Table 1. Stresses and Deflections in Beams Table 1. (Continued) Stresses and Deflections in Beams
Stresses
Deflections
General Formula for Stress at any Point
Stresses at Critical Points General Formula for Deflection at any Point a
Type of Beam
Deflections at Critical Points a
Case 8. — Both Ends Overhanging Supports, Single Overhanging Load
Stress at support adjacent to load, Z Wc If cross section is constant, this is the maximum stress. Stress is zero at other support.
Between load and adjacent support, s Z W c u = − ^ h Between supports, s Zl Wc l x = − ^ h Between unloaded end and adjacent supports, s = 0.
Deflection at load, EI Wc c l 3 2 + ^ h Maximum upward deflection is at x = .42265 l , and is . EI Wcl 1555 2 − Deflection at unloaded end, EI Wcld 6
Between load and adjacent support, y EI Wu cu u cl 6 3 2 2 = − + ^ h Between supports, y EIl Wcx l x l x 6 2 =− − − ^ h^ h Between unloaded end and adjacent support, y EI Wclw 6 =
W
u
x
w
c
d
l
W ( c + l ) l
Wc l
–
Case 9. — Both Ends Overhanging Supports, Symmetrical Overhanging Loads Between each load and adjacent support, s Z W c u = − ^ h Between supports, s Z Wc = Stress at supports and at all points between, Z Wc If cross section is constant, this is the maximum stress.
Between each load and adjacent support, y EI Wu c l u u 6 3 2 = + − ^ h 6 @ Between supports, y EI Wcx l x 2 =− − ^ h
Deflections at loads, EI Deflection at center, EI Wcl 8 2 −
Wc c l 6 2 3 2 + ^ h
W
W
u x
u
The above expressions involve the usual approximations of the theory of flexure, and hold only for small deflections. Exact expressions for deflections of any magnitude are as follows: Between supports the curve is a circle of radius r Wc EI = Deflection at any point x between supports y r l r l x 2 2 2 2 = − − − − a k
c
c
l
W
W
4 1
2 1
1 2 2 − −
Deflection at center, r
l r 4
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