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​ = 80MPaprincipal element

At a point in a loaded part the in-plane principal stresses are σ₁ = 190 MPa and σ₂ = 80 MPa (tension positive). Find the normal stress σavg that acts on the planes of maximum in-plane shear stress.

σ1 = 190 MPaσ2 = 80 MPa
σavg = MPa
2.Easy
σx​ = 50σy​ = 90τxy​ = −50MPa

A point in a loaded part has σₓ = 50 MPa, σᵧ = 90 MPa and τₓᵧ = −50 MPa (tension positive). Find the largest principal stress σ1.

σx = 50 MPaσy = 90 MPaτxy = −50 MPa
σ1 = MPa
3.Medium
σx​ = −40σy​ = −10τxy​ = −50MPa

A point in a loaded part has σₓ = −40 MPa, σᵧ = −10 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 = −40 MPaσy = −10 MPaτxy = −50 MPa
θp = °
4.Medium
σx​ = 90σy​ = 80τxy​ = 40MPa

A point in a loaded part has σₓ = 90 MPa, σᵧ = 80 MPa and τₓᵧ = 40 MPa (tension positive). Find the maximum in-plane shear stress τmax.

σx = 90 MPaσy = 80 MPaτxy = 40 MPa
τmax = MPa
5.Hard
σx​τxy​surface of the shaft

A solid shaft of diameter 40 mm carries an axial tensile force of 130 kN together with a torque of 2000 N·m. At the surface of the shaft, find the largest principal stress σ1.

d = 40 mmF = 130 kNT = 2000 N·m
σ1 = MPa
6.Hard
σx​ = 70σy​ = 90τxy​ = 50MPa

A point in a loaded part has σₓ = 70 MPa, σᵧ = 90 MPa and τₓᵧ = 50 MPa (tension positive). Find the normal stress σθ on a plane whose normal is rotated 25° counter-clockwise from the x-axis.

σx = 70 MPaσy = 90 MPaτxy = 50 MPaθ = 25°
σθ = MPa

Answer key

  1. 1. σavg = 135 MPa
  2. 2. σ1 = 123.9 MPa
  3. 3. θp = −53.35 °
  4. 4. τmax = 40.31 MPa
  5. 5. σ1 = 219.1 MPa
  6. 6. σθ = 111.9 MPa

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