SOLUTION
Compatibility equation.
0=∆CD +FCDfCDCD
(1)
Use the virtual work method:
mM
(0.5)(20x)x
11520
9–37. The king-post trussed beam supports a concentrated
force of 40 k at its center. Determine the force in each of the
three struts. The struts each have a cross-sectional area of 2in2.
Assume they are pin connected at their end points. Neglect
both the depth of the beam and the effect of axial compression
in the beam. Take
E=29(103)
ksi
for both the beam and struts.
Also,
IAB =400
in4.
5 ft
12 ft
40 k
12 ft
A
C
B
D
SOLUTION
Compatibility Equation. Referring to Fig. a, the required
9–41. The two cantilever beams are in contact using the roller
support C. Determine the reactions at the fixed supports A and B
when the load of 9 k is applied. EI is constant.
10 ft
9 k
A
B
C
470
Ans.
FCD =7.48 kip (T)
SOLUTION
Method of Superposition: Using the method of superposition
as discussed in Chapter 4, the required displacements are
∆
C=
PL
3
48EI
=
F
CD
(1203)
48(29)(10
3
)(475)
=0.002613FCD T
Using the axial force formula,
∆
F=
PL
AE
=
F
CD
(50)
p
4
(0.752)(29)(103)
=0.003903FCD
c
The thermal contraction is
∆T=
a
∆TL =6.5(10 –6)(150)(50) =0.04875 in.T
The compatibility condition requires
(+ T)
∆C=∆T+∆F
0.002613FCD =0.04875 +(–0.003903FCD)
FCD =7.48 kip (T)
Ans.
9–42. The beam AB has a moment of inertia of
I=475
in4
and rests on the smooth supports at its ends. A 0.75-in.-diameter
rod CD is welded to the center of the beam and to the fixed
support at D. If the temperature of the rod is decreased by
150°F, determine the force developed in the rod. The beam and
rod are both made of steel for which
E=29 (10
3) ksi
and
a=6.5(10 –6)>°F.
50 in.
5 ft 5 ft
AB
C
D
471
SOLUTION
The primary real beam and qualitative influence line are
shown in Fig. a and the conjugate beam is shown in Fig. b.
Referring to Fig. c,
144
9–43. Draw the influence line for the reaction at C. Plot
numerical values at the peaks. Assume A is a pin and B and C
are rollers. EI is constant.
AC
6 m 6 m
B