**Chapter 1 Principles and Tools for Static Analysis 1**

1.1 How Does Engineering Analysis Fit Into Engineering Practice? 2

1.2 Physics Principles: Newton’s Laws Reviewed 4

1.3 Properties and Units in Engineering Analysis 5

1.4 Coordinate Systems and Vectors 8

1.5 Drawing 11

1.6 Problem Solving 14

1.7 A Map of This Text 18

1.8 Just the Facts 20

**Chapter 2 Forces 23**

2.1 What are Forces? An Overview 24

2.2 Gravitational Forces 25

Example 2.2.1 Gravity, Weight, and Mass 28

Example 2.2.2 Is Assuming Gravity is a Constant Reasonable? 30

Example 2.2.3 Gravitational Force from Two Planets 31

2.3 Contact Forces 32

Example 2.3.1 Identifying Types of Forces 36

2.4 Identifying Forces for Analysis 37

Example 2.4.1 Defining a System for Analysis 40

2.5 Representing Force Vectors 42

Example 2.5.1 Rectangular Components of a Nonplanar Force Given its Line of Action 46

Example 2.5.2 Representing Nonplanar Forces with Rectangular Coordinates 47

Example 2.5.3 Representing a Planar Force in Skewed Coordinate System 49

Example 2.5.4 Representing Direction of a Planar Force 54

Example 2.5.5 Scalar Components of a Planar Force 55

Example 2.5.6 Representing a Planar Force with Spherical Coordinates 58

Example 2.5.7 Representing Nonplanar Forces with Spherical Angles 59

2.6 Resultant Force—Vector Addition 60

Example 2.6.1 Component Addition: Planar 63

Example 2.6.2 Component Addition: Nonplanar 64

Example 2.6.3 Graphical Addition Using Force Triangle 67

Example 2.6.4 Graphical Addition Using Parallelogram Law 69

Example 2.6.5 Resultant of Two Forces Using a Trigonometric Approach 71

Example 2.6.6 Analyzing a System: Trigonometric Addition 73

Example 2.6.7 Analyzing a System: Trigonometric Approach 74

2.7 Angle Between Two Forces— the Dot Product 75

Example 2.7.1 Projection of a Vector in Two Dimensions 78

Example 2.7.2 Projection of a Vector in Three Dimensions 79

Example 2.7.3 Angle Between Two Vectors 80

2.8 Just the Facts 82

**Chapter 3 Moments 87**

3.1 What are Moments? 88

Example 3.1.1 Specifying the Position Vector - Planar 95

Example 3.1.2 Specifying the Position Vector - Nonplanar 96

Example 3.1.3 The Magnitude of a Moment - Planar 97

Example 3.1.4 The Magnitude of a Moment - Nonplanar 98

Example 3.1.5 Moment Center on the Line of Action of Force 100

3.2 Mathematical Representation of a Moment 101

Example 3.2.1 Calculating the Moment About the *z *Axis with a Vector-Based Approach 105

Example 3.2.2 Calculating the Moment About the *z *Axis with the Component of the Force Perpendicular to the Position Vector 106

Example 3.2.3 Calculating the Moment - Nonplanar 107

Example 3.2.4 Calculating the Magnitude and Direction of a Moment - Nonplanar 109

Example 3.2.5 Finding the Force to Create a Moment - Nonplanar 110

3.3 Finding Moment Components in a Particular Direction 111

Example 3.3.1 Finding the Moment About the *z *Axis 113

Example 3.3.2 Finding the Moment in a Particular Direction 114

3.4 When are Two Forces Equal to a Moment? (When They are a Couple) 115

Example 3.4.1 A Couple in the *xy *Plane 117

Example 3.4.2 Working with Couples 118

3.5 Equivalent Loads 120

Example 3.5.1 Equivalent Moment and Equivalent Force - Planar 122

Example 3.5.2 Equivalent Moment and Equivalent Force - Nonplanar 124

Example 3.5.3 Equivalent Load for an Applied Couple 126

3.6 Just the Facts 127

**Chapter 4 Modeling Systems with Free-Body Diagrams 131**

4.1 Types of External Loads Acting on Systems 132

4.2 Planar System Supports 134

Example 4.2.1 Free-Body Diagram of a Planar System 140

Example 4.2.2 Free-Body Diagram of a Planar System with Moment 141

Example 4.2.3 Using Questions to Determine Loads at Supports 142

4.3 Nonplanar System Supports 143

Example 4.3.1 Exploring Single and Double Bearings and Hinges 149

4.4 Modeling Systems as Planar or Nonplanar 150

Example 4.4.1 Identifying Planar and Nonplanar Systems 152

Example 4.4.2 Identifying Planar and Nonplanar Systems with a Plane of Symmetry 153

4.5 A Step-By-Step Approach to Free-Body Diagrams 154

Example 4.5.1 Creating a Free-Body Diagram of an Airplane Wing 156

Example 4.5.2 Creating a Free-Body Diagram of a Ladder 157

Example 4.5.3 Creating a Free-Body Diagram of a Nonplanar System 157

Example 4.5.4 Creating a Free-Body Diagram of a Leaning Person 158

4.6 Just the Facts 159

**Chapter 5 Mechanical Equilibrium 161**

5.1 Conditions of Mechanical Equilibrium 162

5.2 The Equilibrium Equations 163

Example 5.2.1 Using a Free-Body Diagram to Write Equilibrium Equations 165

5.3 Applying the Planar Equilibrium Equations 166

Example 5.3.1 Applying the Analysis Procedure to a Planar Equilibrium Problem 169

Example 5.3.2 Analysis of a Simple Structure 171

Example 5.3.3 Analysis of a Planar Truss 172

5.4 Equilibrium Applied to Four Special Cases 173

Example 5.4.1 Analyzing a Planar Truss Connection as a Particle 174

Example 5.4.2 Two-Force Member Analysis 177

Example 5.4.3 Climbing Cam Analysis 180

Example 5.4.4 Three-Force Member Analysis 182

Example 5.4.5 Ideal Pulley Analysis 185

5.5 Applying the Nonplanar Equilibrium Equations 187

Example 5.5.1 Analysis of a Nonplanar System with Simple Loading 189

Example 5.5.2 Analysis of a Nonplanar System with Complex Loading 192

Example 5.5.3 High-Wire Circus Act 194

Example 5.5.4 Analysis of a Nonplanar System with Unknowns Other than Loads 196

5.6 Zooming in on Subsystems 198

Example 5.6.1 Analysis of a Toggle Clamp 199

Example 5.6.2 Analysis of a Pulley System 202

5.7 Determinate, Indeterminate, and Underconstrained Systems 204

Example 5.7.1 Identify Status of a Structure 206

5.8 Just the Facts 207

**Chapter 6 Distributed Force 211**

6.1 Center of Mass, Center of Gravity, and the Centroid 212

Example 6.1.1 Centroid of a Volume 219

Example 6.1.2 Center of Mass with Variable Density 220

Example 6.1.3 Locating the Centroid of a Composite Volume 221

Example 6.1.4 Finding the Centroid of An Area 223

Example 6.1.5 Center of Mass of a Composite Assembly 225

Example 6.1.6 Centroid of a Built-Up Section 227

6.2 Distributed Force Acting on a Boundary 228

Example 6.2.1 Using Integration to Find Total Force 235

Example 6.2.2 Inclined Beam with Nonuniform Distribution 237

Example 6.2.3 Beam Subjected to Polynomial Load Distribution 239

Example 6.2.4 Using Properties of Standard Shapes to Find Total Force 241

Example 6.2.5 Centroid of Distribution Composed of Standard Line Loads 243

Example 6.2.6 Calculating Center of Pressure of a Pressure Distribution 244

Example 6.2.7 Pressure on a Rectangular Water Gate 245

6.3 Hydrostatic Pressure 247

Example 6.3.1 Proof of Nondirectionality of Fluid Pressure 250

Example 6.3.2 Proof that Hydrostatic Pressure Increases Linearly with Depth 251

Example 6.3.3 Hydrostatic Pressure on Vertical Reservoir Gate 252

Example 6.3.4 Hydrostatic Pressure on Sloped Gate 253

Example 6.3.5 Pressure Distribution Over a Curved Surface 255

Example 6.3.6 Center of Buoyancy and Stability 257

6.4 Area Moment of Inertia 258

Example 6.4.1 Moment of Inertia Using Integration 262

Example 6.4.2 Moment of Inertia Using Parallel Axis Theorem 263

Example 6.4.3 Moment of Inertia of a Composite Area 264

6.5 Just the Facts 265

**Chapter 7 Dry Friction and Rolling Resistance 271**

7.1 Coulomb Friction Model 272

Example 7.1.1 Dry Friction - Sliding or Tipping 275

7.2 Friction in Static Analysis: Wedges, Belts, and Journal Bearings 276

Example 7.2.1 Analysis of a Pulley System with Bearing Friction 281

7.3 Rolling Resistance 283

Example 7.3.1 Rolling Resistance 284

7.4 Just the Facts 285

**Chapter 8 Member Loads in Trusses 289**

8.1 Defining a Truss 290

8.2 Truss Analysis by Method of Joints 293

Example 8.2.1 Truss Analysis Using Method of Joints 296

8.3 Truss Analysis by Method of Sections 298

Example 8.3.1 Method of Sections and Wise Selection of Moment Center Location 300

Example 8.3.2 Method of Sections and Where to Cut 301

Example 8.3.3 Combining Method of Joints and Method of Sections 303

8.4 Identifying Zero-Force Members 305

Example 8.4.1 Identifying Zero-Force Members 306

8.5 Determinate, Indeterminate, and Unstable Trusses 308

Example 8.5.1 Checking the Status of Planar Trusses 310

Example 8.5.2 Checking the Status of Space Trusses 312

8.6 Just the Facts 313

**Chapter 9 Member Loads in Frames and Machines 315**

9.1 Defining and Analyzing Frames 316

Example 9.1.1 Identify Systems as Trusses or Frames 317

Example 9.1.2 Planar Frame Analysis 319

Example 9.1.3 Finding Loads at Frame Supports 321

Example 9.1.4 Analysis of Frame with Friction 323

Example 9.1.5 Nonplanar Frame Analysis 324

9.2 Defining and Analyzing Machines 326

Example 9.2.1 Analysis of a Bicycle Brake 327

Example 9.2.2 Analysis of a Toggle Clamp 329

Example 9.2.3 Analysis of a Frictionless Gear Train 331

Example 9.2.4 Analysis of a Gear Train with Friction 333

9.3 Determinacy and Stability in Frames 335

Example 9.3.1 Determining Status of a Frame 338

9.4 Just the Facts 339

**Chapter 10 Internal Loads in Beams 341**

10.1 Defining Beams and Recognizing Beam Configurations 342

Example 10.1.1 Beam Identification 345

Example 10.1.2 Determine Loads Acting on a Beam 346

10.2 Beam Internal Loads 347

Example 10.2.1 Internal Loads in a Planar Simply Supported Beam 351

Example 10.2.2 Internal Loads in a Planar Cantilever Beam 353

Example 10.2.3 Internal Loads in a Nonplanar Beam 354

10.3 Axial Force, Shear Force, and Bending Moment Diagrams 356

Example 10.3.1 Shear, Moment, and Axial Force Diagram for a Simply Supported Beam 359

Example 10.3.2 A Simple Beam with an Applied Moment 361

Example 10.3.3 Beam with Distributed Load 362

Example 10.3.4 Simply Supported Beam with an Overhang 364

10.4 Bending Moment Related to Shear Force and Normal Stress 365

Example 10.4.1 Using the Relationships Between *ω*, *V*, and *M *368

Example 10.4.2 Calculating Beam Normal Stress 370

10.5 Just the Facts 371

**Chapter 11 Internal Loads in Cables 373**

11.1 Cables with Point Loads 374

Example 11.1.1 Flexible Cable with Concentrated Loads 375

11.2 Cables with Distributed Loads 377

Example 11.2.1 Catenary Curve with Supports at Same Height 381

Example 11.2.2 Catenary with Supports at Different Elevations 382

Example 11.2.3 Uniformly Loaded Cable with Supports at Same Height 384

Example 11.2.4 Uniformly Loaded Cable with Supports at Unequal Heights 385

Example 11.2.5 Catenary Versus Parabolic 387

11.3 Just the Facts 388

Appendix A Selected Topics in Mathematics 393

Appendix B Physical Quantities 397

Appendix C Properties of Areas and Volumes 401

Appendix D Case Study: The Bicycle 407

Appendix E Case Study: The Golden Gate Bridge 419

Index 433