Machinery's Handbook, 31st Edition
268 Rectangular and Round Solid Beams In Table 1, if there are several kinds of loads, as, for instance, a uniform load and a load at any point, or separate loads at different points, the total stress and the total deflection at any point is found by adding together the various stresses or deflections at the point considered due to each load acting by itself. If the stress or deflection due to any one of the loads is negative, it must be subtracted instead of added. Table 2a and Table 2b give expressions for determining dimensions of rectangular and round beams in terms of beam stresses and load. Table 2a. Rectangular Solid Beams
Stress in Extreme Fibers, f lb/in 2 (N/mm 2 )
Breadth of Beam, b inch (mm)
Total Load, W lb (N)
Beam Height, h inch (mm)
Beam Length, l inch (mm)
Style of Loading and Support
Beam fixed at one end, loaded at the other
h
2
2
fh lW b 6 2 =
bh lW f 6 2 =
bf lW h 6 =
W bfh 6
l bfh 6
W
l
=
=
l
Beam fixed at one end, uniformly loaded
h
2
2
fh lW b 3 2 =
bh lW f 3 2 =
bf lW h 3 =
W bfh 3
l bfh 3
W
l
=
=
l
Beam supported at both ends, single load in middle
h
W bfh 3 2 2
l bfh 3 2 2
3
2 3
bf lW h 2 3 =
fh lW b 2 =
bh lW f 2 =
W
l
=
=
2
l
Beam supported at both ends, uniformly loaded
h
l bfh 3 4 2
W bfh 3 4 2
3
4 3
bf lW h 4 3 =
fh lW b 2 =
bh lW f 2 =
W
l
=
=
4
l
Beam supported at both ends, single unsymmetrical load
a
c
h
ac bh fl 6 2
bh l Wac f 6 2 =
fh l Wac b 6 2 =
bfl Wac h 6 =
a + c = l
W
=
l
Beam supported at both ends, two symmetrical loads
a a h
l , any length W bh f a 3 2 =
a bh f 3 2
bh Wa f 3 2 =
fh Wa b 3 2 =
bf Wa h 3 =
W
=
l
Deflection of Beam Uniformly Loaded for Part of Its Length.— In the following for mulas, lengths are in inches, weights in pounds. W = total load; L = total length between supports; E = modulus of elasticity; I = moment of inertia of beam section; a = fraction of length of beam at each end, that is not loaded = b ÷ L ; and f = deflection. f EI a WL a a 384 12 524 16 3 2 4 = − − + ^ ^ h h The expression for maximum bending moment is: M max = 1 ∕ 8 WL (1 + 2 a ).
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