Civil Engineering Reference
In-Depth Information
w
P
P
Δ
FIGURE 8.5
Member subjected to combined compressive axial force and bending.
and the bending moment,
M
, is determined as
5
wL
4
384
+
.
wL
2
8
−
L
2
Δ
P
EI
P
=
M
(8.30)
8
Equation 8.29 indicates that the deflection builds upon itself (a deflection causes more
deflection) and instability may occur due to this
P
-
Δ
effect. Therefore, an iterative
solution to Equation 8.30 is required, which is not efficient for routine design work.
Alternately,forsomeboundaryconditionsandloads,thedifferentialequationforaxial
compressionandflexuremaybesolved.However,forroutinedesignwork,limitations
on combined stresses or semiempirical interaction equations have been developed.
AREMA (2008) uses interaction equations for both the yielding and stability criteria.
8.4.2.1
Differential Equation for Axial Compression and Bending
on a Simply Supported Beam
Consider a member loaded with a general uniform lateral load,
w(z)
, a concentrated
load,
Q
, at a location,
a
, end moments,
M
A
and
M
B
, and compressive axial force,
P
,
as shown in Figure 8.6 . The bending moments at
z
due to loads
M
A
,
M
B
,
Q
, and
w(z)
are combined into a collective bending moment,
M
p
, such that
M
p
(z)
=
M
w(z)
+
M
MA
+
M
MB
+
M
Q(z)
,
(8.31)
where
M
w(z)
is the bending moment at
z
due to
w(z)
,
M
MA
and
M
MB
are the bending
moments due to
M
A
and
M
B
,
M
Q(z)
is the bending moment at
z
due to
Q(z)
.
Q
M
A
M
B
w
(
z
)
P
P
y
z
a
L
FIGURE 8.6
General loading of combined axial compression and bending member.
