Table 1. 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 1. — Supported at Both Ends, Uniform Load
Stress at center, Z Wl 8 −
Maximum deflection, at center, EI Wl 384 5 3
Zl W x l x
Total Load W
EIl Wx l x l x l x 24 2 − + − ^ ^ h 6
s
y
− ^ h
2 =−
@
h
=
W 2
W
x
l
2
If cross section is constant, this is the maximum stress.
W = wl (where w = load per unit length)
Case 2. — Supported at Both Ends, Load at Center
Between each support and load, y EI Wx l x 48 3 4 2 2 = − ^ h
Stress at center, Z Wl 4 −
Maximum deflection, at load, EI Wl 48 3
Between each support and load, s Z Wx 2 =−
W
x
x
W 2
W
1 2
1 2
2
If cross section is constant, this is the maximum stress.
l
Case 3. — Supported at Both Ends, Load at any Point
For segment of length a , s Zl Wbx =− For segment of length b , s Zl Wav =−
Stress at load, Zl Wab − If cross section is constant, this is the maximum stress.
Deflection at load, EIl Wa b 3 2 2
For segment of length a ,
W
y EIl Wbx l x b 6 2 2 2 = − − ^ h
x
x
a
b
For segment of length b ,
Let a be the length of the shorter segment and b of the longer one. The maximum deflection
Wb l
Wa
l
y EIl Wav l v a 6 2 2 2 = − − ^ h
l
a + b = l
1 3
Wav 3
EIl
is in the longer segment, at v b b 3 1 3 2 = + = a v
1
Made with FlippingBook - Share PDF online