Theory of Structures by Ernst & Sohn

Peter Marti

Theory of structures F u n d a m e n ta l s F r a m e d St r u c t u r e s P l a t e s a n d S h e lls W

Q = aT k a V = K v v=a V v T

V=b v s=k v

Q = aT s

v=f s

s s =b Q

V = bT f b Q = F Q

Contents

VII

CONTENTS

Preface ........ V I INTRODUCTION 1

THE PURPOSE AND SCOPE OF THEORY OF STRUCTURES ........ 1

1.1 1.2 1.3 1.4 1.5

General ........ 1 The basis of theory of structures ........ 1 Methods of theory of structures ........ 2 Statics and structural dynamics ........ 3 Theory of structures and structural engineering ........ 3

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

II FUNDAMENTALS 3

3.1 3.2 3.3 3.4 3.5

DESIGN OF STRUCTURES ........ 11 General ........ 11 Conceptual design ........ 11 Service criteria agreement and basis of design ........ 14 Summary ........ 26 Exercises ........ 27

STRUCTURAL ANALYSIS AND DIMENSIONING ........ 29

4.1 4.2 4.2.1 4.2.2 4.3 4.4 4.5 4.6 4.6.1 4.6.2 4.6.3 4.6.4 4.7 4.8

General ........ 29 Actions ........ 29 Actions and action effects ........ 29 Models of actions and representative values ........ 30 Structural models ........ 31 Limit states ........ 31 Design situations and load cases ........ 32 Verifications ........ 33 Verification concept ........ 33 Design values ........ 33 Verification of structural safety ........ 34 Verification of serviceability ........ 35 Commentary ........ 35 Recommendations for the structural calculations ........ 36 Recommendations for the technical report ........ 38 Summary ........ 40 Exercises ........ 41

4.9 4.10 4.11

5.1 5.1.1 5.1.2 5.1.3 5.1.4 5.1.5 5.1.6 5.1.7 5.2 5.2.1 5.2.2 5.2.3 5.2.4 5.3 5.3.1 5.3.2 5.4 5.5

STATIC RELATIONSHIPS ........ 43 Force systems and equilibrium ........ 43 Terminology ........ 43 Force systems ........ 44 Equilibrium ........ 45 Overall stability ........ 45 Supports ........ 47 Hinges ........ 50 Stress resultants ........ 51 Stresses ........ 53 Terminology ........ 53 Uniaxial stress state ........ 53 Coplanar stress states ........ 54 Three-dimensional stress states ........ 57 Differential structural elements ........ 61 Straight bars ........ 61 Bars in single curvature ........ 62 Summary ........ 68 Exercises ........ 69

KINEMATIC RELATIONSHIPS ........ 71

6.1 6.2 6.3 6.4 6.5

Terminology ........ 71 Coplanar deformation ........ 72 Three-dimensional deformation state ........ 74 Summary ........ 76 Exercises ........ 77

CONSTITUTIVE RELATIONSHIPS ........ 79

7.1 7.2 7.3 7.3.1 7.3.2 7.3.3 7.4 7.4.1 7.4.2 7.5 7.6 7.6.1 7.6.2 7.6.3 7.7 7.8

Terminology ........ 79 Linear elastic behaviour ........ 81 Perfectly plastic behaviour ........ 83 Uniaxial stress state ........ 83 Three-dimensional stress states ........ 84 Yield conditions ........ 85 Time-dependent behaviour ........ 90 Shrinkage ........ 90 Creep and relaxation ........ 91 Thermal deformations ........ 94 Fatigue ........ 94 General ........ 94 S-N curves ........ 95 Damage accumulation under fatigue loads ........ 96 Summary ........ 98 Exercises ........ 99

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

VIII

CONTENTS

ENERGY METHODS ........ 101 Introductory example ........ 101 Statically determinate system ........ 101 Statically indeterminate system ........ 103 Work equation ........ 104 Commentary ........ 105 Variables and operators ........ 105 Introduction ........ 105 Plane framed structures ........ 107 Spatial framed structures ........ 109 Coplanar stress states ........ 110 Coplanar strain state ........ 111 Slabs ........ 111 Three-dimensional continua ........ 113 Commentary ........ 114 The principle of virtual work ........ 115 Virtual force and deformation variables ........ 115 The principle of virtual deformations ........ 115 The principle of virtual forces ........ 115 Commentary ........ 116 Elastic systems ........ 118 Hyperelastic materials ........ 118 Conservative systems ........ 119 Linear elastic systems ........ 125 Approximation methods ........ 128 Introduction ........ 128 The RITZ method ........ 129 The GALERKIN method ........ 132 Summary ........ 134 Exercises ........ 135

8.1 8.1.1 8.1.2 8.1.3 8.1.4 8.2 8.2.1 8.2.2 8.2.3 8.2.4 8.2.5 8.2.6 8.2.7 8.2.8 8.3 8.3.1 8.3.2 8.3.3 8.3.4 8.4 8.4.1 8.4.2 8.4.3 8.5 8.5.1 8.5.2 8.5.3 8.6 8.7

III LINEAR ANALYSIS OF FRAMED STRUCTURES 9

STRUCTURAL ELEMENTS AND TOPOLOGY ........ 137

9.1 9.2 9.3 9.3.1 9.3.2 9.3.3 9.3.4

General ........ 137 Modelling of structures ........ 137 Discretised structural models ........ 140 Description of the static system ........ 140 Joint equilibrium ........ 141 Static determinacy ........ 142 Kinematic derivation of the equilibrium matrix ........ 144 Summary ........ 147 Exercises ........ 147

9.4 9.5 10

10.1 10.2 10.3 10.4 10.5 10.6

DETERMINING THE FORCES ........ 149 General ........ 149 Investigating selected free bodies ........ 150 Joint equilibrium ........ 154 The kinematic method ........ 156 Summary ........ 158 Exercises ........ 158

STRESS RESULTANTS AND STATE DIAGRAMS ........ 159

11.1 11.2 11.2.1 11.2.2 11.2.3 11.3 11.3.1 11.3.2 11.3.3 11.3.4 11.3.5 11.3.6 11.4 11.5

General ........ 159 Hinged frameworks ........ 160 Hinged girders ........ 161 Hinged arches and frames ........ 163 Stiffened beams with intermediate hinges ........ 165 Trusses ........ 166 Prerequisites and structural topology ........ 166 Methods of calculation ........ 169 Joint equilibrium ........ 169 CREMONA diagram ........ 171 RITTER method of sections ........ 172 The kinematic method ........ 173 Summary ........ 174 Exercises ........ 175

INFLUENCE LINES ........ 177 General ........ 177 Determining influence lines by means of equilibrium conditions ........ 178 Kinematic determination of influence lines ........ 179 Summary ........ 183 Exercises ........ 183

12.1 12.2 12.3 12.4 12.5 13

13.1 13.2 13.2.1 13.2.2 13.2.3 13.2.4 13.2.5 13.2.6 13.2.7 13.3 13.3.1 13.3.2 13.3.3 13.3.4 13.4 13.4.1 13.4.2 13.4.3 13.4.4 13.5 13.6 14

14.1 14.2 14.2.1 14.2.2 14.2.3 14.2.4

ELEMENTARY DEFORMATIONS ........ 185 General ........ 185 Bending and normal force ........ 185 Stresses and strains ........ 185 Principal axes ........ 187 Stress calculation ........ 189 Composite cross-sections ........ 190 Thermal deformations ........ 192 Planar bending of curved bars ........ 193 Practical advice ........ 194 Shear forces ........ 194 Approximation for prismatic bars subjected to pure bending ........ 194 Approximate coplanar stress state ........ 196 Thin-wall cross-sections ........ 197 Shear centre ........ 199 Torsion ........ 200 Circular cross-sections ........ 200 General cross-sections ........ 201 Thin-wall hollow cross-sections ........ 204 Warping torsion ........ 207 Summary ........ 216 Exercises ........ 218 SINGLE DEFORMATIONS ........ 221 General ........ 221 The work theorem ........ 222 Introductory example ........ 222 General formulation ........ 223 Calculating the passive work integrals ........ 223 Systematic procedure ........ 226

Contents

14.3 14.4 14.5 14.6

Applications ........ 226 MAXWELLâ&#x20AC;&#x2122;s theorem ........ 230 Summary ........ 231 Exercises ........ 231

DEFORMATION DIAGRAMS ........ 233 General ........ 233 Differential equations for straight bar elements ........ 233 In-plane loading ........ 233 General loading ........ 235 The effect of shear forces ........ 235 Creep, shrinkage and thermal deformations ........ 235 Curved bar axes ........ 235 Integration methods ........ 236 Analytical integration ........ 236 MOHRâ&#x20AC;&#x2122;s analogy ........ 238 Exercises ........ 243

15.1 15.2 15.2.1 15.2.2 15.2.3 15.2.4 15.2.5 15.3 15.3.1 15.3.2 15.5 16

16.1 16.2 16.2.1 16.2.2 16.2.3 16.2.4 16.2.5 16.2.6 16.3 16.3.1 16.3.2 16.3.3 16.3.4 16.4 16.5 16.6 17

17.1 17.1.1 17.1.2 17.1.3 17.1.4 17.2 17.2.1 17.2.2 17.2.3 17.2.4 17.3 17.3.1

THE FORCE METHOD ........ 245 General ........ 245 Structural behaviour of statically indeterminate systems ........ 245 Overview ........ 245 Statically determinate system ........ 246 System with one degree of static indeterminacy ........ 247 System with two degrees of static indeterminacy ........ 249 In-depth analysis of system with one degree of static indeterminacy ........ 250 In-depth analysis of system with two degrees of static indeterminacy ........ 253 Classic presentation of the force method ........ 254 General procedure ........ 254 Commentary ........ 255 Deformations ........ 257 Influence lines ........ 259 Applications ........ 262 Summary ........ 272 Exercises ........ 274 THE DISPLACEMENT METHOD ........ 277 Independent bar end variables ........ 277 General ........ 277 Member stiffness relationship ........ 277 Actions on bars ........ 278 Algorithm for the displacement method ........ 280 Complete bar end variables ........ 281 General ........ 281 Member stiffness relationship ........ 282 Actions on bars ........ 283 Support force variables ........ 283 The direct stiffness method ........ 284 Incidence transformation ........ 284

17.3.2 17.3.3 17.4 17.4.1 17.4.2 17.4.3 17.4.4 17.4.5 17.4.6 17.4.7 17.5 17.6

Rotational transformation ........ 285 Algorithm for the direct stiffness method ........ 286 The slope-deflection method ........ 290 General ........ 290 Basic states and member end moments ........ 292 Equilibrium conditions ........ 293 Applications ........ 294 Restraints ........ 298 Influence lines ........ 303 CROSS method of moment distribution ........ 305 Summary ........ 309 Exercises ........ 310

CONTINUOUS MODELS ........ 311 General ........ 311 Bar extension ........ 311 Practical examples ........ 311 Analytical model ........ 312 Residual stresses ........ 314 Restraints ........ 315 Bond ........ 316 Summary ........ 320 Beams in shear ........ 321 Practical examples ........ 321 Analytical model ........ 321 Multi-storey frame ........ 321 VIERENDEEL girder ........ 323 Sandwich panels ........ 324 Summary ........ 326 Beams in bending ........ 326 General ........ 326 Analytical model ........ 327 Restraints ........ 327 Elastic foundation ........ 329 Summary ........ 332 Combined shear and bending response ........ 333 General ........ 333 Shear wall - frame systems ........ 334 Shear wall connection ........ 338 Dowelled beams ........ 342 Summary ........ 344 Arches ........ 345 General ........ 345 Analytical model ........ 345 Applications ........ 346 Summary ........ 350 Annular structures ........ 350 General ........ 350 Analytical model ........ 351 Applications ........ 352 Edge disturbances in cylindrical shells ........ 353 Summary ........ 354 Cables ........ 354 General ........ 354 Analytical model ........ 355 Inextensible cables ........ 357

18.1 18.2 18.2.1 18.2.2 18.2.3 18.2.4 18.2.5 18.2.6 18.3 18.3.1 18.3.2 18.3.3 18.3.4 18.3.5 18.3.6 18.4 18.4.1 18.4.2 18.4.3 18.4.4 18.4.5 18.5 18.5.1 18.5.2 18.5.3 18.5.4 18.5.5 18.6 18.6.1 18.6.2 18.6.3 18.6.4 18.7 18.7.1 18.7.2 18.7.3 18.7.4 18.7.5 18.8 18.8.1 18.8.2 18.8.3

CONTENTS

18.8.4 18.8.5 18.8.6 18.9 18.9.1 18.9.2 18.9.3 18.9.4 18.9.5 18.10

Extensible cables ........ 358 Axial stiffness of laterally loaded cables ........ 360 Summary ........ 360 Combined cable-type and bending response ........ 361 Analytical model ........ 361 Bending-resistant ties ........ 362 Suspended roofs and stress ribbons ........ 363 Suspension bridges ........ 368 Summary ........ 368 Exercises ........ 369

DISCRETISED MODELS ........ 371 General ........ 371 The force method ........ 372 Complete and global bar end forces ........ 372 Member flexibility relation ........ 372 Actions on bars ........ 374 Algorithm for the force method ........ 374 Comparison with the classic force method ........ 376 Practical application ........ 376 Reduced degrees of freedom ........ 376 Supplementary remarks ........ 379 Introduction to the finite element method ........ 381 Basic concepts ........ 381 Element matrices ........ 381 Bar element rigid in shear ........ 381 Shape functions ........ 385 Commentary ........ 386 Summary ........ 386 Exercises ........ 387

19.1 19.2 19.2.1 19.2.2 19.2.3 19.2.4 19.2.5 19.2.6 19.2.7 19.2.8 19.3 19.3.1 19.3.2 19.3.3 19.3.4 19.3.5 19.4 19.5

IV NON-LINEAR ANALYSIS OF FRAMED STRUCTURES 20

20.1 20.2 20.2.1 20.2.2 20.3 20.3.1 20.3.2 20.3.3 20.3.4 20.3.5 20.4 20.5 21

21.1 21.2 21.2.1 21.2.2 21.2.3 21.2.4

ELASTIC-PLASTIC SYSTEMS ........ 389 General ........ 389 Truss with one degree of static indeterminacy ........ 389 Single-parameter loading ........ 389 Dual-parameter loading and generalisation ........ 395 Beams in bending ........ 398 Moment-curvature diagrams ........ 398 Simply supported beams ........ 399 Continuous beams ........ 403 Frames ........ 404 Commentary ........ 405 Summary ........ 406 Exercises ........ 407 LIMIT ANALYSIS ........ 409 General ........ 409 Upper- and lower-bound theorems ........ 410 Basic concepts ........ 410 Lower-bound theorem ........ 410 Upper-bound theorem ........ 411 Compatibility theorem ........ 411

21.2.5 Consequences of the upper- and lower-bound theorems ........ 411 21.3 Static and kinematic methods ........ 412 21.3.1 General ........ 412 21.3.2 Simply supported beams ........ 413 21.3.3 Continuous beams ........ 415 21.3.4 Plane frames ........ 416 21.3.5 Plane frames subjected to transverse loads ........ 421 21.4 Plastic strength of materials ........ 426 21.4.1 General ........ 426 21.4.2 Skew bending ........ 426 21.4.3 Bending and normal force ........ 428 21.4.4 Bending and torsion ........ 432 21.4.5 Bending and shear force ........ 434 21.5 Shakedown and limit loads ........ 435 21.6 Dimensioning for minimum weight ........ 437 21.6.1 General ........ 437 21.6.2 Linear objective function ........ 438 21.6.3 FOULKES mechanisms ........ 438 21.6.4 Commentary ........ 440 21.7 Numerical methods ........ 441 21.7.1 The force method ........ 441 21.7.2 Limit load program ........ 442 21.7.3 Optimum design ........ 444 21.8 Summary ........ 446 21.9 Exercises ........ 447 22

22.1 22.2 22.2.1 22.2.2 22.2.3 22.2.4 22.2.5 22.2.6 22.3 22.3.1 22.3.2 22.3.3 22.4 22.4.1 22.4.2 22.4.3 22.4.4 22.5 22.6

STABILITY ........ 449 General ........ 449 Elastic buckling ........ 449 Column deflection curve ........ 449 Bifurcation problems ........ 453 Approximation methods ........ 454 Further considerations ........ 460 Slope-deflection method ........ 465 Stiffness matrices ........ 469 Elastic-plastic buckling ........ 471 Concentrically loaded columns ........ 471 Eccentrically loaded columns ........ 474 Limit loads of frames according to second-order theory ........ 477 Flexural-torsional buckling and lateral buckling ........ 480 Basic concepts ........ 480 Concentric loading ........ 482 Eccentric loading in the strong plane ........ 483 General loading ........ 485 Summary ........ 488 Exercises ........ 489

V PLATES AND SHELLS PLATES ........ 491 23.1 General ........ 491 23.2 Elastic plates ........ 491 23.2.1 Stress function ........ 491

Contents

23.2.2 Polar coordinates ........ 493 23.2.3 Approximating functions for displacement components ........ 496 23.3 Reinforced concrete plate elements ........ 496 23.3.1 Orthogonal reinforcement ........ 496 23.3.2 General reinforcement ........ 500 23.4 Static method ........ 501 23.4.1 General ........ 501 23.4.2 Truss models ........ 501 23.4.3 Discontinuous stress fields ........ 505 23.4.4 Stringer-panel model ........ 511 23.5 Kinematic method ........ 512 23.5.1 Applications in reinforced concrete ........ 512 23.5.2 Applications in geotechnical engineering ........ 517 23.6 Summary ........ 520 23.7 Exercises ........ 522 24

24.1 24.1.1 24.1.2 24.1.3 24.2 24.2.1 24.2.2 24.2.3 24.2.4 24.2.5 24.2.6 24.3 24.3.1 24.3.2 24.4 24.4.1 24.4.2 24.4.3 24.5 24.5.1 24.5.2 24.5.3 24.6 24.6.1 24.6.2 24.6.3 24.7 24.7.1 24.7.2 24.7.3 24.8 24.9 25

25.1 25.2

SLABS ........ 525 Basic concepts ........ 525 General ........ 525 Static relationships ........ 525 Kinematic relationships ........ 531 Linear elastic slabs rigid in shear with small deflections ........ 533 Fundamental relationships ........ 533 Methods of solution ........ 535 Rotationally symmetric problems ........ 536 Rectangular slabs ........ 539 Flat slabs ........ 543 Energy methods ........ 546 Yield conditions ........ 547 VON MISES and TRESCA yield conditions ........ 547 Reinforced concrete slabs ........ 550 Static method ........ 557 Rotationally symmetric problems ........ 557 Moment fields for rectangular slabs ........ 560 Strip method ........ 563 Kinematic method ........ 567 Introductory example ........ 567 Calculating the dissipation work ........ 568 Applications ........ 569 The influence of shear forces ........ 572 Elastic slabs ........ 572 Rotationally symmetric VON MISES slabs ........ 574 Reinforced concrete slabs ........ 575 Membrane action ........ 575 Elastic slabs ........ 575 Perfectly plastic slab strip ........ 577 Reinforced concrete slabs ........ 578 Summary ........ 581 Exercises ........ 583 FOLDED PLATES ........ 587 General ........ 587 Prismatic folded plates ........ 588

25.2.1 25.2.2 25.2.3 25.3 25.4 25.5

Sawtooth roofs ........ 588 Barrel vaults ........ 589 Commentary ........ 593 Non-prismatic folded plates ........ 594 Summary ........ 594 Exercises ........ 594

SHELLS ........ 595 General ........ 595 Membrane theory for surfaces of revolution ........ 596 Symmetrical loading ........ 596 Asymmetric loading ........ 600 Membrane theory for cylindrical shells ........ 601 General relationships ........ 601 Pipes and barrel vaults ........ 602 Polygonal domes ........ 604 Membrane forces in shells of any form ........ 606 Equilibrium conditions ........ 606 Elliptical problems ........ 607 Hyperbolic problems ........ 608 Bending theory for rotationally symmetric cylindrical shells ........ 613 Bending theory for shallow shells ........ 615 Basic concepts ........ 615 Differential equation for deflection ........ 616 Circular cylindrical shells subjected to asymmetric loading ........ 617 Bending theory for symmetrically loaded surfaces of revolution ........ 620 Basic concepts ........ 620 Differential equation for deflection ........ 620 Spherical shells ........ 621 Approximation for shells of any form ........ 623 Stability ........ 623 General ........ 623 Bifurcation loads ........ 624 Commentary ........ 626 Summary ........ 627 Exercises ........ 628

26.1 26.2 26.2.1 26.2.2 26.3 26.3.1 26.3.2 26.3.3 26.4 26.4.1 26.4.2 26.4.3 26.5 26.6 26.6.1 26.6.2 26.6.3 26.7 26.7.1 26.7.2 26.7.3 26.7.4 26.8 26.8.1 26.8.2 26.8.3 26.9 26.10

APPENDIX A1 A2 A3 A4

DEFINITIONS ........ 631 NOTATION ........ 637 PROPERTIES OF MATERIALS ........ 643 GEOMETRICAL PROPERTIES OF SECTIONS ........ 645

MATRIX ALGEBRA ........ 649

A5.1 A5.2 A5.3 A5.4 A5.5 A5.6

Terminology ........ 649 Algorithms ........ 650 Linear equations ........ 652 Quadratic forms ........ 652 Eigenvalue problems ........ 653 Matrix norms and condition numbers ........ 654

XII

CONTENTS

TENSOR CALCULUS ........ 655

A6.1 A6.2 A6.3 A6.4 A6.5

Introduction ........ 655 Terminology ........ 655 Vectors and tensors ........ 656 Principal axes of symmetric second-order tensors ........ 658 Tensor fields and integral theorems ........ 658

CALCULUS OF VARIATIONS ........ 661

A7.1 A7.2 A7.3

Extreme values of continuous functions ........ 661 Terminology ........ 661 The simplest problem of calculus of variations ........ 662 A7.4 Second variation ........ 663 A7.5 Several functions required ........ 664 A7.6 Higher-order derivatives ........ 664 A7.7 Several independent variables ........ 665 A7.8 Variational problems with side conditions ........ 665 A7.9 The RITZ method ........ 666 A7.10 Natural boundary conditions ........ 667 REFERENCES ........ 669 NAME INDEX ........ 671 SUBJECT INDEX ........ 673

Contents

EXAMPLECOLLECTION Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example

3.1 3.2 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 6.1 7.1 7.2 7.3 7.4 7.5 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11 8.12 8.13 8.14 8.15 8.16 10.1 10.2 10.3 10.4 10.5 10.6 11.1 11.2 11.3 11.4 11.5 11.6 12.1 12.2 12.3 13.1 13.2 13.3 13.4 13.5 13.6 13.7

Service criteria agreement for industrial building XY in Z ........ 15 Basis of design for industrial building XY in Z ........ 19 Cantilever retaining wall ........ 45 Support envelope ........ 47 Steel plate ........ 56 Stress tensor ........ 59 Hoop stress formula ........ 63 Thrust line ........ 63 Three-hinged arch ........ 65 Beam as circular arc ........ 67 Measuring grid ........ 73 Time-independent restraint ........ 93 Time-dependent restraint ........ 93 Prestressing ........ 93 Loss of prestress ........ 93 Fatigue of reinforcing steel ........ 97 Determining internal force variables ........ 116 Determining external deformation variables ........ 116 Geometric and material non-linearity ........ 117 Tie ........ 119 Beam with one degree of static indeterminacy ........ 121 Geometric non-linearity ........ 122 Cantilever beam ........ 122 Cantilever beam ........ 124 Calibration ring ........ 124 Simply supported beam ........ 126 Simply supported beam ........ 128 Tie ........ 129 Cantilever beam ........ 130 Ideal cantilever column ........ 130 Cantilever beam column ........ 131 Simply supported beam column ........ 133 Plane truss ........ 152 Plane frame ........ 153 Plane truss ........ 154 Plane frame ........ 154 Three-hinged arch ........ 156 Plane frame ........ 157 Hinged girder ........ 162 Three-hinged frame with tie ........ 164 Plane truss ........ 169 Plane truss ........ 171 Plane truss ........ 172 Plane truss ........ 172 Hinged girder ........ 180 Three-hinged arch ........ 180 Plane truss ........ 182 Unequal leg angle ........ 188 Rectangular cross-section – kern ........ 190 Reinforced concrete slab – bending ........ 191 Reinforced concrete slab – shrinkage ........ 192 Rectangular cross-section – shear stress distribution ........ 195 Wide-flange beam ........ 197 Unequal leg angle ........ 198

XIII

XIV

CONTENTS

Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example Example

13.8 13.9 13.10 13.11 13.12 13.13 13.14 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 15.1 15.2 15.3 15.4 15.5 15.6 15.7 16.1 16.2 16.3 16.4 16.5 16.6 16.7 16.8 16.9 16.10 16.11 17.1 17.2 17.3 17.4 17.5 17.6 17.7 17.8 17.9 17.10 17.11 17.12 17.13 18.1 18.2 18.3 18.4 18.5 18.6 18.7 18.8 18.9 18.10

Elliptical bar ........ 202 Narrow rectangular cross-section ........ 203 Reinforced concrete box girder ........ 205 Twin-cell box girder ........ 206 Twisted beam – concentrated load ........ 208 Twisted beam – distributed load ........ 209 Reinforced concrete beam ........ 213 SIMPSON’s rule ........ 225 Beam with one degree of static indeterminacy ........ 226 Hinged girder ........ 227 Cantilever beam ........ 228 Cranked cantilever beam ........ 228 Plane truss ........ 229 Rectangular cross-section – area shear factor ........ 229 Thin-wall hollow cross-section ........ 230 Simply supported beam ........ 236 Beam fixed at both ends ........ 236 Beam with one degree of static indeterminacy ........ 237 Beam with spring restraint ........ 239 Cantilever beam ........ 239 Beam with one degree of static indeterminacy ........ 240 Hinged girder ........ 240 Plane frame ........ 257 Bar fixed at both ends ........ 258 Beam with one degree of static indeterminacy ........ 259 Continuous beam ........ 260 Beam fixed at both ends ........ 262 Continuous beam of infinite length ........ 263 Continuous beam – support settlement ........ 267 Arch fixed at both ends ........ 268 Beam on skew supports ........ 269 Beam as circular arc ........ 270 Considering subsystems ........ 271 Cantilever beam rigid in shear ........ 280 Cantilever beam rigid in shear ........ 283 Plane frame ........ 287 Non-sway frame ........ 294 Grandstand frame ........ 295 Multi-storey sway frame ........ 296 Multi-storey non-sway frame ........ 297 Non-sway frame – settlement of supports ........ 299 Non-sway frame – uniform rise in temperature ........ 299 Non-sway frame – temperature difference ........ 301 Sway frame – uniform rise in temperature ........ 301 Three-span frame ........ 304 Continuous beam ........ 306 Bar restrained at both ends ........ 313 Bar with spring restraint at one end ........ 313 Reinforced concrete column – change in temperature ........ 314 Reinforced concrete column – shrinkage ........ 314 Pulling out a reinforcing bar ........ 317 Multi-storey frame ........ 322 Externally statically indeterminate VIERENDEEL girder ........ 323 Plastic panel with bonded sheet steel outer faces ........ 325 Simply supported beam – sinusoidal line load ........ 327 Bar fixed at both ends – linear temperature gradient ........ 328

Contents

18.11 18.12 18.13 18.14 18.15 18.16 18.17 18.18 18.19 18.20 18.21 18.22 18.23 18.24 18.25 18.26 18.27 18.28 18.29 18.30 19.1 19.2 19.3 21.1 21.2 21.3 21.4 21.5 21.6 22.1 22.2 22.3 22.4 22.5 22.6 22.7 22.8 22.9 22.10 22.11 22.12 22.13 22.14 22.14 22.15 22.16 23.1 23.2 23.3 23.4 23.5 23.6 23.7 23.8 23.9 23.10

High-rise building ........ 335 High-rise building with outrigger ........ 337 Shear wall ........ 340 Shear wall – influence of wall extensions ........ 341 Two-hinged arch – uniformly distributed load ........ 347 Two-hinged arch – sinusoidal load ........ 348 Two-hinged arch – constant load segment by segment ........ 349 Displacement of the abutments to a concrete arch ........ 350 Stiffened pipe subjected to internal pressure ........ 354 Single strand – uniformly distributed load ........ 359 Single strand – thermal action ........ 359 Single strand – prestress ........ 359 Single strand – constant loads on both halves of the span ........ 360 Cable with wheel load ........ 362 Stresses in stay cable ........ 363 Suspended roof – uniformly distributed load ........ 364 Suspended roof – asymmetric imposed load ........ 365 Stress ribbon – asymmetric imposed load ........ 365 Suspended roof – central point load ........ 367 Stress ribbon – thermal action ........ 367 Plane frame ........ 374 Orthogonalised restraint states ........ 379 Beam with one degree of static indeterminacy ........ 384 Unequal leg angle ........ 427 Two-span beam – repeated variable actions ........ 436 Plane frame ........ 442 Plane frame – static program ........ 443 Plane frame – kinematic program ........ 444 Plane frame – minimum weight ........ 445 Beam column ........ 451 Cantilever column ........ 455 Ideal column ........ 455 Beam column ........ 456 Ideal column ........ 456 Ideal column with one degree of static indeterminacy ........ 457 Column with abrupt change in stiffness ........ 458 Load applied to top of cantilever column ........ 458 Statically determinate frame ........ 459 Elastically supported inclined leg frame ........ 463 Two-hinged frame ........ 467 Non-sway frame ........ 468 Sway frame ........ 468 Elastically restrained vertical cantilever ........ 468 Vertical cantilever ........ 477 Lateral buckling of an section ........ 486 Lateral buckling – shifting the point of load application ........ 486 Cantilever beam ........ 492 Cylindrical pipe ........ 495 Beam in the form of a circular arc ........ 495 Uniaxial tension ........ 498 Vertical embankment ........ 505 Strip foundation on TRESCA half-space ........ 506 Curtailed reinforcement in tension chord ........ 514 Web crushing failure ........ 516 Dissipation at hyperbolic slip line ........ 517 Strip foundation on TRESCA half-space ........ 519

XVI

CONTENTS

24.1 24.2 24.3 24.4 24.5 24.6 24.7 24.8 24.9 24.10 24.11 24.12 24.13 24.14 24.15 24.16 24.17 24.18 24.19 24.20 24.21 24.22 24.23 24.24 24.25 25.1 25.2 26.1 26.2 26.3 26.4 26.5 26.6 26.7 26.8 26.9 26.10 26.11 26.12 26.13 A7.1 A7.2

Square slab supported at the corners ........ 528 Square slab supported in the centre ........ 529 Rectangular slab supported at the corners ........ 529 Simply supported square slab ........ 547 Fixed square slab ........ 547 Simply supported circular slab ........ 549 Fixed circular slab ........ 550 Reinforced concrete slab – dimensioning for bending ........ 553 Slab element subjected to pure twist ........ 554 Fixed circular slab ........ 558 Annular slab fixed at its inner edge and loaded on its outer edge by mu ........ 558 Simply supported rectangular slab ........ 561 Square slab simply supported along two adjacent edges ........ 561 Simply supported regular polygonal slabs ........ 561 Cantilever slab with point load on edge ........ 563 Fixed rectangular slab ........ 569 Square slab simply supported along two adjacent edges ........ 569 Fixed square slab ........ 570 Slab strip subjected to a central point load ........ 570 Cantilever slab with point load on edge ........ 570 Flat slab ........ 571 Semi-infinite rectangular slab subjected to edge loads ........ 573 Buckling of simply supported rectangular slabs ........ 575 Rectangular slab with initial deformation ........ 576 Square membrane ........ 577 Barrel vault – membrane theory ........ 590 Barrel vault – bending theory ........ 591 Spherical shells ........ 597 Spherical tanks ........ 598 Conical shell ........ 598 Spherical shell – dead load ........ 599 Spherical shell – wind pressure ........ 600 Conical shell – wind pressure ........ 601 Dome with circular cylindrical sectors – self-weight ........ 605 Pipe subjected to end loads ........ 613 Cylindrical tank ........ 614 Pipe subjected to thermal action ........ 614 Chimney subjected to wind pressure ........ 618 Fixed spherical shell subjected to internal pressure ........ 622 Pressure vessel ........ 622 Cantilever beam rigid in shear ........ 666 Cantilever beam – uniformly distributed load plus load at free end ........ 667

5.1 Force systems and equilibrium

STATIC RELATIONSHIPS

5.1

Force systems and equilibrium

5.1.1

Terminology

Forces are perceived through their effects. They correspond to physical interactions that cause or modify states of deformation or motion in material systems. The effect of a 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 with point 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 imagine a body connected to this one plane, then it is clear that F would cause a rotation of the body about the axis n perpendicular to the plane and passing through O. The tendency to rotate is proportional to the magnitude F and the distance a of force F from O. The position vector r of the point of application A of F expresses the tendency to rotate with the moment MwrqF (5:1) correctly in terms of magnitude and direction; jMj = Fa applies and the vectors M, r and F constitute a right-hand screw, see Fig. 5.1(b). As can be seen, the moment M remains unaltered if force F is translated along its line of action f. Every force F has a corresponding reaction â&#x20AC;&#x201C; F with the same line of action. According 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 their reactions; the interaction between two bodies with mass generally takes place without contact. But in the case of contact forces (e. g. support forces), the points of application of forces and reactions are geometrically identical (although not materially identical); the interaction between support and supported body comes about through contact â&#x20AC;&#x201C; if the contact is eliminated, so the contact force disappears as well.

(a) M M

Contact forces are generally in the form of surface forces (surface loads) distributed over a finite area. The contact force related to the unit of surface area, the force per unit area dF (5:2) tw dA 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 called body forces (body loads) with a force per unit volume of dF (5:3) qw dV see Fig. 5.2(b).

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

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

(b)

A f

(M ) (F )

(r)

Fig. 5.1 Force and moment: (a) reference point O, point of application A, line of action f, axis of rotation n, (b) right-hand screw rule

5 STATIC RELATIONSHIPS

(a)

(b)

dF dF dF

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

If, ultimately, a body such as a beam or cable is idealised as one-dimensional and if distributed forces act on this in the form of line forces (line loads), we get a force per unit length dF (5:4) qw ds see Fig. 5.2(c). In the SI or MKS systems, the unit used for specifying the magnitude of a force is the newton [1 N = 1 mkgs–2 ] or [kN] or [MN]. Correspondingly, the unit used for moments 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 per unit 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 application lie within the arbitrary limits of a body or system. A body isolated from a body or system (or part thereof) by means of an imaginary cut is known as a free body (FB). By introducing all the forces that act on the free body, we obtain a free-body diagram (FBD). The resultant force of a force system is obtained by adding together the vectors acting on the free body: X F (5:5) Rw FB

Likewise, the resultant couple of a force system with respect to an arbitrary reference point O is X rqF (5:6) MO w FB

see Fig. 5.3 and (5.1). If instead of O we select a different reference point Ol, then according to Fig. 5.3, with rl = r – rL and considering (5.5) and (5.6), it follows that X X X (5:7) M Ol w rl q F w r q F s rL q F w M O s rL q R FB

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 with respect to an arbitrary reference point. According to (5.7), the equivalence of two force systems need only be verified for one reference point; the identity of the resultant couple is then given for all points.

II FUNDAMENTALS

O y

r ´´

z r

r´

O ´

F Fig. 5.3 Free body diagram with reference points O and Ol

5.1 Force systems and equilibrium

5.1.3

Equilibrium

A force system is in equilibrium when its force-couple system sums to zero: Rw0 , MO w 0 (5:8) The equilibrium conditions (5.8) result in six scalar equations in the case of force systems in three dimensions, i. e. three force-balance equations and three moment-balance 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-balance 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 reference point. Generally, in the three-dimensional case, moment-balance equations can be formulated about six non-collinear axes and in the coplanar case about three points not lying in a straight line. In practice, this is often easier than setting up the force-balance equations. Depending on the particular problem, in the coplanar case only one, and in the three-dimensional case only one or two, force-balance equations are replaced by moment-balance equations, as is explained further in chapter 10. Applying (5.8) to differential structural elements results in differential equations for the equilibrium, as dealt with in section 5.3. When defining free bodies and applying the equilibrium conditions to those bodies, we generally use the so-called free-body principle : if we remove arbitrary parts from a compatibly 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 depending on whether the material point of application of the reaction to a force lies inside or outside 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 be in equilibrium if the free body is in equilibrium in its entirety. This assertion is known as the fundamental theorem of statics. If the equilibrium conditions â&#x20AC;&#x201C; at best following a suitable breakdown of the system â&#x20AC;&#x201C; are sufficient for determining the unknowns in a problem, we speak of a statically determinate 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 surroundings according 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 of the base (G3 and G4), the active and passive earth pressures (Ea and Ep) plus a contact force A acting on the underside of the base. For simplicity, hydrostatic pressures are neglected. Further, the calculation with the earth surcharges G3 and G4 represents a considerable idealisation. Actually, in the event of an overturning failure, a wedge-shaped mass of soil would form in the ground behind the wall. And this would be linked with the mobilisation of further forces that are neglected here; a similar consideration 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 of application, e. g. by setting up the two force-balance equations in the y and z directions and the moment-balance equation about O. Alternatively, A can also be determined graphically. Fig. 5.4(c) shows

5 STATIC RELATIONSHIPS

(d)

(a)

B G4 G3 D

y O

(b)

(c)

G2 Ea E p G3

2 Az 3a

G4 G2

(e)

G1 z

G3 Ep

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

the associated addition of the force vectors in the so-called force polygon; for equilibrium, the force polygon must be closed, which determines the magnitude and direction of A. The point of application of A follows from the funicular polygon according to Fig. 5.4(d). This is done by successively forming the interim resultants (shown as dotted lines in the force polygon) of forces Ea and G4 etc., drawing their lines of action starting from point A, the intersection of Ea and G4 , and extending this to intersect with the next force G1 at B etc. This approach enables us to establish point E, the intersection of the lines of action CE and DE for (Ea , G4 , G1 , G2) or (Ep , G3), and hence determine the line of action 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 the strength of the subsoil is finite. Fig. 5.4(d) shows one practical possibility and Fig. 5.4(e) shows a statically 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 â&#x20AC;&#x201C; 3a i y j â&#x20AC;&#x201C; 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 at the underside of the base solely on the basis of static considerations. For simplicity, a distribution proportional to Az is assumed, which in this particular case means a triangular distribution.

II FUNDAMENTALS

5.1 Force systems and equilibrium

(a)

A G0

h G1 + G2 G0 G1 + G2 C B

(d)

(b) F

H A

I G2 D

D2 G0

D1 G1 C B a1 a0

G0 J

G 1 + G2 A

Fig. 5.5 Upturned right-angled element on horizontal surface: (a) elevation, (b) plan, (c) force polygon, (d) funicular polygon

Example 5.2 Support envelope

The upturned right-angled element supported on a horizontal surface shown in Fig. 5.5(a) and (b) is loaded at Al by a horizontal force Q. In Fig. 5.5(b) we must distinguish between the area of contact ABCDEF and the support envelope ABCEF. The latter is the smallest convex envelope enclosing the former. The overall stability of the right-angled element can be checked with the help of a moment-balance equation about axis CE. The overturning moment Qh due to Q acting about CE may not exceed the resisting 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 funicular polygons. For overall stability, the point of application J of the contact force A must lie within the support envelope. The contact force A is assumed to be distributed equally over the end zones of the two legs of the element. In the limit case, the force is concentrated at points C and E, which means that the local bearing 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 the x, 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 support force-couple system) is known as the determinacy of the support.

x y v

Ď&#x2022;x

Ď&#x2022;y Ď&#x2022;z Fig. 5.6 Displacements and rotations

5 STATIC RELATIONSHIPS

(a)

(b)

(c)

(d)

(e)

z 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

Fig. 5.7 shows a number of different types of support. If the concrete hinge shown in Fig. 5.7(a) is in the form of a linear support (long in the y direction), it inhibits displacements u, v, w and rotations fx , fz ; in the form of a discrete support (short in the y direction), both fy and also fx and fz are practically not inhibited at all. It is notable that 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 the horizontal joint, especially in the z direction, too, because of the reinforcement intersecting the horizontal joint. The steel linear rocker bearing shown in Fig. 5.7(b) acts bilaterally with respect to u and unilaterally with respect to w â&#x20AC;&#x201C; the support would lift up in the z direction when the force tends towards zero; with respect to v, it works bilaterally up to a certain amount, either via friction or with lugs at the sides (after overcoming the play between lug and body of bearing); rotation fy is practically unrestrained, and rotations fx , fz are inhibited. In the case of the steel roller bearing shown in Fig. 5.7(c), u and fy are not inhibited and the support acts unilaterally with 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 laminated elastomeric bearing shown in Fig. 5.7(d) functions unilaterally with respect to w and, depending on the particular type, enables displacements u, v as well as rotations fy , fx . The same is true for the elastomer pot sliding bearing shown in Fig. 5.7(e). A closer look at Fig. 5.7 shows that, depending on the particular design, the displacement and rotation capabilities of supports always lie within certain limits and are never enabled or prevented in absolute terms. Likewise, the components of the support force-couple system associated with the inhibited displacement and rotation capabilities are restricted to certain limit values. In practice, it is certainly necessary to consider these limits carefully every time. In theory of structures, we assume the appropriate idealisations shown in Fig. 5.8 for the coplanar case. Fig. 5.8(a) shows a unilaterally or bilaterally functioning sliding support (hinged support capable of displacement) that only inhibits w and whose support force-couple system is limited to the force component in the z direction. In the case of the hinged support shown in Fig. 5.8(b), u is also inhibited and the corresponding 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. Extending these considerations to the general three-dimensional case is easily possible with the help of Fig. 5.6. Static equivalents to the types of support shown in Fig. 5.8 can be realised according to Fig. 5.9 with pin-jointed members. These are straight, weightless bars connected concentrically on both sides with frictionless hinges. With such assumptions, only forces can be transferred from the bars, whose lines of action coincide with the axes of the bars. 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 inclination of the pin-ended strut as a result of a displacement u is negligible in comparison to the force component in the z direction, assuming infinitesimally small disII FUNDAMENTALS

(a)

(b)

(c)

x z

Fig. 5.8 Support idealisations: (a) sliding support, (b) hinged support, (c) fixed support

5.1 Force systems and equilibrium

(a)

(b)

(c)

z Fig. 5.9 Equivalent supports with pin-jointed members: (a) sliding support, (b) hinged support, (c) fixed support.

placements (first-order theory, see section 6.1). The support force component possible in the x direction with a hinged support requires a corresponding second pin-jointed member, as shown in Fig. 5.9(b). Ultimately, a third pin-jointed member is required to achieve fixity, as shown in Fig. 5.9(c); here, the first two pin-jointed members exhibit different lines of action and the axis of the third may not pass through the intersection of the first two, otherwise the support would not provide fixity, instead function like a hinged support at this point. For the general three-dimensional case, six pin-jointed members are necessary for a braced support. With the force-couple system {R, MO}, the coordinates rij of one point on the axis of the pin-jointed member i and the direction cosine cij of the six pinjointed members and the forces in those members Ni , then according to (5.5) and (5.6) the following applies: 9 38 9 8 2 c1x c2x . . . c6x N1 > > Rx > > > > > > > > > > 7> 6 > > > c1y c2y . . . c6y > > > > N2 > > Ry > 7> 6 > > > > > > > 7> 6 7< N3 = < Rz = 6 c1z c2z . . . c6z 7 6 w 6 r c sr c > r2y c2z s r2z c2y . . . r6y c6z s r6z c6y 7 N4 > 1z 1y > > > > MOx > 7> 6 1y 1z > > > > > > > 7> > > 6 > > > > 4 r1z c1x s r1x c1z r2z c2x s r2x c2z . . . r6z c6x s r6x c6z 5> N M > > > > 5> > Oy > > > ; : ; : r1x c1y s r1y c1x r2x c2y s r2y c2x . . . r6x c6y s r6y c6x N6 MOz (5:9) In order that (5.9) can have a solution N for any force-couple system, the matrix on the left 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 the xz plane) with Rx , Rz , MOy to be the intersection of bars 1 and 2 (r1 = r2 = 0) and consider the point of pin-jointed member axis 3 on the z axis (r3x = 0, r3z 0 0). In this case, removing the second, fourth and sixth rows as well as the fourth, fifth and sixth columns from the matrix in (5.9) gives us the following matrix: 2 3 c3x c1x c2x 4 c1z c2z (5:10) c3z 5 0 0 r3z c3x and hence det w r3z c3x Ă°c1x c2z s c1z c2x Ă&#x17E;00 (5:11) must apply. Without restricting the universal applicability, it is possible to place the x 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 may not be collinear. The above requirements are therefore confirmed. It is often not possible to idealise supports as fully restrained, as has been assumed up to now; instead, it is necessary to consider their flexibility. To this end, in accordance with Fig. 5.10, we use appropriate translational and rotational springs and in the simplest case presume a linear relationship between the components of the support force-couple system and the corresponding displacements and rotations: (5:12) Ax w s kx uA , Az w s kz wA , MA w s ky fyA where kx , kz and ky denote the stiffnesses of the translational and rotational springs.

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