NONLINEAR ANALYSIS OF CABLE STRUCTURES By H. C. T. PEIRIS A THESIS SUBMITTED TO THE FACULTY OF ENGINEERING IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE DEPARTMENT OF CIVIL ENGINEERING UNIVERSITY OF MORATUWA SRI LANKA SEPTEMBER, 2008 - i - DECLARATION I hereby, declare, that the work included in this thesis in part or whole, has not been submitted for any other academic qualification at any institution. H. C. T. Peiris (Author) Certified by Dr. I. R. A. Weerasekera Supervisor/ Senior Lecturer Division of Building & Structural Engineering Department of Civil Engineering University of Moratuwa Sri Lanka - ii - ABSTRACT Large structures are widely used in the modern construction industry for infra-structure facilities development. Among these, long span structures with cables are becoming increasingly popular. In this category of structures deformations are large and estimations based on small deformation theory in the normal analysis are inadequate. The large deformation analysis results in nonlinear behavior where principle of superposition does not hold. In geometrical nonlinear analysis, the equations of equilibrium are based on the deformed geometry after the load application. The length of a curved deflected line is longer than the initial length and the basic assumptions used in linear analysis may cause inaccuracies when the deformations are very large. It is also essential that bending effects of cables are considered. Here we deal with large deformations, but small strain problems with linear stress- strain relationships. Although there are many methods found in literature to analyze cables exhibiting large deformation nonlinear behavior, it is hard to find a universal approach to describe the exact behavior of a cable considering all geometrical nonlinearity issues. The analysis described in this study recognizes all such influences contributing to geometrical non-linearity. The procedure developed is versatile and gives a state-of- the-art analytical tool. This work fills a void in the current practice recognizing large deformation issues without any knowledge of small or large strains as opposed to what is required in commercial software. A numerical solution procedure has been evolved to solve the resulting nonlinear non- homogeneous integral differential equation. The procedure is converging and a computer program has been developed for practical use. The results are compared with those in literature to validate the findings and to ensure the accuracy of the new large deformation nonlinear analysis technique. - iii - ACKNOWLEDGEMENT First and foremost I would like to express my gratitude and deep appreciation to My supervisor Dr. I.R.A Weerasekera, Senior Lecturer, Department of Civil Engineering, University of Moratuwa for his invaluable assistance, advice and guidance throughout this project. This association has been interesting and rewarding. My sincere thanks are also due to Prof. (Mrs) N. Rathnayake, Former Head and Prof. W. P. S. Dias, Present Head, Department of Civil Engineering, University of Moratuwa, for making available all resources and facilities for this research work.. My sincere thanks and gratitude are also due to Dr. G. T. F. De Silva, Former Vice Chancellor and Head, Department of Mathematics, University of Moratuwa, for his professional assistance to overcome the computational difficulties in solving mathematical equations of this research. The Senate Research Committee of University of Moratuwa should also be thanked for supporting and financing my research. I appreciate very much invaluable support, encouragement and understanding shown by my parents. Finally I would like to acknowledge with fraternal love my colleagues and others who have assisted me in various ways whose contribution have led to the successful completion of this thesis. H. C. T. Peiris - iv - DEDICATION To my parents and all those who are interested and committed in advancement of science - v - CONTENTS Declaration ii Abstract iii Acknowledgement iv Dedication v List of Figures x List of Tables xiv Abbreviations xv 1 Introduction 1 1.1 Background 1 1.2 Non-Linear Load – Deflection Response. 2 1.2.1 Material Non-Linearity. 3 1.2.2 Geometric Non-Linearity 3 1.3 Linear vs. Non-Linear Analysis. 3 1.4 Effect of Axial Load on the Stiffness. 6 1.5 Use of Cables in Structures 7 1.6 Objectives of the Research 11 1.7 Scope 11 1.8 Organisation of the Thesis 11 2 Literature Review 13 2.1 Introduction 13 2.2 Historical Aspects of the Large Deformation Structures 14 2.3 Analytical Methods 16 2.3.1 Analysis of a Suspended Cables under Point Loads 16 2.3.2 Analysis of a Suspended Cable Subjected to Uniformly Distributed Loading 18 2.3.2.1 Elastic Catenary 19 2.3.2.2 Elastic Parabola 22 2.3.2.3 Summary 23 2.4 P-Delta Analysis for Large Deformation Structures 24 - vi - 2.4.1 Effect on Stiffness of a Flexural member Subjected to a Tensile Load. 24 2.4.2 Effect on Stiffness of a Flexural member Subjected to Compression Loading 26 2.4.3 Summary on P-delta Analysis 28 2.5 Finite Element Methods 29 2.5.1 Tangent Stiffness Matrix of a member of plane or space truss. 29 2.5.1.1 Solution Procedure by Using Newton-Raphson’s Technique. 34 2.5.1.2 Calculation in One Iteration Cycle 35 2.5.1.3 Convergence Criteria 37 2.5.1.4 Summary 37 2.5.2 Tangent Stiffness Matrix of a Member of Plane Frame. 38 2.5.2.1. Solution Procedure for Plane Frame Elements by Using Newton- Raphson’s Technique. 42 2.5.3 The Total Lagrangian Bar Element 46 2.5.3.1 Element Kinematics 47 2.5.3.2 Strain Measure 49 2.5.3.3 Stress Measure 50 2.5.3.4 The Tangent Stiffness Matrix 51 2.5.3.5 Summary 53 2.5.4 Catenary Cable Element 53 2.5.4.1 Summary 55 2.5.5 Three-Node Isoparametric Cable Finite Element 55 2.5.5.1 Solution Procedure 61 2.5.5.2 Convergence Criteria 62 2.5.5.3 Summary 62 2.6 Nonlinear Analysis Using Integrated Force Method 63 2.6.1 Introduction 63 2.6.2 Basic theory of integrated force method 64 2.6.3 Nonlinear analysis by integrated force method 66 2.6.4 Development of elemental matrices 70 2.6.5 Summary 71 2.7 The Theory of Elastica 72 2.7.1 The Large Deformation of a Cantilever Beam with concentrated Load. 72 - vii - 2.7.2 Flexible Cantilever Beams Subjected to Distributed and Combined Loadings. 76 2.7.3 Uniform Simply Supported Beam Subjected to a Concentrated Load 79 2.7.4 Simply Supported Beams Loaded with a Uniformly Distributed Loading 82 2.7.5 Statically Indeterminate Flexible Bars Subjected to a Concentrated Load. 84 2.7.6 Statically Indeterminate Single Span Beams Subjected to Distributed Loadings 87 2.7.7 Summary 89 2.8 Non-Linear Analysis in SAP 2000 Non-Linear 90 2.8.1 Nonlinear analyze cases 90 2.8.2 P-Delta forces in the Frame Element. 91 2.8.3 Initial P-Delta Analysis 92 2.8.4 Large Displacement Analysis 94 2.8.5 Initial Large –Displacement Analysis 94 2.9 Modified Elastic Stiffness 95 2.9.1 Other Formulae for Equivalent Modulus of Elasticity 97 2.9.2 Summary 98 2.10 Limitations in the Available Methods 99 2.11 Summary 101 3 Methodology 102 3.1 Introduction 102 3.2 Static Behaviour of a Cable 103 3.3 Finite Difference Solution for Differential Equations 108 3.3.1 Introduction 108 3.3.2 Representation of derivatives by finite differences. 109 3.3.3 Errors in finite-difference equations 110 3.4 Solution of Equations Using Gauss-Seidel Method 112 3.4.1 Introduction 112 3.4.2 Gauss-Seidel method 112 3.5 Polynomial Curve Fitting Near the Boundaries 113 3.6 Identification of the Problem 115 - viii - 3.6.1 Cable Properties, Coordinate system, Sign Convention 115 3.6.2 Loading 115 3.6.3 Boundary Conditions 117 3.7 Solution Procedure 117 3.8 Flow Chart for the Computer Programme 134 3.9 Summary 137 4 Results and Analysis 138 4.1 Sensitivity of the Stiffness of Cables against Different Problem parameters 139 4.2 Validity of Superposition 147 4.3 Large Deformation vs Small Deformation. 152 4.4 Bending of Cables 156 4.5 Effect of Sag on Axial Stiffness 161 4.6 Comparison with the Catenary and Parabolic Approximations. 164 4.7 Comparison with the Finite Element Modeling 168 4.8 Summary 172 5 Conclusions and Recommendations 173 5.1 Introduction 173 5.2 Conclusions 173 5.3 Recommendations for Further Research 177 References 178 Appendices A. Estimation of End Forces Caused by End Displacement in P-Delta Analysis 182 B. Computer Program for Large Deformation Geometrically Nonlinear Analysis of Cable Structures. 183 - ix - LIST OF FIGURES Figure 1.1 Load deflection response diagram 2 Figure 1.2 General Deformation of a member 5 Figure 1.3 Deformation of a Cantilever Subjected to an Axial Force and a Point Load 6 Figure 1.4 General representation of a Structure Response Curve 7 Figure 1.5 a) Normandy Bridge b) Ikuchi Bridge, 8 Figure 1.6 a) Akashi-kaikyo bridge b) Humber Bridge 9 Figure 1.7 a) Millennium Dome b) Georgia Dome 9 Figure 1.8 a) Suspending cables in power transmission b) Guyed Towers 10 Figure 1.9 a) Ribbon stressed bridge, b) Lateral stiffening by using cables, c) Support structure for a pipe 10 Figure 2.1 Reference configuration 16 Figure 2.2 Deformed configuration 16 Figure 2.3 Suspended cable subjected to self weight and axial tension 19 Figure 2.4 Elastic Catenary / Parabola 21 Figure 2.5 Effects of tensile forces on bending deformations 24 Figure 2.6 Effects of compressive axial forces on bending deformations 26 Figure 2.7 Variation of S vs µ 27 Figure 2.8 Variation of C vs µ 28 Figure 2.9 Derivation of the geometric stiffness matrices for a truss 30 Figure 2.10 Derivation of tangent stiffness matrix for a plane frame member 40 Figure 2.11 Member end-forces. 43 Figure 2.12 A plane truss structure undergoing large displacements 46 Figure 2.13 The Geometrically nonlinear, two-node, two- dimensional bar element in total Lagrangian description. 47 Figure 2.14 The definition of displacement fields 48 Figure 2.15 Geometric Interpretations of the Quantities used in Element Kinematics 49 Figure 2.16 Forces on a catenary cable finite element 53 Figure 2.17 Geometry of the proposed Cable Finite Element 55 - x - Figure 2.18 (a) General displacement of the element (b) Displacement of a arbitrary differential element 56 Figure2.19 2D element for nonlinear analysis 70 Figure 2.20 (a) Large deformation of a cantilever beam of uniform cross section; (b) Free body diagram of a beam element. 74 Figure 2.21 (a) Uniform cantilever beam loaded with a distributed loading k ; (b) uniform cantilever beam; padded with a distributed load and a concentrated load P at the free end. 76 k Figure 2.22 Simply supported beam loaded with a concentrated load P at any point along its length. 80 Figure 2.23 Straight and deflected configuration of a simply supported beam loaded with a uniformly distributed load. 83 Figure 2.24 Statically indeterminate beam loaded with a concentrated load P at any point along its length. 85 Figure 2.25 Straight and deflected configuration of a statically indeterminate beam loaded with a uniformly distributed load. 87 Figure 2.26 (a) Parabolic profile of the cable (b) Typical element from the cable taken from the node 2 95 Figure 3.1 Cables subjected to different point loads 106 Figure 3.2 Different distributed loads on the cables 106 Figure 3.3 (a) deformed configuration for end rotations (b) Deformed configuration for combined end translation and rotations 107 Figure 3.4 Graph of function y=f(x) 109 Figure 3.5 Loading, Deformed configuration and other problem parameters 116 Figure 3.6 Sub-divisions of the Cable for apply the Finite Difference Schemes 124 Figure 3.7 Applying corrections to the Existing End Moments 126 Figure 3.8 Calculating Axial Force along the Tangential direction 131 Figure 3.9 Flow Chart for Arrangement of the Computer Programme 134 Figure 3.10 Flowchart for Subroutine 01 135 Figure 3.11 Flowchart for Subroutine 02 136 - xi - Figure 4.1 Comparison of the variation of S vs µ for different number of nodes in finite difference scheme 140 Figure 4.2 Comparison of the variation of C vs µ for different number of nodes in finite difference scheme 140 Figure 4.3 Comparison of the variation of S vs µ for members having different lengths 142 Figure 4.4 Comparison of the variation of C vs µ for members having different lengths 142 Figure 4.5 Comparison of the variation of S vs µ for members subjected to different end rotations 143 Figure 4.6 Comparison of the variation of C vs µ for members subjected to different end rotations 143 Figure 4.7 Variation of S vs µ for members subjected to different uniformly distributed loads along the projected length 144 Figure 4.8 Variation of C vs µ for members subjected to different uniformly distributed loads along the projected length 145 Figure 4.9 Variation of S vs µ for members subjected to different uniformly distributed loads along the deformed length (catenary loading) 145 Figure 4.10 Variation of C vs µ for members subjected to different uniformly distributed loads along the deformed length (catenary loading) 146 Figure 4.11 Variation of Fixed End Moment vs µ for different uniformly distributed loads along a projected length of the cable. 148 Figure 4.12 Variation of Fixed End Moment vs µ for different uniformly distributed loads along the Deformed length of the cable. 149 Figure 4.13 Comparison between the superimposed values and directly analyzed values. 149 Figure 4.14 Comparison of Fixed End Moment at node 01 vs µ , between the directly analyzed values and the superposed values 150 Figure 4.15 Comparison of Fixed End Moment at node 02 vs µ between the directly analyzed values and the superposed values for a W=50 kN/m UDL and a Unit rotation at node 02 151 - xii - Figure 4.16 Undefromed lengths vs µ of the member subjected to W=50 kN/m UDL, Unit rotation at node 02, and the combination of loading 151 Figure 4.17 Variation of the support moments vs µ for small and large displacement analysis 153 Figure 4.18 Variation of the undeformed length vs µ for small and large displacement analysis 154 Figure 4.19 Deformed configuration obtained from large deformation analysis and small deformation analysis 155 Figure 4.20 Variation of the Fixed End Moments vs µ for different small and large displacement analysis 156 Figure 4.21 a) deformed configurations for live load and dead loads, b)Variation of bending moments along the length of a cable and a beam. 158 Figure 4.22 a) deformed configurations for dead load and lateral deflection of the support b)Variation of bending moments along the length of a cable and a beam. 160 Figure 4.23 Applying incremental axial deformations to initially loaded cables 161 Figure 4.24 Variation of axial forces and sag vs axial deformation 162 Figure 4.25 Variation of axial forces and sag vs axial deformation 162 Figure 4.26 Comparison of the variation of axial forces vs axial deformation 164 Figure 4.27 a) Definition of the compared parameters. b) Deformed shape of a stay cable of Normandy Bridge from the output of the program. 165 Figure 4.28 Definition of the problem parameters for comparison of finite element analysis verses the developed analytical technique. 168 Figure 4.29 Sensitivity of the sag for the variation of the undeformed length 171 Figure 4.30 Sensitivity of the axial force 171 - xiii - LIST OF TABLES Table 2.1 Maximum Sags calculated using catenary and parabolic methods. 18 Table 4.1 Bending Moments and Displacements Due to Live loads of cable stressed structures 159 Table 4.2 Bending moments at supports due to unit vertical displacement at the support 1 161 Table 4.3 Comparison of the Deformational Characteristics of actual Cables vs the Results obtained by the new technique 166 Table 4.4 representation of the results of deformed cables in (Table ) assuming a UDL along a projected length of the cable by using both analytical techniques and the new technique. 167 Table 4.5 Comparison between finite element method and the new analysis technique in analysis of the cables 169 - xiv - ABBRIVIATIONS A Cross sectional area Aθ Matrix contains the derivatives of the displacements B Strain - displacement gradient relationship matirix / Equilibrium matrix C Carry over factor CC Compatibility conditions [ ]cC Compatibility matrix D Nodal coordinate vector DDR Deformation displacement relationship 1[ ]fD , 2[ ]fD differential operator matrix E Modulus of elasticity ,e ε Material strain 1 2,e e errors in finite difference EQE Equivalent modulus of elasticity f external load vector F Independent member forces rF Nodal forces due to temperature effects GL Green Lagrange [ ]sG Flexibility matrix of the whole structure [ ]tG geometry matrix h Interval of finite difference technique I Second Moment of area i,j,k,it,iii,ii,r integers IE Internal energy IFM Integrated Force Method [J] Jacobian transformation matrix - xv - k Uniformly distributed load along the member.(Catenary udl) [K], [S] Stiffness matrix [KG],[ ]gS Geometrical Stiffness matrix [KM],[ Elastic/ Material Stiffness matrix ]eS l Horizontal distance between the supports L Span CL Curved length of the member 0L Original Undeformed length of the cable UL Undeformed length of the cable (in an iteration) M Bending moment n Number of divisions of the member in finite difference N Axial force Ni / Hi Shape function p surface forces per unit area of the deformed body A , P Force / Point load ( )p x Polynomial curve approximation for the profile eqP equivalent nodal forces due to { }q and { }p . q body forces per unit mass Q Unbalance force/ Out of balance force Radius of curvature r s Length of the member measured along the profile S Rotational stiffness ,s c Sine and cosines [ ]eS elastic stiffness matrix of the node [ ]gS geometric stiffness matrix of the node [ ]tS Tangential stiffness matrix of the node - xvi - Stdv Standard deviation T Tensile force [t] Transformation matrix 1T 2Τ Force components along the axis of the cable TL Total Lagrangian u Displacement within the element V Volume w Uniformly distributed load along a projected line of the member. W Total load along the cable X→ Coordinate axis for x measurements x Measurement along X/ measurement in x direction of the of other points in finite difference technique when considered point is measured as X X measurement in x direction of the considered point in finite difference technique Y→ Coordinate axis for y measurements y Measurement along Y/ measurement in y direction of the of other points in finite difference technique when considered point is measured as Y Y measurement in y direction of the considered point in finite difference technique Z→ Coordinate axis for z measurements Measurement along Z z α /T EI β Constant in finite difference γ Constant in finite difference Small increment ∆ δ Linear displacement of a point ε* effective strain η Inclination of direction of load w to the normal axis of the cable - xvii - θ Angular deformation measurement ϑ generalized internal deformations of the element λ cosines of the angle between the member local axis and the global axes µ Non dimensional stress parameter ξ Isoparametric coordinate π Total potential energy functional ρ mass density and σ Axial stress υ Tolerance value φ Inclination of direction of load k to the normal axis of the cable ϕ Inclination of the cable to the horizontal Φ Scalar function [ ]Ω Compliance matrix ω Angle of deviation at supports of a pinned supported member ( )0 (superscript) Value at the start of the current cycle ( )1 (superscript) Refers to the state at cycle end in , 0 (subscript, superscript) Initial states of the problem e (subscript) use to symbolize elemental properties ,L NL (subscript) Linear, Nonlinear component [] Brackets indicate rectangular or square matrix. {} Curly brackets indicate a vector. ' , " First, Second derivative - xviii -