Examples in Structural Analysis

Contents 

1.  Structural Analysis and Design
1.1  Introduction 
1.2  Equilibrium 
1.3  Mathematical Modelling
1.3.1  Line Diagrams
1.3.2  Load Path 
1.3.3  Foundations 
1.4  Structural Loading
1.5  Statical Indeterminacy
1.5.1  Indeterminacy of Two-Dimensional Pin-Jointed Frames
1.5.2  Indeterminacy of Two-Dimensional Rigid-Jointed Frames
1.6  Structural Degrees-of-Freedom
1.6.1  Problems: Indeterminacy and Degrees-of-Freedom
1.6.2  Solutions: Indeterminacy and Degrees-of-Freedom

2.  Material and Section Properties 

2.1  Introduction
2.1.1  Simple Stress and Strain
2.1.2  Young’s Modulus (Modulus of Elasticity)
2.1.3  Secant Modulus 
2.1.4  Tangent Modulus
2.1.5  Shear Rigidity (Modulus of Rigidity)
2.1.6  Yield Strength
2.1.7  Ultimate Tensile Strength
2.1.8  Modulus of Rupture in Bending
2.1.9  Modulus of Rupture in Torsion
2.1.10  Poisson’s Ratio
2.1.11  Coefficient of Thermal Expansion
2.1.12  Elastic Assumptions
2.2  Elastic Cross-Section Properties
2.2.1  Cross-sectional Area
2.2.2  Centre of Gravity and Centroid
2.2.3  Problems: Cross-sectional Area and Position of Centroid
2.2.4  Solutions: Cross-sectional Area and Position of Centroid
2.2.5  Elastic Neutral Axes  40
2.2.6  Second Moment of Area and Radius of Gyration  41
2.2.6.1  The Parallel Axis Theorem  41
2.2.7  Elastic Section Modulus  43
2.2.8  Problems: Second Moment of Area and Elastic Section Modulii  45
2.2.9  Solutions: Second Moment of Area and Elastic Section Modulii  45
2.3  Plastic Cross-Section Properties  51
2.3.1  Stress/Strain Relationship  51
2.3.2  Plastic Neutral Axis  52
2.3.3  Evaluation of Plastic Moment and Plastic Section Modulus  53
2.3.4  Shape Factor    54
2.3.5  Section Classification  54
2.3.5.1  Aspect Ratio  54
2.3.5.2  Type of Section  55
2.4  Example 2.1: Plastic Cross-section Properties  Section 1  56
2.5  Problems: Plastic Cross-section Properties  57
2.6  Solutions: Plastic Cross-section Properties  58

3.  Pin-Jointed Frames  

3.1  Introduction    62
3.2  Method of Sections   62
3.2.1  Example 3.1: Pin-Jointed Truss  62
3.3  Method of Joint Resolution  65
3.3.1  Problems: Method of Sections and Joint Resolution  67
3.3.2  Solutions: Method of Sections and Joint Resolution   69
3.4  Method of Tension Coefficients  93
3.4.1  Example 3.2: Two-Dimensional Plane Truss  94
3.4.2  Example 3.3: Three-Dimensional Space Truss  95
3.4.3  Problems: Method of Tension Coefficients  98
3.4.4  Solutions: Method of Tension Coefficients  101
3.5  Unit Load for Deflection  113
3.5.1  Strain Energy (Axial Load Effects)  113
3.5.2  Castigliano’s 1 st  Theorem  114
3.5.3  Example 3.4: Deflection of a Pin-Jointed Truss  116
3.5.3.1  Fabrication Errors (Lack-of-fit)  120
3.5.3.2  Changes in Temperature  120
3.5.4  Example 3.5: Lack-of-fit and Temperature Difference  120
3.5.5  Problems: Unit Load Method for Deflection of Pin-Jointed frames  122
3.5.6  Solutions: Unit Load Method for Deflection of Pin-Jointed frames  123
3.6  Unit Load Method for Singly-Redundant Pin-Jointed Frames  135
3.6.1  Example 3.6: Singly-Redundant Pin-Jointed Frame 1  135
3.6.2  Example 3.7: Singly-Redundant Pin-Jointed Frame 2  137
3.6.3  Problems: Unit Load for Singly-Redundant Pin-Jointed Frames  140
3.6.4  Solutions: Unit Load for Singly-Redundant Pin-Jointed Frames  141

4.  Beams

4.1  Statically Determinate Beams  157
4.1.1  Example 4.1: Beam with Point Loads  157
4.1.2  Shear Force Diagrams  159
4.1.3  Bending Moment Diagrams  163
4.1.4  Example 4.2: Beam with a Uniformly Distributed Load  167
4.1.5  Example 4.3: Cantilever Beam  169
4.1.6  Problems: Statically Determinate Beams  Shear Force and Bending Moment  170
4.1.7  Solutions: Statically Determinate Beams Shear Force and Bending Moment  173
4.2  McCaulay’s Method for the Deflection of Beams   183
4.2.1  Example 4.4: Beam with Point Loads   184
4.2.2  Example 4.5: Beam with Combined Point Loads and UDL’s  186
4.3  Equivalent Uniformly Distributed Load Method for the Deflection of Beams 189
4.3.1  Problems:  McCaulay’s and Equivalent UDL Methods for Deflection of Beams  191
4.3.2  Solutions:  McCaulay’s and Equivalent UDL Methods for Deflection of Beams  192
4.4  The Principle of Superposition   202
4.4.1  Example 4.6:   Superposition Beam 1   203
4.4.2  Example 4.7:   Superposition Beam 2   204
4.4.3  Example 4.8:   Superposition Beam 3   205
4.4.4  Example 4.9:   Superposition Beam 4   206
4.4.5  Example 4.10: Superposition  Beam 5   207
4.5  Unit Load for Deflection of Beams   208
4.5.1  Strain Energy (Bending Load Effects)   208
4.5.2  Example 4.11: Deflection and Slope of a Uniform Cantilever   211
4.5.3  Example 4.12: Deflection and Slope of a Non-Uniform Cantilever  212
4.5.4  Example 4.13: Deflection and Slope of a Linearly Varying Cantilever   214
4.5.5  Example 4.14: Deflection of a Non-Uniform, Simply-Supported Beam  216
4.5.6  Example 4.15: Deflection of a Frame and Beam Structure  218
4.5.7  Example 4.16: Deflection Uniform Cantilever using Coefficients   221
4.5.8  Problems: Unit Load Method for Deflection of Beams/Frames  222
4.5.9  Solutions: Unit Load Method for Deflection of Beams/Frames  225
4.6  Statically Indeterminate Beams    252
4.6.1  Unit Load Method for Singly-Redundant Beams  253
4.6.2  Example 4.17: Singly-Redundant Beam 1   253
4.6.3  Example 4.18: Singly-Redundant Beam 2  255
4.6.4  Problems: Unit Load Method for Singly-Redundant Beams  258
4.6.5  Solutions: Unit Load Method for Singly-Redundant Beams  259
4.7  Moment Distribution Method for Multi-Redundant Beams   269
4.7.1  Bending (Rotational) Stiffness  269
4.7.2  Carry-Over Moment  270
4.7.3  Pinned End    270
4.7.4  Free and Fixed Bending Moments  271
4.7.5  Example 4.19:  Single-span Encastre Beam  272
4.7.6  Propped Cantilevers  274
4.7.7  Example 4.20:  Propped Cantilever  275
4.7.8  Distribution Factors  278
4.7.9  Application of the Method  279 viii  Contents
4.7.10  Example 4.21:  Three-span Continuous Beam   280
4.7.11  Problems: Moment Distribution - Continuous Beams   289
4.7.12  Solutions: Moment Distribution - Continuous Beams   290
4.8  Redistribution of Moments  314
4.8.1  Example 4.22:  Redistribution of Moments in a Two-span Beam  314
4.9  Shear Force and Bending Moment Envelopes  317

5.  Rigid-Jointed Frames  

5.1  Rigid-Jointed Frames   318
5.1.1  Example 5.1: Statically Determinate, Rigid-Jointed Frame 1   319
5.1.2  Example 5.2: Statically Determinate, Rigid-Jointed Frame 2  323
5.1.3  Problems: Statically Determinate, Rigid-Jointed Frames  328
5.1.4  Solutions: Statically Determinate, Rigid-Jointed Frames  330
5.2  Unit Load Method for Singly-Redundant, Rigid-Jointed Frames   342
5.2.1  Example 5.3: Singly-Redundant, Rigid-Jointed Frame  344
5.2.2  Problems: Unit Load Method for Singly-Redundant, Rigid-Jointed Frames    350
5.2.3  Solutions: Unit Load Method for Singly-Redundant, Rigid-Jointed Frames    352
5.3  Moment Distribution for No-Sway, Rigid-Jointed Frames  368
5.3.1  Example 5.3: No-Sway, Rigid-Jointed Frame 1   370
5.3.2  Problems: Moment Distribution  No-Sway Rigid-Jointed Frames  376
5.3.3  Solutions: Moment Distribution  No-Sway Rigid-Jointed Frames  378
5.4  Moment Distribution for Rigid-Jointed Frames with Sway  415
5.4.1  Example 5.4: Rigid-Jointed Frame with Sway  Frame 1   417
5.4.2  Problems: Moment Distribution  Rigid-Jointed Frames with Sway  427
5.4.3  Solutions: Moment Distribution  Rigid-Jointed Frames with Sway  429

6.  Buckling Instability  

6.1  Introduction    462
6.1.1  Local Buckling  462
6.1.1.1  Class 1 Sections  464
6.1.1.2  Class 2 Sections  465
6.1.1.3  Class 3 Sections  466
6.1.1.4  Class 4 Sections  466
6.1.1.5  Section Classification  466
6.1.2  Flexural Buckling  467
6.1.2.1  Short Elements  467
6.1.2.2  Slender Elements  468
6.1.2.3  Intermediate Elements  468
6.2  Secondary Stresses  469
6.2.1  Effect on Short Elements   470
6.2.2  Effect on Slender Elements  470
6.2.3  Effect on Intermediate Elements  470
6.3  Critical Stress   470
6.3.1  Critical Stress for Short Columns   471
6.3.2  Critical Stress for Slender Columns  471
6.3.3  Euler Equation  471
6.3.4  Effective Buckling Length   473
6.3.5  Critical Stress for Intermediate Columns  475
6.3.6  Tangent Modulus Theorem  475
6.4  Perry-Robertson Formula  476
6.5  European Column Curves  479
6.5.1  Non-dimensional Slenderness  480
6.6  Example 6.1: Slenderness  487
6.7  Example 6.2: Rolled Universal Column Section  487
6.8  Example 6.3: Compound Column Section  490
6.9  Built-up Compression Members  492
6.9.1  Shear Stiffness for Laced Columns  494
6.10  Example 6.4: Laced Built-up Column  496
6.11  Problems: Buckling Instability   501
6.12  Solutions: Buckling Instability   504

7.  Direct Stiffness Method   

7.1  Direct Stiffness Method of Analysis  516
7.2  Element Stiffness Matrix  516
7.2.1  Beams Elements with Two Degrees-of-Freedom  517
7.2.2  Beams Elements with Four Degrees-of-Freedom  518
7.2.3  Local Co-ordinate System   523
7.2.4  Beams Elements with Six Degrees-of-Freedom  523
7.3  Structural Stiffness Matrix  525
7.4  Structural Load Vector  528
7.5  Structural Displacement Vector  530
7.6  Element Displacement Vector  530
7.7  Element Force Vector  531
7.8  Example 7.1: Two-span Beam  531
7.9  Example 7.2: Rigid-Jointed Frame  537
7.10  Problems: Direct Stiffness Method   546
7.11  Solutions: Direct Stiffness Method   548

8.  Plastic Analysis  

8.1  Introduction    597
8.1.1  Partial Collapse  598
8.1.2  Conditions for Full Collapse  598
8.2  Static Method for Continuous Beams  599
8.2.1  Example 8.1: Encastré Beam  599
8.2.2  Example 8.2: Propped Cantilever 1  600
8.2.3  Example 8.3: Propped Cantilever 2  601
8.3  Kinematic Method for Continuous Beams  602
8.3.1  Example 8.4: Continuous Beam  605
8.4  Problems: Plastic Analysis  Continuous Beams   609
8.5  Solutions: Plastic Analysis  Continuous Beams   610
8.6  Rigid-Jointed Frames  628
8.6.1  Example 8.5: Frame 1  628
8.7  Problems: Plastic Analysis  Rigid-Jointed Frames 1   635
8.8  Solutions: Plastic Analysis  Rigid-Jointed Frames 1   636
8.9  Example 8.6:  Joint Mechanism  653
8.10  Problems: Plastic Analysis  Rigid-Jointed Frames 2   657
8.11  Solutions: Plastic Analysis  Rigid-Jointed Frames 2   659
8.12  Gable Mechanism  690
8.13  Instantaneous Centre of Rotation  691
8.14  Example 8.7:  Pitched Roof Frame  692
8.15  Problems: Plastic Analysis  Rigid-Jointed Frames 3   696
8.16  Solutions: Plastic Analysis  Rigid-Jointed Frames 3   698

9.  Influence Lines for Beams  730 

9.1  Introduction    730
9.2  Example 9.1: Influence Lines for a Simply Supported Beam  730
9.2.1  Influence Lines for the Support Reactions  731
9.2.2  Influence Line for the Shear Force  732
9.2.3  Influence Line for the Bending Moment  734
9.3  Müller-Breslau Principle for the Influence Lines for Beams  737
9.4  Influence Lines for a Statically Determinate Beam  737
9.5  Example 9.3: Influence Line for a Statically Indeterminate Beam  739
9.6  The use of Influence Lines  741
9.6.1  Concentrated Loads  741
9.6.2  Distributed Loads  741
9.6.3  Example 9.4: Evaluation of Functions for Statically Determinate Beam 1  742
9.6.4  Example 9.5: Evaluation of Functions for Statically Determinate Beam 2  743
9.7  Example 9.6: Evaluation of Functions for a Statically Indeterminate Beam  745
9.8  Train of Loads  748
9.8.1  Example 9.7: Evaluation of Functions for a Train of Loads  749
9.9  Problems: Influence Lines for Beams   752
9.10  Solutions: Influence Lines for Beams   754

10. Approximate Methods of Analysis 

10.1  Introduction    762
10.2  Example 10.1: Statically Indeterminate Pin-jointed Plane Frame 1  762
10.3  Example 10.2: Statically Indeterminate Pin-jointed Plane Frame 2  766
10.4  Example 10.3: Statically Indeterminate Single-span Beam  768
10.5  Example 10.4: Multi-span Beam  770
10.6  Rigid-jointed Frames Subjected to Vertical Loads  772
10.6.1  Example 10.5: Multi-storey Rigid-jointed Frame 1  772
10.6.2  Approximate Analysis of Multi-storey Rigid-jointed Frames Using Sub-frames  778
10.6.3  Simple Portal Frames with Pinned Bases Subjected to Horizontal Loads  780
10.6.3.1  Example 10.6: Simple Rectangular Portal Frame – Pinned Bases  780
10.6.4  Simple Portal Frames with Fixed Bases Subjected to Horizontal Loads
10.6.4.1  Example 10.7: Simple Rectangular Portal Frame – Fixed Bases  782
10.7  Multi-storey, Rigid-jointed Frames Subjected to Horizontal Loads  783
10.7.1  Portal Method  783
10.7.1.1  Example 10.8: Multi-storey Rigid-jointed Frame 2  784
10.7.1.2  Approximate Analysis of Vierendeel Trusses using the Portal Method  792
10.7.1.3  Example 10.9: Vierendeel Truss  793
10.7.2  Cantilever Method  796
10.7.2.1  Example 10.10: Multi-storey Rigid-jointed Frame 3  79

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