Mechanics of Materials

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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

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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

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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

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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

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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

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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