sm7 154

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PROBLEM (*7.154) A rectangular beam with b = 3 in. and h = 5 in. (Fig. P7.154) is subjected to
a plastic moment Mp. If y
σ
= 36 ksi for this beam of ductile material, calculate, for the case in which the
loading has been removed, the residual stresses at the upper and lower faces.
*SOLUTION
3
1(3)(5)
12
I=
y
4
31.25 .in=
yy
I
Mc
σ
=
3
31.25(36 10 )
2.5
=×
450kip in.
=
⋅
3(450) 675 .
2
py
M
fM kip in== = ⋅
Elastic rebound stress
3
max 675 10 (2.5)
'31.25
Mc
I
σ
×
==
54 ksi=
The results are sketched below.
5 in.
z
3 in.
M
P
Loading
=
M
P +
M
P =0
Residual
stresses
y
x
-36 ksi 54 ksi
2.5 in.
-18 ksi
18 ksi
36 ksi -54 ksi
-36 ksi
Unloading
36 ksi
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