## Abstract

Safety norms across the world are becoming more and more stringent posing new challenges to achieve lightweight vehicle structures. Structures made of advanced/ultra-high strength steels (AHSS) play a vital role in meeting the vehicle safety targets, by absorbing large amounts of impact energy, as well as by withstanding higher impact loads that occur due to vehicle collisions. Safety simulations usually take longer solution times due to their complexity and nonlinear nature. Engineers often encounter with a problem of quick evaluation of safety performance using different grades of materials to optimize the weight and cost. A new methodology—equivalent energy absorption (EEA)—has been proposed to do a quick trade-off study on performance versus weight for various thicknesses and material combinations. A relationship is established between the gauge and grade of a component to derive an equivalent safety performance so that engineers can make quick decisions by conducting minimal number of simulations. A simple rectangular crush box was considered for study to assess the energy absorption (EA) with various material and thickness combinations. A design of experiments (DOEs) study was done using simulations with many numbers of material grades and gauges to construct a 3D response surface between gauge grade and EA parameters to understand the relationship between each of these parameters. A case study has been discussed in the paper about application of this methodology on a vehicle to evaluate its safety performance. It was found that more than 80% evaluation time is reduced using this methodology.

## 1 Introduction

Vehicle weight reduction, reduced costs, and improved safety performance are the main driving forces behind today’s automotive industry [1]. The continuous enhancements of safety norms across the globe make the vehicles heavier and heavier; on the other hand, every gram of weight saved reduces CO_{2} emissions and thus makes the vehicles eco-friendlier and more fuel efficient. It has been estimated that for every 10% of weight reduction from vehicle’s total weight, fuel economy improves by 6%, in case of electric vehicles, a 14% driving range increase [2].

To achieve both light weight and crash safety, the application of advanced high strength steels (AHSS) is essential [3–5]. The reduction in sheet thickness by the use of high strength steels is being increasingly adopted as an effective means for weight reduction without compromising safety [6,7].

Steels with yield strength levels more than 550 MPa are generally referred to as AHSS. These steels are also sometimes called “ultra-high-strength steels” for tensile strengths exceeding 780 MPa. AHSS with tensile strength of at least 1000 MPa are often called “GigaPascal steel” (1000 MPa = 1 GPa) [8].

As shown in Fig. 1 [9], the AHSS family includes dual phase (DP), complex-phase (CP), ferritic-bainitic (FB), martensitic (MS or MART), transformation-induced plasticity (TRIP), hot-formed (HF), and twinning-induced plasticity (TWIP).

These first and second generation AHSS grades are uniquely qualified to meet the functional performance demands of certain parts. For example, DP and TRIP steels are excellent in the crash zones of the car for their high energy absorption (EA) [8]. For structural elements of the passenger compartment, extremely high strength steels, such as martensitic and boron-based press hardened steels result in improved safety performance [8]. Recently “third generation” of AHSS is being developed with improved strength ductility combinations compared to present grades, with potential for more efficient joining capabilities, at lower costs [9].

Computer aided engineering (CAE) simulations are being used widely in automobile industry to assess different design concepts and prediction of vehicle’s crashworthiness performance [10,11]. The crashworthiness and lightweight of vehicles are generally improved through structural optimization, advanced manufacturing, and lightweight materials [12]. Response surface model (RSM) or meta-model-based design optimization is being commonly used for optimizing large-scale design problems in the automotive industry [13]. Despite their efficiency, metamodeling techniques could still require a significant number of crash simulations, especially when the number of design variables is large [14]. The increasing complexity of simulations as well as huge number of simulations required for crashworthiness optimization has led to a demand for techniques capable of drastically reducing computation time and quick decision-making to optimize the weight and cost. Kanugula and Peddi [15] used an equivalent static load (ESL) method for crash optimization to reduce time and cost. The ESL method divides the original nonlinear dynamic optimization problem into an iterative linear optimization and nonlinear analysis. This process cannot be used for material optimization as the optimization is done in linear domain where material non-linearity is not considered. Fang et al. developed simplified models or reduced finite element models and used in crash simulations [11] to reduce the computational cost. Although this method is helpful in understanding the mechanism of crash and improving vehicle designs, the reduced models have its own limitations in terms of representing accurately full-scale models [14]. There have been many innovations in metamodeling-based approaches like sequential local RSM [16,17], adaptive RSM [18], and trust region-based RSM [19] but these are all statistical methods involving higher number of crash simulations. The focus of the current study is to utilize the fundamental characteristic of the materials, the energy absorption capacity, for quickly arriving at the optimized solution rather than using the complex statistical methods which are more time consuming.

In this paper, a methodology called equivalent energy absorption (EEA) has been proposed to do a quick trade-off study on performance versus weight for various thickness and material combinations. Weight reduction opportunities can be easily explored using this method and quick decisions can be made by conducting minimal number of simulations. Equivalent safety performance is determined by means of establishing a relationship between gauge and grade of a component. The workflow process of the EEA method is depicted in the flowchart shown in Fig. 2.

## 2 Equivalent Energy Absorption Method

The important property which characterizes the crash behavior of the material is the energy absorption capacity of the material [20]. EEA is a method to derive the equivalent energy absorption of a component when it is subjected to impact loading, by varying different thicknesses and material combinations of the component. This is done by establishing a relationship between the gauge and grade of a component so that a quick trade-off study can be conducted on performance versus weight without conducting full CAE simulation.

The area under stress–strain curve represents the energy absorbed by the material during the deformation. Energy absorption capacity (*E*) of materials can be increased by increasing the yield/tensile strength of a material at a given strain. Figure 3 [21] shows the test results, where the absorbed energy for static as well as dynamic (at 50 km/h) conditions has been compared for various steel materials.

*P*and the deformation

*dx*as shown in Fig. 4.

*P*=

*kX*as shown in Fig. 5. Substituting for

*P*in the above equation, we get that the relationship between stiffness (

*k*) and deformation (

*X*) in a linear system is expressed as

*k*(stiffness) is directly proportional to its area moment of inertia and hence to the thickness of the component (assuming all other variables constant)

Thickness of the material plays a significant role and energy absorption is directly proportional to it.

From Eq. (6), we can understand that the thickness and tensile strength of the material are inversely proportional to each other with respect to energy absorption for a given geometry of the component. It may be noted that the current study is done only considering the steel material grades for light weighting applications, however the scope can be extended using alternate materials with the same method.

The equations shown above are indicating only the proportional relationship between two parameters, thickness and material, by considering the linear case for easy understanding. This leads to conclusion that if the material is changed from lower to upper grade, there is an opportunity to reduce the thickness of the material to achieve equivalent energy absorption and vice versa. However, to quantify and establish the exact relationship between the two parameters, it is necessary to consider all the non-linearities, material strain rates effects in crash applications. A design of experiments (DOEs) approach using simulations with all non-linearities and strain rate effects were considered to establish the relation between these parameters with EEA as the criteria between various material and thickness configurations.

## 3 Gauge Versus Grade

It is observed in the previous section that for light weighting solution, the reduction in sheet thickness needs to be always compensated by increasing the yield/tensile strength of material or vice versa to keep the energy absorption constant for a given geometry. Now it is essential to understand the relationship between thickness and yield strength with respect to energy absorption.

To derive the relationship between gauge (thickness) and grade (material) of a component for a given geometry, a simple rectangular crush box with an uniform cross section of 56 × 73 mm and with a length of 200 mm is considered for study. A rigid plate of 500 kg mass is attached rigidly at one end of the box to represent the mass of the system as shown in Fig. 6. LS-DYNA Belytschko-Tsay (ELFORM2) shell elements were used to model the crush box and rigid plate.

Initial thickness of the component was considered as 1.0 mm and the material grade was taken as MAT 10 (ultra-high strength material—Bentler BTR 165 with 1180 MPa yield stress). The number of integration points defined along the thickness of shell elements were 5. An experimentally validated LS-DYNA MAT 24 (MAT_PIECEWISE_LINEAR_PLASTICITY) material model with the included strain rate effects was used. A rigid wall was defined with LS-DYNA RIGIDWALL_PLANAR card on the other side of crush box to carry out the impact. The crush box is impacted to a rigid wall with a velocity of 56 km/h and LS-DYNA solver was used to determine the energy absorption capacity of the box.

Figure 7 shows the crush deformation pattern at various intervals of time. As observed from Fig. 7, after approximately 10 ms of the impact, crush box is deformed and bottomed out to the maximum extent. This can also be seen from the *F* versus *T* curve shown in Fig. 7. The force value is sharply increasing after the bottoming out phenomenon due to high stiffness of stacked up material and the rigid wall.

Figure 8 shows the force versus displacement (*F* versus *D*) curve of the impact. The highlighted area under the *F* versus *D* curve represents the maximum energy absorption (at approximately 10 ms of time) the crush box can absorb. Table 1 shows the rail crush performance parameters obtained from the simulation.

Parameters | Values | Units |
---|---|---|

Rail crush | 146 | mm |

Energy absorption | 13 | kJ |

Average force achieved | 89 | kN |

Parameters | Values | Units |
---|---|---|

Rail crush | 146 | mm |

Energy absorption | 13 | kJ |

Average force achieved | 89 | kN |

A similar exercise has been done using DOEs approach with a greater number of material grades and gauges as indicated in Table 2 to understand the relationship between gauge and grade of the crush box.

S. no. | Material name | Thickness range (mm) |
---|---|---|

1 | MAT 1 to MAT 10 | 0.6–4 mm |

S. no. | Material name | Thickness range (mm) |
---|---|---|

1 | MAT 1 to MAT 10 | 0.6–4 mm |

As shown in Table 2, ten steel materials ranging from mild to ultra-high strength steel grades are considered for this study, with the thickness range from 0.6 to 4 mm. MAT 1 represents mildest steel grade and MAT 10 represents ultra-high strength steels in the ascending order of yield strength ranging from 150 MPa to 1180 MPa.

From these materials and thickness ranges, several DOEs (approximately 1000) are generated using space filler algorithms (uniform Latin hypercube) in order to fill the design space uniformly to get a high-quality RSM. The response surface is a meta-model which derives a mathematical relation between various output parameters with respect to input parameters involved in it.

To have uniformity in comparison of output response, the energy absorption at 60 mm deformation is considered as a response parameter for all the simulations, and it is estimated for each DOE point.

The 3D response surface between gauge, grade, and EA is shown in Fig. 9, and it shows material grade on *X* axis, thickness on *Y* axis, and the energy absorption on *Z* axis. From Fig. 9, it is observed that for each energy absorption value, corresponding combination of thickness and material grades can be found quickly. For example, from Fig. 9, it is noted that MAT 10 with thickness 1 mm shows an energy absorption of ∼5.2 kJ. By looking at the response surface, we can quickly identify that the MAT 7 with approximately ∼1.4 mm thickness gives EEA of 5.2 kJ. Similarly, the MAT 2 is showing an equivalent thickness of ∼1.8 mm to output the energy absorption (EEA) of 5.2 kJ. These observations are tabulated in Table 3 for all the materials.

Details | Energy at 60 mm deformation | |||
---|---|---|---|---|

S. no. | Material | Thickness (mm) | RSM result (kJ) | Simulation result (kJ) |

1 | MAT 1 | 1.78 | 5.30 | 5.26 |

2 | MAT 2 | 1.75 | 5.28 | 5.27 |

3 | MAT 3 | 1.71 | 5.31 | 5.24 |

4 | MAT 4 | 1.57 | 5.26 | 5.25 |

5 | MAT 5 | 1.52 | 5.26 | 5.28 |

6 | MAT 6 | 1.51 | 5.27 | 5.30 |

7 | MAT 7 | 1.38 | 5.28 | 5.28 |

8 | MAT 8 | 1.36 | 5.31 | 5.22 |

9 | MAT 9 | 1.17 | 5.27 | 5.31 |

10 | MAT 10 | 1.00 | 5.27 | 5.29 |

Details | Energy at 60 mm deformation | |||
---|---|---|---|---|

S. no. | Material | Thickness (mm) | RSM result (kJ) | Simulation result (kJ) |

1 | MAT 1 | 1.78 | 5.30 | 5.26 |

2 | MAT 2 | 1.75 | 5.28 | 5.27 |

3 | MAT 3 | 1.71 | 5.31 | 5.24 |

4 | MAT 4 | 1.57 | 5.26 | 5.25 |

5 | MAT 5 | 1.52 | 5.26 | 5.28 |

6 | MAT 6 | 1.51 | 5.27 | 5.30 |

7 | MAT 7 | 1.38 | 5.28 | 5.28 |

8 | MAT 8 | 1.36 | 5.31 | 5.22 |

9 | MAT 9 | 1.17 | 5.27 | 5.31 |

10 | MAT 10 | 1.00 | 5.27 | 5.29 |

Table 3 shows the comparison of RSM and simulation results corresponding to approximately 5.2 kJ of energy absorption with respect to various material and thickness combinations. The results obtained from RSM are verified with actual simulations which are shown in the last column of Table 3. It is observed that there is close degree of correlation between the RSM values and simulation values.

From Table 3, it is observed that by varying different material and thickness combinations, we can arrive at same EEA response. Also, it is noted that the MAT 10, being an ultra-high strength steel material with yield more than 1000 MPa, the thickness required to get the same EEA value is much lesser as compared to mild steel materials like MAT 2, giving rise to huge weight saving opportunity.

To understand the validity of the concept in bending applications, a 100 mm diameter hollow tube with 1000 mm length is considered with three randomly chosen materials, MATERIAL 4, MATERIAL 7, and MATERIAL 9. The thickness of the tube is assigned with corresponding thickness values as shown in Table 3. Bending load is applied by a rigid vertical cylinder of 40 mm diameter as shown in Fig. 10. The rigid cylinder is given with a prescribed displacement of 40 mm and the energy absorbed at the end of the displacement is noted.

Figure 11 shows the energy versus displacement response of all the three materials overlaid. It is observed that the total energy absorption ∼0.62 kJ due to the resistance force is closely matching for all the three materials.

By using the mathematical response surface given above, for any material upgrade or downgrade, designers can quickly select the corresponding equivalent thickness to retain the similar crash performance and arrive at an estimation of weight saving opportunity without conducting a full crash simulation.

Conventionally large number of simulations are required for deriving equivalent performances to evaluate the effect of different material grades.

Once we understand the relationship between various available material grades and thicknesses, a quick material optimization also can be done without conducting DOE study for arriving at optimum solution.

## 4 Equivalent Energy Absorption Method Applied on a Case Study

To demonstrate the benefits of the EEA method, a Monocoque vehicle is considered as a case study. Full vehicle analyses like frontal offset deformable barrier (FODB), side pole impact, and subsystem analysis like side door intrusion load cases (Figs. 12, 16, and 18) are evaluated using the EEA method and compared the results with baseline results. In an offset impact, vehicle is impacted to a deformable barrier at a speed of 64 km/h with an overlap of 40% as shown in Fig. 12 [27].

To quickly estimate the weight saving potential in the vehicle, few critical load path components as shown in Fig. 13 were identified for frontal and side pole impact load cases, for which materials were upgraded from mild steel grades to high strength steel material, without any deterioration from baseline performance.

EEA method has been applied to quickly arrive at thicknesses corresponding to high strength materials for all the components, and CAE evaluations were carried out for the identified load cases. Due to the reduction of thickness of high strength steel material parts, it is observed that there is an approximate reduction of 10 kg of weight in the body in white (BIW).

In an offset impact, the passenger compartment intrusions play a significant role along with the deceleration pulse, in reducing the occupant injuries during the vehicle collisions. For optimal frontal crash behavior, the kinetic energy of the vehicle should be dissipated efficiently by the front end structure without deforming the passenger compartment. The passenger compartment zone should be strong enough to sustain and support the higher impulse forces transmitted from the front end structure

Figure 14 shows FODB simulation baseline CAE performance (mild steel) compared against high strength steel material model CAE performance using the EEA method. The intrusions are measured at critical locations of passenger compartment as per Insurance Institute for Highway Safety (IIHS) guidelines [22] like steering column (SC), footwell (Toepan 1), acceleration (Toepan 2), brake pedal (Toepan 3), instrument panel (IP), and vehicle dynamic crush (structure performance).

It may be noted that the values of response parameters shown in Fig. 14 are scaled values. It is observed that the high strength steel model shows closer results as compared to baseline performance and in fact some of the parameters are better than the baseline performance as lower values are better. Figure 15 shows the vehicle velocity pulse comparison and it is noted that there is a very close correlation between the two models throughout the time range.

Similar exercise has been conducted on the side pole impact load case. In side pole impact analysis, the vehicle impacts a rigid pole of 254 mm diameter with a velocity of 32 km/h at an angle of 75 deg as shown in Fig. 16. The intrusion responses are measured at various locations on B pillar (intrusions 1–6) and the occupant survival space (structural performance) [23]. The scaled results are shown in Fig. 17, and the results comparison shows pretty good correlation between the mild steel model and high strength steel models.

To demonstrate the validity of this method on the subsystem load cases, a front and rear door intrusion analysis has been carried out. In the door intrusion load case, as shown in Fig. 18, a rigid cylinder is given with a prescribed displacement of 450 mm and ramped into the door with vehicle fixed at four suspension mounts. The resisting force of the structure is measured with three parameters as given below [25].

*Parameter 1: Initial crush resistance*: It is the average force required to deform the door over the initial 150 mm of crush distance.

*Parameter 2: Intermediate crush resistance*: The average force required to deform the door over the initial 300 mm of crush distance.

*Parameter 3: Peak crush resistance*: The largest force recorded over the entire 450 mm of crush distance.

The intrusion beams shown in Fig. 19 are the major load carrying members in the door intrusion analysis which are identified to convert from low strength steels to high strength steels.

The analysis has been carried out and the scaled results are shown in Figs. 20 and 22. Figures 21 and 23 show the overlay of force versus displacement curves. It can be observed that the results are closely matching in terms of both the values as well as the *F* versus *D* curves.

Conventionally, it requires trial and error approach or a DOEs-based approach which involves in carrying out large number of iterations to determine the equivalent crash performance by changing different grades of materials and thicknesses. Even with todays advanced computational resources, safety simulations usually take longer solution times as a result of highly nonlinear nature of the analysis and complexity of the simulation models.

In a typical DOE approach for optimization using optimal Latin hypercube method, minimum number of design iterations required with respect to the number of design variables *n* is 3*n* + 1 [26]. For a FODB crash load case like the one shown above with six design variables (critical load path members), it takes minimum 19 iterations of 20 h each to perform a material optimization for arriving at equivalent crash performance with reduced weight.

The EEA method shows huge benefit in terms of arriving at equivalent crash performance with one iteration. Moreover, there is a huge amount of benefits in terms of computational resources and pre-processing efforts. EEA method has an additional advantage in terms considering any number of design variables simultaneously whereas DOE approach is limited by the number of design variables due to the computational cost involved. Table 4 shows the comparison of run times in DOE method versus EEA method. Even in case of manual trial and error approach method for an experienced engineer, it will take minimum 5–10 iterations to come up with a required crash performance.

FODB | DOE method | EEA method |
---|---|---|

No. of design variables | 6 | 6 |

Total no. of runs required | 19 | 1 |

Run time/run in hours | 20 | 20 |

Total run time in hours | 380 | 20 |

FODB | DOE method | EEA method |
---|---|---|

No. of design variables | 6 | 6 |

Total no. of runs required | 19 | 1 |

Run time/run in hours | 20 | 20 |

Total run time in hours | 380 | 20 |

EEA method demonstrated a quick way of achieving equivalent crash performance by means of establishing a relationship between grade and gauge of a component. In the examples presented above, only a single iteration has been conducted as a verification process to compare the results against baseline performance. Our experience confirms the validity of the method to the close degree of correlation as demonstrated above on various crash load cases, tried on different platform vehicles. This gives more confidence in using this method as a decision-making tool to make a quick trade-off study without even conducting a verification run saving huge amount of computational time and cost. At several instances during the critical product development stages of various programs at Mahindra & Mahindra (M&M), this tool has been used for making quick decisions. It is being applied continuously in all the vehicle development programs at M&M for deriving a material strategy of the vehicle design during the development phase.

## 5 Conclusion

A new methodology—EEA—has been demonstrated in the paper for a quick trade-off study on performance versus weight for various thickness and material combinations useful for safety simulations. A relationship is established between the gauge and grade of a component to derive an equivalent safety performance so that engineers can make quick decisions by conducting minimal number of simulations. Case studies have been discussed on full vehicle and subsystem level nonlinear crash load cases to validate the methodology. As seen from the results, it is noted that there is more than 80% time reduction from DOE/manual approach.

Designers often encounter with lots of challenges during manufacturing in terms of unavailability of design intended material grades, cost, feasibility, supply chain issues, etc. Often these issues (problems) occur as a last-minute surprise during the production phase, where quick decisions need to be taken to alter the material grade and gauges without a compromise on the existing (intended) performance of the vehicle. Also, many times during the development phase of the vehicle, platform leaders need to understand weight versus cost impact by means of changing various material grades used in the vehicle without compromising the performance. As demonstrated above, the EEA methodology is very useful in quick identification of various options during the critical stage of the product development cycle, in terms of weight versus cost versus performance by enabling engineers and leaders to take faster decisions.

This methodology can also be used as a quick material optimization tool without conducting a complex vehicle level DOE study for arriving at optimum solution.

## Acknowledgment

The first author would like to thank Dr. Akella Sarma for his invaluable motivation and support provided during this work. The first author would also like to thank Mr. Nitin Nigul of BIW design for his invaluable support during this work and Mr. Deshmukh Chandrakant for developing the EEA calculator used in this study.

## Conflict of Interest

There are no conflicts of interest. This article does not include research in which human participants were involved. Informed consent not applicable. This article does not include any research in which animal participants were involved.

## Data Availability Statement

The authors attest that all data for this study are included in the paper.