Advanced Methods of Structural Analysis

Description

This revised and significantly expanded edition contains a rigorous examination of key concepts, new chapters and discussions within existing chapters, and added reference materials in the appendix, while retaining its classroom-tested approach to helping readers navigate through the deep ideas, vast collection of the fundamental methods of structural analysis. The authors show how to undertake the numerous analytical methods used in structural analysis by focusing on the principal concepts, detailed procedures and results, as well as taking into account the advantages and disadvantages of each method and sphere of their effective application. The end result is a guide to mastering the many intricacies of the range of methods of structural analysis. The book differentiates itself by focusing on extended analysis of beams, plane and spatial trusses, frames, arches, cables and combined structures; extensive application of influence lines for analysis of structures; simple and effective procedures for computation of deflections; introduction to plastic analysis, stability, and free and forced vibration analysis, as well as some special topics. Ten years ago, Professor Igor A. Karnovsky and Olga Lebed crafted a must-read book. Now fully updated, expanded, and titled Advanced Methods of Structural Analysis (Strength, Stability, Vibration), the book is ideal for instructors, civil and structural engineers, as well as researches and graduate and post graduate students with an interest in perfecting structural analysis.  Preface to the Second Edition Preface to the First Edition Contents Introduction The Subject and Purposes of the Structural Mechanics Modeling of Engineering Structures and Principal Assumptions Design Diagram Idealization of Material Properties Load Idealization The Location of Points at Which Forces Are Applied to the Body The Nature of Change of Forces in the Process of Their Application The Nature of the Dependence of Forces on the Deformation of the Structure Duration of the Influence of Loads on Structures Idealization of the Structure as a Whole Types of Structures Modeling of Structural Members Features of Some Loads and Their Load Path Simplest Industrial Building Specific Loads Load Combination Types of Analysis and Particular Assumptions Fundamental Approaches of Linear Static Analysis Part I: Statically Determinate Structures Chapter 1: Kinematical Analysis of Structures 1.1 Classification of Structures by Kinematical Viewpoint 1.2 Generation of Geometrically Unchangeable Structures 1.2.1 Required and Redundant Constraints 1.2.2 Constraint Replacing 1.3 Analytical Criteria of the Instantaneously Changeable Structures 1.4 Degrees of Freedom 1.5 Null Load Method Problems Chapter 2: The Simplest Beams: Theory of Influence Lines 2.1 General 2.2 Analytical Method for Construction of Influence Lines 2.2.1 Influence Lines for Reactions 2.2.1.1 Simply Supported Beam Influence Line for RA Influence Line for RB 2.2.1.2 Simply Supported Beam with Overhang 2.2.1.3 Cantilevered Beam The Vertical Reaction RA The Moment M0 at Support A 2.2.2 Influence Lines for Internal Forces 2.2.2.1 Bending Moment Mk Load P = 1 is Located to the Left of Section k Load P = 1 is Located to the Right of Section k 2.2.2.2 Influence Line Qk 2.2.2.3 Discussion Bending Moment M1 and Shear Force Q1 Shear Force Q2 and Bending Moment M2 (Fig. 2.10) 2.3 Application of Influence Lines for Fixed and Moving Loads 2.3.1 Fixed Loads 2.3.1.1 Concentrated Loads 2.3.1.2 Uniformly Distributed Load 2.3.1.3 Couple 2.3.1.4 Nonuniform Distributed Load Summary 2.3.2 Moving Loads 2.3.2.1 Influence Line Forms a Triangle 2.3.2.2 Influence Line Forms a Polygon 2.3.3 Envelope Diagrams of Internal Forces 2.3.3.1 The Beam is Loaded by Single Moving Load P (Fig. 2.16a) 2.3.3.2 The Beam is Loaded by Set of Connected Moving Loads P 2.3.4 Absolute Maximum of Bending Moment 2.4 Indirect Load Application 2.5 Kinematical Method for Construction of Influence Lines 2.5.1 Influence Line for Bending Moment at Section k, IL(Mk) 2.5.2 Influence Line for Shear Force at Section k, IL(Qk) 2.5.3 Conclusion 2.6 Combining of Fixed and Moving Load Approaches 2.7 Properties of Influence Lines Problems Chapter 3: Multispan Hinged Beams and Frames 3.1 Generation of Multispan Hinged Beams 3.2 Interaction Diagrams and Load Path 3.3 Fixed Load Method for Beams 3.4 Influence Lines for Reactions and Internal Forces 3.4.1 Static Approach 3.4.1.1 Influence Line for RD 3.4.1.2 Influence Lines for Shear Qk, and Bending Moment Mk at Section k 3.4.1.3 Influence Lines for Shear Qn, and Bending Moment Mn at Section n 3.4.1.4 Influence Lines for Shear Qs, and Bending Moment Ms at Section s 3.4.1.5 Summary 3.4.2 Kinematical Approach 3.4.2.1 Influence Line for RA 3.4.2.2 Influence Line for RB 3.4.2.3 Influence Line for Mi 3.4.3 Indirect Load Application 3.4.3.1 Influence Line for RA (Fig. 3.11c) 3.4.3.2 Influence Line for Mk (Fig. 3.11d) 3.4.3.3 Discussion 3.5 Fixed Load Method for Frames Summary Problems Chapter 4: Plane Trusses 4.1 General 4.1.1 Classification of the Plane Trusses 4.1.2 Assumptions and Design Diagram Discussion 4.2 The Generation of Statically Determinate Trusses 4.2.1 Simple Trusses 4.2.2 Compound Trusses 4.2.3 Complex Trusses 4.3 Simple Trusses: Fixed Loads 4.3.1 Analytical Methods of Analysis 4.3.1.1 Method of Joint Isolation 4.3.1.2 Method of Through Sections (Ritter´s Method) 4.3.1.3 Special Cases 4.3.2 Maxwell-Cremona Diagram 4.4 Simple Trusses: Influence Line Method-Static Approach 4.5 Trusses with Subdivided Panels 4.5.1 Main and Auxiliary Trusses and Load Path 4.5.1.1 Kinematical Analysis 4.5.2 Single-Tiered Auxiliary Trusses 4.5.3 Two-Tiered Auxiliary Trusses Discussion 4.6 Special Types of Trusses 4.6.1 Three-Hinged Trusses 4.6.2 Trusses with a Hinged Chain 4.6.2.1 Reaction of Supports and Internal Forces 4.6.2.2 Discussion 4.7 Kinematical Method for Construction of Influence Lines 4.8 Complex Trusses 4.8.1 Substitution Bar Method Discussion 4.8.2 Closed Section Method 4.8.3 Summary Problems Chapter 5: Space Frameworks 5.1 General Assumptions 5.2 Classification of the Space Frameworks 5.2.1 Method of Formation 5.2.2 Three-Dimensional Engineering Rod Structures 5.3 Space Framework Supports 5.4 Kinematical Analysis of Space Frameworks 5.4.1 Attached and Released Frameworks 5.4.1.1 Spatial Trusses Attached to Supports 5.4.1.2 Spatial Trusses Separated from Supports 5.4.2 Improper Connections of 3-D Structures 5.4.3 Meshwork Structures 5.4.3.1 Cauchy Theorem 5.5 Static Analysis of 3-D Structures 5.5.1 General 5.5.1.1 Zero Rod Cases 5.5.1.2 Method of Sections 5.5.2 Meshwork Structures 5.5.3 Compound Space Frameworks 5.5.4 Complex Space Frameworks: The Rod Replacement Method Discussion 5.5.5 Schwedler Dome 5.5.5.1 Design Diagram of Dome 5.5.5.2 Kinematical Analysis 5.5.5.3 Features of Schwedler Structure 5.5.5.4 Decomposing of Space Framework to Plane Trusses Problems Chapter 6: Three-Hinged Arches 6.1 Preliminary Remarks 6.1.1 Design Diagram of Three-Hinged Arch 6.1.2 Peculiarities of the Arches 6.1.3 Geometric Parameters of Circular and Parabolic Arches 6.1.3.1 Circular Arch 6.1.3.2 Parabolic Arch 6.2 Internal Forces 6.2.1 Concept of Substitute Beam 6.2.1.1 Analysis of Formulae (6.8) and (6.10) 6.2.2 Numerical Example 6.2.3 Maximum Economy Arches 6.3 Influence Lines for Reactions and Internal Forces 6.3.1 Influence Lines for Reactions 6.3.2 Influence Lines for Internal Forces 6.3.2.1 Bending Moment 6.3.2.2 Shear Force 6.3.2.3 Axial Force 6.3.2.4 Features of the Influence Lines for Internal Forces 6.3.3 Application of Influence Lines 6.3.3.1 Reactions of Supports 6.3.3.2 Internal Forces in Section k 6.4 Nil Point Method for Construction of Influence Lines 6.4.1 Bending Moment 6.4.2 Shear Force 6.4.3 Axial Force 6.5 Special Types of Arches 6.5.1 Askew Arch 6.5.1.1 Reactions and Bending Moment at Section k 6.5.1.2 Influence Lines for Thrust and Bending Moment Mk 6.5.2 Parabolic Arch with Complex Tie 6.5.2.1 Reactions and Bending Moment at Section k 6.5.2.2 Influence Lines for Thrust and Bending Moment at Section k Problems Chapter 7: Cables 7.1 Preliminary Remarks 7.1.1 Direct and Inverse Problems 7.1.2 Fundamental Relationships 7.2 Cable with Neglected Self-Weight 7.2.1 Cables Subjected to Concentrated Load 7.2.1.1 Thrust-Shape Problem 7.2.1.2 Length-Thrust Problem 7.2.2 Cable Subjected to Uniformly Distributed Load 7.2.2.1 Thrust-Shape Problem 7.2.2.2 Length-Thrust Problem Approximate Solution of Length Determination Exact Solution of the Length Determination 7.3 Effect of Arbitrary Load on the Thrust and Sag 7.4 Cable with Self-Weight 7.4.1 Fundamental Relationships 7.4.2 Cable with Supports Located at the Same Level 7.4.3 Cable with Supports Located on Different Elevations 7.4.3.1 Saddle Point Within the Span 7.4.3.2 Saddle Point Outside of the Span 7.5 Comparison of Parabolic and Catenary Cables 7.6 Effect of Axial Stiffness 7.6.1 Elastic Cable with Concentrated Load 7.6.2 Elastic Cable with Uniformly Distributed Load Problems Chapter 8: Deflections of Elastic Structures 8.1 Introduction 8.2 Double Integration Method Conclusion 8.3 Initial Parameter Method 8.3.1 Universal Equation of Elastic Curve of a Beam 8.3.2 Beam Subjected to Settlement of Supports 8.3.3 Beams with Intermediate Hinge 8.4 Conjugate Beam Method Summary Short History (Bernshtein 1957) 8.5 Strain Energy Methods 8.5.1 Fundamental Concepts 8.5.1.1 The Work Performed by External Forces in Terms of External Forces and Displacements 8.5.1.2 The Work Performed by External Forces in Terms of Internal Forces 8.5.2 Work-Energy Principle Summary 8.5.3 Castigliano´s Theorem Short Historical Remarks 8.5.4 Principle of Least Work (Menabrea Principle) Strain Energy Methods: Summary Short Historical Remarks 8.6 Maxwell-Mohr Method (Dummy Load Method) 8.6.1 Deflections Due to Fixed Loads 8.6.2 Deflections Due to Change of Temperature Summary 8.7 Displacement Due to Settlement of Supports and Errors of Fabrication Discussion 8.8 Graph Multiplication Method Discussion 8.9 Elastic Load Method Discussion Summary 8.10 Reciprocal Theorems 8.10.1 Reciprocal Work Theorem (Betti´s Theorem) 8.10.2 Reciprocal Unit Displacement Theorem (Maxwell´s Theorem) 8.10.3 Reciprocal Unit Reaction Theorem (Rayleigh´s First Theorem) 8.10.4 Reciprocal Unit Displacement and Reaction Theorem (Rayleigh´s Second Theorem) 8.11 Summary Problems Part II: Statically Indeterminate Structures Chapter 9: The Force Method 9.1 Fundamental Idea of the Force Method 9.1.1 Degree of Redundancy, Primary Unknowns, and Primary System 9.1.2 Compatibility Equation in Simplest Case 9.2 Canonical Equations of Force Method 9.2.1 The Concept of Unit Displacements 9.2.2 Coefficients and Free Terms of Canonical Equations 9.3 Analysis of Redundant Beams 9.3.1 Canonical Equation of the Force Method Summary 9.3.2 The Three-Moment Equation (Clapeyron Theorem) 9.3.2.1 Special Cases 9.3.2.2 Discussion 9.3.2.3 Summary 9.3.2.4 Short Historical Remarks 9.3.3 Focal Ratios Method Summary 9.3.4 Redundant Beam with Intermediate Hinge 9.4 Redundant Plane Frames 9.4.1 Frames of the First Degree of Redundancy 9.4.2 Frames of the Second and More Degree of Redundancy Discussion 9.4.3 Frame with Closed Contour. Elastic Center Discussion 9.4.4 Frame with Elastically Compliant Supports and Joints 9.5 Redundant Trusses 9.5.1 Externally Redundant Truss Discussion 9.5.2 Internally Redundant Truss 9.5.3 Some Properties of Redundant Trusses 9.6 Redundant Arches 9.6.1 Parabolic Two-Hinged Arch 9.6.2 Circular Arch with Clamped Supports 9.6.3 Analysis of Parabolic Arch on the Basis of Modified Design Diagram Discussion 9.7 Combined Redundant Structures 9.8 Deflections of Statically Indeterminate Structures 9.9 Settlements of Supports 9.9.1 Internal Forces due to the Settlements of Supports Discussion 9.9.2 Displacements Due to the Settlements of Supports. Modified Approach 9.10 Temperature Changes 9.10.1 General 9.10.2 Redundant Beams Discussion 9.10.3 Redundant Trusses Discussion 9.10.4 Redundant Frames 9.11 Some Features of Redundant Structures 9.12 Comparison of the Redundant and Statically Determinate Structures Problems Additional Problems Chapter 10: The Displacement Method 10.1 Fundamental Idea of the Displacement Method 10.1.1 Kinematical Indeterminacy 10.1.2 Primary System and Primary Unknowns 10.1.3 Compatibility Equation: Concept of Unit Reaction 10.2 Canonical Equations of Displacement Method 10.2.1 Compatibility Equations in General Case 10.2.2 Calculation of Unit Reactions 10.2.3 Properties of Unit Reactions 10.2.4 Procedure for Analysis 10.2.4.1 Continuous Beams 10.2.4.2 Beam with Intermediate Hinge 10.2.4.3 Discussion 10.2.4.4 Summary 10.3 Comparison of the Force and Displacement Methods 10.3.1 Properties of Canonical Equations 10.3.2 Variations of Design Diagrams and Choice of Methods of Analysis 10.4 Sidesway Frames with Absolutely Rigid Crossbars 10.5 Special Types of Exposures 10.5.1 Settlements of Supports Discussion 10.5.2 Errors of Fabrication Discussion 10.6 Analysis of Symmetrical Structures: Combined Method 10.6.1 Symmetrical and Antisymmetrical Loading 10.6.2 Concept of Half-Structure Problems Chapter 11: Mixed Method 11.1 Fundamental Idea of the Mixed Method 11.1.1 General 11.1.2 Mixed Indeterminacy, Primary Unknowns, and Primary System 11.2 Canonical Equations of the Mixed Method 11.2.1 The Matter of Unit Coefficients and Canonical Equations 11.2.2 Calculation of Coefficients and Free Terms 11.2.3 Computation of Internal Forces Problems Chapter 12: Influence Lines Method 12.1 Construction of Influence Lines by the Force Method 12.1.1 General 12.1.2 Continuous Beams 12.1.2.1 Primary System 12.1.2.2 Influence Line for Primary Unknown X1 12.1.2.3 Influence Line for Bending Moment Mk 12.1.2.4 Influence Line for Shear Force Qk 12.1.2.5 Statically Indeterminate Primary System 12.1.3 Hingeless Nonuniform Arches 12.1.3.1 Unit Coefficients 12.1.3.2 Reactions of Support A 12.1.3.3 Bending Moment at Crown C 12.1.3.4 Discussion 12.1.4 Statically Indeterminate Trusses 12.1.4.1 Construction of Function δP1 12.1.4.2 Influence Line for Primary Unknown X1 12.1.4.3 Summary 12.2 Construction of Influence Lines by the Displacement Method 12.2.1 General 12.2.2 Continuous Beams 12.2.2.1 Influence Line for Primary Unknown Z1 12.2.2.2 Influence Line for Bending Moment Mk 12.2.2.3 Influence Line for Shear Force Qk 12.2.2.4 Discussion 12.2.3 Redundant Frames Discussion 12.3 Comparison of the Force and Displacement Methods Notes 12.4 Kinematical Method (Müller-Breslau Principle) 12.4.1 General 12.4.2 Continuous Beams: Analytical Solution 12.4.3 Continuous Beams: Models of Influence lines Summary Problems Chapter 13: Matrix Stiffness Method 13.1 Basic Idea and Concepts 13.1.1 Finite Elements 13.1.2 Global and Local Coordinate Systems 13.1.3 Displacements of Joints and Degrees of Freedom 13.2 Auxiliary Diagrams 13.2.1 Joint-Load (J-L) Diagram 13.2.2 Displacement-Load (Z-P) Diagram 13.2.3 Internal Forces-Deformation (S-e) Diagram Summary 13.3 Initial Matrices 13.3.1 Vector of External Joint Loads 13.3.2 Vector of Internal Unknown Forces Summary 13.4 Resolving Equations 13.4.1 Static Equations and Static Matrix 13.4.2 Geometrical Equations and Deformation Matrix 13.4.3 Physical Equations and Stiffness Matrix in Local Coordinates 13.5 Set of Formulas and Procedure for Analysis 13.5.1 Stiffness Matrix in Global Coordinates 13.5.2 Unknown Displacements and Internal Forces 13.5.3 Matrix Procedures 13.6 Analysis of Continuous Beams 13.6.1 Fixed Loads 13.6.2 Settlements of Supports 13.6.3 Moving Load (Construction of Influence Lines) 13.6.3.1 Load P = 1 in the First Span 13.6.3.2 Load P = 1 in the Second Span 13.6.3.3 Load P = 1 in the Third Span 13.7 Analysis of Redundant Frame 13.8 Analysis of Redundant Trusses 13.9 Stiffness Matrices: Expanded Forms 13.9.1 Truss Element (Pinned-Pinned Element) 13.9.2 Beam Elements 13.9.3 More General Case of a Finite Element 13.10 Summary Remarks Concerning Analysis of the Bending Structures by MSM Problems Part III: Special Topics Chapter 14: Plastic Behavior of Structures 14.1 Idealized Stress-Strain Diagrams 14.2 Direct Method of Plastic Analysis 14.2.1 Elastic Analysis 14.2.2 Plastic Analysis 14.3 Fundamental Methods of Plastic Analysis 14.3.1 Kinematical Method 14.3.2 Static Method 14.3.3 Summary 14.4 Plastic Analysis of Continuous Beams 14.4.1 Static Method 14.4.2 Kinematical Method Discussion 14.5 Plastic Analysis of Frames 14.5.1 Beam Failure 14.5.2 Sidesway Failure 14.5.3 Combined Failure 14.5.4 Limit Combination Diagram Problems Chapter 15: Stability of Elastic Systems 15.1 Fundamental Concepts 15.2 Stability of Structures with Finite Number of Degrees of Freedom 15.2.1 Structures with One Degree of Freedom Discussion 15.2.2 Structures with Two or More Degrees of Freedom 15.3 Stability of Columns with Rigid and Elastic Supports 15.3.1 The Double Integration Method 15.3.1.1 Uniform Clamped-Free Column 15.3.1.2 Uniform Columns with Elastic Supports 15.3.2 Initial Parameters Method 15.3.2.1 Limiting Cases 15.3.2.2 Discussion 15.3.3 Euler´s Solution and Paradox of Critical Load 15.4 Stability of Continuous Beams and Frames 15.4.1 Unit Reactions of the Beam-Columns Some Features of the Solution 15.4.2 Displacement Method Special cases Special Cases 15.4.3 Modified Approach of the Displacement Method 15.5 Stability of Arches 15.5.1 Introduction 15.5.1.1 Forms of the Loss of Stability of the Arches 15.5.2 Circular Arches under Hydrostatic Load 15.5.3 Complex Arched Structure: Arch with Elastic Supports 15.5.4 Parabolic Arch under Gravity Load Arches with Variable Cross Sections Problems Chapter 16: Dynamics of Elastic Systems: Free Vibration 16.1 Fundamental Concepts 16.1.1 Kinematics of Vibrating Processes 16.1.2 Forces Which Arise at Vibrations 16.1.3 Degrees of Freedom 16.1.4 Purpose of Structural Dynamics 16.2 Systems with Finite Number of Degrees of Freedom: Force Method 16.2.1 Differential Equations of Free Vibration in Displacements 16.2.2 Frequency Equation 16.2.3 Mode Shapes of Vibration and Modal Matrix 16.3 Systems with Finite Number of Degrees of Freedom: Displacement Method 16.3.1 Differential Equations of Free Vibration in Reactions 16.3.2 Frequency Equation 16.3.3 Mode Shape of Vibrations and Modal Matrix 16.3.4 Comparison of the Force and Displacement Methods 16.4 Structures with Infinite Number of Degrees of Freedom 16.4.1 Differential Equation of Transversal Vibration of the Beam 16.4.1.1 Boundary Conditions 16.4.1.2 Initial Conditions 16.4.2 Fourier Method 16.4.3 Krylov-Duncan Method 16.4.3.1 Properties of Krylov-Duncan Functions (16.24) 16.4.4 Initial Parameters Method 16.4.5 Transfer Matrices Method 16.4.5.1 Transfer Matrix of Uniform Beam with Distributed Mass 16.4.5.2 Transfer Matrix of Uniform Massless Beam 16.4.5.3 Mathematical Model of a Complex Structure 16.4.6 Displacement Method 16.4.7 Missed (Unaccounted) Frequencies Problems Chapter 17: Dynamics of Elastic Systems: Forced Vibration 17.1 Structures with One Degree of Freedom 17.1.1 Differential Equations: Two Classical Approaches 17.1.1.1 Force Method 17.1.1.2 Displacement Method 17.1.2 Types of Excitations 17.1.3 Duhamel Integral and Some Special Types of Excitation 17.1.4 Harmonic Excitation: Equivalent Design Diagram 17.1.4.1 Equivalent Design Diagram 17.1.5 Kinematical Excitation 17.2 Structures with Finite Number of Degrees of Freedom: The Force Method 17.2.1 Resolving Equation of the Force Method 17.2.2 Harmonic Excitation: Reciprocal Theorems 17.2.3 Impulsive Excitation 17.2.4 General Case of Excitation 17.3 Structures with Finite Number of Degrees of Freedom: Initial Parameters Method 17.3.1 Resolving Equations 17.3.2 Steady-State Vibrations 17.4 Structures with Finite Number of Degrees of Freedom: Displacement Method 17.4.1 The Steady-State Vibration 17.4.2 Group Unknowns Method 17.5 Structures with Distributed Parameters 17.5.1 Initial Parameter Method 17.5.2 Displacement Method 17.5.3 Bolotin Approximate Method Problems Structures with Finite Number of Degrees of Freedom Kinematical Excitation Forced Excitations Structures with Infinite Numbers Degrees of Freedom Chapter 18: Special Topics of Structural Dynamics 18.1 Timoshenko-Ehrenfest Beam Theory 18.2 Flexural Vibration of Compressed Beam 18.2.1 Fundamental Equations 18.2.2 Galef´s Formula 18.3 Traveling Load 18.3.1 General 18.3.2 Quasi-Static Loading 18.3.3 Critical Velocity of Moving Load 18.4 Parametric Vibration 18.4.1 Dynamic Stability of Simply Supported Column 18.4.2 Ince-Strutt Diagram 18.5 Vibration Protection 18.5.1 Vibration Protection Methods 18.5.2 Dynamic Absorber 18.5.3 Lumped Vibration Absorber of the Beams Chapter 19: Nonlinear Structural Analysis 19.1 Introduction and Types of Nonlinearities 19.1.1 Types of Nonlinearities 19.2 Compressed Rods with Lateral Loading 19.2.1 Double Integration Method 19.2.2 Initial Parameters Method 19.2.2.1 Free-Clamped Beam: Precise Solution 19.2.3 P-Delta Analysis 19.3 Static Nonlinearity 19.3.1 Features of the Problem 19.3.2 Transversal Vibration of a Rod 19.4 Physical Nonlinearity 19.4.1 Features of the Problem 19.4.2 Transversal Free Vibration of Uniform Beam 19.5 Geometrical Nonlinearity 19.5.1 General 19.5.2 Stability of a Flexible Rod Problems Chapter 20: Conclusion: Once More About Modeling of Structures 20.1 Some Problems of Structural Modeling 20.2 Common Modeling Problems 20.3 Differences Between Structural Analysis and Direct Design Chapter 21: Fundamental Developments in the History of Structural Mechanics Appendix Tabulated Data for Standard Uniform Beams Bibliography General Textbooks and Manuals Handbooks Related List of Literature History Index Index