## Saturday, 14 October 2017

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 = bt3/12

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 bt3/12
M = Ïƒ x bt2/ 6

Total resisting moment will be given by following formula
= n x M
= Ïƒ x n x bt2/ 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 bt2/ 6

### Ïƒ = 3W.L/ (2n. bt2)

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