Civil Engineering Reference
In-Depth Information
flange bending moments
M
f
=
EI
f
d
2
v
f
d
2
φ
d
x
2
d
x
2
=
EI
z
d
f
2
2
shown in Figure 10.24b. This pair of equal and opposite flange moments
M
f
,
spaced
d
f
apart, form the bimoment
B
=
d
f
M
f
.
(10.46)
The bending stresses induced by the bimoment are the warping normal stresses
2
y
d
2
φ
σ
w
=∓
M
f
y
I
f
=∓
E
d
f
d
x
2
shown in Figure 10.24b.These can also be obtained from equation 10.37.
When the flange moments
M
f
vary along the member, they induce equal and
opposite flange shears
d
3
φ
d
x
3
,
V
f
=−
d
M
f
d
x
=−
EI
z
d
f
2
2
as shown in Figure 10.24b.The warping shear stresses
1.5
−
6
y
2
b
f
τ
w
=±
V
f
b
f
t
f
(inwhich
b
f
istheflangewidth)whicharestaticallyequivalenttotheseshearforces
are also shown in Figure 10.24b.They can also be obtained from equation 10.42.
Theequalandoppositeflangeshears
V
f
arestaticallyequivalenttothewarping
torque
EI
z
d
f
4
d
3
φ
d
x
3
.
T
w
=
V
f
d
f
=−
Ifthisiscomparedwithequation10.44,itcanbededucedthatthewarpingsection
constant for the I-section is
I
z
d
f
4
.
I
w
=
(10.47)
This result can also be obtained from equation 10.40.
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