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 σy↑⇀τyx

σx←τxy

# 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.

## 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). AxialStressSolidMechEx1solution.pdf

3. Computing strain, stress and modulus of elasticity in an axial member Example
4. solution.pdf

5. Computing average shear strain, average shear strain and relative displacement Example
6. solution.pdf

8. solution.pdf

9. Finding Allowable Forces/Stresses using Factor of Safety (FS) Example
10. 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). SolidMechAxialEx1solution.pdf

13. Finding axial stress and elongation in an axial member Example
14. solution.pdf

15. Finding compressive axial stress and displacement in a steel pipe column Example
16. solution.pdf

17. Finding internal forces and displacement in a statically indeterminate structure (axial) Example
18. 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. solution.pdf

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

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

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

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

31. Deriving shear and moment equation in a simply supported beam with triangular Uniform Distributed Load (UDL)
32. 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. OverhangBeam2VandMsolution.pdf

35. Strain-displacement analysis for beams
36. solution.pdf

37. Finding maximum tensile stress, radius of curvature and maximum deflection of a beam
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 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. 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. 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? FlexuralStressEx3solution.pdf

45. Finding maximum compressive and tensile flexural stresses in beams
46. solution.pdf

47. Design problem: choosing lightest wide flange beam given allowable stress
48. solution.pdf

49. Finding maximum flexural stress and maximum shear stress in a beam
50. 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). 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). 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) 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). 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) 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) 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. solution.pdf

65. Finding allowable shear given allowable shear flow in a symmetrical I shape beam and drawing shear flow distribution
66. solution.pdf

67. Boundary Conditions
68. 69. Continuity Conditions
70. ## 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. 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, 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. 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. 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 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. 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 Fixed-FixedBeamDeflectionsolution.pdf

85. Deflections and slopes of beams;Fixed-End Actions page 1
86. 87. Deflections and slopes of beams;Fixed-End Actions page 2
88. 89. Deflections and Slopes of Simply-Supported Uniform Beams page 1
90. 91. Deflections and Slopes of Simply-Supported Uniform Beams page 2
92. 93. Fixed-End Actions for Uniform Beams
94. 95. Finding slope and deflection using superposition for a fixed beam with a uniform distributed load (UDL), concentrated moment, and a point load
96. solution.pdf

97. Finding deflection using superposition example 2
98. solution.pdf

99. Solving statically indeterminate beam (superposition),finding slope and deflection
100. solution.pdf

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

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. 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. 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. MohrsCircleEx3solution.pdf

107. Finding stress resultants and drawing Mohr's circle Example 4
108. 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. PressureVesselEx1solution.pdf

111. Pressure vessels Example 2

112. solution.pdf

113. Pressure vessels Example 3
114. solution.pdf

## Column Buckling

115. Buckling of Columns Example 1
116. solution.pdf

117. Buckling of Columns Example 2
118. solution.pdf

119. Buckling of Columns Example 3
120. solution.pdf

121. Buckling of Columns Example 4
122. solution.pdf

### Videos

1. Stress-Strain Diagram

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