MSX International do Brasil Ltda. Fabio Martins. MSX International do Brasil Ltda. Sidnei Kameoka. Trilogy International do Brasil Ltda. Amir Sadalla Salloum.
SAE TECHNICAL PAPER SERIES
2003-01-3547
E
Structural Optimization of Automotive Components Applied to Durability Problems Wallace Gusmão Ferreira MSX International do Brasil Ltda
Fabio Martins MSX International do Brasil Ltda
Sidnei Kameoka Trilogy International do Brasil Ltda
Amir Sadalla Salloum Ford Motor Company do Brasil
Jorge Teruo Kaeya Ford Motor Company do Brasil
12th International Mobility Technology Congress and Exhibition São Paulo, Brasil 2003, November 18-20
AV. PAULISTA, 2073 - HORSA II - CJ. 2001 - CEP 01311-940 - SÃO PAULO – SP
2003-01-3547
Structural Optimization of Automotive Components Applied to Durability Problems Wallace Gusmão Ferreira MSX International do Brasil Ltda
Fabio Martins MSX International do Brasil Ltda
Sidnei Kameoka Trilogy International do Brasil Ltda
Amir Sadalla Salloum Ford Motor Company do Brasil
Jorge Teruo Kaeya Ford Motor Company do Brasil Copyright © 2003 Society of Automotive Engineers, Inc
ABSTRACT The application of structural optimization methodology in the preliminary design phases allows to combine several component attributes and to obtain the optimal design configuration to satisfy the project requirements. Basic concepts of structural optimization and design for automotive durability are presented in order describe the general aspects of the methodology. Case studies applying structural optimization of components for automotive application are presented and discussed. The optimization criteria are defined in terms of size optimization: component thickness (sheet metal components) are calculated in order to use a minimal amount of material and to provide maximum durability.
INTRODUCTION Structural Optimization is a well known mathematical methodology that can be applied on many design areas in order to obtain the optimal parameters to satisfy the requirements for a determined application [1,2]. In the design of automotive components, attributes such as shape, performance, durability, weight, volume of material, manufacturing process variables, costs, energy economy, customer satisfaction, environmental impacts and others can be improved simultaneously by structural optimization criteria to provide the best design solution.
Conventional design methodologies are based on iterative processes of simulation and tests. A great number of calculation and tests on physical prototypes are performed before to achieve the “optimal” component that satisfies the requirements. The number of design parameters that can be handled at once in these methodologies are limited and the success is strongly dependent on the experience of engineers involved in the project. In most cases, there is no guarantee that component design is the “best” for the application due to the fact that only few requirements are satisfied at once. In addition, when it is necessary to satisfy many requirements simultaneously the problem becomes non linear and the number of iterations in the conventional design process to achieve the best solution increases exponentially with the number of variables involved. There are many examples in the industry that confirm these difficulties. As example, when a component cracks due to excessive loads, the first action is to reinforce the area that fail to improve its mechanical resistance (e.g. increasing cross sectional area or applying local reinforcements). In many cases, the new reinforcements increase excessively the local stiffness and other areas that were not a problem start to crack in future tests. In consequence of this fact, new reinforcement configurations are performed until the cracks are completely controlled. A question arises in this case: is the final reinforcement configuration the best in terms of durability, vibrations (mass and stiffness), safety/crash, manufacturing processes and costs? In this way the answer can only be found after several simulation and test iterations. Clearly in
a high competitive market, the costs and timing demanded by this traditional and widely used design methodology become prohibitive. Structural optimization concepts provide a strong and systematical methodology that permits the reduction of costs and design time. The number of iterations in the conventional design to achieve the best solution to satisfy several requirements simultaneously can be substantially reduced by the application of optimization criteria. In this work the basic concepts of structural optimization theory are described and the application on the design of automotive components to satisfy the minimal amount of material used and durability requirements is performed and discussed.
BACKGROUND
STRUCTURAL OPTIMIZATION – An optimization problem can be mathematically described as follows [1,2]: Minimize f(x),
(1)
subjected to gi(x) – gU ≤ 0
i = 1, 2, . . . n,
hj(x) – hU = 0
j = 1, 2, . . . n
xL ≤ x ≤ xU,
(2) and
(3) (4)
where f(x) is the objective function to be minimized and gi(x) and hj(x) are respectively are the inequality and equality constraints. These functions are dependent on the vector of design variables x, that is restricted by the lower and upper limits xL and xU. In terms of structural design, f(x) should represent a system response or a quantity to be minimized like: volume, thickness, mass, stiffness, strain energy, stress, displacements, cost, etc. These responses are generally restricted by one or more “admissible” engineering parameters such as: rupture or yield stress, fatigue limit, resonance frequency, maximum displacements at some locations, admissible weight, cost, etc, that are represented by the equality and inequality constraints. The design variables x represent system control parameters or properties, that should be changed to define different design configurations in order to achieve the optimum of the objective function. Shape, plate thickness, area of truss bars, moment of inertia of beams and rods,
diameter, length, etc, are common examples of design variables. In this sense, when x assumes admissible engineering values, a feasible design is defined. In the last decades many research have been performed and great advances have been achieved in this area [3,4]. Maney methods and computer algorithms have been proposed and implemented to solve optimization problems in various knowledge areas. There are several types of optimization approaches applied to structural design. Size Optimization defines the design variables in terms of discrete parameters of the system in study. Generally these parameters do not change the overall “shape” of the component (geometrical domain). Only the “size” is modified. Parameters like geometrical properties such as thickness, diameter, area, moment of inertia, etc, generally are used as design variables in size optimization. On the other hand, Shape Optimization and Topological Optimization methods should change appearance of the geometrical domain in study. In shape optimization the boundaries of the domain are mapped by a set of “control variables” that defines the coordinates of the domain borders. These coordinates are changed in order to minimize the objective function. Therefore the final shape of the system is changed to find the optimal design to satisfy the requirements. Topological optimization generally changes drastically the shape of the system. The initial geometrical domain is roughly defined (e.g. a square plate with restrictions and loads at the corners) and the optimization algorithms create several “voids” in the whole system in order to achieve the best material distribution to obtain the optimum design. Figure 1 illustrates the principle of these optimization methods. These are the most common, but several other types of methods and optimization algorithms have been proposed and can be found on the literature [3,4]. CAE technology (Computer Aided Engineering) is supported by a lot of commercial software packages that successfully implement various simulation methods like Finite Element Method, which is widely used in structural design. Most of these packages present integrated optimization algorithms that can be easily used to optimize structures and components. System responses like stress (von Mises, Tresca, principal, normal, shearing, etc), displacements, rotations, natural frequencies, strain energy, mass, volume, etc, should be optimized and/or used as constraints. Normally size, shape and topological optimization algorithms are available.
These methods are generally “collections” of empirical formulae based on physical observation. Although fatigue is a very complex phenomenon (statistical in nature) these and other methods can present acceptable correlation with experiments and are widely used in the industry by engineers in the determination of life of components and structures. The durability of automotive components is determined based on the average time of vehicle usage by end users (consumers). Many simulation and physical tests criteria can be defined in order to guarantee the required vehicle durability. As an example, automobiles should not fail (to present excessive deformation or cracks) before a determined number of kilometers or years of consumer’s usage (e.g. 250 000km or ten years). For pick-ups, trucks and other vehicles another criteria can be defined. Figure 1. Methods of Structural Optimization.
STRUCTURAL DURABILITY – In structural design several criteria are defined to verify the performance of components. Components should be designed to resist to service loads applied, without failure (i.e. excessive deflections, yielding, rupture, wear, etc). In this way, the design is conduced to obtain components with a prescribed mechanical strength in order to be “safe” under service conditions. Mechanical components normally are subjected to dynamical loading (i.e. variable in time). When dynamical loads are present on the structure, mechanical fatigue phenomenon occurs and the component strength should be strongly reduced. In this case the durability or “useful life” of the component under variable loading must be determined.
The major difficult in automotive design for durability is the determination of real service loads (e.g. road loads) to estimate the component life in an acceptable confidence level. Normally “road profiles” are created on proving grounds to subject vehicle prototypes or components to the “most probable” loads that can be found on real usage. Figure 2 shows a typical road load event for a vehicle measured at proving ground. These road profiles can also be inputs on computer simulation to calculate or predict vehicle and component durability based on fatigue life theories. Obviously these input loads are just good estimates (real road loads are random) and could accurately not predict all loads cases that components are subjected in practice. In this case “safety” or “service” factors based on experience are used to prevent components failure in a wide range of precision.
In few words fatigue is characterized by initiation and propagation of cracks in the structure until the complete rupture that determines the component failure. Fatigue phenomenon is influenced by many effects like load variation amplitude, material properties, surface finish, stress concentration factors, average loading (multiaxial and complex loading), environmental effects (e.g. corrosion) and others. There are many theories that deal with fatigue phenomenon. The most common used in design are S-N (stress amplitude versus number of load cycles of loading), ε-N (strain amplitude versus number of load cycles) and da/dN-N (crack size increase rate versus number or load cycles). S-N and ε-N normally are used in crack initiation phase and da/dN-N is used to determine crack propagation behavior [5,6].
Figure 2. Typical road load event for a vehicle measured at proving ground.
Besides durability, automotive components must be designed to be efficient in terms of fast response
(acceleration and brake efficiency), driveability, comfort, vibrations, noise, fuel economy, safety, environmental impacts (low emissions) and low cost. These and several other criteria must be combined in order to develop vehicles to be at the same time “mechanically efficient” and market competitive. If only durability is kept in mind, heavy weight vehicle might be designed with not acceptable performance and prohibitive costs. In this context, structural optimization can be a powerful tool in order to develop components that meet simultaneously several and different criteria. Besides, experience shows that the use of structural optimization in preliminary design phases is a way to reduce calculus and test iterations that are demanded in conventional design methodologies (i.e. heuristically based, strongly dependent on the know how of engineering team).
are calculated using FDynam (in house Ford software for durability calculation) [8]. CASE 1 – Optimize the thickness of a cantilever beam. Problem Description: The objective is to optimize the thickness of a cantilever beam (see Figure 3) in order to obtain a minimal volume of material. Maximum durability (fatigue life) is expected and displacement at the free and has to be lower than 1mm. The dynamical load is a fully reversed bending with constant amplitude F =1000 N.
In other words, structural optimization applied to component design is a real way to reduce simulation and physical tests (decrease the number of prototypes), design, costs and time to market.
This work presents case studies applied in the development of components based on structural optimization using computer simulation resources and finite element modeling techniques.
CASE STUDIES
To exemplify the application of structural optimization in component design, two examples will be performed. The first one is the optimization of the thickness of a cantilever beam to get the minimal volume of material, a maximum displacement at the free end and the maximum fatigue life. The objective of this example is to introduce the main concepts of structural optimization (i.e. objective function, constraints and design variables) in the context of computer simulation. In the second example, the thickness of a trailing arm bracket and its reinforcement are optimized to provide a minimal amount of material and maximum fatigue life. These components are designed for a real application in a Ford SUV (Sport Utility Vehicle), subjected to road loads measured at proving ground. The examples are performed using Altair Optistruct® (a general purpose finite element software and optimization package)[7] and the fatigue life of components
Figure 3. Cantilever beam for case 1. Plane stress state is considered in this case. The problem is modeled with quadrilateral finite elements (see Figure 4). All displacements are constrained at nodes on the left side and a concentrated force is applied at the mid node on free end (node “N”). The optimization problem is solved in two ways. First the thickness is allowed to change along the height of the beam and, in a second case, the thickness should change also along the length. For optimization purposes only the stress and displacements due to the peak load are considered. In this case, the problem is handled as static, with F = 1000 N. Case 1a - Optimization parameters (variation of thickness along height "h" of beam): !
Objective Function: Minimize volume of beam;
!
Constraint 1 – Maximum admissible displacement at free end (node "N", Figure 4): uy ≤ 1mm (absolute value);
!
Constraint 2 – Maximum admissible von Mises stress: σVM ≤ 50 MPa (fatigue limit for fully reversed bending load, material SAE1010);
!
The optimization problem converged after 7 iterations. The reduction in the volume achieved is 46%. The final thickness are: t_upper = 22.1mm, t_middle = 1.90mm and t_lower = 22.1mm. Maximum stress at final iteration is 27.9MPa and vertical displacement at node N is 1mm. The behavior of the objective function and design variables along the iterations can be observed in the figures 6 and 7.
Design variables (thickness, see Figure 5):
1mm ≤ t_upper ≤ 50mm, 1mm ≤ t_middle ≤ 50mm, 1mm ≤ t_lower ≤ 50mm.
Figure 6. Variation of volume along iterations for case 1a.
Figure 4. Finite element model for the cantilever beam with 1000 linear plane stress quadrilateral elements (100 x 10).
Figure 7. Variation of thickness along iterations for case 1a.
Case 1b - Optimization parameters (variation of thickness along height "h" and length “L” of beam):
Figure 5. Design variables for case 1a. Results 1a:
!
Objective Function: Minimize volume of beam;
!
Constraint 1 – Maximum admissible displacement at free end: uy ≤ 1mm (absolute value);
!
Constraint 2 – Maximum admissible von Mises stress: σVM ≤ 50 MPa;
!
Design variables (thickness, see Figure 8):
1mm ≤ ti
≤ 50mm,
i = 1, 2, 3, ..., 15.
Figure 11. Variation of thickness along iterations for case 1b.
Figure 8. Design variables for case 1b. Results 1b: The optimization problem converged after 6 iterations. The reduction in the volume achieved is 58%. The final thicknesses are shown in Figure 9. Maximum stress at final iteration is 28MPa and vertical displacement at node N is 1mm.
Discussion: The optimal thickness “t” (constant along the height and length) for a cantilever beam, considering maximum admissible stress and displacement at free end is a basic problem of solid mechanics. The solution in this case can be easily found by the “flexural formula” for a cantilever beam. For the load, dimensions and constraints considered, the optimum thickness is t ≅ 20mm. The main objective of this example is to illustrate the resources of structural optimization to help engineers to find “non-trivial” solutions for optimum components, based on arbitrary initial parameters, multiple constraints and several design variables.
Figure 9. Thickness optimized for case1b (results in mm). The behavior of the objective function and design variables along the iterations can be observed in the figures 10 and 11.
In the first case studied (case 1a), 3 design variables were considered (thickness) and a constant “I” shape section along the length of the beam was achieved. This shape is an engineering well-known design for “optimum” behavior of beams under bending loads. The stress along the height the section increases linearly from zero at the “neutral line” (centroid of the section) to maximum value (tension or compression) at “outer fibers”. Naturally, a lower amount of material is required at the center of the section to resist to stress and displacements induced by bending. In this way, a beam reduced 46% in volume was achieved to resist the same loads from initial design. Against t = 20mm, the “real” material economy in this case is an order of 33%. On the other hand, the stresses increase linearly along the length (from load point to constraints) if a constant section beam is considered. A lower amount of
Figure 10. Variation of volume along iterations for case 1b.
material is necessary near the free end to resist to bending loads. Therefore if the thickness may also change along the length of the beam (case 1b) a better design should be achieved (58% reduced in volume from initial design). This result was achieved using 15 design variables at the same project. Against t = 20 mm, the reduction in volume is 50%. For comparison of designs studied, 3D shapes for the initial and optimized beams are illustrated on Figure 12. Stress and displacement diagrams along the length for beam designs are shown at figures 13 and 14. It is clearly observed that on the optimized designs the stresses are uniformly distributed along the beam length (stress saturation phenomenon to optimize material distribution). The displacement profile along the length is not quite changed among the designs.
Figure 13. Stress distribution along beam length for initial and optimized designs.
Besides the better material distribution for the beam, the constant stress behavior on the optimized design (low stress concentration) can improve also the fatigue properties for the component.
Figure 14. Displacement profile along beam length for initial and optimized designs.
Figure 12. 3D shape of beam for initial and optimized designs. This is only a simplified example of design optimization. Obviously other factors (e.g. manufacturing feasibility) should be considered to choose the best design configuration for an application. For simplicity, only few constraints (stress and displacement) were used to determine the optimum design for the beam. Many other constraints like natural frequencies, strain energy, buckling, etc, combined to several design variables may be easily introduced in the problem to optimize a wide range of properties and requirements for the component performance.
CASE 2 – Optimize the thickness of a trailing arm bracket component. Problem Description: The objective is to optimize the thickness of a trailing arm bracket (see Figures 15 and 16) in order to obtain a minimal volume of material and maximum durability (fatigue life) for the component. Trailing arm bracket is welded (seam weld) to the reinforcement bracket and bolted to vehicle body rail. Sheet metal components and seam welds are modeled as linear shell finite elements. Load and constraints are applied to the components using “rigid links” (see Figure 17).
!
Fz = 1258 kgf.
Optimization parameters: !
Objective Function: Minimize volume trailing arm bracket components;
!
Constraint – Maximum admissible von Mises stress: σVM ≤ 50 MPa (fatigue limit for fully reversed bending load, material SAE1010);
!
Design variables (thickness of trailing arm bracket, reinforcement and welds):
1mm ≤ t_tarm ≤ 6.5mm, Figure 15. Overview of trailing arm bracket attached to vehicle underbody.
1mm ≤ t_reinf ≤ 6.5mm, 1mm ≤ t_weld ≤ 6.5mm. Obs.: shell elements attached to rigid links were not considered in the domain of optimization due to the high stress values at these locations. Otherwise the optimization problem might lead to wrong results for thickness.
Figure 16. Detail of trailing arm bracket and its reinforcement.
The load input at trailing arm bracket for optimization is a RMS (Root Mean Square) value in x y and z directions of a complete structural durability road load profile measured at proving ground. At the hard point of the trailing arm bracket the RMS load components are: !
Fx = 530 kgf,
!
Fy = 158 kgf,
Figure 17. Finite element model for trailing arm bracket. 1586 elements.
Optimization Results: The optimization problem converged after 7 iterations. 40% reduction in the volume is achieved, considering all shell components with 6.5mm for initial thickness.
After the last iteration the thickness are: t_tarm = 4.3 mm, t_reinf = 1.96mm and t_weld = 1mm. Peak stress for initial design is 16MPa and at final iteration is 50 MPa (see figures 18 and 19). The behavior of the objective function and design variables along the iterations can be observed in the figures 20 and 21.
16 MPa Figure 20. Variation of volume along iterations for trailing arm bracket (all components considered).
Figure 18. Stress at critical points for initial design. Elements attached to rigid links are masked.
50 MPa
Figure 21. Variation of thickness along iterations for trailing arm bracket, reinforcement and welds.
Durability Analysis: In order to certify the results obtained in the local optimization procedure (simplified model with constant amplitude loading), trailing arm bracket components were submitted to fatigue calculation in a full vehicle finite element model (average 300 000 elements) using FDynam on a Cray machine (supercomputer). Figure 19. Stress at critical points at final iteration. Elements attached to rigid links are masked.
Complete road load history for a structural durability route measured at proving ground is load inputs on the model. The nominal thickness considered were (increased by 10% for safety factor due to variation on blank thickness): !
t_tarm = 4.75mm,
!
t_reinf = 2.25mm,
!
t_weld = 2.25mm.
Life factors (reciprocal of total damage accumulated based on Palmgreen-Miner’s Rule) calculated for these components are: !
trailing arm bracket: L = 1.55,
!
bracket reinforcement: L = 0.1,
!
welds: L = 3.43.
Obs.: Normalized life factors. L ≥ 1.0 indicates that the component durability requirement is achieved for the road loads considered. Bracket reinforcement did not achieve the durability requirement. A new iteration of durability analysis was performed using the same thickness and a high strength steel material for the bracket reinforcement (SAEJ1392050). The life factor for this case is still bellow the durability requirement (L = 0.41). Durability results have shown that the bracket reinforcement is overloaded in the full vehicle model with complete road load history. The solution found after a new iteration of durability analysis was to increase the thickness of components using high strength steel (SAEJ1392050) for the bracket reinforcement. This new thickness and material configuration achieve the durability requirements for the components. At the final of durability analysis the design configuration is: !
t_tarm = 4.75mm (SAEJ1010),
!
t_reinf = 3.0mm (SAEJ1392050),
!
t_welds = 3.0mm (SAE1010).
Discussion: The optimized trailing arm bracket components were submitted to durability calculation in a full vehicle model using a complete road load history to verify the results obtained in the optimization procedure. Fatigue calculation with a full vehicle model using road loads (analytical or measured) is the conventional and method for durability analysis of automotive components and the results has a good correlation with physical tests. With the optimized thickness and SAE1010 material (low carbon steel) the bracket reinforcement did not achieve the durability criterion. Further complete durability analyses with thickness and material improvement permitted to obtain a design that satisfies the requirements with an optimized amount of material.
Additional iterations of durability analysis in a full vehicle model (after local optimization) to improve the component performance can be explained by the fact that the RMS loads used in optimization are not sufficiently representative to simulate fatigue phenomenon. Besides, the stress criterion in the local optimization procedure did not include fatigue correction factors (i.e. loading type, stress concentration, mean stress effects, etc) and were considered the same for all components in the model (general fatigue limit for the material). It is well known that dynamical effects in a full vehicle model subjected to real road loads (i.e. inertial forces, random loading, non linear damage accumulation due to variable amplitude and load directions, etc.) may drastically change the stress behavior of the components when compared with the same in a “local” model. These effects can be represented by a uniaxial loading case (e.g. RMS or similar) if correlation of the loads (e.g. uniaxial-multiaxial correlation for damage equivalence) are used to better represent the random loading input in a simplified model. Besides, safety factors and different stress criterion for fatigue analysis for each component in the model can be used to improve the results of the optimization procedure in order to reduce the number of additional full vehicle durability iterations to achieve the optimum design. Full vehicle durability analysis is further costly than local optimization procedure. Computational costs are too high for a full vehicle durability analysis (4 hours of calculation in a Cray computer). The optimization procedure in the examples studied here demanded less than five minutes of calculation (for all iterations) in a conventional UNIX workstation. In addition, if at least a good “starting point” for the design configuration is not known at the preliminary phases of component development, the number of iterations of calculus and tests in the conventional design methodologies may increase significantly. The timing and costs demanded by these conventional methodologies become prohibitive in a competitive market. Besides of that, there is no guarantee that an optimum component will be achieved for all requirements and design variables involved in the traditional design methodologies. Structural optimization procedure, allied to good confidence level input loads and adequate design constraints, is clearly a strong methodology to design competitive vehicles in terms of performance, costs and time.
CONCLUSIONS In this work, basic concepts of Structural Optimization were presented in the context of Vehicle Durability Analysis. The methodology applied in the design of a simple component (cantilever beam) and in an automotive component (trailing arm bracket) permitted to obtain design configurations with minimal amount of material and maximum durability. For the trailing arm bracket the optimized design was achieved after few additional iterations of full vehicle analysis. Further investigation is needed in the area of loading correlation and stress criteria for a better equivalence between local optimization procedure and full vehicle model subjected to road loads. Case studies of thickness optimization of components for minimal volume of material and maximum durability showed that structural optimization methodology is a powerful tool that helps engineers to design components involving several criteria and parameters in a fast and systematical way.
ACKNOWLEDGEMENT The authors would like to thank the CAE Departments of Ford Motor Company do Brasil, MSX International do Brasil and Trilogy International do Brasil for the resources supplied to produce this work.
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