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​ = 90τxy​ = 60MPa

A point in a loaded part has σₓ = 40 MPa, σᵧ = 90 MPa and τₓᵧ = 60 MPa (tension positive). Find the largest principal stress σ1.

σx = 40 MPaσy = 90 MPaτxy = 60 MPa
σ1 = MPa
2.Easy
60 kN60 kNθ = 50°A = 1000 mm²

A bar with cross-sectional area 1000 mm² carries an axial tensile force of 60 kN. Find the normal stress σθ on a section whose normal makes an angle of 50° with the bar axis (tension positive).

P = 60 kNA = 1000 mm²θ = 50°
σθ = MPa
3.Medium
σx​ = −70σy​ = −20τxy​ = −60MPa

A point in a thin plate is in plane stress: σₓ = −70 MPa, σᵧ = −20 MPa and τₓᵧ = −60 MPa, with σz = 0. Find the absolute maximum shear stress τabs at the point (considering all planes in three dimensions).

σx = −70 MPaσy = −20 MPaτxy = −60 MPaσz = 0
τabs = MPa
4.Medium
σx​ = 30σy​ = −80τxy​ = 50MPa

A point in a loaded part has σₓ = 30 MPa, σᵧ = −80 MPa and τₓᵧ = 50 MPa (tension positive). Find the angle θp from the x-axis to the direction of the largest principal stress, between −90° and 90°.

σx = 30 MPaσy = −80 MPaτxy = 50 MPa
θp = °
5.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 250 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 = 250 N·m
σ1 = MPa
6.Hard
σx​ = 80σy​ = 70τxy​ = −20MPa

A point in a loaded part has σₓ = 80 MPa, σᵧ = 70 MPa and τₓᵧ = −20 MPa (tension positive). Find the normal stress σθ on a plane whose normal is rotated 30° clockwise from the x-axis.

σx = 80 MPaσy = 70 MPaτxy = −20 MPaθ = -30°
σθ = MPa

Answer key

  1. 1. σ1 = 130 MPa
  2. 2. σθ = 24.79 MPa
  3. 3. τabs = 65 MPa
  4. 4. θp = 21.14 °
  5. 5. σ1 = 123.1 MPa
  6. 6. σθ = 94.82 MPa

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