We were discussing the basic concept
of spring in strength of material, various
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.

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Stiffness of spring could be easily determined by dividing the load with deflection

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

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**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.

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**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 10

^{0}. 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

Ï„ = Maximum shear stress developed in
the spring wire

Î¸ = Angle of twist in wire of spring

L = Length of the spring

Î´ = 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

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*Strain energy stored in the spring = (Ï„ *^{2}/4C)
x Volume of the spring

*Strain energy stored in the spring = (Ï„*

^{2}/4C) x Volume of the spring
Volume of spring = Area of cross section
(V) x Length of the spring (L)

V = (ÐŸ/4) x d

^{2}
L = 2ÐŸRn

*Strain energy stored in the spring = (Ï„*

^{2}/4C) x Volume of the spring
Strain energy stored in the spring = (Ï„

^{2}/4C) x 2ÐŸRn
Let us recall the expression for shear stress developed in spring under axial loading and we will have following
result for shear stress Ï„.

Therefore, Strain energy stored in the
spring will be given as

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

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

Work done on spring = (1/2) W x Î´

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.

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**Stiffness
of spring**

Stiffness of spring could be easily determined by dividing the load with deflection

Stiffness of spring = Load (W) /
Deflection (Î´)

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.

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**Reference:**

Strength of material, By R. K. Bansal

Image Courtesy: Google

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