Page 249 - Marks Calculation for Machine Design

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PRINCIPAL STRESSES AND MOHR’S CIRCLE
Eq. (5.19) would still yield the maximum value for the maximum shear stress. In any case,
this stress element is called a triaxial stress element. 231
The situation where Eq. (5.20) must be applied is when the third principal stress (σ 3 ) is
less than the minimum principal stress (σ 2 ). For example, if the third principal stress is a
negative value, such as an internal pressure (p i ) on the inside of a thin-walled vessel, the
principal stresses (σ 1 ) and (σ 2 ) are both positive, and as already seen earlier form a biaxial
stress element, then Eq. (5.20) will yield a much larger value for the maximum shear stress
than Eq. (5.19).
This can be seen graphically using the Mohr’s circle process, where the third principal
stress (σ 3 ) is added as a point on the (σ) axis. As can be seen in Fig. 5.51, if (σ 3 ) is less
than (σ 2 ), and particularly if it is negative, then the radius of Mohr’s circle represented by
Eq. (5.20) is much larger than the radius represented by Eq. (5.19).

s
s 3 s 2 s 1

t max

t (2q ccw)

FIGURE 5.51 Mohr’s circle for a triaxial stress element.

Notice that there is a circle represented by the difference between the principal stresses
(σ 1 ) and (σ 2 ), a circle represented by the difference between the principal stresses (σ 2 ) and
(σ 3 ), but the biggest circle isrepresented by thedifference between the principalstresses (σ 1 )
and (σ 3 ) that is the maximum shear stress (τ max ). The importance of finding the maximum
shear stress, especially for ductile materials, will be discussed shortly. For now, it is just
necessary to keep in mind what might be taking place normal to a plane stress element,
even if this third stress is zero.
Let us look at a previous example to see how Eq. (5.20) comes into play.
Consider the following example where the two normal stresses are the axial and hoop
stresses for a thin-walled cylinder under an internal pressure (p i ).