GENERAL EQUATIONS OF PLANE-STRESS TRANSFORMATION

The method of transforming the normal and shear stress components
from the x, y to the x?, y? coordinate axes, as discussed in the previous
section, can be developed in a general manner and expressed as a set of
stress-transformation equations.
Sign Convention. To apply these equations we must first establish
a sign convention for the stress components. As shown in Fig. 9–5, the +x
and +x? axes are used to define the outward normal on the right-hand
face of the element, so that sx and sx? are positive when they act in the
positive x and x? directions, and txy and tx?y? are positive when they act in
the positive y and y? directions.
The orientation of the face upon which the normal and shear stress
components are to be determined will be defined by the angle u, which is
measured from the +x axis to the +x? axis, Fig. 9–5b. Notice that the
unprimed and primed sets of axes in this figure both form right-handed
coordinate systems; that is, the positive z (or z?) axis always points out
of the page. The angle u will be positive when it follows the curl of the
right-hand fingers, i.e., counterclockwise as shown in Fig. 9–5b.
Normal and Shear Stress Components. Using this established
sign convention, the element in Fig. 9–6a is sectioned along the inclined
plane and the segment shown in Fig. 9–6b is isolated. Assuming the
sectioned area is ?A, then the horizontal and vertical faces of the segment
have an area of ?A sin u and ?A cos u, respectively.