Structural Theory and Analysis

Structural Theory and Analysis
J.D. Todd
Fellow of St. Edmund Hall and University Lecturer in Engineering Science
University of Oxford
CONTENTS
Preface ix
Plane statics 1
1.1 Introduction 1
1.2 Equations of equilibrium 2
1.3 Supports, reactions and free-body diagrams 2
1.4 Stability and determinacy of reactions 4
1.5 Calculation of reactions 5
1.6 Equation of condition 8
1.7 Principle of virtual work 9
1.8 Shear force and bending moment 11
1.9 Relations between load, shear force and bending moment 15
1.10 Principle of superposition 16
1.11 The force polygon 20
1.12 The funicular polygon 21
1.13 Graphical determination of reactions 22
1.14 Graphical construction of bending-moment diagrams 23
1.15 Graphics applied to a three-pin arch 25
1.16 The differential equation for a vertical load funicular polygon 27
1.17 Intraduction to influence lines 28
1.18 Influence lines for reactions, shear force and bending moment 29
1.19 Loading systems 30
1.20 Application of virtual-work methods to influence lines 32
1.21 Multi-load systems 35
1.22 Influence lines for girders with floor beams 39
1.23 Influence line for three-pin arch 40
2 Statically determinate structures 45
2.1 Simple plane trusses
2.2 Stability and determinacy
2.3 Resolution at joints
2.4 Graphical method
2.5 Method of sections
2.6 Compound trusses
2.7 Complex trusses
2.8 Virtual work
CONTENTS
2.9 Statically determinate space structures
2.10 Tension coefficients
2.11 Rigid jointed structures
2.12 Influence lines for statically determinate trusses
3 Elementary elasticity, plasticity and bending of beams
3.1 Sign conventions
3.2 Stress
3.3 Strain
3.4 Relations between stress and strain
3.5 Composite rods with axial tension or compression
3.6 Principal stresses in two dimensions
3.7 The Mohr circle diagram for stress
3.8 Principal strain in two dimensions
3.9 The Mohr circle diagram for strain
3.10 Relationships between elastic constants
3.11 Strain energy
3.12 Pure bending of beams
3.13 Bending of a composite beam
3.14 Bending of unsymmetrical sections
3.15 Strain energy due to bending
3.16 Combined bending and axial force
3.17 Plastic bending of beams
3.18 Combined plastic bending and axial force
4 Torsion and shear effects
4.1 Torsion of a circular cross-section
4.2 Strain energy due to torsion
4.3 Combined torsion, bending and axial force
4.4 Plastic torsion of circular rods
4.5 Shear stresses in beams
4.6 Shear-stress distribution in flanges
4.7 Shear centre
4.8 Torsion of thin-walled tubes
4.9 Torsion of a thin rectangular section
5 Deflection of beams
5.1 Introduction 138
5.2 Deflection by direct integration 139
5.3 Deflections using singularity functions or the Macaulay method 141
5.4 Moment-area methods 146
5.5 Use of standard cases 151
5.6 Deflections due to shear 153CONTENTS
6 Virtual work and energy methods
6.1 Introduction
6.2 Strain energy and complementary energy
6.3 Flexibility coefficients
6.4 Maxwell’s reciprocal theorem
6.5 The flexibility approach
6.6 Kinematic indeterminacy
6.7 The stiffness approach
6.8 The method of real work
6.9 Method of virtual work
6.10 Virtual work applied to statically determinate systems
6.11 Energy theorems and their application to statically determinate
structures
6.12 Williot-Mohr displacement diagram
6.13 Virtual work applied to a statically indeterminate truss
6.14 Engesser’s theorem of compatibility
6.15 Trusses with several redundants
6.16 The trussed beam
6.17 Virtual work and energy methods applied to frames
6.18 Ring and arch problems
6.19 Redundant trusses using the stiffness approach
7 Moment distribution and slope deflection
7.1 Moment distribution
7.2 Sign convention
7.3 Stiffness and carry-over factor
7.4 Distribution factor
7.5 Fixed end moments
7.6 Examples
7.7 Modified stiffness
7.8 Deflection of supports and sidesway
7.9 Frames with inclined members
7.10 Rectangular multi-storey frames
7.11 The slope-deflection equation
8 Stiffness and flexibility methods
8.1 Introduction
8.2 Outline of the stiffness method
8.3 Stiffness matrix for a space structure member
8.4 Pin-jointed plane trusses
8.5 Example of a pin-jointed redundant truss
8.6 The space truss
8.7 Continuous beams
8.8 The plane frame
8.9 Grillages
8.10 Special cases
8.11 Outline of the flexibility method
270 viii CONTENTS
8.12 Member flexibility matrix
8.13 Example of a pin-jointed redundant truss
8.14 Example of a plane frame
8.15 Choice of stiffness or flexibility approach
9 Influence lines for statically indeterminate beams
9.1 Beams with two spans
9.2 Betti’s reciprocal theorem
9.3 Applications of Betti’s theorem to influence lines
9.4 Multi-span beams
10 Stability of columns
10.1 Introduction
10.2 Euler critical loads
10.3 Strut with initial deformation
10.4 Struts made from ideal elasto-plastic materials
10.5 Double-modulus theory
10.6 Tangent-modulus theory
10.7 Practical strut formulae
10.8 The Southwell method
10.9 Energy methods
II Plastic analysis of beams and frames
11.1 Introduction
11.2 Collapse of redundant beams
11.3 Load factor
11.4 Basic theorems
11.5 Graphical analysis
11.6 Virtual-work approach
11.7 Combination of mechanisms
SuggestionsforjUrtherreading
Answers to problems
Index
INDEX
Antisymmetry, stiffness factor 224
Arches, funicular polygon 25
influence line 40
rib shortening 208
temperature effects 208
three-hinge 7
two-hinge 206
Area-moment method 146
Axes, local 244
member 246
system 245
Axial force, composite rods 80
sign convention 12
with bending 108
Beams, bending deflection 138
bending stresses 97
collapse of redundant 319
composite 102
influence lines 29, 287
moment distribution 220
plastic bending 109
shear centre 131
shear deflection 153
shear stress 124
shear stress in flanges 129
slope deflection 237
stiffness matrix 261
unsymmetrical section I 04
Bending moment, calculation of 12
envelope 39
graphical method 23
influence lines 30
rigid frames 62
sign convention 12
Bending of beams, composite 102
Mohr circle 1OS
neutral axis 98
350
plastic bending 109
plastic bending with axial force 114
principal axis 99
second moment of area 99
section modulus 101
strain energy 107
unsymmetric section 104
with axial force 108
with torsion and axial force 122
Betti’s reciprocal theorem 289
Bow’s notation 22, 50
Buckling of struts 299
Bulk modulus 93
Carry-over factor 215
Castig1iano’s theorem 177
applied to non-linear members 183
applied to redundant trusses 21 0
Centre of shear 131
Collapse, of beams 319
of frames 325
Combination of mechanisms 334
Combined, bending with axial force 108
plastic bending with axial force 114
torsion, bending and axial force 122
Compatibility 164
Engesser’s theorem of 192
Complementary energy 157
applied to beams 182
applied to frames 200
applied to rings and arches 204
applied to trusses 180
Complementary shear stress 76
Complex trusses 53
Composite beams I 02
Composite rods, axial forces 80
temperature effects 81
Compound trusses 53INDEX 351
Conjugate beam 150
Core of section I 08
Criticalload 299
Deflection of beams 138
complementary energy 182
conjugate beam 150
direct integration 139
duetoshear 153
Macaulay’s method 141
moment area 146
singularity functions 141
standard cases 151
variable section 150, 182
virtual work I 75
Deflection of trusses, Castigliano’s
theorem 183
complementary energy 179
non-linear members 176
real work 168
virtual work 171
Williot-Mohr diagram 185
Determinacy of reactions 4
Direct integration 139
Direction cosines 259
Direct stress 75
Displacement diagram 56
Distribution factor 217
Double modulus theory 307
Electrical resistance strain gauge 89
Energy methods applied to struts 313
Energy theorems, Castigliano’s theorem
177
complementary energy theorem
179
Engesser’s theorem 192
Engesser’s theorem 192
Equation of condition 5, 8
Equilibrium equations 2
Equivalent section I 03
Euler critical load 299
Fixed-end moments 217
Flexibility coefficients 159
Flexibility matrix for space member
274
Flexibility method, outline of 270
plane frame 279
redundant plane truss 275
Flexural rigidity 139
Force polygon 20
Form factor 110
Frames, energy methods 199
flexibility matrix method 279
inclined members 233
moment-distribution method 230
multi-storey rectangular 235
slope-deflection method 239
stiffness-matrix method 264
virtual-work method 199
Free-body diagram 3
Fully plastic moment 110
Funicular polygon, bending moment 23
description 21
differential equation 27
reactions 22
three-pin arch 25
Graphical analysis, plastic collapse 325
Grillages 269
Hooke’s law 78
generalised 80
Hydrostatic pressure 93
Influence lines, applications of Betti’s
theorem 290
beams with two spans 287
bending moments 30, 33, 36
envelope 36, 39
girders with floor beams 39
multi-load systems 35
multi-span beams 294
reactions 29, 32
shear force 30, 33, 35
three-pin arch 40
trusses 66
virtual work 3 2
Initial lack of fit 192
Instability 298
Interaction diagram 115, 327
Kinematic indeterminacy 165
Lack of fit 192
Load factor 324
Local coordinates 244
Lower-bound theorem 325
Macaulay’s method 141
Matrix analysis of structures 244
Maximum bending moment 36
Maximum shear force 35352 INDEX
Maxwell’s reciprocal theorem 151, 161
Modified stiffness, antisymmetry 224
pinned end 222
symmetry 223
Modular ratio 103
Modulus, bulk 93
elasticity 78
elastic section 101
plastic section 11 0
rigidity 80
Mohr circle, second moment of area
105
strain 89
stress 85
Mohr equation of virtual work 170
Moment-area method, 146
Moment distribution, deflection of
supports 227
distribution factor 217
frames with inclined members 233
fixed-end moments 21 7
modified stiffness 222
rectangular multi-storey frames 23~
sign convention 215
stiffness and carry-over factor 215
Miiller-Breslau principle 287
Neutral axis 98
Non-linear members 176
Normal strain 77
Normal stress 7 5
No-sidesway distribution 232
Permanent set 112
Perry-Robertson formula 311
Plane stress 83
Plastic bending of beams 109
Plastic bending with axial force 114
Plastic collapse, combination of
mechanisms 334
deflections 321
graphical analysis 325
interaction diagram 327
load factor 324
redundant beams 319
theorems 325
virtual-work method 330
Plastic section modulus 110
Plastic torsion of circular rods 122
Poisson’s ratio 78
Principal axes 99
Principal plane 84
Principal strain 87
Principal stress 83
Principle of superposition 16
Principle of virtual work 9
Product second moment of area 99
Rayleigh method 314
Reactions, influence lines 29, 32
moments and resolution 6
statical determinacy 4
virtual work I0
Real work, method of 168
Reciprocal theorem, Betti’s 289
Maxwell’s 161
Relationships between elastic constants
92
Rib shortening 208
Rigidity, modulus of 78
Rigid jointed structures 60
Rings 203
Safe theorem 325
St Vernant’s principle 79
Second moment of area 99
Section modulus 101
Sections, method of 51
Shape factor 110
Shear centre 131
Shear deflection 153
Shear force, calculation 12
influence lines 30, 33, 35
sign 12
Shear strain 77
Shear stress, complementary 76
in beams 124
in flanges 129
Sidesway 227
Sign convention, general 73
moment distribution 215
shear force and bending moment 12
Singularity functions 141
Slenderness ratio 305
Slope-deflection method 236
Southwell method for struts 311
Space truss 58
Squash load 114
Stability of columns, double modulus
theory 307
Euler critical loads 299
ideal elasto-plastic material 305
initially deformed strut 304
practical strut formulae 310INDEX 353
Stability of columns-continued
Rayleigh method 314
Southwell method 311
tangent modulus theory 309
Timoshenko method 314
Statical determinacy, reactions 4
rigid jointed frames 61
space trusses 57
trusses 46
Stiffness coefficients 167
Stiffness factor 215
Stiffness matrix, continuous beams 261
grillages 269
pinned members 250
plane frames 264
space structure member 259
transformation matrix 251
Stiffness method 244
Strain 76
Mohr circle for 89
normal 77
principal 87
shear 77
Strain energy 9 5
axial load 9 5
bending 107
torsion 121
Stress 74
complementary shear 76
hoop 94
Mohr circle for 85
normal 75
principal 83
shear 75
String polygon, see Funicular polygon
Struts, see Stability of columns
Superposition, principle of 16
Supports 2
Symmetry, stiffness factor 223
System coordinates 245
Tangent modulus theory 309
Temperature effects, arches 208
composite rods 81
redundant trusses 192
Tension coefficients 58
Three-pin arch, determination of
reactions 7
funicular polygon 25
influence line 40
Timoshenko method 314
Torsion. circular rods 118
multi-cell tubes 133
plastic 122
strain energy 121
thin rectangular section 134
thin-walled tubes 132
with bending and axial force 122
Transformation matrix 250
Transformed section, beams 103
composite rods 81
Trussed beam 198
Trusses, complex 53
compound 53
deflections, see Deflection of trusses
matrix methods applied 250
redundant 189
simple plane 45
space 57
Uniqueness theorem 325
Unit-load method 171
Unsafe theorem 325
Upper-bound theorem 325
Virtual displacement 10
Virtual work, applied to plastic
collapse 330
deflections by 170
influence lines by 32, 68
Mohr equation of 170
principle of 9
Warping 127, 154
Williot-Mohr displacement diagram
185
Young’s modulus 78
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