The creative engineers in the leaf spring manufacturing field seek to optimize the design in a service of a leaf spring they want during its conception. In this paper, we use a probabilistic approach to design an Equal-Leaves Spring (ELS) by verifying the design criterion for a given reliability that allows introducing design conservator diagrams for several ELS characteristics. The reliability of the design is determined using the performance function with Monte Carlo Simulation (MCS). In addition, we study the effect of each parameter expressing the design criterion of such a spring and we also study the interaction between these parameters using the Design of Experiment method (DoE). The Surface Response method (SR) is used to analyze the effect of design criterion parameters on the security level response. This proposed method leads to a more reliable design of an ELS compared to deterministic design criterion

BEN SGHAIER, Rabï

DOUKH, Akram

I-Revues

21ème Congrès Français de Mécanique                                                                  Bordeaux, 26 au 30 août 2013 
  
1
Probabilistic approach for the design of an Equal-Leaf Spring  
A. DOUKHa, R. BEN SGHAIERb  
a. Ecole Nationale d’Ingénieurs de Sousse, BP 264 Sousse Erriadh, 4023 (Tunisie).  
b. LMS Laboratoire de mécanique de Sousse, Ecole Nationale d’Ingénieurs de Sousse, BP 264 Sousse 
Erriadh, 4023 (Tunisie). Institut Supérieur des Sciences Appliquées et de Technologie de Sousse, Cité 
Taffala (Ibn Khaldoun), 4003 Sousse, Tunisie. 
 
Résumé :  
Les ingénieurs créatifs dans le domaine de construction  des éléments de suspension, cherchent à optimiser 
le dimensionnement en service des ressorts à lames durant la conception. Dans ce papier, on utilise une 
approche probabiliste pour le dimensionnement d’un ressort à lames égales en vérifiant le critère de 
dimensionnement à une fiabilité donnée ce qui nous permet d’introduire des diagrammes de 
dimensionnement conservateurs. La fiabilité de dimensionnement est déterminée en utilisant la fonction de 
performance avec simulation numérique à travers la méthode de Monte Carlo. La méthode proposée permet 
d’obtenir un dimensionnement d’un ressort à lames égales plus fiable que le dimensionnement basé sur le 
critère déterministe. 
Abstract : 
The creative engineers in the suspension parts manufacturing field seek to optimize the design in a service of 
leaf springs they want during its conception. In this paper, we use a probabilistic approach to design an 
Equal-Leaf Spring (ELS) by verifying the design criterion for a given reliability that allows introducing 
design conservator diagrams for several ELS characteristics. The reliability of the design is determined 
using the performance function with numerical Monte Carlo Simulation (MCS). This proposed method leads 
to a more reliable design of an ELS compared to deterministic design criterion. 
Mots clefs: ressort à lames, dimensionnement, fiabilité, méthode de Monte Carlo. 
1 Introduction 
A spring is a mechanical part that uses elastic properties of some materials to absorb mechanical energy. It 
produces a movement or exerts an effort or a couple over a system. They are variously employed in motor 
vehicle suspension systems, the actual choice made being determined largely by versatility in application and 
best economy of material in terms of energy storage per unit volume. The goal of the use of such a spring is 
to fulfill a best suspension: the stability of vehicle, the bearing of contact of the wheel on the ground, the 
safety of the automotive passenger and transported material against the impacts and vibrations that can be 
found during the path. We distinguish many types of springs: leaf spring, helical spring and torsion spring. 
Leaf springs are among springs that are used for the suspension in a wheeled vehicle because of the 
flexibility of one or many super-imposed metallic leaves. This flexibility orders the oscillation of spring due 
to the vehicle load. There are many types of leaf springs [1]: laminated leaf spring; parabolic leaf spring; 
single, equal or multi-leaf spring. The equal-leaf spring (ELS) is constituted of super-imposed leaves having 
the same dimension parameters.  
The modelisation of this spring is basically carried out from the modelisation of a flexible beam with 
uniform rectangular section with a fixed-one-end point (a cantilevered beam) which serves to find the design 
in a service criterion for this spring [1]. The design in a service criterion is expressed by a mathematical 
inequality. When it is verified, the leaf spring keeps its form and performance. This criterion depends on 
many parameters defining the dimension and the metallurgy of the leaf spring. Varying the parameters 
values, the service behaviour of the leaf changes; these parameters influence the leaf flexibility (oscillation 
period). We consider the parameters defining the design in a service criterion of a leaf as random variables. 
They are expressed by a sampling drawn following a probabilistic distribution[3,4]. In this work, the service 
reliability is computed by using Monte Carlo method as the ratio of the secured events number and the total 
21ème Congrès Français de Mécanique                                                                  Bordeaux, 26 au 30 août 2013 
  
2
P/2 
l1 l2 
l 
P/2 
la 
Bridle 
Inserter 
size. In order to stabilize the reliability response of the modelling, the size number should be optimized.  
2 Mathematical modelling of an equal-leaf spring 
In our work, we consider an ELS defined by the following dimensional data: length l, width B, leaf thickness 
e and number of equal leaves n .   
 
 
 
 
Figure 1. Equal-leaf spring 
A section part of the ELS is fitted in the bridle stopping the flexure and is then eliminated from the length 
calculation as expressed in [1]: 
1 2 aL=l +l = l-l    (1) 
The load P is uniformly distributed on the whole section of the leaves. However, the first leaf called master 
leaf, which is directly rested on the supports, has to rest on the longitudinal loading due to the engine 
acceleration and breaking, as well as on the lateral loading. The master leaf contains reinforced extremities 
called eyes which allow the spring handling.                                                                                             
3 Design criterion of an equal-leaf spring 
 In order to ensure a good design of an ELS, the maximal stress must be always inferior to the admissible 
stress. This condition is given by the following inequality:    
max 2
3. .
2. . .
= < adm
P L
B n e
σ σ           (2)     
admσ is the admissible stress which defines the state just before failure and it is expressed as follow: 
=
e
adm
R
s
σ             (3)                                                                                                           
where s  is the safety coefficient and eR  is the yield stress of the material.  
In this paper, we consider the 55Cr3 steel that is the most useful in the leaf spring construction. This material 
has the yield stress of 1250N/mm2 at 0.2%. We use a safety coefficient of 5 corresponding for this case of 
material and application [2]. The applied stress is expressed as follows: 
          
  
.
app
P
B L
σ =                        (4)                
 the criterion is equivalent to the strict inequality below:  
2
max 2
3. .
2. .
app
adm
L
n e
σ
σ σ= <                                                     (5) 
This design criterion in a service may be interpreted graphically on the figure 2: 
21ème Congrès Français de Mécanique                                                                  Bordeaux, 26 au 30 août 2013 
  
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Figure 2. The curve of the maximal stress as a function of the applied stress. 
The horizontal straight line represents the admissible stress and the inclined straight line represents the 
maximal stress as a function of the applied stress. The zone in which the maximal stress is strictly less than 
the admissible stress (the point is situated below the horizontal straight line) is called design in a service 
security zone. The ELS is then designed and it is performant. Whereas, the zone in which the maximal stress 
is strictly superior to the admissible stress is called the weakness zone and the spring is defective. 
The equality condition is a condition of doubt: we cannot decide whether it is the state of security or 
weakness. The slope of the oblique straight line is 
2
2
3.
2. .
L
n e
.                                                                 
The variation of the slope is deduced from the design variation of the ELS. When varying the values of e and 
L parameters, we get a beam of straight lines with different slopes dispersed from the origin instead of one 
inclined straight line. This is geometrically interpreted by the figure 3: 
 
Figure 3. Curve of the maximal stress as a function of applied stress for different design types. 
 
Higher values of a slope mean that the length value of the spring is big and the thickness value is small. A 
small load applied on the ELS produces a big deflection and strong oscillations. The behavior of the spring in 
a service rapidly decreases with increasing the slope. Otherwise, when the slope is small it means that the 
length value is small and the thickness one is big. A heavy load applied on the ELS lead to a small deflection 
and the spring will be stable. Then, the behavior in a service of the ELS improves.     
4 Probabilistic approach for design an equal-leaf spring 
The aim of this probabilistic approach is to eliminate the use of safety coefficient (s) and to replace it by 
defining different reliability values with corresponding confidence levels. 
4.1 Reliability computation 
 Many experiences have been carried out on the ELS. We vary the dimensions of the ELS small variations; 
we verify therefore the design criterion and we test the behaviour of the leaves spring. In this paper, we 
verify the design criterion and then we compute the reliability using the Monte Carlo method [4]. This 
 n=3; Re=1250N/mm2 
e=5:3:20mm; s=5 
L=1000:100:2000mm  
B=20:10:120mm 
P=1:49:2500N 
n=3 ; e=11mm 
L=1500mm  
B=50mm; s=5 
Re=1250N/mm2 
 
 (MPa) 
 
(M
Pa
) 
(MPa) 
 
(M
Pa
) 
 
21ème Congrès Français de Mécanique                                                                  Bordeaux, 26 au 30 août 2013 
  
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probabilistic approach is used to find the dimensions that carry a way to the best reliability: we speak about 
design criterion with a given reliability. 
Then, we define the following random variables: eR , L , e  with a coefficient of variation (cov) for each one. 
These variables can be represented by a vector. For each random variable, we carry out an N random 
sampling following a Gaussian normal distribution defined by a mean value and a coefficient of variation. 
These vectors allow the computation of the reliability in a service.   
The proposed probabilistic approach for this problem of design is based on the  couple « maximal stress-
yield stress » that consist in determining the probability that the yield stress of ELS will be superior to the 
maximal stress. For a function of yield stress we have 
 YieldS= eR                                                                                                                                                      (6) 
and for a function of maximal stress we have Sigmax=
2
max 2
3. .
2. .
=
app L
n e
σ
σ                                                     (7) 
We define a performance function G that separates the design in a service field: a zone where the spring 
fulfills the best suspension and the weakness area by i i iG (x )=YieldS(x )-Sigmax(x ) , where xi is an 
element of random vector {X}, having [3,4] a joint probability distribution function if{x }({X}) . 
iG (x )>0 presents a best suspension of the spring field (spring designed in a service) Fs for the ELS; 
iG (x )<0 presents a weakness field Fd for the same spring.  
The security design probability PS is determined by: 
{ }{ ( ) 0} 11 ({ }) ... Pr ( ( ) max( )) Pr ( ( ) 0)i i
n
S G x n i i ixP f X dx dx ob YieldS x Sig x ob G x>= = > = >∫
ℝ
           (8)                                                                                                                                   
When n is the number of parameters defining the function G.  
This probability can be computed by numerical integration which remains most of time complicated due to 
the existence of many integration variables. Consequently, the Monte Carlo method is very practical and 
easy for use. In this paper, the design probability PS is approximated by numerical resolution using the 
Monte Carlo method. 
For a total number of simulations N, we admit that the frequency of performance events of the spring where 
G (xi)>0 stretch to design probability PS  when → +∞N  : 
( ( ) 0)
limS iP
N
Number of performant events G x
Total number of events= →+∞
>                                                                             (9) 
This procedure presents the simplicity of the reliability computing. The speed of the method convergence is 
very slow as it is proportional [5] at N . A number of simulations equal to 104 is required in order to ensure 
the estimation [6] of the reliability which represents the security design probability PS with a variation 
coefficient of 10%. The reliability response begins to stabilize from the number N=105 which will be chosen 
as a standard size applied in our work [4]. The used random variables for the following application are 
defined by a cov of 2%. In order to explore the proposed design service criterion, we use a graphic 
representation (figure 4) showing the yield stress as a function of the applied stress which both are 
considered as random variables. 
 
Figure 4. Surface of dispersion and cloud of points of an ELS. 
N=300; n=3; cov=2% 
L=1500mm; e=5mm 
B=50mm; Re=1250N/mm2 
σapp=0.0275MPa 
(MPa) 
(M
Pa
) 
21ème Congrès Français de Mécanique                                                                  Bordeaux, 26 au 30 août 2013 
  
5
 
The straight line representing the limit state of the design criterion ( iG(x )=0 ) is transformed into a beam of 
straight lines. Instead of the loading representative point defined by the applied stress, we get a cloud of 
points. The surface dispersion corresponding to the criterion and to the loading depends on the fixed size of 
the random sampling. The obtained beam of lines defines a new zone called the uncertainty zone which 
separates both security and weakness zones.   
4.2 Iso probabilistic design in a service criterion 
  In order to obtain an iso probabilistic design in a service criterion for a given reliability, we increment the 
stress amplitude loading and we compute the reliability in each incrementation step until obtaining the target 
one [4]. The whole random variables are used with a cov of 2%. Several reliability values 
(99,90,75,50,25,10,1%) for the iso probabilistic design in a service criterion are determined in this 
application. These results are illustrated in figure 5: 
 
Figure 5. Design diagrams of a given reliability. 
By applying the iso probabilistic design in a service approach for different reliability values, we obtain 
several confidence level zones instead of both unique security and weakness zones, characterizing the classic 
deterministic approach. Particularly, the result of this last application is fundamentally based on the 
hypothesis that iso reliability criteria, in-sighted by horizontal lines (figure 5), are obtained for a specific 
ELS leading to increment the loading amplitude according to a unique direction. In order to obtain an iso 
probabilistic design criterion valid to many different types of ELS, many increment directions have been 
used corresponding each one to specific design characteristics. In each direction, we increment the load until 
obtaining the desirable reliability value. We record the points corresponding each one a particular direction. 
Fitting these recorded points, we obtain the iso probabilistic design in a service criterion curve. 
An application for 99% reliability is shown on figure 6: 
 
Figure 6.  99%  iso reliability diagram for many ELS characteristics. 
n=3; cov=2% 
L=1500mm; e=5mm 
B=50mm; Re=1250N/mm2 
n=3; cov=2% 
Re=1250N/mm2 
(MPa) 
(M
Pa
) 
(MPa) 
(M
Pa
) 
99% Reliability 
21ème Congrès Français de Mécanique                                                                  Bordeaux, 26 au 30 août 2013 
  
6
5  Conclusions 
The proposed probabilistic approach permits improving the design in a service criterion of an ELS, in the 
sense that we increase the security confidence level. Geometrically, this can be interpreted as: the 
probabilistic approach allows obtaining a beam of straight lines and a cloud of points leading to get an 
uncertainty zone. Furthermore, the probabilistic design in a service criterion for a given reliability allows 
plotting iso probabilistic diagrams (conservative diagrams). Two applications of the iso probabilistic design 
criterion are studied: the first one is for a specific dimension of ELS following then a unique increment 
direction, and the second one is for several ELS characteristics following so many increment directions. 
 
References 
[1] M. Duchemin, Techniques de l’ingénieur Ressorts contraintes de flexion. Réf. B5432. 
[2] Y. Xiong "Formulaire de mécanique-Pièces de constructions" Groupe Eyrolles, 2007, ISBN : 978-2-212-
12045-5. 
[3] Ch. Bouraoui R. Ben Sghaier R. Fathallah "Effets des dispersions des paramètres de condition de 
surfaces sur la fiabilité des composants mécaniques grenaillés sollicités à la fatigue polycyclique" 18ème 
Congrès Français de Mécanique Grenoble, 27-31 Aôut CFM'07 2007, France. 
[4] R. Ben Sghaier, Ch Bouraoui, R. fathallah et al. Probabilistic high cycle fatigue behaviour prediction 
based on global approach criteria International Journal of Fatigue 29 2007.  
[5] Lemaire M, Chateauneuf- A. & Mitteau J.C. Fiabilité des structures: couplage mécano-fiabiliste statique, 
Edit. Hermes Paris, 2005. 
[6] Zhao Y.G, Ono T. Moment for structural reliability. Structural safety. 23 (2001) pg: 47-75.  
 
 


Probabilistic approach for the design of an equal-leaf spring

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