Analysis of Stress and Strain

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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?
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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

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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
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Transformation Equations
angle between x1 and x axes, measured
counterclockwise
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Plane Stress Special Cases
Uniaxial Stress
Pure Shear
Biaxial Stress
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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.
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Principal Stresses
Principal stresses maximum and minimum normal
stresses. Principal planes the planes on which
the principal stresses act
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Principal Stresses
Shear stress on the principal planes
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Principal Stresses
Example 2 Find principal stresses of an element
which is in pure shear.
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Maximum Shear Stresses
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Plane Stress
Example 3 Find the principal stresses and
maximum shear stresses and
show them on a sketch of a properly oriented
element.
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Mohrs Circile For Plane Stress Equations of
Mohrs Circle
Transformation equations
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Two Forms of Mohrs Circle
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Construction of Mohrs Circle
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Applications of Mohrs Circle

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

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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.
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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.
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Plane Strain
Transformation Equations
Principal Strains
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Measurement of Strain Electrical Resistance
Strain Gages
Principle length changes
electrical resistance change
Measurement of Principal Strains need three
gages!
Why?
How?
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Hookes Law For Plane Stress
Assumptions - -
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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.
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Strain Energy Density
Strain Energy