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 13. — Fixed at One End, Supported at the Other, Load at Center
Maximum stress at point of fixture, 16 3 Z Wl Stress is zero at x = 3 ∕ 11 l Greatest negative stress at center, 32 5 − Z Wl Greatest positive stress, at point of fixture, Zl Wab l b 2 2 + ^ h Greatest negative stress, at load, Zl Wa b l a 2 3 3 2 − − ^ h If a < 0.5858 l , the first is the maximum stress. If a = 0.5858 l , the two are equal and are . Z Wl 583 ! If a > 0.5858 l , the second is the maximum stress. Stress is zero at x m n =
Between point of fixture and load, y EI Wx l x 96 9 11 2 = − ^ h Between support and load, y EI Wv l v 96 3 5 2 2 = − ^ h
Maximum deflection is at v = 0.4472 l , and is . EI Wl 10733 3 Deflection at load, 768 7 EI Wl 3
Between point of fixture and load, s Z W l x 16 3 11 = − ^ h Between support and load, s 16 5 =− Z Wv Between point of fixture and load, s Zl Wb n mx 2 3 = − ^ h Between support and load, s Zl Wa v l a 2 3 3 2 =− − ^ h
3 16
W
Wl
x
v
W 5
1 2
1 2
l
16
11 16
W
Case 14. — Fixed at One End, Supported at the Other, Load at any Point
Between point of fixture and load, y EIl Wx b n mx 12 3 3 2 = − ^ h Between support and load, y EIl Wa v l b v l a 12 3 3 3 2 2 2 = − − ^ h 6 @
Deflection at load, EIl Wa b l b 12 3 3 3 2 + ^ h If a < 0.5858 l , maximum
deflection is
m l a l b al n al l b = + + + = + ^ ^ h^ h h
EI Wa b 6 2
l b b
2
+ and located between load and support, at v l l b b 2 = + If a = 0.5858 l , maximum deflection is at load and is . EI Wl 1019 3 If a > 0.5858 l , maximum deflection is Im E l Wbn 3 2 3 3 and located between load and point of fixture, at x m n 2 =
Wab ( l + b ) 2 l 2
W
x
v
a
b
l
Wa 2 (3 l – a ) 2 l 3
a 2 W [2 – (3 l – a )] 2 l 3
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