The tool MC-03 computes maximum applied load, maximum deflection, and spring rate of a compression spring.

In addition, a tutorial on how to use the tool in computation is provided and a subject review on fundamental theories and useful formulas is presented.

**Example: A Helical Compression Spring**

The input in the tool is

After clicking "Run", the results will be shown as follows in another page.

Maximum applied load (F) = 22.00685412 newton(N)

Maximum deflection (δ) = 0.00362228 m

Spring rate (K) = 6075.41349199 N/m

Total number of active coils (N_{a}) = 8

Solid length (L_{s}) = 0.495 m

Pitch at free length = 0.118125 m

### Helical Compression Springs

**Fig. 1** A helical compression spring

Fig. 1 shows a typical helical compression spring. A helical compression spring is subjected to both transverse and torsional shear. The total shear stress
is in the form of

where

d: Wire diameter

D: Spring mean diameter

F: Applied axial load

K_{d} = (2C+1)/(2C) : Shear stress multiplication factor

with C = D/d being spring index

Due to the curvature effect, the stress at the inside surface is higher than that at the outside surface. If the curvature effect is included, the shear
stress multiplication factor is replaced by the Bergstrasser factor

The spring deflection is

where

G: Elastic shear modulus

N_{a}: Number of active coils (See Table 1)

**Table 1** Formulas for compression springs with different end conditions

Then the spring rate is given by

For springs in parallel, the spring rate is

For springs in series, the spring rate is

### References

1. R.G. Budynas and J.K. Nisbett, Mechanical Engineering Design, 2011, McGraw-Hill.

2. S.R. Schmid, B.J. Hamrock and B.O. Jacobson, Fundamentals of Machine Elements, 2014, CRC Press Taylor & Francis Group.

3. A.C. Ugural, Mechanical Design: An Integrated Approach, 2004, McGraw-Hill.