# Compound Stress and Strain Part 2

**Contents**

- Introduction
- Mohr's Stress Circle
- Pure Compression.
- Principal Stresses Equal Tension And Compression.
- A Two-dimensional Stress System.
- Principal Strains In Three Dimensions.
- Principal Stresses Determined From Principal Strains.
- Analysis Of Strain.
- Mohr's Strain Circle.
- Volumetric Strain.
- Strain Energy
- Shear Strain Energy.
- Page Comments

## Introduction

This is the second part in our discussion on the topic of Compound Stress and Strain. In this section we analyse the state of Stress at a point with a graphical representation using Mohr's Circle. In the latter half, we also look at how Mohr's circle can be adapted to represent direct or linear strain, and shear strain.*See Also*the section on Compound Stress and Strain Part 1 .

## Mohr's Stress Circle

**Mohr**'

**s circle**is a two-dimensional graphical representation of the state of stress at a point. The abscisa and ordinate of each point on the circle are the normal stress and shear stress components respectively, acting on a particular cut plane with a unit vector with components , , . In other words, the circumference of the circle is the locus of points that represent the state of stress on individual planes at all their orientations.

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**diameter**of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle.

- Draw a line such that represents and . Note that the positive direction (Tension) is to the right.
- On as a Diameter draw a Circle with centre
- On this drawing ,
**but**this is not a necessary condition.

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- The radius represents the plane of
- The radius represents the plane of
- The Plane is obtained by rotating through and if on the Stress Circle is rotated through in the same direction, then the radius is obtained. This will be shown to represent the plane . (Note that could equally well be obtained by rotating clockwise through corresponding to rotating clockwise through )
- Draw perpendicular to

**shear stress**is defined as the component of stress coplanar with a material cross section. Shear stress arises from a force vector perpendicular to the surface normal vector of the cross section.

- The Stresses on the plane , perpendicular to , are obtained from the radius ' which is at to .

- The Maximum Shear Stress occurs when (i.e. ) and is equal in magnitude to
- The maximum Value of is obtained when is a tangent to the Stress Circle.

## Pure Compression.

If is a Compressive Stress then the other Principle Stress is zero.##### MISSING IMAGE!

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## Principal Stresses Equal Tension And Compression.

Let be the angle measured anticlockwise from the Plane of Tensile.##### MISSING IMAGE!

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##### Example - Example 1

**position**of the Plane across which the resultant Stress is most inclined to the Normal and determine the

**value**of this resultant Stress.

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## A Two-dimensional Stress System.

It has been shown that every system can be reduced to the action of pure normal Stresses on the Principal Planes.##### MISSING IMAGE!

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- Strain in the direction of
- Strain in the direction of

- Strain in the direction of
- Strain in the direction of

- Strain in the direction of

- Strain in the direction of

## Principal Strains In Three Dimensions.

Using a similar argument to that used in the previous paragraph t can be shown that the Principal Strains in the direction , and are given by :##### MISSING IMAGE!

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##### Example - Example 4

**ratio**of the Stresses and their

**value**if the greatest is

- The
**ratio**of the Stresses are - , and

## Analysis Of Strain.

If and are the linear and Shear Strains in the plane , then we require an expression for , the linear Strain in a direction inclined at an angle to in terms of and .##### MISSING IMAGE!

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**Shear strain**refers to a

**deformation**of a solid body in which a plane in the body is displaced parallel to itself relative to parallel planes in the body; quantitatively, it is the displacement of any plane relative to a second plane, divided by the perpendicular distance between planes.

##### Example - Example 5

**magnitude**and

**direction**of the Principal Strains in this plane. If there is no Stress perpendicular to the given Plane, determine the

**Principal Stresses**at the point. and

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## Mohr's Strain Circle.

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## Volumetric Strain.

A rectangular solid of sides , , is under the action of three principal Stresses , and .##### MISSING IMAGE!

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**Principal Stresses**in which case: Volumetric Strain =

## Strain Energy

The Strain Energy is the Work done by the Stresses in Straining material.It is sufficiently general to consider a unit cube acted upon by the three Principal Stresses , and . If the corresponding Strains are , and then, since the Stresses are applied gradually from zero, the Total work done = .##### Example - Example 6

**volumetric Strain**and the

**resilience**. and

- The
**volumetric Strain**is - The
**resilience**is

## Shear Strain Energy.

**volumetric Strain Energy**

**Shear strain Energy**is defined as the Total Strain Energy and the Volumetric Strain Energy.

**Note:**The relationship between and will be discussed in "Elastic Constants".