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Equation Sheets for ME 381Principles of Design Spring 2015
Chapter 3.9 Uniformly Distributed Stresses
Normal Stress:
3-22
Shear Stress: 3-23Chapter 3.10 Normal Stresses for Beams in
Bending
Bending stress: 3-24Second-area moment about the z-axis:
3-25Maximum magnitude of the bending stress: 3-26a)Two plane
bending: 3-27Chapter 3.11 Shear Stresses for Beams in Bending
Transverse shear stress: 3-31Maximum transverse shear stress:
3-34
Chapter 3.12 Torsion
Angle of twist for a solid round bar 3-35Shear stress for a
solid round bar
3-36
Max shear stress for a solid round bar 3-37Polar second moment
of area for a solid round section: 3-38Polar second moment of area
for a hollow round section: 4 4 3-39
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Max shearing stress in a rectangular bx csection bar: 3 .8/
3-40Horse power
6: T(lbf*in), n(rpm), F(lbf), V(ft/min) 3-42
SI Units: 3-43Chapter 3.13 Stress Concentrations
Geometric stress-concentration factor: 3-48Net tensile force for
a thin plate in tension or compression: Nominal stress for a thin
plate in tension or compression: Chapter 3.14 Stresses in
Pressurized Cylinders
Tangential stress for a cylinder: / 3-49Radial stress for a
cylinder: +/ 3-49Tangential stress for a cylinder, 0: 1 3-50Radial
stress for a cylinder, 0: 1 3-50Longitudinal stress for a cylinder,
0: 3-51Average tangential stress for thin-walled cylindrical
pressure vessel:
3-52
Maximum tangential stress for thin-walled cylindrical pressure
vessel: + 3-53Longitudinal stress for thin-walled cylindrical
pressure vessel: 4 3-54
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Chapter 3.15 Stresses in Rotating Rings
Tangential stress for a rotating ring: +8 ++ 3-55Radial stress
for a rotating ring: +8 3-55Chapter 3.16 Press and Shrink Fits
Interference contact pressure: [ (+)+ ] 3-56
Interference contact pressure if the two members are of the same
material:
3-57Tangential stress for inner member at transition radius: +
3-58Tangential stress for outer member at transition radius: +
3-59Chapter 3.18 Curved Beams in Bending
Location of neutral axis: 3-63Stress distribution: 3-64Critical
stresses at inner and outer surfaces: 3-65Approximation for small
values of e: 3-66Approximation for stress with small values of e:
3-67
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Chapter 3.19 Contact Stresses
Spherical Contact
Radius of the circular contact area:
8
+
+
3-68
Maximum pressure at center of contact area: 3-69Principal
stresses along thez-axis: -[1 |/| 1 +] 3-70 + 3-71Maximum shear
stress:
=
3-72
Cylindrical Contact
Half-width of the rectangular contact area: +
+ 3-73
Maximum pressure: 3-74Stress state along thez-axis:
21 3-75 ++ 2
3-76 + 3-77
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Chapter 4.17 Shock and Impact for Suddenly Applied Loading
Equation for maximum deflection of spring: 2 2 0 4-57(a)Maximum
deflection of spring: 1 4-58Maximum force acting on the spring: 1
4-59
Chapter 5.4 Maximum-Shear-Stress Theory for Ductile
Materials
Maximum shear stress theory predicts yielding when: 5-1Yield
strength in shear: 0.5 5-2Maximum shear stress modified to include
factor of safety: 5-3Yield condition for Case 1: 0 5-4Yield
condition for Case 2: 0 5-5Yield condition for Case 3: 0 5-6
Chapter 5.5 Distortion-Energy Theory for Ductile Materials
Average stress: ++ (a)Yield is predicted if
++
(5-10)
Von Mises stress: ++ (5-12)Von Mises stress for plane stress:
(5-13)
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Von Mises stress usingxyzcomponents of three-dimensional
stress:
+++6 + + (5-14)And for plane stress:
3 (5-15)
Von Mises stress modified to include factor of safety:
(5-19)Shear yields strength predicted by the distortion-energy
theory: 0.577 (5-21)Chapter 5.6 Coulomb-Mohr Theory for Ductile
Materials
Coulomb-Mohr theory predicts the following failure boundary:
1 (5-22)Modified to include a safety factor:
(5-26)Yield condition for Case 1: 0 (5-23)Yield condition for
Case 2: 0 1 (5-24)Yield condition for Case 3: 0 (5-25)Shear yields
strength predicted by the Coulomb-Mohr theory: + (5-27)Chapter 5.8
MaximumNormal-Stress Theory for Brittle Materials
Maximum-Normal-Stress theory predicts the following failure
boundary: or (5-28)For plane stress:
or
(5-29)
Two sets of design equations: / or / (5-30)
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Chapter 5.9 Modifications of the Mohr Theory for Brittle
Materials
Brittle-Coulomb-Mohr
/
0 (5-31a)
0 (5-31b) / 0 (5-31c)Modified Mohr
/ 0 (5-32a)
0 and 1 0 and > 1 (5-32b) / 0 (5-32c)Chapter 5.12
Introduction to Fracture Mechanics
Stress field on a dx dyelement in the vicinity of the crack
tip:
1 (5-34a) 1 (5-34b) (5-34c)Plane stress: 0 Plane strain:
(5-34d)Stress near the tip: = (a)Stress intensity factor (mode I):
(5-37)Factor of safety: / (5-38)
