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

Since the sum of the four interior angles of a quadrilateral must equal
360° and since the angles at B 1 and B 2 are each 90°, the acute angle at B′
must equal 360° − 90° − 90° − 137.39° = 42.61°.
Using this deformation diagram, we can determine the horizontal
and vertical distances between initial joint position B and displaced joint
position B′.
42.61°
B 2
δ 2 = – 0.6782 mm
B δ 1 = 0.5227 mm
137.39°
B 1
(d) Joint Displacement
The deformation diagram can now be analyzed to determine the location of
B′, which is the final position of joint B. By inspection, the horizontal translation
Dx of joint B is
42.61°
D x = δ 1 = 0.5227 mm = 0.523 mm
Ans.
Computation of the vertical translation Dy requires several intermediate
steps. From the deformation diagram, the distance labeled b is simply equal
to the magnitude of deformation δ 2 ; therefore, b = δ 2 = 0.6782 mm. The
distance a is found from
cos 42.61°=
a
0.5227 mm
∴ a = (0.5227 mm)cos42.61°=
0.3847 mm
42.61°
B 2
δ 2 = – 0.6782 mm
B
δ 1 = 0.5227 mm
b
a
B
B 1
The vertical translation Dy can now be computed as
Δy
sin 42.61° =
a + b
Dy
42.61°
∴D y =
a + b 0.3847 mm 0.6782 mm
sin 42.61° = +
sin 42.61°
= 1.570 mm Ans.
B
By inspection, joint B is displaced downward and to the right.
Δx
pRoBLEmS
p5.16 Rigid beam ABC shown in Figure P5.16 is supported by
rods (1) and (2) that have identical lengths L = 7.0 m. Rod (1) is
made of steel [E = 200 GPa] and has a cross-sectional area of
175 mm 2 . Rod (2) is made of aluminum [E = 70 GPa] and has a
cross-sectional area of 300 mm 2 . Assume dimensions of a = 1.5 m
and b = 3.0 m. For a uniformly distributed load w = 15 kN/m, determine
the deflection of the rigid beam at point B.
(1) (2)
w
A
a
B
b
FIGURE p5.16
C
L
p5.17 In Figure P5.17, the horizontal rigid beam ABCD is supported
by vertical bars (1) and (2) and is loaded at points A and D
by vertical forces P = 30 kN and Q = 40 kN, respectively. Bar (1) is
made of aluminum [E = 70 GPa] and has a cross-sectional area
of 225 mm 2 and a length L 1 = 8.0 m. Bar (2) is made of steel [E =
200 GPa] and has a cross-sectional area of 375 mm 2 and a length
L 2 = 5.0 m. Assume dimensions a = 2.5 m, b = 2.0 m, and
c = 1.50 m. Determine the deflection of the rigid beam (a) at point A
and (b) at point D.
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