The following list gives a summary of the techniques

The following list gives a summary of the techniques pre-
sented in this section; it also outlines the general procedure
for applying the techniques to a given stress analysis problem.
➭ Angle for Principal Stress Element
uni03D5
σ
=
1
2
arctan [2 τ
xy
/( σ
x
- σ
y
)] (4–3)
The angle uni03D5
σ
is measured from the positive x -axis of the
original stress element to the maximum principal stress, σ
1
.
Then the minimum principal stress, σ
2
, is on the plane 90°
from σ
1
.
When the stress element is oriented as discussed so that
the principal stresses are acting on it, the shear stress is zero.
The resulting stress element is shown in Figure 4–4.
Maximum Shear Stress
On a different orientation of the stress element, the maxi-
mum shear stress will occur. Its magnitude can be computed
from
➭ Maximum shear stress
τ
max
=
C
a
σ
x
- σ
y
2
b
2
+ τ
xy
2
(4–4)
The angle of inclination of the element on which the maxi-
mum shear stress occurs is computed as follows:
➭ Angle for Maximum Shear Stress Element
uni03D5
τ
=
1
2
arctan [ - ( σ
x
- σ
y
)/2 τ
xy
] (4–5)
The angle between the principal stress element and the max-
imum shear stress element is always 45°.
On the maximum shear stress element, there will be
normal stresses of equal magnitude acting perpendicular to
the planes on which the maximum shear stresses are acting.
These normal stresses have the value
➭ Average Normal Stress
σ
avg
= ( σ
x
+ σ
y
)/2 (4–6)
Note that this is the average of the two applied normal
stresses. The resulting maximum shear stress element is
shown in Figure 4–5. Note, as stated above, that the angle
between the principal stress element and the maximum
shear stress element is always 45°.
FIGURE 4–4
Principal stress element
y
x
H9268
avg
H9268
avg
H9268
avg
H9268
avg
H9270
max
– H9270
max
H9270
max
– H9270
max
H9278
H9270
FIGURE 4–5
Maximum shear stress element
GENERAL PROCEDURE FOR COMPUTING
PRINCIPAL STRESSES AND MAXIMUM
SHEAR STRESSES
1. Decide for which point you want to compute the stresses.
2. Clearly specify the coordinate system for the object, the
free-body diagram, and the magnitude and direction of
forces.
3. Compute the stresses on the selected point due to the
applied forces, and show the stresses acting on a stress
element at the desired point with careful attention to
directions. Figure 4–3 is a model for how to show these
stresses.
4. Compute the principal stresses on the point and the di-
rections in which they act. Use Equations (4–1), (4–2),
and (4–3).
5. Draw the stress element on which the principal stresses
act, and show its orientation relative to the original x -axis.
It is recommended that the principal stress element be
drawn beside the original stress element to illustrate the
relationship between them.
6. Compute the maximum shear stress on the element and
the orientation of the plane on which it acts. Also, com-
pute the normal stress that acts on the maximum shear
stress element. Use Equations (4–4), (4–5), and (4–6).