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​ = 70σy​ = 20MPaprincipal element

At a point in a loaded part the in-plane principal stresses are σ₁ = 70 MPa and σ₂ = 20 MPa (tension positive). Find the normal stress σavg that acts on the planes of maximum in-plane shear stress.

σ1 = 70 MPaσ2 = 20 MPa
σavg = MPa
2.Easy
σx​ = −50σy​ = 40τxy​MPa

A point has σₓ = −50 MPa and σᵧ = 40 MPa, plus some shear stress. On an element rotated by 25° the normal stress in the rotated x′ direction is −26.3 MPa. Find the normal stress σy′ in the rotated y′ direction.

σx = −50 MPaσy = 40 MPaσx′ = −26.3 MPaθ = 25°
σy′ = MPa
3.Medium
σx​ = −60σy​ = 80τxy​ = 30MPa

A point has σₓ = −60 MPa, σᵧ = 80 MPa and τₓᵧ = 30 MPa (tension positive). The element is rotated 25° 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 = 30 MPaθ = 25°
σy′ = MPa
4.Medium
σx​ = −70σy​ = 20τxy​ = −50MPa

A point in a loaded part has σₓ = −70 MPa, σᵧ = 20 MPa and τₓᵧ = −50 MPa (tension positive). Find the angle θp from the x-axis to the direction of the largest principal stress, between −90° and 90°.

σx = −70 MPaσy = 20 MPaτxy = −50 MPa
θp = °
5.Hard
σx​ = 0σy​ = 70τxy​ = 60MPa

A point in a loaded part has σₓ = 0 MPa, σᵧ = 70 MPa and τₓᵧ = 60 MPa (tension positive). Find the normal stress σθ on a plane whose normal is rotated 25° counter-clockwise from the x-axis.

σx = 0 MPaσy = 70 MPaτxy = 60 MPaθ = 25°
σθ = MPa
6.Hard
σx​ = −20σy​ = −80τxy​ = −60MPa

A point in a thin plate is in plane stress: σₓ = −20 MPa, σᵧ = −80 MPa and τₓᵧ = −60 MPa (σz = 0). The material has a yield strength of 250 MPa. Find the factor of safety n against yielding by the Tresca (maximum shear stress) criterion.

σx = −20 MPaσy = −80 MPaτxy = −60 MPaσY = 250 MPa
n =

Answer key

  1. 1. σavg = 45 MPa
  2. 2. σy′ = 16.3 MPa
  3. 3. σy′ = 32.01 MPa
  4. 4. θp = −65.99 °
  5. 5. σθ = 58.47 MPa
  6. 6. n = 1.863

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