4.1 The Basics of a Timoshenko Beam
93
Table 4.3 Different boundary conditions and their corresponding reactions for a continuum Timoshenko beam (bending occurs in the X -Z plane)
Case
Boundary condition
Reaction
. . .
X
Z
u Z (0) = 0, φ Y (0) = 0
. . .
R
R
X
F
M Y
Y
Z
. . .
u Z (0) = 0, M Y (0) = 0
. . .
R
F
. . .
u Z (0) = 0, M Y (0) = 0
. . .
R
F
. . .
φ Y (0) = 0, Q Z (0) = 0
Y
. . .
R
M
0
. . .
u
L
u Z (L) = u 0 , M Y (L) = 0
. . .
L
R
F
0
F
. . .
L
Q Z (L) = F 0 , M Y (L) = 0
. . .
L
u
0
φ
. . .
L
φ Y (L) = φ 0 , Q Z (L) = 0
. . .
L
Y
R
M
0
M
. . .
L
M Y (L) = M 0 , Q Z (L) = 0
. . .
L
Y
ϕ
. . .
L
M Y (L) = 0, Q Z (L) = 0
. . .
L
Fig. 4.2 Internal reactions
for a continuum Timoshenko
beam (bending occurs in the
X -Z plane)