Engineering Statics with MATLAB

Engineering Statics with MATLAB
Lester W. Schmerr Jr.
Contents
Preface .ix

1. Introduction to Statics and the Concept of Force 1
   1.1 About This Book 1
   1.2 Quantities and Units . 2
   1.3 Numerical Calculations 4
   1.4 Forces and Vectors . 5
   1.5 Scalars, Vectors, and Matrices . 10
   1.6 Unit Vectors and Cartesian (Rectangular) Components 12
   1.7 Oblique Components 18
   1.8 The Dot Product and Direction Cosines 24
   1.8.1 The Dot Product . 24
   1.8.2 Direction Cosines . 28
   1.9 Position Vectors 31
   1.10 Problems 34
   1.10.1 Review Problems 41
2. Moments and Couples 44
   2.1 Moment of a Force about a Point . 44
   2.1.1 The Vector Cross Product . 47
   2.2 Couples 57
   2.3 Moment of a Force about a Line 61
   2.4 Problems 68
   2.4.1 Review Problems 73
3. Equivalent Systems and Resultants 76
   3.1 Resultants of Two-Dimensional Force Systems . 78
   3.2 Resultants of Three-Dimensional Force Systems . 81
   3.3 Reduction to a Wrench . 84
   3.4 Resultants and the Center of Gravity 87
   3.5 Resultants of Distributed Forces . 92
   3.6 Problems 95
   3.6.1 Review Problems 100
4. Equilibrium . 103
   4.1 Free Body Diagrams 104
   4.2 Reaction Forces at Supports . 110
   v4.3 Solving Equilibrium Problems – I 116
   4.4 Alternate Systems of Equilibrium Equations 135
   4.5 Problems 138
   4.5.1 Review Problems 146
5. Trusses 149
   5.1 The Method of Pins . 151
   5.1.1 Summary 157
   5.2 Zero-Force Members . 157
   5.3 The Method of Sections 159
   5.4 Space Trusses 161
   5.5 Problems 165
   5.5.1 Review Problems 171
6. Frames and Machines . 173
   6.1 Solving Equilibrium Problems – II . 173
   6.2 Problems 196
   6.2.1 Review Problems 201
7. Centroids 204
   7.1 Centroids of Volumes, Areas, and Lines . 204
   7.1.1 Integration with Strips 210
   7.2 Composite Areas 215
   7.3 Distributed Line Loads . 218
   7.4 Problems 220
   7.4.1 Review Problems 227
   7.5 Tables of Centroids 228
8. Beams 232
   8.1 Internal Forces and Moments 232
   8.2 Singularity Functions 246
   8.3 Problems 256
   8.3.1 Review Problems 259
9. Frictional Forces . 261
   9.1 Slipping and Tipping 261
   9.2 Flat Belts or Ropes in Contact with Rough Surfaces 281
   9.3 Wedges . 286
   9.4 Problems 287
   9.4.1 Review Problems 291
10. Statically Indeterminate Structures . 294
    10.1 Solving a Statically Indeterminate Problem 295
    vi Contents10.2 A Matrix-Vector Force-Based Method . 299
    10.3 Steps in the Force-Based Method . 309
    10.4 Examples of the Force-Based Method 311
    10.5 Problems 332
    10.5.1 Review Problems 336
11. Area Moments and Mass Moments of Inertia 339
    11.1 Area Moments 340
    11.2 Polar Area Moment and the Radius of Gyration 347
    11.2.1 Polar Area Moment . 347
    11.2.2 Radius of Gyration . 347
    11.3 Composite Areas 348
    11.4 Rotation of Coordinates 350
    11.5 Principal Area Moments and Principal Axes . 351
    11.6 Area Moment Matrices . 356
    11.7 Mass Moments of Inertia 363
    11.7.1 Parallel Axis Theorem . 365
    11.7.2 Radius of Gyration . 367
    11.8 Problems 367
    11.8.1 Review Problems 371
    11.9 Tables of Mass Moments of Inertia and Area
    Moments 372
    Appendices . 375
    A MATLAB® Overview . 375
    A.1 Constants, Operations, and Built-in Functions . 375
    A.2 Vectors 377
    A.3 Matrices 382
    A.4 Solutions of Simultaneous Linear Equations . 388
    A.4.1 Numerical Solution 389
    A.4.2 Symbolic Solution in Terms of Numerical
    Variables 390
    A.4.3 Symbolic Solution in Terms of Symbolic
    Variables 391
    A.5 Element-by-Element Operations, Plotting, and
    Logical Vectors 397
    A.5.1 Element-by-Element Operations 397
    A.5.2 Plotting . 398
    A.5.3 Logical Vectors . 401
    A.5.4 Constants eps and inf 402
    A.6 Functions and Scripts . 403
    A.7 Anonymous Functions . 405
    Contents viiA.8 Symbolic Calculations 406
    A.8.1 Symbolic Integration 406
    A.8.2 Symbolic Differentiation . 407
    A.8.3 Substitution . 408
    B MATLAB® Files . 409
    Problem Answers 411
    Index 417
    Index
    alternate equilibrium equations, 135
    anonymous functions (MATLAB®),
    252, 405
    area moments, 340
    area moment matrix, 356
    beams, 232
    bending moment, 233
    bending moment diagram, 238
    Cartesian components, 12
    Cartesian coordinates, 12
    center of gravity, 87
    center of mass, 216, 365
    centroid (line, area, volume), 204
    compatibility equations, 297
    compatibility matrix, 300
    composite body (line, area, volume),
    215, 216
    constants, operations, and built-in
    functions (MATLAB®), 375
    coplanar force system, 77
    couple, 57
    cross product, 47
    deformation, 297
    deformation-displacement
    relationship, 301
    determinant, 50, 51
    direction cosines, 5, 24
    direction cosine matrix, 357
    displacements, 294
    distributed line loads, 92, 218
    dot product, 24
    eigenvalue, 359
    eigenvalue problem, 359, 361
    eigenvector, 359
    elastic spring constant (see stiffness)
    element-by-element operations
    (MATLAB®), 397
    equationsToMatrix function
    (MATLAB®), 322, 396
    equilibrium, 103
    equilibrium equations, 103
    equilibrium matrix, 121
    equipollent system (see equivalent
    system)
    equivalent system, 76
    finite element method, 305
    first moments (area, volume), 206, 209
    flat belt or rope, 281
    flexibility, 294, 298
    flexibility matrix, 300
    force-based method, 299
    force system, 76
    frames and machines, 173
    free body diagram, 105
    free vector, 58
    frictional forces, 261
    functions (MATLAB®), 403
    fzero function (MATLAB®), 253
    homogeneous solutions, 301, 394
    internal forces and moments, 104
    International System of Units, 3
    kinetic coefficient of friction, 263
    law of cosines, 6
    law of sines, 6
    linear equations solution
    (MATLAB®), 388
    logical vectors (MATLAB®), 401
    Macaulay bracket function
    (see singularity function)
    magnitude of a vector, 10, 15
    mass moments of inertia matrix,
    339, 363
    matrices (MATLAB®), 382
    417matrix determinant (see determinant)
    matrix multiplication (MATLAB®), 386
    method of pins (joints), 151
    method of sections, 159
    mixed area moment, 340
    moment about a line, 61
    moment about a point, 44
    Navier’s table problem, 311
    null function (MATLAB®), 394
    oblique force components, 18
    parallel axis theorem, 342, 343
    parallel force system, 81
    parallelogram law of addition, 6
    planar truss, 149
    plotting (MATLAB®), 398
    polar area moment, 347
    position vector, 31
    principal area moments, 351
    principal directions (principal axes or
    coordinates), 352
    radius of gyration, 347, 367
    reaction forces, 104, 110
    rectangular components (see Cartesian
    components)
    rectangular coordinates (see Cartesian
    coordinates)
    resultant force, 6
    resultant (of a force system), 76
    right-handed coordinate system, 48, 49
    right-hand-rule, 46
    rotation of coordinates, 350
    scalar product (see dot product)
    scientific notation, 4
    scripts (MATLAB®), 403
    second moments (see area moments)
    self-locking, 272
    shear force, 232
    shear force diagram, 232
    SI System (see International System of
    Units)
    significant digits, 4
    simple truss, 149
    singularity functions, 232, 246
    sliding vector, 52
    slipping, 261
    solution of linear equations (MATLAB®)
    (see linear equations solution
    (MATLAB®))
    solve function (MATLAB®), 390
    space truss, 149
    static coefficient of friction, 262
    statically determinate system, 116,
    149, 150
    statically indeterminate system, 117,
    149, 150, 294
    statically overdetermined, 149, 150
    stiffness, 297
    stiffness matrix, 304
    strip integrals, 210, 344
    symbolic calculations (MATLAB®), 406
    three-force member, 115
    tipping, 261
    transpose (matrix), 383
    trusses, 149
    two-dimensional force system
    (see coplanar force system)
    two-force member, 114
    underdetermined system of equations
    (see statically indeterminate)
    unit vector, 12
    U.S. Customary System of Units, 3
    vector cross product (see cross product)
    vectors (MATLAB®), 377
    wedge, 286
    weightless link (see two-force member)
    wrench, 84
    Young’s modulus, 256, 297
    zero-force member, 157

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