Functionally Graded Thickness Research Articles

Crush Analysis of External Inversion of Tubes with Variable Thickness Distribution

Axial crush tests on Uniform Thickness (UT) and Functionally Graded Thickness (FGT) tubes have been performed for external inversion process, and crashworthiness performance, and benefits of FGT tubes compared with their UT equivalents. Furthermore, in order to find detail verification about crush process of FGT tubes, Finite element (FE) models developed using explicit FE code under axial loading. Then, these FE models validated by experimental tests to ensure that they can accurately predict the responses of FGT inversion tubes. Our study shows that tube thickness distribution, die radius, and coefficient of friction between die and tube have great influence on the responses of FGT tubes inversion. The results give new design ideas to improve crashworthiness performance of inversion tubes. DOI: https://doi.org/10.52783/pst.272

Optimizing functionally graded hexagonal crash boxes with honeycomb filler for enhanced crashworthiness

A crash box is typically designed to enhance crashworthiness by absorbing impact energy and reducing the crushing force that can cause injury or fatality to passengers. However, the structural integrity of conventional crash boxes is often compromised due to buckling deformations during collisions. This study introduces a novel approach utilizing functionally graded thickness (FGT) in hexagonal crash box design with honeycomb fillers to improve crashworthiness through multi-objective optimization. Finite element analysis using LS-DYNA was performed for axial and oblique impacts. Mesh sensitivity study indicated that constructing the FGT crash box with 41 layers yielded favourable outcomes, with simulation results differing by a mere 2.5% compared to experimental data. A surrogate model based on response surface methodology was employed to mathematically formulate the objective functions, which aimed to maximize specific energy absorption (SEA) and minimize initial peak crash force (IPF). Subsequently, a multi-objective particle swarm optimization was used to determine the optimal grading exponent of both outer and filler structures. The optimized FGT crash box demonstrated superior control over progressive deformation, exhibiting high SEA and reducing IPF compared to the uniform thickness crash box. Particularly, under oblique impact loads, using two grading exponents in optimizing the FGT crash box potentially achieved significant improvements in crashworthiness within the design procedure.

Nondeterministic multi-objective and multi-case discrete optimization of functionally-graded front-bumper structures for pedestrian protection

Pedestrian lower-leg protection and lower-speed crashworthiness often present two important yet competing criteria on the design of front-bumper structures. Conventional design optimization is largely focused on a single loading condition without considering multiple impact cases. Furthermore, design of front-bumper structures is usually discrete in engineering practice and impacting conditions are commonly random. To cope with such a sophisticated nondeterministic design problem, this study aimed to develop a successive multiple attribute decision making (MADM) algorithm for optimizing a functionally graded thickness (FGT) front-bumper structure subject to multiple impact loading cases. The finite element (FE) model of front-end vehicle was constructed and validated with the in-house experimental tests under the loads of both Flexible Pedestrian Legform Impactor (Flex-PLI) impact and lower-speed impact. In the proposed successive MADM algorithm, the order preference by similarity to ideal solution (TOPSIS) based upon relative entropy was coupled with the analytic hierarchy process (AHP) to develop a MADM model for converting multiple conflicting objectives into a unified single cost function. The presented optimization procedure is algorithmically iterated using the successive Taguchi method to deal with a large number of design variables and design levels. The results showed that not only the algorithm enabled to generate an optimal design efficiently, but also the robustness of Flex-PLI impact is significantly enhanced. The proposed algorithm can be potentially used for other engineering design problems with similar complexity.

Deformation and energy absorption of additively manufactured functionally graded thickness thin-walled circular tubes under lateral crushing

Functionally graded thickness (FGT) is an innovative concept to create light-weight structures with better material distribution and promising energy absorption characteristics suitable for vehicle crashworthiness applications. Accordingly, this paper suggests innovative circular tubes with in-plane thickness gradient along their perimeter and assesses their crashworthiness behaviour under lateral loading. Three different designs of circular tubes with thickness gradient were considered in which the locations of maximum and minimum thicknesses are varied. Selective laser melting method of additive manufacturing was used to manufacture the different tubes. Two different bulk powders including titanium (Ti6Al4V) and aluminium (AlSi10Mg) were used in the manufacturing process. Quasi-static crush experiments were conducted on the laser melted tubes to investigate their crushing and energy absorption behaviour. The energy absorption characteristics of the different FGT tubes were calculated and compared against a uniform thickness design. The results revealed that the best crashworthiness metrics were offered by FGT titanium tube in which the maximum thickness regions were along the horizontal and vertical directions while the minimum thickness regions were at an angle of 45° with respect to the loading direction. The aforementioned tube was found to absorb 79% greater energy per unit mass than its uniform thickness counterpart. Finally, with the aid of numerical simulations and surrogate modelling techniques, multi-objective optimisation and parametric analysis were conducted on the best FGT tube. The influences of the geometrical parameters on the crashworthiness responses of the best FGT structure were explored and the optimal thickness gradient parameters were determined. The results reported in this paper provide valuable guidance on the design of FGT energy absorption tubes for lateral deformation.

Study on rollover crashworthiness of a bus with local functional graded thickness sidewall pillar

There are serious traffic accidents and large angle impact on vehicle body when a bus rolls over. In this paper, a novel local functionally graded thickness (LFGT) sidewall pillar and a minimum deformation search (MDS) method are investigated to improve the bus rollover crashworthiness. According to ECE R66 standard, there was most serious deformation of the traditional side column of the bus when the bus rolls over with large angle impact. To reduce the deformation, a LFGT tube is designed to apply to sidewall pillar. In addition, the characteristics of the LFGT tube under large angle impact are analyzed by radial basis function. From the characteristics, a MDS method is proposed to improve the rollover crashworthiness of bus body. Then, the simulation experiments are conducted to validate the influence of LFGT tube on bus rollover crashworthiness. The results show that the MDS method meets the machining accuracy and ensures the computational efficiency for bus rollover collision optimization, and LFGT sidewall pillar optimised by MDS method can improve the rollover crashworthiness of bus body.

Energy-absorbing analysis and reliability-based multiobjective optimization design of graded thickness B pillar with grey relational analysis

Abstract The functionally graded property structures, as a relatively novel configuration with higher material utilization rate, have been increasingly captured researcher's attention to date. In this paper, a functionally graded thickness (FGT) B pillar, which is characterized by a thicker wall thickness in load-bearing regions and a thinner wall thickness in other areas, is introduced to investigate its energy-absorbing and bending performance. And the reliability-based multiobjective optimization with grey relational analysis is carried out to solve infeasibility created by violation of constraint for conventional deterministic optimization via considering uncertainties of design parameters. Based on the validated finite element model, the FGT B pillar with three configurations and the corresponding uniform thickness (UT) B pillar are compared to explore the benefits of B pillar with different thickness layouts. And it reveals that the FGT B pillars, especially double functionally graded thickness (DFF) with different gradient exponents n for upper and lower part, exhibit superior crashworthiness to the UT B pillar. In addition, in order to research the distinctness between deterministic and reliable optimization in its entirety, multi-response optimization problems, coupling the high-accuracy Radical Basis Functions Network (RBF) and Monte Carlo Simulation (MCS), are established to produce a set of non-dominated solutions. The optimization results demonstrate that reliable Pareto fronts slightly are shifted away with diminishing performance from their deterministic counterpart attributable to its considered uncertainties, but significantly improve its reliability. Beyond that, a grey relational analysis combined with grey entropy is proposed to strike a desirable balance to the specific energy absorption (SEA) and the peak crashing force (Fmax) from Pareto-set under the condition of satisfied intrusion (Ma_In) constraint. Finally, the optimized result illustrates that the thickness variations of the upper and lower parts of the B column are subjected to concave and linear function distribution, respectively, which can provide some guidance for practical engineering with insightful design information.

Crashworthiness Design and Multiobjective Optimization for Hexagon Honeycomb Structure with Functionally Graded Thickness

Higher energy absorption efficiency and better crashworthiness performance are always the key objectives for different energy absorbing structures applied in numerous industries including aerospace, rail equipment transportation, and automotive. In this study, a functionally graded thickness (FGT) design method is introduced in the design of a hexagon honeycomb structure to improve energy absorbing efficiency on the basis of a traditional honeycomb with uniform thickness (UT). The validation of a numerical analysis model for a UT honeycomb under axial loading is implemented by a nonlinear finite element code LS-DYNA (V971). Furthermore, the multiobjective crashworthiness optimization of an FGT honeycomb subjected to axial quasi-static compression is conducted to maximize specific energy absorption (SEA) and minimize peak crashing force (PCF). In addition, three surrogate models, including radial basis function (RBF), response surface method (RSM), and kriging (KRG), are compared in the accuracy of predicting SEA and PCF and capacity for optimization design of FGT honeycomb structure; the Nondominated Sorting Genetic Algorithm (NSGA-II) is applied to obtain the Pareto optimal solutions for the maximum thickness, minimum thickness, and thickness variation gradient exponent of a honeycomb wall. The optimal points obtained by different surrogate models subjected to an SEA value of 18.5 kJ/kg, 20 kJ/kg, 22 kJ/kg, and 24 kJ/kg are validated, and corresponding optimal parameters are compared; RBF and RSM are more suitable in crashworthiness optimization design of the FGT honeycomb structure. It is indicated that the FGT honeycomb with optimal geometrical parameters presents remarkable enhancement and energy absorbing potential compared to the traditional honeycomb structure.

A comparative study on crashworthiness of thin-walled tubes with functionally graded thickness under oblique impact loadings

The main objective of this study is to investigate the effects of functionally graded thickness (FGT) patterns and cross-sectional shapes (i.e. circular, square and hexagonal) on crashworthiness performance of thin-walled tubes under multiple impact loading angles (0°–30°) by using the nonlinear explicit finite-element (FE) method. In order to show the efficiency of FGT tubes under different impact loading angles, the crashworthiness performances of the FGT tubes are also compared with their uniform thickness (UT) counterparts. At this point, the FGT and UT tubes are designed to have the same height, average cross-section area and weight. In addition, a multigene genetic programming (MGP)-based procedure is first time presented in literature for crashworthiness prediction of thin-walled structures under different impact loadings. To ensure the accuracy of the numerical models, the FE models are validated against both theoretical and experimental results in literature. The results demonstrated that the cross-sectional shapes, gradient exponents and impact loading angles effect the crashworthiness performances of thin-walled tubes, significantly. The simulation results showed that the FGT tubes have a superior crashworthiness performance compared to their UT counterparts especially at high impact loading angles due to the fact that FGT makes possible more folds to be formed and significantly increases the global buckling resistance of tubes. In particular, the SEA values of FGT tubes can reach 93% higher values than that of UT counterparts. The results also showed that the FGT tubes with square cross-section have generally lower energy absorption performance compared with circular and hexagonal ones. Especially, the square FGT tubes have up to 31% lower the SEA values than hexagonal and circular tubes. It is also revealed that the proposed MGP approach is able to predict the crashworthiness parameters with high accuracy.

An Overview on Impact Behaviour and Energy Absorption of Collapsible Metallic and Non-Metallic Energy Absorbers used in Automotive Applications

Collapsible impact energy absorbers play an important role of protecting automotive components from damage during collision. Collision of the two objects results into the damage to one or both of them. Damage may be in the form of crack, fracture and scratch. Designers must know about how the material and object behave under impact event. Owing to above reasons different types of collapsible impact energy absorbers are developed. In the past different studies were undertaken to improve such collapsible impact energy absorbers. This article highlights such studies on common shapes of collapsible impact energy absorber and their impact behaviour under the axial compression. The literature based on studies and analyses of effects of different geometrical parameters on the crushing behaviour of impact energy absorbers is presented in detail. The energy absorber can be of different shape such as circular tube, square tube, and frustums of cone and pyramids. The crushing behaviour of energy absorbers includes studies on crushing mechanics, modes of deformation, energy absorbing capacity, effect on peak and mean crushing load. In this work efforts are made to cover major outcomes from past studies on such behavioural parameters. Even though the major literature reviewed is related to metallic energy absorbers, emphasis is also laid on covering literature on use of composite tube, fiber metal lamination (FML) member, honeycomb plate and functionally graded thickness (FGT) tube as a collapsible impact energy absorber.

Analytical study on functionally graded thickness tubes under external inversion process

The axial crushing of Functionally Graded Thickness (FGT) tubes under external inversion process is studied analytically, and their results have been checked with experimental and Finite Element (FE) simulation results. Employing doubly curved shell theory and using rigid-perfectly plastic (R-PP) material idealization, a theoretical solution is derived for the crushing inversion load of a variable thickness distribution tube. Experimental tests are carried out by employing tubes with linearly variable thickness distribution, and their results compared with analytical model. It is seen that the obtained results from experiments and theoretical model show excellent agreement. To evaluate nonlinear thickness patterns behavior, numerical simulations are carried out using the explicit FE code LSDYNA. It was found that FE simulations and theoretical analysis have reasonable correlation. Finally, the study was extended to find out the most accurate loads of thickness functions that the proposed theoretical model is predicted. The results show that the predicted theoretical loads of concave and linear thickness distribution have more precision with respect to convex function based on the FE and experimental results.

Theoretical analysis of functionally graded thickness tubes under dynamic external inversion loading

The effect of strain rate on dynamic external inversion loading of Functionally Graded Thickness (FGT) tubes are studied analytically and their results are compared with Finite Element (FE) results. Based on the deformation theory and using rigid-perfectly plastic (R-PP) material idealization, a theoretical solution for dynamic inversion crushing load is derived by dividing the contact region of circular sections into three different sections. Comparing analytical and FE results indicates that prediction of crashworthiness loading for different thickness distribution of FGT tubes are satisfactory. Also, it is found that the concave distribution of thickness for FGT tubes improves the energy absorption more efficient than other geometries.

Design optimization of thin-walled circular tubular structures with graded thickness under later impact loading

In order to improve the crashing performance under lateral impact scenario, a thin-walled circular tube with functionally graded thickness (FGT) is introduced with its superior performance in this paper. The wall thickness of the FGT tubual structure is graded along the axial direction. Based on the assumed graded thickness function, several important parameters (such as the graded exponent, the tube diameter and yield stress) are selected and their effects on dynamic energy absorption characteristics are discussed. The analyzed results show that the FGT has better crashworthiness in special energy absorption (SEA) and crash force efficiency (CFE) than uniform thickness (UT) tube. Then, the optimization design is further employed to obtain the Pareto fronts of the graded configuration under lateral impact loading. Note that the specific energy absorption (SEA) and crashing force efficiency (CFE) are regarded as the objectives, and the grading exponent, yield stress and diameter are defined as the design variables. The surrogate model with the best accuracy is chosen by error analysis for improving the accuracy of optimization process. Thus, the optimal solution is reasonably obtained and analyzed. The optimal results indicate that the FGT structures have significant potential applications into vehicle body especially under later impacting event.

Optimal shape design of functionally graded thickness inversion tubes subjected to oblique loading

This paper aims to understand and optimize the crush response of Functionally Graded Thickness (FGT) tubes with various thickness distributions subjected to oblique loading using multi-objective optimization method. Hence, finite element (FE) models are established and their results are validated by experimental tests. Two objective functions (specific energy absorption and peak load) are approximated by four different multi-objective optimization models: the weighted average, multi-design optimization (MDO) technique, constrained single-objective optimization, and geometrical average methods. The optimum design results demonstrate that the selection of appropriate inversion tube parameters such as the die radius, the coefficient of friction between the die and tube, and thickness distribution function have significant roles in the crashworthiness design. The results give new ideas to improve the crashworthiness performance of inversion tubes under oblique loading conditions.

Crashworthiness analysis of octagonal multi-cell tube with functionally graded thickness under multiple loading angles

This paper provides a study on a novel octagonal multi-cell tube with functionally graded thickness (FGT) under multiple loading angles. First, comparative analysis on the FGT tube and the counterpart tube with uniform thickness(UT) under multiple loading angles reveal that the energy absorption is more superior for the FGT tube when the loading angle exceeds the lower bound of the transition range of the UT tube. Second, parametric study on the FGT tube indicates that thickness gradient exponent and thickness range have significant effect on its crashworthiness. Third, multiobjective optimizations of the FGT tube are conducted, aiming to maximize specific energy absorption(SEA) and minimize initial force(IPF) under multiple loading angles, based upon the Non-dominated sorting genetic algorithm(NAGA-Ⅱ) and RBF technique. The optimized FGT tube demonstrates better crashworthiness than the UT tube in all design cases. These findings can provide valuable guidelines for the design of multi-cell tube with functionally graded thickness under multiple loading angles.

On design of graded honeycomb filler and tubal wall thickness for multiple load cases

Circular tube filled with cellular materials becomes a fairly attractive structural option in energy absorbing devices, such as crash box and front rail in vehicle. This paper introduces a novel configuration, namely double functionally graded (DFG) structure, which comprises of a functionally graded honeycomb filler in a functionally graded thickness (FGT) tube. Based on the validated finite element (FE) models, a comparative study on the DFG tube, single functionally graded (SFG) tube, and traditional uniform honeycomb filled uniform thickness (H-UT) tube were carried out to explore the crashing behaviors of different structures under multiple load cases. It is found that as crushing displacement increases, DFG structure exhibits superior capacity of energy absorption over other configurations and this trend is positively related to the impact angles. In addition, the comparisons of deformation modes and critical crushing angles clearly indicate that the DFG structure is of better and more stable crashing characteristics, being a crashworthy structure. Following the configurational comparison, further parametric studies on the DFG structures were conducted to explore the effects of tubal thickness range and honeycomb thickness range on the crashworthiness. It is found that the tube thickness range is more important to crashworthiness, which provides a basis for structural optimization.

Multi-objective optimization of functionally graded thickness tubes under external inversion over circular dies

This paper optimizes the crushing response and energy absorption of functionally graded thickness (FGT) inversion tubes using the multi-objective optimization method. The numerical results, which are validated by the experimental tests, confirm that optimizing geometry of FGT inversion tube improves energy absorption capacity with respect to ordinary uniform tube UT with the same weight. Our study shows that the die radius r, coefficient of friction μ d between the die and tube, and thickness distribution function have great influence on the responses of FGT inversion tubes. The specific energy absorption (SEA), peak crushing force $$ {P}_{max} $$ , and dynamic amplification factor (DAF) are selected as the objectives of crashworthiness optimal design. Finally, the weighted average method, multi design optimization (MDO) technique, constrained single-objective optimization, and geometrical average method were employed to find the optimal configuration of the proposed inversion tube. The results give new design ideas to improve crashworthiness performance of inversion tubes.

On energy absorption of functionally graded tubes under transverse loading

As a relatively new configuration with higher efficiency of material utilization, functionally graded thickness (FGT) structure has become more and more attractive. In order to understand transverse load bearing behaviors, three circular FGT tubes with different linear thickness gradients from one end to its nearest indenter are investigated through three-point and four-point bending tests in this study. A series of loading conditions, namely different loading offset for three-point bending and different loading span for four-point bending tests, are considered to explore the energy absorption characteristics of the FGT tubes. The experimental deformation modes and force-displacement curves of FGT tubes are first compared with corresponding uniform thickness (UT) tubes. Second, the finite element (FE) models of circular FGT tubes are established for comparisons with the experimental tests. Third, a dimensionless analysis is conducted to compare the deflections in the three-point and four-point bending tests of the circular FGT tubes with different thickness gradient. It is shown that the FGT tubes have more regions to participate in deformation than the UT counterparts to bear the transverse loads. Based on the validated FE models, a parametric study is finally carried out to explore the influence of thickness gradient or thickness difference on the three-point and four-point bending behaviors of the FGT tubes in comparison with the same mass UT counterparts. The results show that a proper selection of thickness gradient can effectively enhance the energy absorption of tubal structures under transverse loading, exhibiting that the FGT tubes are considerably more advantageous over the uniform tubes.

Crashworthiness optimization of circular tubes with functionally-graded thickness

Purpose – The purpose of this paper is to develop a new multi-objective optimization procedure for crashworthiness optimization of thin-walled structures especially circular tubes with functionally graded thickness. Design/methodology/approach – The proposed optimization approach is based on finite element analyses for construction of sample design space and verification; gene-expression programming (GEP) for generating algebraic equations (meta-models) to compute objective functions values (peak crash force and specific energy absorption) for design parameters; multi-objective genetic algorithms for generating design parameters alternatives and determining optimal combination of them. The authors have also utilized linear and non-linear least square regression meta-models as a benchmark for GEP. Findings – It is shown that the proposed approach is able to generate Pareto optimal designs which are in a very good agreement with the actual results. Originality/value – The paper presents the application of a genetic programming-based method, namely, GEP first time in the literature. The proposed approach can be used to all kinds of related crashworthiness problems.

Quasi-static Axial Crushing Behavior of Thin-walled Circular Aluminum Tubes with Functionally Graded Thickness

In this paper, the crushing behavior of circular aluminum thin-walled tubes with functionally graded thickness (FGT) are investigated experimentally. To this aim, three different thickness grading patterns are introduced to the longitudinal direction of tubes using a grading function, and then specimens are tested by universal testing machine under quasi-static axial compression loading. In addition, to show the efficiency of FGT tubes, the quasi-static axial crushing performance of FGT tubes are compared with their uniform thickness (UT) counterparts at the same weight. The test specimens have been made of commercial quality 6060 aluminum alloy and manufactured with computerized numerical controlled (CNC) machining workbench. Initial crush force (ICF), total energy absorption (EA), specific energy absorption (SEA) and mean crush force (MCF) values are measured from the tests to infer the quasi-static axial crushing performance of the tubes. The test results showed that the energy absorption characteristics of FGT tubes are better than the UT counterparts and FGT tubes allow effective control of the energy absorption parameters when compared with the UT tubes due to their varying stiffness along their longitudinal direction.

Enhancing material efficiency of energy absorbers through graded thickness structures

At present, there has been constant aspiration of advanced thin-walled structures in vehicular industries for more efficient usage of materials to achieve much lighter weight and even higher energy absorption. In this paper, functionally graded thickness (FGT) tubes with a varying wall thickness are introduced and their energy-absorbing efficiency is enhanced. Apart from the geometrical parameters such as diameter and length, the gradient exponent that controls the variation of thickness distributions has also a significant effect on the increase in absorbed-energy. Numerical model is validated by performed crashing experiments of FGT tube. Parametric analysis demonstrates that the FGT column is superior to the uniform thickness (UT) column. Further, the multiobjective optimization (MOO) of FGT tubes is conducted for axial impacting by considering specific energy absorption (SEA) and crashing force efficiency (CFE) as objectives, and the diameter, initial length and gradient exponent of thickness variation as the design variables. The multiobjective particle swarm optimization algorithm (MOPSO) is applied to obtain the Pareto optimal solutions. In addition, a comparative study on different surrogate models, such as response surface method (RSM), Kriging method (KRM), and radial basis function (RBF), is also carried out to gain insights into their relative performance and features in computational modeling and design optimization. It is indicated that the performance of FGT tubes can be significantly improved by optimizing the geometrical parameters and gradient exponent.