Initial Local Stress Element


For a particular point in a structure, the local stresses were found to
be 30 psi and 10 psi in the horizontal (x) and vertical (y) directions, respectively.
The shear stress was found to be 20 psi.
To help understand this stress state, a Mohr's circle will be constructed and
used to find the 1) principal direction and principal stresses, 2) maximum shear
stress direction and the maximum shear stress, and 3) the stress state if the
element is rotated 15 degrees (counterclockwise).

Basic Mohr Circle for Stress 

To construct a Mohr circle for a given stress state, first find the average
normal stress, which will be the location of the circle's center. For this problem,
it is
σ_{avg} = (σ_{x} + σ_{y})/2 = (30 + 10)/2 = 20 psi
This value is plotted on the graph at the left. Remember, the average normal
stress, σ_{avg}, is always on the horizontal
axis.
Next, at least one point on the circle's edge is needed to define the circle's
radius. There are two points that can be found immediately without any calculation,
(σ_{x}, τ_{xy})
or (σ_{y}, τ_{xy}).
Note that both σ_{x} and σ_{y} are plotted on the same horizontal axis. This may seem strange
at first, but it works well since they are both normal stresses.
Generally, a line is drawn between the two points and the center. All three
should be in a straight line. If they are not, this is an early indication something
is wrong.
The circle radius can be determined by using the right triangle with vertices
(20, 0), (30, 0) and (30, 20).

Principal Stresses and Direction 

The principal direction is where the normal stresses, σ_{x} and σ_{y} are at a maximum or minimum. This condition is represented
by the intersection of the circle and the horizontal axis (shown in the diagram
as a orange line). To get to this condition, the current stress state (blue line),
needs to be rotated to the horizontal by an angle of 2θ_{p} in
the counterclockwise direction. Using geometry in the diagram, this
gives
tan(2θ_{p})
= 20/(30  20) = 2
θ_{p} = 31.72^{o}
The actual principal stresses can be found using the circle center and the
radius. They are,
σ_{1} = σ_{avg} +
r = 20 + 22.36 = 42.36 psi
σ_{2} = σ_{avg} 
r = 20  22.36 = 2.36 psi

Maximum Shear Stresses and Direction 

The maximum shear stress occurs at the top or bottom of the circle. Thus, the
current stress state, blue line, needs to be rotated clockwise (negative direction)
by angle 2θ_{τ}.
From the circle geometry, this angle is
tan(2θ_{τmax})
= (30  20)/20 = 0.5
θ_{τmax}= 13.28^{o}
The maximum shear stress (or minimum) is simply the radius of the circle which
was found previously to be
τ_{max} =
r = 22.36 psi

Stress State at an Arbitrary Angle


Mohr's circle can also be used to find a new stress state for an arbitrary rotation
angle. The new stress state is identified by rotating the current stress state
(blue line) by twice the angle, 2θ. The new stress state
is shown on the diagram as a green line. Positive rotation angles are counterclockwise.
Using geometry, the stress points on the circle are
σ_{x′} =
σ_{avg} + r cos(2θ_{p} 
2θ)
= 20 + 22.36 cos(63.44
 30) = 38.66
psi
σ_{y′} = σ_{avg} 
r cos(2θ_{p}  2θ)
= 20  22.36 cos(63.44
 30) = 1.34 psi
τ_{x′y′} = r sin(2θ_{p}  2θ)
= 22.36 sin(63.44  30) = 12.32 psi
The initial stress state (blue line) and rotated stress state (green line) are also shown in the diagram as a small stress element to understand the orientation of the Mohr's circle stresses.
