Except for a brief diversion in Chapter 1, we have not considered stresses in directions other than the axial and tangential directions. In Chapter 1, you saw that a shearing stress could exist even though the only force was axial, and that ductile materials fail on a 45 degree surface where the shearing stress is a maximum.
Let's consider the figures below, which
were taken from Figure 7.23 in your text. The orientation of the element
in the figure on the left is the orientation that has been considered so
far. The element in the right figure has been rotated and, as a result,
shearing stresses have appeared which were not present on the original
element. This was covered in Chapter 1.
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How can we develop equations which will provide the normal and shearing stresses for any angle of rotation? Even more importantly, can we determine if there is an angle of rotation for which those stresses are maximized?
The derivation is not very difficult. On
page 426, the authors take the sum of the forces on a small wedge of the
element, where 
is the counterclockwise angle of rotation between the y and y', and x and
x', axes. Without repeating the derivation here, the results are on the
previous page.
As stated on the previous page, the equations
are parametric equations for a circle, where
is the parameter. For example, the following equations are parametric equations
for a circle centered at (0,0):
If the equations were
the circle would be centered at (h,k).
Suppose, instead of x and y, the plot is made using 
.
Suppose further that the center is at zero on the tau axis but at the average
value of the normal stresses 
:
You can see that the first equation is similar to equation 7.5, and the second is similar to equation 7.6, provided that
If you care to square these two equations and add them, you will get the radius of the circle described in the text.