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 εa = +400 × 10^-6 in./in. and εb = -50 × 10^-6 in./in. Knowing that Poisson’s ratio for the given material

Question

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 [latex]\epsilon_{a}=+400 imes 10^{-6}[/latex] in./in. and [latex]\epsilon_{b}=-50 imes 10^{-6}[/latex] 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.

Accepted Answer

(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:

[latex]g _{\max ( ext { in plane) }}=400 imes 10^{-6}+50 imes 10^{-6}=450 imes 10^{-6} rad[/latex]

(b) Maximum Shearing Strain. We first determine the third principal strain [latex]\epsilon_{c}[/latex]. Since we have a state of plane stress on the surface of the machine component, we use Eq. (7.59) and write

[latex]\epsilon_{c}=-\frac{ n }{1- n }\left(\epsilon_{a}+\epsilon_{b} ight)[/latex] (7.59)

[latex]=-\frac{0.30}{0.70}\left(400 imes 10^{-6}-50 imes 10^{-6} ight)=-150 imes 10^{-6} ext { in. } / in.[/latex]

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:

[latex]g _{\max }=400 imes 10^{-6}+150 imes 10^{-6}=550 imes 10^{-6} rad[/latex]

We note that, even though [latex]\epsilon_{a}[/latex] and [latex]\epsilon _{b}[/latex] have opposite signs, the maximum in-plane shearing strain does not represent the true maximum shearing strain.

Question 7.5: As a result of measurements made on the surface of a machine...

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