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​ = 170σy​ = −20MPaprincipal element

At a point in a loaded part the in-plane principal stresses are σ₁ = 170 MPa and σ₂ = −20 MPa (tension positive). Find the normal stress σavg that acts on the planes of maximum in-plane shear stress.

σ1 = 170 MPaσ2 = −20 MPa
σavg = MPa
2.Easy
τxy​ = −70MPapure shear

A point on the surface of a shaft in torsion is in pure shear with τₓᵧ = −70 MPa (σₓ = σᵧ = 0). Find the normal stress σθ on a plane whose normal is rotated 25° counter-clockwise from the x-axis.

τxy = −70 MPaθ = 25°
σθ = MPa
3.Medium
σx​ = 30σy​ = 70τxy​ = −20MPa

A point in a thin plate is in plane stress: σₓ = 30 MPa, σᵧ = 70 MPa and τₓᵧ = −20 MPa, with σz = 0. Find the absolute maximum shear stress τabs at the point (considering all planes in three dimensions).

σx = 30 MPaσy = 70 MPaτxy = −20 MPaσz = 0
τabs = MPa
4.Medium
σx​ = 30σy​ = 80τxy​ = −10MPa

A point has σₓ = 30 MPa, σᵧ = 80 MPa and τₓᵧ = −10 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 = 30 MPaσy = 80 MPaτxy = −10 MPa
θs = °
5.Hard
σx​ = −40σy​ = 50τxy​ = 30MPa

A point in a loaded part has σₓ = −40 MPa, σᵧ = 50 MPa and τₓᵧ = 30 MPa (tension positive). Find the shear stress τθ on a plane whose normal is rotated 50° counter-clockwise from the x-axis (positive in the same sense as τₓᵧ).

σx = −40 MPaσy = 50 MPaτxy = 30 MPaθ = 50°
τθ = MPa
6.Hard
σx​ = 100σy​ = −10τxy​ = −60MPa

A point in a loaded part has σₓ = 100 MPa, σᵧ = −10 MPa and τₓᵧ = −60 MPa (tension positive). Find the normal stress σθ on a plane whose normal is rotated 20° counter-clockwise from the x-axis.

σx = 100 MPaσy = −10 MPaτxy = −60 MPaθ = 20°
σθ = MPa

Answer key

  1. 1. σavg = 75 MPa
  2. 2. σθ = −53.62 MPa
  3. 3. τabs = 39.14 MPa
  4. 4. θs = 55.9 °
  5. 5. τθ = 39.11 MPa
  6. 6. σθ = 48.57 MPa

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