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 developed in the plate of leaf spring in our previous posts.

Today we will derive here the expression for central
deflection developed in the plate of leaf spring with the help of this post.

Let us go ahead step by step for easy understanding,
however if there is any issue we can discuss it in comment box which is
provided below this post.

###
**So,
what is a leaf spring?**

A leaf or laminated spring is basically a simple
type of suspension spring usually used for absorbing the shocks in heavy
vehicles such as Lorries, railway wagons, cars, trailers and trucks.

Leaf spring will be made by number of parallel metal
strips of having identical width but different lengths placed one over another
as displayed in following figure. As we have seen that leaf springs are made by
flat plates and therefore leaf springs are also called as flat springs.

In initial situation, all plates of leaf spring will
be bent in same radius and will be free to slide one over the other. Above
figure indicates the initial condition i.e. before loading of leaf spring,
there is some amount of deflection i.e. Î´ as displayed in above figure.

Once leaf spring will be loaded with rated load,
central deflection will be disappeared and all plated will become flat.

###
**Expression
for central deflection developed in the plate of leaf spring**

We have seen above the basic construction and
definition of a leaf spring. Now we will derive here the expression central
deflection developed in the plate of leaf spring.

Let us consider

b = Width of each plate

n = Number of plates

L = Leaf spring span

t = Thickness of each plate of leaf spring

Î´ = Deflection of the top spring

R = Radius of the plate in which plates are bent
initially

W = Point load acting at the center of the leaf
spring

Ïƒ = Maximum bending stress developed in the plate of
leaf spring

A and B = Two ends of the leaf spring

C = Center point of the leaf spring

Let us consider here the triangle AOD; we will have
following equation as mentioned here

AO

^{2}= OD^{2 }+ AD^{2}
R

^{2}= (R - Î´)^{ 2}+ (L/2)^{2}
R

^{2}= R^{2}+ Î´^{2}- 2R. Î´ + L^{2}/4
R

^{2}= R^{2}- 2R. Î´ + L^{2}/4
We have neglected small term i.e. Î´

^{2}
2R. Î´ = L

^{2}/4
Î´ = L

^{2}/8R
Let us remind here the bending equation and we can
write here following equation as mentioned here

Ïƒ/y = E/R

R = (E x y) / Ïƒ

R = (E x t) / 2Ïƒ

Now we will use the above value of R in deflection
equation in order to secure the expression for central deflection developed in
the plate of leaf spring.

Î´ = 2Ïƒ x L

^{2}/8(E x t)###
**Î´
= Ïƒ x L**^{2}/4E.t

^{2}/4E.t

Above equation secured here represent the expression
for central deflection developed in the plate of leaf spring.

Do you have suggestions? Please write in comment
box.

We will now discuss another topic, spring materials and their properties, in the category
of strength of material, in our next post.

###
**Reference:**

Strength of material, By R. K. Bansal

Image Courtesy: Google