Transcript In general stress in a material can be defined by... stresses. For many cases we can simplify to three...

In general stress in a material can be defined by three normal stresses and three shear
stresses. For many cases we can simplify to three stress components. If no stresses
act on the outside surfaces we call the state of stress plane stress. In this case we can
treat the problem in 2 dimensions.
To determine the maximum normal and shear stresses it is necessary that we can
transform stresses from one co-ordinate system to another.
To transform stress from one orientation to
another we need to take into account
1)
magnitude of each stress
2)
orientation of each stress
3)
the orientation of each area the stress
acts on
So the transformation is similar to that used for
force components but is a little more involved.
Take sections
Draw the free
body diagram
Force
equilibrium
The state of plane stress at a point on the surface of the airplane fuselage is
represented on the element oriented as shown in the figure. Represent the state of
stress at the point on an element that is oriented 30° clockwise from the position.
Step 1
Take sections
A
B
Step 2
Draw the two free body diagrams.
A
B
Step 3
Force Equilibrium for A
A
F
x
0
 x A  50A cos 30 cos 30  25A cos 30sin 30
 80A sin 30 sin 30  25A sin 30 cos 30  0
 x  4.15MPa
B
A
F
y
0
 xy A  50A cos 30sin 30  25A cos 30 cos 30
 80A sin 30 cos 30  25A sin 30sin 30  0
 xy  68.8MPa
B
General Equations of Plane-Stress
Transformation
 x 
 y 
 x  y
2
 x  y
2
 xy  


 x  y
2
 x  y
2
 x  y
2
cos 2   xy sin 2
cos 2   xy sin 2
sin 2   xy cos 2
 y 
 x  y
2

 x  y
2
cos 2   xy sin 2
 80  50  80  50
 y 

cos  60  25 sin  60
2
2
 25.85MPa
B
Step 3
Force Equilibrium for A
Answer
 y  0.2 x
 xy  0.2 x
Graph showing the variation of normal stress
and shear stress with angle.
The principal in-plane stresses are the
maximum and minimum stresses. These will
occur at the maxima and minima of the
curve.
d x1
  x   y sin 2  2 xy cos 2  0
d
2 xy
tan 2 p 
 x  y
 1, 2 
 x  y
2
  x  y 
2
   xy
 
 2 
2
The magnitude and direction of the principal stresses can therefore be
defined as a circle.
Mohr’s circle can be used as a graphical
technique to transform stress and/or strain.
Stress transformations
  x  y    x  y 
  
 cos 2   xy sin 2
 x  
 2   2 
 xy
 x  y 
 sin 2   xy cos 2
 
 2 
Combine and add
2

  x   y 
  x  y 
2
2
2
















R
 x 

xy 
xy

 2 
2





Circle with radius R
  x  y 
   xy 2
R  
 2 
2
2
Christian Otto Mohr (October 8, 1835 - October 2, 1918) was a
German civil engineer, one of the most celebrated of the nineteenth
century.
Starting in 1855, his early working life was spent in railroad engineering
for the Hanover and Oldenburg state railways, designing some famous
bridges and making some of the earliest uses of steel trusses.
Even during his early railway years, Mohr's interest had been attracted
by the theories of mechanics and the strength of materials, and in 1867,
he became professor of mechanics at Stuttgart Polytechnic and, in
1873, at Dresden Polytechnic. Mohr had a direct and unpretentious
lecturing style that was popular with his students.
Mohr was an enthusiast for graphical tools and developed the method,
for visually representing stress in three dimensions, previously
proposed by Carl Culmann. In 1882, he famously developed the
graphical method for analysing stress known as Mohr's circle and used
it to propose an early theory of strength based on shear stress.
Constructing Mohr’s Circle
1) Co-ordinate system with normal stress +ve to right and shear
stress +ve down
2) Using the sign convention shown plot the center of the circle C at
the average normal stress on the x axis
3) Plot a reference point at x xy – this represents =0
4) Connect A with C and determine the radius.
5) Draw the circle
Principle stresses are at B and D
p is measured counter-clockwise from AC to BC & DC
s is measured clockwise from AC to EC or FC