Civil Engineering Reference
In-Depth Information
2.3 The eigenvalue equation
Equation (2.17) is sometimes integrated directly (Chapter 11) but is also the starting point
for derivation of the eigenvalues or natural frequencies of single elements or meshes of
elements.
Suppose the elastic rod element is undergoing free harmonic motion. Then all nodal
displacements will be harmonic, of the form,
{
u
} = {
a
}
sin
(ωt
+
ψ)
(2.18)
where
are amplitudes of the motion,
ω
its frequency and
ψ
its phase shift. When (2.18)
is substituted in (2.17), the equation
{
a
}
2
[
m
m
]
[
k
m
]
{
a
} −
ω
{
a
} = {
0
}
(2.19)
is obtained, which can easily be rearranged as a standard eigenvalue equation. Chapter 10
describes solution of equations of this type.
2.4 Beam element
2.4.1 Beam element stiffness matrix
As a second one-dimensional solid element, consider the slender beam in Figure 2.2. The
end nodes 1 and 2 are subjected to shear forces and moments which result in translations
and rotations. Each node, therefore, has 2 “degrees of freedom”.
The element shown in Figure 2.2 has length
L
, flexural rigidity
EI
, and carries a uniform
transverse load of
q
(units of force/length). The well known equilibrium equation for this
system is given by,
EI
d
4
w
d
x
=
q
(2.20)
4
distributed
load
beam flexural stiffness
EI
1
2
q
loads
a)
m
2
m
1
x
d
x
f
z
1
f
z
2
L
b)
displacements
q
1
q
2
w
1
w
2
Figure 2.2 Slender beam element
