Robust Design of S-Shaped Box Beams Subjected to Compressive Load

In order to design the automotive components considering probabilistic uncertainties in the decision variables, it is desired to perform a robust design process. The peak crushing force of the energy absorber components is one of the important objectives of the design of such components. In this paper, at first, the peak crushing force of the S-shaped box beams, as a highly simplified model of front member of a vehicle body, is extracted mathematically. Using such obtained mathematical model and Monte Carlo simulation, genetic algorithm is then used for the robust design of the S-shaped box beams having probabilistic uncertainties in material and geometrical parameters. In this way the variance-per-mean ratio of the peak crushing force Pmax is considered as the objective function. It is shown that some interesting relationships as useful robust design principles involved in the performance of the S-shaped box beams can be discovered by the robust design of the obtained mathematical model.


Introduction
Higher-speed transportation increases the probability of traffic accidents which in turn cause serious damages to passengers.Design of auxiliary metal structure or structural components capable of sustaining prescribed loads and absorbing crushing energies during plastic deformation has become a special task in design to ensure the occupant safety.Therefore the crash characteristic of energy absorbing component has received considerable attention over the past decades 1-7 .As a highly simplified model of the front-side member of a vehicle body, which plays an important role in absorbing energy during collision, various investigators in previous works 8-14 have studied the crushing behavior of the S-shaped structures.
In real engineering practices, there exist various sources of uncertainty which have to be compensated through optimal robust design approach 15-17 .Those uncertainties include model parameter variations due to environmental conditions, incomplete knowledge of parameters including material and geometry variables, age, and so forth 18-20 .In conventional optimum system design, uncertainties are not addressed and the optimization process is accomplished deterministically.In fact, it has been shown that optimization without considering uncertainty generally leads to nonoptimal and potentially highrisk solution 15, 21-25 .Therefore, it is very desirable to find a robust design whose performance variation in the presence of uncertainties is not high.Generally, there exist two approaches addressing the stochastic robustness issue, namely, robust design optimization RDO and reliability-based design optimization RBDO 26 .Both approaches represent nondeterministic optimization formulations in which the probabilistic uncertainty is incorporated into the robust optimal design process.Therefore, the propagation of a prior knowledge regarding the uncertain parameters through the system provides some probabilistic metrics such as random variable and random process 21 .In RDO approach, the robust performance is required to be less sensitive to the random variation induced by uncertain parameters so that the performance degradation from ideal deterministic behavior is minimized.In RBDO approach, some evaluated reliability metrics subjected to probabilistic constraints are satisfied so that the violation of design requirement is minimized.Regardless of the choice of any of these two approaches, the objective function e.g., peak crushing force and the constraints of the optimal design should be evaluated reflecting the effect of probabilistic nature of uncertain parameters in the performance of the system.With the aid of ever-increasing computational power, there have been a great amount of research activities in the field of robust analysis and design devoted to the use of Monte Carlo simulation 21, 22 .In fact, MCS has also been used to verify the results of other methods in RDO or RBDO problems when sufficient number of sampling is adopted.In the Monte Carlo simulation MCS method, random samples are generated assuming pre-defined probabilistic distributions for uncertain parameters.
Basically, the optimization process is defined as finding a set of values for a vector of design variables so that it leads to an optimum value of an objective or cost function.In such single-objective optimization problems, there may or may not exist some constraint functions on the design variables and they are, respectively, referred to as constrained or unconstrained optimization problems.There are many calculus-based methods including gradient approaches to search for mostly local optimum solutions and these are well documented 27, 28 .However, some basic difficulties in the gradient methods such as their strong dependence on the initial guess can cause them to find a local optimum rather than a global one.This has led to other heuristic optimization methods, particularly Genetic Algorithms GAs that are being used extensively during the last decade 29, 30 .Such nature-inspired evolutionary algorithms differ from other traditional calculus-based techniques 31, 32 .The main difference is that GAs work with a population of candidate solutions and not with a single solution in the search space.This helps significantly to avoid being trapped in local optima 33 as long as the diversity of the population is well preserved.
In this paper, peak crushing force of the S-shaped box beams is mathematically modeled.The obtained model is then used in a combined robust and reliability-based design approach to find a reliable and robust design.In this way, to minimize the variation of the peak crushing force P max subjected to probabilistic constraint, genetic algorithm is used.In this single-objective GA optimization problem, variance-per-mean ratio of P max is considered as objective function and for each generation the value of mean and variance of objective function and the value of probability of failure are calculated using the Monte Carlo simulation MSC method.The value of energy absorption of each obtained design point under specific load and boundary conditions is then carried out using finite-element commercial software ABAQUS/Explicit.Results are informative and maybe effectively used for the design of S-shaped square tubes.

Extraction of Peak Crushing Force
The front-and rear-side members of vehicle, which play an important role in absorbing energy during collision, usually have a curved shape to avoid interference with other components like engine, driving system and fuel tank, and so forth.The S-shaped box beam which is an idealized model of front-side member of a vehicle body is depicted in Figure 1.Any variation of geometric parameters which are denoted as curve radius R , curve angle θ , web width C , wall thickness t , and offset of two-end part D will lead to new design and new behavior.In this study, the total length of the structure L is fixed equal to 1 m.It should be noted that both the straight lengths l ab and l ef and oblique length l cd are derived variables, depending on the values of the curve angle and curve radius.The mechanical properties of employed material can be specified with independent parameters, namely, elastic modulus E , Poisson's ratio υ , yielding stress σ 0 , and density ρ .
Due to geometrical restriction, the upper bound of curve radius depends on the offset of the two-end part D and the curve angle θ and is given by Loading conditions considered in the mathematical model are depicted in Figure 2. The front end of the beam is considered free just in the direction of the external load, and the rear end has been fully clamped.Free-body diagram of the S-beam under the applied load P is depicted in Figure 3; Q a , P f , and Q f are the reaction forces, and M a and M f are the reaction moments.The equilibrium equations for the S-beam are expressed as follows:

Mathematical Problems in Engineering
For determining unknown reactions, two additional equations are needed.Equations for the deflection along direction y and the slope of beam at the point a can be used to solve this indeterminate system.Using Castigliano's theorem, these equations can be written as follows: where U is the strain energy of the beam under the applied load P. For simplicity, in this study, only the strain energy due to bending has been considered and formulated as follows: where, M s is the cross-sectional bending moment along the beam.To determine M s , the five different sections depicted in Figure 3 along the beam have been considered.The equation of the equilibrium of moment for these sections is expressed as follows For Section 1.
where x 1 is the distance from a along the first straight part ab.For Section 2, where θ 1 denotes meridian coordinate system along the first curve part bc.
For Section 3, where x 2 is the distance from c along the oblique part cd.For Section 4,

2.10
where θ 2 denotes the meridian coordinate system along the second curve part de.For Section 5, where x 3 is the distance from f along the second straight part ef.Substituting 2.6 -2.11 and 2.3 in 2.4 ,

2.12
Mathematical Problems in Engineering which can be evaluated as fM a − gQ a hP 0, 2.13 where

2.15
Mathematical Problems in Engineering 7 which can be evaluated as where 2.17 From 2.13 and 2.16 , Q a and M a can be obtained from where Substituting values of M a and Q a obtained from 2.18 and 2.19 in the 2.7 -2.11 , bending moment will be determined for any section along the beam.Similarly, the equilibrium equation of the axial force for sections shown in Figure 3 can be written as follows: 21 22 23 a, g, h, and f in Figure 7, schematically.Substituting 2.18 and 2.19 in 2.8 and 2.23 , the axial force and bending moment at the point g can be obtained from N g L g P, M g K g P,

2.28
When the external load P is applied, the S-beam deforms elastically until yield is reached in the extreme fibers on the most stressed sections.Assuming that the beam is made of elasticperfectly plastic material, increasing the external load, the plastic region in the cross section will be increased.In the limit, the whole section becomes plastic and then P P max ; P max which is the collapse load is also called peak crushing load.
The stress distribution across the section of a fully plastic cross section is depicted in longitudinal stress from the central axis of the beam, the bending moment and axial force are related to c by where Y is the yielding stress of the material.Eliminating c from the two expressions of 2.29 leads to where N 0 and M 0 are the fully plastic bending moment and fully plastic axial force, respectively, and can be obtained from

2.31
Considering sections a and f depicted in Figure 7 as the most stressed sections and substituting 2.18 in 2.30 , P max can be obtained from Otherwise, considering sections g and h as the most stressed sections and substituting 2.18 in 2.30 , P max can be obtained from Therefore, the peak crushing force of S-shaped box beams can be obtained from P max min{P max,1 , P max,2 }. 2.34

Stochastic Robust Analysis
In the robust design approach, it is desired to minimize the variability of a random process due to the uncertain probabilistic parameters about a deterministic behavior.Therefore, the conventional robust optimization problem can be formulated as follows: Minimize μ f x, d, p , υ f x, d, p , 3.1 where f x, d, p is the performance or the cost function, μ is the mean value, and υ is one of the dispersion measure operators such as variance σ 2 , standard deviation σ , or coefficient of variation C v = σ/ μ .In this study, x is the vector of design variables which are uncertain, d is the vector of deterministic design variables, and p is the vector of uncertain parameters which are not design variables.
In the reliability-based design approach, it is required to define some reliability metrics via some inequality constraints.Let us consider a deterministic constraint of the form g i (x,d,p)≤ g i , where g i is the limiting value of the ith constraint.This constraint can be transformed into a probabilistic constraint using the definition of a random process The typical probability constraint is then represented as where P i f denotes the probability of failure of the ith reliability measure and m is the number of inequality constraints i.e., limit-state functions and ε i is the highest value of the desired admissible probability of failure.It is clear that the ideal value of each P i f is zero.In the reliability-based robust design process presented in this paper, an approach that simultaneously considers reliability and robustness is proposed.This methodology can be formulated as follows:

3.4
Taking into consideration the stochastic distribution of uncertain parameters, the probability of failure, P G x, d, p ≤ 0 , can now be evaluated for each probability function as where f X is the probability density function of X x, p .This integral is, in fact, very complicated particularly for systems with complex G i x, d, p 34 , and Monte Carlo simulation is alternatively used to approximate 3.5 .In this case, a binary indicator function is defined such that it has the value of 1 in the case of failure G i X, d ≤ 0 and the value of zero, otherwise:

3.6
Consequently, the integral of 3.5 can be rewritten as Based on Monte Carlo simulation 35, 36 , the probability using sampling technique can be estimated using

3.8
In other words, the probability of failure is equal to the number of samples in the failure region divided by the total number of samples.Evidently, such estimation of P f approaches the actual value in the limit as N → ∞ 34 .However, there have been many research activities on sampling techniques to reduce the number of samples keeping a high level of accuracy.A newer method that has become more widely used is Hammersley Sequence Sampling HSS .HSS is considered a quasi-MC sampling method because deterministic points are used instead of random points.Hammersley points are used to divide a unit hypercube, providing uniform sample points across the sample space.Since the points are chosen on a unit hypercube, they are transformed to the given parameter distributions providing sample points for simulation.This method produces good coverage of the distribution with a greatly reduced set of sample points 37-39 .

Robust Design of the Box Beam
The mathematical model of peak crushing force P max obtained in previous section is now employed as model in the reliability-based robust design process.The desired value of P max is considered less than 70 KN due to occupant safety and more than 35 KN due to vehicle safety.Therefore, optimization of the mean value of peak crushing force is not the aim of this paper.The goal of the robust design approach presented in this study is to minimize the variation of P max subjected to probabilistic constraint considering uncertain design variables.This reliability-based robust design process can be formulated as:

4.1
In the case of robust design, parameters like density and yield stress vary according to a priori known probabilistic distribution functions around a nominal set of parameters.In this study, the uncertain design parameters, namely, σ 0 and t are varied with the Gaussian distributions.The standard deviations of the Gaussian distributions are considered equal to 5.5 and 0.05 for σ 0 and t, respectively.The evolutionary process of multiobjective optimization is accomplished by 1000 Monte Carlo evaluations using HSS Hammersley Sequence Sampling distribution.In the optimization process, the population size, the crossover probability P c , and the mutation probability P m are considered equal to 100, 0.7, and 0.07, respectively.
Results have been produced for the S-beams with various values of the offset of twoend parts D 150, 200, 250, 300 and various values of the web width C 50, 55, 60, 65, 70 .In this way, for each specific value of D and C, genetic algorithm is used to find optimal values of other design parameters: R, θ, and t.The variance-per-mean ratios of all optimum design points are shown in Table 1.For choosing an optimum design point, additional competing criteria functions such as weight and energy absorption capacity of S-beams are considered as objective functions.In this way, 13 various finite-element analyses have been performed due to those different design points Table 1 under the same load and boundary conditions.4-node doubly curved thin-shell elements S4R with reduced integration were employed for the finite-element analysis.The material of S-shaped energy absorbers has been considered as elastic-plastic steel with linear strain hardening.The mechanical properties are assumed as follows: Young's modulus E 206 GPa, Poisson's ratio υ 0.3, plastic modulus E T 1.4 GPa, yielding stress σ 0 162 MPa, and density ρ 7800 kg/m 3 .In order to simulate the dynamic crushing condition, a 500 kg lumped mass with an initial velocity of V 0 10 m/s is attached to the end of the beam, as shown in Figure 9.The rear end of the beam is considered free just in the direction of impact whilst the front end has been fully clamped.

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The computed values of the absorbed energy and the weight of the structure are depicted in Table 1.The results clearly reveal that the designs B1 and C1 have maximum absorbed energy and minimum weight, respectively.It is now desired to find a tradeoff optimum design point compromising both objective functions Absorbed energy & Weight .This can be achieved by the method employed in this paper, namely, the mapping method.In the mapping method, the values of objective functions of all design points are mapped into intervals 0 and 1. Mapped values of each objective function and sum of them are shown in Table 2. Using the sum of the mapped values for each design point, the tradeoff point simply is the one having the minimum sum of those values.Consequently, optimum design point, C1, is the tradeoff point which has been obtained from the mapping method.

Conclusion
An analytical model based on energy method has been employed for the peak crushing force of the S-shaped box beams.Using obtained analytical model and Monte Carlo simulation, genetic algorithm has been used for the robust design of the S-shaped box beams having probabilistic uncertainties in material and geometrical parameters.In this way, 13 design points with minimum variance-per-mean ratio of peak crushing force have been obtained.In order to choose the tradeoff optimum design point, finite-element study using the S4R element using the commercial software ABAQUS has been performed and energy absorption capacity of all 13 design points has been computed.The mapping method has been proposed and used in this paper to find the tradeoff optimum design point and one of 13 design points has been chosen using this method.The very good behavior of chosen design point indicates the worthiness of the approach of this paper.

Figure 1 :
Figure 1: The geometric parameters of S-shaped box beams.

Figure 2 :Figure 3 :
Figure 2: Loading conditions considered in the mathematical model.

Figure 4 :Figure 5 :Figure 6 :Figure 7 :
Figure 4: Cross sectional bending moment diagram for the S-shaped box beams with three different curve radii.

Figure 8 .Figure 8 :
Figure 8: The stress distribution across the section of a fully plastic cross section.

Figure 9 :Figure 10 :
Figure 9: The impact model of the S-shaped box beam with loading and boundary conditions.

Table 1 :
The values of objective functions and their associated design variables of the obtained design points.

Table 2 :
The mapped values of objective functions and sum of them for each design point.