1 th International LS-DYNA Users Conference Siulation() Rando Vibration Fatigue Analysis with LS-DYNA Arnaud Ringeval 1, Yun Huang 1 CIMES, France 36 rue Marc Lefrancq, Les Ateliers Nuériques, 593 Valenciennes, France Liverore Software Technology Corporation 7374 Las Positas Road, Liverore, CA, 94551, United States Abstract Fatigue daage assessent for coponents under rando cyclic loading is an iportant concern in engineering. A new feature of rando vibration fatigue analysis has been ipleented to LS-DYNA, to perfor the structural fatigue analysis in a rando vibration environent. This feature coputes cuulative daage ratio and expected fatigue life for structures, based on the Palgren-Miner s rule of cuulative daage ratio and aterial s S-N fatigue curve. A series of fatigue analysis ethods have been ipleented. They include the Steinberg s three band ethod, Dirlik ethod, Narrow band ethod, Wirsching ethod, Chaudhury and Dover ethod, Tunna ethod and Hancock ethod. Brief introduction of the analysis ethods is provided. To facilitate post-processing of the fatigue analysis, a new binary plot file d3ftg has been ipleented in LS-DYNA. This binary plot file provides fatigue analysis inforation including cuulative daage ratio, expected life, zerocrossing frequency, peak-crossing frequency and irregularity factor for the structure, based on the stress index adopted in the analysis and the load period. This file is accessible to LS-PREPOST. Several exaples are given to deonstrate the effectiveness of the rando vibration fatigue analysis feature with LS-DYNA. Soe preliinary discussions on the different fatigue analysis ethods are included. Introduction Machines and echanical parts are often subjected to cyclic loads which are lower than the aterial s strength (e.g. tensile stress liit, or yield stress liit), but fail early due to fatigue. Fatigue is defined as the progressive and localized structural daage that occurs when a aterial is subjected to cyclic loading [1]. Industrial data show that, about 8% to 95% of all structural failures occur through a fatigue echanis. Thus it is iportant to study the fatigue life in the early design phase of new products. This paper introduces a new feature of LS-DYNA: rando vibration fatigue analysis. This feature is an iportant tool in structural durability analysis and has wide application in various industries. Fatigue analysis can be conducted in tie doain and frequency doain. In tie doain fatigue analysis, people usually use rain-flow counting algorith to get the nuber of cycles at each stress/strain level, based on the stress/strain tie history. In any situations, a description in frequency doain is ore practical. This is because 1) the load for the structure ay be rando in nature, for exaple, the wind load on wind turbine, or wave load on an offshore structure in this case, the best approach for fatigue analysis is to use the statistical ethod; ) it ay be too intensive to calculate the fatigue life in tie doain for large scale structures with long tie 1
Siulation() 1 th International LS-DYNA Users Conference history of load. This is the otivation for ipleenting the frequency doain fatigue analysis tool in LS-DYNA. Ipleentation of the feature in LS-DYNA A new keyword *FREQUENCY_DOMAIN_RANDOM_VIBRATION has been introduced in LS-DYNA to perfor rando vibration analysis since version 971 R5 []. Through the keyword, user provides inforation about the location, direction, range of frequencies for the rando excitation. Daping inforation is also provided through the keyword. The location of the excitation and response area can be given as node, set of nodes, set of segents, or part. The direction of load can be in any of the x, y, z directions or given as a vector by using *DEFINE_VECTOR. Load curve IDs for the Power Spectral Density (PSD) loads in rando coputation are also specified under the keyword. The feature of rando vibration fatigue is ipleented as an option of the keyword *FREQUENCY_DOMAIN_RANDOM_VIBRATION, as it is a natural extension of the rando analysis procedure. The ethod for perforing fatigue analysis is defined by the paraeter MFTG in card 1. An additional card (Card 6) is needed when the {FATIGUE} option appears in the keyword. This card defines the parts or eleents where the fatigue analysis is needed, the aterial s S-N fatigue curve ID, and soe other options. The exposure tie is defined by the paraeter TEXPOS in Card 4. As the stress state for a physical proble is 3D in the for of stress tensor, a stress invariant is needed to perfor fatigue analysis. The typical choices include axiu Principal stress, axiu shear stress and Von-Mises stress, which can be obtained fro the stress coponents. Material s fatigue characteristics is featured by an S-N (E-N) fatigue curve, which depicts the fatigue life (no. of cycles) for a given cyclic stress (strain) level. The plots are usually given in logarithic scale. For high cycle, low stress level fatigue, it is ore appropriate to use stress index. The S-N fatigue curve is obtained by a large aount of fatigue testing experients under different stress levels. The S-N fatigue curve can also be given in the for of analytical equations or N S a (1) log( S) a log( N) () where N is the nuber of cycles for fatigue failure and S is the stress aplitude, and a and are aterial paraeters deterined by experients. Particularly is the slope of the S-N curve. Please note that stress variation range (fro tension to copression) is used in the paper so S is ties of the aplitude if the stress is given as a cyclic function. Since odal analysis is the first step for running this feature, the keywords *CONTROL_IMPLICIT_GENERAL and *CONTROL_IMPLICIT_EIGENVALUE ust be included in the input. Soe other keywords related to iplicit solution ay also be needed, depending on the type of analysis. The results are given in binary plot file d3ftg which is accessible to LS-PREPOST. Five plot states are included in d3ftg: State 1: Cuulative daage ratio
1 th International LS-DYNA Users Conference Siulation() State : Expected fatigue life State 3: Zero-crossing frequency State 4: Peak-crossing frequency State 5: Irregularity factor Frequency doain fatigue analysis ethods A series of frequency doain fatigue analysis ethods have been ipleented in LS-DYNA. They are all based on Palgren-Miner s rule of cuulative daage ratio ni E [ D] (3) i N i where E[D] is the expected daage ratio, n i is the nuber of cycles at stress level S i, and N i is the nuber of cycles for failure at stress level S i, given by aterial s S-N curve. To get n i fro the PSD of the rando stress response and further copute E[D], a variety of approaches have been proposed. They are briefly introduced in this section. Dirlik ethod The Dirlik ethod was developed during the 198 s [3]. This ethod was found to have wider applications than other ethods and to be very accurate coparing with other ethods. The ethod uses an epirical closed-for expression for the Probability Density Functions (PDF) of stress aplitude, based on the Monte Carlo technology. The ethod consists of a series of calculations which are based on the oents of the PSD functions. The n-th oent of the PSD stress is coputed as n n f G( f ) df (4) where f is the frequency and G(f) is the PSD stress at frequency f. Based on the oents of the PSD stress, soe useful paraeters can be calculated: E[] (5) is the nuber of upward zero crossing; 4 E[ P] (6) is the nuber of peaks; E[] E[ P] (7) 4 is the irregularity factor. The irregularity factor, which varies between and 1, is a useful ter when interpreting the type of the rando stress signal. It approaches 1 as the stress signal 3
Siulation() 1 th International LS-DYNA Users Conference approaches narrow band (e.g. a single sine wave for = 1). It approaches as the stress signal approaches wide band (e.g. for =, the stress signal is white noise). Another useful paraeter is calculated as 1 (8) which is called the bandwidth paraeter, an alternative version of the irregularity factor. Then, the cuulative daage ratio is given as E n N i ( D) (9) i i i i p[ S ] E[ P] T ds N where T is the exposure tie and the PDF function is expressed by i Where Z Z Z D1 Q D e e D 3Z e Q R p( S) (1) 4 S ( X ) Z ; D 1 ; 1 1 1 1 X D R 1 D D ; 1 D 5 ( D1 D Q 4D 1 D1 D 1 R R) ; 1 X ; D3 1 D1 D As can be seen fro the equations, X, D 1, D, D 3, Q and R are all functions of the PSD oents, 1, and 4. Steinberg s three band ethod For Steinberg s ethod [4], the expected daage ratio is given as E D) E[] T.683 ( ).71(4 ).43 (6 ) / a (11) ( where E() is given by equation (5). The solution is based on the assuption that stress levels occur for 68.3% at RMS (= ), 7.1% at 4 RMS, and 4.3% at 6 RMS. The Steinberg s approach leads to a very siple solution based on the assuption that no stress cycles occur with ranges greater than 6 RMS. The ethod is used for testing electronic equipent in USA. Narrow band ethod The narrow band ethod was presented by Bendat [5]. Bendat showed that the PDF of peaks for a narrow band signal tended towards Rayleigh distributions as the bandwidth reduced. According to Bendat s theory, the expected cuulative daage can be written as E[ P] T E[ D] S p( S) ds (1) a 1 4 ;
1 th International LS-DYNA Users Conference Siulation() Or E[ P] T E[ D] S eqv (13) a 1/ where the equivalent stress is defined as For narrow band process, S 8 S eqv S p( S) ds (14) S p( S) e (15) 4 and equation (13) can be siplified as (1 / ) E[ P] T E[ D] (16) a where, (.) is the gaa function. Many expressions have been proposed to correct the conservatis associated with this solution. Most were developed with reference to offshore industry. The solutions of Wirsching, Chaudhury and Dover, Tunna and Hancock were all derived using this approach. They are all expressed in ters of the spectral oents up to 4. Wirsching ethod Wirsching s equation is given as [6]: E [ D] E[ D] (17) W W NB where, W is the rain-flow correction factor. It is an epirical factor derived fro extensive Monte Carlo siulations that include a variety of spectral density functions. It is expressed as follows b W W a W ( 1 aw )(1 ) (18) where a W and b W are best fitting paraeters expressed as a W.96. 33 (19) b W 1.587.33 () Chaudhury and Dover ethod For Chaudhury and Dover ethod [7], the expected daage ratio is expressed by equation (13), with the equivalent stress given as Where, S eqv erf 1 erf ( ) 3 4 5 6 7 ( ).31.4916.9181.354 3.337 15.654 1.7846 1/ (1) () 5
Siulation() 1 th International LS-DYNA Users Conference Tunna ethod The procedure for Tunna ethod is siilar to that for Dirlik ethod. The PDF is given as [8] S p( S) 4 S exp 8 For = 1., this forula becoes the narrow band forula given earlier (equation (15)). Tunna s equation was developed with specific reference to the railway industry. Hancock ethod For Hancock ethod, the expected daage ratio is expressed by equation (13), with the equivalent stress given as [9] S eqv 1/ (1 / ) (3) (4) Nuerical exaples Several exaples are presented in this section. For the first one, a panel bracket structure is considered and the Steinberg s three-band ethod is used to calculate the cuulative daage ratio; for the second one, an aluinu bea given in the literature [1] is studied, for which the theoretically estiated fatigue life is copared with the results given by experients; the third exaple under consideration is an industrial odel: an antenna support ounted on a bogie frae of railway vehicle. It is subjected to acceleration PSD defined by the International Standard for railway applications IEC61373 [11]. The nuerical results given by LS-DYNA are copared with the observation. Exaple 1: a panel bracket Consider a panel bracket structure in Figure 1. Material properties are given as density = 8 kg/ 3, Young s odulus E = 7.41 9 Pa, Poisson s ratio =.33. Shell eleent type 18 (Fully integrated linear DK quadrilateral/triangular shell) is adopted. Totally 197 nodes and 1865 shell eleents are used. The structure is fixed to shaker table. Constant base acceleration PSD. g /Hz for the range 1- Hz is applied. The structure is exposed to the rando vibration environent for 4 hours (144 seconds). Nodes constrained to shaker table 6
Stress (Ksi) 1 th International LS-DYNA Users Conference Siulation() Figure 1 A rectangular plate with free boundaries The aterial s S-N fatigue curve is given as Figure. 5 45 4 35 3 5 15 1 5 1.E+3 1.E+4 1.E+5 1.E+6 1.E+7 1.E+8 1.E+9 No. of cycles Figure S-N fatigue curve Steinberg s three-band ethod is used for this proble. The RMS of Von-Mises stress is given in Figure 3 (d3rs) and the cuulative daage ratio is given in Figure 4 (d3ftg). It is seen that the highest cuulative daage ratio (.78) takes place at the sae location as the highest RMS Von-Mises stress, which is on the edge with sharp curvature. Meanwhile, the expected fatigue life for the structure can be coputed as 144 /.78 = 19774.79 seconds. Figure 3 RMS of Von-Mises stress Exaple : aluinu bea Figure 4 cuulative daage ratio A siple cantilever aluinu bea [1] subjected to base accelerations is considered. The nuerical values are copared with the experiental results. Different fatigue failure theories are used to predict fatigue life. The odel is coposed of 5 nodes and 39 4-node type 18 shell eleent (fully integrated linear DK quadrilateral and triangular shell). Aluinu alloy 5754 is adopted for the aterial odel and its properties are given as density = 7 kg/ 3, Young's odulus E = 7, MPa and Poisson's ratio =.33. The odel is shown in Figure 5. The bea is designed such that failure occurs at a predeterined location where failure can be observed anually. 7
Siulation() 1 th International LS-DYNA Users Conference Figure 5 aluinu bea using for the shaker table experient (all diensions are in illieter) The attachent point of the bea is subjected to base Acceleration Spectral Density (ASD) for the range of frequency 1-3 Hz, shown in Figure 6. A constant odal daping ratio.4 is adopted. The bea is exposed to the rando vibration load for 3 inutes (18 seconds). The fatigue assessent of the notch area is suggested by the European Standard Eurocode 9 [1]. According to this code the fatigue strength of plain aterial is described by the detail category FAT1 calculated for a probability of survival equal to 97.7%, a reference fatigue strength = 1 MPa at.1 6 cycles and a single inverse constant slope =7.. The first 1 natural odes are required for the eigenvalue analysis. To get stress on shell surface it is necessary to use Lobatto's integration rule with three integration points through the thickness of the aluinu sheet (INTGRD=1 in the keyword *CONTROL_SHELL). Figure 6 - input acceleration PSD Figure 7 bea on the shaker table 8
1 th International LS-DYNA Users Conference Siulation() No. of cycles Figure 8 S-N fatigue curve used for the notched aluinu bea The response stress PSD easured at the critical point shows that several natural frequencies are excited by the input loading in the range 1-3 Hz (Figure 1). The natural frequencies of the bea, i.e. 13 Hz, 64 Hz and 185 Hz, obtained with LS-DYNA, atch reasonably well with the experiental results: 1Hz, 58 Hz and 18 Hz. Three saples of the aluinu bea are tested in the experients. The average fatigue life value observed is 7 n 5 s with a iniu of 5 n and a axiu of 1 n 3 s. Table 1 suarizes the fatigue life observed in the experient and the nuerical predictions archived with LS-DYNA. The results obtained provide satisfactory atch with the experiental results, although the results depend on the ethod used to interpret the RMS results. Dirlik, Wirsching and Chaudhury & Dover ethods give the best theoretical results. Steinberg gives conservative results while Tunna's predictions are copletely off. These different observations and results are entirely consistent with those found in the literature on the sae subject. Also, the sae proble has been siulated with ANSYS and RADIOSS BULK and the axiu RMS stress S x coputed by LS-DYNA (35. MPa) is in good agreeent with the results by the others coercial software : 33.5 MPa for the first and 35.7 MPa for the second. RMS S x = 35. MPa at critical point Figure 9 RMS of Sx stress in the local eleent axis 9
Siulation() 1 th International LS-DYNA Users Conference Figure 1 Stress PSD at critical point Theoretical ethods Fatigue life (n, s) Theoretical fatigue daage Experient 7n 5s - Steinberg 4n 1s 7.19 Dirlik 5n 5s 5.54 Narrow Band n 5s 14.41 Wirsching 5n 45s 5.8 Chaudhury and Dover 6n 3s 6.3 Hancock 4n 6s 7.31 Tunna n 18s 1.35 Table 1 Experiental and theoretical fatigue life Daage ratio = 5.54 Figure 11 - Cuulative daage ratio by Dirlik ethod 1
1 th International LS-DYNA Users Conference Siulation() Daage ratio = 7.19 Figure 1 - Cuulative daage ratio by Steinberg s ethod Exaple 3: support antenna Figure 13 - Failure at the notched point in experient The antenna support is ounted on a bogie frae of a railway vehicle and required a rando fatigue analysis to deterine the fatigue life before its failure. The odel is shows in Figures 14 (a) and (b) and it's coposed of 38,199 nodes and 14,993 4-node shell eleent type (fully integrated linear assued C shell). This eleent is based on thick plate theory and is recoended for thin and thick plates. The antenna is built up using 76,51 tetrahedron eleents with an eleent solid foration type 1 (one point tetrahedron). Its ass is.7 Kg. The total ass of the support is 3.7 Kg. The aterial properties of P75NL1 grade steel are given as density = 785 kg/ 3, Young's odulus E = 1, MPa and Poisson's ratio =.3. The aterial failed due to dynaic fatigue loading after only a few years in service. Before proposing a new design it is necessary to estiate the safe life of the actual configuration. The crack started around a echanically fastened joint between support and transverse stop and spread through the support to break it (Figure 15). 11
Siulation() 1 th International LS-DYNA Users Conference The structure is subjected to base acceleration defined by the International Standard IEC61373 to siulate the long-life test. This standard intends to highlight any weakness which ay result in proble as a consequence of operation under environent where vibrations are known to occur in service on a railway vehicle. Acceleration levels depend only upon the equipent's location within the vehicle. Coponent which is to be ounted on the bogie will be tested as category with acceleration spectral density 6.1 (/s ) /Hz for the range of frequency 5-5 Hz. The test duration is 5 hours (18, seconds). Constant odal daping ratio. is adopted and the first 15 natural odes are eployed in the odal superposition. Broken support Transverse stop antenna sensor Figure 14 (a) global view of support antenna odel 1
1 th International LS-DYNA Users Conference Siulation() 4 points are claped Figure 14 (b) back view of the support antenna odel Initial failure area Figure 15 broken support The fatigue life calculation should be based on the use of the European Standard Eurocode 3 [13] which defines the acceptable fatigue strength. Mechanically fastened joints are assued to be described by the detail category FAT11: probability of survival equal to 97.7%, reference fatigue strength = 11 MPa at 1 6 cycles and = 45.3 MPa for cut-off liit at 1 8 cycles. The fatigue analysis is based on the RMS of Von-Mises stress and the Steinberg's ethod. Figure 16 (a) and Figure 16 (b) provide respectively contour plot of the cuulative daage ratio observed on the support and a photograph of the broken support. Results given by LS-DYNA are in good agreeent with the expected results. 13
Siulation() 1 th International LS-DYNA Users Conference Area - daage ratio = 3.45 Area 1 - daage ratio = 5.5 Figure 16 (a) cuulative daage according Steinberg's ethod Area Area 1 Figure 16 (b) broken support 14
1 th International LS-DYNA Users Conference Siulation() Conclusion A new feature of LS-DYNA, rando vibration fatigue analysis is introduced in the paper. The feature provides cuulative daage ratio calculation and also fatigue life prediction for structures subjected to rando vibration excitations, based on various theories. The feature has wide application in durability analysis for various industries. Three exaples, including one fro industry, are adopted to deonstrate the effectiveness of the new feature. The nuerical prediction atches reasonably well with experiental results, or observations. Different ethods of fatigue analysis are discussed. References 1. http://en.wikipedia.org/wiki/fatigue_(aterial).. LS-DYNA Keyword User's Manual, Version 971, Liverore Software Technology Corporation, 1. 3. Dirlik T. Application of Coputers in Fatigue Analysis, Ph.D. thesis, University of Warwick, 1985. 4. Steinberg, D.S., Vibration Analysis for Electronic Equipent ( nd edition), John Wiley & Sons, New York, 1988. 5. Bendat J.S., Probability Functions for Rando Responses. NASA report on Contact NASA-5-459, 1964. 6. Wirshing, P.H., Paez, T.L., and Ortiz K., Rando Vibration, John Wiley & Sons Inc., New York, 1995. 7. Chaudhury, G.K., and Dover, W.D., Fatigue analysis of offshore platfors subjected to sea wave loadings, International Journal of Fatigue, Vol. 7, No 1, pp 13-19. 1985. 8. Tunna, J.M., Fatigue life Prediction for Gaussian Rando Loads at the Design Stage, Fatigue and Fracture of Engineering Materials & Structures. Vol. 9 No. 3, pp. 169-184, 1986. 9. Gall DS, Hancock JW. Fatigue crack growth under narrow and broad band stationary loading. Glasgow University, Marine Technology Centre; 1985. 1. Vinod Kuar Nagulpalli, Abhijit Gupta and Shaofeng Fan, Estiation of Fatigue Life of Aluinus Beas subjected to Rando Vibration, Departent of Mechanical Engineering, Northern Illinois University, DeKalb, IL. 11. International Standard IEC 61373 - railway applications Rolling stock equipent Shock and vibration tests. 1. EUROCODE 9: Design of aluinu structures Structures susceptible to fatigue. 13. Steel construction EUROCODE3 "Design of steel structutre" Part 1-1: general rules for building Chapter 9: fatigue. 15
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