JOURNAL OF THEORETICAL AND APPLIED MECHANICS 44, 3, pp. 691-712, Warsaw 2006 NUMERIAL IMPLEMENTATION OF MULTIAXIAL HIGH-CYCLE FATIGUE CRITERION TO STRUCTURAL OPTIMIZATION Mirosław Mrzygłód Institute of Rail Vehicles, Cracow University of Technology e-mail: mrzyglod@mech.pk.edu.pl Andrzej P. Zieliński Institute of Machine Design, Cracow University of Technology e-mail: apz@mech.pk.edu.pl Modern multiaxial high-cycle fatigue criteria were investigated with re- spect to their application in structural optimization procedures coupled with finite element codes. As a result of tests carried out for several fati- gue criteria, theDangVanhypothesiswasused for thedetailednumerical study. Away of respective adapting the high-cycle load history was also suggested. The complete algorithm of the fatigue optimization was illu- strated by applying the proposed procedures to vehicle parts which are subject to high-cycle loadings. The finite element code ANSYS R© was used in the structural modeling. Key words: high-cycle fatigue, multi-axial fatigue criterion, structural optimization, finite element method 1. Introduction The optimization of structures subjected to Multiaxial High-Cycle Fatigue (MHCF) is a very important problem for both aircraft and ground vehicle design. Among problems, which impede development of this area of study, is the lack of a unique hypothesis for theMHCFphenomena. Equally important seems to be the problem of suitable adaptation of an experimental high-cycle load history. It is also necessary to note that the optimization of this type of problems enounters a barrier of long computational times. However, this 692 M. Mrzygłód, A.P. Zieliński last obstacle seems to be less important, considering the rapid development of computing power. Before starting the investigations, certain necessary assumptions were made. They can be divided into two groups, the first of which concerns the range of the analyzed phenomena: • only high-cycle fatigue and infinite fatigue life are taken into considera- tion, • examined hypotheses concern multi-axial loading conditions. The second group takes into account a method of modelling and a level of simplification: • state of stresses and deformations in structures is modelled using the finite element method, • particular components of loads have in-phase or reverse character, • frequency of loads is considerably lower than the first eigen-frequency of studied structures, • inertial effects are not taken into account, • real time load history is transformed into a form admissible for the ana- lysis and optimization, • possible short term violations of the established load regime are accoun- ted for by a suitable safety factor. The above mentioned simplifications are not very restrictive and can some- times be easily weakened, for example, by adding inertial forces to the load history.However, taking into account less important influence of the simplified phenomena on the behavior of a large family of the considered structures, they were not taken into account in the present analysis. 2. Multiaxial high-cycle fatigue criteria Experimental investigations in recent decades resulted innumeroushypotheses of the Multiaxial High-Cycle Fatigue (MHCF). In literature, many proposals of such criteria can be found (Garud, 1981; Ballard et al., 1995; Papadopoulos et al., 1997; Dang Van and Papadopoulos, 1999). From the point of view of applications to numerical optimizations, the most convenient are two groups: criteria based on stress state invariants and criteria using average stresses or Numerical implementation of multiaxial high-cycle... 693 deformations in an elementary volume. For the final selection of a suitable fatigue hypothesis, the following three formulaewere taken into consideration: Sines (1959), Crossland (1956) and Dang Van et al. (1989). The first two be- long to the invariant group and the third one is based on the average value concept.The components of these criteria are easy to obtain fromFEAsoftwa- re. The invariant formulae usually consist of quantities related to hydrostatic and octahedral stresses. The use of these hypotheses allows one to determi- ne the initiation point of fatigue cracks. However, the orientation of potential crackswith these criteria cannot be defined. Sines (1959) andPapadopoulos et al. (1997) analyzed the influence of different combinations of variable bending and torsion stresses on the fatigue life of a structure. On this basis, the for- mulated a criterion including the amplitude of octahedral shear stresses and an average value of normal hydrostatic stresses. The Sines hypothesis shows satisfactory correlations with experimental investigations. It has the following form √ J2,a+κσH,m ¬λ (2.1) where κ and λ are material parameters, J2 is the second invariant of stress deviator tensor J2,a = 1 6 [(σ1a−σ2a)2+(σ2a−σ3a)2+(σ3a−σ1a)2] and σH,m = 1 3 [σ1m+σ2m+σ3m] σia, i = 1,2,3, are amplitude-type principal stresses; symbol a means corre- sponding amplitudes of loads, σim, i=1,2,3 aremean-type principal stresses; symbol mmeans the average (mean) value of loads. A criterion formulatedbyCrossland (1956) andBallard et al. (1995) is very close to the Sines formula. The difference in the approach of both researchers concerns the hydrostatic stresses σH which, according to Crossland, should be represented by their maximum value √ J2,a+κσH,max ¬λ (2.2) where σH,max =σH,a+σH,m Criteria based on average stresses in an elementary volume V take into consi- deration the average value of shear and normal stresses in this volume. Dang Van (1989) and Dang Van and Papadopoulos (1999) formulated his hypo- thesis by observing local plastic deformations on a microscopic scale, on the 694 M. Mrzygłód, A.P. Zieliński level of crystallites. They can initiate micro-cracks even then, when a studied structure remains in themacroscopic scale in a range of elastic strains. Accor- ding to Dang Van, the fatigue damage appears in a definite time, when the combination of local shear stresses τ(t) and the hydrostatic stress σH(t) cuts the borders of an admissible fatigue area. Initially complicated, the calculation method of shear stress amplitude became simplified in a new formof theDang Van criterion (Ballard et al., 1995) max A [τ(t)+κσH(t)]¬λ (2.3) where A is the area of the studied object τ(t)= σ1(t)−σ3(t) 2 σH(t)= 1 3 [σ1(t)+σ2(t)+σ3(t)] Thematerial parameters can, inprinciple, be expressedbydata fromtwohigh- cycle fatigue tests: reversed bending (fatigue limit f −1) and reversed torsion (fatigue limit t −1). For all the criteria λ= t−1. However, κ is different for the particular hypothesis: — Sines κ= 3t −1(σf +f−1) f −1σf − √ 6 (2.4) where σf is the ultimate tensile strength of a material, —Crossland κ= 3t −1 f −1 − √ 3 (2.5) —Dang Van κ= 3t −1 f −1 − 3 2 (2.6) 3. Load time history transformation The Sines and Crossland criteria require calculations of appropriate stress invariants in the whole structure for equivalent average and amplitude loads. The equivalent loads could be determined in different ways. In our case, we decided to base the computations on extreme values of these loads from the available time load history of investigated objects. This would increase the safety factor of the new, improved structures. Dang Van criterion (2.3) proposes the analysis with respect to variable time. For optimization the following pattern of reduction of the load time history was proposed: Numerical implementation of multiaxial high-cycle... 695 • determination of the equivalent amplitude and mean load values in the way described above, • calculation of five load levels, which would be checked in every single fatigue analysis (see Fig.1). Fig. 1. An example of the load time history q(t) (a) transformed to a convenient form for the fatigue Dang Van criterion (b) The whole fatigue analysis of five load cases according to the proposed rules was carried out in an automatic way. 4. Numerical implementation of fatigue criteria For computer calculations, a convenient notion of the von Mises equivalent stress can be introduced to Sines criterion (2.1) 1 √ 3 σeq,a(von Mises)+κσH,m ¬λ (4.1) Regarding (2.4), the Sines equation has now the following form 1 √ 3 σeq,a(von Mises)+ (3t −1 f −1 + 3t −1 σf − √ 6 ) σH,m ¬ t−1 (4.2) Analogically, the Crossland criterion with the vonMises equivalent stress can be written as follows 1 √ 3 σeq,a(von Mises)+κσH,max ¬λ (4.3) and, taking into account material parameters (2.5), it has the final form 1 √ 3 σeq,a(von Mises)+ (3t −1 f −1 − √ 3 ) σH,max ¬ t−1 (4.4) 696 M. Mrzygłód, A.P. Zieliński The numerically convenient form of the Dang Van equation uses the Tresca equivalent stress max A [1 2 σeq(Tresca)(t)+κσH(t) ] ¬λ (4.5) After introduction the material parameters (2.6) max A [1 2 σeq(Tresca)(t)+ (3t −1 f −1 − 3 2 ) σH(t) ] ¬ t −1 (4.6) For the numerical implementation of the criteria, a scripting language APDL of the programANSYS R© (Ansys Inc. 2004) was used. APDL makes it possible to buildmacros with the following functions: • to record current values of stress tensor components and a certain pre- defined quantity (e.g. an equivalent stress), • to calculate the value of the hydrostatical stress σH, • to calculate the material parameters κ and λ, • to calculate the value of the equivalent stresses according to Sines (4.2), Crossland (4.4) and Dang Van (4.6) criteria, • to automatically record the results of analysis in the form of contour maps. Two versions of the fatigue analysis programs were prepared. The first version, for the Sines and Crossland criteria, contained two series of static calculations: for the equivalent average and amplitude loads. From the FEM analysis, the invariants (i.e. the criteria components) were obtained in the whole structure. The second version of the program, for the Dang Van crite- rion, calculated five load cases, according to the accepted assumptions (Fig.1). The whole fatigue program was executed automatically, including generation of the graphical map for each criterion. Full listings of fatigue APDL macros of the Sines, Crossland and Dang Van criteria are enclosed in Appendices A and B. 5. Test of fatigue criteria To compare the fatigue criteria discussed above, certain FEM models were proposed and suitable analysis programs were prepared. The programs we- re introduced into the FE code ANSYS v.8.1. A car front suspension arm (Fig.2a) was chosen as a test example. The Multi-Body Simulation (MBS) Numerical implementation of multiaxial high-cycle... 697 using the program CarDyn 1.0 was conducted for determination of the load history (Walczak, 2003). From these results, the equivalent loads for fatigue analysis were prepared according to the rules assumed in the previous section. Fig. 2. The front suspension arm of a vehicle (a) and its FEmodel (b) A three-dimensional shape of the structure was simplified to a 2D model with varying thickness. The analysis was carried out with the help of the ANSYS SHELL63 element (see the model in Fig.2b). For the assumed material model (f −1 = 260MPa, t−1 = 160MPa, σf = 580MPa), the analyzed fatigue criteria take the following form (com- pare Eqs (2.4)-(2.6)): — Sines σeqv,S =0.58σeq,a(von Mises)+0.224σH,m ¬ 160MPa (5.1) —Crossland σeqv,C =0.58σeq,a(von Mises)+0.114σH,max ¬ 160MPa (5.2) —Dang Van σeqv,DV =0.5σeq(Tresca)(t)+0.346σH(t)¬ 160MPa (5.3) The results of conducted fatigue analysis of the studied structure are presented in the form of contour-maps of equivalent stresses accordingly to the investigated criteria (Fig.3). The contour-map of the Dang Van criterion (Fig.3c) represents the maximum value of the equivalent stress obtained in the five-step load history. As it resulted from the numerical investigations, the Dang Van hypothesis turned out to be the most rigorous formulation. It leads to the most deman- ding requirements ensuring, however, the highest safety level. The Dang Van 698 M. Mrzygłód, A.P. Zieliński Fig. 3. Distribution of equivalent stresses [Pa] according to the Sines, Crossland and Dang Van criteria; SMX –maximal equivalent stress [Pa] hypothesis is used in the next stage of the investigations concerning the opti- mization of a structure working in high-cycle load conditions. For the process of optimization, the method of probabilistic search based on evolutionary algorithmswas chosen and the software package Evolutionary Optimization System(EOS) (Osyczka, 2000)was applied. It shouldbealso no- ted that the parallel computing used by the authors to accelerate calculations presented in this paper (Mrzygłód, 2005) was an effective and very convenient tool in the optimization algorithms. 6. Examples of structural optimization 6.1. Example I A front suspension arm of a car was chosen as the first example of fatigue optimization.The regimeofworkaswell as geometry of thispartwasdiscussed in the previous chapter. The structure presented in Fig.2a was subjected to Numerical implementation of multiaxial high-cycle... 699 variable, high-cycle load conditions. The objective of the optimization was to decrease the mass of the suspension arm by parametric adaptation of the geometry of the object with maintaining its fatigue endurance on the same level. The safety factor for the optimized part was accepted by assuming, as the admissible limit, the extreme equivalent stress obtained in the fatigue test of the initial form of the structure (Fig.3c). Considering the geometrical form of the part as well as the way of its loading, it was possible to simplify the examined problem to two dimensions. This permitted a considerable acceleration of the optimization process with a minor decrease in the quality of the results. As a FEM model of the examined component, the earlier preparedmodel of the fatigue criteria testwas adapted.To the non-parametric fatigue analysis of the suspension arm an option of a parametric change of the geometry was introduced.Moreover, functions of automatic choosing themaximum value of the state parameter were added. This means investigation of five load cases and calculation of five values of the equivalent stresses according to DangVan equation (4.5). The accepted state parameter bound 18Mpa was taken from the preliminary analysis of the original formof the examinedpart (see Fig.3c). The optimization problemwas formulated as follows: • Eight decision variables were considered (four radii of holes in the left side of the suspension arm, position of the first hole on a secant line and, respectively, three distances between edges of the remaining holes, see Fig.4) x= [x1,x2, . . . ,xn] n=8 (6.1) The decision variables were limited by their upper and lower bounds xi ¬xi ¬xi i=1,2, . . . ,8 (6.2) • Onestate variable (themaximumequivalent stressaccording to theDang Van criterion) was considered, with the upper limit max A gi(x)¬ gi i=1 (6.3) • Themass of the arm was chosen as the main objective function W(x). In this example of optimization, a decrease of about 4.11% mass of the examined structure was obtained (in reference to the original FEM model). For a comparative optimization procedure of the considered structure, the von Mises criterion was applied. In this case, a mass reduction of 6.14% was 700 M. Mrzygłód, A.P. Zieliński Fig. 4. Description of design variables in the first optimization example obtained. The results of the optimization processes for the Dang Van and von Mises criteria are compared in Fig.5. A considerably bigger value of the von Mises stress (Fig.5a) in relation to theDangVanequivalent stress (Fig.5b) can be observed. However, it should be noted that the von Mises stress is usually compared to the yield stress limit σy, whereas the fatigue equivalent stress – to the reversed torsion fatigue limit t −1. In Fig.5, a low level of admissible stresses can be noticed. This is due to the large safety factor, whichwas taken from the analysis of the original suspension arm. Fig. 5. Comparison of results of the first optimization example between the Dang Van (a) and vonMises (b) criteria; maps of equivalent stresses [Pa]; SMX –maximal equivalent stress [Pa] Numerical implementation of multiaxial high-cycle... 701 6.2. Example II A more complex object – a rear suspension arm of a car (Fig.6a and Fig.7a) was chosen as a second numerical example. In a vehicle, this element is also subjected to high-cycle load conditions (the load scheme is presented in Fig.4b, see alsoWalczak (2003)). In the example, themass of the structurewas optimized with the Dang Van fatigue equivalent stress as a state parameter. For the examined element, MBS simulations were also carried out to obtain the load history of close to standard work conditions (Walczak, 2003). Based on this simulation, the history of 12 components of loads was determined. Fig. 6. A car suspension systemwith the optimized part (a) and a scheme of loads acting on the optimized suspension arm (b) (Walczak, 2003) The optimized suspension arm is built from two thin sheet draw pieces assembled by welding. Hence, the shell-type finite elements (SHELL63) were usedhere in theFEMmodeling.This allowed us to obtain acceptable results in a relatively small computational time.Using theAPDLscripting language, the parametric FE model of the armwas coded. Because of the limited modeling capabilities of the ANSYS R© program, the original form of the suspension arm had to undergo certain simplifications (Fig.7b). The process of simplification was carried out to obtain results on the safe side for the studied object. The Dang Van fatigue subroutine was added to the FEA program. Also the automatic recording sequences of the state parameter and the objective functionwere included into the textfile.The load timehistorywas transformed according to the assumptions of the Dang Van criterion. The five cases from the equivalent time history (see Fig.1b) were calculated for all twelve load components. Moreover, to find the limit value of the state parameter (the 702 M. Mrzygłód, A.P. Zieliński Fig. 7. The rear suspension arm (a) and its FEmodel with loads and boundary conditions (b) Dang Van equivalent stress), a preliminary fatigue analysis for the original shape of the suspension armwas done. The optimization task was formulated in the following form: • Five decision variables were selected x= [x1,x2, . . . ,xn] n=5 (6.4) Remark: Initially selected twelve (see Fig.8) decision variables were reduced to five during the process of sensitivity analysis (Mrzygłód and Zieliński, 2005). • The decision variables were limited by their upper and lower bounds xi ¬xi ¬xi i=1,2, . . . ,5 (6.5) Numerical implementation of multiaxial high-cycle... 703 • Onestate variable (themaximumequivalent stressaccording to theDang Van criterion) was considered, with the upper limit max A gi(x)¬ gi i=1 (6.6) • Themass of the arm was chosen as the main objective function W(x). Fig. 8. Decision variables initially accepted for the optimization process; variables selected by sensitivity analysis marked with the boxes As a result of the optimization, a decrease of ca 11.3%mass of the studied structure was obtained. Figure 9 presents the initial and optimal forms of the investigated structure. 704 M. Mrzygłód, A.P. Zieliński Fig. 9. Comparison of the initial (upper) and optimal (bottom) geometry of the structure; maps of equivalent stresses [Pa]; SMX –maximal equivalent stress [Pa] 7. Conclusions In this paper, proposals ofmodelling ofmultiaxial high-cycle fatigue phenome- na with the use of the finite element method are presented. As a result of the discussed tests, theDangVan fatigue criterionwas chosen and implemented to investigate the structural optimization process. The proposed strategy of ad- aptation of the time load history considerably accelerated thewhole numerical cycle of the fatigue analysis and optimization. The examples of structural optimization described in thework refer to the use of the fatigue criterion as a constraint (state parameter). However, this is only one of the possible applications. The examples of numerical implemen- tations of MHCF criteria in the ANSYS R© APDL code (enclosed) show the possibility of their easy application to any fatigue numerical analysis. Hence, the proposedmethodology of fatigue analysis and optimization can be used in a wide spectrum of high-cycle fatigue problems. Numerical implementation of multiaxial high-cycle... 705 Appendix A. The ANSYS APDL macro of fatigue analysis according to the Sines and Crossland MHCF criteria !============================== ! ANSYSAPDLmacro of fatigue analysis according ! Sines & CrosslandMHCF criteria ! CracowUniversity of Technology ! MiroslawMrzyglod 2005 (C) ! mrzyglod@mech.pk.edu.pl !============================== /prep7 ! preprocessing /SHOW,WIN32C ! definition of material model ! e.g. : Rm=580e6 zgo=260e6 zso=160e6 ! definition of geometry, FEmesh and boundary conditions finish !============================== !—– ! 1 !—– /solution ! definition of loads: mean values ! e.g. Fk,43,Fx,(forcex m/2) Fk,40,Fx,(forcex m/2) Fk,41,Fy,(forcey m/2) Fk,42,Fy,(forcey m/2) SOLVE finish !============================== /post1 ! postprocessing /CONT,1,25, ! Hydrostatic mean stress etable,m s 1,s,1 ! S 1 etable,m s 2,s,2 ! S 2 etable,m s 3,s,3 ! S 3 706 M. Mrzygłód, A.P. Zieliński SADD,m s12,m s 1,m s 2„, ! S 1 + S 2 SADD,m s123,m s12,m s 3„,! S 1 + S 2+ S 3 ! SigmaH,m= 1/3 * (S 1 + S 2+ S 3) (hydrostatic mean ! stress) sadd,m s h,m s123„(1/3), ! print of contourmap of stress SigmaH,m to screen and ! ! to file /title, SigmaH,m PLetable,m s h, /ui,copy,save,tiff,graph,color,reverse,portrain,yes„ finish !============================== !—– ! 2 !—– /solution FDELE,ALL,ALL ! definition of loads: amplitude ! e.g. Fk,43,Fx,(forcex a/2) Fk,40,Fx,(forcex a/2) Fk,41,Fy,(forcey a/2) Fk,42,Fy,(forcey a/2) SOLVE finish !============================== /post1 /CONT,1,25, ! Amplitude of vonMises eqv. stress etable,a s hmh,s,eqv /title, Sigma EQV,a (vonMises) PLetable,a s hmh, /ui,copy,save,tiff,graph,color,reverse,portrain,yes„ ! Amplitude of hydrostatic stress etable,m s 1a,s,1 ! S 1a etable,m s 2a,s,2 ! S 2a etable,m s 3a,s,3 ! S 3a SADD,m s12a,m s 1a,m s 2a„, !S 1a + S 2a SADD,m s123a,m s12a,m s 3a„,! S 1a + S 2a+ S 3a ! SigmaH,a = 1/3 * (S 1a + S 2a+ S 3a) Numerical implementation of multiaxial high-cycle... 707 sadd,m s ha,m s123a„(1/3), ! Maximal hydrostatic stress ! SigmaH,max= SigmaH,a + SigmaH,m SADD,max s h,m s ha,m s h„, /title, SigmaH,max PLetable,max s h, /ui,copy,save,tiff,graph,color,reverse,portrain,yes„ !—————————————————— ! Sines criterion !—————————————————— sines kappa=(((3*zso)/zgo)+((3*zso)/Rm)-(sqrt(6))) sadd,sin eqv,a s hmh,m s h,(1/(sqrt(3))),sines kappa, /title, Sines Eqv PLETAB,sin eqv, /ui,copy,save,tiff,graph,color,reverse,portrain,yes„ !—————————————————— !Crossland criterion !—————————————————— crossland kappa=(((3*zso)/zgo)-(sqrt(3))) sadd,cros eqv,a s hmh,max s h,0.577350269,crossland kappa, /title, Crossland Eqv PLETAB,cros eqv, /ui,copy,save,tiff,graph,color,reverse,portrain,yes„ !—————————————————— finish Appendix B. The ANSYS APDL macro of fatigue analysis according to the Dang Van MHCF criterion !============================== ! ANSYSAPDLmacro of fatigue analysis according ! Dang VanMHCF criterion ! CracowUniversity of Technology ! MiroslawMrzyglod 2005 (C) ! mrzyglod@mech.pk.edu.pl !============================== /prep7 ! preprocessing 708 M. Mrzygłód, A.P. Zieliński /SHOW,WIN32C ! definition of material model ! e.g. : zgo=260e6 zso=160e6 ! definition of geometry, FEmesh and boundary conditions finish !—————-1—————– /solution ! definition of loads: level 1 !e.g.: F,29003,FX,(Fox level1) ! etc. SOLVE /post1 etable,max s 1,s,1 ! sigma 1 etable,max s 2,s,2 ! sigma 2 etable,max s 3,s,3 ! sigma 3 etable,max s TG,s,int ! Tresca stress intensity SADD,max s12,max s 1,max s 2„, SADD,max s123,max s12,max s 3„, sadd,max s h,max s123„(1/3), ! SigmaH,max ! Dang Van criterion dang van kappa=(((3*zso)/zgo)-(3/2)) sadd,dv eqv,max s tg,max s h,0.5,dang van kappa, ESORT,ETAB,dv eqv„1 *GET,SMAX 1,SORT„MAX !—————-2—————– /solution FDELE,ALL,ALL ! definition of loads: level 2 !e.g.: F,29003,FX,(Fox level2) ! etc. SOLVE /post1 etable,max s 1,s,1 ! sigma 1 etable,max s 2,s,2 ! sigma 2 etable,max s 3,s,3 ! sigma 3 etable,max s TG,s,int ! Tresca stress intensity SADD,max s12,max s 1,max s 2„, Numerical implementation of multiaxial high-cycle... 709 SADD,max s123,max s12,max s 3„, sadd,max s h,max s123„(1/3), ! SigmaH,max ! Dang Van criterion dang van kappa=(((3*zso)/zgo)-(3/2)) sadd,dv eqv,max s tg,max s h,0.5,dang van kappa, ESORT,ETAB,dv eqv„1 *GET,SMAX 2,SORT„MAX !—————-3—————– /solution FDELE,ALL,ALL ! definition of loads: level 3 !e.g.: F,29003,FX,(Fox level3) ! etc. SOLVE /post1 etable,max s 1,s,1 ! sigma 1 etable,max s 2,s,2 ! sigma 2 etable,max s 3,s,3 ! sigma 3 etable,max s TG,s,int ! Tresca stress intensity SADD,max s12,max s 1,max s 2„, SADD,max s123,max s12,max s 3„, sadd,max s h,max s123„(1/3), ! SigmaH,max ! Dang Van criterion dang van kappa=(((3*zso)/zgo)-(3/2)) sadd,dv eqv,max s tg,max s h,0.5,dang van kappa, ESORT,ETAB,dv eqv„1 *GET,SMAX 3,SORT„MAX !—————-4—————– /solution FDELE,ALL,ALL ! definition of loads: level 4 !e.g.: F,29003,FX,(Fox level4) ! etc. SOLVE /post1 etable,max s 1,s,1 ! sigma 1 etable,max s 2,s,2 ! sigma 2 etable,max s 3,s,3 ! sigma 3 etable,max s TG,s,int ! Tresca stress intensity 710 M. Mrzygłód, A.P. Zieliński SADD,max s12,max s 1,max s 2„, SADD,max s123,max s12,max s 3„, sadd,max s h,max s123„(1/3), ! SigmaH,max ! Dang Van criterion dang van kappa=(((3*zso)/zgo)-(3/2)) sadd,dv eqv,max s tg,max s h,0.5,dang van kappa, ESORT,ETAB,dv eqv„1 *GET,SMAX 4,SORT„MAX !—————-5—————– /solution FDELE,ALL,ALL ! definition of loads: level 5 !e.g.: F,29003,FX,(Fox level5) ! etc. SOLVE /post1 etable,max s 1,s,1 ! sigma 1 etable,max s 2,s,2 ! sigma 2 etable,max s 3,s,3 ! sigma 3 etable,max s TG,s,int ! Tresca stress intensity SADD,max s12,max s 1,max s 2„, SADD,max s123,max s12,max s 3„, sadd,max s h,max s123„(1/3), ! SigmaH,max ! Dang Van criterion dang van kappa=(((3*zso)/zgo)-(3/2)) sadd,dv eqv,max s tg,max s h,0.5,dang van kappa, ESORT,ETAB,dv eqv„1 *GET,SMAX 5,SORT„MAX !————————————— smax=smax 1 *IF,smax,LT,smax 2,THEN smax=smax 2 *ENDIF *IF,smax,LT,smax 3,THEN smax=smax 3 *ENDIF *IF,smax,LT,smax 4,THEN smax=smax 4 *ENDIF *IF,smax,LT,smax 5,THEN smax=smax 5 *ENDIF ! print of maximum value of Dang Van equivalent stress in 5 Numerical implementation of multiaxial high-cycle... 711 ! steps fatigue analysis /OUTPUT,par state var,txt„ max eq stress=smax /OUTPUT References 1. ANSYS Inc., 2004, Release 8.1 Documentation 2. Ballard P., Dang Van K., Deperrois A., Papadopoulos Y. V., 1995, High cycle fatigue and a finite element analysis,Fatigue and Fracture of Engi- neering Materials and Structures, 18, 397-411 3. 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Walczak S., 2003, Analysis of Dynamic Loads Occurring in Different Types of Independent Car Suspension Systems, PhD Thesis, Cracow University of Technology, Cracow Numeryczne badania kryterium wieloosiowego wysoko-cyklowego zmęczenia pod kątem zastosowania do optymalizacji konstrukcji Streszczenie W artykule przedstawiono badania nad zastosowaniem nowoczesnych kryteriów wieloosiowego,wysoko-cyklowegozmęczeniamateriałudoproceduroptymalizacji kon- strukcji analizowanych metodą elementów skończonych. Po przeprowadzeniu testów kilku kryteriów zmęczeniowych do szczegółowych badań numerycznych wybrana zo- stałahipotezaDangVana.Wpracy zaproponowanotakżemetodę odpowiedniej adap- tacji wysoko-cyklowychprzebiegów obciążeń. Całkowity algorytmoptymalizacji zmę- czenia został zilustrowany przez zastosowanie proponowanych procedur do optyma- lizacji części pojazdu, które są szczególnie narażone na obciążenia wysoko-cyklowe. Do modelowania konstrukcji został użyty program metody elementów skończonych ANSYS R©. Manuscript received December 21; accepted for print March 15, 2006