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Finite Elements in Analysis and Design 45 2009 456 462 Contents lists available at ScienceDirect FiniteElementsinAnalysisandDesign journal homepage Simplified modelling of joints and beam like structures for BIW optimization in a conceptphaseofthevehicledesignprocess D Mundoa R Hadjitb S Dondersb M Brughmansb P Masb W Desmetc aDepartment of Mechanical Engineering University of Calabria 87036 Arcavacata di Rende Italy bLMS International Interleuvenlaan 68 B 3001 Leuven Belgium cDepartment of Mechanical Engineering Katholieke Universiteit Leuven Division PMA B 3001 Leuven Belgium A R T I C L EI N F OA B S T R A C T Article history Received 3 March 2008 Received in revised form 3 December 2008 Accepted 10 December 2008 Available online 7 February 2009 Keywords Beam Joint Conceptual design NVH Vehicle body The paper proposes an engineering approach for the replacement of beam like structures and joints in a vehicle model The final goal is to provide the designer with an effective methodology for creating a concept model of such automotive components so that an NVH optimization of the body in white BIW can be performed at the earliest phases of the vehicle design process The proposed replacement method ology is based on the reduced beam and joint modelling approach which involves a geometric analysis of beam member cross sections and a static analysis of joints The first analysis aims at identifying the beam center nodes and computing the equivalent beam properties The second analysis produces a simplified model of a joint that connects three or more beam members through a static reduction of the detailed joint FE model In order to validate the proposed approach an industrial case study is presented where beams and joints of the upper region of a vehicle s BIW are replaced by simplified models Two static load cases are defined to compare the original and the simplified model by evaluating the stiffness of the full vehicle under torsion and bending in accordance with the standards used by automotive original equipment manufac turer OEM companies A dynamic comparison between the two models based on global frequencies and modal shapes of the full vehicle is presented as well 2009 Elsevier B V All rights reserved 1 Introduction In highly competitive markets design engineers face the chal lenging problem of developing products which must fulfil complex and even conflicting design criteria In the field of automotive indus try the task of improving various functional performance attributes such as safety noise and vibrations ecological impact etc is made more and more difficult by the necessity of launching new products or renewing existing models in an increasingly short time frame In order to make the complexity of the design criteria compatible with the necessity of reducing the time to market predictive computer aided engineering CAE methods must be already available in the early phases of the design process Traditional computer aided design CAD software packages have a very limited applicability in early design stages since they require detailed data of the vehicle Besides they are based on the traditional definition of geometry via points lines and surfaces thus making Corresponding author Fax 39984494673 E mail address d mundo unical it D Mundo 0168 874X see front matter 2009 Elsevier B V All rights reserved doi 10 1016 j finel 2008 12 003 the parameterization of models difficult and time consuming 1 As a result the experience of engineers is a key factor for the selection of proper structural concepts at the beginning of the design process Recently research efforts have been spent to enable designers to use CAE as a support in the conceptual phase of the design process when functional performance targets are defined while detailed ge ometrical data are still unavailable The objective is to improve the initial CAD design hence shortening the design cycle 2 5 In the field of NVH and crashworthiness prediction several con cept modelling approaches have been proposed by researchers They can be classified into three categories methods based on predecessor FE models methods from scratch and methods concurrent with CAD Methods belonging to the first category which includes mesh morphing and concept modification approaches 6 8 are used to de sign a variant or incremental improvement of an existing vehicle model By using a predecessor FE model early CAE predictions can be performed to identify issues and to include possible countermea sures already in the initial CAD design If a new car concept is to be designed and a predecessor FE model is not available methods from scratch can be used to support the design process during the early design phases Two classes of meth ods are distinguished The first class is topology design optimization D Mundo et al Finite Elements in Analysis and Design 45 2009 456 462457 where material is eliminated from an initial admissible design domain in order to make the structure lighter without violating functional requirements 9 11 Performance based on the opti mized topological information is usually improved by optimizing shape and size The second class of methods from scratch known as functional layout design aims at building a simplified concept model consisting of beams joints and panels which represents the functional layout and which is used to predict the performance of the model 12 Methods concurrent with CAD are CAE tools available in an early phase of the design process These methods provide simulation re sults as soon as component level CAD models are available while vehicle level models are still unavailable 13 Among the methods based on predecessor FE models the reduced beam and joint modelling approach has been recently proposed by Donders et al 14 to improve the fundamental NVH behavior of a vehicle BIW The proposed approach creates a reduced modal model at the beam center nodes to which beam elements and joint superelements can be added thus enabling a concept modification of the body and an accurate prediction of dynamic NVH performance The commercial software program LMS Virtual Lab 15 includes a user friendly implementation of the reduced beam and joint modelling approach Design engineers can define a beam and joint layout calculate the body reduced modal model and perform efficient design modification and optimization of the body beam like sections and joint connections In this paper the reduced beam and joint modelling approach is employed to replace beams and joints of the predecessor FE model with concept models After identifying the beam center nodes as the geometric center of the cross sections the equivalent beam properties are calculated through a geometric analysis and applied to simplified beam elements that connect the beam center nodes The stiffness parameters of thin walled beams as computed by means of a geometric approach need a correction that takes into account section variations and discontinuities holes spot welds stiffeners 16 17 For this purpose proper correction factors are defined and estimated for each beam member by means of an iterative model updating procedure In a next step a simplified model of joints connecting two or more beam members is then obtained through a static reduction of the detailed FE model of the joint In order to validate the proposed approach a case study is pre sented in which beams and joints of the upper region of a vehicle BIW are replaced by simplified models A static comparison between the original and the simplified model is performed by evaluating the static stiffness of the full FE vehicle BIW under torsion and bend ing A dynamic comparison between the two models based on the global frequencies and mode shapes of the full vehicle is performed as well 2 The reduced beam and joint modelling approach The reduced beam and joint modelling approach is proposed by Donders et al 14 for efficient modification of beams and joints of a vehicle based on the reduced modal model of the nominal vehicle The basic idea is to identify the so called beam center nodes and to create a reduced modal model at these beam center nodes Subsequently the mass and stiffness properties of the structure are modified by connecting the beam center nodes through simple beam elements and joint superelements In this paper simplified beam and joint models are created to completely replace the original FE model without the necessity of the reduced modal model so that an optimization of the vehicle can be performed in the early phase of conceptual design when a detailed model of the structure is not yet available yi zi xi x B C N z y Fig 1 Schematic representation of a beam end section In this section an overview of the procedure that is used to es timate the mass and stiffness properties of the simplified beam and joint models is provided 2 1 Beam property estimation Beam like members i e structures for which the dimension in the longitudinal direction is much larger than the characteristic di mension of the cross sectional area are the primary structural ele ments in a BIW They strongly influence the natural frequencies of the vehicle body In the FE model of a vehicle beam like members are typically thin walled structures formed by shell elements In order to replace the detailed mesh of such components by sim plified beam elements a number of beam cross sections are consid ered and the equivalent beam properties are computed for each of them For this purpose the following procedure is implemented 1 a cut node is selected in the region of the beam member where an intersection plane is to be applied 2 an axis system that defines the approximate beam direction and intersection plane is defined 3 the primary member s shell elements along the intersection plane are cut and analyzed to locate the beam center node in the geometric center of the original cross section 4 the following equivalent beam properties w r t the beam center node are computed A cross section area Ixx torsional moment of inertia Iyy Izz moments of inertia of area and Iyz product of inertia of area Here x denotes the beam direction and the y z plane is the intersection plane as shown in Fig 1 For an arbitrary cross section the calculation of the properties can be implemented by computing the equivalent beam properties for each shell element that belongs to the cross section according to the local principal axes xi yi zi Then a transformation from the local axis system to that of the intersection plane x y z is performed Finally a summation over all shell elements is performed to find the global properties for that cross section 5 the beam center node is connected to the surrounding mesh by means of interpolation relations Nastran superelements RBE3 These relations are defined between each beam center node and a particular node group formed by all nodes of the shell ele ments that are defined at the intersection plane at the consid ered cross section Typically along each primary beam member a number of intersec tion planes are defined for which equivalent beam properties are computed The entire beam member can then be represented as a series of linear beam elements taken from a standard FE library An example is shown in Fig 2 where both the original detailed and the simplified FE model of B pillars of a vehicle BIW are represented 458D Mundo et al Finite Elements in Analysis and Design 45 2009 456 462 Fig 2 a Original and b conceptual models of a BIW B pillars Fig 3 Original FE model of a joint group extracted from the vehicle model for static reduction 2 2 Joint property estimation Complementary to the simplified beam modelling approach de scribed in Section 2 1 a procedure for simplifying joints connecting beam like structural members in a vehicle body is proposed After evaluating the equivalent beam properties of all beam members con nected by the joint a joint group is created that includes the inter polation elements to the beam center nodes at the joint ends 15 In Fig 3 an example is shown in which the mesh of the joint that con nects the left B pillar of the vehicle to the roof rails is extracted from the rest of the vehicle body For this isolated joint model Guyan re duction is used to calculate a small sized representation of the joint Guyan reduction 18 also known as static condensation is a method to reduce the finite element stiffness and mass matrices of structures For an arbitrary structure the basic static FE matrix equation is given by K x F 1 where K is the stiffness matrix F and x are the force and the dis placement vectors respectively By identifying ntboundary degrees of freedom DOFs which must be retained in the solution and no internal DOFs which are to be removed by static condensation the system of Eq 1 can be partitioned as follows K oo Kot KtoKtt x o xt F o Ft 2 where subscripts t and o are used to designate the boundary and the internal DOFs respectively From the first line of Eq 2 the internal displacement vector can be determined as xo K 1 oo Fo Kot xt 3 By introducing the static reduction matrix Got K 1 ooKotand substi tuting Eq 3 into the second line of Eq 2 the following equation is obtained Ktt red xt Ft red 4 where Ft red Ft GT otFo is the reduced loading vector while Ktt red KtoGot Kttis the ntx ntreduced stiffness matrix Physically this matrix represents the stiffness values between each pair of boundary DOFs This way the stiffness of the structure has been condensed to the boundary DOFs The same transformation can be used to condense the mass ma trix on the boundary DOFs to obtain a reduced system also for dy namic analyses However while exact for the stiffness matrix the Guyan reduction is an approximation for the mass matrix By re ducing the mass matrix it is assumed for the considered structure that inertia forces on internal DOFs are less important than elastic forces transmitted by the boundary DOFs This is true for very stiff components or in cases where local dynamic effects can be ignored Therefore the accuracy of the result is case dependent For each isolated joint model a Guyan reduction is performed with the DOF of the joint s end nodes i e beam center nodes as the boundary DOFs to be retained in the solution The FE model of the joint is thus reduced to a small superelement consisting of a reduced stiffness and mass matrix For typical automotive joints the stiffness relations between the end points of the joint have a much stronger influence on the global body behavior than the exact distribution of mass on the joint For this reason Guyan reduction of the joint structure to its joint end nodes i e beam center nodes seems an appropriate choice to create a small sized representation of the actual joint 14 D Mundo et al Finite Elements in Analysis and Design 45 2009 456 462459 3 Case study 3 1 Model description Fig 4 shows an industrial BIW model consisting of 123 panels that are modelled with linear shell elements The constituent panels are assembled by means of about 3000 spot weld connections 19 which are represented in the FE model by means of Hexa solid ele ments 15 Inordertovalidatethereducedbeamandjointmodelling approach as described in the previous section a group of beam like structures labelled in Fig 4 as B1 B5 are selected and replaced by equivalent simple beams In total 10 beams are selected for the replacement namely the A and B pillars and the longitudinal and transversal roof rails Four joints symmetrically arranged w r t the longitudinal plane of the vehicle connecting these beams are labelled in Fig 4 as J1 J2 J3 and J4 are statically reduced Fig 5 shows the simplified BIW model where the detailed shell models of the beam like structures have been replaced by simple two node beam elements The number and length have been selected based on the geometric characteris tics i e length and cross section variations of the original mesh The original FE joint models have been removed from the BIW FE model and the joints have been represented by static superelements i e the equivalent mass and stiffness matrices of each joint 3 2 Static comparison To validate the proposed approach static and dynamic indica tors of the full vehicle performance are considered These indicators are evaluated for both the original BIW model and the simplified or conceptual model To assess the static behavior the torsional and bending stiffness of the BIW are calculated The body is clamped at the rear suspensions while static vertical forces are applied at the Fig 4 Original FE model of the BIW Fig 5 Conceptual FE model of the BIW The original meshes of 10 beam members and four joints are replaced by simplified beam elements and joint superelements Fig 6 Static load cases defined to estimate the BIW stiffness under a torsion and b bending front suspensions points A and B in Fig 6 Based on the estimation of the vertical displacements vAand vBat the excitation points the bending and torsion deflection angles band tare determined as b arctan v A vB 2L 5 t arctan v A vB W 6 where L and W denote the wheelbase and the width of the car respectively measured at the front suspension points Based on the torsional deflection t the torsional stiffness Ktis determined as Kt M t 7 where M F W is the moment applied at the front suspension resulting from two oppositely oriented forces F Similarly the bending stiffness Kbis determined from bas Kb 2FL b 8 where F is the vertical force applied at the frontal suspension loca tion The stiffness properties of the BIW are estimated for both the models in Figs 4 and 5 by performing a static FE analysis Nastran Sol 101 with both models In Table 1 the torsional and bending stiffness indicators are listed as well as the approximation involved by the simplified model w r t the original model The results show that the bending stiffness of the original vehicle model is accurately predicted by the model with the replaced simplified beams and joints while a significant discrepancy between t