Question 7.5: As a result of measurements made on the surface of a machine...
As a result of measurements made on the surface of a machine component with strain gages oriented in various ways, it has been established that the principal strains on the free surface are \epsilon_{a}=+400 \times 10^{-6} in./in. and \epsilon_{b}=-50 \times 10^{-6} in./in. Knowing that Poisson’s ratio for the given material is n = 0.30, determine (a) the maximum in-plane shearing strain, (b) the true value of the maximum shearing strain near the surface of the component.
(a) Maximum In-Plane Shearing Strain. We draw Mohr’s circle through the points A and B corresponding to the given principal strains (Fig. 7.76). The maximum in-plane shearing strain is defined by points D and E and is equal to the diameter of Mohr’s circle:
g _{\max (\text { in plane) }}=400 \times 10^{-6}+50 \times 10^{-6}=450 \times 10^{-6} rad
(b) Maximum Shearing Strain. We first determine the third principal strain \epsilon_{c}. Since we have a state of plane stress on the surface of the machine component, we use Eq. (7.59) and write
\epsilon_{c}=-\frac{ n }{1- n }\left(\epsilon_{a}+\epsilon_{b}\right) (7.59)
=-\frac{0.30}{0.70}\left(400 \times 10^{-6}-50 \times 10^{-6}\right)=-150 \times 10^{-6} \text { in. } / in.
Drawing Mohr’s circles through A and C and through B and C (Fig. 7.77), we find that the maximum shearing strain is equal to the diameter of the circle of diameter CA:
g _{\max }=400 \times 10^{-6}+150 \times 10^{-6}=550 \times 10^{-6} rad
We note that, even though \epsilon_{a} and \epsilon _{b} have opposite signs, the maximum in-plane shearing strain does not represent the true maximum shearing strain.
