SP - 5

Problem :
Residual stresses in beams
(Strength of Materials - II, Midterm Exam-42-5)
1. Elasto-plastic beam in pure bending
A rectangular box beam with height h =12in. and width b =8 in. has a constant wall thickness
t =0.50 in. It is made of structural steel with σY =36ksi and E =30× 10 3 ksi. Calculate numerical
values for the yield moment MY , the plastic moment MP , and the residual stress distribution after
unloading the plastic moment.
Solution :
The centroidal principal moment of inertia of the beam section is
The maximum elastic moment is :
Izz =
8 × 123 7 × 113
−
12 12
Izz = 375. 58 in 4
Onset of yielding
σY = MY × ¡ ¢
h
2
Izz
36 × 10 3 = MY × (6)
375. 58
MY = 2254 kip-in
Since F1 = F4 and F2 = F3 the plastic moment is
Dr. M. Kemal Apalak 1

Mp = σY
2. Fully plastic case
X (Ai × ¯yi) =F1 × d1 + F2 × d2
Mp = 2× 36 × 10 3 × (0.5 × 8) × (6 − 0.25) + 2 × 2 × 36 × 10 3 ×
Mp = 2 745 kip-in
The maximum reverse stress is
σ 0 = Mp × ¡ h
2
Izz
¢
= 2 745 × 103 × 6
, σ
375. 58
0 =43.852 ksi
Residual stress distribution
The residual stresses at some locations are
σ R A = σY + σ 0 A = −36 + 43.852 = 7. 852 ksi
σ R B = σY + σ 0 B = −36 + 43.852 × 11
σ
=4. 197 7 ksi
12 R C = σY + σ 0 C = −36 + 0 = −36 ksi
The zero residual stress occurs at
7. 852 + 36
=
36
6
,
y
y =4. 925 6 in
µ
0.5 × 11
×
2
11
4
Dr. M. Kemal Apalak 2