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

Case 2—Cantilever Beam with Concentrated load
The tie rod provides the reaction force at B for the cantilever
beam. Consider the cantilever beam subjected to this upward reaction
force B y . From Appendix C, the maximum beam deflection
(which occurs at B) due to a concentrated load applied at the tip
of the cantilever is given by
v
max
PL (
= v = − = − − By)
L BL y
B
=
3EI
3EI
3EI
3 3 3
Compatibility Equation
The expressions developed for v B from the two cases [Equations
(a) and (b)] are combined in a compatibility equation:
(b)
v
(1)
9 m
v
B
4 3
wL BL y
=− + ≠ 0
8EI
3EI
(c)
A
x
B
v B
In this instance, however, the beam deflection at B will not equal
zero, as it would if there were a roller support at B. The beam is
supported at B by an axial member that will stretch; consequently,
we must determine how much the rod will stretch in this situation.
Consider a free-body diagram of the aluminum tie rod. In
general, the elongation produced in rod (1) is given by
6 m
B y
δ
1
1 1
= FL
AE
The tie rod exerts an upward force B y on the cantilever beam. In
turn, the beam exerts a force of equal magnitude, but opposite in
direction, on the tie rod. Therefore, the deformation of rod (1) can
be stated in terms of the unknown reaction force B y as
δ
1
1 1
= BL y
AE
As rod (1) elongates because of the force it carries, the lower end
of the rod deflects downward. Since the beam is supported by the
rod, the beam also deflects downward at this point. The compatibility
equation [Equation (c)] must therefore be adjusted to account
for the elongation of the tie rod:
1
1 1
v
A
40 kN/m
6 m
x
(1)
B
B y
9 m
v
B
4 3
wL BL y
=− + =
8EI
3EI
BL
y
AE
1
1 1
(d)
B y
this equation is not quite correct. The error is a subtle, but
important, one. How is Equation (d) incorrect?
The upward direction has been defined as positive for beam
deflections. When tie rod (1) elongates, point B (the lower end of
the rod) moves downward. Since the compatibility equation pertains
to deflections of the beam, the tie-rod term on the right-hand
side of the equation should have a negative sign:
v
B
4 3
wL BL y BL y
=− + = −
8EI
3EI
AE
1
1 1
(e)
v
A
40 kN/m
6 m
x
(1)
B
9 m
v B
469