Engineer's Calculator

Mohr's Circle Calculator

Enter the stresses on an element in plane stress to get the principal stresses, the maximum shear stress, the angles where they act, and the stresses on the element turned to any angle. The circle and the element are drawn as you type. Free, no sign-up.

1. Enter the stresses

Tension is positive. τxy is positive when it points up (+y) on the right-hand (+x) face. θ is counterclockwise from the x axis. The units are labels only: every result is in the units you pick.

2. Mohr's circle and the stress element

Point X is the x face (σx, τxy) and Y is the y face (σy, −τxy). X′ and Y′ are the faces of the turned element. τ is plotted positive downward, so turning the element θ counterclockwise moves its point 2θ counterclockwise around the circle.
As entered
Turned

3. Results

Stresses on the turned element

The largest shear in any direction includes the planes out of the page. In plane stress σz = 0 is the third principal stress, so when σ1 and σ2 have the same sign the largest shear is half the larger of them, not R.

Formulas

C= σx + σy2
R= √( (σx − σy2)² + τxy² )
σ1, σ2= C ± R
tan 2θp= 2τxyσx − σy
σx′= C + σx − σy2 cos 2θ + τxy sin 2θ
σy′= C − σx − σy2 cos 2θ − τxy sin 2θ
τx′y′= −σx − σy2 sin 2θ + τxy cos 2θ
σvm= √( σ1² − σ1σ2 + σ2² )

tan 2θp gives two angles 90° apart. The calculator picks the one where σ1 acts.

Working out the stresses in a beam? Get the moment and shear from the beam calculator and the section properties from the cross section calculator, or check a steel or wood beam to code in the WebStructural beam designer.

Open the beam designer

What Mohr's circle shows

The stresses on a small element depend on which way the element is turned. Mohr's circle draws every possible orientation at once: each point on the circle is one face, with its normal stress across and its shear stress down. The two points where the circle crosses the σ axis have no shear; those are the principal stresses. The bottom and top of the circle are the largest shear, R.

Drawing it by hand

  1. Plot X at (σx, τxy) and Y at (σy, −τxy), with τ positive downward.
  2. Join them. The line crosses the σ axis at the center, C = (σx + σy) / 2.
  3. Draw the circle through X and Y. Its radius is R.
  4. Read σ1 = C + R and σ2 = C − R where the circle meets the σ axis.
  5. The angle on the circle from X to σ1 is 2θp. Turn the element half that much, the same way, to reach the principal faces.

Worked example

Take σx = −20 MPa, σy = 90 MPa and τxy = 60 MPa, the values the calculator starts with. C = (−20 + 90) / 2 = 35 MPa and R = √(55² + 60²) = 81.4 MPa, so σ1 = 116.4 MPa and σ2 = −46.4 MPa. 2θp = atan2(120, −110) = 132.5°, so σ1 acts on the face turned θp = 66.3° counterclockwise from x. The largest in-plane shear is 81.4 MPa, on faces turned 21.3°.

Questions

Which way is positive shear on the circle?

Books differ. This calculator plots τ positive downward, the convention in Hibbeler's Mechanics of Materials, so the circle turns the same way as the element. If your class plots τ upward, the circle is mirrored top to bottom; the principal stresses and the angles on the element are the same.

Is this the same as the transformation equations?

Yes. The circle is a picture of the σx′, σy′ and τx′y′ equations above. The calculator uses the equations for its numbers and the circle to show them.

What about three-dimensional stress?

This calculator is for plane stress, where σz, τxz and τyz are zero, which covers thin plates, beam webs and the surface of most members. It still accounts for the out-of-plane shear in the largest shear in any direction.