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

mecmovies
ExAmpLE
m5.14 A rectangular bar 30 mm wide and 24 mm thick
made of aluminum [E = 70 GPa; α = 23.0 × 10 −6 /°C] and two
rectangular copper [E = 120 GPa; α = 16.0 × 10 −6 /°C] bars
30 mm wide and 12 mm thick are connected by two smooth
11 mm diameter pins. When the pins are initially inserted
into the bars, both the copper and aluminum bars are stress
free. After the temperature of the assembly has increased by
65°C, determine
(a) the internal axial force in the aluminum bar.
(b) the normal strain in the copper bars.
(c) the shear stress in the 11 mm diameter pins.
ExAmpLE 5.8
A pin-connected structure is loaded and supported as shown. Member
BCDF is a rigid plate. Member (1) is a steel [E = 200 GPa; A 1 =
310 mm 2 ; α = 11.9 × 10 −6 /°C] bar, and member (2) is an aluminum
[E = 70 GPa; A 2 = 620 mm 2 ; α = 22.5 × 10 −6 /°C] bar. A load of
6 kN is applied to the plate at F. If the temperature increases by
20°C, compute the normal stresses in members (1) and (2).
A
500 mm 350 mm
(1)
B
100 mm
C
F
Plan the Solution
The five-step procedure for solving indeterminate problems will
be used. Since the rigid plate is pinned at C, it will rotate about C.
A deformation diagram will be sketched to show the relationship
between the rigid-plate deflections at joints B and D, based
on the assumption that the plate rotates clockwise about C. The
joint deflections will be related to the deformations δ 1 and δ 2 ,
which will lead to a compatibility equation expressed in terms of
the member forces F 1 and F 2 .
E
300 mm
(2)
400 mm
D
350 mm
6 kN
SOLUTION
Step 1 — Equilibrium Equations:
∑ M = F (100 mm) −F
(300 mm) − (6kN)(350 mm) = 0
C 1 2 (a)
A
F 1 (1)
100 mm
C x
B
C
F
Step 2 — Geometry of Deformation: Sketch the deflected
position of the rigid plate. Since the plate is pinned at C, the
plate will rotate about C. The relationship between the deflections
of joints B and D can be expressed by similar triangles:
vB
vD
= (b)
100 mm 300 mm
E
F 2
300 mm
(2)
Cy
D
6 kN
How are the deformations in members (1) and (2) related to
the joint deflections at B and D?
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