Analysis of Stress and Strain
Review:
- Axially loaded Bar
- Torsional shaft
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
x 
x 
1

x
y1
1 y1

y

2
y

2
 x 
2
 x 
2

y
cos 2   xy sin 2
y
cos 2   xy sin 2
2
x 

 x 
y
x 
1
y1
x 
sin 2   xy cos 2
angle between x1 and x axes, measured counterclockwise
y
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
1 
2 
x 
y

 x 


2


 x 


2

2
x 
2
y
2
y

2
   xy


2
y

2
   xy


Shear stress on the principal planes:
1   2
Principal Stresses
Example 2: Find principal stresses of an element which is in pure shear.
Maximum Shear Stresses
 max 
 x 


2

2
y

 2
2
   xy  1

2

 s  p 
1
 s  p 
2
 
x  y
2
1
2

4

4
Plane Stress
Example 3: Find the principal stresses and maximum shear stresses and
show them on a sketch of a properly oriented element.
Mohr’s Circile For Plane Stress –
Equations of Mohr’s Circle
Transformation equations:
x 
x 
1
x
1 y1

y

2
 x 
2
 x 
2
y
y
cos 2   xy sin 2
sin 2   xy cos 2

x1
  ave
 ave 

2
  x1 y 1  R
x 
2
2
y
,
2
R 
 x 


2

2
y

2
   xy


Two Forms of Mohr’s Circle
Construction of Mohr’s Circle
Applications of Mohr’s Circle
- Stresses on inclined element
- Principal stresses and maximum shear stresses
Applications of Mohr’s Circle
Example: An element in plane stress at the surface of a large machine
is subjected to stresses  x  15000 psi ,  xy  5000 psi
Using Mohr’s 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:
x 
x y
1
y 
2
1 y1
2

x y
2

cos 2 
2
x y
1
x

x y
cos 2 
sin 2 
2
Principal Strains:
sin 2
2
2
x y
 xy
 xy
x y  x y
sin 2
1
1
2
 xy
cos 2
2
1 
2 
x y
2
2
2
2


  xy 
 


 2 




x y


2


  xy 
 


 2 



2
x y
2
x y


2

Measurement of Strain – Electrical
Resistance Strain Gages
Principle: length changes
electrical resistance change
Measurement of Principal Strains: need three gages!
Why?
How?
x 
x y
1
y 
2
1 y1
2

x y
2

cos 2 
2
x y
1
x

x y
2
sin 2
2
cos 2 
2
x y
 xy
sin 2 
 xy
2
 xy
2
cos 2
sin 2
Hooke’s Law For Plane Stress
Assumptions:
x 
y 
 xy 
1
E
1
E

x
 
 
x
y


x 
y


 xy
y

1 
G 
E
2 1  


2

E
1 
 xy  G  xy
G
2
E
x
 
x
y

y
Hooke’s 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:
u 
1
2

x
 x   y  y   xy  xy 