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}=0 \mu \varepsilon \quad \varepsilon_{y}=320 \mu \varepsilon \quad \gamma_{x y}=260 \mu rad[/latex]

Accepted Answer

The basic Mohr’s circle is shown.

[latex]\begin{aligned}&C=\frac{(0 \mu)+(320 \mu)}{2}=160 \mu \varepsilon \&R=\sqrt{(-160 \mu)^{2}+(130 \mu)^{2}}=206.1553 \mu \varepsilon\end{aligned}[/latex]

[latex]\begin{aligned}\varepsilon_{p 1} &=C+R=160 \mu \varepsilon+206.1553 \mu \varepsilon=366 \mu \varepsilon \\varepsilon_{p 2} &=C-R=160 \mu \varepsilon-206.1553 \mu \varepsilon=-46.2 \mu \varepsilon \\gamma_{\max } &=2 R=412 \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{260 \mu}{|(0 \mu)-(-320 \mu)|}=\frac{260 \mu}{320 \mu}=0.81250 \quad herefore 2 heta_{p}=39.0939^{\circ} \quad ext { thus, } heta_{p}=19.55^{\circ}[/latex]

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

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

[latex]\gamma_{ abs \max }=\gamma_{\max }=412 \mu rad[/latex]

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

Question 13.37: The strain components εx, εy, and γxy are given for a point ...

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