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}=670 \mu \varepsilon \quad \varepsilon_{y}=455 \mu \varepsilon \quad \gamma_{x y}=-900 \mu rad[/latex]

Accepted Answer

The basic Mohr’s circle is shown.

[latex]\begin{aligned}&C=\frac{(670 \mu)+(455 \mu)}{2}=562.5 \mu \varepsilon \&R=\sqrt{(107.5 \mu)^{2}+(450 \mu)^{2}}=462.6621 \mu \varepsilon\end{aligned}[/latex]

[latex]\begin{aligned}&\varepsilon_{p 1}=C+R=562.5 \mu \varepsilon+462.6621 \mu \varepsilon=1,025 \mu \varepsilon \&\varepsilon_{p 2}=C-R=562.5 \mu \varepsilon-462.6621 \mu \varepsilon=99.8 \mu \varepsilon \&\gamma_{\max }=2 R=925 \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{900 \mu}{|(670 \mu)-(455 \mu)|}=\frac{900 \mu}{215 \mu}=4.18605 \quad herefore 2 heta_{p}=76.5645^{\circ} ext { thus, } heta_{p}=38.3^{\circ}[/latex]

By inspection, the angle [latex] heta_{p}[/latex] from point x to point 1 is turned clockwise.

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}=1,025.1621 \mu-(0 \mu)=1,025 \mu rad[/latex]

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

Question 13.36: The strain components εx, εy, and γxy are given for a point ...

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