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Mechanics of Materials
i
Contents
1. Introduction and Review
1.1 Determinate and Indeterminate Structure 1-5
1.2 Units and Conversions 1-6
1.3 Review of Statics 1-7
1.3.1 Equilibrium Equations and Free Body Diagram 1-7
1.3.2 Two Force and Multiforce Body Equilibrium 1-9
1.3.3 Analysis of Trusses 1-10
1.3.4 Analysis of Frames 1-12
1.3.5 Centroids and Applications 1-15
1.3.6 Moments of Inertia 1-19
1.4 Summary 1-22
1.5 Problems 1-23
2. Stress and Strain
2.1 Introduction 2-3
2.2 Stress 2-3
2.2.1 Axial or Normal Stress 2-4
2.2.2 Shear Stress 2-5
2.2.3 Bearing Stress 2-7
2.2.4 Practical Units of Stress 2-9
2.2.5 Dependence of Stress on Force and Area 2-9
2.2.6 Force, Traction and Stress 2-12
2.3 Deformation 2-16
2.3.1 Axial Displacement, Elongation and Axial Strain 2-16
2.3.2 Shear Strain 2-21
2.4 Hooke’s Law Relating Stresses to Strains 2-22
2.4.1 Young’s Modulus and Poisson’s Ratio 2-22
2.4.2 Shear Modulus 2-24
2.5 Design Issues 2-26
2.6 Summary 2-30
2.7 Problems 2-31
Mechanics of Materials
ii
3. Axial Force Diagram
3.1 Introduction 3-3
3.2 Axial Force Diagram 3-3
3.2.1 Method of Sections 3-3
3.3 Displacements and Elongations in Axially Loaded Member 3-5
3.3.1 General Expression for Elongation 3-8
3.4 Temperature Related Deformations 3-10
3.5 Design Considerations in Axially Loaded Members 3-11
3.6 Statically Indeterminate Problems 3-13
3.6.1 Definition of Indeterminacy 3-13
3.6.2 Equilibrium and Compatibility 3-14
3.6.3 Flexibility and Stiffness 3-15
3.6.4 Statically Indeterminate Thermal Problem 3-25
3.7 Summary 3-32
3.8 Problems 3-33
4. Multi-axial Loading
4.1 Introduction 4-3
4.2 Definition of State of Stress 4-3
4.3 Stresses in Solid Subjected to Multi-Axial Normal Stresses 4-5
4.3.1 Superposition Principle 4-6
4.4 Shear Stress-Shear Strain Relationship 4-8
4.5 Generalized Hooke’s Law Equations 4-9
4.6 Relationships between Three Elastic Constants 4-10
4.7 Bulk Modulus 4-15
4.8 Summary 4-18
4.9 Problems 4-20
5. Torsion and Shear Stress
5.1 Introduction 5-3
5.2 Torque and Shear Stresses in Circular Shafts 5-3
5.2.1 Shear Stress due to a Twisting Moment on a Circular Shaft 5-4
5.2.2 Shear Stresses on Hollow Circular Cross-Section 5-9
5.3 Shear Strain and Angle of Twist 5-11
Table of Contents
iii
5.4 Statically Indeterminate Torsion Problems 5-15
5.4.1 Indeterminate Reactions 5-15
5.4.2 Composite Shafts 5-19
5.5 Torsion in Shafts Made of Non-Circular Cross-Section 5-22
5.6 Thin Walled Cross-Section 5-22
5.6.1 Shear Stresses in the Shaft 5-22
5.6.2 Shear Flow 5-23
5.6.3 Torque and Shear flow 5-25
5.6.4 Angle of Twist and Torque 5-25
5.7 Design of Shafts 5-26
5.8 Summary 5-29
5.9 Problems 5-30
6. Shear Force and Bending Moment Diagrams
6.1 Introduction 6-3
6.2 Definitions of Bending Moment and Shear Force 6-3
6.3 Method of Sections 6-5
6.3.1 Shear and Moment Diagrams 6-5
6.4 Method of Integration 6-16
6.5 Summation Method 6-19
6.6 Shear Force and Bending Moment Diagrams for Cantilever Beams 6-37
6.7 Summary 6-45
6.8 Problems 6-46
7. Bending (Flexural) Stresses in Beams
7.1 Introduction 7-3
7.2 Phenomenon of Bending 7-3
7.2.1 Kinematics of Bending 7-4
7.2.2 Connecting Kinematics to Material Behavior 7-6
7.2.3 Relating the Bending Moment to Axial Stress 7-6
7.2.4 Summary of Equations 7-8
7.3 Steps for the Determination of Maximum Bending Stress in a Beam 7-10
7.4 Forces due to Flexural Stresses 7-11
7.4.1 Basic Idea of Force Computation 7-11
Mechanics of Materials
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7.4.2 Forces due to Bending Stress 7-12
7.4.3 Location of Compressive or Tensile Force 7-12
7.4.4 Summary of Equations 7-13
7.5 Design for Bending 7-19
7.5.1 Allowable Stress and Section Modulus 7-19
7.6 Beams Made of Two Materials – Composite Beams 7-22
7.6.1 Principles of Composite Beams 7-23
7.6.2 Location of Neutral Axis and Stress Computation 7-24
7.6.3 Alternative Method: Transformed Area Method 7-25
7.7 Summary 7-29
7.8 Problems 7-30
8. Shear Stresses in Beams
8.1 Introduction 8-3
8.1.1 Concept of Shear Stresses in Perspective 8-3
8.2 Shear Flow 8-4
8.2.1 Horizontal Shear Flow and Horizontal Shear Force 8-4
8.2.2 Shear Connectors and Design of Connectors 8-6
8.3 Shear Stress in Beams 8-11
8.3.1 Shear Stress Distribution Formula 8-11
8.3.2 Rectangular Cross Section 8-12
8.3.3 I Shaped Section 8-13
8.3.4 Maximum Shear Stress 8-17
8.4 Shear in Thin Walled Cross-section 8-17
8.5 Shear Center for Singly Symmetric Cross Sections 8-19
8.6 Summary 8-31
8.7 Problems 8-32
9. Transformation of Stresses
9.1 Introduction 9-3
9.1.1 Purpose of this Chapter 9-3
9.1.2 Maximum Normal and Shear Stresses 9-3
9.1.3 Mathematical Representation of Stress 9-4
9.1.4 Common Examples of Stresses in Cubic Elements 9-4
Table of Contents
v
9.2 Stress Transformation for Plane Stress 9-6
9.2.1 Stress on an Arbitrary Plane 9-7
9.2.2 Maximum Normal Stress and Maximum Shear Stress 9-10
9.2.2.1 Principal Stress and Principal Angle 9-10
9.2.3 Maximum Shear Stress and Associated Plane 9-11
9.2.4 Relation between orientations of Principal Stress and Maximum
Shear Stress 9-12
9.3 Mohr’s Circle Method 9-12
9.3.1 Drawing a Mohr’s Circle 9-13
9.3.2 Solution Using Mohr’s Circle Method 9-15
9.4 Design Issues and Failure of Ductile and Brittle Materials 9-22
9.5 Summary 9-34
9.6 Problems 9-35
10. Deflection in Beams
10.1 Introduction 10-3
10.2 Differential Equations of Deflection 10-4
10.2.1 Second Order Differential Equation 10-4
10.2.2 Fourth Order Differential Equation 10-4
10.3 Solution of Differential Equations 10-5
10.3.1 Second Order Differential Equation 10-5
10.3.2 Fourth Order Differential Equation 10-5
10.4 Boundary Conditions and Application of Boundary Conditions 10-6
10.5 Second Order Versus Fourth Order Differential Equation Solution Approaches 10-8
10.6 Statically Indeterminate Beams 10-22
10.7 Determination of Reactions of Statically Indeterminate Beams 10-23
10.8 Method of Superposition 10-27
10.9 Summary 10-29
10.10 Problems 10-30
11. Stability of Structures
11.1 Introduction 11-3
11.2 Euler Buckling Load and Buckling Stress 11-4
11.2.1 Euler Buckling Load and its Physical Significance 11-6
Mechanics of Materials
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11.2.2 Critical Buckling Stress 11-8
11.3 Boundary Conditions 11-9
11.4 Eccentric Loading 11-19
11.4.1 Maximum Load and Deflection for a given Eccentricity 11-21
11.5 Summary 11-26
11.6 Problems 11-27
Index