# Derivation of spring constant k: Where can I find it?

• redoubt
In summary, the spring constant k of a coiled, ideal spring can be derived from the equation k = G*d^4/(8*n*D^3) using Castigliano's theorem as presented in A. C. Ugural's book "Mechanical Design: An Integrated Approach."
redoubt

Folks---

I want a reference(text or journal article) that presents the mathematical
derivation of the following formula for the spring constant k of a coiled,
ideal spring.

k = G*d^4/(8*n*D^3)

Spring Constant Derivation: Where can I find it?

Folks---

I want a reference(textbook or journal article) that presents the mathematical derivation of the following formula for the spring constant k of a coiled, ideal spring.

k=G*d^4/(8*n*D^3)

When this question was asked in the General Physics Forum several weeks ago, there were 81 views, but no replies. I believe this derivation is a classic in linear elasticity theory, but I can't locate it.

If this is not a homework type question, you might try Eng-Tips to see if anyone on there knows.

Where did you find the formula?

I found this:
http://www.sussex.ac.uk/engineering/documents/sm_lecture_19.pdf

but no obvious derivations...

Last edited by a moderator:
redoubt said:
Folks---

I want a reference(textbook or journal article) that presents the mathematical derivation of the following formula for the spring constant k of a coiled, ideal spring.

k=G*d^4/(8*n*D^3)

When this question was asked in the General Physics Forum several weeks ago, there were 81 views, but no replies. I believe this derivation is a classic in linear elasticity theory, but I can't locate it.

This constant comes from the derivation of the deflection of a coiled spring. By deriving the deflection, one can extract the spring constant k from the equation.

The derivation of the equation for the deflection of a spring comes from Castigliano's theorem.

See A. C. Ugural "Mechanical Design: An Integrated Approach" McGraw-Hill Professional (2003): pg. 565

## 1. What is the spring constant k?

The spring constant, denoted as k, is a measure of the stiffness of a spring. It represents the amount of force required to stretch or compress a spring by a certain distance.

## 2. How is the spring constant k derived?

The spring constant k can be derived by dividing the applied force by the resulting displacement. This relationship is expressed as k = F/x, where F is the applied force and x is the displacement.

## 3. Can the spring constant k be calculated for all types of springs?

Yes, the spring constant k can be calculated for all types of springs as long as the material and dimensions of the spring are known. However, the formula for calculating k may vary depending on the type of spring (e.g. compression, tension, torsion).

## 4. Where can I find the spring constant k for a specific spring?

The spring constant k can be found in the manufacturer's specifications for the spring. It can also be calculated experimentally by measuring the force and displacement of the spring.

## 5. Does the spring constant k change for different lengths or sizes of a spring?

Yes, the spring constant k is affected by the length and size of the spring. Generally, a longer or thicker spring will have a higher spring constant compared to a shorter or thinner spring.

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