Mechanics of Materials For Dummies

Mechanics of Materials For Dummies
اسم المؤلف
James H. Allen
22 يناير 2018

Mechanics of Materials For Dummies
by James H. Allen III, PE, PhD
Table of Contents
Introduction . 1
About This Book 1
Conventions Used in This Book .2
What You’re Not to Read 3
Foolish Assumptions .3
How This Book Is Organized 4
Part I: Setting the Stage for Mechanics of Materials 4
Part II: Analyzing Stress 4
Part III: Investigating Strain 5
Part IV: Applying Stress and Strain 5
Part V: The Part of Tens 5
Icons Used in This Book .6
Where to Go from Here .6
Part I: Setting the Stage for Mechanics of Materials . 7
Chapter 1: Predicting Behavior with Mechanics of Materials .9
Tying Statics and Mechanics Together .10
Defning Behavior in Mechanics of Materials .11
Stress .11
Strain .11
Using Stresses to Study Behavior 12
Studying Behavior through Strains .12
Incorporating the “Material” into Mechanics of Materials .13
Putting Mechanics to Work 13
Chapter 2: Reviewing Mathematics and Units Used
in Mechanics of Materials .15
Grasping Important Geometry Concepts .15
Tackling Simultaneous Algebraic Equations 16
Taking On Basic Trig Identities 18
Covering Basic Calculus .18
Integration and differentiation of polynomials 18
Defning maximum and minimum values with calculus 20viii Mechanics of Materials For Dummies
Working with Units in Mechanics of Materials .21
SI units .21
U.S. customary units 22
Micro and kip: Noting two exceptions 22
All the derived mechanics units you’ll ever need .23
Converting angular units from degrees
to radians (and back again) 24
Chapter 3: Brushing Up on Statics Basics 25
Sketching the World around You with Free-Body Diagrams .25
External loads .26
Internal loads on two-dimensional objects 27
Support reactions 29
Self weight .29
Reviewing Equilibrium for Statics .30
Locating Internal Forces at a Point .31
Finding Internal Loads at Multiple Locations 32
Writing generalized equations .33
Drawing simple shear and moment diagrams
by using area calculations 35
Chapter 4: Calculating Properties of Geometric Areas .41
Determining Cross-Sectional Area .41
Classifying cross-sectional areas .42
Computing cross-sectional areas .43
Considering prismatic members 45
Defning symmetry of cross sections 46
Finding the Centroid of an Area .47
Making discrete region calculations 47
Working with continuous (general) regions .51
Using symmetry to avoid centroid calculations 54
Chapter 5: Computing Moments of Area
and Other Inertia Calculations 55
Referencing with the Centroidal Axis .56
Computing Q, the First Moment of Area .57
Establishing the equations for Q .58
Revisiting centroid calculations with frst moment of area .58
Determining Q within a cross section .59
Creating a table for calculating Q about a centroidal axis .61
Encore! Encore! I, a Second Moment of Area .63
Conceptualizing on area moments of inertia .64
Categorizing area moments of inertia .65
Calculating Basic Area Moments of Inertia 66
Keeping inertia simple with basic shapes and centroidal axes 66
Transferring reference locations with the parallel axis theorem 70Table of Contents ix
Having It Both Ways with Product Moments of Area .73
Including x- and y-axes for product moment calculations .74
Computing the product moment of area 74
Putting a Twist on Polar Moments of Inertia .76
Computing Principal Moments of Inertia .78
Calculating principal moments of inertia .79
Finding the principal orientation angles .79
Determining moments of area at specifc orientation angles 80
Rounding Up the Radius of Gyration 82
Part II: Analyzing Stress . 83
Chapter 6: Remain Calm, It’s Only Stress! 85
Dealing with a Stressful Relationship 85
Calculating stress .86
Defning the types of stress 87
Understanding the units of stress .88
Remaining Steady with Average Stress .88
Computing average normal stress for axial loads .89
Determining average shear stress .90
Developing Stress at a Point 92
Deriving stresses at a single point by using force components .92
Looking at useful shear stress identities for stress at a point .95
Containing Plane Stress 97
Chapter 7: More than Meets the Eye: Transforming Stresses .99
Preparing to Work with Stresses .99
Building a stress block diagram .100
Identifying basic states of stress .100
Establishing a sign convention for stresses .101
Stress Transformation: Finding Stresses at a Specifed
Angle for One Dimension 104
Extending Stress Transformations to Plane Stress Conditions .106
Displaying the Effects of Transformed Stresses 109
Wedging in on the action with stress wedges 109
Rotating the basic stress element .111
When Transformed Stresses Aren’t Big Enough: Principal Stresses .113
Defning the principal normal stresses .114
Orienting the angles for principal normal stresses .114
Calculating principal shear stresses .117
Finding the principal shear stress orientation angle 118
Distinguishing between in-plane and out-of-plane
maximum shear stresses 120x Mechanics of Materials For Dummies
Utilizing Mohr’s Circle for Plane Stress 120
Establishing basic assumptions and requirements
for Mohr’s circle .121
Constructing the Mohr’s circle 121
Computing coordinates and other important
values on Mohr’s circle .122
Determining principal normal stresses and angles .124
Calculating other items with Mohr’s circle 125
Finding stress coordinates at arbitrary angles on Mohr’s circle .126
Adding a third dimension to Mohr’s circle .128
Chapter 8: Lining Up Stress Along Axial Axes .131
Defning Axial Stress 131
Getting Your Bearings about Bearing Stresses 133
Exploring bearing stresses on ?at surfaces 134
Perusing bearing stresses on projected planes .135
Containing Pressure with Pressure Vessels .136
Differentiating between thin- and thick-walled pressure vessels 137
Taking a closer look at thin-walled pressure vessels 138
When Average Stresses Reach a Peak: Finding Maximum Stress 141
Explaining gross versus net areas for average
normal stress calculations 141
Using the force lines to locate maximum stress 144
Concentrating on normal stress concentrations .145
Chapter 9: Bending Stress Is Only Normal:
Analyzing Bending Members 149
Explaining Bending Stress 149
Handling Stresses in Bending .150
Solving Pure Bending Cases .152
Establishing basic assumptions .152
Computing stresses in pure-bending applications 153
Looking at pure bending of symmetrical cross sections 155
Bending of Non-Prismatic Beams 158
Chapter 10: Shear Madness: Surveying Shear Stress .161
It’s Not Sheer Folly: Examining Shear Stress 161
Working with Average Shear Stresses 162
Shear on glue or contact surfaces .163
Shear for bolts and shafts .163
Punching shear .166
Exploring Shear Stresses from Flexural Loads 168
Determining the shear stress distribution
in uniform cross sections 168
Handling shear stresses in nonuniform cross sections 170
Calculating Shear Stresses by Using Shear Flow .171
Going with the shear ?ow .171Table of Contents xi
Chapter 11: Twisting the Night Away with Torsion .177
Considering Torsion Characteristics 177
Working with Shear Stresses Due to Torsion 178
Defning the shear stress element for torsion 179
Computing the torsional constant .180
Computing Shear Stress from Torsion 182
Tackling torsion of circular shafts .182
Determining torsion of non-circular cross sections 183
Applying shear ?ow to torsion problems
in thin-walled sections .184
Using shear ?ow to analyze torsion of multicell cross sections .186
Part III: Investigating Strain . 189
Chapter 12: Don’t Strain Yourself: Exploring Strain
and Deformation .191
Looking at Deformation to Find Strain 192
Strained relationships: Comparing lengths 192
Examining units of strain 193
Using formulas for engineering and true strains 193
Normal and Shear: Seeking Some Direction on the Types of Strain .194
Getting it right with normal strain .194
Finding a new angle with shear strain .196
Expanding on Thermal Strains .198
Considering Plane Strains .200
Chapter 13: Applying Transformation Concepts to Strain 201
Extending Stress Transformations to Plane Strain Conditions .201
Transforming strains .202
Sketching a rotated strain element 204
Calculating and Locating Principal Strain Conditions 205
Defning the principal normal strains 206
Determining the angles for principal normal strains 206
Computing the principal shear strain .207
Exploring Mohr’s Circle for Plane Strain 209
Gauging Strain with Strain Rosettes 212
Chapter 14: Correlating Stresses and Strains
to Understand Deformation .215
Describing Material Behavior 216
Elastic and plastic behavior: Getting back in shape? 216
Ductile and brittle materials: Stretching or breaking 217
Fatigue: Weakening with repeated loads 218xii Mechanics of Materials For Dummies
Creating the Great Equalizer: Stress-Strain Diagrams 219
Justifying stress-strain relationships 220
Describing materials with stress versus strain 220
Exploring Stress-Strain Curves for Materials .221
Defning the regions of a stress- strain curve .222
Site-seeing at points of interest on a stress-strain diagram .223
Knowing Who’s Who among Material Properties 224
Finding stiffness under load: Young’s modulus of elasticity 224
Getting longer and thinner (or shorter and fatter)
with Poisson’s ratio .226
Relating Stress to Strain 227
Making assumptions in stress versus strain relationships 227
Hooke springs eternal! Using Hooke’s law for one dimension 228
Developing a generalized relationship for Hooke’s la
in two or three dimensions .230
Calculating stress from known strain values .232
Part IV: Applying Stress and Strain . 233
Chapter 15: Calculating Combined Stresses .235
Understanding the Principle of Superposition:
A Simple Case of Addition .236
Setting the Stage for Combining Stresses .237
Following some simple rules 237
Establishing a few handy conventions 238
Handling Multiple Axial Effects 239
Including Bending in Combined Stresses .241
Bending biaxially from inclined point loads .241
Combining ?exural shear and bending stresses 244
Acting eccentrically about axial loads 247
Putting a Twist on Combined Stresses of Torsion and Shear 249
Chapter 16: When Push Comes to Shove:
Dealing with Deformations .251
Covering Deformation Calculation Basics 252
Defning stiffness 252
Making some key assumptions 253
Addressing Displacement of Axial Members .253
Computing axial deformations .254
Determining relative displacements 255
Handling non-prismatic sections under axial load .258
Discovering De?ections of Flexural Members .259
Setting up ?exural assumptions .259
Defning the elastic curve for displacements .260
Integrating the load distribution to solve for
beam displacements 265Table of Contents xiii
Angling for a Twist Angle 268
Measuring the angle of twist in prismatic shafts .269
Measuring the angle of twist in compound torsion problems 270
Chapter 17: Showing Determination When Dealing
with Indeterminate Structures .273
Tackling Indeterminate Structures 273
Categorizing indeterminate structures .274
Clarifying assumptions for indeterminate methods 275
Withdrawing Support: Creating Multiple Redundant Systems 275
Axial bars with indeterminate supports .276
Systems of axial members 278
Flexural members of multiple supports 281
Torsion of shafts with indeterminate supports .285
Dealing with Multiple Materials .287
Axial bars of multiple materials .287
Flexure of multiple materials 290
Torsion of multiple materials .294
Using Rigid Behavior to Develop Compatibility 296
Rigid bar problems 296
Rigid end cap problems for axial and torsion cases .299
Chapter 18: Buckling Up for Compression Members 301
Getting Acquainted with Columns .301
Considering column types 302
Calculating a column’s slenderness ratio .302
Classifying columns with slenderness ratios .304
Determining the Strength of Short Columns 304
Buckling Under Pressure: Analyzing Long, Slender Columns 305
Determining column capacity 305
Computing the elastic buckling load .306
Computing elastic buckling stress .308
Incorporating support reactions into buckling calculations .308
Working with Intermediate Columns 310
Incorporating Bending Effects .311
Chapter 19: Designing for Required Section Properties .313
Structural Adequacy: Adhering to Formal Guidelines
and Design Codes .314
Exploring Principles of the Design Process .315
Explaining member strength and design loads 316
Creating a design criteria 317
Developing a Design Procedure .318
Outlining a basic design procedure .318
Determining design requirements from modes of failure .319
Designing Axial Members .320
Calculating for simple tension members 321
Guessing a column classifcation for compression loads .321xiv Mechanics of Materials For Dummies
Designing Flexural Members 323
Planning for bending moments with the elastic
section modulus .324
Accounting for ?exural shear .327
Designing for Torsion and Power 328
Interacting with Interaction Equations .329
Chapter 20: Introducing Energy Methods 331
Obeying the Law of Conservation of Energy 332
Working with Internal and External Energy .333
Finding the internal strain energy .334
Setting the internal strain energy equal to
the external work energy 337
Brace Yourself: Figuring Stresses and Displacements from Impact 339
Determining impact from kinetic energy 339
Determining energy relationships through
vertical impact factors 341
Part V: The Part of Tens 343
Chapter 21: Ten Mechanics of Materials Pitfalls to Avoid 345
Failing to Watch Your Units .345
Not Determining Internal Forces First 345
Choosing the Wrong Section Property .346
Forgetting to Check for Symmetry in Bending Members .346
Carelessly Combining Stresses and Strains .346
Ignoring Generalized Hooke’s Law in Three Dimensions .347
Classifying Columns Incorrectly 347
Overlooking that Principal Normal Stresses Have No Shear .347
Neglecting to Test the Principal Angle after You Calculate It 348
Falling Victim to Tricky Issues with Mohr’s Circle 348
Chapter 22: Ten Tips to Solving Mechanics of Materials Problems .349
Do Your Statics 349
Expose Internal Forces .350
Identify How the Object Can Break .350
Compute Appropriate Section Properties 351
Sketch Combined Stress Elements 351
Transform Those Stresses! .352
Have Your Material Properties Handy 352
Apply Factors of Safety and Local Code Requirements 353
Compute Strains and Deformations for Your Stress Elements 353
Design for De?ections .353
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