2.2.2 Procedure 196 7.2.2.2.1 Simplification of the 3-D space system to 2-D planar system 196 7.2.2.2.2 Analysis of the 2-D system . 199 7.2.2.3 Numerical verification 208 7.3 Dynamic analysis 215 7.3.1 Free vibration analysis 215 7.3.1.1 Simplified solution 215 7.3.1.2 Numerical verification 217 7.3.2 Earthquake analysis 219 7.3.2.1 Simplified solution 219 7.3.2.2 Numerical verification 220 7.3.3 Blast analysis 228 7.3.3.1 Blast loading . 228 7.3.3.2 Blast response analysis . 229 7.3.3.2.1 Introduction . 229 7.3.3.2.2 Elastic single degree freedom (SDOF) system . 230 7.3.3.2.3 Elastic Multi-degree freedom (MDOF) system 235 7.3.3.2.4 Numerical verification . 238 7.4 Summary . 245 CHAPTER CONCLUSIONS AND RECOMMENDATIONS . 247 8.1 8.2 Conclusions . 247 Recommendations for further work 249 REFERENCES . 251 APPENDIX: SIMULATION MOVIES FOR TYPE-C ENHANCED DEPLOYABLE CUBIC TRUSS SYSTEM 258 v SUMMARY Simplified analysis and deployable development of cable strut systems are conducted in this thesis. There are two objectives. The first one is to propose efficient simplified analysis methods for the preliminary design of cable strut roofs under both static and dynamic loads. The second one is to propose a novel deployable cable strut system which has better structural behavior and simpler stabilizing procedure than existing systems. Various types of cable strut systems are investigated and generally classified into two categories: tension and free standing systems. For the first category, radially arranged cable truss with parabolic shape is chosen for study; for the second category, a novel deployable cable strut system is proposed and chosen for study. Concerning radially arranged cable truss, improved simplified solution is proposed for calculating static response by considering inner ring effect. It is more accurate than the existing solution. An empirical formula for predicting natural vibration frequency and mode sequence is proposed based on membrane analogy method. The predicted results are much closer to the numerical solutions when compared with classical approach. A hand calculation formula for estimating the maximum earthquake responses is proposed based on many important findings. Numerical verification suggests that it can be adopted in preliminary design. A novel deployable cable strut system named as cubic truss system is proposed. It has basic and enhanced forms. The basic system is suitable for small span and load condition, vi while the enhanced system is developed for large span and load condition. To verify the deployment and stabilization of the two systems, a prototype model is built for basic cubic system and a computer simulation is conducted for enhanced system. Comparison on structural efficiency is made between the proposed and existing deployable cable strut systems. It is demonstrated that the proposed system has both easier stabilization procedure and higher structure efficiency than existing cable strut systems. The optimal depth/span ratio and module width/span ratio of the proposed system are investigated and found to agree with the previous published results for other cable strut systems. A rapidly assembled shelter formed by five deployable cubic panels is proposed. Simplified analysis methods for truss systems are proposed based on studies on the novel cubic truss system. For static analysis, plate analogy method is adopted by deriving the equivalent stiffness expressions for the novel cubic truss system. A novel simplified analysis method based on 2-D planar truss is proposed for the analysis of orthogonal truss systems with aim to overcome the boundary limitation of the plate analogy method. Both methods are verified by finite element method. For dynamic analysis, frequency formulae to cover all common boundary conditions are established. A hand calculation formula similar to that for cable truss is proposed for estimating earthquake response. Diagrams for estimating maximum blast response under different frequencies and weights are established based on Dynamic load factor (DLF) method. Numerical verification suggests the proposed simplified methods for calculating frequency and dynamic responses can be adopted in the preliminary design. vii NOMENCLATURE a ratio of radius of inner ring to radius of cable truss a1 beam spacing at x direction, or modular width A area A1, A2, . An areas for different members b ratio of axial stiffness of inner ring / axial stiffness of radial cable b1 beam spacing at y direction br ratio of axial stiffness of cable / axial stiffness of strut c spacing angle between radial cable C , C1,C2, .Cn coefficients used to calculate equivalent loads in Chapter [C ] damping matrix d sag of cable dx,dy,dz derivative in x,y,z direction respectively D flexural stiffness e ratio of strut spacing to half span used in cable truss es ratio of strut spacing to the radial distance between inner and outer ring er aspect ratio E Yong’s modulus E1, E2, .En Yong’s modulus for different members f frequency f1, f 2, . f n frequency corresponding to 1st, 2nd,…nth mode f a , fb, fc, f d coefficients used to calculate displacement and reaction forces in Chapter sag ratio used in cable truss fs viii fr generalized force for mode r F pretension force of strut in cable truss F1 pretension force of strut not connected to inner ring in cable truss F2 pretension force of strut connected to inner ring in cable truss {F } force vector g gravity acceleration h additional horizontal pretension force in cable h′ dimensionless additional horizontal pretension force in cable h1 modular or structural height H horizontal pretension force in cable i, i1 coefficients I moment of inertia j coefficient J Bessel function of the first kind k coefficient used in Chapter and k1 , k2 coefficients used in Chapter k3 coefficient used in Chapter ka unit axial stiffness of truss member in Chapter kb equivalent lateral stiffness of supporting beam in Chapter kc equivalent lateral stiffness of supporting column in Chapter kH coefficient used for calculating equivalent pretension force in Chapter kr generalized mass associated with mode r ks equivalent lateral stiffness of supporting structure in Chapter ix Figure 7.23 Reaction force history at the middle support joint Figure 7.24 Stress history for central top strut 243 Figure 7.25 Stress history for central bottom strut Figure 7.26 Stress history for boundary vertical strut 244 Figure 7.27 Stress history for boundary cable 7.4 Summary Simplified analysis of the truss system is investigated in this chapter through the study on the novel cubic truss system. For static analysis, first, plate analogy method based on both thin and thick plate theory is applied to the novel system by deriving its equivalent flexural and shear stiffness. Numerical verification suggests that shear deformation should be considered for calculating displacement of truss system. Secondly, a novel simplified method based on 2-D planar truss is proposed for perimeter-supported orthogonal truss system. Solution for cable strut cube system is derived and numerical verification shows that the error is below 10%. Both of the methods are easy to use but the former can be only applied to roller support condition while the latter can cover all common boundary conditions, whether or not the support is restrained in horizontal direction. 245 Concerning dynamic analysis, first, frequency formula to cover all common boundary conditions is established based on plate analogy method and static solutions. The error is found to be very small compared to numerical solution. Secondly, a hand calculation procedure similar as that for cable truss is proposed for estimating earthquake response of such systems. Numerical verification suggests that the proposed simplified procedure can be adopted in preliminary design with acceptable error. Thirdly, diagrams for estimating maximum blast response of truss system under different frequencies and self weights are established based on dynamic load factor (DLF) method. Numerical verification suggests the DLF method can be adopted at least in preliminary design. 246 CHAPTER CONCLUSIONS AND RECOMMENDATIONS 8.1 Conclusions Two kinds of systems are studied in this thesis. One is the radially arranged cable truss representing the tension cable strut system and the other one is deployable cable-strut truss system representing free standing cable strut systems. The main contributions are summarized below. (1) An improved simplified analysis is proposed for calculating static response of radially arranged cable truss by considering inner ring effect. It is more accurate than the existing simplified solution. Investigation suggests that the simplified solution can be applied to this system in most common cases. (2) Empirical formula are proposed for predicting frequency and mode sequence of radially arranged cable truss with parabolic shape based on membrane analogy method. It is much closer to the numerical solution than the existing simplified solution. (3) A hand calculation formula is proposed for estimating the maximum earthquake responses of radially arranged cable truss based on many important findings: the vertical response is dominated by the first vertical symmetrical mode, the vertical component is the main contributor of total earthquake response and the mode shape of the first vertical symmetrical mode is similar in shape to static deflection curve and 247 thus can be modeled by deflection function. Numerical verification suggests that the proposed method can be adopted in preliminary design. (4) Two novel deployable cubic truss systems are proposed. They have much simpler stabilization procedure and much better structural behavior than existing deployable cable strut systems. Due to the use of cables, the cubic truss system can have similar structural behavior to space truss. They are classified as basic and enhanced systems which are suitable for small span and load, and large span and load, respectively. A rapidly assembled shelter formed by five deployable cubic panels is also proposed. (5) A simplified method is proposed for the static analysis of orthogonal truss systems. It overcomes the difficulties of existing simplified methods and combines their advantages together. It can be applied to all common support conditions, no matter how the supports are restrained in the horizontal direction. Solutions based on plate analogy method are obtained for the novel cubic system by deriving the equivalent flexural and shear stiffness. (6) Simplified dynamic analysis methods are proposed for truss system. Frequency formulae to cover all common boundary conditions are established based on proposed static solution and plate analogy method. A hand calculation formula is proposed for estimating the maximum earthquake response. Diagrams for estimating maximum blast response under different frequencies and self weights are established based on Dynamic load factor (DLF) method. 248 8.2 Recommendations for further work The following works are recommended for future study: (1) Investigation on wind response of radially arranged cable truss. A brief literature review has been conducted for simplified wind analysis and it is concluded from the experimental work that equivalent static load approach can be applied to normal truss system. For relatively flexible radially arranged cable truss, no detailed reports have been found. To investigate the validity and efficiency of the equivalent static load approach on this system, nonlinear time history analysis combined with wind tunnel experiment or computational fluid dynamic (CFD) analysis should be conducted. (2) Inelastic and post-buckling analysis. Elastic analysis and design is adopted in the current study where the structure’s failure is controlled by elastic capacity of the member, cable slackening and displacement limit. This approach is adopted in many design codes and can satisfy the need for the normal design of cable and truss systems. For radially arranged cable truss, strut buckling or cable slacking will make a whole structure unstable and collapse, and breakage of cable will not happen since the design code requires the maximum stress in cable to be less than half of the breaking strength of the cable. Thus nothing concerning inelastic analysis needs to be considered for cable truss. For truss system, one member’s failure usually cannot lead to the collapse of whole system (but leads to a large displacement which is not allowed in the design). The system may still be subjected to increased load when one strut begins to buckle or yield. To capture the 249 whole path of the structure response, inelastic and post-buckling analysis may need to be conducted. (3) Experimental study for novel cubic truss system. It is recommended that an experimental study on a prototype novel cubic truss system, especially for Type-C enhanced system, should be conducted before applying in the actual construction. The purpose of this work is to test the deployment, stabilizing and erection ability of the system with massive modules, and also to test joint strength and structure response under both static and dynamic aspects. (4) Exploration of new vibration control measure for spatial structures There are many discussions and devices concerning the vibration control of high –rise buildings where horizontal vibration control is the main task, but only a few is found for spatial structures where the main aim is to control vertical vibration. 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(1966). “A continuum analysis for double layer space frame shells.” Journal of IABSE, 26, 593–610. Yang, Y.B. and Kuo, S.R. (1994), Theory and analysis of nonlinear frame strutures, Printice hall, Singapore. 257 APPENDIX: SIMULATION MOVIES FOR TYPE-C ENHANCED DEPLOYABLE CUBIC TRUSS SYSTEM Simulation movies for Type-C enhanced deployable cubic truss system proposed in Chapter are contained in the CD attached. There are a total of six files: (1) Compact-plan.avi (2) Compact-elev.avi (3) Compact-perspective.avi (4) Deploy-plan.avi (5) Deploy-elev.avi (6) Deploy–perspective.avi The first three files record the compact process from deployed form to compact form in plan, elevation and perspective view respectively. The other three files record the deployment process from compact form to deployed form in plan, elevation and perspective view respectively. 258 [...]... some deployable cable strut systems based 2 on free standing cable strut systems have been proposed and studied, free standing cable strut systems can be further divided into non -deployable and deployable systems All these systems are briefly introduced below 1.2.2 Tension cable strut systems (1) Cable truss Cable truss has the longest history among all the cable strut systems One of the first such systems. .. not, cable strut systems can be broadly classified into two categories: tension cable strut systems and free standing cable strut systems Each category includes three kinds of system Cable truss, cable dome and suspend dome belong to the tension cable strut system; while hybrid truss cable system, tensegrity system, and some newly developed cable strut systems belong to the free standing cable systems. .. higher than cable structures due to the lower strength of strut In addition, from aesthetical point of view, its geometrical shape is less fascinating than cable structures By combination of cables and struts, cable strut systems are formed Cables are subjected to tension force while struts are subjected to compression by design Due to the combination effect, some disadvantages of the former two systems. .. two systems can be overcome and a better structural behavior can be achieved in cable strut systems Available cable strut systems are reviewed in the following sections It should be noted that cable supported truss roof is not included because cables only provide additional support for elements which themselves carry a major part of the load 1.2 Cable strut systems 1.2.1 Structural types Based on whether... researchers like Emmerich (1990), Hanaor (1992) and Motro (1992 and 1996) An example is shown in Figure 1.5 (3) Cable- strut truss The cable- strut truss denotes a group of newly developed cable strut systems developed by Wang (1998), Lee (2001) and Liew et al (2003) The main feature of these systems is that both cables and struts are continuous, and the whole system can be constructed side by side with simple... unit of cable truss is formed by two cables in opposite curvature counter-tensioned one against the other The shape is achieved by struts that keep the cables apart By arranging these units in a different way, three kinds of cable truss can be constructed: parallelly arranged cable truss, radically arranged cable truss and orthogonally arranged cable truss Since cable truss has many merits: simple and. .. diameter of 80m built in Denmark in 1974 and Guanhan Stadium with a diameter of 44m built in China in 1991 More details can be found in the books by Krishna (1978) and Buchholdt (1999) 3 (2) Cable dome Cable dome was first proposed by Geiger (1986) in his patent file It consists of ridge cables, diagonal cables, cable hoops and struts An outer ring beam is needed to balance the tension force in the ridge and. .. cable strut systems 1.2.3.1 Non -deployable structures (1) Hybrid truss cable system Hybrid truss cable system is a special case of beam string structure (BSS) BSS is firstly proposed by Saitoh (1987) It is a hybrid system formed by bottom flexible cables, upper stiff beam and middle connecting struts It is a combination of tension and stiff structure 4 for the purpose of overcoming the weakness of each... derived and numerical solution for M8-9 (under the loading of 0.5kN/m2) 46 Table 2.5 Comparison of strut force between derived and numerical solution for M10-6 (under the loading of 0.5kN/m2) 46 Table 2.6 Comparison of strut force between derived and numerical solution for M6-6 (under the loading of 10kN/m2) 47 Table 2.7 Comparison of strut force between derived and numerical... Figure 2.13 Three types of cable truss (struts are only shown in half of span) 53 xiv Figure 2.14 Comparison of analytical solution with various types of cable truss 54 Figure 2.15 Effects of sag ratio 55 Figure 2.16 Effects of ratio a 55 Figure 2.17 Effects of ratio b 55 Figure 2.18 Effects of spacing angle c 56 Figure 2.19 Effects of ratio es . Tension cable strut systems 3 1.2.3 Free standing cable strut systems 4 1.2.3.1 Non -deployable structures 4 1.2.3.2 Deployable structures 6 1.3 Structural analyses of the two focused systems. STRUCTURAL ANALYSIS AND DEPLOYABLE DEVELOPMENT OF CABLE STRUT SYSTEMS SONG JIANHONG NATIONAL UNIVERSITY OF SINGAPORE 2007 STRUCTURAL. 12 1.3.1 Analysis types and methods 12 1.3.2 Simplified analysis of cable truss 16 1.3.3 Simplified analysis of cable- strut truss 19 1.4 Objectives and scope 23 1.5 Organization of thesis