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 24. — Continuous Beam, with Two Unequal Spans, Unequal Loads at any Point of Each
Between R 1 and W 1 , s Z wr 1 =− Between R and W 1 , s = l Z m l u W a u 1 1 1 1 1 − − ^ h 6 @ Between R and W 2 , s = l Z m l x W a x 1 2 2 2 2 − − ^ h 6 @ Between R 2 and W 2 , s Z vr 2 =−
Stress at load W 1 , Z a r 1 1 − Stress at support R , Z m Stress at load W 2 , Z a r 2 2 − The greatest of these is the maximum stress.
Between R 1 and W 1 , y EI w l w l w r 1 1 = Between R and W 1 , y EIl u W a b l a 6 1 − + − ^ h^ h '
Deflection at load W 1 ,
l W b
EIl a b a b W m l a 6 2 1 1 1 1 1 1 1 1 − + ^ h 6 @
1 1 1 3
1
1 2 ( l 1 + l 2 )
W 1 a 1 b 1 l 1
W 2 a 2 b 2 l 2
m =
( l 1 + a 1 ) +
( l 2 + a 2 )
Deflection at load W 2 ,
W a u m l u l u 6 2 1 1 1 1 1 1 1 1 2 1 1 = + − − − − ^ ^ h^ h h 6 @ Between R and W 2 y EIl x W a b l a W a x m l x l x 6 2 2 2 2 2 2 2 2 2 2 2 2 = + − − − − ^ ^ h^ h h 6 @ Between R 2 and W 2 , y EI v l v l v r l W b 6 2 2 2 2 2 2 3 = − + − ^ h^ h ' 1
EIl a b a b W m l a 6 2 2 2 2 2 2 2 2 2 − + ^ h 6 @ This case is so complicated that convenient general expressions for the maximum deflections cannot be obtained.
W 1
W 2
R
R
R 2
1
w u
x
v
a 1
a 2
b 1
b 2
l 1
l 2
W 1 b 1 – m l 1
W 1 a 1 + m l 1
W 2 a 2 + m l 2
W 2 b 2 – m l 2
+
= r 1
= r
= r 2
a The deflections apply only to cases where the cross section of the beam is constant for its entire length. In the diagrammatical illustrations of the beams and their loading, the values indicated near, but below, the supports are the “reactions” or upward forces at the supports. For Cases 1 to 12, inclusive, the reactions, as well as the formulas for the stresses, are the same whether the beam is of constant or variable cross section. For the other cases, the reactions and the stresses given are for constant cross section beams only. The bending moment at any point in inch-pounds (newton-meters if metric units are used) is s 3 Z and can be found by omitting the divisor Z in the formula for the stress given in the tables. A positive value of the bending moment denotes tension in the upper fibers and compression in the lower ones. A negative value denotes the reverse, The value of W corresponding to a given stress is found by transposition of the formula. For example, in Case 1, the stress at the critical point is s = − Wl ÷ 8 Z . From this formula we find W = − 8 Zs ÷ l . Of course, the negative sign of W may be ignored.
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