Civil Engineering Reference
In-Depth Information
(It must be emphasized that, although an equality is indicated by Equation 2.17,
the relation, for finite elements in general, is an approximation. For the bar ele-
ment, the relation, in fact, is exact.) To determine the interpolation functions, we
require that the boundary values of
u
(
x
)
(the nodal displacements) be identically
satisfied by the discretization such that
u
(
x
u
2
(2.18)
Equations 2.17 and 2.18 lead to the following boundary (nodal) conditions:
N
1
(0)
=
0)
=
u
1
u
(
x
=
L
)
=
=
1
N
2
(0)
=
0
(2.19)
1
(2.20)
which must be satisfied by the interpolation functions. It is required that the dis-
placement expression, Equation 2.17, satisfy the end (nodal) conditions identi-
cally, since the nodes will be the connection points between elements and the
displacement continuity conditions are enforced at those connections. As we
have two conditions that must be satisfied by each of two one-dimensional func-
tions, the simplest forms for the interpolation functions are polynomial forms:
N
1
(
x
)
N
1
(
L
)
=
0
N
2
(
L
)
=
=
a
0
+
a
1
x
(2.21)
b
1
x
(2.22)
where the polynomial coefficients are to be determined via satisfaction of the
boundary (nodal) conditions. We note here that any number of mathematical
forms of the interpolation functions could be assumed while satisfying the
required conditions. The reasons for the linear form is explained in detail in
Chapter 6.
Application of conditions represented by Equation 2.19 yields
a
0
=
1,
b
0
=
N
2
(
x
)
=
b
0
+
0
while Equation 2.20 results in
a
1
=−
(1
/
L
)
and
b
1
=
x
/
L
. Therefore,
the interpolation functions are
N
1
(
x
)
=
1
−
x
/
L
(2.23)
L
(2.24)
and the continuous displacement function is represented by the discretization
u
(
x
)
N
2
(
x
)
=
x
/
L
)
u
2
(2.25)
As will be found most convenient subsequently, Equation 2.25 can be expressed
in matrix form as
=
(1
−
x
/
L
)
u
1
+
(
x
/
N
2
(
x
)]
u
1
(2.26)
u
(
x
)
=
[
N
1
(
x
)
=
[
N
]
{
u
}
u
2
where
[
N
]
is the row matrix of interpolation functions and {
u
} is the column
matrix (vector) of nodal displacements.
Having expressed the displacement field in terms of the nodal variables, the
task remains to determine the relation between the nodal displacements and
applied forces to obtain the stiffness matrix for the bar element. Recall from
