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​ = 190σy​ = −100MPaprincipal element

At a point in a loaded part the in-plane principal stresses are σ₁ = 190 MPa and σ₂ = −100 MPa (tension positive). Find the normal stress σavg that acts on the planes of maximum in-plane shear stress.

σ1 = 190 MPaσ2 = −100 MPa
σavg = MPa
2.Easy
150 kN150 kNθ = 50°A = 500 mm²

A bar with cross-sectional area 500 mm² carries an axial tensile force of 150 kN. Find the normal stress σθ on a section whose normal makes an angle of 50° with the bar axis (tension positive).

P = 150 kNA = 500 mm²θ = 50°
σθ = MPa
3.Medium
σx​ = −60σy​ = 80τxy​ = 50MPa

A point has σₓ = −60 MPa, σᵧ = 80 MPa and τₓᵧ = 50 MPa (tension positive). The element is rotated 40° counter-clockwise. Find the normal stress σy′ on the faces of the rotated element that are perpendicular to the rotated y′-axis.

σx = −60 MPaσy = 80 MPaτxy = 50 MPaθ = 40°
σy′ = MPa
4.Medium
σx​ = 10σy​ = 120τxy​ = −30MPa

A point in a loaded part has σₓ = 10 MPa, σᵧ = 120 MPa and τₓᵧ = −30 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 = 120 MPaτxy = −30 MPa
θp = °
5.Hard
σx​ = 160σy​ = −50τxy​ = 50MPa

A point in a loaded part has σₓ = 160 MPa, σᵧ = −50 MPa and τₓᵧ = 50 MPa (tension positive). Find the normal stress σθ on a plane whose normal is rotated 45° clockwise from the x-axis.

σx = 160 MPaσy = −50 MPaτxy = 50 MPaθ = -45°
σθ = MPa
6.Hard
σx​ = −40σy​ = 60τxy​ = 30MPa

A point in a thin plate is in plane stress: σₓ = −40 MPa, σᵧ = 60 MPa and τₓᵧ = 30 MPa (σz = 0). The material has a yield strength of 355 MPa. Find the factor of safety n against yielding by the Tresca (maximum shear stress) criterion.

σx = −40 MPaσy = 60 MPaτxy = 30 MPaσY = 355 MPa
n =

Answer key

  1. 1. σavg = 45 MPa
  2. 2. σθ = 124 MPa
  3. 3. σy′ = −27.09 MPa
  4. 4. θp = −75.69 °
  5. 5. σθ = 5 MPa
  6. 6. n = 3.044

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