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​ = 160σy​ = −80τxy​ = 10MPa

A point in a loaded part has σₓ = 160 MPa, σᵧ = −80 MPa and τₓᵧ = 10 MPa (tension positive). Find the smallest principal stress σ2.

σx = 160 MPaσy = −80 MPaτxy = 10 MPa
σ2 = MPa
2.Easy
σx​ = 60σy​ = −80MPaprincipal element

At a point in a loaded part the in-plane principal stresses are σ₁ = 60 MPa and σ₂ = −80 MPa (tension positive). Find the normal stress σavg that acts on the planes of maximum in-plane shear stress.

σ1 = 60 MPaσ2 = −80 MPa
σavg = MPa
3.Medium
σx​ = −10σy​ = 100τxy​ = −60MPa

A point in a loaded part has σₓ = −10 MPa, σᵧ = 100 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 = −10 MPaσy = 100 MPaτxy = −60 MPa
θp = °
4.Medium
σx​ = −40τxy​ = 30MPa

A point has σₓ = −40 MPa, σᵧ = 0 MPa and τₓᵧ = 30 MPa (tension positive). Find the angle θs, between −90° and 90°, from the x-axis to the normal of the plane on which the shear stress has its maximum positive value +R (shear positive in the same sense as τₓᵧ).

σx = −40 MPaσy = 0 MPaτxy = 30 MPa
θs = °
5.Hard
σx​ = 20σy​ = 120τxy​ = −60MPa

A point in a loaded part has σₓ = 20 MPa, σᵧ = 120 MPa and τₓᵧ = −60 MPa (tension positive). Find the normal stress σθ on a plane whose normal is rotated 60° counter-clockwise from the x-axis.

σx = 20 MPaσy = 120 MPaτxy = −60 MPaθ = 60°
σθ = MPa
6.Hard
σx​ = 160σy​ = 0τxy​ = 60MPa

A point in a loaded part has σₓ = 160 MPa, σᵧ = 0 MPa and τₓᵧ = 60 MPa (tension positive). Find the normal stress σθ on a plane whose normal is rotated 35° counter-clockwise from the x-axis.

σx = 160 MPaσy = 0 MPaτxy = 60 MPaθ = 35°
σθ = MPa

Answer key

  1. 1. σ2 = −80.42 MPa
  2. 2. σavg = −10 MPa
  3. 3. θp = −66.26 °
  4. 4. θs = 16.85 °
  5. 5. σθ = 43.04 MPa
  6. 6. σθ = 163.7 MPa

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