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In many situations physical constraints prevent strain
Chapter 2, Problem 2.74(choose chapter or problem)
In many situations physical constraints prevent strain from occurring in a given direction. For example, \(\epsilon_{z}=0\) in the case shown, where longitudinal movement of the long prism is prevented at every point. Plane sections perpendicular to the longitudinal axis remain plane and the same distance apart. Show that for this situation, which is known as plane strain, we can express \(\sigma_{z}, \epsilon_{x}, \text { and } \epsilon_{y}\) as follows:
\(\begin{aligned} \sigma_{z} & =\nu\left(\sigma_{x}+\sigma_{y}\right) \\ \epsilon_{x} & =\frac{1}{E}\left[\left(1-\nu^{2}\right) \sigma_{x}-\nu(1+\nu) \sigma_{y}\right] \\ \epsilon_{y} & =\frac{1}{E}\left[\left(1-\nu^{2}\right) \sigma_{y}-\nu(1+\nu) \sigma_{x}\right] \end{aligned}\)
Questions & Answers
QUESTION:
In many situations physical constraints prevent strain from occurring in a given direction. For example, \(\epsilon_{z}=0\) in the case shown, where longitudinal movement of the long prism is prevented at every point. Plane sections perpendicular to the longitudinal axis remain plane and the same distance apart. Show that for this situation, which is known as plane strain, we can express \(\sigma_{z}, \epsilon_{x}, \text { and } \epsilon_{y}\) as follows:
\(\begin{aligned} \sigma_{z} & =\nu\left(\sigma_{x}+\sigma_{y}\right) \\ \epsilon_{x} & =\frac{1}{E}\left[\left(1-\nu^{2}\right) \sigma_{x}-\nu(1+\nu) \sigma_{y}\right] \\ \epsilon_{y} & =\frac{1}{E}\left[\left(1-\nu^{2}\right) \sigma_{y}-\nu(1+\nu) \sigma_{x}\right] \end{aligned}\)
ANSWER:
Step 1 of 4
The diagram can be given as,
The strains in the z-direction can be given as,
