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}=515 \mu \varepsilon \quad \varepsilon_{y}=-265 \mu \varepsilon \quad \gamma_{x y}=-1,030 \mu rad[/latex]

Accepted Answer

The basic Mohr’s circle is shown.

[latex]\begin{aligned}&C=\frac{(515 \mu)+(-265 \mu)}{2}=125 \mu \varepsilon \&R=\sqrt{(390 \mu)^{2}+(515 \mu)^{2}}=646.0070 \mu \varepsilon\end{aligned}[/latex]

[latex]\begin{aligned}\varepsilon_{p 1} &=C+R=125 \mu \varepsilon+646.0070 \mu \varepsilon=771 \mu \varepsilon \\varepsilon_{p 2} &=C-R=125 \mu \varepsilon-646.0070 \mu \varepsilon=-521 \mu \varepsilon \\gamma_{\max } &=2 R=1,292 \mu rad\end{aligned}[/latex]

The magnitude of the angle [latex]2 heta_{p}[/latex] between point x and point 1 (i.e., the principal plane associated with [latex]\varepsilon_{p 1}[/latex]) is found from:

[latex] an 2 heta_{p}=\frac{1,030 \mu}{|(515 \mu)-(-265 \mu)|}=\frac{1,030 \mu}{780 \mu}=1.32051 herefore 2 heta_{p}=52.8640^{\circ} ext { thus, } heta_{p}=26.4^{\circ}[/latex]

By inspection, the angle [latex] heta_{p}[/latex] from point x to point 1 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 }=1,292 \mu rad[/latex]

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

Question 13.34: The strain components εx, εy, and γxy are given for a point ...

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