Mohr’s circle & stress transformation worksheet with answers

Free printable Mohr’s circle & stress transformation worksheet: 6 problems from easy to exam level, with an answer key. Print it or save it as a PDF.

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1.Easy
σx​ = 120σy​ = 70MPaprincipal element

At a point in a loaded part the in-plane principal stresses are σ₁ = 120 MPa and σ₂ = 70 MPa (tension positive). Find the normal stress σavg that acts on the planes of maximum in-plane shear stress.

σ1 = 120 MPaσ2 = 70 MPa
σavg = MPa
2.Easy
σx​ = −40σy​ = −40τxy​ = 20MPa

A point in a loaded part has σₓ = −40 MPa, σᵧ = −40 MPa and τₓᵧ = 20 MPa (tension positive). Find the largest principal stress σ1.

σx = −40 MPaσy = −40 MPaτxy = 20 MPa
σ1 = MPa
3.Medium
σx​ = −40σy​ = −10τxy​ = 40MPa

A point has σₓ = −40 MPa, σᵧ = −10 MPa and τₓᵧ = 40 MPa (tension positive). The element is rotated 15° counter-clockwise. Find the normal stress σy′ on the faces of the rotated element that are perpendicular to the rotated y′-axis.

σx = −40 MPaσy = −10 MPaτxy = 40 MPaθ = 15°
σy′ = MPa
4.Medium
σx​ = −50σy​ = −20τxy​ = −10MPa

A point in a thin plate is in plane stress: σₓ = −50 MPa, σᵧ = −20 MPa and τₓᵧ = −10 MPa, with σz = 0. Find the absolute maximum shear stress τabs at the point (considering all planes in three dimensions).

σx = −50 MPaσy = −20 MPaτxy = −10 MPaσz = 0
τabs = MPa
5.Hard
σx​τxy​top surface of the shaft

A solid shaft of diameter 25 mm carries a bending moment of 100 N·m together with a torque of 250 N·m. At the top surface, where the bending stress is tensile and largest, find the largest principal stress σ1.

d = 25 mmM = 100 N·mT = 250 N·m
σ1 = MPa
6.Hard
σx​ = 10σy​ = 70τxy​ = −10MPa

A point in a thin plate is in plane stress: σₓ = 10 MPa, σᵧ = 70 MPa and τₓᵧ = −10 MPa (σz = 0). Find the von Mises equivalent stress σvM.

σx = 10 MPaσy = 70 MPaτxy = −10 MPa
σvM = MPa

Answer key

  1. 1. σavg = 95 MPa
  2. 2. σ1 = −20 MPa
  3. 3. σy′ = −32.01 MPa
  4. 4. τabs = 26.51 MPa
  5. 5. σ1 = 120.4 MPa
  6. 6. σvM = 67.82 MPa

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