Static Analysis of Leaf Spring for Automobile Suspension Systems

STATIC ANALYSIS OF LEAF SPRING
G HARINATH GOWD 1*
Associate Professor, Department of Mechanical Engineering
Sri Krishnadevaraya Engineering college,
NH-7, Gooty, Anantapur Dist, PIN 515 401
Andhra Pradesh., INDIA
[e-mail]
E VENUGOPAL GOUD
Associate Professor, Department of Mechanical Engineering
Pullareddy Engineering college,
Kurnool, Anantapur Dist, PIN 515 401
Andhra Pradesh., INDIA
[e-mail]
1* Corresponding author, Email: [e-mail].
ABSTRACT
Leaf springs are special kind of springs used in automobile suspension systems. The advantage of leaf
spring over helical spring is that the ends of the spring may be guided along a definite path as it deflects to act as
a structural member in addition to energy absorbing device. The main function of leaf spring is not only to
support vertical load but also to isolate road induced vibrations. It is subjected to millions of load cycles leading
to fatigue failure. Static analysis determines the safe stress and corresponding pay load of the leaf spring and
also to study the behavior of structures under practical conditions. The present work attempts to analyze the safe
load of the leaf spring, which will indicate the speed at which a comfortable speed and safe drive is possible. A
typical leaf spring configuration of TATA-407 light commercial vehicle is chosen for study. Finite element
analysis has been carried out to determine the safe stresses and pay loads.
Keywords: Leaf spring, Geometric modeling, Static analysis.
1. INTRODUCTION
A spring is defined as an elastic body, whose function is to distort when loaded and to recover its
original shape when the load is removed. Leaf springs absorb the vehicle vibrations, shocks and bump loads
(induced due to road irregularities) by means of spring deflections, so that the potential energy is stored in the
leaf spring and then relieved slowly [1]. Ability to store and absorb more amount of strain energy ensures the
comfortable suspension system. Semi-elliptic leaf springs are almost universally used for suspension in light and
heavy commercial vehicles. For cars also, these are widely used in rear suspension. The spring consists of a
number of leaves called blades. The blades are varying in length. The blades are us usually given an initial
curvature or cambered so that they will tend to straighten under the load. The leaf spring is based upon the
theory of a beam of uniform strength. The lengthiest blade has eyes on its ends. This blade is called main or
master leaf, the remaining blades are called graduated leaves. All the blades are bound together by means of
steel straps.
The spring is mounted on the axle of the vehicle. The entire vehicle load rests on the leaf spring. The
front end of the spring is connected to the frame with a simple pin joint, while the rear end of the spring is
connected with a shackle. Shackle is the flexible link which connects between leaf spring rear eye and frame.
When the vehicle comes across a projection on the road surface, the wheel moves up, leading to deflection of
the spring. This changes the length between the spring eyes. If both the ends are fixed, the spring will not be
able to accommodate this change of length. So, to accommodate this change in length shackle is provided at one
end, which gives a flexible connection. The front eye of the leaf spring is constrained in all the directions, where
as rear eye is not constrained in X-direction. This rare eye is connected to the shackle. During loading the spring
deflects and moves in the direction perpendicular to the load applied.
When the leaf spring deflects, the upper side of each leaf tips slides or rubs against the lower side of the
leaf above it. This produces some damping which reduces spring vibrations, but since this available damping
may change with time, it is preferred not to avail of the same. Moreover, it produces squeaking sound. Further if
moisture is also present, such inter-leaf friction will cause fretting corrosion which decreases the fatigueG Harinath Gowd et al. / International Journal of Engineering Science and Technology (IJEST)ISSN : 0975-5462Vol. 4 No.08 August 20123794

Strength of the spring, phosphate paint may reduce this problem fairly. The elements of leaf spring are shown in
Figure 1. Where t is the thickness of the plate, b is the width of the plate and L is the length of plate or distance
of the load W from the cantilever end.
Fig. 1 Elements of Leaf Spring
Bending Stress of Leaf Spring
Leaf springs (also known as flat springs) are made out of flat plates. The advantage of leaf spring over
helical spring is that the ends of the spring may be guided along a definite path as it deflects to act as a structural
member in addition to energy absorbing device. Thus the leaf springs may carry lateral loads, brake torque,
driving torque etc., in addition to shocks. Consider a single plate fixed at one end and loaded at the other end.
This plate may be used as a flat spring.
Let t = thickness of plate
b = width of plate, and
L = length of plate or distance of the load W from the cantilever end, as shown in the Figure 1.
We know that the maximum bending moment at the cantilever end
M = W.L
And section modulus,
Z = y
I where I = (b.t3 / 12) and Y = t/2
So Z = b.t2 / 6
The bending stress in such a spring,
f = M / Z = (6W.L) / b.t2 ....................................... .. (i)
We know that the maximum deflection for a cantilever with concentrated load at free end is given by
 = W.L 3 / 3.E.I = 2f.L 2 / 3.E.t ......................... (ii)
It may be noted that due to bending moment, top fibers will be in tension and bottom fibers are in
compression, but the shear stress is zero at the extreme fibers and the maximum at centre, hence for analysis,
both stresses need not to be taken into account simultaneously. We shall consider bending stress only.
If the spring is not of cantilever type but it is like a simply supported beam, with length 2L and load
2W in the centre
Maximum bending moment in the centre,
M = W.L
Section modulus
Z = b.t2 / 6
Bending stress
f = 6W.L /b.t2
We know that maximum deflection of a simply supported beam loaded in the centre is given by
 = W.L 3 / 3.E.IG Harinath Gowd et al. / International Journal of Engineering Science and Technology (IJEST)ISSN : 0975-5462Vol. 4 No.08 August 20123795

From above we see that a spring such as automobile spring (semi-elliptical spring) with length 2L and
load in the centre by a load 2W may be treated as double cantilever. If the plate of cantilever is cut into a series
of n strips of width b and these are placed as shown in Figure 1, then equations (i) and (ii) may be written as
f = 6W.L / n.b.t2 ................................ (iii)
 = 4.W.L3 / n.E.b.t3 = 2.f.L2 /3.E.t ............... (iv)
The above relation gives the bending stress of a leaf spring of uniform cross- section and is given in
Table 1 at various loads. The stress at such a spring is maximum at support.
Table 1 variation of Bending Stress with load
Load
( Newton )
Bending Stress
N/mm2
1000 145.507
2000 291.015
3000 436.522
4000 582.0302
5000 727.540
6000 873.045
7000 1018.550
8000 1164.060
9000 1309.570
10000 1455.076
11000 1600.583
12000 1746.091
13000 1891.598
14000 2037.106
15000 2182.613
Length of Leaf Spring Leaves
The length of the leave springs are calculated by using the formulas given below
Length of smallest leaf = lengtheIneffectiv1
1n
lengthEffective 

Length of next leaf = lengtheIneffectiv2
1n
lengthEffective 

Similarly,
Length of (n-1) th leaf = lengtheIneffectiv1)-(n
1n
lengthEffective 

Length of master leaf = 2L1 + 2  ( d + t )
Where 2L1 = Length of span or overall length of the spring,
l=distance between centers of U-bolts (ineffective length (I.L) of the leaf spring),
n F = Number of full length leaves,
n G = Number of graduated leaves,
n = Total number of leaves = nF + n G,
E.L = Effective length of the spring = 2L1 – (2/3)l,
d = Inside diameter of eye and
t = Thickness of master leaf.G Harinath Gowd et al. / International Journal of Engineering Science and Technology (IJEST)ISSN : 0975-5462Vol. 4 No.08 August 20123796