3.3 Varying Material and Geometry Parameters
45
E
dI Y (X )
dX
d
2 u
dX 2 + I Y (X )
d
3 u
dX 3
= −Q Z (X ) .
(3.24)
Introduction of centered difference schemes for the second- and third-order derivative
according to Table 1.1 gives:
E
dI Y (X )
dX
i
u i+1−2u i +u i−1
X 2
+ I Y,i
u i+2 − 2u i+1 + 2u i−1 − u i−2
2X 3
= −Q Z ,i ,
(3.25)
where the gradient of the second moment of area function can be approximated based
on a centered difference scheme as
dI Y (X )
dX
=
I Y,i+1 − I Y,i−1
2X
.
(3.26)
Another approach can be based on the moment formulation of the partial differential
equation according to (3.16) to give for node i:
E i I Y,i
d
2 u
dX 2
i
= E i I Y,i
u i+1 − 2u i + u i−1
X 2
= −M Y,i .
(3.27)
3.4 Solved Problems
3.1 Example: Finite difference approximation of a simply supported and cantilevered beam loaded by a single force
Given is an Euler–Bernoulli beam with different supports as shown in Fig. 3.5. The
bending stiffness E I Y is constant and the length is equal to L. The simply supported
beam (a) is loaded in the middle by a single force F y while the cantilevered beam
(b) is loaded at its right-hand end by a single force F 0 . Derive for both cases a finite
difference approximation based on five grid points, i.e. an equidistant spacing of
X =
L
4
.
Determine for both cases
• the displacement of the beam at the force application point,
• the analytical solution and
• calculate the relative error between the analytical and finite difference solution.
3.1 Solution
(a) The finite difference discretization of the simply supported beam is shown in
Fig. 3.6 where at first only the inner nodes will be considered.
It should be noted here that the fourth-order differential equation according to
Eq. (3.9) does not allow to account for external single forces (in our case: F 0 ).
45
E
dI Y (X )
dX
d
2 u
dX 2 + I Y (X )
d
3 u
dX 3
= −Q Z (X ) .
(3.24)
Introduction of centered difference schemes for the second- and third-order derivative
according to Table 1.1 gives:
E
dI Y (X )
dX
i
u i+1−2u i +u i−1
X 2
+ I Y,i
u i+2 − 2u i+1 + 2u i−1 − u i−2
2X 3
= −Q Z ,i ,
(3.25)
where the gradient of the second moment of area function can be approximated based
on a centered difference scheme as
dI Y (X )
dX
=
I Y,i+1 − I Y,i−1
2X
.
(3.26)
Another approach can be based on the moment formulation of the partial differential
equation according to (3.16) to give for node i:
E i I Y,i
d
2 u
dX 2
i
= E i I Y,i
u i+1 − 2u i + u i−1
X 2
= −M Y,i .
(3.27)
3.4 Solved Problems
3.1 Example: Finite difference approximation of a simply supported and cantilevered beam loaded by a single force
Given is an Euler–Bernoulli beam with different supports as shown in Fig. 3.5. The
bending stiffness E I Y is constant and the length is equal to L. The simply supported
beam (a) is loaded in the middle by a single force F y while the cantilevered beam
(b) is loaded at its right-hand end by a single force F 0 . Derive for both cases a finite
difference approximation based on five grid points, i.e. an equidistant spacing of
X =
L
4
.
Determine for both cases
• the displacement of the beam at the force application point,
• the analytical solution and
• calculate the relative error between the analytical and finite difference solution.
3.1 Solution
(a) The finite difference discretization of the simply supported beam is shown in
Fig. 3.6 where at first only the inner nodes will be considered.
It should be noted here that the fourth-order differential equation according to
Eq. (3.9) does not allow to account for external single forces (in our case: F 0 ).