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 6. — Both Ends Overhanging Supports Unsymmetrically, Uniform Load For overhanging end of length c , s ZL W c u 2 2 = − ^ h Between supports, s ZL W c l l x d l x x l x 2 2 2 = − + − − ^ a h k & & For overhanging end of length d , s ZL W d w 2 2 = − ^ h Stress at support next to end of length c , ZL Wc 2 2 Critical stress between supports is at x l l c d x 2 2 2 2 1 = + − = and is ZL W c x 2 2 1 2 − ^ h Stress at support next to end of length d , ZL Wd 2 2
Deflection at end c ,
For overhanging end of length c , y EIL Wu l d c c u u c u l 24 2 2 6 4 2 2 2 2 3 = + + − − − ^ ^ h h 6 @ Between supports, y EIL Wx l x x l x l d c l d x c l x 24 2 2 2 2 2 2 2 = − − + − + − + − ^ ^ ^ ^ h h h h 6 @ " , For overhanging end of length d , y EIL Ww l c d d w w d w l 24 2 2 6 4 2 2 2 2 3 = + + − − − ^ ^ h h 6 @
EIL Wc l d c c l 24 2 2 3 2 2 3 3 + + − ^ h 6 @
Total Load W
Deflection at end d ,
EIL Wd l c d d l 24 2 2 3 2 2 3 3 + + − ^ h 6 @ This case is so complicated that convenient general expressions for the critical deflections between supports cannot be obtained.
u
w
x
c
d
l
L
W 2 l
W 2 l
( l – d + c )
( l + d – c )
If cross section is constant, the greatest of these three is the maximum stress. If x 1 > c , the stress is zero at points x c 1 2 2 − on both sides of x = x 1 . Stress at load, Zl Wab − If cross section is constant, this is the maximum stress.
Case 7. — Both Ends Overhanging Supports, Load at any Point Between
Between supports: For segment of length a , s Zl Wbx =− For segment of length b , s Zl Wav =− Beyond supports s = 0.
Between supports, same as Case 3. Deflection at end c , EIl Wabc l b 6 − + ^ h Deflection at end d , EIl Wabd l a 6 − + ^ h
Between supports, same as Case 3. For overhanging end of length c , y EIl Wabu l b 6 =− + ^ h For overhanging end of length d , y EIl Wabw l a 6 =− + ^ h
W
u x
v w
c
a
b
d
l ( a + b = l )
Wb l
Wa l
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