Page 153 - Shigley's Mechanical Engineering Design

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128 Mechanical Engineering Design
3–9 Repeat Prob. 3–5 using singularity functions exclusively (including reactions).

3–10 Repeat Prob. 3–6 using singularity functions exclusively (including reactions).

3–11 Repeat Prob. 3–7 using singularity functions exclusively (including reactions).
3–12 Repeat Prob. 3–8 using singularity functions exclusively (including reactions).
3–13 For a beam from Table A–9, as specified by your instructor, find general expressions for the
loading, shear-force, bending-moment, and support reactions. Use the method specified by your
instructor.
3–14 A beam carrying a uniform load is simply supported with the supports set back a distance a from
the ends as shown in the figure. The bending moment at x can be found from summing moments
to zero at section x:
1 1
2
M = M + w(a + x) − wlx = 0
2 2
or
w 2
M = [lx − (a + x) ]
2
where w is the loading intensity in lbf/in. The designer wishes to minimize the necessary weight
of the supporting beam by choosing a setback resulting in the smallest possible maximum bend-
ing stress.
(a) If the beam is configured with a = 2.25 in, l = 10 in, and w = 100 lbf/in, find the magnitude
of the severest bending moment in the beam.
(b) Since the configuration in part (a) is not optimal, find the optimal setback a that will result in
the lightest-weight beam.

x
w(a + x)
w, lbf/in

M
Problem 3–14
V
a a x
wl
l
2
3–15 For each of the plane stress states listed below, draw a Mohr’s circle diagram properly labeled,
find the principal normal and shear stresses, and determine the angle from the x axis to σ 1 . Draw
stress elements as in Fig. 3–11c and d and label all details.
(a) σ x = 20 kpsi,σ y =−10 kpsi,τ xy = 8 kpsi cw
(b) σ x = 16 kpsi,σ y = 9 kpsi,τ xy = 5 kpsi ccw
(c) σ x = 10 kpsi,σ y = 24 kpsi,τ xy = 6 kpsi ccw
(d) σ x =−12 kpsi,σ y = 22 kpsi,τ xy = 12 kpsi cw
3–16 Repeat Prob. 3–15 for:
(a) σ x =−8MPa,σ y = 7MPa,τ xy = 6MPa cw
(b) σ x = 9MPa,σ y =−6MPa,τ xy = 3MPa cw
(c) σ x =−4MPa,σ y = 12 MPa,τ xy = 7MPa ccw
(d) σ x = 6MPa,σ y =−5MPa,τ xy = 8MPa ccw