Understanding Shear Strain in Mechanics of Materials, Study Guides, Projects, Research of Acting

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Mechanics of Materials

CIVL 3322 / MECH 3322

Shear Strain

¢ Axial strain is the ratio of the deformation of

a body along the loading axis to the original

un-deformed length of the body

¢ The units of axial strain are length per

length and are usually given without

dimensions

Shear Strain

¢ If we look at a material undergoing a shear

stress, we have to go back to statics to start

¢ We need to remember about couples

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Shear Strain

We can start by looking at an element of material undergoing a shear stress (the red arrows)

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Shear Strain So the bottom face has a force

directed parallel to it that it equal and opposite to the force that is acting parallel to the top face. This assumes that the area of the top and bottom face are equal.

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Shear Strain These two forces equal in

magnitude but opposite in direction generate a couple on the element. Since the element is in equilibrium, something must offset the moment produced by this couple.

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Shear Strain This will always be the case

when an element in under shear.

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Shear Strain The shear acting on the four

faces of the element cause a deformation of the element. If the lower left corner is considered stationary, we can look at how much the upper left corner of the element moves.

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Shear Strain This is not the angle θ’.

The sine of this angle is δ

x

L

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Shear Strain If δ

x is very small with respect to L, which is generally the case, then the value of the angle in radians is approximately equal to the sign of the angle. δ

x

L

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Shear Strain It is labeled with an xy subscript

because we are looking at the shear strain in the xy plane I have labeled it with a y subscript because it is the angle made with the y-axis. γ

y

= δ

x

L

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Shear Strain The shear is actually the

difference between the original angle between the side along the y-axis and the side along the x-axis and the angle after loading θ’ γ

y

= δ

x

L

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Shear Strain If there is also a deformation

above or below the x-axis, it must also be included to solve for θ’ θ ' = π 2 − γ

xy

• P02_
• Problem 2.