40
3 Investigation of Euler–Bernoulli Beams in the Elastic Range
Table 3.2 Different formulations of the partial differential equation for an Euler–Bernoulli beam
in the X -Z plane (X -axis: right facing; Z -axis: upward facing)
Configuration
Partial differential equation
E I Y
d 4 u Z
dX 4 = 0
d 2
dX 2
E(X )I Y (X )
d 2 u Z
dX 2
= 0
E I Y
d 4 u Z
dX 4 = q Z (X )
E I Y
d 4 u Z
dX 4 =
dm Y (X )
dX
E I Y
d 4 u Z
dX 4 = −k(X )u Z
Fig. 3.3 Different stress distributions of an Euler–Bernoulli beam with rectangular cross section
and linear-elastic material behavior: a normal stress and b shear stress (bending occurs in the X -Z
plane)
Application of Hooke’s law (i.e., σ X = Eε X and τ X Z = Gγ X Z ) allows us to calculate
the normal and shear strains. Typical distributions of the two stress components in a
beam element are shown in Fig. 3.3. It can be seen that normal stress distribution is
linear while the shear stress distribution is parabolic over the cross section.
Finally, it should be noted here that the one-dimensional Euler–Bernoulli beam
theory has its two-dimensional analogon in the form of Kirchhoff plates
2 [3–5, 7, 9,
14].
2 Also called thin or shear-rigid plates.