Civil Engineering Reference
In-Depth Information
or in matrix form
.
/
!
α
1
α
2
α
3
α
4
{
ε
}=
[0026
x
]
(17.51)
0
"
which we write as
{
ε
}=
[
C
]
{
α
}
(17.52)
Substituting for
{
α
}
in Eq. (17.52) from Eq. (17.47) we have
[
C
][
A
−
1
]
δ
e
{
ε
}=
{
}
(17.53)
Step five relates the internal stresses in the element to the strain
{
ε
}
and hence, using
{
δ
e
Eq. (17.53), to the nodal displacements
. In our beam-element the stress distribu-
tion at any section depends entirely on the value of the bending moment
M
at that
section. Thus we may represent a 'state of stress'
}
at any section by the bending
moment
M
, which, from simple beam theory, is given by
{
σ
}
EI
∂
2
v
∂
x
2
M
=−
or
{
σ
}=
[
EI
]
{
ε
}
(17.54)
which we write as
{
}=
[
D
]
{
}
σ
ε
(17.55)
The matrix [
D
] in Eq. (17.55) is the 'elasticity' matrix relating 'stress' and 'strain'. In
this case [
D
] consists of a single term, the flexural rigidity
EI
of the beam. Generally,
however, [
D
] is of a higher order. If we now substitute for
in Eq. (17.55) from
Eq. (17.53) we obtain the 'stress' in terms of the nodal displacements, i.e.
{
ε
}
[
D
][
C
][
A
−
1
]
δ
e
{
σ
}=
{
}
(17.56)
The element stiffness matrix is finally obtained in step six in which we replace the
internal 'stresses'
F
e
, thereby relat-
ing nodal loads to nodal displacements (from Eq. (17.56)) and defining the element
stiffness matrix [
K
e
]. This is achieved by employing the principle of the stationary
value of the total potential energy of the beam (see Section 15.3) which comprises the
internal strain energy
U
and the potential energy
V
of the nodal loads. Thus
{
σ
}
by a statically equivalent nodal load system
{
}
1
2
T
δ
e
T
F
e
U
+
V
=
vol
{
ε
}
{
σ
}
d(vol)
−{
}
{
}
(17.57)
