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Calculating Torsional Stiffness, Displacement, and Stress

  • This sheet goes through the process of calculating the stiffness and maximum shear stress for a structural member loaded in torsion. The stiffness and peak stress have a dependence on the cross-section shape. In general, circular cross sections are the most efficient at resisting torsional loads. However, there are situations where other cross-sections need to be used based on other design considerations. This sheet allows for the analysis of several common cross-sectional shapes as discussed below.


    The torsional stiffness of a structure depends on its length, l, and its cross-sectional properties. Additionally, the torsional deflection and stress will depend on the applied torque, T.

    The length, l, and applied torque, T, are defined below:

  • Two constants, K and Q, are used to calculate the stiffness and peak shear stress for a structural member loaded in torsion. Choose the row in the table below for the desired cross-section to be analyzed. Each section has a different set of parameters to defined its dimensions (d, a, b, etc.). Edit the appropriate columns for the desired cross section based on the diagram that appears above the table for each cross-section.


    Source for section torsion constants: Young, W. C., and R. G. Budynas. "Roark's formulas for stress and strain." (2002).

  • Max stress occurs at section boundary

    Circle
    Hollow Circle
    Square
    Hollow Square
    Rectangle
    Hollow Rectangle
    Ellipse
    Hollow Ellipse
    Open Circle
    Open Arbitrary
    Closed Arbitrary

  • The torsional stiffness and deflection will depend on the shear modules of the material, which will be computed from the elastic modules and Poisson's ratio provided in the table below.


    Source for material properties: Norton, R. L. "Machine design. A integrated approach, 5th Editi." (2013).

  • Aluminum Alloys
    Beryllium Copper
    Brass, Bronze
    Copper
    Iron, Cast, Gray
    Iron, Cast, Ductile
    Iron, Cast, Malleable
    Magnesium Alloys
    Nickel Alloys
    Steel, Carbon
    Steel, Alloys
    Steel, Stainless
    Titanium Alloys
    Zinc Alloys

  • The shear modules can be calculated as a function of the elastic modulus and the Poisson's ratio using the following equation:

  • We can now calculate the torsional stiffness. Torsional stiffness has the units of torque per unit angle. Note that we need to multiply the denominator by 1 [radian] so that we can later convert the torsional stiffness into the desired units. Dividing by 1 [radian] does not change the numerical value of the expression.

  • We can also query the torsion stiffness using degrees for the angular unit:

  • The actual torsional displacement can be obtained by dividing the applied torque, T, by the torsional stiffness:

  • The stress for a section loaded in torsion is shear stress. The peak shear stress can be calculated using the sections Q value:

Use of this software is subject to these (updated 1/10/2024)