Closed Coiled helical springs

Springs are energy absorbing units whose function is to store energy and to restore it slowly or rapidly depending on the particular application.

A spring may be defined as an elastic member whose primary function is to deflect or distort under the action of applied load; it recovers its original shape when load is released.

Helical spring: They are made of wire coiled into a helical form, the load being applied along the axis of the helix. In these type of springs the major stresses is torsional shear stress due to twisting. They are both used in tension and compression.

Derivation of the Formula :

In order to derive a formula which governs the deflection and stress of closed coil helical springs, consider a closed coiled spring subjected to an axial load W. with the fallowing Assumptions:

1. The Bending and shear Force effects may be neglect
2. For the purpose of derivation of formula, the helix angle is considered to be so small that it may be neglected.

Let
W = axial load , D = mean coil diameter,
d = diameter of spring wire, n = number of active coils ,
l = length of spring wire, G = modulus of rigidity, X = deflection of spring
o = Angle of twist

when the spring is being subjected to an axial load to the wire of the spring gets be twisted like a shaft.

If o is the total angle of twist along the wire and x is the deflection of spring under the action of load W along the axis of the coil, so that

x = (D / 2) .O

[ consider ,one half turn of a close coiled helical spring ]

Any one coil of a such a spring will be assumed to lie in a plane which is nearly perpendicular to the axis of the spring. This requires that adjoining coils be close together. With this limitation, a section taken perpendicular to the axis the spring rod becomes nearly vertical. Hence to maintain equilibrium of a segment of the spring, only a shearing force F = W and Torque T =W.D/2 are required at any X – section.

In the analysis of springs it is customary to assume that the shearing stresses caused by the direct shear force is uniformly distributed and is negligible.
Using the torsion formula i.e

O = 2x / D

Twisting moment in spring coil,  T = WxD/2  

The SPRING DEFLECTION ' x' will be,

Shear stress in the spring:

Stiffness of Spring :

The stiffness is defined as the load per unit deflection therefore

Springs in Series:

If two springs of different stiffness are joined end on and carry a common load W, they are said to be connected in series and the combined stiffness and deflection are given by the following equation.

Springs in parallel:

If the two spring are joined in such a way that they have a common deflection ‘x' ; then they are said to be connected in parallel.In this care the load carried is shared between the two springs and total load W = W₁ + W₂