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

An in-depth explanation of shear strain, its definition, and its relationship with shear stress. It covers the concept of couples, the generation of moments, and the calculation of shear strain using the angle of deformation. The document also includes problem-solving examples and homework questions.

What you will learn

  • How do forces acting on opposite faces of an element produce a shear strain?
  • How can the angle of deformation be used to calculate shear strain?
  • How does shear strain relate to shear stress?
  • What is a couple in the context of shear strain?
  • What is shear strain and how is it defined?

Typology: Study Guides, Projects, Research

2021/2022

Uploaded on 09/27/2022

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Mechanics of Materials
CIVL 3322 / MECH 3322
Shear Strain
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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

F02_

Shear Strain

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

F02_

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.

F02_

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.

F02_

Shear Strain This will always be the case

when an element in under shear.

F02_

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.

F02_

Shear Strain This is not the angle θ’.

The sine of this angle is δ

x

L

F02_

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

F02_

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

F02_

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

F02_

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.