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Analysis of Stress and Strain

Review

- Torsional shaft

- Axially loaded Bar

Questions (1) Is there any general method

to determine stresses on any arbitrary plane

at one point if the stresses at this point

along some planes are known? (2) For a

general loaded member, how many planes on which

stresses are known are required to

determine the stresses at any plane at one point?

Analysis of Stress and Strain

State of stress at one point

Stress element

- Use a cube to represent stress element. It is

infinitesimal in size. - (x,y,z) axes are parallel to the edges of the

element - faces of the element are designated by the

directions of their - outward normals.

Sign Convention

- Normal stresses tension -

compression. - Shear stresses the directions

associated with its subscripts are - plus-plus

or minus-minus - - the

directions associated with its subscripts are - plus-minus

or minus-plus

Plane Stress

Definition Only x and y faces are subjected to

stresses, and all stresses are

parallel to the x and y axes.

Stresses on inclined planes

Transformation Equations

angle between x1 and x axes, measured

counterclockwise

Plane Stress Special Cases

Uniaxial Stress

Pure Shear

Biaxial Stress

Plane Stress

Example 1 A plane-stress condition exists at a

point on the surface of a loaded structure,

where the stresses have the magnitudes and

directions shown on the stress element of the

following figure. Determine the stresses acting

on an element that is oriented at a clockwise

angle of 15o with respect to the original

element.

Principal Stresses

Principal stresses maximum and minimum normal

stresses. Principal planes the planes on which

the principal stresses act

Principal Stresses

Shear stress on the principal planes

Principal Stresses

Example 2 Find principal stresses of an element

which is in pure shear.

Maximum Shear Stresses

Plane Stress

Example 3 Find the principal stresses and

maximum shear stresses and

show them on a sketch of a properly oriented

element.

Mohrs Circile For Plane Stress Equations of

Mohrs Circle

Transformation equations

Two Forms of Mohrs Circle

Construction of Mohrs Circle

Applications of Mohrs Circle

- Stresses on inclined element
- Principal stresses and maximum shear stresses

Applications of Mohrs Circle

Example An element in plane stress at the

surface of a large machine is subjected to

stresses

Using Mohrs circle, determine the

following quantities (a) the stresses acting on

an element inclined at an angle of 40o, (b) the

principal stresses and (c) the maximum shear

stress.

Plane Strain

Definition Only x and y faces are subjected to

strains, and all strains are

parallel to the x and y axes.

Note Plane stress and plane strain do not occur

simultaneously.

Plane Strain

Transformation Equations

Principal Strains

Measurement of Strain Electrical Resistance

Strain Gages

Principle length changes

electrical resistance change

Measurement of Principal Strains need three

gages!

Why?

How?

Hookes Law For Plane Stress

Assumptions - -

Hookes Law For Plane Stress

Example A 450 strain rosette consists of three

strain gages arranged to measure strain in two

perpendicular directions and also at a 450 angle

between them, as shown in the Following figure.

The rosette is bounded to the surface before it

is loaded. It is measured that the strains in

three gages (A,B,C) are 0.0003, 0.0001 and

0.00011 respectively. Find principal stresses at

this point.

Strain Energy Density

Strain Energy

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