# Structural Analysis

1 Introduction to Structural Analysis

CLASSIFICATION OF STRUCTURES

2.1 Dead Loads

2.2 Live Loads

2.4 Wind Loads

2.5 Snow Loads

2.6 Earthquake Loads

2.7 Hydrostatic and Soil Pressures

3.1 Equilibrium of Structures

3.2 External and Internal Forces

3.3 Types of Supports for Plane Structures

3.4 Static Determinacy, Indeterminacy, and Instability

3.5 Computation of Reactions

3.5.1 Computation of Reactions Problems and Solutions

3.6 Principle of Superposition

3.7 Reactions of Simply Supported Structures Using Proportions

4.1 Assumptions for Analysis of Trusses

4.2 Arrangement of Members of Plane Trusses-Internal Stability

4.3 Equations of Condition for Plane Trusses

4.4 Static Determinacy, Indeterminacy, and Instability of Plane Trusses

4.5 Analysis of Plane Trusses by the Method of Joints

4.5.1 Plane Trusses by the Method of Joints Problems and solutions

4.6 Analysis of Plane Trusses by the Method of Sections

4.6.1 Plane Trusses by the Method of Sections Problems and Solutions

4.7 Analysis of Compound Trusses

4.7.1 Compound Trusses Problems ans Solutions

4.8 Complex Trusses

4.9 Space Trusses

5.1 Axial Force, Shear, and Bending Moment

5.2 Shear and Bending Moment Diagrams

5.3 Qualitative Deflected Shapes

5.4 Relationships between Loads, Shears, and Bending Moments

5.5 Static Determinacy, Indeterminacy, and Instability of Plane Frames

5.6 Analysis of Plane Frames

6 Deflections of Beams: Geometric Methods 226

6.1 Di¤erential Equation for Beam Deflection 227

6.2 Direct Integration Method 230

6.3 Superposition Method 233

6.4 Moment-Area Method 234

6.5 Bending Moment Diagrams by Parts 248

6.6 Conjugate-Beam Method 253

Summary 269

Problems 269

7 Deflections of Trusses, Beams, and Frames: Work–Energy Methods 275

7.1 Work 276

7.2 Principle of Virtual Work 278

7.3 Deflections of Trusses by the Virtual Work Method 282

7.4 Deflections of Beams by the Virtual Work Method 293

7.5 Deflections of Frames by the Virtual Work Method 301

7.6 Conservation of Energy and Strain Energy 312

7.7 Castigliano’s Second Theorem 316

7.8 Betti’s Law and Maxwell’s Law of Reciprocal Deflections 325

Summary 326

Problems 328

8 Influence Lines 337

8.1 Influence Lines for Beams and Frames by Equilibrium

Method 338

8.2 Mu¨ eller-Breslau’s Principle and Qualitative Influence Lines 353

8.3 Influence Lines for Girders with Floor Systems 367

8.4 Influence Lines for Trusses 377

8.5 Influence Lines for Deflections 389

Summary 392

Problems 393

9 Application of Influence Lines 401

9.1 Response at a Particular Location Due to a Single Moving

Concentrated Load 401

9.2 Response at a Particular Location Due to a Uniformly

Distributed Live Load 403

9.3 Response at a Particular Location Due to a Series of Moving

Concentrated Loads 408

9.4 Absolute Maximum Response 415

Summary 421

Problems 422

10 Analysis of Symmetric Structures 425

10.1 Symmetric Structures 426

10.2 Symmetric and Antisymmetric Components of Loadings 432

10.3 Behavior of Symmetric Structures under Symmetric and

Antisymmetric Loadings 443

10.4 Procedure for Analysis of Symmetric Structures 447

Summary 455

Problems 456

PART THREE ANALYSIS OF STATICALLY INDETERMINATE STRUCTURES 459

11 Introduction to Statically Indeterminate Structures 461

11.1 Advantages and Disadvantages of Indeterminate Structures 462

11.2 Analysis of Indeterminate Structures 465

Summary 470

12 Approximate Analysis of Rectangular Building Frames 471

12.1 Assumptions for Approximate Analysis 472

12.2 Analysis for Vertical Loads 475

12.3 Analysis for Lateral Loads—Portal Method 481

12.4 Analysis for Lateral Loads—Cantilever Method 497

Summary 504

Problems 505

13 Method of Consistent Deformations—Force Method 508

13.1 Structures with Single Degree of Indeterminacy 509

13.2 Internal Forces and Moments as Redundants 531

13.3 Structures with Multiple Degrees of Indeterminacy 544

13.4 Support Settlements, Temperature Changes and Fabrication

Errors 568

Summary 577

Problems 578

14 Three-Moment Equation and the Method of Least Work 586

14.1 Derivation of Three-Moment Equation 587

14.2 Application of Three-Moment Equation 592

14.3 Method of Least Work 599

Summary 606

Problems 607

15 Influence Lines for Statically Indeterminate Structures 609

15.1 Influence Lines for Beams and Trusses 610

15.2 Qualitative Influence Lines by Mu¨ ller-Breslau’s Principle 627

Summary 631

Problems 632

16 Slope-Deflection Method 635

16.1 Slope-Deflection Equations 636

16.2 Basic Concept of the Slope-Deflection Method 644

16.3 Analysis of Continuous Beams 651

16.4 Analysis of Frames without Sidesway 673

16.5 Analysis of Frames with Sidesway 681

Summary 702

Problems 702

17 Moment-Distribution Method 707

17.1 Definitions and Terminology 708

17.2 Basic Concept of the Moment-Distribution Method 717

17.3 Analysis of Continuous Beams 725

17.4 Analysis of Frames without Sidesway 741

17.5 Analysis of Frames with Sidesway 744

Summary 761

Problems 762

18 Introduction to Matrix Structural Analysis 767

18.1 Analytical Model 768

18.2 Member Sti¤ness Relations in Local Coordinates 772

18.3 Coordinate Transformations 780

18.4 Member Sti¤ness Relations in Global Coordinates 786

18.5 Structure Sti¤ness Relations 787

18.6 Procedure for Analysis 795

Summary 813

Problems 814

Appendix A Areas and Centroids of Geometric Shapes 817

Appendix B Review of Matrix Algebra 821

B.1 Definition of a Matrix 821

B.2 Types of Matrices 822

B.3 Matrix Operations 824

B.4 Solution of Simultaneous Equations by the Gauss-Jordan

Method 831

Problems 835

Appendix C Computer Software 837

Starting the Computer Software 837

Inputting Data 837

Results of the Analysis 844

Problems 849