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​ = 40σy​ = 120τxy​ = −30MPa

A point in a loaded part has σₓ = 40 MPa, σᵧ = 120 MPa and τₓᵧ = −30 MPa (tension positive). Find the largest principal stress σ1.

σx = 40 MPaσy = 120 MPaτxy = −30 MPa
σ1 = MPa
2.Easy
60 kN60 kNθ = 45°A = 500 mm²

A bar with cross-sectional area 500 mm² carries an axial tensile force of 60 kN. Find the normal stress σθ on a section whose normal makes an angle of 45° with the bar axis (tension positive).

P = 60 kNA = 500 mm²θ = 45°
σθ = MPa
3.Medium
σx​ = 120σy​ = −40τxy​ = −60MPa

A point in a loaded part has σₓ = 120 MPa, σᵧ = −40 MPa and τₓᵧ = −60 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 = −40 MPaτxy = −60 MPa
θp = °
4.Medium
σx​ = −70σy​ = −80τxy​ = 10MPa

A point in a thin plate is in plane stress: σₓ = −70 MPa, σᵧ = −80 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 = −70 MPaσy = −80 MPaτxy = 10 MPaσz = 0
τabs = MPa
5.Hard
σx​τxy​top surface of the shaft

A solid shaft of diameter 60 mm carries a bending moment of 1000 N·m together with a torque of 1500 N·m. At the top surface, where the bending stress is tensile and largest, find the largest principal stress σ1.

d = 60 mmM = 1000 N·mT = 1500 N·m
σ1 = MPa
6.Hard
σx​τxy​top surface of the shaft

A solid shaft of diameter 60 mm carries a bending moment of 500 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 = 60 mmM = 500 N·mT = 750 N·m
σ1 = MPa

Answer key

  1. 1. σ1 = 130 MPa
  2. 2. σθ = 60 MPa
  3. 3. θp = −18.43 °
  4. 4. τabs = 43.09 MPa
  5. 5. σ1 = 66.09 MPa
  6. 6. σ1 = 33.04 MPa

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