84
3 Investigation of Euler–Bernoulli Beams in the Elastic Range
Fig. 3.33 Simply supported Euler–Bernoulli beam on elastic foundation loaded by a single force
Fig. 3.34 a Simply supported Euler–Bernoulli beam loaded by a distributed load; b function of
the varying bending stiffness E I Y = E I Y (X )
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.
3.17 Finite difference approximation of a simply supported beam with varying
bending stiffness – constant bending moment
Given is a simply supported Euler–Bernoulli beam of length L with varying bending
stiffness E I Y (X ) =
E I 0
1+(2X/L−1) 2 as shown in Fig. 3.35. The beam is loaded by single
moments M 0 at its ends. 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 , to determine
• the vertical displacement in the middle of the beam, i.e. X =
L
2
,
• the analytical solution and