sm7 3

PROBLEM (7.3) A steel beam having the modulus elasticity E and allowable normal stress
all
σ
experiences pure bending (Fig. P7.3). Calculate the maximum moment M that may be
applied and the corresponding radius of curvature
ρ
for the two cases:
(a) The cross section is a circle of diameter d.
(b) The cross section is an equilateral triangle of sides b.
Given: d = 12 mm, b = 15 mm, E = 200 GPa, all
σ
= 160 MPa
SOLUTION
(a) 3
32
all
M
d
σπ =
or
3
32
all d
M
σ
π
=
63
(160 10 ) (0.012)
32
M
π ×
=
D
=2c=12 m
m
27.14Nm
=
⋅
Equation (7.4):
61
9
1 160 10 0.133
200 10 (0.006)
all m
Ec
σ
ρ
−
×
== =
×
7.52 m
ρ
=
Alternatively, Eq. (7.6) yields the same result.
(b) 3
3
1 15(12.99)
36 36
Ibh==
12 4
913 10 m
−
=×
all
M
cI
σ
=
or
(2 3)
all
M
Ih
σ
=
612
3
160 10 (913 10 ) 16.87
8.66 10
M
Nm
−
−
××
==
×⋅
Equation (7.9):
61
93
1 160 10 0.0924
200 10 (8.66 10 )
all m
Ec
σ
ρ
−
−
×
== =
××
or
10.82 m
ρ
=
2
=3
ch
h=12.99 mm
B
=15 mm
C
60o
60
o
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