Combined Bending, Torsion and Shear Stress Calculator for a Round Bar or Tube

How this calculator works

Enter the diameter (and the inside diameter for a tube) and the bending moment, torque, shear force and axial force at the section. The calculator gives the bending stress Mc/I, the torsional shear Tc/J, the maximum transverse shear (4V/3A for a solid bar, with an exact formula for a tube) and the axial stress P/A.

The worst points are the outer fibre, where bending is greatest and the transverse shear is zero, and the neutral axis, where the transverse shear is greatest. At each point it works out the principal stresses, the maximum shear stress and the von Mises stress, so you can compare with yield.

Use it for bent shafts, cantilever bars with an offset load, sign posts and poles, and for checking exam problems.

Formulas used

• \sigma_b = \frac{Mc}{I}
• \tau_t = \frac{Tc}{J}
• \sigma_{1,2} = \frac{\sigma}{2} \pm \sqrt{\left(\frac{\sigma}{2}\right)^2 + \tau^2}
• \sigma_{vM} = \sqrt{\sigma^2 + 3\tau^2}

Worked example

With the default numbers in the calculator above:

1. Area, second moment of area and polar moment: A = 2827.43 mm²
2. Bending stress at the extreme fibre: 113.18 MPa
3. Torsional shear stress at the surface: 70.74 MPa
4. Maximum transverse shear stress (at the neutral axis): 0.94 MPa
5. Extreme fibre (shear is torsion only): normal and shear stress: 113.18 MPa and 70.74 MPa
6. Extreme fibre (shear is torsion only): principal stresses: 147.17 and -34.00 MPa
7. Extreme fibre (shear is torsion only): maximum shear stress: 90.59 MPa
8. Extreme fibre (shear is torsion only): von Mises stress: 166.79 MPa
9. Neutral axis (torsion plus transverse shear): normal and shear stress: 0.00 MPa and 71.68 MPa
10. Neutral axis (torsion plus transverse shear): principal stresses: 71.68 and -71.68 MPa
11. Neutral axis (torsion plus transverse shear): maximum shear stress: 71.68 MPa
12. Neutral axis (torsion plus transverse shear): von Mises stress: 124.15 MPa

Change the numbers in the calculator to see the full working, every step, for your own case.

A second worked example

A different case, with these numbers: Outside diameter 150 mm; Inside diameter (0 for a solid bar) 120 mm; Bending moment at the section 8.584 kN·m; Torque at the section 18.375 kN·m; Transverse shear force at the section 10.5 kN; Axial force (tension positive, compression negative) -4.905 kN.

1. Area, second moment of area and polar moment: A = 6361.73 mm²
2. Bending stress at the extreme fibre: 43.88 MPa
3. Torsional shear stress at the surface: 46.97 MPa
4. Maximum transverse shear stress (at the neutral axis): 3.27 MPa
5. Thin-wall estimate of the transverse shear stress: 3.30 MPa
6. Axial stress: -0.77 MPa
7. Extreme fibre (shear is torsion only): normal and shear stress: 43.11 MPa and 46.97 MPa
8. Extreme fibre (shear is torsion only): principal stresses: 73.23 and -30.12 MPa
9. Extreme fibre (shear is torsion only): maximum shear stress: 51.68 MPa
10. Extreme fibre (shear is torsion only): von Mises stress: 92.06 MPa
11. Neutral axis (torsion plus transverse shear): normal and shear stress: -0.77 MPa and 50.24 MPa
12. Neutral axis (torsion plus transverse shear): principal stresses: 49.86 and -50.63 MPa

Common questions

Related calculators

• Principal Stress and Mohr's Circle Calculator with von Mises
• Shaft Torsion Calculator: Shear Stress and Angle of Twist
• Section Properties Calculator: Area, Second Moment and Section Modulus

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Hibbeler or Gere, Mechanics of Materials (combined loading); Shigley, Mechanical Engineering Design (von Mises).

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