Civil Engineering Reference
In-Depth Information
2.5 Beam with an axial force
If the beam element in Figure 2.2 is subjected to an additional axial force
P
, as shown in
Figure 2.3, a simple modification to (2.20) results in the differential equation
EI
d
4
d
2
w
d
x
w
d
x
±
P
=
q
(2.31)
4
2
where the positive sign corresponds to a compressive axial load and vice versa.
Finite element discretisation and application of Galerkin's method leads to an additional
matrix associated with the axial force contribution,
w
1
θ
1
w
2
θ
2
L
d
N
i
d
x
d
N
j
d
x
∓
P
d
x
(2.32)
0
i,j
=
1
,
2
,
3
,
4
On discretising
w
in space by finite elements as before, the first term in (2.31) clearly
leads again to [
k
m
]. The second term from (2.32) takes the form for compressive
P
,
36 3
L
−
36
3
L
w
1
θ
1
w
2
θ
2
1
30
L
2
2
4
L
−
3
L
−
L
P
(2.33)
36
−
3
L
2
symmetrical
4
L
The matrix is sometimes called the
beam geometric matrix
, since it is a function only
of the length of the beam, given by,
36 3
L
−
36
3
L
k
g
=
1
30
L
2
2
4
L
−
3
L
−
L
(2.34)
36
−
3
L
2
symmetrical
4
L
and the equilibrium equation can be written as:
(
[
k
m
]
−
P
[
k
g
]
)
{
w
} = {
f
}
(2.35)
Buckling of a member can be investigated by solving the eigenvalue problem where
{
, or by increasing the compressive force
P
on the element until large deformations
result or in simple cases, by determinant search. Equations (2.34) and (2.35) represent
an approximation of the approach to modifying the element stiffness involving stability
f
} = {
0
}
Beam flexural stiffness
EI
q
P
P
L
Figure 2.3 Beam with an axial force
