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Variable Cross-sectional Rectangular Beam and Sensitivity Analysis for Lightweight Design of Bus Frame
Abstract and Figures
Timoshenko beam element of variable cross-section rectangular tube is developed and applied in the lightweight design of bus frame in this paper. Firstly, the finite element formulations of variable cross-section beam (VCB) are derived under the loadsteps of axial deformation, torsional deformation and bending deformation. Secondly, bending deformation experiment and its detailed shell finite element model (FEM) simulation of variable cross-section rectangular tube were conducted; and the proposed VCB, detailed shell FEM and experimental results can be highly consistent. Thirdly, VCBs are used to substitute for parts of the uniform ones in a bus frame. An innovatively lightweight bus frame is obtained and all the performance responses are improved simultaneously. Finally, rollover analysis further shows the advantage of variable cross-section bus frame in crashworthiness design.
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In this paper an ordered multi-material SIMP (solid isotropic material with penalization) interpolation is proposed to solve multi-material topology optimization problems without introducing any new variables. Power functions with scaling and translation coefficients are introduced to interpolate the elastic modulus and the cost properties for multiple materials with respect to the normalized density variables. Besides a mass constraint, a cost constraint is also considered in compliance minimization problems. A heuristic updating scheme of the design variables is derived from the Kuhn-Tucker optimality condition (OC). Since the proposed method does not rely on additional variables to represent material selection, the computational cost is independent of the number of materials considered. The iteration scheme is designed to jump across the discontinuous point of interpolation derivatives to make stable transition from one material phase to another. Numerical examples are included to demonstrate the proposed method. Because of its conceptual simplicity, the proposed ordered multi-material SIMP interpolation can be easily embedded into any existing single material SIMP topology optimization codes.
In usual automobile practice, the structural optimization of a car body is based on changes in the topology, the shape, the cross-sectional shape and finally the thickness size successively depending on the development stages. However, design engineers mostly rely on their experience, intuition and data accumulation when making decisions on the cross-sectional shape of a car body frame. Therefore, a bi-level structural optimization model is proposed to determine the optimal cross-sectional shapes of each thin-walled beam and to achieve a lightweight car body frame with a high stiffness. Intermediate box sections are introduced to bridge the relationships between the two levels. At level 1, the car body frame with static stiffness and dynamic frequency stiffness constraints is optimized to obtain the optimal sizes of the box section, using a sequential approximate optimization method. At level 2, arbitrarily shaped cross-sections constrained with cross-sectional properties are optimized using a genetic algorithm. Component sensitivity analysis and manufacturing constraints promote this model to be closer to actual automobile practice. Finally, a numerical example, solved by the developed software Vehicle Body - Forward Design and Optimization, verifies that the presented method is effective in guiding the conceptual design of the car body frame.
At conceptual design stage, engineers mostly rely on their experience, intuition, and data accumulation when making decisions on conceptual vehicle body (CVB). This paper presents a component sensitivity method for the lightweight design of the CVB. Firstly, CVB is simplified as a frame structure consisting of box beams. Secondly, torsional stiffness, bending stiffness and frequencies are adopted to evaluate the global stiffness performances of the CVB frame. Thirdly, component sensitivity formulas are derived. Fourthly, two application examples of car and bus frame verify that the proposed method is efficient to guide the cross-sectional sizes modification. The sufficient condition to obtain positive frequency sensitivity is also discussed. And finally Vehicle Body-FDO software is released for free to implement this method.
Presented manuscript discusses the usage of multi-attribute decision making tools to assist in the material selection for vehicular structures; mainly the automotive Body-In-White (BiW) panels at the conceptual design stage using Quality Function Deployment (QFD) and Analytical Hierarchy Process (AHP). The main advantage of using QFD and AHP is their abilities to rank choices in the order of their effectiveness in meeting the functional objective. AHP discriminates between competing options where interrelated objectives need to be met; AHP is based on straightforward mathematical formulations. QFD on the other side is a customer focused method that usually starts by collecting customer needs and tries to integrate these needs into the product. In this study, following classes of engineering materials are analyzed; forming grade Bake Harden-able steel (BH), Dual Phase steel (DP), High Strength Low Alloy Steel (HSLA), Martenistic steel, Aluminum 5xxx, 6xxx sheets, Magnesium sheets, Titanium sheets, Carbon Fiber Reinforced Plastic (CFRP) and High Density Polyethylene (HDPE). The presented study showed that the different grades of steel gained the first ranks in the selection process for almost most of the BiW panels; however other alternatives could work in trade-off with cost and manufacturability.
A method is presented for synthesizing multi-component structural assemblies with maximum structural performance and manufacturability. The problem is posed as a relaxation of decomposition-based assembly synthesis (1,2,3), where both topology and decomposition of a structure are regarded as variables over a ground structure with non-overlapping beams. A multi-objective genetic algorithm (4,5) with graph-based crossover (6,7,8), coupled with FEM analyses, is used to obtain Pareto optimal solutions to this problem, exhibiting trade-offs among structural stiffness, total weight, component manufacturability (size and simplicity), and the number of joints. Case studies with a cantilever and a simplified automotive floor frame are presented, and representative designs in the Pareto front are examined for the trade-offs among the multiple criteria.
A method for optimally synthesizing multicomponent structural assemblies of an alumi-num space frame (ASF) vehicle body is presented, which simultaneously considers struc-tural stiffness, manufacturing and assembly costs and dimensional integrity under a unified framework based on joint libraries. The optimization problem is posed as a simultaneous determination of the location and feasible types of joints in a structure selected from the predefined joint libraries, combined with the size optimization for the cross sections of the joined structural frames. The structural stiffness is evaluated by finite element analyses of a beam-spring model modeling the joints and joined frames. Manufacturing and assembly costs are estimated based on the geometries of the compo-nents and joints. Dissimilar to the enumerative approach in our previous work, dimen-sional integrity of a candidate assembly is evaluated as the adjustability of the given critical dimensions, using an internal optimization routine that finds the optimal subas-sembly partitioning of an assembly for in-process adjustability. The optimization problem is solved by a multiobjective genetic algorithm. An example on an ASF of the midsize passenger vehicle is presented, where the representative designs in the Pareto set are examined with respect to the three design objectives.
Optimal design of launch vehicles is a complex problem which requires the use of specific techniques called Multidisciplinary Design Optimization (MDO) methods. MDO methodologies are applied in various domains and are an interesting strategy to solve such an optimization problem. This paper surveys the different MDO methods and their applications to launch vehicle design. The paper is focused on the analysis of the launch vehicle design problem and brings out the advantages and the drawbacks of the main MDO methods in this specific problem. Some characteristics such as the robustness, the calculation costs, the flexibility, the convergence speed or the implementation difficulty are considered in order to determine the methods which are the most appropriate in the launch vehicle design framework. From this analysis, several ways of improvement of the MDO methods are proposed to take into account the specificities of the launch vehicle design problem in order to improve the efficiency of the optimization process.
A higher-order beam theory is proposed for the analysis of a thin-walled beam with a generally shaped cross section, which consists of straight cross-section edges and is non-uniform along the axial direction. To derive cross-sectional shape functions for the higher-order deformation modes, a new approach is introduced using a set of beam frame models. The distortions with inextensional cross-sectional walls are determined by solving an eigenvalue problem of a beam frame model under inextensional wall constraints. Subsequently, the distortions with extensional cross-sectional walls are evaluated by considering orthogonality with respect to the inextensional distortions. Moreover, the extensional distortions due to the Poisson effect, which is generated due to the uniform axial strain of the rigid-body cross-sectional deformations, are considered. Warpings induced by the inextensional and extensional distortions are consistently defined based on the orders of the tangential displacements of their corresponding distortions. To deal with the varying cross section, three-dimensional displacements at an arbitrary point are interpolated using those at the cross sections of the nodes, where the beam frame analyses are performed. The proposed method is validated by performing static and vibration analyses of beams with varying single- and multi-cell cross sections.
A thin-walled beam finite element with a varying quadrilateral cross section is formulated based on a higher order beam theory. For the calculation of distortions, the beam frame approach, which models the cross section by using two-dimensional Euler beams, is used. Distortions induced by the Poisson’s effect and warpings are analytically derived. Three-dimensional displacements at an arbitrary point of a present beam element can be described by interpolating three-dimensional displacements at the end sections. Straight and curved thin-walled beams with varying cross sections are solved to show the validity of the proposed approach.
At conceptual design stage, automotive body is usually simplified as a frame structure, which consists of thinwalled beams (TWBs). Therefore, the most important issue is to determine the cross-sectional shape of TWBs under the requirement of mechanical properties. However, design engineers mostly depend on their experience or repeated modification to design the cross-sectional shape of TWBs. So this paper presents a rapidly cross-sectional shape design and optimization method to satisfy the demand of mechanical properties and meanwhile minimize the weight of TWB. Firstly, cross-sectional mechanics property formulations are summarized. Especially, the torsional rigidity formulation of three-cell cross section is derived for the first time in this paper. Secondly, the shape optimization model is created to minimize the weight of TWB and improve the mechanical properties, which is solved by genetic algorithm. Moreover, three stamping constraints, draft angle, chamfer radius and assembly, are introduced to promote the cross-sectional shape more practice. Lastly, numerical examples verify the effectiveness of the optimization model and show the application in structural modification of automotive frame.
At conceptual design stage, beam element is extensively used to create the frame structure of automobile body, which can not only archive the accurate stiffness but also reduce much computational cost. However, the stress definition of beam element is very complex so that the stress sensitivity and optimization are difficult to analytically derive and numerically program. This paper presents an solution to this problem and an application in the lightweight optimization design of automobile frame. Firstly, maximal Von Mises stress of rectangular tube is calculated by using the superposition of stress, which is together induced by the axial force, bending moments, torsional moment and shear force. Secondly, the sensitivity of Von Mises Stress with respect to size design variables: breadth, height and thickness are derived, respectively. Thirdly, an optimal criterion is constructed by Lagrangian multiplier method to solve the frame optimization with stress constraints. Lastly, numerical example of car frame proves that the proposed method can guarantee the stress of each beam element almost fully reaches at the yielding stress.
In natural environment, many biological structures are tubular and exhibit excellent mechanical properties that can reduce self-weight effectively and transport more water and nutrients, such as bamboo. In this paper, the structure of bamboo was introduced to increase the axial and lateral energy absorption of thin-walled tubes by using bionic design method. Energy absorption ability of bamboo was tested by drop-weight experiments. The results showed that the energy absorption was excellent due to the gradient distribution of vascular bundles, nodes and density. These advantages of the bamboo make it possible to design of bionic structure which composed of 1 bionic node and 3 bionic inner tubes with 18, 9 and 4 bionic elements in each inner tube. Numerical examples of bionic structures under axial/lateral impacts were solved with nonlinear finite element method (FEM). The results indicated that the bionic design enhances the specific energy absorption (SEA) of tubes. Thus, the bionic structure is exactly excellent energy absorption under lateral/axial impact and can be used in the future.
Abstract The energy absorption characteristics and material parameters of the bamboo species Phyllostachys pubescens are studied by drop-weight and dynamic tensile tests. The dynamic tensile test shows that 1-year-old bamboo has the greatest tensile strength (251 MPa), which is 1.85 times that of a 2A12 aluminium alloy. The drop-weight test shows that the energy absorption of nodal samples is greater than that of the internode samples, and the specific energy absorption (SEA) values of the nodal and internode samples are 11.85 kJ/kg and 9.78 kJ/kg, respectively. The SEA of nodal samples is close to that of aluminium alloy and steel tube, and the SEA of the internode samples is close to that of copper tube. The energy absorption increases with increasing moisture content and decreases with growth age. Analyses show that the bamboo nodes and the vascular bundle are the main factors affecting bamboo energy absorption. The results indicate that bamboo is a tubular structure with excellent mechanical and energy absorption properties.
This paper presents a new one-dimensional theory for static and dynamic analysis of thin-walled closed beams with general cross-sections. Existing one-dimensional approaches are useful only for beams with special cross-sections. Coupled deformations of torsion, warping and distortion are considered in the present work and a new approach to determine sectional warping and distortion shapes is proposed. One-dimensional C0 beam elements based on the present theory are employed for numerical analysis. The effectiveness of the present theory is demonstrated in the analysis of thin-walled beams having pillar sections of automobiles and excavators.
A semianalytical procedure is presented for the axial buckling of elastic columns with step-varying profiles. Profiles with continuous variations can be approximated, to any desired degree of accuracy, by a series of step variations. The formulation leads to a general procedure that can be applied to any continuous or discontinuous profile regardless of the number of steps. Since the step changes of the profile are represented by distributions, the differential equation cannot be solved in the ordinary sense. The differential equation is therefore converted to an integral equation. The solution of the integral equation is obtained by polynomial functions. The formulation involves a significant amount of algebraic manipulations. This problem is alleviated by using a symbolic manipulation system to carry out the algebraic manipulations. The integral kernels are obtained for the different boundary conditions. Formulas for buckling loads for members with variable profiles and different boundary conditions can be obtained in terms of the section and profile parameters. The example problems present the solution of some common columns with variable cross sections. The procedure can be used to derive the elastic and geometric stiffness matrices for beam columns with variable cross sections as well as for other elements.
At the conceptual design stage, automobile body is evaluated by simplified frame structure, consisting of thin-walled beams (TWBs). In the automobile practice, design engineers mostly rely on their experience and intuition when making decisions on cross-sectional shape of TWBs. So this paper presents a cross-sectional shape optimization method in order to achieve a high stiffness and lightweight TWB. Firstly, cross-sectional property formulations is summarized and reviewed. Secondly, we build up a shape optimization model to minimize the cross-sectional area and satisfy the stiffness and manufacturing demands. The objective and constraints are nonlinear polynomial functions of the point coordinates defining the cross-sectional shape. Genetic algorithm is introduced to solve this nonlinear optimization problem. Thirdly, object-oriented programming and design patterns are adopted to design and implement the software framework. Lastly, numerical example is used to verify the presented method. This software, “SuperBeam” for short, is released for free and does speed up the conceptual design of automobile body.
End effects due to sectional deformations of thin-walled beams with closed cross section are analysed by a one-dimensional theory. In particular, end effects associated with warping (out of plane m otion) and distortion (in plane motion) are investigated. The exact solutions as a vector form are newly derived to reveal slowly-decaying nature of the end effects in a thin-walled beam loaded by a couple. Several examples of thin-walled beams under various loading conditions indicate that the local end effect zone due to warping and distortion is approximately ten times the typical beam width.
Approximate formulas are derived for upper and lower values of the three‐lowest natural frequencies for transverse vibrations of cantilever bars of variable cross section. For cones, truncated cones, wedges, and truncated wedges, as well as bars of constant cross section, the effects of rotatory inertia and shear are taken into account also and compared with the results obtained by neglecting these effects according to the elementary Bernoulli‐Euler's theory of bending of beams. Numerical results for the first three natural frequencies are presented in tabular form for different inertia characteristics.
This paper concerns the formulation of a new finite element method for the solution of beams with variable cross section. Using only one element, it is possible to derive the exact static and dynamic stiffness matrices (up to the accuracy of the computer) for any polynomial variation of axial, torsional, and bending stiffnesses along the beam. Examples are given for the accuracy and efficiency of the method.
The Timoshenko beam model incorporates the effect of shear deformations and rotary inertia in the vibration response of beams. For constant cross-section beams it was shown that the effect is dependent on the aspect ratio of the beams, and for beams with large ratio the effect is small. In this work exact vibration frequencies for variable cross-section beams are found using the dynamic stiffness matrix approach. An example is given and compared with known results of beams with various taper ratios.
Commercial optimization software development has a different set of goals and constraints than the development of academic
or industrial research codes. Commercial codes must be all things to all people. They must contain a wide range of analysis
options and be able to handle large, real world, industrial analysis models. As most of the users of the software in industry
come from analysis, rather than design optimization backgrounds, the codes must perform in a robust manner. Inconsistent input
data must be detected. Optimization methods must be automatically chosen by the program. Optimization parameters need to be
adjusted automatically by the program. Another very important aspect is ease of use. A very intuitive and easy to use GUI
(Graphical User Interface) should be developed. This work describes some of the development objectives and concerns that are
essential to the development of commercial optimization software products.
In this study, an adaptive space mapping technique based on response of objective was suggested for solving practical engineering
problems. Response surface methodology was engaged in approximation of objective and constraint functions based on coarser
model. The fine simulation model is not only used for correction of the coarser simulation model and validation of final solution
but also for applied construction of space mapping expression. Finally, genetic algorithm (GA) was used to optimize updated
metamodel according to coarse model. The proposed method combines the space mapping technology based on response of coarse
model and modification of design of experiment. It guarantees that metamodel based coarser model is stepwise updated in the
right searching direction. For demonstrating practicability of developed method, it was applied for optimization of geometric
parameters of addendum surface, blank holder force and drawbead restraining force in sheet forming problems. It was confirmed
that the corresponding problem can be optimized successfully in remarkably short computing time by proposed optimization method.
The majority of analyses of thin-walled beam-columns in the linear and nonlinear ranges of structural response have been for prismatic sections. This paper presents a theory for the nonlinear axial strain and Kirchhoff stress resultants for a thin-walled beam-column whose cross-section is tapered. An expression for the first variation of the Total Potential is derived, that may be used in a nonlinear equilibrium analysis, and an expression for the second variation of the Total Potential is derived, that may be used in a stability analysis. These variations are used as the basis for a finite element analysis, as described in the companion paper. The results are discussed in light of a previous study of tapered monosymmetric I-beams, and for a linear analysis it is shown that the results of this independent study agree with those presented in this paper.
Structural optimization of a single product is in this paper extended to optimization of a family of products where size, shape, and topology optimization are accomplished simultaneously. The design variables are either variables used only locally in a product or variables globally shared among all or several of the other products in the family. A mathematical formulation for optimization, based on classical performance and cost measures such as compliance and mass, is given. A comprehensive product family optimization application is discussed, where three products exposed to seven load cases each, are studied. These are existing products. The split into modules is governed by the true commonality between components used for manufacturing of these products. An accurate mass distribution in the products has been found to be of great importance to be able to simulate the different load cases as correctly as possible. The methodology described in this paper can serve as a tool for investigations of what the reduction of some typical optimal properties of a specific product will be if some components have to be shared by other similar products. Another important aspect of this methodology is that the optimal topology and load transfer through a single product can be compared to what is optimal if the product has to be a member of a product family.
In this study, the previously developed adaptive response surface method (ARSM) is suggested for construction of metamodel for highly non-linear responses. In order to develop the accuracy and efficiency of metamodel, the particle swarm optimization intelligent sampling (PSOIS) scheme is developed. This kind of intelligent method can guarantee the sampling search in right direction and constraint the bounds of design variables in feasible region. For validation of developed method, the Rosenbrock function is successfully approximated by proposed method; corresponding metamodel appropriateness can be well predicted by analysis of variance (ANOVA). Metamodel by ARSM with PSOIS are employed for optimization of initial blank shape and blank hold force (BHF) in sheet forming process, with validations by finite element simulations using LSDYNA970 commercial code. The results show that developed method is able to produce remarkable metamodels for highly non-linear problems with multi-parameter.
This work gives exact solutions for the buckling loads of variable cross-section columns, loaded by variable axial force, for several boundary conditions. Both the cross-section bending stiffness and the axial load can vary along the column as polynomial expressions. The proposed solution is based on a new method that enables one to get the stiffness matrix for the member including the effects of the axial loading. The buckling load is found as the load that makes the determinant of the stiffness matrix equal zero. Several examples are given and compared to published results to demonstrate the accuracy and flexibility of the method. New exact results are given for several other cases.
At the conceptual design stage, simplified finite element (FE) model of body-in-white (BIW) structure focuses on its specific merit to provide early-stage predictions for detailed FE model of that. This paper exploits a semi-rigid beam element (SRBE) that consists of a beam element with two semi-rigid connections at the ends to simulate the flexibility of joint. Guyan reduction method condenses the SRBE as a super element. A special finite element software for structural modeling and analysis of BIW (SMAB) is developed in .NET framework. The Unified Modeling Language is employed to depict the classes and their relationship. The design patterns are identified and applied in the framework design to facilitate communication and system expansion. Microsoft DirectX and GDI+ implement graphics display of spatial BIW frame and planar thin-walled cross section. Based on multi-threaded technology in .NET, subspace iteration method is paralleled to speed up the mode analysis. As a result, the efficiency of the SRBE is demonstrated by a benchmarking automotive body. Multi-threaded parallel is effective and useful, especially for frequency optimization.
This paper discusses a new structural optimization method, based on topology optimization techniques, using frame elements
where the cross-sectional properties can be treated as design variables. For each of the frame elements, the rotational angle
denoting the principal direction of the second moment of inertia is included as a design variable, and a procedure to obtain
the optimal angle is derived from Karush–Kuhn–Tucker (KKT) conditions and a complementary strain energy-based approach. Based
on the above, the optimal rotational angle of each frame element is obtained as a function of the balance of the internal
moments. The above methodologies are applied to problems of minimizing the mean compliance and maximizing the eigen frequencies.
Several examples are provided to show the utility of the presented methodology.