Multiobjective crashworthiness optimization of multi-layer honeycomb energy absorber panels under axial impact

Thin-Walled Structures 107 (2016) 197–206

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Multiobjective crashworthiness optimization of multi-layer honeycomb energy absorber panels under axial impact Jamshid Fazilati n, Maryam Alisadeghi Aerospace Research Institute, Tehran, Iran

art ic l e i nf o

a b s t r a c t

Article history: Received 24 September 2015 Received in revised form 11 June 2016 Accepted 11 June 2016

In this study, the design and optimization of a multi-layer configuration of hexagonal metal honeycomb energy absorber is performed using the genetic algorithm. It is assumed that the body structure with a predefined velocity impacts with barrier and the design objectives are to absorb whole kinetic energy besides limiting impact shock force. The response surfaces of honeycomb impact characteristics are extracted using finite element approach and then a honeycomb energy absorber is sized for case of a presumed impact problem. A multi-objective optimization technique is adopted to maximize the energy absorption capacity and to minimize the impact shock level while minimizing the total absorber size. A factorial design of experiment and response surface method is utilized to solve the optimization problem. The geometric specifications of honeycomb panels including the cell size, foil thickness, height and absorber face area for each layer of honeycomb panel are assumed as the design variables. Some optimization problems are handled and the optimized designs are compared to those from the literature wherever available. & 2016 Elsevier Ltd. All rights reserved.

Keywords: Multi-layer honeycomb absorber Crashworthiness Optimization Design of experiment Response surface method

1. Introduction Due to the increasing demand for safety in land, sea and air transportation, the low-cost and low-price impact protection is among the indispensable fields of study. To avoid loss of life, any injury or damage during an accident, there is a need for components that can absorb and dissipate the colliding bodies’ energy. The energy absorbing substructures are the components used to reduce the unwelcomed effects in accidents. In energy absorbing devices, the kinetic energy prior to impact is transformed into other kinds of energy majorly to plastic strain energy through large deformations of material [1–3]. Honeycomb structures are widely used in various engineering applications (e.g. space, aeronautics, high speed trains and automotive industry) as energy absorbers and protective components, because of their good crashworthiness characteristics of high energy absorption capacity and high strength-to-weight ratio. Due to its low density, high transverse strength and crashworthiness characteristics, the honeycomb structures are recognized as lightweight structural components. In out-of-plane impact direction these structures are more effective in terms of energy absorption while in in-plane directions could be used as soft dampers [2–7]. Metal honeycombs have been successfully used as shock absorbers for example in n

Corresponding author. E-mail address: [email protected] (J. Fazilati).

http://dx.doi.org/10.1016/j.tws.2016.06.008 0263-8231/& 2016 Elsevier Ltd. All rights reserved.

Apollo 11 moon landing modulus legs [8,9]. A knowledge of structural dynamics besides material mechanical properties is fundamental in the design of components required in withstanding and mitigating mechanical impact conditions. In recent years, many theoretical as well as experimental studies have been published to uncover the mechanical behavior of thin-walled and honeycomb structures during external mass impacts. Most of these investigations have been reported for single layer honeycomb panels. Yasui [10] has experimentally studied the quasi-static as well as dynamic impact crushing behavior of single and multi-layer honeycomb sandwich panels. The crushed multi-layer panel assemblies have been arranged in uniform type as well as in the pyramid type. In his study, the pyramid-type panel assemblies from two or three basic panel rows have been observed to be the most effective design from the standpoint of energy absorption capacity. Yamashita and Gotoh in 2005 [11] have employed finite element method (FEM) simulation besides experimental methods to investigate the out-of-plane impact properties of aluminum alloy hexagonal honeycomb cores under crushing impact. The numerical simulation has been conducted on a repeatable "Y" cross section column. Deqiang et al. [12] have been studied the functionality of configuration parameters of double-walled hexagonal honeycomb cores (DHHCs) and their out-of-plane dynamic plateau stresses for impact velocity range of 3 to 350 m/s by using ANSYS/ LS-DYNA FE simulations. They suggested some empirical expressions on the out-of-plane dynamic mean plateau stresses of

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DHHCs. They showed that in aluminum double walled hexagonal honeycomb cores, for a given impact velocity out of plane dynamic plateau mean stress is related to the ratio between cell wall thickness and edge length. Li et al. [8] have used metal honeycomb panels as buffering and crushable structures in a lunar lander system and have performed size optimization by using response surface method (RSM). Besides the design optimization procedure, some parametric studies have been carried out to investigate the influence of foil thickness and cell length on the metal honeycombs crushing characteristics. In their study, explicit solver of LSDYNA FE code has been employed to perform the crashworthiness analyses. Yin et al. [13] have optimized energy absorbers of honeycomb type with various cell specifications. In their optimizing process, the specific energy, and peak crush stress has been aimed as objective functions while the single wall thickness, double wall width and branch angle of the “Y” column have been set as the optimization variables. Xu et al. [6] have experimentally investigated the out-of-plane crushing behavior of four types of aluminum hexagonal honeycomb panels over a wide range of impact strain rates with each test has been conducted at constant compressive velocity. The effects of specimen dimensions, relative density, strain rate, and honeycomb cell size on the mechanical properties of honeycombs have been studied. Also they presented Semi-empirical relations describe the effects of a defined coefficient of relative density and strain rate on the plateau stress. Meran et al. [3] have investigated crashworthiness characteristics of aluminum hexagonal honeycomb structures under impact loads both numerically and experimentally. In the present paper, sizing of an energy absorber component for a sample moving vehicle (mass) is considered. The manuscript deals with the multi-objective mechanical as well as crashworthiness optimization of the energy absorber sizing. The capability of absorption of energy and simultaneously limiting the shock force besides observing the volume constraints are decided as the key design goals. The energy absorbing facility is supposed as multi-layer aluminum hexagonal honeycomb brick. Firstly, by employing finite element analysis, and a design of experiments approach, the response surfaces of crashworthiness characteristics of the metallic honeycomb (e.g. mean and peak crushing stress) are extracted. The optimization process is then handled by using a genetic algorithm tool to fulfill the objectives of minimizing the absorber volumetric size subjected to peak stress and energy absorption capacity limitation constraints. Geometry specifications including the honeycomb cell size, cell wall thickness, layer heights, and absorber panels’ face area are problem variables that will be sized through the optimization process. Since the objective of the optimization is to find the best and smallest absorber unit with highest energy absorption and lowest possible shock level, the multi-objective optimization procedure is formulated for minimization of occupied space subjected to equality constraint of energy to be dissipated and inequality condition of upper shock level limit. It is to be noted that to the best of the authors’ knowledge the design and optimization of multi-layer absorber configurations has not been attempted using similar methods elsewhere.

Fig. 1. Honeycomb structural pattern and a repeatable unit-cell with “Y” cross section.

each other in a mutual edge. A single “Y” cross-section shell column that is exhibited to appropriate boundary conditions at the side edges is considered as a representative numerical model for complete reticulated construction of whole honeycomb layer. One of column legs is doubled in thickness in order to demonstrate a real honeycomb wall typically made of two perfectly bonded walls [11,14]. This modeling approach ignores the rare delamination of bonded interfaces and considers the strength of the adhesive bond as infinite. As depicted in Fig. 2, the crushable shell model is positioned between two parallel rigid plates. The top massed plate is moving downward with an initial kinetic energy (velocity of 10 m/s) and is applied to compress the Y-shape column in the axial direction. The bottom rigid surface is totally fixed. Due to the repeatability, there is no difference or priority between any two adjacent cell models from mechanical properties as well as deformation points of view. This fact may be fulfilled through limiting translation in local y direction besides constrained rotation about the local x and z local directions at three non-intersecting edge nodes on all column walls. As mentioned in Fig. 2, this type of end conditions is called local y direction symmetry. Furthermore, all degrees of freedom

2. Numerical simulation The single-layer crushable honeycomb structure is numerically simulated. In order to reduce the calculation cost with a complete honeycomb model, the periodicity of the structural pattern is benefited to model simplifying into a variety of unit repeatable cells. The very simplest unit-cell could be handled is a column of “Y”-shape section within a repeating triangular unit-cell area highlighted in Fig. 1. The model consists of three walls intersecting

Fig. 2. “Y” column geometrical model, loading and constraints setup.

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Table 1 Aluminum alloy 5052-H39 material properties [15]. Property

Value

Density Elastic modulus Poisson’s ratio Yield stress Hardening modulus

2680 70 0.33 265 700

CFE =

kg/m3 GPa MPa MPa

(translational as well as rotational) of Y-shape column top and bottom nodes are kept fixed while just axial displacement at its top end is allowed. An elastic-linear hardening material model is assumed to simulate cell-wall material mechanical behavior. The shell model is assumed to be made from aluminum alloy 5052H39 foils with the material properties presented in Table 1. The explicit nonlinear finite element code LS-DYNA is employed in order to numerically simulate the axial crushing process of honeycomb structure. The Y-shape column is meshed using shell elements of Belytschko-Tsay formulation with three integration points through the thickness. An automatic single surface to surface contact algorithm is applied to avoid penetration of column elements which might contact each other during deformation and folding process. An automatic nodes-to-surface contact formulation is defined between column mesh and the top and bottom rigid plate surfaces in order to avoid any penetrations. The coefficient of friction for all sliding and sticking conditions is set to 0.2 and 0.3, respectively. A mesh study for the case of a model with cell size of 3.175 mm and foil thickness of 0.0254 mm showed that using elements with size of 0.07 mm leaves converged and accurate-enough results of peak and mean forces.

199

Fmean Fpeak

(5)

3.1. Design of experiments and response surface method The design of experiment (DoE) technique is providing a valuable tool for efficient selection of the sampling points in the design space. The DoE allows for maximized achieving information through running of minimum number of selected experimental attempts. Different types of design of experimental techniques including the factorial, Koshal, composite, Latin hypercube and D-optimal design etc. are available. The rn factorial design utilized here generates a mesh of sampling points in shape of n dimensional hypercube, consisting of r points spaced at regular intervals in each variable direction. To create an approximation of order r, at least (r þ 1)n factorial design points are necessitated. [13,17,18]. The response surface method (RSM) is a combination of mathematical as well as statistical techniques useful for the analysis of response optimizing problems in which a response of interest is induced by some changes in the design variables. The original RSM has been initially proposed by Box and Wilson [19] for case of experimental responses modeling. Nowadays RSM is considered to be a proper optimization technique in numerical simulations, too [8]. RSM is also considered appropriate in the design optimization problems especially in complex nonlinear mechanics where contact impact occurs [17]. In this approach, an approximation of the function yr(x) to the structural response is assumed a priori in terms of the basis function in a series form as in Eq. (6). N

yr ( x ) =

∑ βiφi( x )

(6)

i=1 n

3. Crashworthiness study Some important parameters for comparing the performance of energy absorption are briefly defined in this section: Total energy absorption (E) is the area under the force–displacement curve represents the total energy absorption capacity of the absorber.

E=

∫0

δmax

F (δ )dδ

(1)

Mean crushing force (Fmean) is the value whom the crushing force values fluctuate around. The mean crushing force is the ratio of total absorbed energy to the maximum shortening taken place. δ

Fmean =

∫0 max F (δ )dδ δmax

(2)

Specific energy absorption (SEA) is the total energy absorbed per unit mass or per unit volume of the absorber element.

where N represents the number of basis functions φi( x ) on x∈R . A typical, mathematically simple and wide used class of basis functions is the polynomial algebraic where its terms in full quartic form is given below.

1, x1, x2 , …, x n , x12 , x1x2 , … x1x n , …, x n2 , x13, x12x2 , … x12x n , x1x22 , …, x1x n2 , …, x n3, x14 , x13x2 , …, x1x n3 , x12x22 , …, x12x n2 , …, x1x23, …, x1x n3, …, x n4

Generally, the selection of basis functions should ensure sufficient accuracy and fast convergence. The basis equations (Eq. (7)) involves full cross-term functions. Through the use of the leastsquare method, the weight coefficients of the polynomial functions b ¼(β1,β2,...,βN) may be obtained via the definition of the following vector function [8].

(

b = ΦT Φ

−1

) ( Φ y) T

E V

(3)

SEA m =

E m

(4)

Peak crushing force (Fpeak) is the maximum value of force in the force–displacement curve. Crushing force efficiency (CFE) is the ratio of crushing mean force to the crushing peak force [3,16].

(8)

where the basis function matrix ϕ( x ) is defined as

⎡ ( 1) ⎢ φ1( x ϕ = ⎢⎢ ⋮ ⎢ φ ( x ( M) ⎣ 1

)

SEAV =

(7)

)

⎤ ⋯ φN ( x ( 1) ⎥ ⎥ ⋱ ⋮ ⎥ M ⋯ φN ( x ( ) ⎥⎦ M×N

)

)

(9)

where M is the number of numerical sample points. Through the handling of Eqs. (8) and (9), the approximation functions’ weight coefficient vector could be discovered. In order to determine the unknown parameters b¼(β1,β2,...,βN), a number of design sampling points x(i)(i¼1,2,…,M) are required (M 4N) [8,9,17].

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In order to evaluate the accuracy of the developed approximating metamodels, a relative error (RE) index could be defined as a comparison between the FEA result and y(x) as Eq. (10).

RE =

yi ( x)−y^i ( x) yi ( x)

(10)

RMSE =

2 and Radj have values in [0,1] interval and represent the capability of the stepwise regression models to identify the variability of the design responses (Eqs. (11)–(13)). In general, the larger the values 2 of Radj and R2, the better the approximation function is fitted [8,9,17,20].

R2 = 1 −

SSE SST

2 Radj = 1 −

(11)

m−1 (1 − R2) m−n

(12)

Table 2 Impact event simulation results. C (mm)

t (mm)

sp (MPa)

sm (MPa)

SEAm (kj/kg)

SEAv (kj/m3)

3.175 3.175 3.175 3.175 3.175 3.9688 3.9688 3.9688 3.9688 3.9688 4.7625 4.7625 4.7625 4.7625 4.7625 5.5562 5.5562 5.5562 5.5562 5.5562 6.35 6.35 6.35 6.35 6.35

0.0254 0.0381 0.0508 0.0635 0.0762 0.0254 0.0381 0.0508 0.0635 0.0762 0.0254 0.0381 0.0508 0.0635 0.0762 0.02540 0.0381 0.0508 0.0635 0.0762 0.0254 0.0381 0.0508 0.0635 0.0762

7.0309 10.9758 14.4168 17.9242 21.4866 5.6154 8.8204 11.5724 14.3537 17.1540 4.6806 7.3595 9.6952 11.9637 14.3056 4.0074 6.3032 8.3335 10.293 12.3133 3.4971 5.5131 7.3023 9.0148 10.7845

1.7128 3.1743 5.3575 7.6174 10.5127 1.1426 2.1985 3.5830 5.4039 7.6150 0.8498 1.6466 2.7081 3.9746 5.6101 0.6631 1.2725 2.1784 3.2323 4.5536 0.5455 1.0568 1.7776 2.7792 3.6662

29.9588 37.0138 46.8531 53.2933 61.2915 24.9824 32.0455 39.1692 47.2599 55.4975 22.2959 28.7994 35.5244 41.7109 49.0621 20.2968 25.9662 33.3389 39.5747 46.4596 19.0809 24.6461 31.0907 38.8874 42.7498

80289.6186 99196.8943 125566.4250 142826.0879 164261.2941 66952.7985 85881.9714 104973.3964 126656.6186 148733.1839 59753.0112 77182.4826 95205.3650 111785.1832 131486.3153 54395.3557 69589.3523 89348.2538 106060.0896 124511.6084 51136.8404 66051.5065 83323.1737 104218.3431 114569.5516

(13)

SSE and SST are the sum of squared errors and the total sum of squares, respectively and are calculated as: m

∑ ( yi

SST =

−y

2

)

(14)

i=1

In addition, the accuracies of these metamodels can be evaluated based on the root mean squared error function (RMSE), the coefficient of multiple determination (R2), and also the adjusted 2 coefficient of multiple determination (Radj ). The indicators R2

SSE m

m

SSE =

∑ ( yi

− y^i

2

)

(15)

i=1

where y̅ is the mean value of FEA results, yi and y^i is the predicted response obtained from RSM.

3.2. Response surface extraction Enough sample simulation runs is required obtaining the polynomial response surface. The five-level full factorial design is employed here due to its uniformity in sampling in an unknown design space, which resulted in 25 (or 5  5) evenly distributed design sample points. Honeycomb sample geometries have been selected from Hexcel products datasheet [21]. A total number of 25 honeycomb sample geometries with the same height but various cell size and cell wall thicknesses are modeled and simulated using explicit finite element calculations. The crashworthiness data resulted from FEM simulations are tabulated in Table 2. The results of model accuracy check related to using different approximation basis functions are presented in Table 3. By comparing the results listed in Table 3, it could be found that the quartic polynomial basis functions provide the most accurate approximation among all tried metamodels. From the simulation results shown in Table 2, the empirical model of speak and smean based on a quartic order approximation can be expressed as Eqs. (16) and (17), respectively. σp = 1.61 − 6.757x + 1275y + 2.393x 2 − 300.1xy − 1.129e4y 2 − 0.3506x 3 + 41.09x 2y+400. 4xy 2 + 1. 178e5y 3 + 0. 019 x 4 −2. 316x 3y + 17. 52 x 2y 2−3570xy 3−4. 136e5 y 4 (16)

σm = − 6.897 + 5.533x + 346.6y − 1.399x2 − 230.7xy + 5406y2 + 0.1769x 3 + 33.58x2y +225. 8xy2 −5. 835e4y 3 −0. 009185x

4

−1. 652 x 3y −6. 548 x2y2 −2903 xy 3 + 3. 442e5y 4

(17)

Response surfaces of peak and mean stresses are graphically depicted in Figs. 3 and 4, respectively. Table 3 Accuracy check for the smean and s peak response surface metamodels.

Peak Crush Stress

Mean Crush Stress

Approximation order

R2 index

R2adj index

RSME

RE%

Linear Quadratic Cubic Quartic Linear Quadratic Cubic Quartic

0.9469 0.9976 0.9999 1.0000 0.8817 0.9924 0.9995 0.9997

0.9421 0.9970 0.9998 1.0000 0.8710 0.9904 0.9993 0.9993

1.1020 0.2507 0.05656 0.0206 0.8990 0.2455 0.0672 0.0669

[  22.5593, 54.2441] [  10.0628, 5.8281] [  1.7033, 1.5453] [  0.2571, 0.3241] [  45.3254, 305.7322] [  79.5059, 35.7174] [  10.1237, 9.3055] [  4.7980, 3.9299]

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⎧ optimize F ( x) = F ( f ( x), f ( x),…,f ( x)) 1 2 q ⎪ n L U ⎪ S.t. x R and X X X ∈ ≤ ≤ ⎨ gu( x)≥0, u=1, . .s ⎪ ⎪ hv ( x)=0, v=1, . .p < n ⎩

Fig. 3. response surface for peak stress.

201

(18)

where q is the number of objective functions while s and p are the number of inequality-type and equality-type constraints, respectively. XL denotes the lower bound and XU the upper bound for n design variables. The multi-objective genetic algorithm is utilized to find the optimized solution of the crashworthiness problem. The genetic algorithm (GA) is a heuristic optimization method that its scheme brings from the process of biological generation evolution models. In GA, a random initial population of the design variables is generated and progressively new populations are produced using previous individuals. Each newly generated population is modified such that finally converges towards an optimal population which is accepted as the problem solution. Unlike other standard optimization methods, the GA is found to be sufficiently accurate in case of crashworthiness studies where the objective functions are behave highly non-linear with respect to the design variables [22]. In the present research single to multi-layer honeycomb energy absorbers are considered as the dissipater of the kinetic energy and the attenuator of the impact mechanical shock in the event of a typical moving mass impacting a rigid barrier. The moving mass

Fig. 4. response surface of mean stress.

4. Optimization process The effects of thickness as well as cell size of honeycomb structures have been widely studied in the literature. But there are always unsolved challenging problems of which configuration leads to more energy absorption in any special application. Furthermore the limitations of minimum possible absorber occupied space as well as minimum arisen impact shocks is among all design preferences. Due to the opposing nature of the above mentioned phenomena, it must be an optimal compromise to be looked for. The RSM helps finding the functionality of energy absorption capability over the geometric factors of thickness and cell size of honeycomb as explicit mathematical relationships. Many different parameters influence the response of the structure under dynamic loading. Therefore a variety of different optimization problems may be introduced [22]. This means that in a real problem, it is necessary to consider several objectives and constraints of the problem at the same time through a multi-objective approach. Mathematically, a general multi-objective optimization problem, including the maximization and minimization, can be expressed as [23]: Table 4 Impact problem specifications. Traveling Mass Mass velocity prior to impact Max admissible stop distance Allowable impact shock level

1000 kg 10 m/s 20 cm 40g

Fig. 5. flowchart of design and optimization process using response surface method.

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specifications are assumed as remarked in Table 4. A multi-objective optimization procedure for a multi-layer arrangement of honeycomb impact shock absorber is performed. For sake of finding the optimized solution, the obtained response surfaces of the main absorbing parameters of the honeycomb layers are utilized as objective functions while some constraint equations are also considered. The flowchart of the design and optimization process of the multi-layered honeycomb energy absorber involving the RSM and design of experiment steps is shown in Fig. 5.

5.2. Solutions to the problem In the first example, a single layer honeycomb panel is assumed to be subjected to a mass striker as a bumper. As it is shown in Table 6, using the weakest honeycomb structure, the striker’s kinetic energy of 50 kj could be absorbed during the absorption stroke of 20 cm if a panel with an area of 0.61 m2 is provided. In case of using the honeycomb layer with maximum strength, a minimum needed panel area of 0.03 m2 will be required. So this

5. Results and discussion 5.1. Algorithm validation A thin-walled cylindrical column with hexagonal cross section with length of 400 mm is considered impacting a rigid wall at the speed of 10 m/s. A lumped mass of 500 kg is attached to the free end of the column. The optimization problem of choosing the best sectional foil thickness (t) as well as section edge length (a) for maximized specific energy absorption is studied in both cases of no constraint and existing load constraint. The problems are defined as in Eqs. (19) and (20), respectively [17].

Fig. 6. Typical force–displacement diagram for two layer honeycomb arrangement.

Table 7 The optimized two-layer honeycomb absorber design geometry specifications.

⎧ Maximize : y = SEA( t, a) ⎪ PrI:⎨ 1.4 mm ≤ t ≤ 3.0 mm ⎪ ⎩ 30 mm ≤ a ≤ 60 mm

c (mm)

(19)

t (mm)

A (m2)

h (mm)

Layer 1 3.1750002 0.07619 0.01789 0.0405 Layer 2 3.17500003 0.07619 0.02684 0.1498

⎧ Maximize : y = SEA( t, a) ⎪ ⎪ S.t. Max PL( t, a) ≤ 70 kN PrII:⎨ ⎪ 1.4 mm ≤ t ≤ 3.0 mm ⎪ ⎩ 30 mm ≤ a ≤ 60 mm

Total volume (m3)

Total height (m)

0.004745

0.1903

(20)

The specific energy absorption (SEA) and peak load of the collapse (PL) are defined in [17]. Using initial population of 50 with the genetic algorithm process, the optimization results are tabulated in Table 5. It is shown that the optimization calculations of the present developed process are in very good agreement with the reported ones.

Table 8 The optimized two-layer honeycomb absorber design behavior characteristics in impact.

Layer 1 Layer 2

Peak crush force (kN)

Mean crush force (kN)

383.8065 575.8877

188.4636 282.7827

Table 5 optimization problem of thin-walled hexagonal cylinder impact. Case study

Method

Optimized Optimized edge thickness (mm) length (mm)

Maximized SEA (kj/kg)

Peak load (kN)

PrI

[17] present [17] present

3.0 3.0 2.29 2.3

12.6140 12.6140 12.0472 12.07

79.05 79.05 69.9 69.99

PrII

30 30 30 30

Fig. 7. The optimized two-layer energy absorber typical configuration.

Table 6 Single-layer energy absorber behavior characteristics limitations. Honeycomb

Honeycomb layer with lowest strength Honeycomb layer with highest strength

Strength (MPa)

0.5455 10.5127

Peak stress (MPa)

3.5 21.5

Stroke ¼15 cm

Stroke ¼20 cm

A (m2)

Peak force (kN)

t (s)

A (m2)

Peak force (kN)

t (s)

0.61 0.03

2133.231 644.598

0.0124 0.0124

0.46 0.024

1610 516

0.0166 0.0166

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problem is not the case as an optimization problem nature. The minimum peak load is 516 kN equal to about 52g's that is more than the allowable defined shock level according to problem definition in Table 4. Accordingly, there is not any single-layer solution that could meet the defined problem constraints and conditions. Hence, to solve this problem, a two layer honeycomb panel is considered instead. While considering a two-layer honeycomb panel energy absorber arrangement, it is assumed that each layer has its own specifications including the panel area, layer height, cell size and foil thickness. A typical simplified force–displacement diagram for case of a two layer honeycomb panel is depicted in Fig. 6 which it shows that the response characteristics of each layer could be defined using three parameters of peak crush force, mean crushing load and the layer height (h). Due to its importance in the design configuration of any transport vehicle, the main objective of the optimization procedure is set to minimization of the volume of the absorber arrangement. Two other main objectives e.g. maximized energy absorption and limited crush shock load are behaved as main equality and inequality constraints of the problem, respectively. Another inequality constraint on the maximum stopping distance (absorber height) is also included. A number of four parameters including the foil thickness, cell size, layer height, and panel area for each layer are handled as the optimization variables. The optimization problem with the main objective and its constraints are

203

expressed in Eq. (21).

min: V = A1*h1 + A2 *h2

objective function

s.t.: Fp1 = σp1 × A1 ≤ 400 kN

inequality constraints

Fp2 − Fm1 = σp2 × A2 − σm1 × A1 ≤ 400 kN h1 + h2 ≤ 200 mm E = σm1 × A1 × h1 + σm2 × A2 × h2 = 50 kj equality constraints 0.0254 mm ≤ t1, t2 ≤ 0.0762 mm

variables bounds

3.175 mm ≤ c1 , c2 ≤ 6.35mm 0.01 m2 ≤ A1 , A2 ≤ 0.62 m2 20 mm ≤ h1 , h2 ≤ 150 mm

(21)

In the genetic algorithm procedure, the peak stress and mean stress equations derived from the response surface method are employed. The cell size (c), foil thickness (t), layer height (h), and panel area (A) variables change limits are defined based on the available products limitations. The optimization process using the genetic algorithm is handled with an initial population size of 50. According to the optimization results, the feasible two-layer honeycomb configuration and its behavior characteristics are presented in Tables 7 and 8, respectively. The obtained configuration is also graphically depicted in Fig. 7. Both layers are picked

Fig. 8. Force–displacement diagram for two-layer honeycomb absorber crushing.

Fig. 9. Deceleration time-history of the moving mass during impact with two-layer absorber.

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from the same honeycomb cell size and foil thickness geometry but are different in face area and layer heights. A total minimized essential occupied volume of 4.745 liters is calculated for the absorber. The force–displacement as well as acceleration–time history diagrams of mass during the impact are presented in Figs. 8 and 9, respectively. The results from the presented tables and figures show that the total crushing distance is about 19 cm while the shock level is limited to 38.38g in layer 1 crush initiate time and 38.75g in layer 2 crush jump both are lower than the problem upper constraint of 40g. The total calculated time interval of the impact event is about 20 ms.

For sake of comparison, a single-layer honeycomb energy absorber of the same cell size, foil thickness, total height, and total volume is considered and its response diagrams is depicted in Figs. 8 and 9. It shows a mean force of 262.5 N and a shock level jump of 53.5g which is higher than the limitation of 40g. The impact event also reduced to 18.6 ms. For the case where using an energy absorber with three-layer honeycomb configuration, the same design and optimization process is conducted. In such a design paradigm case, there is a total of 12 design variables (4 variable per layer). The optimization problem is formulated as in Eq. (22). min: V = A1*h1 + A2 *h2 + + A3 *h3

objective function

s.t.: Table 9 The optimized three-layer honeycomb absorber design geometry and the equivalent single layer specifications.

Fp1 = σp1 × A1 ≤ 400 kN

inequality constraints

Fp2 − Fm1 = σp2 × A2 − σm1 × A1 ≤ 400 kN Fp3 − Fm2 = σp3 × A3 − σm2 × A2 ≤ 400 kN

c (mm)

t (mm)

A (m2)

h (mm)

Layer 1 3.17505 0.07619 0.01864 0.02155 Layer 2 3.175 0.07619 0.02779 0.041165 Layer 3 3.175 0.07619 0.03199 0.100

Total volume (m3)

Total height (m)

h1 + h2 + h3 ≤ 200 mm

0.004745

0.16271

0.0254 mm ≤ t1 , t2 , t3 ≤ 0.0762 mm

E = σm1 × A1 × h1 + σm2 × A2 × h2 + σm3 × A3 × h3 = 50 kj 3.175 mm ≤ c1 , c2 , c3 ≤ 6.35 mm 0.01 m2 ≤ A1 , A2 , A3 ≤ 0.62 m2 20 mm ≤ h1 , h2 , h3 ≤ 150 mm

Table 10 The optimized three-layer honeycomb absorber design behavior characteristics during impact.

Layer 1 Layer 2 Layer 3

Peak crush force (kN)

Mean crush force (kN)

399.9970 596.3070 686.6344

196.4126 292.8093 337.1635

Fig. 10. The optimized configuration.

three-layer

configuration

equality constraints

variables bounds

energy absorber

typical

(22)

The design optimization process results for the variables, geometries and crushing behavior of all layers are presented in Tables 9 and 10. A typical graphical view of the design optimized three-layer energy absorber with square face is also shown in Fig. 10 where the layers 1 and 3 are located in top and bottom positions, respectively. The force–displacement as well as acceleration–time history diagrams of the impacting mass are then presented in Figs. 11 and 12, respectively. The results show that the optimized absorber’s volume does not changed during transformation from a two-layer absorber design to a three-layer one whilst the total crushing distance in case of three-layer absorber has met a reduction of about 27 mm in comparison with two-layer design. The shock level is also limited to 40g for the first and second crush jumps and to 34.4g in the last crush in third layer which all remain below the problem allowable upper constraint of 40g. For sake of comparison, crushing of a single-layer honeycomb energy absorber with the same cell size, foil thickness, total height, and total volume as those of the 3-layer design configuration is considered. The calculations show single shock level jump of 62.6g

Fig. 11. Force–displacement diagram for three-layer honeycomb during impact.

J. Fazilati, M. Alisadeghi / Thin-Walled Structures 107 (2016) 197–206

which is much higher than 40g limitation followed by a mean force of 307.2 N as depicted in Figs. 11 and 12. A study on results from three-layer absorber optimized

205

configuration shows that increasing the layer number reduces the total stopping distance while has no significant effect on the required absorber volume. It is also observed that optimized design

Fig. 12. Deceleration time–history of the moving mass for three layer honeycomb during impact.

Fig. 13. Comparison of design behavior in a) 2-layer design and b) 3-layer design casestudies and equivalent single layer absorber.

206

J. Fazilati, M. Alisadeghi / Thin-Walled Structures 107 (2016) 197–206

layers with smaller face area are also comparatively thinner in height. Impact time duration of about 24 ms is calculated in case of a 3-layer absorber design that is longer than that of a 2-layer one by about 20 percents. Fig. 13 collects the shock levels as well as impact duration time for two design case studies of 2- and 3- layer absorbers that is compared to their equivalent single-layer absorber parameters. It shows that in case of using equivalent single-layer absorbers, the impact duration is reduced to 18.6 and 15.9 ms for 2- and 3-layer design optimization problems, respectively.

6. Conclusions In this study, for the first time, the theoretical design and optimization of a multi-layer honeycomb energy absorber is performed by using the response surface method and genetic algorithm. Through the use of finite element analysis, the design of experiment approach and response surface method, the behavior of honeycomb structure crashworthiness parameters including the mean crushing load and the peak crush force has been extracted as explicit functions of the structure’s geometrical variables. According to typical design requirements, a multi-layer honeycomb panel is designed and optimized to absorb the total impact energy of impact while controls the maximum shock transmitted to the moving mass. The absorber volume is set as the main objective of the optimization problem subjected to equality energy and inequality shock level constraints with number of four geometrical design variables (e.g. panel area, layer height, cell size and foil thickness) per layer. Applying the method for cases of two- and three-layer absorbers shows that in spite of limitations in singlelayer configuration, it is possible to find a desired design point meeting all of the design limitations in multi-layer designs. The results also show that, by using the multi-layer configuration and increasing the number of layers, the energy absorption performance could be improved. Increasing the number of layers from two to three has negligible effect on total required absorber volume but could reduce the maximum stopping distance while increases the duration of impact event. The results also shows that the optimizing algorithm tend to use the advantages of the most strengthened honeycomb layers with highest thickness and smallest cell size in all layers which means that the optimizing variables could be reduced to just the face area and height variables per layer. Comparing with a single layer brick absorber with equivalent volume, height and cell specifications, it is shown that using multi-layer absorber designs could notably weaken shock levels experienced by the mass while simultanously expand the time to stop interval.

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