Numerical Modelling and Multi Objective Optimization Analysis of Heavy Vehicle Chassis

: The primary supporting structure of an automobile and its other vital systems is the chassis. The chassis structure is required to bear high shock, stresses, and vibration, and therefore it should possess adequate strength. The objective of current research is to analyze a heavy motor vehicle chassis using numerical and experimental methods. The CAD design and FE analysis is conducted using the ANSYS software. The design of the chassis is then optimized using Taguchi design of Experiments (DOE); the optimization techniques used are the central composite design (CCD) scheme and optimal space ﬁlling (OSF) design. Thereafter, sensitivity plots and response surface plots are generated. These plots allow us to determine the critical range of optimized chassis geometry values. The optimization results obtained from the CCD design scheme show that cross member 1 has a higher effect on the equivalent stresses as compared to cross members 2 and 3. The chassis mass reduction obtained from the CCD scheme is approximately 5.3%. The optimization results obtained from the OSF scheme shows that cross member 2 has a higher effect on equivalent stress as compared to cross members 1 and 3. The chassis mass reduction obtained from optimal space ﬁlling design scheme is approximately 4.35%.


Introduction
The two important parts of a vehicle body are the chassis and bodywork, which constitute the maximum volume and weight. The chassis structure is comprised of longitudinal and lateral members placed at critical stress concentration regions [1]. All the components and loads of the vehicle are supported by the chassis. Such loads include the weight of the components and loads due to motion of vehicle (acceleration and retardation and cornering). Vehicle chassis should be rigid enough to absorb the shocks, twists, vibrations, and other stresses experienced by a vehicle when in operation. The critical considerations for a good chassis design is its ability to resist bending, to have torsional stiffness and strength for good handling characteristics [2,3].
Commercial vehicle chassis are designed to bear heavy payloads and are meant to provide durability and versatility. The design of the chassis should be based on various load considerations (payload), laden weight and dynamic loading conditions as mentioned above. The fatigue loads acting on the chassis (caused by road conditions and engine weight) also affect its life and durability and thereby its necessary to analysis chassis under fatigue loading conditions [4]. The three major types of chassis used in the automobile industry are spaceframe chassis, ladder frame chassis and Monocoque chassis [5]. That said, heavy-duty trucks (e.g., the TATA1612) employ ladder frame type chassis shown in Figure 1 [6]. The structure is comprised of longitudinal and transverse members arranged systematically in the given ladder form. The longitudinal and transverse members are made of channeled sections (C shape or square shape).
In last two decades, the advancement in safety features has steadily increased the weight of the chassis [7]. By using advanced optimization techniques, significant improvements in the design of chassis could be achieved in the early design stage [8,9]. Many researchers like Chiandussi et al. [10], Pedersen [11], and Duddeck [12] have worked in optimizing the design of vehicle suspensions and other body parts, which aided in reducing the weight of the vehicle. The FEA analysis conducted on heavy vehicle chassis has shown the zones of high stresses that are induced by the applied heavy loads when vehicles are in operation. The self-weight of chassis frames also adds to the stresses [13]. Thus, the vehicle chassis design should also include self-weight considerations [14].
Yang and Chahande [15] have conducted a space frame analysis using NASTRAN FEA simulation package. Kang et al. [16] have optimized the design of heavy vehicle chassis using the analytical target cascading (ATC) method. The findings have shown that the ATC method is a viable tool in improving the design of existing chassis. E.R. Deore et al. [17] conducted numerical investigations on low-loader chassis to reduce the weight and cost of chassis by optimizing the side member thickness and positional variation of cross members. Their findings show that the chassis cross member thickness has significant effects on the deformation and stresses generated on chassis.
Patel et al. [18] optimized the TATA 2516TC truck chassis by reducing the weight using the Pro Mechanica software. The research would serve as base model for further research. P. K. Sharma et al. [19] conducted FE simulation on the TATA turbo SE 1613 chassis to determine stresses and deformation under heavy loading conditions. The numerical results obtained from the simulation were in close agreement with analytical results. Rajasekar et al. [20] optimized the design of an on-road heavy vehicle using the genetic algorithm (GA) by varying dimensions of cross sections, and this resulted in a reduced weight of chassis.
Guosheng Feng [21] conducted vibrational analysis on a chassis using FEA tool. The modal and response spectrum analysis conducted on the chassis enabled them to determine the natural frequency, deformation mode shape and maximum amplitude of the chassis. Marathe and Tadamalle [22] optimized the chassis of a trolley, under heavy loading conditions, using numerical methods. It was observed that web thickness, upper flange thickness and lower flange thickness have significant effects on the equivalent stresses and the deformation of the chassis. Jay Prakash Srivastava [23] conducted FE simulation on a go-kart chassis using ANSYS 16 FEA software. Their findings show that by changing the material and design of the chassis, the safety factor can be significantly improved. S. Prabhakaran et al. [24] conducted analytical and numerical investigation on a chassis to reduce its weight. The analytical calculations were based on beam bending the- Figure 1. A ladder chassis frame (truck) view [6].
The structure is comprised of longitudinal and transverse members arranged systematically in the given ladder form. The longitudinal and transverse members are made of channeled sections (C shape or square shape).
In last two decades, the advancement in safety features has steadily increased the weight of the chassis [7]. By using advanced optimization techniques, significant improvements in the design of chassis could be achieved in the early design stage [8,9]. Many researchers like Chiandussi et al. [10], Pedersen [11], and Duddeck [12] have worked in optimizing the design of vehicle suspensions and other body parts, which aided in reducing the weight of the vehicle. The FEA analysis conducted on heavy vehicle chassis has shown the zones of high stresses that are induced by the applied heavy loads when vehicles are in operation. The self-weight of chassis frames also adds to the stresses [13]. Thus, the vehicle chassis design should also include self-weight considerations [14].
Yang and Chahande [15] have conducted a space frame analysis using NASTRAN FEA simulation package. Kang et al. [16] have optimized the design of heavy vehicle chassis using the analytical target cascading (ATC) method. The findings have shown that the ATC method is a viable tool in improving the design of existing chassis. E.R. Deore et al. [17] conducted numerical investigations on low-loader chassis to reduce the weight and cost of chassis by optimizing the side member thickness and positional variation of cross members. Their findings show that the chassis cross member thickness has significant effects on the deformation and stresses generated on chassis.
Patel et al. [18] optimized the TATA 2516TC truck chassis by reducing the weight using the Pro Mechanica software. The research would serve as base model for further research. P. K. Sharma et al. [19] conducted FE simulation on the TATA turbo SE 1613 chassis to determine stresses and deformation under heavy loading conditions. The numerical results obtained from the simulation were in close agreement with analytical results. Rajasekar et al. [20] optimized the design of an on-road heavy vehicle using the genetic algorithm (GA) by varying dimensions of cross sections, and this resulted in a reduced weight of chassis.
Guosheng Feng [21] conducted vibrational analysis on a chassis using FEA tool. The modal and response spectrum analysis conducted on the chassis enabled them to determine the natural frequency, deformation mode shape and maximum amplitude of the chassis. Marathe and Tadamalle [22] optimized the chassis of a trolley, under heavy loading conditions, using numerical methods. It was observed that web thickness, upper flange thickness and lower flange thickness have significant effects on the equivalent stresses and the deformation of the chassis. Jay Prakash Srivastava [23] conducted FE simulation on a go-kart chassis using ANSYS 16 FEA software. Their findings show that by changing the material and design of the chassis, the safety factor can be significantly improved. S. Prabhakaran et al. [24] conducted analytical and numerical investigation on a chassis to reduce its weight. The analytical calculations were based on beam bending theory and the results obtained were validated with FE simulation results. Based on FE simulation results, the necessary modifications were made on the chassis structure, which resulted in a 6.7% weight reduction. Fernando et al. [25] worked on the optimization of a 3D vehicular structure using MATLAB ® and ANSYS ® software. The author looked at variables like the position of the center of gravity, the dynamic and manufacture constraints in their design criteria. A multi-objective function was defined which considered mass reduction and chassis stiffness as objectives. They were able to reduce the weight of 3D structure by 5.31 kg [25].
It has been observed that the determination of the stresses on a truck chassis before production/manufacturing is key to future design improvements as per user requirements and quality purposes. A limited number of studies have been conducted on the design and optimization of heavy commercial vehicles chassis frames. With the increase in fuel costs, the automotive industry demands lighter weighing components. The designing of components is quite a critical process. It requires optimization, which is often a trade-off between weight savings and the dynamic performance of those components.
The objective of this current research is to analyze a heavy vehicle (truck) chassis using numerical and experimental methods. We then seek to optimize this chassis by looking at a number of chassis variables. The next section describes the methodology we followed in achieving the above.

Methodology
The chassis is firstly designed in a CAD software. The FEA on the designed chassis is then conducted in ANSYS ® software. The modelled chassis is then optimized using the Taguchi design of Experiments (D.O.E). The optimization techniques used are the central composite design (CCD) scheme and the optimal space filling (OSF) design. Sensitivity plots and response surface plots are then generated. The critical range of optimized variables is then determined. The equivalent stress, deformation, mass and surface response are evaluated for each design points. The optimization techniques used in this research have not been implemented in chassis design and could be helpful in the development of future chassis designs.
The FEA pre-processing stage CAD modeling, meshing, applying loads and boundary conditions on the model [26]. The solution stage involves element matrix formulations, assemblage of global stiffness elements, followed by inversions and multiplications. The final stage is the post-processing, which involves the interpretation of the analysis results. The chassis specifications were obtained from [27].

The Simulation Environment
The simulation package is ANSYS ® V18.1. The interaction with the software is through graphical user interface (GUI). However, the inbuilt language of the software is C++. The chassis is analyzed as a static structure. A finite element model analysis of the chassis is then performed. We initially evaluate the chassis deformations and the stress distributions on a given standard chassis [27]. We then optimize this chassis, in terms of reducing the deformations and the stress distributions, using the Taguchi design of experiments method. The next subsection details the chassis geometry, loading, and boundary conditions.

Chassis Specifications
The CAD model of the chassis is developed as per the specification tabulated Table 1:

CAD Modelling
The CAD model of HMV (Heavy Motor Vehicle) chassis is developed in ANSYS ® design modeler using the sketch and extrude tool. The dimensions of chassis are taken from literature as per Table 1 [27]. Two longitudinal members and eight lateral members are modeled as shown in Figure 2.

CAD Modelling
The CAD model of HMV (Heavy Motor Vehicle) chassis is developed in ANSYS ® design modeler using the sketch and extrude tool. The dimensions of chassis are taken from literature as per Table 1 [27]. Two longitudinal members and eight lateral members are modeled as shown in Figure 2. The right hand side of the chassis in Figure 2 is the vehicle frontal portion. The model of chassis is discretized using tetrahedral elements as shown in Figure 3. The number of maximum layers is set to 5, the transition is set to smooth and inflation set to normal. The number of elements generated is 20080 and number of nodes generated is 42840. The tetrahedral element has 4 nodes with 3 degrees of freedom at each node. The final meshed model of HMV chassis in shown by Figure 4.  The right hand side of the chassis in Figure 2 is the vehicle frontal portion. The model of chassis is discretized using tetrahedral elements as shown in Figure 3. The number of maximum layers is set to 5, the transition is set to smooth and inflation set to normal. The number of elements generated is 20,080 and number of nodes generated is 42,840. The tetrahedral element has 4 nodes with 3 degrees of freedom at each node. The final meshed model of HMV chassis in shown by Figure 4.

CAD Modelling
The CAD model of HMV (Heavy Motor Vehicle) chassis is developed in ANSYS ® design modeler using the sketch and extrude tool. The dimensions of chassis are taken from literature as per Table 1 [27]. Two longitudinal members and eight lateral members are modeled as shown in Figure 2. The right hand side of the chassis in Figure 2 is the vehicle frontal portion. The model of chassis is discretized using tetrahedral elements as shown in Figure 3. The number of maximum layers is set to 5, the transition is set to smooth and inflation set to normal. The number of elements generated is 20080 and number of nodes generated is 42840. The tetrahedral element has 4 nodes with 3 degrees of freedom at each node. The final meshed model of HMV chassis in shown by Figure 4.               After defining the loads and boundary conditions, the simulation is run using a sparse matrix solver. This is done as it can also reuse the same matrix structure throughout the simulation and avoids the sequential step that orders equations at every iteration [28]. The matrices are formulated for each element. The deformation and stresses are calculated at nodes, and these results interpolated for the entire element edge length. The next section presents the results of the finite element simulations on the standard chassis.

FE Results on a Standard Chassis
The FE simulation is conducted on the chassis to determine the deformation and equivalent stresses. The regions of longitudinal members near the supported end have higher equivalent stress as shown in Figure 7. The mid-section of the chassis also has high stress. This means the center of chassis and corner ends of chassis can be subject to failure under extra loading conditions. The failure of the chassis can occur in the form of crack initiation, which may propagate under repeated loading.
Processes 2021, 9, x FOR PEER REVIEW 6 of 20 After defining the loads and boundary conditions, the simulation is run using a sparse matrix solver. This is done as it can also reuse the same matrix structure throughout the simulation and avoids the sequential step that orders equations at every iteration [28]. The matrices are formulated for each element. The deformation and stresses are calculated at nodes, and these results interpolated for the entire element edge length. The next section presents the results of the finite element simulations on the standard chassis.

FE Results on a Standard Chassis
The FE simulation is conducted on the chassis to determine the deformation and equivalent stresses. The regions of longitudinal members near the supported end have higher equivalent stress as shown in Figure 7. The mid-section of the chassis also has high stress. This means the center of chassis and corner ends of chassis can be subject to failure under extra loading conditions. The failure of the chassis can occur in the form of crack initiation, which may propagate under repeated loading. The deformation plot of chassis is shown in Figure 8. The maximum deformation (of approximately 347 mm) occurs at the mid-section/mid-length of the chassis. The deformation decreases and is least at the transverse members placed at the ends. In order to improve the chassis' response to such loads, we need to improve it. The deformation plot of chassis is shown in Figure 8. The maximum deformation (of approximately 347 mm) occurs at the mid-section/mid-length of the chassis. The deformation decreases and is least at the transverse members placed at the ends. In order to improve the chassis' response to such loads, we need to improve it.
The next section presents the optimization scheme, the Taguchi design of experiments (DOE), that we used to improve the chassis. The next section presents the optimization scheme, the Taguchi design of experiments (DOE), that we used to improve the chassis.

Optimization of Standard Chassis
The optimization variables selected for the optimization of the chassis are the dimensions of the cross members. These are defined in the ANSYS ® design modeler. These variables are the widths of cross member 1, 2 and 3, as shown in Figure 9 (H6), Figure 10 (H12) and Figure 11 (H14). The width dimensions are tabulated in Figure 12 below. The distance between these cross members is 1461.6 mm and 1724 mm, respectively. The three (3) selected cross sections are from rightmost (front) transverse member of chassis.

Optimization of Standard Chassis
The optimization variables selected for the optimization of the chassis are the dimensions of the cross members. These are defined in the ANSYS ® design modeler. These variables are the widths of cross member 1, 2 and 3, as shown in Figure 9 (H6), Figure 10 (H12) and Figure 11 (H14). The width dimensions are tabulated in Figure 12 below. The distance between these cross members is 1461.6 mm and 1724 mm, respectively. The three (3) selected cross sections are from rightmost (front) transverse member of chassis. The next section presents the optimization scheme, the Taguchi design of experiments (DOE), that we used to improve the chassis.

Optimization of Standard Chassis
The optimization variables selected for the optimization of the chassis are the dimensions of the cross members. These are defined in the ANSYS ® design modeler. These variables are the widths of cross member 1, 2 and 3, as shown in Figure 9 (H6), Figure 10 (H12) and Figure 11 (H14). The width dimensions are tabulated in Figure 12 below. The distance between these cross members is 1461.6 mm and 1724 mm, respectively. The three (3) selected cross sections are from rightmost (front) transverse member of chassis.    The first optimization scheme used is the central composite design (CCD). This design scheme has the following design points; axial, cube and center points, as shown in Figure 13 below.   The first optimization scheme used is the central composite design (CCD). This design scheme has the following design points; axial, cube and center points, as shown in Figure 13 below.   The first optimization scheme used is the central composite design (CCD). This design scheme has the following design points; axial, cube and center points, as shown in Figure 13 below. The first optimization scheme used is the central composite design (CCD). This design scheme has the following design points; axial, cube and center points, as shown in Figure 13 below.  Table 2 below.  Table 3 shows three input points (columns B, C and D). The software evaluated the output parameters at these design points using Finite Element Analysis (FEA). These output parameters are equivalent stress and the total deformation as shown in column E and column F.   Table 2 below.  Table 3 shows three input points (columns B, C and D). The software evaluated the output parameters at these design points using Finite Element Analysis (FEA). These output parameters are equivalent stress and the total deformation as shown in column E and column F.
The second optimization method used is space fill design. Space fill designs are recommended for testing with deterministic models because the design points are evenly distributed in the design area as shown in Figure 14. To use these design options, an important assumption is necessary: the computer simulation must reflect the actual physical system [30]. The second optimization method used is space fill design. Space fill designs are recommended for testing with deterministic models because the design points are evenly distributed in the design area as shown in Figure 14. To use these design options, an important assumption is necessary: the computer simulation must reflect the actual physical system [30]. Different design points are generated using the optimal space filling design scheme. These points are shown in column B, C and D of Table 4. The output parameters, i.e., equivalent stress (column E), total deformation (F) and solid mass (G) are evaluated for each design point.  Different design points are generated using the optimal space filling design scheme. These points are shown in column B, C and D of Table 4. The output parameters, i.e., equivalent stress (column E), total deformation (F) and solid mass (G) are evaluated for each design point.

Total Deformation
The variation of the total deformation versus the design points is shown in Figure 15. In the CCD optimization scheme, the maximum deformation (347.89 mm) is observed for design point two (2), and minimum deformation (340.78 mm) is observed for design point seven (7). That is a 2% difference in deformation. The dimensions corresponding to design points number two (2) and seven (7), respectively, are: • 58.5 mm (−10%) for cross member 1, 65 mm for member 2 and 65 mm for member 3. • 65 mm for cross member 1, 65 mm for member 2 and 71.5 mm (+10%) for member 3. In the OSF optimization scheme, the maximum deformation (351.44 mm) is observed for design point number 12 and minimum deformation (341.16 mm) is observed for design point number 10. That is 3% difference in deformation. The dimensions corresponding to design points 12 and 10 and the percentage change from the initial dimension are: • 60.67 mm (−6.6%) for cross member 1, 65 mm for member 2 and 61.53 (−5.3%) mm for member 3. • 65 mm for cross member 1, 67.6 mm (+4%) for member 2 and 71.07 (+9.3%) mm for member 3. Figure 16 shows the variation of equivalent stress versus the design points. In the CCD optimization scheme, the maximum equivalent stress (3527.5 MPa) is observed for design point 5 and the minimum equivalent stress (3220.3 MPa) is observed for point 7. That result a 9.5% difference in equivalent stresses. The dimensions corresponding to design point number 5 and 7, respectively, are:
(a) (b) Figure 16. Equivalent stress vs. design points (a) for CCD scheme (b) for optimal space filling design.

Solid Mass
The variation of solid mass vs. design points is shown in Figure 17. In the CCD scheme, the maximum (226.1 kg) solid mass obtained at design point number 15 and minimum (203.1 kg) solid mass at design point number 8, as shown in Figure 17a.
The dimensions corresponding to design point number 15 is 70.285 mm (+8.13 %) for all cross members.  In the next section, we present the response surface plots of the optimization variables under discussion.

Solid Mass
The variation of solid mass vs. design points is shown in Figure 17. In the CCD scheme, the maximum (226.1 kg) solid mass obtained at design point number 15 and minimum (203.1 kg) solid mass at design point number 8, as shown in Figure 17a. (a) (b) Figure 16. Equivalent stress vs. design points (a) for CCD scheme (b) for optimal space filling design.

Solid Mass
The variation of solid mass vs. design points is shown in Figure 17. In the CCD scheme, the maximum (226.1 kg) solid mass obtained at design point number 15 and minimum (203.1 kg) solid mass at design point number 8, as shown in Figure 17a.
The dimensions corresponding to design point number 15 is 70.285 mm (+8.13 %) for all cross members.  In the next section, we present the response surface plots of the optimization variables under discussion. In the next section, we present the response surface plots of the optimization variables under discussion.

Surface Response
The response surface plot aids in determining the combined functional response of a range of optimization variables values. Figure 18a shows the response surface plot for the CCD optimization scheme for cross members 1 and 2.
Two peaks of equivalent stress are observed as shown in the red colored region. It is clear in Figure 18a that the first peak appears along cross member 1, the equivalent stress is maximum for cross member 1 in the range between 62 mm and 67 mm and between 69 mm and 71 mm for cross member 2. The second peak appears along cross member 2, the equivalent stress is maximum for cross member 1 in the range between 69 mm to 71 mm and between 61 mm and 67 mm for cross member 2. One of the advantages of a response surface plot is that it clearly shows regions of interest for the particular objective and optimization variables. Figure 18b shows the response surface plot of the OSF method for cross members 1 and 2. The first peak occurs along cross member 2, on this peak, the equivalent stress is maximum for cross member 1 dimension ranging from 59 mm to 62 mm and between 59 mm and 62 mm for cross member 2. The second lower peak occurs along cross member 1. At this peak, the equivalent stress is maximum for cross member 1 in the range between 61 mm to 63 mm and between 69 mm and 71 mm for cross member 2. The equivalent stress is minimum for other dimensions (Refer Tables 3 and 4) of cross member 1 and cross member 2, as shown in dark the blue colored region.
(a) (b) Figure 18. Response surface plot of equivalent stress vs. cross member 1 and cross member 2 (a) for CCD scheme, and (b) for optimal space filling (OSF) design.
The response surface plot of equivalent stress vs. cross member 2 and cross member 3 is shown in Figure 19. A single peak of equivalent stress is observed as shown in the red colored region. The dimensions corresponding to maximum equivalent stress are obtained using the interpolation method. The maximum equivalent stress is observed for cross member 3 value ranging from 61 mm to 67 mm and between 69 mm and 71 mm for cross member 2 (see Figure 19a). Figure 19b shows that the maximum equivalent stress is observed for cross member 3 in the range from 64 mm to 71 mm and 66 mm to 71 mm for cross member 2. The equivalent stress is minimum for other values of cross member 2 and cross member 3, which is represented by the dark blue colored region.

Equivalent Stresses Surface Response
Two peaks of equivalent stress are observed as shown in the red colored region. It is clear in Figure 18a that the first peak appears along cross member 1, the equivalent stress is maximum for cross member 1 in the range between 62 mm and 67 mm and between 69 mm and 71 mm for cross member 2. The second peak appears along cross member 2, the equivalent stress is maximum for cross member 1 in the range between 69 mm to 71 mm and between 61 mm and 67 mm for cross member 2. One of the advantages of a response surface plot is that it clearly shows regions of interest for the particular objective and optimization variables. Figure 18b shows the response surface plot of the OSF method for cross members 1 and 2. The first peak occurs along cross member 2, on this peak, the equivalent stress is maximum for cross member 1 dimension ranging from 59 mm to 62 mm and between 59 mm and 62 mm for cross member 2. The second lower peak occurs along cross member 1. At this peak, the equivalent stress is maximum for cross member 1 in the range between 61 mm to 63 mm and between 69 mm and 71 mm for cross member 2. The equivalent stress is minimum for other dimensions (Refer Tables 3 and 4) of cross member 1 and cross member 2, as shown in dark the blue colored region.
The response surface plot of equivalent stress vs. cross member 2 and cross member 3 is shown in Figure 19. A single peak of equivalent stress is observed as shown in the red colored region. The dimensions corresponding to maximum equivalent stress are obtained using the interpolation method. The maximum equivalent stress is observed for cross member 3 value ranging from 61 mm to 67 mm and between 69 mm and 71 mm for cross member 2 (see Figure 19a). Figure 19b shows that the maximum equivalent stress is observed for cross member 3 in the range from 64 mm to 71 mm and 66 mm to 71 mm for cross member 2. The equivalent stress is minimum for other values of cross member 2 and cross member 3, which is represented by the dark blue colored region. The variation of equivalent stress with respect to cross member 1 is shown in Figure  20. The graph in Figure 20a shows a gradual increase in equivalent stress up to 65 mm. The equivalent stress then increases exponentially and reaches a maximum value of 3570 Mpa at a cross member dimension of 71 mm. Figure 20b shows that the equivalent stress initially decreases and reaches a minimum value at cross member 1 (66 mm). The equivalent stress then increases linearly and reaches a maximum value at cross member 1 (71 mm). The initial maximum equivalent stress is obtained for cross member 1 (at 58.5 mm). The variation of equivalent stress with respect to cross member 3 and member 2 is shown in Figure 21a and 21b, respectively. The graph in Figure 21a shows the gradual increase in the equivalent stress up to 63 mm for cross member 3 then a linear decrease with an increase in the width this cross member. The minimum equivalent stress is observed at 71.5 mm.
In Figure 21b, the equivalent stress decreases with an increase in cross member 2's width and reaches a minimum value of 64 mm. Thereafter, a linear increase of equivalent stress is observed and reaches maximum value at 71 mm. The variation of equivalent stress with respect to cross member 1 is shown in Figure 20.
The graph in Figure 20a shows a gradual increase in equivalent stress up to 65 mm. The equivalent stress then increases exponentially and reaches a maximum value of 3570 Mpa at a cross member dimension of 71 mm. Figure 20b shows that the equivalent stress initially decreases and reaches a minimum value at cross member 1 (66 mm). The equivalent stress then increases linearly and reaches a maximum value at cross member 1 (71 mm). The initial maximum equivalent stress is obtained for cross member 1 (at 58.5 mm). The variation of equivalent stress with respect to cross member 1 is shown in Figure  20. The graph in Figure 20a shows a gradual increase in equivalent stress up to 65 mm. The equivalent stress then increases exponentially and reaches a maximum value of 3570 Mpa at a cross member dimension of 71 mm. Figure 20b shows that the equivalent stress initially decreases and reaches a minimum value at cross member 1 (66 mm). The equivalent stress then increases linearly and reaches a maximum value at cross member 1 (71 mm). The initial maximum equivalent stress is obtained for cross member 1 (at 58.5 mm). The variation of equivalent stress with respect to cross member 3 and member 2 is shown in Figure 21a and 21b, respectively. The graph in Figure 21a shows the gradual increase in the equivalent stress up to 63 mm for cross member 3 then a linear decrease with an increase in the width this cross member. The minimum equivalent stress is observed at 71.5 mm.
In Figure 21b, the equivalent stress decreases with an increase in cross member 2's width and reaches a minimum value of 64 mm. Thereafter, a linear increase of equivalent stress is observed and reaches maximum value at 71 mm. The variation of equivalent stress with respect to cross member 3 and member 2 is shown in Figure 21a and Figure 21b, respectively. The graph in Figure 21a shows the gradual increase in the equivalent stress up to 63 mm for cross member 3 then a linear decrease with an increase in the width this cross member. The minimum equivalent stress is observed at 71.5 mm.

Mass Surface Response
The variation of mass with respect to cross member 1 and 2 is shown in Figure 22. The maximum mass is represented in the red colored region, whereas the minimum mass is represented by the blue region.
The maximum mass in the CCD optimization, Figure 22b, is observed for cross member 1 in the range 68 mm to 71 mm and for cross member 2 between 68 mm to 71 mm. In the optimal space filling method, the maximum mass on cross member 1 is between 68 mm to 71 mm and between 65 mm and 71 mm for cross member 2. The minimum mass is observed for cross member 1 and cross member 3 dimensions ranging from 58.5 mm to 61 mm.
(a) (b) Figure 22. Three-dimensional response surface plot of solid mass (a) for CCD scheme (b) for optimal space filling design.
The variation of chassis mass with respect to cross member 1 and cross member 3 dimensions is shown in Figure 23 a, b and Figure 24 a, b respectively. Unsurprisingly, the solid mass of the chassis is observed to increase linearly with the increase in all the cross members dimensions. In Figure 21b, the equivalent stress decreases with an increase in cross member 2's width and reaches a minimum value of 64 mm. Thereafter, a linear increase of equivalent stress is observed and reaches maximum value at 71 mm.

Mass Surface Response
The variation of mass with respect to cross member 1 and 2 is shown in Figure 22. The maximum mass is represented in the red colored region, whereas the minimum mass is represented by the blue region.

Mass Surface Response
The variation of mass with respect to cross member 1 and 2 is shown in Figure 22. The maximum mass is represented in the red colored region, whereas the minimum mass is represented by the blue region.
The maximum mass in the CCD optimization, Figure 22b, is observed for cross member 1 in the range 68 mm to 71 mm and for cross member 2 between 68 mm to 71 mm. In the optimal space filling method, the maximum mass on cross member 1 is between 68 mm to 71 mm and between 65 mm and 71 mm for cross member 2. The minimum mass is observed for cross member 1 and cross member 3 dimensions ranging from 58.5 mm to 61 mm.
(a) (b) Figure 22. Three-dimensional response surface plot of solid mass (a) for CCD scheme (b) for optimal space filling design.
The variation of chassis mass with respect to cross member 1 and cross member 3 dimensions is shown in Figure 23 a, b and Figure 24 a, b respectively. Unsurprisingly, the solid mass of the chassis is observed to increase linearly with the increase in all the cross members dimensions. The maximum mass in the CCD optimization, Figure 22b, is observed for cross member 1 in the range 68 mm to 71 mm and for cross member 2 between 68 mm to 71 mm. In the optimal space filling method, the maximum mass on cross member 1 is between 68 mm to 71 mm and between 65 mm and 71 mm for cross member 2. The minimum mass is observed for cross member 1 and cross member 3 dimensions ranging from 58.5 mm to 61 mm.
The variation of chassis mass with respect to cross member 1 and cross member 3 dimensions is shown in Figure 23a The minimum mass of the chassis is observed for 58.5 mm cross member 3 and cross member 1 dimension 71.5 mm.
(a) (b) Figure 24. Solid mass vs. cross member 3 (a) for CCD scheme, and (b) for optimal space filling design.

Sensitivity
Sensitivity plots are given in Figure 25a and 25b, respectively. For the chassis deformation, the maximum sensitivity percentage is shown by cross member 3, and the minimum sensitivity percentage is shown by cross member 1. This signifies that cross member 3 has the maximum effect on the total deformation. For solid mass, all three variables show the same sensitivity percentage, which signifies that all the three optimization variables have the same effect on mass of chassis.  The minimum mass of the chassis is observed for 58.5 mm cross member 3 and cross member 1 dimension 71.5 mm.
(a) (b) Figure 24. Solid mass vs. cross member 3 (a) for CCD scheme, and (b) for optimal space filling design.

Sensitivity
Sensitivity plots are given in Figure 25a and 25b, respectively. For the chassis deformation, the maximum sensitivity percentage is shown by cross member 3, and the minimum sensitivity percentage is shown by cross member 1. This signifies that cross member 3 has the maximum effect on the total deformation. For solid mass, all three variables show the same sensitivity percentage, which signifies that all the three optimization variables have the same effect on mass of chassis.  The minimum mass of the chassis is observed for 58.5 mm cross member 3 and cross member 1 dimension 71.5 mm.

Sensitivity
Sensitivity plots are given in Figure 25a and Figure 25b, respectively. For the chassis deformation, the maximum sensitivity percentage is shown by cross member 3, and the minimum sensitivity percentage is shown by cross member 1. This signifies that cross member 3 has the maximum effect on the total deformation. For solid mass, all three variables show the same sensitivity percentage, which signifies that all the three optimization variables have the same effect on mass of chassis. The minimum mass of the chassis is observed for 58.5 mm cross member 3 and cross member 1 dimension 71.5 mm.
(a) (b) Figure 24. Solid mass vs. cross member 3 (a) for CCD scheme, and (b) for optimal space filling design.

Sensitivity
Sensitivity plots are given in Figure 25a and 25b, respectively. For the chassis deformation, the maximum sensitivity percentage is shown by cross member 3, and the minimum sensitivity percentage is shown by cross member 1. This signifies that cross member 3 has the maximum effect on the total deformation. For solid mass, all three variables show the same sensitivity percentage, which signifies that all the three optimization variables have the same effect on mass of chassis.
(a) (b) Figure 25. Sensitivity plot (a) for CCD scheme, and (b) for optimal space filling design. Figure 25. Sensitivity plot (a) for CCD scheme, and (b) for optimal space filling design. Figure 25a indicates for equivalent stress; the maximum sensitivity percentage is shown by cross member 1 (84.019%), and the minimum sensitivity percentage is shown by cross member 3 (21.231%), which signifies that cross member 1 has the maximum effect on the equivalent stress and cross member 3 has minimal effect. Figure 25b shows that the maximum sensitivity percentage is shown by cross member 2 and the minimum sensitivity percentage is shown by cross member 3, which signifies that cross member 2 has the maximum effect on equivalent stress.
The experimental investigation of the chassis was conducted under flexural loading conditions. The test results are in close agreement with the finite element simulation results. The experimental testing results are shown in Appendix A.

Conclusions
The dimensions of the chassis have a significant effect on its dynamics and load bearing characteristics. An optimized design can reduce weight and improve these characteristics. For example, a reduced mass of chassis will reduce the material required in its manufacturing.
In this paper, we applied an optimization method in order to improve the design characteristics of the chassis. This was the Taguchi design of experiments (DOE). The Taguchi DOE presented a wide range of dimensions for which equivalent stress, safety factor and mass are maximum or minimum. The optimization is conducted using central composite design (CCD) scheme and optimal space filling (OSF) design scheme. The individual effect of each variable is studied using 2D graphs and 3D response surface plots. This enabled us to determine the range of values (for the optimization variables) for which equivalent stress, mass and total deformation are maximum and or are minimum.
The optimization results obtained from the CCD scheme shows that cross member 1 has higher effect on equivalent stress as compared to cross member 2 and member 3. For total deformation, cross member 3 has a higher effect as compared to cross member 1 and 2. The mass reduction of the chassis obtained from central composite design scheme is nearly 5.3%. The optimization results obtained from optimal space filling design scheme shows that cross member 2 has higher effect on equivalent stress as compared to cross member 1 and 3. For total deformation, cross member 3 has a higher effect as compared to cross member 1 and 2.
The cost factor is quite significant in the decision-making process in industry. A chassis with a reduced weight signifies a reduction in the production, manufacturing, and material cost. Therefore, the optimization scheme that gives us higher weight reduction is desired. The mass reduction of the chassis obtained from optimization is approximately 5.3% when using the CCD method and 4.35% when using the optimal space filing method. Therefore, the central composite design scheme is preferable in this case.

Future Work
In the pursuit of improving vehicle chassis designs, a number of other considerations can be made. The design of vehicle chassis can be improved by perhaps changing the position of the cross members and/or by changing the orientation of cross sections of the chassis members.
Furthermore, the effects of material choices on the strength and rigidity of the heavy vehicle chassis structure requires investigation. This could have a significant effect on the strength to weight ratio of chassis.

Conflicts of Interest:
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

Appendix A
Processes 2021, 9, x FOR PEER REVIEW 18 of 20

Conflicts of Interest:
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.
Appendix A Figure A1. Experimental testing results.