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Helical Compression Spring Calculations



  • This sheet shows how to calculate the spring rate, shut height, and the max stress in a helical compression spring. The figure below shows the parameters that define the geometry of a helical compression spring.

    We'll now define the spring geometry parameters that will be used in the following calculations:











  • There are various end treatments that are possible with helical compression springs. The end treatments impact the number of active coils, N_a, that contribute to the stiffness of the spring, where N_t is the total number of coils in the spring. Squared and squared-ground ends are the most common with squared-ground being typically preferred unless the wire diameter, d, is less than 0.5 mm [1]. The following table allows the end treatment to be specified. The end treatment also impacts the shut length, L_s, of the spring which is the height where all of the coils are closed and the spring cannot be compressed any further. The shut length is also sometimes referred to as the solid length. The equations used in this sheet are from the Norton [1] and Shigley [2] machine design books.



  • Plain Ends
    Plain-Ground Ends
    Squared Ends
    Squared-Ground Ends


  • The shut length can now be queried:





  • The stiffness of a spring will depend on the material used to construct the spring. The following table allows the material to be selected. The material properties are summarized from Norton [1].



  • Spring Steel
    Stainless 301
    Brass CA 260
    Phosphor Bronze


  • The shear modules can be calculated from the elastic modules and the Poisson's ratio:





  • The spring index C is the ratio of coil diameter to wire diameter:





  • Now the spring rate can be calculated:







  • The stresses in a helical compression string are pure shear. The curvature of the spring coil causes a stress concentration at the inside edge of the spring wire. Some situations require accounting for this stress concentration (see Norton for details [1]). The following table allows the shear factor, K, to be chosen with and without accounting for this stress concentration.



  • No stress concentration
    With stress concentration (Wahl's factor)


  • The stress depends on the compression force, F, applied to the spring and can be calculated as:





  • One stress value that is important when designing a compression spring is the stress at the shut length (L_s) of the spring. The spring is not operated at the shut length, but, in order to set a spring, it is loaded to the shut length. Setting a spring introduces beneficial residual stress that improve the static strength of the spring, see Norton [1]. The force required to get to the shut length is the spring constant times the free length minus the shut length. Calling the max shear stress equation as a function the compression force, F, gives the following max shear stress at the shut length:







  • [1] Norton, R.L. "Machine design, A integrated approach, 5th Edition." (2013).

    [2] Budynas, Richard G., and J. Keith Nisbett. "Shigley’s Mechanical Engineering Design, 9th Edition." (2011).