STRESS TRANSFORMATION

PLANE-STRESS TRANSFORMATION
It was shown in Sec. 1.3 that the general state of stress at a point is
characterized by six normal and shear-stress components, shown in
Fig. 9–1a. This state of stress, however, is not often encountered in
engineering practice. Instead, most loadings are coplanar, and so the
stress these loadings produce can be analyzed in a single plane. When
this is the case, the material is then said to be subjected to plane stress.
CHAPTER OBJECTIVES
n In this chapter, we will show how to transform the stress
components acting on an element at a point into components
acting on a corresponding element having a different orientation.
Once the method for doing this is established, we will then be
able to find the maximum normal and maximum shear stress at
the point, and find the orientation of the elements upon which
they act.
General state of stress
(a)
t
yz
t
yz
t
xy
t
xy
t
xz
t
xz
s
z
The general state of plane stress at a point, shown in Fig. 9–1b, is
therefore represented by a combination of two normal-stress components,
s
x , sy , and one shear-stress component, txy, which act on only four faces
of the element. For convenience, in this text we will view this state of
stress in the x–y plane, as shown in Fig. 9–2a. Realize, however, that if this
state of stress is produced on an element having a different orientation u,
as in Fig.9–2b, then it will be subjected to three different stress components,
s
x?, sy?, tx?y?, measured relative to the x?, y? axes. In other words, the state
of plane stress at the point is uniquely represented by two normal-stress
components and one shear-stress component acting on an element. To be
equivalent, these three components will be different for each specific
orientation U of the element at the point.
If these three stress components act on the element in Fig. 9–2a, we
will now show what their values will have to be when they act on the
element in Fig. 9–2b. This is similar to knowing the two force
components Fx and Fy directed along the x, y axes, and then finding the
force components Fx? and Fy? directed along the x?, y? axes, so they
produce the same resultant force. The transformation of force must
only account for the force component’s magnitude and direction. The
transformation of stress components, however, is more difficult since it
must account for the magnitude and direction of each stress and the
orientation of the area upon which it acts.
(a)
y
x
(b)

y?
x?
s
y
s
x
s
y?
s
x?
t
xy
t
x?y?
u
u
Fig