Table 1. Stresses and Deflections in Beams Table 1. (Continued) Stresses and Deflections in Beams
Deflections
Stresses
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 20. — Fixed at Both Ends, Uniform Load
Maximum stress, at ends, Z Wl 12 Stress is zero at x = 0.7887 l and at x = 0.2113 l Greatest negative stress, at center, Z Wl 24 −
Maximum deflection, at center, EI Wl 384 3
2 6 1
s Z Wl =
l x
l x
2
EIl Wx l x 24 2 − ^ h
− + ` j & 0
2
y
=
Total Load W
Wl 12
Wl 12
x
l
W 2
W 2
Case 21. — Continuous Beam, with Two Unequal Spans, Unequal, Uniform Loads
Between R 1 and R , s Z l x l 2 1 1 = − − ^ ( Between R 2 and R , s Z l u l 2 2 = − ^ (
Stress at support R ,
Between R 1 and R ,
This case is so complicated that convenient general expressions for the critical deflections cannot be obtained.
y EI x l x l x R W l W l x 24 2 4 1 1 1 1 1 1 1 2 = − − − − − ^ ^ ^ ^ h h h h " " Between R 2 and R , y EI u l u l u R W l W l u 24 2 4 2 2 2 2 2 2 2 2 = − − − − − ^ ^ ^ ^ h h h h " "
l x W h
Z l l W l 8 1 2 1 1 2 + + ^ h Greatest stress in the first x W l W R 1 1 1 1 = − ^ h and is ZW R l 2 1 W l 2 2 2 1 2 1 − Greatest stress in the u W l W R 2 2 2 2 = − ^ h
1
R
2
−
1
1
Total Load W 1
Total Load W 2
span is at
l u W 2 − h
R
2
R
2
−
R 1
R 2
2
2
x u
l 1
l 2
8 l 1 ( l 1 + l 2 ) l 1 W 1 (3 l 1 + 4 l 2 ) – W 2 l 2 2 2 W 1 + W 2 + 1 8
8 l 2 ( l 1 + l 2 ) l 2 W 2 (3 l 2 + 4 l 1 ) – W 1 l 2 1
W 1 l 1
l 1 W 2 l 2
l 2
+
second span is at
ZW R l 2 2 2 2 2
and is,
−
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