42
3 Investigation of Euler–Bernoulli Beams in the Elastic Range
Fig. 3.4 Euler–Bernoulli
beam loaded with a
distributed load q Z (X )
In the general case, a variable distributed load q Z (X ) is acting along the length of
the beam.
The problem under consideration can be described according to Table 3.2 by the
following fourth-order differential equation
E I Y
d
4 u Z (X )
dX 4 = q Z (X )
(3.7)
and appropriate boundary conditions. Taking from Table 1.1 the centered difference
scheme of second order accuracy, a finite difference approximation of the fourthorder differential equation (3.7) can be written for node i as:
E I Y
u i+2 − 4u i+1 + 6u i − 4u i−1 + u i−2
X 4
= q i ,
(3.8)
or as
E I Y
X 3 (u i+2 − 4u i+1 + 6u i − 4u i−1 + u i−2 ) = R i ,
(3.9)
where the equivalent nodal force R i , resulting from a distributed load q(X ), is in
general given for an inner node i as: R i =
− q( ˆ
X )d ˆ
X . In this integral, the local
coordinate X has its origin at the location of node i. In the case of the boundary nodes,
the equivalent nodal loads must be calculated as R 1 =
0
q( ˆ
X )d ˆ
X (left-hand
node) or R n =
0
− q( ˆ
X )d ˆ
X (right-hand node) in order to completely distribute
the entire load q(X ) to the nodes. It should be noted here that—as in the case of
the finite element method—the finite difference method allows the action of forces
only at nodes. If a single force F 0 is acting at a node i, then this force should not be
considered on the right-hand side of Eq. (3.9).
The derivation of the second-order derivative was demonstrated in Chap. 1 and can
be used to calculate the centered difference scheme for the fourth-order derivative
as follows. The approximation is according to Table 1.1 given for node i as:
d
2 u
dX 2
i
≈
u i+1 − 2u i + u i−1
X 2
.
(3.10)
3 Investigation of Euler–Bernoulli Beams in the Elastic Range
Fig. 3.4 Euler–Bernoulli
beam loaded with a
distributed load q Z (X )
In the general case, a variable distributed load q Z (X ) is acting along the length of
the beam.
The problem under consideration can be described according to Table 3.2 by the
following fourth-order differential equation
E I Y
d
4 u Z (X )
dX 4 = q Z (X )
(3.7)
and appropriate boundary conditions. Taking from Table 1.1 the centered difference
scheme of second order accuracy, a finite difference approximation of the fourthorder differential equation (3.7) can be written for node i as:
E I Y
u i+2 − 4u i+1 + 6u i − 4u i−1 + u i−2
X 4
= q i ,
(3.8)
or as
E I Y
X 3 (u i+2 − 4u i+1 + 6u i − 4u i−1 + u i−2 ) = R i ,
(3.9)
where the equivalent nodal force R i , resulting from a distributed load q(X ), is in
general given for an inner node i as: R i =
− q( ˆ
X )d ˆ
X . In this integral, the local
coordinate X has its origin at the location of node i. In the case of the boundary nodes,
the equivalent nodal loads must be calculated as R 1 =
0
q( ˆ
X )d ˆ
X (left-hand
node) or R n =
0
− q( ˆ
X )d ˆ
X (right-hand node) in order to completely distribute
the entire load q(X ) to the nodes. It should be noted here that—as in the case of
the finite element method—the finite difference method allows the action of forces
only at nodes. If a single force F 0 is acting at a node i, then this force should not be
considered on the right-hand side of Eq. (3.9).
The derivation of the second-order derivative was demonstrated in Chap. 1 and can
be used to calculate the centered difference scheme for the fourth-order derivative
as follows. The approximation is according to Table 1.1 given for node i as:
d
2 u
dX 2
i
≈
u i+1 − 2u i + u i−1
X 2
.
(3.10)