Page 154 - Shigley's Mechanical Engineering Design

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Load and Stress Analysis 129
3–17 Repeat Prob. 3–15 for:
(a) σ x = 12 kpsi,σ y = 6 kpsi,τ xy = 4 kpsi cw
(b) σ x = 30 kpsi,σ y =−10 kpsi,τ xy = 10 kpsi ccw
(c) σ x =−10 kpsi,σ y = 18 kpsi,τ xy = 9 kpsi cw
(d) σ x = 9 kpsi,σ y = 19 kpsi,τ xy = 8 kpsi cw
3–18 For each of the stress states listed below, find all three principal normal and shear stresses. Draw
a complete Mohr’s three-circle diagram and label all points of interest.
(a) σ x =−80 MPa,σ y =−30 MPa,τ xy = 20 MPa cw
(b) σ x = 30 MPa,σ y =−60 MPa,τ xy = 30 MPa cw
(c) σ x = 40 MPa,σ z =−30 MPa,τ xy = 20 MPa ccw
(d) σ x = 50 MPa,σ z =−20 MPa,τ xy = 30 MPa cw
3–19 Repeat Prob. 3–18 for:
(a) σ x = 10 kpsi,σ y =−4 kpsi
(b) σ x = 10 kpsi,τ xy = 4 kpsi ccw
(c) σ x =−2 kpsi,σ y =−8 kpsi,τ xy = 4 kpsi cw
(d) σ x = 10 kpsi,σ y =−30 kpsi,τ xy = 10 kpsi ccw
3–20 The state of stress at a point is σ x =−6,σ y = 18,σ z =−12,τ xy = 9,τ yz = 6, and τ zx =
−15 kpsi. Determine the principal stresses, draw a complete Mohr’s three-circle diagram, label-
ing all points of interest, and report the maximum shear stress for this case.
√
3–21 Repeat Prob. 3–20 with σ x = 20,σ y = 0,σ z = 20,τ xy = 40,τ yz =−20 2, and τ zx = 0 kpsi.
3–22 Repeat Prob. 3–20 with σ x = 10,σ y = 40,σ z = 40,τ xy = 20,τ yz =−40, and τ zx =−20 MPa.
3
3–23 A -in-diameter steel tension rod is 5 ft long and carries a load of 15 kip. Find the tensile stress,
4
the total deformation, the unit strains, and the change in the rod diameter.
3–24 Repeat Prob. 3–23 except change the rod to aluminum and the load to 3000 lbf.
3–25 A 30-mm-diameter copper rod is 1 m long with a yield strength of 70 MPa. Determine the axial
force necessary to cause the diameter of the rod to reduce by 0.01 percent, assuming elastic defor-
mation. Check that the elastic deformation assumption is valid by comparing the axial stress to
the yield strength.
3–26 A diagonal aluminum alloy tension rod of diameter d and initial length l is used in a rectangular frame
to prevent collapse. The rod can safely support a tensile stress of σ allow . If d = 0.5 in, l = 8 ft, and
σ allow = 20 kpsi, determine how much the rod must be stretched to develop this allowable stress.
3–27 Repeat Prob. 3–26 with d = 16 mm, l = 3 m, and σ allow = 140 MPa.
3–28 5 in, l = 10 ft, and σ allow = 15 kpsi.
Repeat Prob. 3–26 with d =
8
3–29 Electrical strain gauges were applied to a notched specimen to determine the stresses in the notch.
The results were x = 0.0019 and y =−0.00072. Find σ x and σ y if the material is carbon steel.
3–30 Repeat Prob. 3–29 for a material of aluminum.

3–31 The Roman method for addressing uncertainty in design was to build a copy of a design that was
satisfactory and had proven durable. Although the early Romans did not have the intellectual
tools to deal with scaling size up or down, you do. Consider a simply supported, rectangular-cross-
section beam with a concentrated load F, as depicted in the figure.
(a) Show that the stress-to-load equation is
2
σbh l
F =
6ac