Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

v
P
v
P
v
A
L—
2
L—
2
(a) Actual beam
x
B
A
Substituting these expressions into Equation (e) gives the following compatibility equation,
which can be solved for the unknown redundant magnitude:
2
PL MAL
3
− + = 0 ∴ MA
= PL
(g)
16EI
3EI
10
The value for M A is the same result computed previously. Once M A has been determined,
the remaining reactions can be computed from the equilibrium equations.
Examples 11.5–11.9 illustrate the application of the superposition method to determine
support reactions for statically indeterminate beams.
θ
L—
2
A
L—
2
(b) Simply supported released beam
FIGURE 11.7 Superposition method for a propped cantilever beam, using a simply supported released beam.
x
B
M A
A
θ′ A
L
x
B
(c) Redundant M A applied to released beam
mecmovies
ExAmpLE
m11.3 Use two different approaches of the superposition
method to determine the roller reaction at A.
ExAmpLE 11.5
For the beam and loading shown, derive an expression for the
reaction at support B. Assume that EI is constant for the beam.
Plan the Solution
The propped cantilever has four unknown reaction forces
(horizontal and vertical reaction forces at fixed reaction A, a A
moment reaction at A, and a vertical reaction force at roller B).
L
Since only three equilibrium equations can be written for the
beam, one additional equation must be developed in order to solve this problem. The
additional equation will be developed by considering the deflected shape of the beam and,
in particular, the known beam deflection at roller B. The roller support at B will be chosen
as the redundant; therefore, the released beam will be a cantilever supported at A. The
analysis will be subdivided into two cases. In the first case, the deflection at B produced
by the concentrated moment M 0 will be determined. In the second case, the unknown
roller reaction force will be applied to the cantilever beam at B and an expression for the
v
B
L—
2
C
M 0
x
463