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​ = 180σy​ = −110MPaprincipal element

At a point in a loaded part the in-plane principal stresses are σ₁ = 180 MPa and σ₂ = −110 MPa (tension positive). Find the normal stress σavg that acts on the planes of maximum in-plane shear stress.

σ1 = 180 MPaσ2 = −110 MPa
σavg = MPa
2.Easy
40 kN40 kNθ = 40°A = 800 mm²

A bar with cross-sectional area 800 mm² carries an axial tensile force of 40 kN. Find the normal stress σθ on a section whose normal makes an angle of 40° with the bar axis (tension positive).

P = 40 kNA = 800 mm²θ = 40°
σθ = MPa
3.Medium
σx​ = −20σy​ = 80τxy​ = 20MPa

A point in a loaded part has σₓ = −20 MPa, σᵧ = 80 MPa and τₓᵧ = 20 MPa (tension positive). Find the maximum in-plane shear stress τmax.

σx = −20 MPaσy = 80 MPaτxy = 20 MPa
τmax = MPa
4.Medium
σx​ = 20σy​ = 50τxy​ = −10MPa

A point in a loaded part has σₓ = 20 MPa, σᵧ = 50 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 = 20 MPaσy = 50 MPaτxy = −10 MPa
θp = °
5.Hard
σx​ = 60σy​ = 120τxy​ = 60MPa

A point in a loaded part has σₓ = 60 MPa, σᵧ = 120 MPa and τₓᵧ = 60 MPa (tension positive). Find the normal stress σθ on a plane whose normal is rotated 45° counter-clockwise from the x-axis.

σx = 60 MPaσy = 120 MPaτxy = 60 MPaθ = 45°
σθ = MPa
6.Hard
σx​ = −50σy​ = −80τxy​ = 50MPa

A point in a loaded part has σₓ = −50 MPa, σᵧ = −80 MPa and τₓᵧ = 50 MPa (tension positive). Find the normal stress σθ on a plane whose normal is rotated 25° counter-clockwise from the x-axis.

σx = −50 MPaσy = −80 MPaτxy = 50 MPaθ = 25°
σθ = MPa

Answer key

  1. 1. σavg = 35 MPa
  2. 2. σθ = 29.34 MPa
  3. 3. τmax = 53.85 MPa
  4. 4. θp = −73.15 °
  5. 5. σθ = 150 MPa
  6. 6. σθ = −17.06 MPa

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