We were discussing the basic concept of spring in strength of material, various definitions and terminology used in springs and importance of spring index in our previous posts.

Today we will start here with the basic concept of
laminated or leaf spring and we will also derive here the expression for
maximum bending stress developed in the plate of leaf spring.

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 maximum bending stress**

We have seen above the basic construction and
definition of a leaf spring. Now we will derive here the expression for maximum
bending stress developed in the plates 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

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

Bending moment at the center = (W/2) x L/2

Bending moment at the center = WL/4

Let us remind here the concept of

*moment of inertia*and we will have following expression for moment of inertia for each plate of leaf spring
I = bt

^{3}/12
As we know that bending stress will be given by
following formula as mentioned here

M/I = Ïƒ/y

M = (Ïƒ/y) x I

M = (2 x Ïƒ/t) x bt

^{3}/12Â
M = Ïƒ x bt

^{2}/ 6
Total resisting moment will be given by following
formula

= n x M

= Ïƒ x n x bt

^{2}/ 6
Maximum bending moment due to load will be equal to
the total resisting moment and from here we will have the expression for
maximum bending stress developed in the plate of leaf spring

WL/4 = Ïƒ x n x bt

^{2}/ 6###
**Ïƒ
= 3W.L/ (2n. bt**^{2})

^{2})

This is 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, in the category
of strength of material, in our next post.

###
**Reference:**

Strength of material, By R. K. Bansal

Image Courtesy: Google

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