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 18. — Fixed at Both Ends, Load at Center
Maximum deflection, at load, EI Wl 192 3
Between each end and load, s Z W l x 2 4 1 = − a k
Stress at ends, Z Wl 8 Stress at load, 8 − These are the maximum stresses and are equal and opposite. Stress is zero at x = 1 ∕ 4 l Z Wl
EI Wx l x 48 3 4 2 − ^ h
y
=
Wl 8
Wl 8
W
x
x
1 2
1 2
l
W 2
W 2
Case 19. — Fixed at Both Ends, Load at any Point
For segment of length a , s Zl Wb al x l a 2 3 2 = − + ^ h 6 @ For segment of length b , s Zl Wa bl v l b 2 3 2 = − + ^ h 6 @
Stress at end next to
For segment of length a , y EIl Wx b a l x l a x 6 2 3 2 2 = − + − ^ h ^ h 6 @ For segment of length b , y EIl Wv a b l v l b v 6 2 3 2 2 = − + − ^ h ^ h 6 @
EIl Wa b 3 3
3 3
Deflection at load,
Zl Wab 2
2
segment of length a ,
Let b be the length of the longer segment and a of the shorter one. The maximum deflection is in the longer segment, at v l b bl 2 2 = + and is EI l b Wa b 3 2 2 2 2 3 + ^ h
Stress at end next to Zl Wa b 2 2 Maximum stress is at end next to shorter segment. Stress is zero at x l a al 2 = + and v l b bl 2 = + Greatest negative stress, at load, Zl Wa b 2 3 2 2 − segment of length b ,
Wab 2 l 2
Wa 2 b l 2
W
x
v
a
b
l
Wb 2 l 3
Wa
2
( l + 2 a )
( l + 2 b )
l 3
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