We were discussing the basic concept of spring in strength of materialvarious definitions and terminology used in springs,  importance of spring index  and expression for maximum bending stress, deflection developed in the plate of leaf spring and basic difference between open coiled and closed coiled helical spring in our previous posts.

Today, we will find out here the expression for deflection of spring under the applied load with the help of this post.

### Let us brief here first helical spring

Helical springs are usually used in number of applications due to their shock absorption and load bearing properties. There are two types of helical spring i.e. Open coiled helical spring and closed coiled helical spring. We will be concentrated here on closed coil helical spring.

### Closed coiled helical spring

Closed coiled helical springs are also termed as tension springs as such springs are designed to resist the tensile load and twisting load. In simple, we can say that closed coiled helical springs are those springs which are used in such applications, where tensile or twisting loads are present.

In case of closed coiled helical spring, spring wires are wound tightly. Hence such springs will have very small pitch. Closed coiled helical springs wires are very close to each other and hence, spring turns or coils will lie in same plane.

In case of closed coiled helical spring, turns or coils of such spring will be located at right angle to the helical axis.

Closed coiled helical spring, as displayed here, carrying an axial load W. In case of closed coiled helical spring, helix angle will be small and it will be less than 100. Therefore, we will neglect the bending effect on spring and we will only consider the effect of torsional stresses on the coils of closed coiled helical spring.
Let us consider the following terms from above figure of closed coil helical spring.
d = Diameter of spring wire or coil
p = Pitch of the helical spring
D = Mean diameter of spring
R = Mean radius of spring
n = Number of spring coils
W = Load applied on spring axially
C = Modulus of rigidity
τ = Maximum shear stress developed in the spring wire
θ = Angle of twist in wire of spring
L = Length of the spring
δ = Deflection of spring under axial load

As spring is loaded by an axial load W, therefore work will be done over the spring and this work done will be stored in the form of energy in spring.

So we will determine here the work done by axial load W over the spring and we will also determine the strain energy stored in the spring.

Expression for deflection developed in spring under axial load could be derived by equating the energy stored in spring with work done on spring.

Each section of spring will be subjected with torsion and hence strain energy stored in the spring will be determined as mentioned here

#### Strain energy stored in the spring = (τ 2/4C) x Volume of the spring

Volume of spring = Area of cross section (V) x Length of the spring (L)
V = (П/4) x d2
L = 2ПRn

Strain energy stored in the spring = (τ 2/4C) x Volume of the spring
Strain energy stored in the spring = (τ 2/4C) x 2ПRn

Let us recall the expression for shear stress developed in spring under axial loading and we will have following result for shear stress τ.
Therefore, Strain energy stored in the spring will be given as

### Work done on spring could be determined as mentioned here

Work done on spring = (1/2) W x δ

As we know that expression for deflection developed in spring under axial load could be derived by equating the energy stored in spring with work done on spring and therefore we will have following equation as mentioned here.

### Stiffness of spring

Stiffness of spring could be easily determined by dividing the load with deflection
Stiffness of spring = Load (W) / Deflection (δ)
Do you have suggestions? Please write in comment box.

We will now discuss another topic i.e. thin cylindrical and spherical shells, in the category of strength of material, in our next post.

You must find out and read the following quite useful engineering articles and these are as mentioned below.

### Reference:

Strength of material, By R. K. Bansal