The strain components εx, εy, and γxy are given for a point in a body subjected to plane strain. Using Mohr’s circle, determine the principal strains, the maximum in-plane shear strain, and the absolute maximum shear strain at the point. Show the angle θp, the principal strain deformations, and the

Question

The strain components [latex]\varepsilon_{x}, \varepsilon_{y} ext {, and } \gamma_{x y}[/latex] are given for a point in a body subjected to plane strain. Using Mohr’s circle, determine the principal strains, the maximum in-plane shear strain, and the absolute maximum shear strain at the point. Show the angle [latex] heta_{p}[/latex], the principal strain deformations, and the maximum in-plane shear strain distortion in a sketch.
[latex]\varepsilon_{x}=475 \mu \varepsilon \quad \varepsilon_{y}=685 \mu \varepsilon \quad \gamma_{x y}=-150 \mu rad[/latex]

Accepted Answer

The basic Mohr’s circle is shown.

[latex]\begin{aligned}&C=\frac{(475 \mu)+(685 \mu)}{2}=580 \mu \varepsilon \&R=\sqrt{(-105 \mu)^{2}+(75 \mu)^{2}}=129.0349 \mu \varepsilon\end{aligned}[/latex]

[latex]\begin{aligned}\varepsilon_{p 1} &=C+R=580 \mu \varepsilon+129.0349 \mu \varepsilon=709 \mu \varepsilon \\varepsilon_{p 2} &=C-R=580 \mu \varepsilon-129.0349 \mu \varepsilon=451 \mu \varepsilon \\gamma_{\max } &=2 R=709 \mu rad\end{aligned}[/latex]

The magnitude of the angle [latex]2 heta_{p}[/latex] between point x and point 2 (i.e., the principal plane associated with [latex]\varepsilon_{p 2}[/latex]) is found from:

[latex] an 2 heta_{p}=\frac{150 \mu}{|(475 \mu)-(685 \mu)|}=\frac{150 \mu}{210 \mu}=0.71429 \quad herefore 2 heta_{p}=35.5377^{\circ} \quad ext { thus, } heta_{p}=17.77^{\circ}[/latex]

By inspection, the angle [latex] heta_{p}[/latex] from point x to point 2 is turned counterclockwise.

Since both [latex]\varepsilon_{p 1} ext { and } \varepsilon_{p 2}[/latex] are positive, the absolute maximum shear strain is greater than the maximum in-plane shear strain:

[latex]\gamma_{ abs \max }=\varepsilon_{p 1}-\varepsilon_{p 3}=709.0349 \mu-(0 \mu)=709 \mu rad[/latex]

A sketch of the principal strain deformations and the maximum in-plane shear strain distortions is shown below.

Question 13.35: The strain components εx, εy, and γxy are given for a point ...

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