Theory of sTrucTures

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CONTENTS

Preface ........ V

I INTRODUCTION

1 THE PURPOSE AND SCOPE OF THEORY OF

STRUCTURES ........ 11.1 General ........ 11.2 The basis of theory of structures ........ 11.3 Methods of theory of structures ........ 21.4 Statics and structural dynamics ........ 31.5 Theory of structures and structural

engineering ........ 3

2 BRIEF HISTORICAL BACKGROUND ........ 5

II FUNDAMENTALS

3 DESIGN OF STRUCTURES ........ 113.1 General ........ 113.2 Conceptual design ........ 113.3 Service criteria agreement and basis of

design ........ 143.4 Summary ........ 263.5 Exercises ........ 27

4 STRUCTURAL ANALYSIS AND

DIMENSIONING ........ 294.1 General ........ 294.2 Actions ........ 294.2.1 Actions and action effects ........ 294.2.2 Models of actions and representative values ........ 304.3 Structural models ........ 314.4 Limit states ........ 314.5 Design situations and load cases ........ 324.6 Verifications ........ 334.6.1 Verification concept ........ 334.6.2 Design values ........ 334.6.3 Verification of structural safety ........ 344.6.4 Verification of serviceability ........ 354.7 Commentary ........ 354.8 Recommendations for the structural

calculations ........ 364.9 Recommendations for the technical report ........ 384.10 Summary ........ 404.11 Exercises ........ 41

5 STATIC RELATIONSHIPS ........ 435.1 Force systems and equilibrium ........ 435.1.1 Terminology ........ 435.1.2 Force systems ........ 445.1.3 Equilibrium ........ 455.1.4 Overall stability ........ 455.1.5 Supports ........ 475.1.6 Hinges ........ 505.1.7 Stress resultants ........ 515.2 Stresses ........ 535.2.1 Terminology ........ 535.2.2 Uniaxial stress state ........ 535.2.3 Coplanar stress states ........ 545.2.4 Three-dimensional stress states ........ 575.3 Differential structural elements ........ 615.3.1 Straight bars ........ 615.3.2 Bars in single curvature ........ 625.4 Summary ........ 685.5 Exercises ........ 69

6 KINEMATIC RELATIONSHIPS ........ 716.1 Terminology ........ 716.2 Coplanar deformation ........ 726.3 Three-dimensional deformation state ........ 746.4 Summary ........ 766.5 Exercises ........ 77

7 CONSTITUTIVE RELATIONSHIPS ........ 797.1 Terminology ........ 797.2 Linear elastic behaviour ........ 817.3 Perfectly plastic behaviour ........ 837.3.1 Uniaxial stress state ........ 837.3.2 Three-dimensional stress states ........ 847.3.3 Yield conditions ........ 857.4 Time-dependent behaviour ........ 907.4.1 Shrinkage ........ 907.4.2 Creep and relaxation ........ 917.5 Thermal deformations ........ 947.6 Fatigue ........ 947.6.1 General ........ 947.6.2 S-N curves ........ 957.6.3 Damage accumulation under fatigue loads ........ 967.7 Summary ........ 987.8 Exercises ........ 99

VIIContents

Theory of Structures. First Edition. Peter Martic 2013 Ernst & Sohn GmbH & Co. KG. Published 2013 by Ernst & Sohn GmbH & Co. KG.

8 ENERGY METHODS ........ 1018.1 Introductory example ........ 1018.1.1 Statically determinate system ........ 1018.1.2 Statically indeterminate system ........ 1038.1.3 Work equation ........ 1048.1.4 Commentary ........ 1058.2 Variables and operators ........ 1058.2.1 Introduction ........ 1058.2.2 Plane framed structures ........ 1078.2.3 Spatial framed structures ........ 1098.2.4 Coplanar stress states ........ 1108.2.5 Coplanar strain state ........ 1118.2.6 Slabs ........ 1118.2.7 Three-dimensional continua ........ 1138.2.8 Commentary ........ 1148.3 The principle of virtual work ........ 1158.3.1 Virtual force and deformation variables ........ 1158.3.2 The principle of virtual deformations ........ 1158.3.3 The principle of virtual forces ........ 1158.3.4 Commentary ........ 1168.4 Elastic systems ........ 1188.4.1 Hyperelastic materials ........ 1188.4.2 Conservative systems ........ 1198.4.3 Linear elastic systems ........ 1258.5 Approximation methods ........ 1288.5.1 Introduction ........ 1288.5.2 The RITZ method ........ 1298.5.3 The GALERKIN method ........ 1328.6 Summary ........ 1348.7 Exercises ........ 135

III LINEAR ANALYSIS OF FRAMED STRUCTURES

9 STRUCTURAL ELEMENTS AND

TOPOLOGY ........ 1379.1 General ........ 1379.2 Modelling of structures ........ 1379.3 Discretised structural models ........ 1409.3.1 Description of the static system ........ 1409.3.2 Joint equilibrium ........ 1419.3.3 Static determinacy ........ 1429.3.4 Kinematic derivation of the equilibrium

matrix ........ 1449.4 Summary ........ 1479.5 Exercises ........ 147

10 DETERMINING THE FORCES ........ 14910.1 General ........ 14910.2 Investigating selected free bodies ........ 15010.3 Joint equilibrium ........ 15410.4 The kinematic method ........ 15610.5 Summary ........ 15810.6 Exercises ........ 158

11 STRESS RESULTANTS AND

STATE DIAGRAMS ........ 15911.1 General ........ 15911.2 Hinged frameworks ........ 16011.2.1 Hinged girders ........ 16111.2.2 Hinged arches and frames ........ 16311.2.3 Stiffened beams with intermediate hinges ........ 16511.3 Trusses ........ 16611.3.1 Prerequisites and structural topology ........ 16611.3.2 Methods of calculation ........ 16911.3.3 Joint equilibrium ........ 16911.3.4 CREMONA diagram ........ 17111.3.5 RITTER method of sections ........ 17211.3.6 The kinematic method ........ 17311.4 Summary ........ 17411.5 Exercises ........ 175

12 INFLUENCE LINES ........ 17712.1 General ........ 17712.2 Determining influence lines by means of

equilibrium conditions ........ 17812.3 Kinematic determination of influence lines ........ 17912.4 Summary ........ 18312.5 Exercises ........ 183

13 ELEMENTARY DEFORMATIONS ........ 18513.1 General ........ 18513.2 Bending and normal force ........ 18513.2.1 Stresses and strains ........ 18513.2.2 Principal axes ........ 18713.2.3 Stress calculation ........ 18913.2.4 Composite cross-sections ........ 19013.2.5 Thermal deformations ........ 19213.2.6 Planar bending of curved bars ........ 19313.2.7 Practical advice ........ 19413.3 Shear forces ........ 19413.3.1 Approximation for prismatic bars subjected to

pure bending ........ 19413.3.2 Approximate coplanar stress state ........ 19613.3.3 Thin-wall cross-sections ........ 19713.3.4 Shear centre ........ 19913.4 Torsion ........ 20013.4.1 Circular cross-sections ........ 20013.4.2 General cross-sections ........ 20113.4.3 Thin-wall hollow cross-sections ........ 20413.4.4 Warping torsion ........ 20713.5 Summary ........ 21613.6 Exercises ........ 218

14 SINGLE DEFORMATIONS ........ 22114.1 General ........ 22114.2 The work theorem ........ 22214.2.1 Introductory example ........ 22214.2.2 General formulation ........ 22314.2.3 Calculating the passive work integrals ........ 22314.2.4 Systematic procedure ........ 226

VIII CONTENTS

14.3 Applications ........ 22614.4 MAXWELL’s theorem ........ 23014.5 Summary ........ 23114.6 Exercises ........ 231

15 DEFORMATION DIAGRAMS ........ 23315.1 General ........ 23315.2 Differential equations for straight bar

elements ........ 23315.2.1 In-plane loading ........ 23315.2.2 General loading ........ 23515.2.3 The effect of shear forces ........ 23515.2.4 Creep, shrinkage and thermal

deformations ........ 23515.2.5 Curved bar axes ........ 23515.3 Integration methods ........ 23615.3.1 Analytical integration ........ 23615.3.2 MOHR’s analogy ........ 23815.5 Exercises ........ 243

16 THE FORCE METHOD ........ 24516.1 General ........ 24516.2 Structural behaviour of statically indeterminate

systems ........ 24516.2.1 Overview ........ 24516.2.2 Statically determinate system ........ 24616.2.3 System with one degree of static

indeterminacy ........ 24716.2.4 System with two degrees of static

indeterminacy ........ 24916.2.5 In-depth analysis of system with one degree of

static indeterminacy ........ 25016.2.6 In-depth analysis of system with two degrees of

static indeterminacy ........ 25316.3 Classic presentation of the force method ........ 25416.3.1 General procedure ........ 25416.3.2 Commentary ........ 25516.3.3 Deformations ........ 25716.3.4 Influence lines ........ 25916.4 Applications ........ 26216.5 Summary ........ 27216.6 Exercises ........ 274

17 THE DISPLACEMENT METHOD ........ 27717.1 Independent bar end variables ........ 27717.1.1 General ........ 27717.1.2 Member stiffness relationship ........ 27717.1.3 Actions on bars ........ 27817.1.4 Algorithm for the displacement method ........ 28017.2 Complete bar end variables ........ 28117.2.1 General ........ 28117.2.2 Member stiffness relationship ........ 28217.2.3 Actions on bars ........ 28317.2.4 Support force variables ........ 28317.3 The direct stiffness method ........ 28417.3.1 Incidence transformation ........ 284

17.3.2 Rotational transformation ........ 28517.3.3 Algorithm for the direct stiffness method ........ 28617.4 The slope-deflection method ........ 29017.4.1 General ........ 29017.4.2 Basic states and member end moments ........ 29217.4.3 Equilibrium conditions ........ 29317.4.4 Applications ........ 29417.4.5 Restraints ........ 29817.4.6 Influence lines ........ 30317.4.7 CROSS method of moment distribution ........ 30517.5 Summary ........ 30917.6 Exercises ........ 310

18 CONTINUOUS MODELS ........ 31118.1 General ........ 31118.2 Bar extension ........ 31118.2.1 Practical examples ........ 31118.2.2 Analytical model ........ 31218.2.3 Residual stresses ........ 31418.2.4 Restraints ........ 31518.2.5 Bond ........ 31618.2.6 Summary ........ 32018.3 Beams in shear ........ 32118.3.1 Practical examples ........ 32118.3.2 Analytical model ........ 32118.3.3 Multi-storey frame ........ 32118.3.4 VIERENDEEL girder ........ 32318.3.5 Sandwich panels ........ 32418.3.6 Summary ........ 32618.4 Beams in bending ........ 32618.4.1 General ........ 32618.4.2 Analytical model ........ 32718.4.3 Restraints ........ 32718.4.4 Elastic foundation ........ 32918.4.5 Summary ........ 33218.5 Combined shear and bending response ........ 33318.5.1 General ........ 33318.5.2 Shear wall - frame systems ........ 33418.5.3 Shear wall connection ........ 33818.5.4 Dowelled beams ........ 34218.5.5 Summary ........ 34418.6 Arches ........ 34518.6.1 General ........ 34518.6.2 Analytical model ........ 34518.6.3 Applications ........ 34618.6.4 Summary ........ 35018.7 Annular structures ........ 35018.7.1 General ........ 35018.7.2 Analytical model ........ 35118.7.3 Applications ........ 35218.7.4 Edge disturbances in cylindrical shells ........ 35318.7.5 Summary ........ 35418.8 Cables ........ 35418.8.1 General ........ 35418.8.2 Analytical model ........ 35518.8.3 Inextensible cables ........ 357

IXContents

18.8.4 Extensible cables ........ 35818.8.5 Axial stiffness of laterally loaded cables ........ 36018.8.6 Summary ........ 36018.9 Combined cable-type and bending response ........ 36118.9.1 Analytical model ........ 36118.9.2 Bending-resistant ties ........ 36218.9.3 Suspended roofs and stress ribbons ........ 36318.9.4 Suspension bridges ........ 36818.9.5 Summary ........ 36818.10 Exercises ........ 369

19 DISCRETISED MODELS ........ 37119.1 General ........ 37119.2 The force method ........ 37219.2.1 Complete and global bar end forces ........ 37219.2.2 Member flexibility relation ........ 37219.2.3 Actions on bars ........ 37419.2.4 Algorithm for the force method ........ 37419.2.5 Comparison with the classic force method ........ 37619.2.6 Practical application ........ 37619.2.7 Reduced degrees of freedom ........ 37619.2.8 Supplementary remarks ........ 37919.3 Introduction to the finite element method ........ 38119.3.1 Basic concepts ........ 38119.3.2 Element matrices ........ 38119.3.3 Bar element rigid in shear ........ 38119.3.4 Shape functions ........ 38519.3.5 Commentary ........ 38619.4 Summary ........ 38619.5 Exercises ........ 387

IV NON-LINEAR ANALYSIS OF FRAMED

STRUCTURES

20 ELASTIC-PLASTIC SYSTEMS ........ 38920.1 General ........ 38920.2 Truss with one degree of static

indeterminacy ........ 38920.2.1 Single-parameter loading ........ 38920.2.2 Dual-parameter loading and generalisation ........ 39520.3 Beams in bending ........ 39820.3.1 Moment-curvature diagrams ........ 39820.3.2 Simply supported beams ........ 39920.3.3 Continuous beams ........ 40320.3.4 Frames ........ 40420.3.5 Commentary ........ 40520.4 Summary ........ 40620.5 Exercises ........ 407

21 LIMIT ANALYSIS ........ 40921.1 General ........ 40921.2 Upper- and lower-bound theorems ........ 41021.2.1 Basic concepts ........ 41021.2.2 Lower-bound theorem ........ 41021.2.3 Upper-bound theorem ........ 41121.2.4 Compatibility theorem ........ 411

21.2.5 Consequences of the upper- and lower-boundtheorems ........ 411

21.3 Static and kinematic methods ........ 41221.3.1 General ........ 41221.3.2 Simply supported beams ........ 41321.3.3 Continuous beams ........ 41521.3.4 Plane frames ........ 41621.3.5 Plane frames subjected to transverse loads ........ 42121.4 Plastic strength of materials ........ 42621.4.1 General ........ 42621.4.2 Skew bending ........ 42621.4.3 Bending and normal force ........ 42821.4.4 Bending and torsion ........ 43221.4.5 Bending and shear force ........ 43421.5 Shakedown and limit loads ........ 43521.6 Dimensioning for minimum weight ........ 43721.6.1 General ........ 43721.6.2 Linear objective function ........ 43821.6.3 FOULKES mechanisms ........ 43821.6.4 Commentary ........ 44021.7 Numerical methods ........ 44121.7.1 The force method ........ 44121.7.2 Limit load program ........ 44221.7.3 Optimum design ........ 44421.8 Summary ........ 44621.9 Exercises ........ 447

22 STABILITY ........ 44922.1 General ........ 44922.2 Elastic buckling ........ 44922.2.1 Column deflection curve ........ 44922.2.2 Bifurcation problems ........ 45322.2.3 Approximation methods ........ 45422.2.4 Further considerations ........ 46022.2.5 Slope-deflection method ........ 46522.2.6 Stiffness matrices ........ 46922.3 Elastic-plastic buckling ........ 47122.3.1 Concentrically loaded columns ........ 47122.3.2 Eccentrically loaded columns ........ 47422.3.3 Limit loads of frames according to second-order

theory ........ 47722.4 Flexural-torsional buckling and lateral

buckling ........ 48022.4.1 Basic concepts ........ 48022.4.2 Concentric loading ........ 48222.4.3 Eccentric loading in the strong plane ........ 48322.4.4 General loading ........ 48522.5 Summary ........ 48822.6 Exercises ........ 489

V PLATES AND SHELLS

23 PLATES ........ 49123.1 General ........ 49123.2 Elastic plates ........ 49123.2.1 Stress function ........ 491

X CONTENTS

23.2.2 Polar coordinates ........ 49323.2.3 Approximating functions for displacement

components ........ 49623.3 Reinforced concrete plate elements ........ 49623.3.1 Orthogonal reinforcement ........ 49623.3.2 General reinforcement ........ 50023.4 Static method ........ 50123.4.1 General ........ 50123.4.2 Truss models ........ 50123.4.3 Discontinuous stress fields ........ 50523.4.4 Stringer-panel model ........ 51123.5 Kinematic method ........ 51223.5.1 Applications in reinforced concrete ........ 51223.5.2 Applications in geotechnical engineering ........ 51723.6 Summary ........ 52023.7 Exercises ........ 522

24 SLABS ........ 52524.1 Basic concepts ........ 52524.1.1 General ........ 52524.1.2 Static relationships ........ 52524.1.3 Kinematic relationships ........ 53124.2 Linear elastic slabs rigid in shear with small

deflections ........ 53324.2.1 Fundamental relationships ........ 53324.2.2 Methods of solution ........ 53524.2.3 Rotationally symmetric problems ........ 53624.2.4 Rectangular slabs ........ 53924.2.5 Flat slabs ........ 54324.2.6 Energy methods ........ 54624.3 Yield conditions ........ 54724.3.1 VON MISES and TRESCA yield

conditions ........ 54724.3.2 Reinforced concrete slabs ........ 55024.4 Static method ........ 55724.4.1 Rotationally symmetric problems ........ 55724.4.2 Moment fields for rectangular slabs ........ 56024.4.3 Strip method ........ 56324.5 Kinematic method ........ 56724.5.1 Introductory example ........ 56724.5.2 Calculating the dissipation work ........ 56824.5.3 Applications ........ 56924.6 The influence of shear forces ........ 57224.6.1 Elastic slabs ........ 57224.6.2 Rotationally symmetric VON MISES slabs ........ 57424.6.3 Reinforced concrete slabs ........ 57524.7 Membrane action ........ 57524.7.1 Elastic slabs ........ 57524.7.2 Perfectly plastic slab strip ........ 57724.7.3 Reinforced concrete slabs ........ 57824.8 Summary ........ 58124.9 Exercises ........ 583

25 FOLDED PLATES ........ 58725.1 General ........ 58725.2 Prismatic folded plates ........ 588

25.2.1 Sawtooth roofs ........ 58825.2.2 Barrel vaults ........ 58925.2.3 Commentary ........ 59325.3 Non-prismatic folded plates ........ 59425.4 Summary ........ 59425.5 Exercises ........ 594

26 SHELLS ........ 59526.1 General ........ 59526.2 Membrane theory for surfaces of revolution ........ 59626.2.1 Symmetrical loading ........ 59626.2.2 Asymmetric loading ........ 60026.3 Membrane theory for cylindrical shells ........ 60126.3.1 General relationships ........ 60126.3.2 Pipes and barrel vaults ........ 60226.3.3 Polygonal domes ........ 60426.4 Membrane forces in shells of any form ........ 60626.4.1 Equilibrium conditions ........ 60626.4.2 Elliptical problems ........ 60726.4.3 Hyperbolic problems ........ 60826.5 Bending theory for rotationally symmetric

cylindrical shells ........ 61326.6 Bending theory for shallow shells ........ 61526.6.1 Basic concepts ........ 61526.6.2 Differential equation for deflection ........ 61626.6.3 Circular cylindrical shells subjected to

asymmetric loading ........ 61726.7 Bending theory for symmetrically loaded

surfaces of revolution ........ 62026.7.1 Basic concepts ........ 62026.7.2 Differential equation for deflection ........ 62026.7.3 Spherical shells ........ 62126.7.4 Approximation for shells of any form ........ 62326.8 Stability ........ 62326.8.1 General ........ 62326.8.2 Bifurcation loads ........ 62426.8.3 Commentary ........ 62626.9 Summary ........ 62726.10 Exercises ........ 628

APPENDIX

A1 DEFINITIONS ........ 631A2 NOTATION ........ 637A3 PROPERTIES OF MATERIALS ........ 643A4 GEOMETRICAL PROPERTIES OF

SECTIONS ........ 645

A5 MATRIX ALGEBRA ........ 649A5.1 Terminology ........ 649A5.2 Algorithms ........ 650A5.3 Linear equations ........ 652A5.4 Quadratic forms ........ 652A5.5 Eigenvalue problems ........ 653A5.6 Matrix norms and condition numbers ........ 654

XIContents

A6 TENSOR CALCULUS ........ 655A6.1 Introduction ........ 655A6.2 Terminology ........ 655A6.3 Vectors and tensors ........ 656A6.4 Principal axes of symmetric second-order

tensors ........ 658A6.5 Tensor fields and integral theorems ........ 658

A7 CALCULUS OF VARIATIONS ........ 661A7.1 Extreme values of continuous functions ........ 661A7.2 Terminology ........ 661A7.3 The simplest problem of calculus of

variations ........ 662A7.4 Second variation ........ 663A7.5 Several functions required ........ 664A7.6 Higher-order derivatives ........ 664A7.7 Several independent variables ........ 665A7.8 Variational problems with side conditions ........ 665A7.9 The RITZ method ........ 666A7.10 Natural boundary conditions ........ 667

REFERENCES ........ 669

NAME INDEX ........ 671

SUBJECT INDEX ........ 673

XII CONTENTS

EXAMPLECOLLECTION

Example 3.1 Service criteria agreement for industrial building XY in Z ........ 15Example 3.2 Basis of design for industrial building XY in Z ........ 19Example 5.1 Cantilever retaining wall ........ 45Example 5.2 Support envelope ........ 47Example 5.3 Steel plate ........ 56Example 5.4 Stress tensor ........ 59Example 5.5 Hoop stress formula ........ 63Example 5.6 Thrust line ........ 63Example 5.7 Three-hinged arch ........ 65Example 5.8 Beam as circular arc ........ 67Example 6.1 Measuring grid ........ 73Example 7.1 Time-independent restraint ........ 93Example 7.2 Time-dependent restraint ........ 93Example 7.3 Prestressing ........ 93Example 7.4 Loss of prestress ........ 93Example 7.5 Fatigue of reinforcing steel ........ 97Example 8.1 Determining internal force variables ........ 116Example 8.2 Determining external deformation variables ........ 116Example 8.3 Geometric and material non-linearity ........ 117Example 8.4 Tie ........ 119Example 8.5 Beam with one degree of static indeterminacy ........ 121Example 8.6 Geometric non-linearity ........ 122Example 8.7 Cantilever beam ........ 122Example 8.8 Cantilever beam ........ 124Example 8.9 Calibration ring ........ 124Example 8.10 Simply supported beam ........ 126Example 8.11 Simply supported beam ........ 128Example 8.12 Tie ........ 129Example 8.13 Cantilever beam ........ 130Example 8.14 Ideal cantilever column ........ 130Example 8.15 Cantilever beam column ........ 131Example 8.16 Simply supported beam column ........ 133Example 10.1 Plane truss ........ 152Example 10.2 Plane frame ........ 153Example 10.3 Plane truss ........ 154Example 10.4 Plane frame ........ 154Example 10.5 Three-hinged arch ........ 156Example 10.6 Plane frame ........ 157Example 11.1 Hinged girder ........ 162Example 11.2 Three-hinged frame with tie ........ 164Example 11.3 Plane truss ........ 169Example 11.4 Plane truss ........ 171Example 11.5 Plane truss ........ 172Example 11.6 Plane truss ........ 172Example 12.1 Hinged girder ........ 180Example 12.2 Three-hinged arch ........ 180Example 12.3 Plane truss ........ 182Example 13.1 Unequal leg angle ........ 188Example 13.2 Rectangular cross-section – kern ........ 190Example 13.3 Reinforced concrete slab – bending ........ 191Example 13.4 Reinforced concrete slab – shrinkage ........ 192Example 13.5 Rectangular cross-section – shear stress distribution ........ 195Example 13.6 Wide-flange beam ........ 197Example 13.7 Unequal leg angle ........ 198

XIIIContents

Example 13.8 Elliptical bar ........ 202Example 13.9 Narrow rectangular cross-section ........ 203Example 13.10 Reinforced concrete box girder ........ 205Example 13.11 Twin-cell box girder ........ 206Example 13.12 Twisted beam – concentrated load ........ 208Example 13.13 Twisted beam – distributed load ........ 209Example 13.14 Reinforced concrete beam ........ 213Example 14.1 SIMPSON’s rule ........ 225Example 14.2 Beam with one degree of static indeterminacy ........ 226Example 14.3 Hinged girder ........ 227Example 14.4 Cantilever beam ........ 228Example 14.5 Cranked cantilever beam ........ 228Example 14.6 Plane truss ........ 229Example 14.7 Rectangular cross-section – area shear factor ........ 229Example 14.8 Thin-wall hollow cross-section ........ 230Example 15.1 Simply supported beam ........ 236Example 15.2 Beam fixed at both ends ........ 236Example 15.3 Beam with one degree of static indeterminacy ........ 237Example 15.4 Beam with spring restraint ........ 239Example 15.5 Cantilever beam ........ 239Example 15.6 Beam with one degree of static indeterminacy ........ 240Example 15.7 Hinged girder ........ 240Example 16.1 Plane frame ........ 257Example 16.2 Bar fixed at both ends ........ 258Example 16.3 Beam with one degree of static indeterminacy ........ 259Example 16.4 Continuous beam ........ 260Example 16.5 Beam fixed at both ends ........ 262Example 16.6 Continuous beam of infinite length ........ 263Example 16.7 Continuous beam – support settlement ........ 267Example 16.8 Arch fixed at both ends ........ 268Example 16.9 Beam on skew supports ........ 269Example 16.10 Beam as circular arc ........ 270Example 16.11 Considering subsystems ........ 271Example 17.1 Cantilever beam rigid in shear ........ 280Example 17.2 Cantilever beam rigid in shear ........ 283Example 17.3 Plane frame ........ 287Example 17.4 Non-sway frame ........ 294Example 17.5 Grandstand frame ........ 295Example 17.6 Multi-storey sway frame ........ 296Example 17.7 Multi-storey non-sway frame ........ 297Example 17.8 Non-sway frame – settlement of supports ........ 299Example 17.9 Non-sway frame – uniform rise in temperature ........ 299Example 17.10 Non-sway frame – temperature difference ........ 301Example 17.11 Sway frame – uniform rise in temperature ........ 301Example 17.12 Three-span frame ........ 304Example 17.13 Continuous beam ........ 306Example 18.1 Bar restrained at both ends ........ 313Example 18.2 Bar with spring restraint at one end ........ 313Example 18.3 Reinforced concrete column – change in temperature ........ 314Example 18.4 Reinforced concrete column – shrinkage ........ 314Example 18.5 Pulling out a reinforcing bar ........ 317Example 18.6 Multi-storey frame ........ 322Example 18.7 Externally statically indeterminate VIERENDEEL girder ........ 323Example 18.8 Plastic panel with bonded sheet steel outer faces ........ 325Example 18.9 Simply supported beam – sinusoidal line load ........ 327Example 18.10 Bar fixed at both ends – linear temperature gradient ........ 328

XIV CONTENTS

Example 18.11 High-rise building ........ 335Example 18.12 High-rise building with outrigger ........ 337Example 18.13 Shear wall ........ 340Example 18.14 Shear wall – influence of wall extensions ........ 341Example 18.15 Two-hinged arch – uniformly distributed load ........ 347Example 18.16 Two-hinged arch – sinusoidal load ........ 348Example 18.17 Two-hinged arch – constant load segment by segment ........ 349Example 18.18 Displacement of the abutments to a concrete arch ........ 350Example 18.19 Stiffened pipe subjected to internal pressure ........ 354Example 18.20 Single strand – uniformly distributed load ........ 359Example 18.21 Single strand – thermal action ........ 359Example 18.22 Single strand – prestress ........ 359Example 18.23 Single strand – constant loads on both halves of the span ........ 360Example 18.24 Cable with wheel load ........ 362Example 18.25 Stresses in stay cable ........ 363Example 18.26 Suspended roof – uniformly distributed load ........ 364Example 18.27 Suspended roof – asymmetric imposed load ........ 365Example 18.28 Stress ribbon – asymmetric imposed load ........ 365Example 18.29 Suspended roof – central point load ........ 367Example 18.30 Stress ribbon – thermal action ........ 367Example 19.1 Plane frame ........ 374Example 19.2 Orthogonalised restraint states ........ 379Example 19.3 Beam with one degree of static indeterminacy ........ 384Example 21.1 Unequal leg angle ........ 427Example 21.2 Two-span beam – repeated variable actions ........ 436Example 21.3 Plane frame ........ 442Example 21.4 Plane frame – static program ........ 443Example 21.5 Plane frame – kinematic program ........ 444Example 21.6 Plane frame – minimum weight ........ 445Example 22.1 Beam column ........ 451Example 22.2 Cantilever column ........ 455Example 22.3 Ideal column ........ 455Example 22.4 Beam column ........ 456Example 22.5 Ideal column ........ 456Example 22.6 Ideal column with one degree of static indeterminacy ........ 457Example 22.7 Column with abrupt change in stiffness ........ 458Example 22.8 Load applied to top of cantilever column ........ 458Example 22.9 Statically determinate frame ........ 459Example 22.10 Elastically supported inclined leg frame ........ 463Example 22.11 Two-hinged frame ........ 467Example 22.12 Non-sway frame ........ 468Example 22.13 Sway frame ........ 468Example 22.14 Elastically restrained vertical cantilever ........ 468Example 22.14 Vertical cantilever ........ 477Example 22.15 Lateral buckling of an section ........ 486Example 22.16 Lateral buckling – shifting the point of load application ........ 486Example 23.1 Cantilever beam ........ 492Example 23.2 Cylindrical pipe ........ 495Example 23.3 Beam in the form of a circular arc ........ 495Example 23.4 Uniaxial tension ........ 498Example 23.5 Vertical embankment ........ 505Example 23.6 Strip foundation on TRESCA half-space ........ 506Example 23.7 Curtailed reinforcement in tension chord ........ 514Example 23.8 Web crushing failure ........ 516Example 23.9 Dissipation at hyperbolic slip line ........ 517Example 23.10 Strip foundation on TRESCA half-space ........ 519

XVContents

Example 24.1 Square slab supported at the corners ........ 528Example 24.2 Square slab supported in the centre ........ 529Example 24.3 Rectangular slab supported at the corners ........ 529Example 24.4 Simply supported square slab ........ 547Example 24.5 Fixed square slab ........ 547Example 24.6 Simply supported circular slab ........ 549Example 24.7 Fixed circular slab ........ 550Example 24.8 Reinforced concrete slab – dimensioning for bending ........ 553Example 24.9 Slab element subjected to pure twist ........ 554Example 24.10 Fixed circular slab ........ 558Example 24.11 Annular slab fixed at its inner edge and loaded on its outer edge by mu ........ 558Example 24.12 Simply supported rectangular slab ........ 561Example 24.13 Square slab simply supported along two adjacent edges ........ 561Example 24.14 Simply supported regular polygonal slabs ........ 561Example 24.15 Cantilever slab with point load on edge ........ 563Example 24.16 Fixed rectangular slab ........ 569Example 24.17 Square slab simply supported along two adjacent edges ........ 569Example 24.18 Fixed square slab ........ 570Example 24.19 Slab strip subjected to a central point load ........ 570Example 24.20 Cantilever slab with point load on edge ........ 570Example 24.21 Flat slab ........ 571Example 24.22 Semi-infinite rectangular slab subjected to edge loads ........ 573Example 24.23 Buckling of simply supported rectangular slabs ........ 575Example 24.24 Rectangular slab with initial deformation ........ 576Example 24.25 Square membrane ........ 577Example 25.1 Barrel vault – membrane theory ........ 590Example 25.2 Barrel vault – bending theory ........ 591Example 26.1 Spherical shells ........ 597Example 26.2 Spherical tanks ........ 598Example 26.3 Conical shell ........ 598Example 26.4 Spherical shell – dead load ........ 599Example 26.5 Spherical shell – wind pressure ........ 600Example 26.6 Conical shell – wind pressure ........ 601Example 26.7 Dome with circular cylindrical sectors – self-weight ........ 605Example 26.8 Pipe subjected to end loads ........ 613Example 26.9 Cylindrical tank ........ 614Example 26.10 Pipe subjected to thermal action ........ 614Example 26.11 Chimney subjected to wind pressure ........ 618Example 26.12 Fixed spherical shell subjected to internal pressure ........ 622Example 26.13 Pressure vessel ........ 622Example A7.1 Cantilever beam rigid in shear ........ 666Example A7.2 Cantilever beam – uniformly distributed load plus load at free end ........ 667

XVI CONTENTS

5 STATIC RELATIONSHIPS

5.1 Force systems and equilibrium

5.1.1 Terminology

Forces are perceived through their effects. They correspond to physical interactionsthat cause or modify states of deformation or motion in material systems. The effect ofa force depends on its point of application, its magnitude and its direction. Therefore,according to Fig. 5.1(a), a force can be represented as a point-based vector F withpoint of application A, magnitude F and line of action f.

The line of action f and an arbitrary reference point O define one plane. If we imaginea body connected to this one plane, then it is clear that F would cause a rotation of thebody about the axis n perpendicular to the plane and passing through O. The tendencyto rotate is proportional to the magnitude F and the distance a of force F from O. Theposition vector r of the point of application A of F expresses the tendency to rotatewith the moment

M w rqF (5:1)

correctly in terms of magnitude and direction; jMj = Fa applies and the vectors M, rand F constitute a right-hand screw, see Fig. 5.1(b). As can be seen, the moment Mremains unaltered if force F is translated along its line of action f.

Every force F has a corresponding reaction –F with the same line of action. Accord-ing to this so-called reaction principle, a force without its reaction cannot exist.

Remote forces (e. g. gravity) exhibit points of application different to those of theirreactions; the interaction between two bodies with mass generally takes place withoutcontact. But in the case of contact forces (e. g. support forces), the points of applica-tion of forces and reactions are geometrically identical (although not materially iden-tical); the interaction between support and supported body comes about through con-tact – if the contact is eliminated, so the contact force disappears as well.

The inertial forces that must be considered in dynamics (see section 8.3.4) do not haveany reactions. They do not correspond to any physical interactions, instead are math-ematical auxiliary variables.

Contact forces are generally in the form of surface forces (surface loads) distributedover a finite area. The contact force related to the unit of surface area, the force per unitarea

t wdF

dA(5:2)

is also known as a stress vector, see Fig. 5.2(a) and section 5.2.1.

Similarly, remote forces distributed over a finite three-dimensional space are calledbody forces (body loads) with a force per unit volume of

qwdF

dV(5:3)

see Fig. 5.2(b).

435.1 Force systems and equilibrium

M

F

rn

a

M

f

A

M( )F( )

r( )

(a)

(b)

O

Fig. 5.1 Force and moment: (a) reference point O,

point of application A, line of action f, axis of ro-

tation n, (b) right-hand screw rule

Theory of Structures. First Edition. Peter Martic 2013 Ernst & Sohn GmbH & Co. KG. Published 2013 by Ernst & Sohn GmbH & Co. KG.

If, ultimately, a body such as a beam or cable is idealised as one-dimensional and ifdistributed forces act on this in the form of line forces (line loads), we get a force perunit length

qwdF

ds(5:4)

see Fig. 5.2(c).

In the SI or MKS systems, the unit used for specifying the magnitude of a force is thenewton [1N = 1mkgs–2] or [kN] or [MN]. Correspondingly, the unit used for mo-ments is [Nm] or [kNm] or [MNm]. To distinguish between forces and moments,we indicate the latter with double arrows, see Fig. 5.1(a). The units for forces perunit length, area and volume are therefore [Nm–1], [Nm–2 = Pa] and [Nm–3].

5.1.2 Force systems

We shall now consider force systems (groups of forces) whose material points of ap-plication lie within the arbitrary limits of a body or system. A body isolated from abody or system (or part thereof) by means of an imaginary cut is known as a freebody (FB). By introducing all the forces that act on the free body, we obtain afree-body diagram (FBD).

The resultant force of a force system is obtained by adding together the vectors actingon the free body:

Rw

XFB

F (5:5)

Likewise, the resultant couple of a force system with respect to an arbitrary referencepoint O is

MO w

XFB

rqF (5:6)

see Fig. 5.3 and (5.1).

If instead of O we select a different reference point Ol, then according to Fig. 5.3, withrl = r – rL and considering (5.5) and (5.6), it follows that

MOl w

XFB

rlq Fw

XFB

rqFs rLqXFB

FwMO s rLqR (5:7)

The pair of vectors {R, MO} or {R, MOl} is called the force-couple system of the force

system at O or Ol.

Two force systems are equivalent when their force-couple systems are identical withrespect to an arbitrary reference point. According to (5.7), the equivalence of two forcesystems need only be verified for one reference point; the identity of the resultantcouple is then given for all points.

44 5 STATIC RELATIONSHIPS

II FUNDAMENTALS

tdF

d

q

dF

q

dF

(a) (b) (c)

A

dVds

Fig. 5.2 Distributed forces: (a) force per unit area, (b) force per unit volume, (c) force per unit length

y zx

´r

´r ´

´r

O

F

FB

O

Fig. 5.3 Free body diagram with

reference points O and Ol

5.1.3 Equilibrium

A force system is in equilibrium when its force-couple system sums to zero:

Rw 0 , MO w 0 (5:8)

The equilibrium conditions (5.8) result in six scalar equations in the case of force sys-tems in three dimensions, i. e. three force-balance equations and three moment-bal-ance equations. In the case of coplanar force systems, this number is reduced to three,i. e. two force-balance equations in the plane of the force system and one moment-bal-ance equation perpendicular to that plane.

If (5.8) applies, then according to (5.7), MOl= 0. Consequently, the force-balance

equations can be replaced by moment-balance equations about a second referencepoint. Generally, in the three-dimensional case, moment-balance equations can be for-mulated about six non-collinear axes and in the coplanar case about three points notlying in a straight line. In practice, this is often easier than setting up the force-balanceequations. Depending on the particular problem, in the coplanar case only one, and inthe three-dimensional case only one or two, force-balance equations are replaced bymoment-balance equations, as is explained further in chapter 10.

Applying (5.8) to differential structural elements results in differential equations forthe equilibrium, as dealt with in section 5.3.

When defining free bodies and applying the equilibrium conditions to those bodies, wegenerally use the so-called free-body principle: if we remove arbitrary parts from acompatibly deformed body or system in equilibrium by way of imaginary cuts,each one of those parts is in equilibrium and compatibly deformed.

Forces acting on arbitrary free bodies are known as internal or external forces depend-ing on whether the material point of application of the reaction to a force lies inside oroutside the free body.

According to the reaction principle, the internal forces form an equilibrium system(i. e. a force system in equilibrium), and so the external forces must themselves bein equilibrium if the free body is in equilibrium in its entirety. This assertion is knownas the fundamental theorem of statics.

If the equilibrium conditions – at best following a suitable breakdown of the system –are sufficient for determining the unknowns in a problem, we speak of a staticallydeterminate system, otherwise a statically indeterminate system.

5.1.4 Overall stability

Structures must be stable, i. e. they must not fail in their entirety (e. g. due to buoyancy,sliding or overturning). Their rigid body equilibrium, or rather their overall stability,must be assured (see section 4.4, limit state type 1).

Example 5.1 Cantilever retaining wall

The cantilever retaining wall shown in Fig. 5.4(a) is to be investigated for overturning about its toe O.To do this, we consider the cantilever retaining wall as a free body isolated from its surroundingsaccording to Fig. 5.4(b) and add all the forces acting on it in order to create a free body diagram.Those forces are the dead loads of the base (G1) (related to the unit length perpendicular to the yz

plane) and the vertical stem (G2), the surcharges due to the earth above the cantilevering parts ofthe base (G3 and G4), the active and passive earth pressures (Ea and Ep) plus a contact force A actingon the underside of the base. For simplicity, hydrostatic pressures are neglected. Further, the calcula-tion with the earth surcharges G3 and G4 represents a considerable idealisation. Actually, in the eventof an overturning failure, a wedge-shaped mass of soil would form in the ground behind the wall. Andthis would be linked with the mobilisation of further forces that are neglected here; a similar consid-eration applies to the soil in front of the wall.The contact force A can easily be determined with (5.8) according to magnitude, direction and point ofapplication, e. g. by setting up the two force-balance equations in the y and z directions and the mo-ment-balance equation about O. Alternatively, A can also be determined graphically. Fig. 5.4(c) shows

455.1 Force systems and equilibrium

the associated addition of the force vectors in the so-called force polygon; for equilibrium, the forcepolygon must be closed, which determines the magnitude and direction of A. The point of applicationof A follows from the funicular polygon according to Fig. 5.4(d). This is done by successively form-ing the interim resultants (shown as dotted lines in the force polygon) of forces Ea and G4 etc., draw-ing their lines of action starting from point A, the intersection of Ea and G4, and extending this tointersect with the next force G1 at B etc. This approach enables us to establish point E, the intersectionof the lines of action CE and DE for (Ea,G4,G1,G2) or (Ep,G3), and hence determine the line ofaction of A.For overall stability, A must act on the base, i. e. 0 J a J b, see Fig. 5.4(d). For the limit case a = 0(or a = b), the bearing pressure would be infinitely large, which is of course impossible because thestrength of the subsoil is finite. Fig. 5.4(d) shows one practical possibility and Fig. 5.4(e) shows astatically equivalent linear bearing pressure distribution with a maximum value of 2Az/(3a) at O.As can be seen, 3a I b, i. e. in the range – 3a i y j –b the foundation experiences partial uplift

with the contact force tending towards zero.It is not possible to reach any conclusion about the distribution of the horizontal component Ay of A atthe underside of the base solely on the basis of static considerations. For simplicity, a distributionproportional to Az is assumed, which in this particular case means a triangular distribution.

46 5 STATIC RELATIONSHIPS

II FUNDAMENTALS

O

(a)

Ea

G4

G3

G1

G2

A

Az

AyEp

G2

Ea

G4G3

z

y O

Ep

A

(b)

EaA

CE

G2

G1

G4

zy O

ab

G3

Ep

A

a3

a

A2

3z

B

(c)

(d)

(e)

G1

D

zy

Fig. 5.4 Overall stability of a cantilever retaining wall: (a) overview, (b) free body diagram, (c) force polygon, (d) funicular polygon,

(e) possible distribution of ground pressure

Example 5.2 Support envelope

The upturned right-angled element supported on a horizontal surface shown in Fig. 5.5(a) and (b) isloaded at Al by a horizontal force Q. In Fig. 5.5(b) we must distinguish between the area of con-

tact ABCDEF and the support envelope ABCEF. The latter is the smallest convex envelope enclos-ing the former.The overall stability of the right-angled element can be checked with the help of a moment-balanceequation about axis CE. The overturning moment Qh due to Q acting about CE may not exceed theresisting moment G0 a0 + (G1 + G2)a1 due to the dead load components G0(AD1DD2), G1(D1BCD)and G2(DEFD2), otherwise the element will overturn.Fig. 5.5(c) and (d) show the alternative graphical examination with the help of the force and funicularpolygons. For overall stability, the point of application J of the contact force A must lie within thesupport envelope.The contact force A is assumed to be distributed equally over the end zones of the two legs of theelement. In the limit case, the force is concentrated at points C and E, which means that the localbearing pressure is then infinitely large.

5.1.5 Supports

Supports correspond to the locally inhibited displacement and rotation capabilities(degrees of freedom) of structures. They can be classified according to the inhibited(restrained) displacement and rotation capabilities or the passive degrees of freedom,i. e. according to whether the displacements u, v, w and the rotations fx, fy , fz in thex, y, z directions are possible or prevented, see Fig. 5.6. The number of passive(restrained) degrees of freedom (or the number of components in the supportforce-couple system) is known as the determinacy of the support.

475.1 Force systems and equilibrium

h

Q

Q

A

G2+G1

G0

C B AD

G0

G2+G1

G2

F

E

D

D

DA

B

C

QG0

2

G1

G0

Q

AH

G2+G1

A

(a) (c)

(b) (d)

aa

1

0

C B AD

I

J

1

Fig. 5.5 Upturned right-angled element on horizontal surface:

(a) elevation, (b) plan, (c) force polygon, (d) funicular polygon

y zx u ϕx

w

ϕz

v

ϕy

Fig. 5.6 Displacements and rotations

Fig. 5.7 shows a number of different types of support. If the concrete hinge shown inFig. 5.7(a) is in the form of a linear support (long in the y direction), it inhibits dis-placements u, v, w and rotations fx, fz; in the form of a discrete support (short in they direction), both fy and also fx and fz are practically not inhibited at all. It is notablethat with respect to the forces, the support acts bilaterally in all three directions, i. e.positive and negative forces can occur in the body of the bearing sliced through at thehorizontal joint, especially in the z direction, too, because of the reinforcement inter-secting the horizontal joint. The steel linear rocker bearing shown in Fig. 5.7(b) actsbilaterally with respect to u and unilaterally with respect to w – the support would liftup in the z direction when the force tends towards zero; with respect to v, it worksbilaterally up to a certain amount, either via friction or with lugs at the sides (afterovercoming the play between lug and body of bearing); rotation fy is practically un-restrained, and rotations fx, fz are inhibited. In the case of the steel roller bearingshown in Fig. 5.7(c), u and fy are not inhibited and the support acts unilaterallywith respect to w; guide rails at the side inhibit displacement v and rotation fz;rotation fx is inhibited because of the long roller in the y direction. The laminatedelastomeric bearing shown in Fig. 5.7(d) functions unilaterally with respect to wand, depending on the particular type, enables displacements u, v as well asrotations fy , fx. The same is true for the elastomer pot sliding bearing shown inFig. 5.7(e).

A closer look at Fig. 5.7 shows that, depending on the particular design, the displace-ment and rotation capabilities of supports always lie within certain limits and are neverenabled or prevented in absolute terms. Likewise, the components of the supportforce-couple system associated with the inhibited displacement and rotation capabil-ities are restricted to certain limit values. In practice, it is certainly necessary to con-sider these limits carefully every time.

In theory of structures, we assume the appropriate idealisations shown in Fig. 5.8 forthe coplanar case. Fig. 5.8(a) shows a unilaterally or bilaterally functioning slidingsupport (hinged support capable of displacement) that only inhibits w and whose sup-port force-couple system is limited to the force component in the z direction. In thecase of the hinged support shown in Fig. 5.8(b), u is also inhibited and the correspond-ing force component in the x direction is added to the support force-couple system.Considering the fixed support shown in Fig. 5.8(c), fy is finally inhibited as well;the support force-couple system also exhibits a moment about the y axis. Extendingthese considerations to the general three-dimensional case is easily possible with thehelp of Fig. 5.6.

Static equivalents to the types of support shown in Fig. 5.8 can be realised according toFig. 5.9 with pin-jointed members. These are straight, weightless bars connected con-centrically on both sides with frictionless hinges. With such assumptions, only forcescan be transferred from the bars, whose lines of action coincide with the axes of thebars. So a statically equivalent substitute for a sliding support, as shown in Fig. 5.9(a),could be a pin-ended strut; the force component in the x direction caused by the in-clination of the pin-ended strut as a result of a displacement u is negligible in com-parison to the force component in the z direction, assuming infinitesimally small dis-

48 5 STATIC RELATIONSHIPS

II FUNDAMENTALS

(e)(d)(c)(b)(a)

x

z

y

Fig. 5.7 Types of bearing: (a) concrete hinge, (b) steel linear rocker bearing, (c) steel roller bearing, (d) laminated elastomeric bearing,

(e) elastomer pot sliding bearing

(c)(b)(a)

x

z

Fig. 5.8 Support idealisations: (a) sliding support,

(b) hinged support, (c) fixed support

placements (first-order theory, see section 6.1). The support force component possiblein the x direction with a hinged support requires a corresponding second pin-jointedmember, as shown in Fig. 5.9(b). Ultimately, a third pin-jointed member is required toachieve fixity, as shown in Fig. 5.9(c); here, the first two pin-jointed members exhibitdifferent lines of action and the axis of the third may not pass through the intersectionof the first two, otherwise the support would not provide fixity, instead function like ahinged support at this point.

For the general three-dimensional case, six pin-jointed members are necessary for abraced support. With the force-couple system {R, MO}, the coordinates rij of one pointon the axis of the pin-jointed member i and the direction cosine cij of the six pin-jointed members and the forces in those members Ni, then according to (5.5) and(5.6) the following applies:

c1x c2x . . . c6x

c1y c2y . . . c6y

c1z c2z . . . c6z

r1yc1z s r1zc1y r2yc2z s r2zc2y . . . r6yc6z s r6zc6y

r1zc1x s r1xc1z r2zc2x s r2xc2z . . . r6zc6x s r6xc6z

r1xc1y s r1yc1x r2xc2y s r2yc2x . . . r6xc6y s r6yc6x

2666666664

3777777775

N1

N2

N3

N4

N5

N6

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;

w

Rx

Ry

Rz

MOx

MOy

MOz

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;

(5:9)

In order that (5.9) can have a solution N for any force-couple system, the matrix on theleft must be invertible, i. e. its determinant may not be zero.

Let us select the origin of coordinates O for the coplanar case (three bars in thexz plane) with Rx, Rz, MOy to be the intersection of bars 1 and 2 (r1 = r2 = 0) and con-sider the point of pin-jointed member axis 3 on the z axis (r3x = 0, r3z 0 0). In thiscase, removing the second, fourth and sixth rows as well as the fourth, fifth and sixthcolumns from the matrix in (5.9) gives us the following matrix:

c1x c2x c3x

c1z c2z c3z

0 0 r3zc3x

24

35 (5:10)

and hence

det w r3zc3x c1xc2z s c1zc2xð Þ00 (5:11)

must apply. Without restricting the universal applicability, it is possible to place thex axis in the direction of bar axis 1, i. e. c1x = 1, c1z = 0. Consequently, c3x 0 0,c2z 0 0 must be true, i. e. bar axis 3 may not pass through O and bars 1 and 2 maynot be collinear. The above requirements are therefore confirmed.

It is often not possible to idealise supports as fully restrained, as has been assumed upto now; instead, it is necessary to consider their flexibility. To this end, in accordancewith Fig. 5.10, we use appropriate translational and rotational springs and in thesimplest case presume a linear relationship between the components of the supportforce-couple system and the corresponding displacements and rotations:

Ax ws kxuA , Az ws kzwA , MA ws kyfyA (5:12)

where kx, kz and ky denote the stiffnesses of the translational and rotational springs.

495.1 Force systems and equilibrium

(c)(b)(a)

x

z

Fig. 5.9 Equivalent supports with pin-jointed members: (a) sliding support,

(b) hinged support, (c) fixed support.

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