Civil Engineering Reference
In-Depth Information
NA
x
M
x
x
y
p
M
y
y
FIGURE 8.1
Biaxial bending of unsymmetrical cross section.
The bending moments,
M
x
and
M
y
, can then be written as
κ
y
xy
d
A
k
x
x
2
d
A)
M
y
=
σ
x
d
A
=−
E(
+
=−
E(
κ
y
I
xy
+ κ
x
I
y
)
,
(8.2)
κ
y
y
2
d
A
k
x
xy
d
A)
M
x
=
σ
y
d
A
=−
E(
+
=−
E(
κ
y
I
x
+ κ
x
I
xy
)
,
(8.3)
where
I
x
is the moment of inertia about the
x
axis and
I
y
is the moment of inertia
about the
y
axis.
Equations 8.2 and 8.3 may be solved simultaneously for
κ
x
and
κ
y
and substituted
into Equation 8.1 to obtain
M
x
I
y
−
M
y
I
x
−
M
y
I
xy
I
x
I
y
−
M
x
I
xy
I
x
I
y
−
σ =
y
+
x
.
(8.4)
I
xy
I
xy
Steel members in railway superstructures subjected to biaxial bending usually have
two axes of symmetry. Therefore,
I
xy
=
0
(8.5)
and Equation 8.4 becomes
M
y
I
y
x
M
x
I
x
y
σ =
+
=
f
b
x
+
f
b
y
≤
F
b
,
(8.6)
where
f
b
x
is the normal stress from bending moment,
M
x
, about the
x
axis,
f
b
y
is the
normal stress from bending moment,
M
y
, about the
y
axis, and
F
b
is the allowable
bending stress.
