We were discussing the basicÂ concept of spring in strength of material,Â various definitions and terminology used in springs, Â importance of spring indexÂ  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
AO2 = OD2 + AD2
R2 = (R - Î´) 2 + (L/2)2
R2 = R2 + Î´2 - 2R. Î´ + L2/4
R2 = R2 - 2R. Î´ + L2/4

We have neglected small term i.e. Î´2
2R. Î´ = L2/4
Î´ = L2/8R

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 L2/8(E x t)

Î´ = Ïƒ x L2/4E.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,Â Â in the category of strength of material, in our next post.

Reference:

Strength of material, By R. K. Bansal