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

p12.35 A shear wall in a reinforced-concrete building is subjected
to a vertical uniform load of intensity w and a horizontal
force H, as shown in Figure P12.35a. As a consequence of these
loads, the stresses at point A on the surface of the wall have the
magnitudes σ y = 115.0 MPa and τ xy = 60.0 MPa, acting in the directions
shown on the stress element in Figure P12.35b.
(a) Determine the largest tension normal stress that acts at point A.
(b) What is the orientation of this stress with respect to the x axis?
A
w
FIGURE p12.35a
p12.36 A 2 in. diameter shaft is supported by bearings at A and
D, as shown in Figure P12.36. The bearings provide vertical reactions
only. The shaft is subjected to an axial load P = 3,600 lb and
a transverse load Q = 250 lb. The shaft length is L = 48 in. For point
B, located on top of the shaft, determine
(a) the principal stresses.
(b) the maximum in-plane shear stress.
Show these results on a stress element.
y
H
y
τ xy
x
A
σ y
FIGURE p12.35b
p12.37 The principal compressive stress on a vertical plane
through a point in a wooden block is equal to three times the principal
compression stress on a horizontal plane through the same
point. The plane of the grain in the block is 25° clockwise from the
vertical plane. If the normal and shear stresses must not exceed
400 psi (C) and 90 psi shear, respectively, determine the maximum
allowable compressive stress on the horizontal plane.
p12.38 At a point on the free surface of a stressed body, a normal
stress of 64 MPa (C) and an unknown positive shear stress exist
on a horizontal plane. One principal stress at the point is 8 MPa (C).
The absolute maximum shear stress at the point has a magnitude of
95 MPa. Determine the unknown stresses on the horizontal and vertical
planes and the unknown principal stress at the point.
p12.39 At a point on the free surface of a stressed body, the
normal stresses are 20 ksi (T) on a vertical plane and 30 ksi (C) on
a horizontal plane. An unknown negative shear stress exists on the
vertical plane. The absolute maximum shear stress at the point has
a magnitude of 32 ksi. Determine the principal stresses and the
shear stress on the vertical plane at the point.
p12.40 At a point on the free surface of a stressed body, a normal
stress of 75 MPa (T) and an unknown negative shear stress
exist on a horizontal plane. One principal stress at the point is
200 MPa (T). The maximum in-plane shear stress at the point has a
magnitude of 85 MPa. Determine the unknown stresses on the vertical
plane, the unknown principal stress, and the absolute maximum
shear stress at the point.
p12.41 For the state of plane stress shown in Figure P12.41,
determine (a) the largest value of τ xy for which the maximum inplane
shear stress is equal to or less than 150 MPa and (b) the corresponding
principal stresses.
70 MPa
P
A B C D P
x
τ xy
120 MPa
L
2
L
4
Q
L
4
FIGURE p12.36 FIGURE p12.41
12.10 mohr’s circle for plane Stress
The process of changing stresses from one set of coordinate axes (i.e., x–y–z) to another set
of axes (i.e., n–t–z) is termed stress transformation, and the general equations for plane stress
transformation were developed in Section 12.7. The equations for computing the principal
stresses and the maximum in-plane shear stress at a point in a stressed body were developed
in Section 12.8. In the current section, a graphical procedure for plane stress transformations
will be developed. In comparison with the various equations derived in Sections 12.7 and
12.8, this graphical procedure is much easier to remember and it provides a functional depiction
of the relationships between stress components acting on different planes at a point.
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