Civil Engineering Reference
In-Depth Information
From Eqs (17.41) and (17.43) we can write down expressions for the nodal displace-
ments
v
i
,
θ
i
and
v
j
,
θ
j
at
x
=
0 and
x
=
L
, respectively. Hence
!
v
i
=
α
1
θ
i
=
α
2
(17.44)
α
3
L
2
α
4
L
3
"
v
j
=
α
1
+
α
2
L
+
+
3
α
4
L
2
θ
j
=
α
2
+
2
α
3
L
+
Writing Eq. (17.44) in matrix form gives
.
/
!
.
/
!
v
i
θ
i
v
j
θ
j
100 0
010 0
1
LL
2
L
3
01
L
3
L
2
α
1
α
2
α
3
α
4
=
(17.45)
0
"
0
"
or
δ
e
{
}=
[
A
]
{
α
}
(17.46)
The third step follows directly from Eqs (17.45) and (17.42) in that we express the
displacement at any point in the beam-element in terms of the nodal displacements.
Using Eq. (17.46) we obtain
[
A
−
1
]
δ
e
{
α
}=
{
}
(17.47)
Substituting in Eq. (17.42) gives
[
f
(
x
)][
A
−
1
]
δ
e
{
v
(
x
)
}=
{
}
(17.48)
where [
A
−
1
] is obtained by inverting [
A
] in Eq. (17.45) andmay be shown to be given by
1
0
0
0
0
1
0
0
[
A
−
1
]
=
(17.49)
3/
L
2
3/
L
2
−
−
2/
L
−
1/
L
2/
L
3
1/
L
2
2/
L
3
1/
L
2
−
In step four we relate the strain
{
ε
(
x
)
}
at any point
x
in the element to the displacement
δ
e
{
. Since we are concerned here with
bending deformations only we may represent the strain by the curvature
∂
2
v/∂
x
2
.
Hence from Eq. (17.41)
v
(
x
)
}
and hence to the nodal displacements
{
}
∂
2
v
∂
x
2
=
2
α
3
+
6
α
4
x
(17.50)
