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

corresponding beam deflection will be derived. These two deflection expressions will be
added together in a compatibility equation to express the total beam deflection at B, which
must equal zero, since B is a roller support. From this compatibility equation, the magnitude
of the unknown roller reaction force at B can be determined.
v
SolutioN
The beam will be analyzed as two cantilever beam cases. In both cases, the roller support
at B will be removed, reducing the propped cantilever beam to a cantilever beam. In the
first case, the concentrated moment M 0 acting at the tip of the
cantilever will be considered. In the second case, the deflection
M 0 caused by the roller reaction force at B will be considered.
A
L
B
v B
2 L —
C
x
Case 1—Concentrated Moment at tip of Cantilever
Remove the roller support at B and consider the cantilever beam
ABC. From Appendix C, the elastic curve equation for a cantilever
beam subjected to a concentrated moment acting at its free
end is given by
2
Mx
v =− (a)
2EI
Use the elastic curve equation to compute the beam deflection at B. In Equation (a), let
M = M 0 and x = L, and assume that EI is a constant for the beam. Substitute these values
into Equation (a) to derive an expression for the beam deflection at B:
v
B
ML
=−
2EI
0 2
(b)
v
A
L
B
v B
B y
2 L —
C
x
Case 2—Concentrated Force at Roller Support location
By applying only redundant B y to the cantilever beam, an expression
for the resulting deflection at B is derived. From Appendix C,
the maximum cantilever beam deflection produced by a concentrated
force acting at the tip of the cantilever is given by the
expression
v
max
3
PL
=−
3EI
(c)
In Equation (c), let P = −B y and L = L. Note that B y is negative, since it acts upward, opposite
to the direction assumed in the beam table. Substitute these values into Equation (c) to
obtain an expression for the beam deflection at B in terms of the unknown roller reaction
force B y :
v
B
3 3
( By)
L BL y
=− − = (d)
3EI
3EI
Compatibility Equation
Two expressions [Equations (b) and (d)] have been developed for the beam deflection at B.
Add these two expressions, and set the result equal to the beam deflection at B, which is
known to be zero at the roller support:
v
B
ML 0 2 BL y
3
=− + = 0
2EI
3EI
(e)
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