Machinery's Handbook, 31st Edition
256
Beams BEAMS Beam Calculations
Reaction at the Supports.— When a beam is loaded by vertical loads or forces, the sum of the reactions at the supports equals the sum of the loads. In a simple beam, when the loads are symmetrically placed with reference to the supports, or when the load is uniformly distributed, the reaction at each end will equal one-half the sum of the loads. When the loads are not symmetrically placed, the reaction at each support may be ascertained from the fact that the algebraic sum of the moments must equal zero. In the accompanying illustration, if moments are taken about the support to the left, then: R 2 3 40 − 8000 3 10 − 10,000 3 16 − 20,000 3 20 = 0; R 2 = 16,000 pounds. In the same way, moments taken about the support at the right give R 1 = 22,000 pounds.
10 ’
6 ’ 4 ’
40 ’
R 1
R 2
The sum of the reactions equals 38,000 pounds, which is also the sum of the loads. If part of the load is uniformly distributed over the beam, this part is first equally divided between the two supports, or the uniform load may be considered as concentrated at its center of gravity. If metric SI units are used for the calculations, distances may be expressed in meters or millimeters, provided the treatment is consistent, and loads in newtons. Note: If the load is given in kilograms, the value referred to is the mass. A mass of M kilograms has a weight (applies a force) of Mg newtons, where g = approximately 9.81 m/s 2 . Stresses and Deflections in Beams.— On the following pages Table 1 gives an extensive list of formulas for stresses and deflections in beams, shafts, etc. It is assumed that all the dimensions are in inches, all loads in pounds, and all stresses in pounds per square inch. The formulas are also valid using metric SI units, with all dimensions in millimeters, all loads in newtons, and stresses and moduli in newtons per millimeter 2 (N/mm 2 ). Note: A load due to the weight of a mass of M kilograms is Mg newtons, where g = approximately 9.81 m/s 2 . In the tables: E = modulus of elasticity of the material I = moment of inertia of the cross section of the beam Z = section modulus of the cross section of the beam = I ÷ distance from neutral axis to extreme fiber W = load on beam s = stress in extreme fiber, or maximum stress in the cross section considered, due to load W . A positive value of s denotes tension in the upper fibers and com pression in the lower ones (as in a cantilever). A negative value of s denotes the reverse (as in a beam supported at the ends). The greatest safe load is that value of W which causes a maximum stress equal to, but not exceeding, the greatest safe value of s y = deflection measured from the position occupied if the load causing the deflec tion were removed. A positive value of y denotes deflection below this position; a negative value, deflection upward u, v, w, x = variable distances along the beam from a given support to any point
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