Mohr's Circle and Combined Stress Calculator

What is Mohr's circle?

The stress at a point depends on the orientation of the plane you cut through it. Rotate the same element and the normal and shear stresses on its faces change. Mohr's circle shows that change in one picture: normal stress on the horizontal axis, shear stress on the vertical axis, and every possible orientation lying on a circle.

The centre of the circle is the average stress: σ_avg = (σx + σy) / 2. Its radius is R = √(((σx − σy) / 2)² + τxy²).

Principal stresses and maximum shear

The two points where the circle crosses the horizontal axis are the principal stresses: σ1 = σ_avg + R and σ2 = σ_avg − R. The shear stress on those planes is zero. The angle of the principal direction follows from tan 2θp = 2 × τxy / (σx − σy).

The top and bottom of the circle give the maximum in plane shear stress: τmax = R. Those planes are at 45° to the principal planes and carry a normal stress equal to the average stress.

Sign convention

• Tensile stress is entered positive, compressive stress negative.
• The shear stress τxy is positive when it acts upward on the right hand face of the element.
• The angle θ is positive counterclockwise from the x axis.
• The shear axis is drawn positive downward. Rotating the element by θ then means moving 2θ around the circle in the same direction.

Example

Take σx = 50 MPa, σy = −10 MPa and τxy = 40 MPa. The average stress is 20 MPa and the radius √(30² + 40²) = 50 MPa. The principal stresses are 70 MPa and −30 MPa and the maximum shear stress is 50 MPa. Since tan 2θp = 80 / 60, the principal direction is 26.6° counterclockwise from the x axis.

The third principal stress

In plane stress the principal stress normal to the plane is zero. When the two in plane principal stresses have the same sign, the true maximum shear stress does not occur in the plane but out of it, and equals half the larger principal stress. The calculator reports this absolute value separately.

Limits

The tool performs the stress transformation only. To decide whether the result is safe, the principal stresses must be assessed against the material strength with a yield or fracture criterion.