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​ = 80σy​ = −120MPaprincipal element

At a point in a loaded part the in-plane principal stresses are σ₁ = 80 MPa and σ₂ = −120 MPa (tension positive). Find the normal stress σavg that acts on the planes of maximum in-plane shear stress.

σ1 = 80 MPaσ2 = −120 MPa
σavg = MPa
2.Easy
σx​ = 110σy​ = −60τxy​ = −60MPa

A point in a loaded part has σₓ = 110 MPa, σᵧ = −60 MPa and τₓᵧ = −60 MPa (tension positive). Find the smallest principal stress σ2.

σx = 110 MPaσy = −60 MPaτxy = −60 MPa
σ2 = MPa
3.Medium
σx​ = −20σy​ = −80τxy​ = −30MPa

A point has σₓ = −20 MPa, σᵧ = −80 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 = −20 MPaσy = −80 MPaτxy = −30 MPa
θs = °
4.Medium
σx​ = −80σy​ = −50τxy​ = 60MPa

A point in a loaded part has σₓ = −80 MPa, σᵧ = −50 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 = −80 MPaσy = −50 MPaτxy = 60 MPa
θp = °
5.Hard
σx​ = −80σy​ = −70τxy​ = 40MPa

A point in a loaded part has σₓ = −80 MPa, σᵧ = −70 MPa and τₓᵧ = 40 MPa (tension positive). Find the normal stress σθ on a plane whose normal is rotated 30° counter-clockwise from the x-axis.

σx = −80 MPaσy = −70 MPaτxy = 40 MPaθ = 30°
σθ = MPa
6.Hard
σx​ = 70σy​ = 20τxy​ = 40MPa

A point in a thin plate is in plane stress: σₓ = 70 MPa, σᵧ = 20 MPa and τₓᵧ = 40 MPa (σz = 0). The material has a yield strength of 275 MPa. Find the factor of safety n against yielding by the Tresca (maximum shear stress) criterion.

σx = 70 MPaσy = 20 MPaτxy = 40 MPaσY = 275 MPa
n =

Answer key

  1. 1. σavg = −20 MPa
  2. 2. σ2 = −79.04 MPa
  3. 3. θs = −67.5 °
  4. 4. θp = 52.02 °
  5. 5. σθ = −42.86 MPa
  6. 6. n = 2.915

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