# Advanced Methods of Structural Analysis Book

## Advanced Methods of Structural Analysis Book

 Advanced Methods of Structural Analysis Book

Introduction ........................................................................ xxi
Part I    Statically Determinate Structures
1      Kinematical Analysis of Structures .........................................    3
1.1      Classiﬁcation of Structures by Kinematical
Viewpoint ...............................................................    3
1.2      Generation of Geometrically Unchangeable
Structures ...............................................................    5
1.3      Analytical Criteria of the Instantaneously
Changeable Structures .................................................    7
1.4      Degrees of Freedom ....................................................  11
Problems .......................................................................  13
2      General Theory of Inﬂuence Lines .........................................  15
2.1      Analytical Method for Construction of Inﬂuence
Lines ....................................................................  15
2.1.1      Inﬂuence Lines for Reactions ................................  16
2.1.2      Inﬂuence Lines for Internal Forces ..........................  20
2.2      Application of Inﬂuence Lines for Fixed and
2.4      Combining of Fixed and Moving Load Approaches..................  35
2.5      Properties of Inﬂuence Lines ..........................................  36
Problems .......................................................................  37
3      Multispan Beams and Trusses. ..............................................  39
3.1      Multispan Statically Determinate Beams .............................  39
3.1.1      Generation of Multispan Statically
Determinate Hinged Beams ..................................  39
3.1.2      Interaction Schemes and Load Path .........................  40
xiiixiv Contents
3.1.3      Inﬂuence Lines for Multispan Hinged
Beams .........................................................  42
3.1.4      Summary ......................................................  45
3.2      The Generation of Statically Determinate Trusses ...................  47
3.2.1      Simple Trusses ................................................  47
3.2.2      Compound Trusses ...........................................  48
3.2.3      Complex Trusses .............................................  49
3.3      Simple Trusses .........................................................  49
3.4      Trusses with Subdivided Panels .......................................  54
3.4.1      Main and Auxiliary Trusses and Load Path .................  55
3.4.2      Baltimore and Subdivided Warren Trusses ..................  57
3.5      Special Types of Trusses ...............................................  61
3.5.1      Three-Hinged Trusses ........................................  61
3.5.2      Trusses with a Hinged Chain.................................  64
3.5.3      Complex Trusses .............................................  68
3.5.4      Summary ......................................................  70
Problems .......................................................................  72
4      Three-Hinged Arches ........................................................  77
4.1      Preliminary Remarks ...................................................  77
4.1.1      Design Diagram of Three-Hinged Arch .....................  77
4.1.2      Peculiarities of the Arches ...................................  78
4.1.3      Geometric Parameters of Circular
and Parabolic Arches .........................................  79
4.2      Internal Forces ..........................................................  80
4.3      Inﬂuence Lines for Reactions and Internal Forces....................  86
4.3.1      Inﬂuence Lines for Reactions ................................  88
4.3.2      Inﬂuence Lines for Internal Forces ..........................  88
4.3.3      Application of Inﬂuence Lines ...............................  92
4.4      Nil Point Method for Construction of Inﬂuence
Lines ....................................................................  94
4.4.1      Bending Moment .............................................  94
4.4.2      Shear Force ...................................................  95
4.4.3      Axial Force....................................................  96
4.5      Special Types of Arches ...............................................  97
4.5.2      Parabolic Arch with Complex Tie ...........................100
Problems .......................................................................103
5      Cables ..........................................................................109
5.1      Preliminary Remarks ...................................................109
5.1.1      Direct and Inverse Problems .................................110
5.1.2      Fundamental Relationships ..................................111
5.2      Cable with Neglected Self-Weight ....................................113
5.2.1      Cables Subjected to Concentrated Load .....................113Contents xv
5.2.2      Cable Subjected to Uniformly Distributed
5.3      Effect of Arbitrary Load on the Thrust and Sag ......................122
5.4      Cable with Self-Weight ................................................125
5.4.1      Fundamental Relationships ..................................125
5.4.2      Cable with Supports Located at the Same
Level ..........................................................127
5.4.3      Cable with Supports Located
on the Different Elevations ...................................130
5.5      Comparison of Parabolic and Catenary Cables .......................135
5.6      Effect of Axial Stiffness ...............................................137
5.6.1      Elastic Cable with Concentrated Load.......................137
5.6.2      Elastic Cable with Uniformly Distributed
Problems .......................................................................140
6      Deﬂections of Elastic Structures ............................................145
6.1      Introduction .............................................................145
6.2      Initial Parameters Method .............................................147
6.3      Maxwell–Mohr Method ................................................159
6.3.1      Deﬂections Due to Fixed Loads .............................159
6.3.2      Deﬂections Due to Change of Temperature .................165
6.3.3      Summary ......................................................170
6.4      Displacement Due to Settlement of Supports
and Errors of Fabrication ..............................................170
6.5      Graph Multiplication Method..........................................176
6.7      Reciprocal Theorems...................................................189
6.7.1      Theorem of Reciprocal Works
(Betti Theorem) ...............................................189
6.7.2      Theorem of Reciprocal Unit Displacements
(Maxwell Theorem) ..........................................190
6.7.3      Theorem of Reciprocal Unit Reactions
(Rayleigh First Theorem) ....................................192
6.7.4      Theorem of Reciprocal Unit Displacements
and Reactions (Rayleigh Second Theorem) .................193
6.7.5      Summary ......................................................193
Problems .......................................................................195
Part II    Statically Indeterminate Structures
7      The Force Method ............................................................211
7.1      Fundamental Idea of the Force Method ...............................211
7.1.1      Degree of Redundancy, Primary Unknowns
and Primary System ..........................................211
7.1.2      Compatibility Equation in Simplest Case ...................214xvi Contents
7.2      Canonical Equations of Force Method ................................217
7.2.1      The Concept of Unit Displacements .........................217
7.2.2      Calculation of Coefﬁcients and Free Terms
of Canonical Equations .......................................219
7.3      Analysis of Statically Indeterminate Structures.......................222
7.3.1      Continuous Beams............................................222
7.3.2      Analysis of Statically Indeterminate
Frames.........................................................224
7.3.3      Analysis of Statically Indeterminate
Trusses ........................................................233
7.3.4      Analysis of Statically Indeterminate Arches ................237
7.4      Computation of Deﬂections of Redundant Structures ................243
7.5      Settlements of Supports ................................................246
7.6      Temperature Changes ..................................................251
Problems .......................................................................259
8      The Displacement Method ...................................................271
8.1      Fundamental Idea of the Displacement Method ......................271
8.1.1      Kinematical Indeterminacy ..................................272
8.1.2      Primary System and Primary Unknowns ....................274
8.1.3      Compatibility Equation. Concept of Unit
Reaction .......................................................275
8.2      Canonical Equations of Displacement Method .......................276
8.2.1      Compatibility Equations in General Case ...................276
8.2.2      Calculation of Unit Reactions ................................277
8.2.3      Properties of Unit Reactions .................................279
8.2.4      Procedure for Analysis .......................................280
8.3      Comparison of the Force and Displacement Methods ................291
8.3.1      Properties of Canonical Equations ...........................292
8.4      Sidesway Frames with Absolutely Rigid Crossbars ..................294
8.5      Special Types of Exposures ............................................296
8.5.1      Settlements of Supports ......................................296
8.5.2      Errors of Fabrication .........................................300
8.6      Analysis of Symmetrical Structures ...................................302
8.6.2      Concept of Half-Structure ....................................303
Problems .......................................................................305
9      Mixed Method.................................................................313
9.1      Fundamental Idea of the Mixed Method ..............................313
9.1.1      Mixed Indeterminacy and Primary
Unknowns .....................................................313
9.1.2      Primary System ...............................................314
9.2      Canonical Equations of the Mixed Method ...........................316
9.2.1      The Matter of Unit Coefﬁcients
and Canonical Equations .....................................316Contents xvii
9.2.2      Calculation of Coefﬁcients and Free Terms .................317
9.2.3      Computation of Internal Forces ..............................318
Problems .......................................................................319
10    Inﬂuence Lines Method ......................................................323
10.1    Construction of Inﬂuence Lines by the Force
Method ..................................................................323
10.1.1    Continuous Beams............................................325
10.1.2    Hingeless Nonuniform Arches ...............................331
10.1.3    Statically Indeterminate Trusses .............................339
10.2    Construction of Inﬂuence Lines
by the Displacement Method ..........................................344
10.2.1    Continuous Beams............................................346
10.2.2    Redundant Frames ............................................353
10.3    Comparison of the Force and Displacements
Methods .................................................................355
10.4    Kinematical Method for Construction of Inﬂuence
Lines ....................................................................358
Problems .......................................................................364
11    Matrix Stiffness Method .....................................................369
11.1    Basic Idea and Concepts ...............................................369
11.1.1    Finite Elements ...............................................370
11.1.2    Global and Local Coordinate Systems.......................370
11.1.3    Displacements of Joints and Degrees
of Freedom ....................................................371
11.2    Ancillary Diagrams ....................................................372
11.2.1    Joint-Load (J -L) Diagram ...................................372
11.2.2    Displacement-Load (Z-P ) Diagram ........................376
11.2.3    Internal Forces-Deformation (S-e)
Diagram .......................................................377
11.3    Initial Matrices .........................................................379
11.3.1    Vector of External Joint Loads ...............................379
11.3.2    Vector of Internal Unknown Forces..........................380
11.4    Resolving Equations ...................................................381
11.4.1    Static Equations and Static Matrix ...........................381
11.4.2    Geometrical Equations and Deformation
Matrix .........................................................386
11.4.3    Physical Equations and Stiffness Matrix
in Local Coordinates .........................................387
11.5    Set of Formulas and Procedure for Analysis ..........................390
11.5.1    Stiffness Matrix in Global Coordinates......................390
11.5.2    Unknown Displacements and Internal
Forces .........................................................391
11.5.3    Matrix Procedures ............................................392xviii Contents
11.6    Analysis of Continuous Beams ........................................393
11.7    Analysis of Redundant Frames ........................................404
11.8    Analysis of Statically Indeterminate Trusses .........................410
11.9    Summary................................................................414
Problems .......................................................................415
Part III    Special Topics
12    Plastic Behavior of Structures...............................................423
12.1    Idealized Stress–Strain Diagrams .....................................423
12.2    Direct Method of Plastic Analysis .....................................427
12.3    Fundamental Methods of Plastic Analysis ............................430
12.3.1    Kinematical Method ..........................................430
12.3.2    Static Method .................................................430
12.4    Limit Plastic Analysis of Continuous Beams .........................432
12.4.1    Static Method .................................................433
12.4.2    Kinematical Method ..........................................435
12.5    Limit Plastic Analysis of Frames ......................................441
12.5.1    Beam Failure ..................................................442
12.5.2    Sidesway Failure..............................................444
12.5.3    Combined Failure.............................................444
12.5.4    Limit Combination Diagram .................................444
Problems .......................................................................445
13    Stability of Elastic Systems ..................................................449
13.1    Fundamental Concepts .................................................449
13.2    Stability of Structures with Finite Number Degrees
of Freedom..............................................................453
13.2.1    Structures with One Degree of Freedom ....................453
13.2.2    Structures with Two or More Degrees
of Freedom ....................................................458
13.3    Stability of Columns with Rigid and Elastic
Supports ................................................................461
13.3.1    The Double Integration Method .............................461
13.3.2    Initial Parameters Method ....................................466
13.4    Stability of Continuous Beams and Frames ...........................471
13.4.1    Unit Reactions of the Beam-Columns .......................471
13.4.2    Displacement Method ........................................473
13.4.3    Modiﬁed Approach of the Displacement
Method ........................................................481
13.5    Stability of Arches .....................................................483
13.5.1    Circular Arches Under Hydrostatic Load ...................484
13.5.2    Complex Arched Structure:
Arch with Elastic Supports ...................................490Contents xix
13.6.1    Double Integration Method ..................................492
13.6.2    Initial Parameters Method ....................................494
13.6.3    P-Delta Analysis of the Frames ..............................499
13.6.4    Graph Multiplication Method
for Beam-Columns ...........................................502
Problems .......................................................................504
14    Dynamics of Elastic Systems ................................................513
14.1    Fundamental Concepts .................................................513
14.1.1    Kinematics of Vibrating Processes ..........................513
14.1.2    Forces Which Arise at Vibrations............................513
14.1.3    Degrees of Freedom ..........................................515
14.1.4    Purpose of Structural Dynamics .............................519
14.1.5    Assumptions ..................................................519
14.2    Free Vibrations of Systems with Finite Number
Degrees of Freedom: Force Method ...................................520
14.2.1    Differential Equations of Free Vibration in Displacements 520
14.2.2    Frequency Equation ..........................................521
14.2.3    Mode Shapes Vibration and Modal Matrix ..................522
14.3    Free Vibrations of Systems with Finite Number
Degrees of Freedom: Displacement Method ..........................530
14.3.1    Differential Equations of Free Vibration
in Reactions ...................................................530
14.3.2    Frequency Equation ..........................................532
14.3.3    Mode Shape Vibrations and Modal Matrix ..................532
14.3.4    Comparison of the Force and Displacement
Methods .......................................................538
14.4    Free Vibrations of One-Span Beams with Uniformly
Distributed Mass .......................................................538
14.4.1    Differential Equation of Transversal Vibration
of the Beam ...................................................540
14.4.2    Fourier Method ...............................................541
14.4.3    Krylov–Duncan Method......................................543
Problems .......................................................................546
Appendix ............................................................................551
Bibliography ........................................................................587