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​ = 130σy​ = −80MPaprincipal element

At a point in a loaded part the in-plane principal stresses are σ₁ = 130 MPa and σ₂ = −80 MPa (tension positive). Find the normal stress σavg that acts on the planes of maximum in-plane shear stress.

σ1 = 130 MPaσ2 = −80 MPa
σavg = MPa
2.Easy
σx​ = −30σy​ = −50τxy​MPa

A point has σₓ = −30 MPa and σᵧ = −50 MPa, plus some shear stress. On an element rotated by 15° the normal stress in the rotated x′ direction is −1.3 MPa. Find the normal stress σy′ in the rotated y′ direction.

σx = −30 MPaσy = −50 MPaσx′ = −1.3 MPaθ = 15°
σy′ = MPa
3.Medium
σx​ = 60σy​ = 90τxy​ = −40MPa

A point in a loaded part has σₓ = 60 MPa, σᵧ = 90 MPa and τₓᵧ = −40 MPa (tension positive). Find the angle θp from the x-axis to the direction of the largest principal stress, between −90° and 90°.

σx = 60 MPaσy = 90 MPaτxy = −40 MPa
θp = °
4.Medium
σx​ = 120σy​ = −20τxy​ = −40MPa

A point in a loaded part has σₓ = 120 MPa, σᵧ = −20 MPa and τₓᵧ = −40 MPa (tension positive). Find the angle θp from the x-axis to the direction of the largest principal stress, between −90° and 90°.

σx = 120 MPaσy = −20 MPaτxy = −40 MPa
θp = °
5.Hard
σx​ = 50σy​ = −40τxy​ = −40MPa

A point in a loaded part has σₓ = 50 MPa, σᵧ = −40 MPa and τₓᵧ = −40 MPa (tension positive). Find the shear stress τθ on a plane whose normal is rotated 15° counter-clockwise from the x-axis (positive in the same sense as τₓᵧ).

σx = 50 MPaσy = −40 MPaτxy = −40 MPaθ = 15°
τθ = MPa
6.Hard
σx​τxy​top surface of the shaft

A solid shaft of diameter 50 mm carries a bending moment of 1500 N·m together with a torque of 400 N·m. At the top surface, where the bending stress is tensile and largest, find the largest principal stress σ1.

d = 50 mmM = 1500 N·mT = 400 N·m
σ1 = MPa

Answer key

  1. 1. σavg = 25 MPa
  2. 2. σy′ = −78.7 MPa
  3. 3. θp = −55.28 °
  4. 4. θp = −14.87 °
  5. 5. τθ = −57.14 MPa
  6. 6. σ1 = 124.4 MPa

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