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Mechanics of Materials/Solid Mechanics/ Strength of Materials/ Mechanics of Deformable Bodies




Solid Mechanics is one of the branches of deformable-body mechanics that focuses on the study of solid objects that can change shape or size as a result of loading or thermal effects.


    Worked out problems with solutions in pdf files


    Allowable Load (Pallow),Factor of Safety (FS), Axial Stresses/Elongation, and Strain


  1. Finding internal stress in an axial member Example
  2. A tension test is performed on a circular cylinder shaped steel specimen as shown below, compute the axial stress in the member (in MPa).

    Tension test in a cylinder/Axial stress
    Figure 21:Tension test on a circular cylinder shaped steel specimen

    AxialStressSolidMechEx1solution.pdf

  3. Computing strain, stress and modulus of elasticity in an axial member Example
  4. Figure 22:

    solution.pdf

  5. Computing average shear strain, average shear strain and relative displacement Example
  6. Figure 23:

    solution.pdf

  7. Finding allowable load Example
  8. Figure 24:

    solution.pdf

  9. Finding Allowable Forces/Stresses using Factor of Safety (FS) Example
  10. Figure 25:

    solution.pdf

  11. Finding axial elongation in an axial structure Example 1
  12. The axial structure depicted below consists of circular steel members of different areas. Compute the following:
    (a) the stresses in each member, and indicate whether the stresses are tensile (T) or compressive (C),
    (b) the axial displacement at B (UB), at C (UC) and at D (UD).

    Axial displacement example 1
    Figure : Axial displacement

    SolidMechAxialEx1solution.pdf

  13. Finding axial stress and elongation in an axial member Example
  14. Figure 26:

    solution.pdf

  15. Finding compressive axial stress and displacement in a steel pipe column Example
  16. Figure 27:

    solution.pdf

  17. Finding internal forces and displacement in a statically indeterminate structure (axial) Example
  18. Figure 28:

    solution.pdf


    Torques, Shear Stress and Relative Rotation in a Shaft


  19. Finding internal torques, maximum shear stress and angle of rotation in a hollow shaft Example (statically determinate)
  20. Figure 29:

    solution.pdf

  21. Finding internal torques, maximum shear stress, and relative rotation in a shaft Example 1 (statically determinate)
  22. Figure 30:

    solution.pdf

  23. Finding internal torques, maximum shear stress, and relative rotation in a shaft Example 2 (statically determinate)
  24. Figure 31:

    solution.pdf

  25. Finding internal torques, maximum shear stress, and relative rotation in a shaft Example 1 (statically indeterminate)
  26. Figure 32:

    solution.pdf

  27. Finding internal torques, maximum shear stress, and relative rotation in a shaft Example 2 (statically indeterminate)
  28. Figure 33:

    solution.pdf


    Internal Forces, Shear Force Diagram (SFD) and Bending Moment Diagram (BMD) Review


  29. Finding internal shear and moment at a point (in a beam)
  30. Figure 34:

    solution.pdf

  31. Deriving shear and moment equation in a simply supported beam with triangular Uniform Distributed Load (UDL)
  32. Figure 35:

    solution.pdf

  33. Drawing shear and moment diagrams Review 1
  34. Draw the shear (V) and moment (M) diagram for this overhang beam with a triangular distributed load, point load and concentrated moment.

    overhang beam with a triangular distributed load, point load and concentrated moment
    Figure 36: Overhang beam with a triangular distributed load, point load and concentrated moment.

    OverhangBeam2VandMsolution.pdf

  35. Strain-displacement analysis for beams
  36. Figure 37:

    solution.pdf

  37. Finding maximum tensile stress, radius of curvature and maximum deflection of a beam
  38. Figure 38:

    solution.pdf


    Normal/Flexural/Bending Stress, Shear Stress and Shear Flow Distribution in a Beam


    Normal and Shear Stress Distribution in a rectangular beam

    Normal and Shear Stress Distribution in a rectangular beam
    Figure 39: Normal and Shear Stress Distribution in a rectangular beam

  39. Finding flexural stresses in beams Example 1
  40. A fixed wide-flange beam has a uniform distributed loading acting throughout half of the span, a point load and concentrated moment. Cross-section dimensions of the beam is given. Compute the maximum flexural tensile and compressive stresses.

    Flexural Stress Example 1-Fixed beam with UDL,concentrated load and moment
    Figure : Fixed I-beam with UDL, concentrated load and moment

    FlexuralStressEx1solution.pdf

  41. Finding flexural stresses in beams Example 2
  42. This simply supported beam is supported by a pin at A , a roller at B, and has a triangular distributed load acting throughout the entire span. Cross-section dimensions of the beam is given. Compute the maximum flexural tensile and compressive stresses.

    Flexural Stress Example 2-Simply supported beam with triangular distibuted loading
    Figure : Simply supported beam with triangular distributed loading

    FlexuralStressEx2solution.pdf

  43. Finding flexural stresses in beams Example 3
  44. This overhang beam is supported by a pin at A , a rocker at B, and has a uniform distributed load acting in between the supports. Cross-section dimensions of the beam is given. Compute:
    (a)the flexural/bending stress at C, D, E and F;
    (b)compute the flexural/bending stress and draw the stress distribution at the left support (A) ,
    (c) from part (b), what are the maximum flexural tensile and compressive stresses at support A
    (d) what are the maximum flexural tensile and compressive stresses that occurs throughout the entire span of the beam?

    Overhang beam with a UDL, point load and concentrated moment
    Figure 39: Overhang beam with a UDL, point load and concentrated moment

    FlexuralStressEx3solution.pdf

  45. Finding maximum compressive and tensile flexural stresses in beams
  46. Figure 40:

    solution.pdf

  47. Design problem: choosing lightest wide flange beam given allowable stress
  48. Figure 41:

    solution.pdf

  49. Finding maximum flexural stress and maximum shear stress in a beam
  50. Figure 42:

    solution.pdf

  51. Finding shear stress at different locations in an inverted C channel shape
  52. Compute the shear stress, τ, at points A, B, C, D and E for the figure shown below.Then, draw the shear stress and shear flow distribution. Also, indicate where the maximum shear stress occurs (V = 25 kN).

    inverted C-Channel
    Figure 43: Inverted C-Channel Section

    InvertedChannelShearStressDistributionsolution.pdf

  53. Finding shear stress at different locations in a C channel shape
  54. Compute the shear stress, τ, at points A, B, C, and D for the figure shown below.Then, draw the shear stress and shear flow distribution. Also, indicate where the maximum shear stress occurs (V = 65 kN).

    C-Channel beam
    Figure 43: C-Channel Section

    ChannelShearStressDistributionsolution.pdf

  55. Finding shear stress at different locations in an unsymmetrical I shape beam and drawing shear stress distribution
  56. Compute the shear stress, τ, at points A, B, C, D and E for the figure shown below.Then, draw the shear stress and shear flow distribution. (V = 50 kN)

    Unsymmetrical Wide Flange Beam
    Figure: Unsymmetrical Wide Flange Beam

    UnsymmetricalWideFlangeBeamShearDistributionsolution.pdf

  57. Finding maximum shear stress in a hollow circle and drawing shear flow distribution
  58. Compute the maximum shear stress, τMax, for the tubular steel beam as shown below. Then, draw the shear flow distribution(q).(V = 75 kN).

    Hollow circular steel beam
    Figure 45:Tubular Steel Beam

    HollowCircleShearStresssolution.pdf

  59. Finding shear stress at different locations in a hollow rectangular shape beam and drawing shear stress distribution
  60. Compute the shear stress, τ, at points A, B, C, D and E for the figure shown below.Then, draw the shear stress and shear flow distribution. (V = 80 kN)

    Hollow Rectangular beam
    Figure: Hollow Rectangular Shaped Beam

    HollowRectangleShearStresssolution.pdf

  61. Finding shear stress at different locations in a T shape beam
  62. Compute the shear stress, τ, at points A, B, C and D for the figure shown below. Then, draw the shear stress and shear flow distribution. Also, indicate where the maximum shear stress occurs (V = 40 kN)

    T-beam
    Figure 43: T beam

    TbeamShearStressDistributionsolution.pdf

  63. Given allowable shear stress in a T shape beam, find maximum load intensity(Wo) and drawing shear flow/shear stress/flexural stress distribution
  64. Figure 46:

    solution.pdf

  65. Finding allowable shear given allowable shear flow in a symmetrical I shape beam and drawing shear flow distribution
  66. Figure 47:

    solution.pdf

  67. Boundary Conditions
  68. Boundary Conditions
    Figure 48:

  69. Continuity Conditions
  70. Continuity Conditions
    Figure 49:

    Deflection, Slope, Curvature Equations (Double Integration Method, Euler Bernoulli Beam Equation), Deflection Shape and Superposition Method


  71. Drawing the deflection shape
  72. Draw the deflection shape and indicate where the maximum deflection will occur for this simply supported beam with Uniformly Distributed Load (UDL) as shown below.

    Simply supported beam with Uniformly Distributed Load (UDL)
    Figure :Simply supported beam with Uniformly Distributed Load (UDL)

    DrawingDeflectionShapeEx1solution.pdf

  73. Finding slope and deflection in a cantilever beam with triangular distributed load
  74. Determine:
    (a)the equation of the slope, Θ(x), and deflection, v(x), and
    (b)the maximum deflection, vMax and draw the deflected shape,
    for the fixed beam with triangular distributed loading as shown below.

    Fixed beam with triangular distributed loading
    Figure :Fixed Beam With Triangular Distributed Loading

    FixedBeamWithTriangularLoadDeflectionandSlopesolution.pdf

  75. Finding slope, deflection equation and maximum deflection in a simply supported beam with triangular distributed load
  76. Determine:
    (a)the equation of the slope, Θ(x), and deflection, v(x), and
    (b)the maximum deflection, vMax, and draw the deflected shape,
    for the simply supported beam with triangular distributed loading as shown below.

    Simply supported beam with triangular distributed loading
    Figure 47: Simply Supported Beam With Triangular Distributed Loading

    SimplySupportedBeamTriangularUDLDeflectionsolution.pdf

  77. Finding deflection equation and reactions in a propped cantilever beam with rectangular distributed load
  78. Using superposition method, determine:
    (a)the reactions,
    (b)the equation of the slope, Θ(x), and deflection, v(x), and
    (c) the location where the maximum deflection occurs (in meters) and the maximum deflection (in mm) if E = 200 GPa and I = 600 x 105 mm4,
    (d) draw the shear, moment diagrams and the deflection shape
    for the propped cantilever beam with rectangular distributed loading as shown below.

    propped cantilever beam with rectangular distributed load
    Figure 48: Propped Cantilever Beam with Rectangular Distributed Load

    ProppedCantileverBeamDeflectionsolution.pdf

  79. Finding deflection equation and reactions in a continuous beam with rectangular distributed load
  80. Using superposition method, determine:
    (a)the reactions,
    (b)the equation of the slope, Θ(x), and deflection, v(x), and
    (c) the location where the maximum deflection occurs (in meters) and the maximum deflection (in mm) if E = 200 GPa and I = 450 x 105 mm4,
    (d) draw the shear, moment diagrams and the deflection shape
    for the continuous beam with rectangular distributed loading as shown below.

    continuous beam with rectangular distributed load
    Figure 49: Continuous Beam with Rectangular Distributed Load

    ContinuousBeamDeflectionsolution.pdf

  81. Finding deflection equation and reactions in a double propped beam with triangular distributed load
  82. Using superposition method, determine:
    (a)the reactions,
    (b)the equation of the slope, Θ(x), and deflection, v(x), and
    (c) the location where the maximum deflection occurs (in meters) and the maximum deflection (in mm) if E = 200 GPa and I = 700 x 105 mm4,
    (d) draw the shear, moment diagrams and the deflection shape
    for the cantilever double propped beam with triangular distributed loading as shown below.

    cantilever double propped beam with triangular distributed load
    Figure 50: cantilever double propped beam with triangular distributed load

    CantileverDoubleProppedBeamDeflectionsolution.pdf

  83. Finding deflection equation and reactions in a fixed-fixed beam with rectangular distributed load
  84. Using superposition method, determine:
    (a)the reactions,
    (b)the equation of the slope, Θ(x), and deflection, v(x), and
    (c) the location where the maximum deflection occurs ,
    (d) draw the shear, moment diagrams and the deflection shape
    for the fixed-fixed beam with rectangular distributed loading as shown below.

    fixed-fixed beam with rectangular distributed load
    Figure 50: Fixed-fixed beam with rectangular distributed load

    Fixed-FixedBeamDeflectionsolution.pdf

  85. Deflections and slopes of beams;Fixed-End Actions page 1
  86. Deflections and slopes of beams;Fixed-End Actions page 1
    Figure 49:

  87. Deflections and slopes of beams;Fixed-End Actions page 2
  88. Deflections and slopes of beams;Fixed-End Actions page 2
    Figure 50:

  89. Deflections and Slopes of Simply-Supported Uniform Beams page 1
  90. Deflections and Slopes of Simply-Supported Uniform Beams page 1
    Figure 51:

  91. Deflections and Slopes of Simply-Supported Uniform Beams page 2
  92. Deflections and Slopes of Simply-Supported Uniform Beams page 2
    Figure 52:

  93. Fixed-End Actions for Uniform Beams
  94. Fixed-End Actions for Uniform Beams
    Figure 53:

  95. Finding slope and deflection using superposition for a fixed beam with a uniform distributed load (UDL), concentrated moment, and a point load
  96. Figure 54:

    solution.pdf

  97. Finding deflection using superposition example 2
  98. Figure 55:

    solution.pdf

  99. Solving statically indeterminate beam (superposition),finding slope and deflection
  100. Figure 56:

    solution.pdf


    Stress Block, Stress Resultants, Mohr's Circle, and Principal Stresses


    Stress block sign convention

    Stress block sign convention
    Figure 57: Stress block sign convention


  101. Finding stress resultants Example 1
  102. For the given stress block below, find the principal stresses, and draw Mohr's Circle.

    Stress block example 1
    Figure 58: Drawing Mohr's Circle for a stress block example 2

    MohrsCircleEx1solution.pdf

  103. Finding stress resultants (stress block) Example 2 and drawing Mohr's Circle
  104. For the given stress block below, find the principal stresses, and draw Mohr's Circle.

    Stress block example 2
    Figure 58: Drawing Mohr's Circle for a stress block example 2

    MohrsCircleEx2solution.pdf

  105. Finding stress resultants and drawing Mohr's circle Example 3
  106. For the given stress block below, find the principal stresses, and draw Mohr's Circle.

    Stress block example 3
    Figure 59: Drawing Mohr's Circle for a stress block example 3

    MohrsCircleEx3solution.pdf

  107. Finding stress resultants and drawing Mohr's circle Example 4
  108. Figure 60:

    solution.pdf


    Axial Stress and Hoop Stress in Pressure Vessels


  109. Pressure vessels Example 1
  110. For the cylindrical thin-walled steel pressure vessel with inner diameter, di = 10 in, and a wall thickness, t = 0.75 in, calculate:
    (a) the axial stress,σa,and,
    (b) the hoop stress, σh.

    Figure 61:Cylindrical thin-walled steel pressure vessel

    PressureVesselEx1solution.pdf

  111. Pressure vessels Example 2

  112. Figure 62:

    solution.pdf

  113. Pressure vessels Example 3
  114. Figure 63:

    solution.pdf


    Column Buckling


  115. Buckling of Columns Example 1
  116. Figure 64:

    solution.pdf

  117. Buckling of Columns Example 2
  118. Figure 65:

    solution.pdf

  119. Buckling of Columns Example 3
  120. Figure 66:

    solution.pdf

  121. Buckling of Columns Example 4
  122. Figure 67:

    solution.pdf

    Videos

  1. Stress-Strain Diagram








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