Springs in Series and in Parallel Calculator

The combined stiffness of up to four springs in series or in parallel.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

Springs side by side, sharing the movement, are in parallel: the stiffnesses add. Springs end to end, sharing the force, are in series: it is the reciprocals that add, so the combined spring is softer than the softest one.

Enter the rates in newtons per millimetre. If you give a force, the calculator also gives the deflection for each arrangement.

Formulas used

  • k_{parallel} = k_1 + k_2 + \dots
  • \frac{1}{k_{series}} = \frac{1}{k_1} + \frac{1}{k_2} + \dots

Worked example

With the default numbers in the calculator above:

  1. In parallel (side by side, sharing the movement): 30 N/mm
  2. In series (end to end, sharing the force): 6.667 N/mm
  3. Deflection under the force, parallel: 3.333 mm
  4. Deflection under the force, series: 15 mm

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

Common questions

When are springs in parallel?

When they all move the same distance, for example nested coil springs or two springs under one plate.

When are springs in series?

When the force passes through each one in turn, for example two springs end to end.

Is a spring and a beam in series?

Yes, when the force goes through both. Use the beam stiffness 3EI/L³ as another k.

Related calculators

Further reading: Meriam and Kraige, Engineering Mechanics: Dynamics; Hibbeler, Engineering Mechanics: Dynamics; Rao, Mechanical Vibrations.

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