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

p12.9 A steel pipe with an outside diameter of 10.75 in. and an
inside diameter of 9.50 in. is subjected to forces P x = 16 kips, P z =
5 kips, and T = 23 kip ⋅ ft, acting as shown in Figure P12.9. Assume
that a = 4 ft. Determine the normal and shear stresses on horizontal
and vertical planes at (a) point H and (b) point K. For each point,
show these stresses on a stress element.
z
y
P x
z
y
P y
H K
a
a
a
a
P z a
x
T
H
K
FIGURE p12.10a
x
P x
FIGURE p12.9
P z
a
y
b
5
b
3
H
K
p12.10 Concentrated loads P x = 3,300 N, P y = 2,100 N, and
P z = 2,800 N are applied to the cantilever beam in the locations and
directions shown in Figure P12.10a. The beam cross section shown
in Figure P12.10b has dimensions b = 100 mm and d = 40 mm. Using
the value a = 75 mm, determine the normal and shear stresses at
(a) point H and (b) point K. Show these stresses on a stress element.
d
FIGURE p12.10b
b
x
y t
F n
F R
F y F t
F x
FIGURE 12.9
488
x
n
12.6 Equilibrium method for plane Stress
Transformations
As discussed in Sections 1.5 and 12.2, stress is not simply a vector quantity: Stress is dependent
on the orientation of the plane surface upon which the stress acts. As shown in
Section 12.2, the state of stress at a point in a material object subjected to plane stress is
completely defined by three stress components—σ x , σ y , and τ xy —acting on two orthogonal
planes x and y defined with respect to x–y coordinate axes. The same state of stress at
a point can be represented by different stress components—σ n , σ t , and τ nt —acting on a
different pair of orthogonal planes n and t, which are rotated with respect to the x and y
planes. In other words, there is only one unique state of stress at a point, but the state of
stress can have different representations, depending on the orientation of the axes used.
The process of changing stresses from one set of coordinate axes to another is termed
stress transformation.
In some ways, the concept of stress transformation is analogous to vector addition.
Suppose that there are two force components F x and F y , which are oriented parallel to the
x and y axes, respectively (Figure 12.9). The sum of these two vectors is the resultant force
F R . Two different force components F n and F t , defined in an n–t coordinate system, could