Helical (Coil) Springs

The figure shows a helical coil spring with ends adapted to support a compressive load. The notation applied are:


P = axial load, N

D = mean diameter of coil, mm

= (Do + Di) / 2

d = diameter of wire, mm

= (Do Di) / 2

p = pitch of coils, mm

δ = deflection of spring, mm

n = number of active coils

C = spring index = D/d = (Do-d)/d

= (4 : 12); less than 4; it is difficult to manufacture, more than 12; it is likely to buckle.

G = torsional modulus of

elasticity, N/mm2

τs = shearing stress, N/mm2


Stresses on coil spring (t):

The coil spring wire is subjected to the following stresses:

a-     Torsional stress due to the load P

b-    Direct shear stress due to load P

c-     Torsional stress due to wire curvature


Torsional stress due to load P:



In order to include the effects of both direct shear and wire curvature, a stress factor (Kw) had been determined by the use of approximate analytical methods developed by A. M. Wahl:



which may be used in the above equation to determine the maximum shearing stress in the wire as follows:




Various lengths associated with a spring:


Lf = Free length

La = Installed length

Lm = Operating length

Ls = Shut height or Solid length


        Compression springs in which the free length is more than four times the mean diameter of the coils may fail by sidewise buckling.


Spring deflection (d):


Spring rate (stiffness) (k):


Spring ends conditions:

Helical springs ends may be either plain, plain ground, squared, or squared and ground as shown in the figure below. This results in a decrease of the number of active coils and affects the free length and solid length of the spring as shown below.

p=(D/3 : D/4), n= ( 3: 15)



Finding spring stress and deflection:

a-     Using the above equations of tmax , and d.

b-    Using the nomogram

c-     Using the Excel program given


Explanation of the nomogram:

The nomogram applies to cylindrical helical extension and compression springs made of round steel wire (shear modulus G=81,400 N/mm2). In the case of materials with a different shear modulus G spring deflection must be multiplied by G/G.

The nomogram indicates the deflection s of one coil. Total deflection s (d) is obtained by multiplying (s) by the number of active coils n:


s (d) = n . s.



Find the stress and the deflection of a spring with the following dimensions:

D = 30 mm

d = 4 mm

F = P = 100 N


Solution steps:

1-   enter the value of D (point 1), and d (point 2) into the nomogram

2-   connect point 1, 2 with the line A and extend the line to points 3 and 3 on either end.

3- enter the value of F (point 4) into the nomogram

4- connect point 3 to the value of the force F (point 4) with line B, extend the line B to point (5)

5- calculate the value of C and enter its value into the nomogram (point 6)

6- connect point 6 and 5 by line C to obtain point (7)

7- connect point 5 to point 3 to obtain point (8)



-         point 5 represents the value of tt (MPa)

-         point 7 represents the value of tmax (MPa)

-         point 8 represents the value of d = d/n (mm)


c- Excel Program:

- enter the values of d, D, P, n, G.

The program will give you the values of C, Kw, tt (Tau) , tmax (Tau), d/n (Delta), d (Delta), and K