3.2 Constant Material and Geometry Parameters
41
Table 3.3 Different boundary conditions and corresponding reactions for a continuum Euler–
Bernoulli beam (bending occurs in the X -Z plane)
Case
Boundary condition
Reaction
u Z (0) = 0, ϕ Y (0) = 0
u Z (0) = 0, M Y (0) = 0
u Z (0) = 0, M Y (0) = 0
ϕ Y (0) = 0, Q Z (0) = 0
u Z (L) = u 0 , M Y (L) = 0
Q Z (L) = F 0 , M Y (L) = 0
ϕ Y (L) = ϕ 0 , Q Z (L) = 0
M Y (L) = M 0 , Q Z (L) = 0
M Y (L) = 0, Q Z (L) = 0
3.2 Constant Material and Geometry Parameters
Let us consider in the following a loaded Euler–Bernoulli beam of length L as shown
in Fig. 3.4. The bending stiffness E I Y , i.e. the material and geometric parameters, are
considered to be constant in the following derivations. The boundary conditions are
not shown in the figure and will be chosen in different ways throughout this chapter.