MECH 321 - Week 09 Lecture 1 - Department of Mechanical and

MECH 321 - Solid Mechanics II
Week 9, Lecture 1
Plane Strain and General Equations of
Plane Strain Transformations
September 2, 2014
Page 1
Plane Strain
The general state of strain at a point in a body is represented
by a combination of three components of normal strain,
 x ,  y and  z
And three components of shear strain.
 x ,  y and  z
These six components tend to deform each face of an element
of the material.
The normal and shear strain components at a point will vary
according to the orientation of the element.
September 2, 2014
Page 2
Plane Strain
Strain gauges are often used to measure these components in
specified directions.
In order to perform analysis, design engineers must sometimes
transform these measurements in order to obtain the components
in other directions.
If we restrict our initial investigation to a study of “plane strain”, we
need not consider the effects of the components  z ,  xz and  yz .
This leaves a plane-strained element as being subjected to two
components of normal strain and one component of shear strain.
 x ,  y and  xy
September 2, 2014
Page 3
Plane Strain
The deformations of an element caused by these strains…
Note that, although plane strain and plane stress each have
three components acting in the same plane, plane stress does
not necessarily cause plane strain and vice versa.
The reason for this is the Poisson effect.
September 2, 2014
Page 4
Plane Strain
If an element is subjected to plane stress σx and σy, not only are
normal strains εx and εy produced, but there is also an
associated normal strain, εz.
This is obviously not a case of plane strain.
Unless v = 0 the Poisson effect will prevent the simultaneous
occurrence of plane strain and plane stress.
It should also be noted that since shear stress and shear strain
are not affected by Poisson’s ratio, a condition of
 xz   yz  0
requires that
September 2, 2014
 xz   yz  0
Page 5
General Equations of Plane Strain
Transformation
Sign convention: Normal strains are positive if they
cause elongation along the x and y axes, respectively.
The shear strain is positive if the interior angle AOB
becomes smaller than 90°.
September 2, 2014
Page 6
General Equations of Plane Strain
Transformation
Normal strains and shear strain shown together.
September 2, 2014
Page 7
General Equations of Plane Strain
Transformation
Normal and shear strain equations
 x 
 y 
x y
2
x y
2
 x y 


x y
2
x y
2
x y
  
2
2

September 2, 2014
cos 2 
cos 2 
 xy
2
 xy
2
sin 2
sin 2
 xy

 sin 2 
cos 2
2

Page 8
General Equations of Plane Strain
Transformation
Principal strains.
Like stress, the orientation of an element at a point can be
determined such that the element’s deformation is represented
by normal strains only (no shear strains).
When this occurs the normal strains are referred to as “principal
strains”.
If the material is isotropic, the axes along which the strains occur
coincide with the axes that define the planes of principal stress.
September 2, 2014
Page 9
General Equations of Plane Strain
Transformation
The direction of the axes and the two values of the
principal strains are...
 xy
tan 2 p 
x y
 1, 2 
September 2, 2014
x  y
2

x y

2


  xy 
  

 2 

2
2
Page 10
General Equations of Plane Strain
Transformation
Maximum In-Plane Shear Strain (orientation and
magnitude).
x y
tan 2 s   
 
xy

 max
in  plane
2

x y

2

 avg 
September 2, 2014




  xy 


  
 2 

2
2
x  y
2
Page 11
Next Time
Mohr’s Circle for Plane Strain
September 2, 2014
Page 12

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