1. Chapter Outline
Shigley’s Mechanical Engineering Design

2. Free-Body Diagram Example 3-1
Shigley’s Mechanical Engineering Design

3. Free-Body Diagram Example 3-1
Fig. 3-1
Shigley’s Mechanical Engineering Design

4. Free-Body Diagram Example 3-1
Shigley’s Mechanical Engineering Design

5. Free-Body Diagram Example 3-1
Shigley’s Mechanical Engineering Design

6. Free-Body Diagram Example 3-1
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7. Shear Force and Bending Moments in Beams
Cut beam at any location x1
Internal shear force V and bending moment M must ensure equilibrium
Fig. 3−2
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8. Sign Conventions for Bending and Shear
Fig. 3−3
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9. Distributed Load on Beam
Distributed load q(x) called load intensity
Units of force per unit length
Fig. 3−4
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10. Relationships between Load, Shear, and Bending
The change in shear force from A to B is equal to the area of the loading diagram between xA and xB.
The change in moment from A to B is equal to the area of the shear-force diagram between xA and xB.
Shigley’s Mechanical Engineering Design

11. Shear-Moment Diagrams
Fig. 3−5
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12. Moment Diagrams – Two Planes
Fig. 3−24
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13. Combining Moments from Two Planes
Add moments from two planes as perpendicular vectors
Fig. 3−24
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14. Singularity Functions
A notation useful for integrating across discontinuities
Angle brackets indicate special function to determine whether forces and moments are active
Table 3−1
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15. Example 3-2
Fig. 3-5
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16. Example 3-2
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17. Example 3-2
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18. Example 3-3
Fig. 3-6
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19. Example 3-3
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20. Example 3-3
Fig. 3-6
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21. Normal stress is normal to a surface, designated by s
Tangential shear stress is tangent to a surface, designated by t
Normal stress acting outward on surface is tensile stress
Normal stress acting inward on surface is compressive stress
U.S. Customary units of stress are pounds per square inch (psi)
SI units of stress are newtons per square meter (N/m2)
1 N/m2 = 1 pascal (Pa)
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22. Represents stress at a point Coordinate directions are arbitrary
Stress element
Represents stress at a point
Coordinate directions are arbitrary
Choosing coordinates which result in zero shear stress will produce principal stresses
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23. Two bodies with curved surfaces pressed together
Contact Stresses
Two bodies with curved surfaces pressed together
Point or line contact changes to area contact
Stresses developed are three-dimensional
Called contact stresses or Hertzian stresses
Common examples
Wheel rolling on rail
Mating gear teeth
Rolling bearings
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24. Spherical Contact Stress
Two solid spheres of diameters d1 and d2 are pressed together with force F
Circular area of contact of radius a
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25. Spherical Contact Stress
Pressure distribution is hemispherical
Maximum pressure at the center of contact area
Fig. 3−36
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26. Spherical Contact Stress
Maximum stresses on the z axis
Principal stresses
From Mohr’s circle, maximum shear stress is
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27. Spherical Contact Stress
Plot of three principal stress and maximum shear stress as a function of distance below the contact surface
Note that tmax peaks below the contact surface
Fatigue failure below the surface leads to pitting and spalling
For poisson ratio of 0.30, tmax = 0.3 pmax at depth of z = 0.48a
Fig. 3−37
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28. Cylindrical Contact Stress
Two right circular cylinders with length l and diameters d1 and d2
Area of contact is a narrow rectangle of width 2b and length l
Pressure distribution is elliptical
Half-width b
Maximum pressure
Fig. 3−38
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29. Cylindrical Contact Stress
Maximum stresses on z axis
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30. Cylindrical Contact Stress
Plot of stress components and maximum shear stress as a function of distance below the contact surface
For poisson ratio of 0.30, tmax = 0.3 pmax at depth of z = 0.786b
Fig. 3−39
Shigley’s Mechanical Engineering Design