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

B
y
2 6 4
3
6(200,000 N/mm )(351 × 10 mm ) ⎡
11(240,000 N)(5,000 mm) ⎤
=
⎢− 5mm +
3
2 6 4
(5,000 mm)
⎣ 96(200,000 N/mm )(351 × 10 mm )
⎥
⎦
= (3,369.6 N/mm)[ − 5mm + 48.967 mm]
3
= 148.152 × 10 N = 148.2 kN
Ans.
The 5 mm settlement at support B decreases the reaction force B y from 165 kN to
148.2 kN. The bending moments in the beam will also change because of the support
settlement. If the roller support at B does not settle, the maximum positive bending
moment in the beam is 243.75 kN · m and the maximum negative bending moment is
−112.5 kN · m. A 5 mm settlement at roller B changes the maximum positive bending
moment to 264.81 kN · m (an 8.6 percent increase) and the maximum negative bending
moment to −70.38 kN · m (a 37 percent decrease). These values show that a relatively
small differential settlement can produce significant changes in the bending moments
produced in the beam. The engineer must be attentive to these potential variations.
ExAmpLE 11.8
v
A
40 kN/m
6 m
(1)
9 m
A structural steel [E = 200 GPa; I = 300 × 10 6 mm 4 ] beam supports
a uniformly distributed load of 40 kN/m. The beam is fixed
at the left end and supported by a 30 mm diameter, 9 m long solid
aluminum [E 1 = 70 GPa] tie rod. Determine the tension in the tie
rod and the deflection of the beam at B.
Plan the Solution
The cantilever beam is supported at B by a tie rod. Unlike a roller
x
support, the tie rod is not rigid: It stretches in response to its internal
tension force. Supports such as this are termed elastic sup-
B
ports. The beam deflection at B will not be zero in this instance;
rather, the beam deflection will equal the elongation of the tie
rod. To analyze this beam, select the reaction force provided by the tie rod as the redundant
reaction. Removing this redundant leaves a cantilever as the released beam. Two
cantilever beam cases will then be considered. In the first case, the downward deflection
of the cantilever beam at B due to the distributed load will be calculated. The second case
will consider the upward deflection at B produced by the internal force in the tie rod. The
resulting two expressions will be added together in a compatibility equation, with the sum
set equal to the downward deflection of the lower end of the tie rod—a deflection that is
simply equal to the elongation of the rod. Since the elongation of the rod depends on the
internal force in the rod, the compatibility equation will contain two terms that include the
unknown tie rod force. Once the tie rod force has been computed from the compatibility
equation, the deflection of the beam at B can be calculated.
v
A
40 kN/m
6 m
x
B
v B
SolutioN
Case 1—Cantilever Beam with uniformly Distributed load
Remove the redundant tie rod support at B, and consider a cantilever
beam subjected to a uniformly distributed load. The deflection of this
beam at B must be determined. From Appendix C, the maximum
beam deflection (which occurs at B) is given by
v
max
4
wL
= vB
= −
8EI
(a)
468