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

Notice that EI appears in both terms; hence, it cancels out upon multiplying both sides of
Equation (e) by EI. In other words, the specific value of EI has no effect on the magnitude
of the roller force for this particular beam. The roller reaction B y is the only unknown
quantity in the compatibility equation, and thus, the roller reaction at B is
B
y
3M0
= Ans.
2L
Once the reaction force at B is known, the beam is no longer statically indeterminate. The
three remaining unknown reactions at fixed support A can be determined from the equilibrium
equations.
ExAmpLE 11.6
For the beam and loading shown, derive an expression for the
reaction at support B. Assume that EI is constant for the beam.
v
P
Plan the Solution
The beam considered here has four unknown reaction forces
(horizontal and vertical reaction forces at pin A and vertical A
B
reaction forces at rollers B and D). Since there are only three
equilibrium equations, a fourth equation must be developed.
L
Although there are several approaches that could be used to
develop this fourth equation, we will focus our attention on
the roller at B. This roller will be chosen as the redundant reaction. Removing this redundant
leaves a released beam that is simply supported at A and D. Two cases will then be
analyzed. The first case consists of a simple beam AD subjected to a load P. The second
case consists of a simple beam AD loaded at B with the unknown roller reaction. In both
cases, equations for the beam deflection at B will be developed. These equations will be
combined in a compatibility equation by using the fact that the beam deflection at B is
known to be zero. From this compatibility equation, an equation for the unknown reaction
force at B can be derived.
L—
2
C
L—
2
D
x
SolutioN
Case 1—Simply Supported Beam with a Concentrated load at C
Remove the roller support at B, and consider the simply supported beam AD with a
concentrated load at C. The deflection of this beam at B must be determined. From
Appendix C, the elastic curve equation for the beam is given as
Pbx
v
6LEI L 2
( b 2 x 2
=− − − )
(a)
v
P
In this equation, the following values will be used:
x
P = P
b = L/2
x = L
A
x = L
3L
2
B
v B
C
L—
2
D
L = 2L
EI = constant
465