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​ = 30σy​ = −40MPaprincipal element

At a point in a loaded part the in-plane principal stresses are σ₁ = 30 MPa and σ₂ = −40 MPa (tension positive). Find the normal stress σavg that acts on the planes of maximum in-plane shear stress.

σ1 = 30 MPaσ2 = −40 MPa
σavg = MPa
2.Easy
σx​ = −20σy​ = −70τxy​ = 40MPa

A point in a loaded part has σₓ = −20 MPa, σᵧ = −70 MPa and τₓᵧ = 40 MPa (tension positive). Find the smallest principal stress σ2.

σx = −20 MPaσy = −70 MPaτxy = 40 MPa
σ2 = MPa
3.Medium
σx​ = 70σy​ = −40τxy​ = 10MPa

A point in a loaded part has σₓ = 70 MPa, σᵧ = −40 MPa and τₓᵧ = 10 MPa (tension positive). Find the angle θp from the x-axis to the direction of the largest principal stress, between −90° and 90°.

σx = 70 MPaσy = −40 MPaτxy = 10 MPa
θp = °
4.Medium
σx​ = 30σy​ = −10MPaτxy​ = ?

A point has σₓ = 30 MPa and σᵧ = −10 MPa. How large can the shear stress τxy (magnitude) become before the largest principal stress reaches 75 MPa?

σx = 30 MPaσy = −10 MPaσ1 = 75 MPa
τxy = MPa
5.Hard
σx​τxy​top surface of the shaft

A solid shaft of diameter 30 mm carries a bending moment of 200 N·m together with a torque of 750 N·m. At the top surface, where the bending stress is tensile and largest, find the largest principal stress σ1.

d = 30 mmM = 200 N·mT = 750 N·m
σ1 = MPa
6.Hard
σx​ = −80σy​ = 80τxy​ = 40MPa

A point in a loaded part has σₓ = −80 MPa, σᵧ = 80 MPa and τₓᵧ = 40 MPa (tension positive). Find the normal stress σθ on a plane whose normal is rotated 45° counter-clockwise from the x-axis.

σx = −80 MPaσy = 80 MPaτxy = 40 MPaθ = 45°
σθ = MPa

Answer key

  1. 1. σavg = −5 MPa
  2. 2. σ2 = −92.17 MPa
  3. 3. θp = 5.152 °
  4. 4. τxy = 61.85 MPa
  5. 5. σ1 = 184.1 MPa
  6. 6. σθ = 40 MPa

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