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 (counter-clockwise).
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,
σavg = (σx + σy)/2 = (30 + 10)/2 = 20 psi
This value is plotted on the graph at the left. Remember, the average normal
stress, σavg, is always on the horizontal
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,
or (σy, -τxy).
Note that both σx and σy 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
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, σx and σy 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 2θp in
the counter-clockwise direction. Using geometry in the diagram, this
= 20/(30 - 20) = 2
θp = 31.72o
The actual principal stresses can be found using the circle center and the
radius. They are,
σ1 = σavg +
r = 20 + 22.36 = 42.36 psi
σ2 = σavg -
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 2θτ.
From the circle geometry, this angle is
= (30 - 20)/20 = 0.5
The maximum shear stress (or minimum) is simply the radius of the circle which
was found previously to be
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 counter-clockwise.
Using geometry, the stress points on the circle are
σavg + r cos(2θp -
= 20 + 22.36 cos(63.44
- 30) = 38.66
σy′ = σavg -
r cos(2θp - 2θ)
= 20 - 22.36 cos(63.44
- 30) = 1.34 psi
τx′y′ = r sin(2θp - 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.