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Finite Elements in Analysis and Design 39 (2003) 433–443www.elsevier.com/locate/�nel

On optimization of a car rim using �nite element methodH. Akbulut

Department of Mechanical Engineering, Atat�urk University, 25240 Erzurum, Turkey

Abstract

This paper presents the optimization of an octopus-type car rim for which critical zones were found �rstand then optimum thickness was investigated using an elasto-plastic analysis. In this study, three-dimensional�nite element method was used for conducting elasto-plastic analysis. In the �nite elements analysis, theelements forming the meshes are hexahedral linear elements with eight nodes. Twelve di.erent meshes wereused. A quadrant of the rim was utilized due to its symmetric shape. The theoretical results were comparedwith experimental ones. It seems that the theoretical results are in agreement with the experimental ones. Theresults are presented in tabular and graphical forms. ? 2002 Elsevier Science B.V. All rights reserved.

Keywords: Rim; Finite element method; Elasto-plastic analysis; Optimization

1. Introduction

As part of technological improvement, comfort and safety have become essential demands ofhuman beings. This comes not only from market-oriented competition but also from legislation thatmay seek some certain standards. Cars, important favours of technological development, are widelyused in daily life. It seems that mankind no longer lives without them. Therefore, these wonderfulmachines should be safe and economical so that people could use them safely and more people couldpurchase them. Since rims, on which cars move, are the most vital elements in a vehicle, they mustbe designed carefully. The rim type examined in this study has some trouble when touching anycurb or entering a sharp curve. The rims manufactured by various methods are made of either steelor cast aluminium alloys. In particular, rims made of aluminium casting alloys are more preferredbecause of the weight and the cost e.ectiveness.Each part of a vehicle is certainly important for various reasons, such as safety and cost e.ective-

ness. Therefore, many works on di.erent parts of vehicles have been carried out until now. However,to the best of the author’s knowledge, there is no paper in the open literature on the optimizationof the rim and the present paper attempts to �ll this gap. For example, Rusin and his group [1]

E-mail address: [email protected] (H. Akbulut).

0168-874X/03/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved.PII: S 0168-874X(02)00091-4

434 H. Akbulut / Finite Elements in Analysis and Design 39 (2003) 433–443

studied railroad car wheel by using rectangular elements, while BagcB[2] investigated rotating wheel.Levis examined diesel engine piston by using three-dimensional �nite element method (FEM) [3].Weisheng et al. investigated S-shape web plate wheel by FEM [4]. In that study, the structure char-acteristics of S-shape web plate wheel are described. Seireg [5], Berger [6], Bhavikatti [7] and Ray[8] worked on optimum designs of di.erent rotating discs. Ogut [9] studied another kind of car rim.But in that study, only stress analysis was carried out. In another study [10], a design optimizationprocess for elasto-plastic material behaviour of laminate composite structures, made of thermoplasticresins, was described, considering two optimization levels: (1) the geometric linear behaviour underelasto-plastic loading conditions and (2) the change of the ply thickness of the plates or shallowshells and the height and width of the reinforcement beams to structure weight minimization, underthe constraints of maximum allowed displacement or maximum strain–stress level related with theamount of plastic zone, without structural plastic collapse or geometric instability in plastic loadingconditions.Also, there are some studies of elasto-plastic analysis, which contribute to this study. Two- and

three-dimensional �nite elements were used for three elasto-plastic analyses of composite materialsand structures. These studies were carried out for thick and thin plates and shells [11–13]. Guoet al. [14] developed a simpli�ed eHcient FEM called the inverse approach (IA) to estimate thelarge elasto-plastic strains in thin metallic panels obtained by deep drawing. In that paper an iterativescheme was used to �nd the original position of each material point in the initial Iat blank afterwhich it is possible to estimate the strains and stresses in the �nal workpiece. Also, Toparli et al.[15] studied the e.ect of the residual stresses on the fatigue crack growth behaviour at fastener holesusing FEM with the initial stress method.The primary objective of this study is to carry out the optimization of a car rim by using

elasto-plastic analysis with initial stress method. In the solution of the problem, FEM, whose meshesinclude hexahedral linear elements with eight nodes, was used. The load applied to the rim wasassumed to be steady, that is, time independent. In the future, a study with impact load will becarried out using the same model. For analysis, a quarter of the rim was utilized. The elements inthe critical zones were taken to be smaller than others in order to obtain stresses more accurately.

2. Elasto-plastic stress analysis

Various computational procedures are used with success for a limit range of elasto-plastic problemsutilizing FEM. Here, after the sti.ness matrix of the structure was obtained applying boundaryconditions, the system equation based on the minimum potential energy principle was solved byusing Gauss elimination method, which results in the element nodal displacements. By means ofthese displacements, the stresses in the nodal points were calculated. At each nodal point, there aresix stress components.Elasto-plastic stress analysis was carried out by using initial stress method, an e.ective method

developed by Zienkiewicz [12]. This method (also known as modi�ed Newton–Raphson method)is illustrated in Fig. 1. In this �gure, the empirical equation proposed by Ludwik is used in theelasto-plastic region [16]:

�f = �0 + K�np; (1)

H. Akbulut / Finite Elements in Analysis and Design 39 (2003) 433–443 435

Fig. 1. Modi�ed Newton–Raphson method (initial stress method).

where �f is the elasto-plastic stress and �p (or �) is the plastic strain corresponding to the �f, and�0 (=�y) is the yield stress, K and n are the hardening parameter and the strain-hardening exponent,respectively. By using of the elasto-plastic stress analysis, it is determined whether the nodal pointsare included in the plastic zone, comparing the equivalent stresses with the material yielding stressobtained from experiments. If the nodal points are included in the plastic zone, �01 from Fig. 1 canbe obtained as below:

�01 = �1 − �f1: (2)

Then, by adding �01 to �1, the increasing stress value �2 can be obtained as

�2 = �1 + �01: (3)

In a general form, it can be rewritten as

�n = �1 + �0n−1 ; (4)

where the stress �n corresponds to �n in the elasto-plastic region. As an illustration, at the strainpoint �2, the stress di.erence between �2 and real stress results in �02. The stress �3 can be obtainedby replacing �02 with �01 in Eq. (3). The following analogue iteration steps lead to the pointcorresponding to the elasto-plastic strain �n and stress �1, where �0i is the initial stress.For an iteration step, the loading stress components occurring during the deformation are as

{�}= {�x �y �z �xy �yz �xz}T: (5)

By using {�}, the initial stress can be calculated as below:{�0}= {�}�0

�; (6)


where � is the equivalent initial stress which is calculated according to the von-Mises yield criterionas follows [16]:

� =1√2

√(�x − �y)2 + (�y − �z)2 + (�z − �x)2 + 6(�2xy + �2yz + �2xz) (7)

and �0 is the equivalent initial stress, and it is calculated by using the following formula:

�0 = � − �f; (8)

where �f is obtained from �–� diagram relating to uniaxially loaded tensile specimen. For thethree-dimensional analysis, the initial stress can be given in vector form as follows:

{�0}= {�0x �0y �0z �0xy �0yz �0xz}T; (9)

where �0x; �0y; �0z; �0xy; �0yz; �0xz are the components of the initial stress in the three-dimensional case.The force corresponding to the initial stress is as given below [11]:

{F}�0 =∫[B]T {�0} dV; (10)

where [B] is the strain–displacement transformation matrix, and V is the volume. For the elasto-plasticanalysis, �rst, the solution displacement vector {�1} is calculated from {F}�01 as follows:

{�}1 = [K]−1({F}�01 − {F}); (11)

where [K] and {F} are the sti.ness matrix and the loading vector, respectively.After the following iteration steps (�i; i= 1; 2; : : : ; n) are computed until there is no or negligible

di.erence between {�}i and {�}i+1, the displacement vector is calculated as

{�}n = [K]−1({F} − {F}�0n): (12)

Finally, the elasto-plastic stress {�f}n corresponding to {�}n is calculated as

{�f}n = [C][B]{�}n − {�0}n; (13)

where [C] is the material property matrix.In the calculation of the residual stresses {�}res [15], the following equation is used:

{�}res = {�f}n − {�}el; (14)

where {�}n are the elasto-plastic stresses given above, and {�}el are the elastic stresses correspondingto {�}1 resulting from the external load P which is applied to the rim [17]. The components ofresidual stress can be given in vector form as

{�res}= {�rx �ry �rz �rxy �ryz �rxz}T: (15)


Fig. 2. Three-dimensional view of the rim.

3. De�nition of the problem

In this paper, an octopus-type car rim, shown in Fig. 2, was studied. Its dimensions are given inFigs. 6 and 7. Three-dimensional FEM was exploited to determine the stress distribution within therim body. The elements used in the mesh are hexahedral linear elements with eight nodes consistingof three degrees of freedom.A quarter of the rim consisting of 269 elements and 529 nodes is suHcient. The analysis of the

whole or half-rim is unnecessary. It is assumed that the displacements in the y–z plane approachzero because the nodes in this plane are too far from the nodes on which forces were applied. Theapproach described above was veri�ed with the experiments.In Figs. 4–7, the three-dimensional �nite element models of the structure are presented in various

viewpoints. Twelve di.erent meshes, whose element and node numbers were the same but thicknesseswere di.erent, were investigated. However, in this paper, four of them were taken into considerationbecause the results found from these four meshes comprised the others. To obtain the optimum rimshape, its strong parts was gradually diminished. The decrease in the amounts of thicknesses of therim are given in Table 3. Its strong parts determined according to the stress analysis case in theoriginal mesh are indicated by the lines in Figs. 6 and 7.Fig. 4 also illustrates both boundary conditions and loading case of the rim. Here, u; v and w are

the displacements in the directions x; y and z, respectively. The boundary conditions were appliedto both the plane x–z (v = 0) and the plane y–z (u = w = 0). The body was loaded convenientlywith the testing loading. The load P having the angle 13◦ with z-axis consists of two componentsacting in x and z directions, which is a critical position for the rim (Figs. 3–8).The mechanical properties of the rim material G-Al Si7 Mg in the norm of DIN (see the norms

of A356), an aluminium casting alloy and (Si: 7.0; Mg: 0.5), were determined in a universal In-stron machine by testing the specimens cut out from the rim. These properties are presented inTable 1.


Fig. 3. Testing con�guration of the rim.

Fig. 4. Mesh generation, boundary conditions and loading.

Table 1Properties of the rim material

Material properties

Elasticity modulus, E 70 GPaPoisson’s ratio, � 0.30Yield stress, �y 70 MPaHardening parameter, K 513 MPaStrain hardening exponent, n 0.545Density, � 2700 kg=m3

The testing con�guration of the rim recommended by the producer �rm is presented in Fig. 3,which is the most extreme position during the car movement. Strain measurements were made usingstrain gauges stuck on a few points outside the yielded zone along the line 111-201 on the surface ofthe rim body (Fig. 5). In order to determine the experimental stresses, one-dimensional stress–strain


Fig. 5. The front view of the �nite element model on the x–y plane.

Fig. 6. The back view of the �nite element model on the x–y plane and the yielding points.


Fig. 7. The view of the �nite element model on the x–z plane and the yielding points.

0

10

20

30

40

50

60

70

80

90

4 6 8 10 12 13 16 18

x (cm)

Equ

ival

ent s

tres

s (M

Pa)

mesh-d

mesh-c

mesh-b

mesh-a

Fig. 8. For four meshes, comparison between material yielding stress and the equivalent stresses occurring at the somenodal points versus the radius along the x-axis.

relationship was used as:

� = �E; (16)

where � and � are the calculated experimental stress and the measured strain along the line 111-201,respectively, and E is the elasticity modulus of the rim material. To be able to compare the exper-imental stresses with theoretical ones, the theoretical stresses calculated at the points on which the


Table 2Volumes and weights of meshes a–d and original rim mesh

Meshes Volumes Weights(cm3) (N)

Original 632.570 66.70Mesh-a 574.109 60.86Mesh-b 564.818 59.84Mesh-c 547.222 57.98Mesh-d 545.762 56.90

Table 3Decrease in the amounts of rim thicknesses along the lines in Figs. 6 and 7

Meshes Lines (mm)

I–I II–II III–III IV–IV V–V VI–VI VII–VII

a 2.5 5 5 3 2 — —b 5 6 5 3 3 — —c 7.5 8 7 3 2.5 2 2d 8.5 9 8 4 3 2.5 2.5

Table 4Residual stress components at the nodal points 192 and 193 for P = 16 kN

Yielded points P = 16 kN

�rx (MPa) �ry (MPa) �rz (MPa) �rxy (MPa) �ryz (MPa) �rxz (MPa)

192 − 80.1 − 26.1 − 20.7 12.0 − 0.6 2.8193 − 67.6 − 23.7 − 18.7 17.1 − 1.8 8.7

strain gauges were stuck were reduced to the line 111-201. The comparison of the stresses computedtheoretically and measured experimentally are given in Table 6.

4. Results and conclusions

In this study, it was found that nodes 192 and 193 yielded �rst when the loading applied to therim was increased up to P = 16 kN. Therefore, it can be certainly said that the neighbourhood ofthose nodes is a critical zone. This zone exists at the joints which connect the rim centre to theouter circle, as shown in Figs. 6 and 7. When the force was increased up to P = 17 kN, plasticdeformation expanded to nodes 162 and 163, as shown in the same �gures (Tables 2–5).In order to optimize the rim body, its thick parts were suHciently weakened with regard to the

lines I–I, II–II, III–III, IV–IV, V–V, VI–VI and VII–VII, shown in Figs. 6 and 7. Fig. 8 showsthe equivalent stresses calculated according to Eq. (7) along the line 21-69-193-421-313 for the


Table 5Residual stress components at the nodal points 162, 163, 192 and 193 for P = 17 kN

Yielded points P = 17 kN

�rx (MPa) �ry (MPa) �rz (MPa) �rxy (MPa) �ryz (MPa) �rxz (MPa)

162 − 60.0 − 20.6 − 17.0 19.2 − 3.3 3.8163 − 62.3 − 22.2 − 21.5 11.0 − 1.5 14.0192 − 88.1 − 28.8 − 22.9 13.3 − 0.6 2.8193 − 74.3 − 26.2 − 20.7 18.8 − 1.9 9.4

Table 6Comparison of the stresses computed theoretically and measured experimentallyalong the line 111-201 for P = 16 kN

Stresses (MPa) 100 mm 120 mm 140 mm

Theoretical 23.70 26.67 34.07Experimental 24.21 27.32 34.35

four meshes. During the calculation, attention was paid to see that the equivalent stresses withinthe rim body did not exceed the yielding stress. Therefore, of the four meshes, only mesh-d isunsuitable because the equivalent stress in the critical zone is higher than the yielding stress andplastic deformation occurs in this zone. But it is clearly seen that mesh-c is the most convenientone. In Table 2, it is seen that the weight of the rim was decreased from 66.7 to 57:98 N, whichmeans a saving of 14.69%.While Tables 4 and 5 present the residual stress components occurring at nodes 192, 193, 162

and 163 for P = 16 and 17 kN, Figs. 6 and 7 illustrate the expansion of the plastic region nearthose nodes. Since these residual stresses in the tables have opposite signs, they caused a consid-erable increase in the strength of the rim. Or instead, for a good design, this region should bestrengthened.Table 6 presents a comparison between theoretical and experimental results at certain points along

the line 111-201 (x = 100; 120and 140 mm). It is seen that theoretical results are very well inagreement with experimental ones. This case proves that the quarter rim is enough for examination.In conclusion, this study shows that stress distribution on rims varies from one region to another.

Based on this type of FEM analysis, one can decide as to which parts are critical, then, can strengthenthose zones. On the other hand, stress distribution may not be that high on some other parts,hence, excess can be removed from these regions to prevent material extravagance. Furthermore, ifthe residual stresses remain in the critical zones of the rim, it should be taken into considerationthat these parts will be more enduring, hence safer. Therefore, this study contributes much to rimdesign concept with the above-mentioned points, which are not possible with conventional rim designtechniques.


References

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[3] D.J. Levis, Taking the stress out of stress analysis using the FEM, CEGB Research, Berkeley, 1985.[4] Z. Weisheng, L. Huiying, Investigation on S-shape web plate wheel, National Conference Publication, No. 92, Part

10, Institution of Engineers, Australia, Barton, Australia, 1992, pp. 225–228.[5] A. Seireg, K.A. Surana, Optimum design of rotating disks, J. Eng. Ind. (1970) 1–10.[6] M. Berger, I. Porat, Optimum design of a rotating disc for kinetic energy storage, J. Appl. Mech. 55 (1988) 164–170.[7] S.S. Bhavikatti, N. Ramakrishnan, Optimum shape design of rotating discs, Comput. Struct. 11 (1979) 397–401.[8] G.S. Ray, B.K. Sinha, Pro�le optimisation of variable thickness rotating discs, Comput. Struct. 42 (1992) 809–813.[9] T. OTgUut, Some examples of application done in Turkey by using ANSYS, MUuhendis ve Makina Dergisi 427 (1995)

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under elasto-plastic loading conditions, Struct. Multidis. Optim. 19 (2000) 50–63.[11] R. Karakuzu, O. Sayman, Elasto-plastic �nite element analysis of orthotropic rotating discs with holes, Comput.

Struct. 51 (1993) 695–703.[12] O.C. Zeinkiewicz, S. Valliappand, I.P. King, Elasto-plastic solutions of Engineering problems with initial stress,

�nite element approach, J. Numer. Methods Eng. 1 (1969) 75–100.[13] D.R.J. Owen, J.A. Figuearias, Anisotropic elasto-plastic �nite element analysis of thick and thin plates and shells,

Internat. J. Numer. Methods Eng. 19 (1983) 541–566.[14] Y.Q. Guo, et al., Recent developments on the analysis and optimum design of sheet metal forming parts using a

simpli�ed inverse approach, Comput. Struct. 78 (2000) 133–148.[15] M. Toparli, A. Ozel, T. Aksoy, E.ect of the residual stresses on the fatigue crack growth behavior at fastener holes,

Mater. Sci. Eng. A 225 (1997) 196–203.[16] A. Mendelson, Plasticity: Theory and Application, Macmillan, New York, 1968.[17] K.J. Bathe, Finite Element Procedures in Engineering Analysis Prentice-Hall, Inc. Englewood Cli.s, NJ, 1982, 07632.

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