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).

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).

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 counter-clockwise 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,

     σ₁ = σ_avg + r = 20 + 22.36 = 42.36 psi

     σ₂ = σ_avg - r = 20 - 22.36 = -2.36 psi

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_τ-max) = (30 - 20)/20 = 0.5

     θ_τ-max= 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 counter-clockwise. 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.