3.5 Supplementary Problems
81
Fig. 3.28 Cantilevered
Euler–Bernoulli beam
loaded by a singe force at the
position X =
3L
4
• the vertical displacement at the end of the beam, i.e. X = L,
• the analytical solution and
• calculate the relative error between the analytical and finite difference solution.
3.10 Finite difference approximation of a cantilevered beam based on five
domain nodes—backward scheme
Reconsider Problem 3.9 and apply a backward finite difference approximation of the
third order derivative where the truncation error is of order
2 to formulate the
equilibrium between the internal shear force and external load:
(a) at the free end, X = L,
(b) at the loading point, X =
3L
4
.
3.11 Finite difference approximation of a cantilevered beam based on five
domain nodes—conversion of tip load into distributed load
Reconsider example Problem 3.1 where a cantilevered beam loaded by a single force
was investigated, see Fig. 3.29a. For five domain nodes and a centered difference
scheme, a relative error of 54.689% for the vertical deformation at X = L was
obtained. In order to improve the centered difference approach, the single force
F 0 can be converted in an equivalent distributed load q 0 = F 0 //X between the two
last nodes, see Fig. 3.29b. Calculate the relative error of the displacement at X = L.
3.12 Finite difference approximation of a simply supported beam based on five
domain nodes – bending moment approach
Given is a simply supported Euler–Bernoulli beam of length L with constant bending
stiffness E I Y as shown in Fig. 3.30. The beam is loaded by a constant distributed
load q 0 . Use five domain nodes of equidistant spacing, i.e. =
L
4
, for the finite
difference approximation. Use the bending differential equation in the form of the
moment, i.e. E I Y
d
2 u Z
dX 2 = −M Y (X ), to determine
• the vertical displacement in the middle of the beam, i.e. X =
L
2
,
• the analytical solution and
• calculate the relative error between the analytical and finite difference solution.
81
Fig. 3.28 Cantilevered
Euler–Bernoulli beam
loaded by a singe force at the
position X =
3L
4
• the vertical displacement at the end of the beam, i.e. X = L,
• the analytical solution and
• calculate the relative error between the analytical and finite difference solution.
3.10 Finite difference approximation of a cantilevered beam based on five
domain nodes—backward scheme
Reconsider Problem 3.9 and apply a backward finite difference approximation of the
third order derivative where the truncation error is of order
2 to formulate the
equilibrium between the internal shear force and external load:
(a) at the free end, X = L,
(b) at the loading point, X =
3L
4
.
3.11 Finite difference approximation of a cantilevered beam based on five
domain nodes—conversion of tip load into distributed load
Reconsider example Problem 3.1 where a cantilevered beam loaded by a single force
was investigated, see Fig. 3.29a. For five domain nodes and a centered difference
scheme, a relative error of 54.689% for the vertical deformation at X = L was
obtained. In order to improve the centered difference approach, the single force
F 0 can be converted in an equivalent distributed load q 0 = F 0 //X between the two
last nodes, see Fig. 3.29b. Calculate the relative error of the displacement at X = L.
3.12 Finite difference approximation of a simply supported beam based on five
domain nodes – bending moment approach
Given is a simply supported Euler–Bernoulli beam of length L with constant bending
stiffness E I Y as shown in Fig. 3.30. The beam is loaded by a constant distributed
load q 0 . Use five domain nodes of equidistant spacing, i.e. =
L
4
, for the finite
difference approximation. Use the bending differential equation in the form of the
moment, i.e. E I Y
d
2 u Z
dX 2 = −M Y (X ), to determine
• the vertical displacement in the middle of the beam, i.e. X =
L
2
,
• the analytical solution and
• calculate the relative error between the analytical and finite difference solution.