A thin plate of width b, thickness t, and length L is placed...

A thin plate of width $b$, thickness $t$, and length $L$ is placed between two frictionless rigid walls a distance $b$ apart and is acted on by an axial force $P$. The material properties are Young's modulus $E$ and Poisson's ratio $ u$. (a) Find the stress and strain components in the $xyz$ coordinate system. (b) Find the displacement field.

Transcript

00:01 Hello students, in the first part of the question we have to find the stress and strain along the x, y and z direction.

00:07 According to the figure in the first part of the question we can write the stress along the x axis will be equal to the we know that stress is equals to load or we can say that the force per unit area or load upon area.

00:25 So along the x direction the load is given as the p and the figure we can see that and the cross -sectional area along the x axis will be equal to the p into t.

00:39 P is the width and t is the thickness.

00:42 So along the x axis stress formula will be this.

00:47 But along the z direction, stress is equals to the along the z direction area will be b into l but the applied force is zero so this will be equal to the zero.

01:06 And one more thing we can see that the strain, these are the stress and strain along the y axis is equals to the zero because y axis sheet along the sheet in according to the y axis is confined.

01:31 So there will be no change in in the direction of y so the strain will be zero in this direction.

01:39 We know the formula of the strain along the y direction is equals to stress along the y direction upon young modulus minus poisson's ratio, stress along the x direction upon young modulus minus poisson's ratio, stress along the z direction upon young modulus.

02:02 So we can see that the stress along the z direction is zero and the strain along the y direction is also zero.

02:10 So we can say that sigma y that means stress along the y direction will be the poisson's ratio into strain stress along the x direction.

02:26 Now we can write this is equals to the p mu upon p t.

02:38 Now we will write again the formula strain along the x direction sigma x upon e minus nu sigma y upon e minus nu sigma z upon e...

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