Stresses Produced by Shocks Table 1. Stresses Produced in Beams by Shocks Machinery's Handbook, 31st Edition
279
Method of Support and Point Struck by Falling Body
Fiber (Unit) Stress p Produced by Weight Q Dropped Through a Distance h
Approximate Value of p
Supported at both ends; struck in center. Fixed at one end; struck at the other. Fixed at both ends; struck in center.
QhE 6
p I QaL =
96
hEI
p a LI =
+ + c
m
4 1 1
QL
3
QhE 6
p I QaL =
QL hEI 1 1 6 3 + + c
p a LI =
m
QhE 6
p I QaL =
384
hEI
p a LI =
+ + c
m
8 1 1
QL
3
I = moment of inertia of section; a = distance of extreme fiber from neutral axis; L = length of beam; E = modulus of elasticity. If metric SI units are used, p is in newtons per square millimeter; Q is in newtons; E = modulus of elasticity in N/mm 2 ; I = moment of inertia of section in mm 4 ; and h , a , and L in mm. Note: If Q is given in kilograms, the value referred to is mass. The weight Q of a mass M kilograms is Mg newtons, where g = approximately 9.81 m/s 2 . Examples of How Formulas for Stresses Produced by Shocks are Derived: The general formula from which specific formulas for shock stresses in beams, springs, and other machine and structural members are derived is: (1) In this formula, p = stress in psi due to shock caused by impact of a moving load; p s = stress in psi resulting when moving load is applied statically; h = distance in inches that load falls before striking beam, spring, or other member; y = deflection in inches resulting from static load. As an example of how Formula (1) may be used to obtain a formula for a specific ap plication, suppose that the load W shown applied to the beam in Case 2 on page 257 were dropped on the beam from a height of h inches instead of being gradually ap- plied (static loading). The maximum stress p s due to load W for Case 2 is given as Wl ÷ 4Z and the maximum deflection y is given as Wl 3 ÷ 48 EI . Substituting these values in Formula (1), (2) If in Formula (2) the letter Q is used in place of W and if Z , the section modulus, is re- placed by its equivalent, I ÷ distance a from neutral axis to extreme fiber of beam, then Formula (2) becomes the first formula given in the accompanying Table 1, Stresses Pro- duced in Beams by Shocks . Stresses in Helical Springs Produced by Shocks.— A load suddenly applied on a spring will produce the same deflection, and, therefore, also the same unit stress, as a static load twice as great. When the load drops from a height h , the stresses are as given in the accom panying Table 2. The approximate values are applicable when the deflection is small as compared with the height h . The formulas show that the fiber stress for a given shock will be greater in a spring made from a square bar than in one made from a round bar, if the diameter of coil is the same and the side of the square bar equals the diameter of the round p p y h 1 1 2 s = + + c m p Z Wl = Wl EI h Z Wl Wl hEI 4 1 1 48 2 4 1 1 96 3 3 ' + + = + + c c m m
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