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

p12.66 The stresses at point O in a solid body that is subjected
to plane stress are σ x = 16.5 ksi, σ y = −5.8 ksi, and τ xy = 8.7 ksi.
Consider Figure P12.66/67. The normal to the inclined surface is
defined by vector OA, where point A has coordinates a = 1.0 in.,
b = 1.4 in., and c = 2.5 in. Determine
(a) the direction cosines of vector OA.
(b) the resultant stress S on the inclined plane.
(c) the normal and shear stresses on the inclined plane.
z
A
(a,b,c)
FIGURE p12.66/p12.67
p12.67 The stresses at point O in a solid body are σ x = −80 MPa,
σ y = −40 MPa, σ z = −40 MPa, τ xy = 23 MPa, τ xz = −65 MPa, and
τ yz = 49 MPa. Consider Figure P12.66/67. The normal to the
inclined surface is defined by vector OA, where point A has coordinates
a = 90 mm, b = 160 mm, and c = 220 mm. Determine
(a) the direction cosines for vector OA.
(b) the resultant stress S on the inclined plane.
(c) the normal and shear stresses on the inclined plane.
y
O
x
p12.68 At a point in a stressed body, the known stresses are
σ x = 10 MPa, σ y = 0, σ z = 0, τ xy = 24 MPa, τ xz = 38 MPa, and τ yz =
15 MPa. Determine the normal and shear stresses on a plane whose
outward normal makes equal angles with the x, y, and z axes.
p12.69 At a point in a stressed body, the known stresses are
σ x = 9 ksi (C), σ y = 13 ksi (T), σ z = 22 ksi (T), τ xy = 4 ksi, τ xz = −19 ksi,
and τ yz = 8 ksi. Determine
(a) the stress invariants.
(b) the principal stresses and the absolute maximum shear stress
at the point.
p12.70 At a point in a stressed body, the known stresses are
σ x = 200 MPa (C), σ y = 90 MPa (C), σ z = 140 MPa (C), τ xy = −24 MPa,
τ xz = 120 MPa, and τ yz = 56 MPa. Determine
(a) the stress invariants.
(b) the principal stresses and the absolute maximum shear stress
at the point.
p12.71 At a point in a stressed body, the known stresses are
σ x = 75 MPa (T), σ y = 30 MPa (C), σ z = 60 MPa (T), τ xy = 35 MPa,
τ xz = 15 MPa, and τ yz = −20 MPa. Determine
(a) the stress invariants.
(b) the principal stresses and the absolute maximum shear stress
at the point.
(c) the orientation of the plane on which the maximum tensile
normal stress acts.
p12.72 At a point in a stressed body, the known stresses are
σ x = 80 MPa (C), σ y = 80 MPa (C), σ z = 120 MPa (C), τ xy = 40 MPa,
τ xz = −60 MPa, and τ yz = 50 MPa. Determine
(a) the stress invariants.
(b) the principal stresses and the absolute maximum shear stress
at the point.
(c) the orientation of the plane on which the maximum compressive
normal stress acts.
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