Review-Stresses

Equilibrium of Deformable Body
Review – Static Equilibrium
If a body is in static equilibrium under the action applied
external forces, the Newton’s Second Law provides us
six scalar equations of equilibrium
 F  0;  F  0;  Fz  0
 M  0;  M  0;  M  0
x
y
x
y
z
•The sum of all the forces in x, y, and z coordinate
directions are zero
•The sum of moments of all the forces about axes x,y,
and z are zero
Considering Static Equilibrium of the element, we have
*
Normal and shear stress components
p
actingg on opposite
pp
sides of an element must be equal in magnitude and
opposite in direction
*
Shear stress components satisfy moment equilibrium
 xy   yx ;  xz   zx ;  yz   zy
PRINCIPAL STRESSES
State of stress at a point in a material is
completely defined by the stresses acting on
the planes of a cubical volume element whose
edges are parallel to coordinate directions.
Often,, it is important
p
to determine the state of stress on a pplane
at some angle to the coordinate axes. Mostly, the state of stress
on a plane on which no shear stresses act is important for design
purposes.
purposes
A plane, where no shear stresses act is called a principal
plane and the normal stress that acts on such a plane is
called principal stress.
Mohr’s Circle - Its Use and Limitations
Consider a stress state at a point is such that on one of the
coordinate plane, there are no shear stresses and only normal
stress (may be zero or not) is present. That normal stress is one
off the
th principal
i i l stress
t
as shown
h
i figure.
in
fi
You can see that on an plane perpendicular to z-axis only sz ( may
be tensile, compressive or zero) is present. Then remaining two
principal
p
c p stresses
s esses ccan be de
determined
e
ed us
using
g bbiaxial sstress
ess field
e d and
d
Mohr Circle.
Biaxial stress field in xy-plane can be shown as above.
Using stress transformation equation, the normal stress  x along x
axis and  xy the shear stress (from last terms Mech.
Mech Of solids) can
be written as.
1
 x   y   12  x   y  cos 2   xy sini 2
2
1
 xy    x   y  sin 2   xy cos 2
2
 x 
(A)
 x will become principal stress if
 xy= 0. Using this condition,
the remaining principal stresses can be determined as.
p 
1
1
x y  

2
2

  y    xy2
2
x
and the orientation of the principal planes with respect to xaxis is given by
tan 2 
2 xy
x  y
The same results can be expressed graphically using Mohr’s
Circle. Rearranging
g g the equations
q
(A),
( ), we obtain
1

 1
 x  2  x   y    2  x   y  cos 2   xy sin 2
1
 xy    x   y  sin 2   xy cos 2
2
Squaring the above equations, then adding, gives
2
2
1


1

2
2
 x  2  x   y     xy   2  x   y     xy
This is the equation of a circle whose center is at
1/ 2
2
andd whose
h
radius
di is
i given
i
by
b  1


2
1

 2  x   y  ,0 
   x   y     xy 

2

Every point on the circle defines the stress state acting on
planes at any angle q from the original x or y axis.
For the correct construction of Mohr’s circle, certain rules are followed and a consistent
handling of positive and negative stress is essential, only if proper orientation of planes is
desired. No such concern is required
q
if only
y the magnitudes
g
of the p
principal
p stresses are
sought.
Although various conventions are in use, we follow the convention given in Hibbler’ Book.
1.Normal stresses are plotted to scale along the abscissa (horizontal axis) with tensile
stresses considered positive and compressive stresses negative.
2.Shear stresses are plotted along the ordinate (vertical axis) with positive direction
downward to the same scale as used for normal stresses.
A shear stress that would tend to cause counter-clock wise rotation of the stress element in
the physical plane is considered positive while negative shear stress tend to cause clockwise
rotation.
3.Angle between lines of direction on the Mohr plot are twice the
indicated angle
g on the physical
p y
plane.
p
The angle ‘2' on the Mohr circle is measured in the same direction
as the angle  for the orientation of the plane in physical plane.
According definition, the values of  corresponding to the
points D and B (where  = 0) are the principal stresses. The
radius of the Mohr’s circle gives the maximum in-plane shear
stress
Three -Dimensional Mohr’s Plot
• As mentioned previously, Mohr’s circle can be drawn to determine principal stresses
only if one of the three principal stresses is known.
• Since the known principal stress is also a normal stress, it can be plotted on s axis and
circles can be drawn between all the principal stresses as shown.
then the maximum absolute shear stress is equal to radius of largest Mohr’s circle.
Absolute maximum shear stress is given as
 max 
1
 pmax   p min 
2
Static Failure Theories
x 
M *(d / 2) M *(d / 2) 32 M 32*6000



 18108lb / in 2
4
3
3
 d / 64
d
 *(1.5)
I
 zx 
Tr
T (d / 2)
16T 16*8000



 48288lb / in 2
4
3
3
J ( d / 32))  d
 (1.5)
( )
v 
4V
4V
4*1000


 754.5lb / in 2
2
2
3 A 3( d / 4) 33*(( *(1.5)
(1.5) / 4)
At Point A
 z
1, 2   x
 2

  x  z 
2
2



   xz  58184, 40076lb / in

 2 
 max  49130lb / in 2
2
At point B
 xy  48288  754  49042lb / in 2
 1 ,  2  49042lb / in 2
 max  49042lb / in 2

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