Mean stress effects on random fatigue of nonlinear structures

Transcription

1 Rocheter Intitute of Technology RIT Scholr Work Preenttion nd other cholrhip 5 Men tre effect on rndom ftigue of nonliner tructure Krl Sweitzer.S. Ferguon Follow thi nd dditionl work t: Recommended Cittion Sweitzer, Krl nd Ferguon,.S., "Men tre effect on rndom ftigue of nonliner tructure" (5). Acceed from Thi Conference Proceeding i brought to you for free nd open cce by RIT Scholr Work. It h been ccepted for incluion in Preenttion nd other cholrhip by n uthorized ditrtor of RIT Scholr Work. For more informtion, plee contct ritcholrwork@rit.edu.

2 Twelfth Interntionl Congre on Sound nd Vibrtion MEA STRESS EFFECTS O RADOM FATIGUE OF OLIEAR STRUCTURES Krl A. Sweitzer nd. S. Ferguon ITT Indutrie Spce Sytem Diviion, Rocheter, Y USA Intitute of Sound nd Vibrtion Reerch, Univerity of Southmpton, UK emil: krl.weitzer@itt.com Abtrct Rndom ftigue nlyi technique hve concentrted on liner tructure tht exhibit ymmetric tre rnge probbility denity function (PDF). In nrrowbnd repone nd brodbnd rndom repone it i umed tht the bending tre rnge PDF i ymmetric nd h zero men tre. Structure tht cn exhibit geometricl nonlinerity, under lrge deflection, typiclly introduce membrne effect, which re dditionl poitive tenile tree tht tend to kew the tre rnge PDF. Thi pper invetigte ftigue nlyi pproche tht include men tre effect nd how how they cn be pplied to nonliner rndom vibrtion problem. ITRODUCTIO Recent dvnce in nonliner rndom vibrtion nlyi technique for tructure hve identified need for ftigue nlyi filure criterion tht include the tenile membrne tree. The tenile membrne tree combine with the cyclic bending tree to form kewed tre rnge PDF. Thi pper review inuoidl nd rndom ftigue reltionhip nd extend thi theory to include nonliner men tree. Sinuoidl Ftigue Life The tndrd inuoidl tre life power lw ftigue model i: ( ) ' b b r = f f = A f A = ' f b (1) 1

3 where r i the fully revered lternting tre nd f i the medin cycle to filure. The power lw lope b (ftigue trength exponent) nd the tre intercept t one reverl (ftigue trength coefficient), or A t one cycle, re detered ' f experimentlly. The ymbol i ued here to denote tre inted of σ, which i ued to denote the tndrd devition. The life f tht i expected for given lternting tre under zero men tre i: 1 b r f = Sinuoidl Ftigue Life with Men Stre A recent pper by Dowling [1] decribe four method tht hve been developed over the pt ~1 yer to detere ftigue life for lternting tree combined with men tree. Thi ection ummrize the currently ccepted method. The Morrow eqution for the number of cycle to filure with men tre i: fmi = A 1 ' f For nonferrou mteril, modifiction uing the true tre t frcture The Wlker eqution i: fmb fw A = A 1 % fb γ mx 1 R = A % fb i: where the rtio R of imum nd mximum tre vlue t the trough nd pek of the cycle i: R = (6) mx The fully revered zero men ce i when R = -1. ote, (5) reduce to () when the exponent γ =1.. The pecil ce of the Wlker eqution for γ =.5 give the Smith- Wton-Topper [5] eqution:.5 mx 1 R mx 1 R fwt = = (7) A A A modified verion of the Wlker eqution i ued in MMPDS-1 [], (the replcement for MIL-HDBK-5), bed on the tre rtio R nd n equivlent tre, where: eq () (3) (4) (5)

4 log eq mx ( 1 R) = A3 ( f 5) = A1+ Alog ( eq A4) (Section from [] give thorough dicuion of (8)). The term A 4 repreent the ftigue limit tre or endurnce limit. Rndom Ftigue & Dmge The number of cycle to filure for nrrow bnd rndom proce (ee [6]) i: 1 b fnr (8) = A p d (9) where p ( ) i the PDF of tre pek. For the nrrow bnd repone umption, p ( ) i umed to be Ryleigh. It h been hown [6] tht (9) reduce to: σ β fnr = Γ 1+ A whereσ i the tndrd devition of tre, The Plmgren-Miner liner cumultive dmge rule i: n D = k k = 1 fk 1 b (1) β = 1 b, nd Γ i the Gmm Function. where D i the ummtion of dmge, k re the number of cycle for lod ce k nd fk re the number of cycle to filure for ce k. D=1 indicte probbility of filure =.5 (i.e. hlf of popultion i expected to hve filed). When one conider nrrow bnd rndom proce the expected vlue of dmge for given rndom lod ce reduce to: ED [ n] E[] T ( A) p( ) d E[] T β σ β = = Γ 1+ (1) A where E[] i the expected rte of zero croing nd T i the durtion of the lod: = E[] T (13) A widebnd rndom proce i more intereting; the ditribution of pek mplitude diverge from the Ryleigh PDF nd the repone will include poitive pek with negtive mgnitude (ee [7] nd the cited reference). The de-fcto method for evlution of widebnd rndom repone i the time domin rinflow rnge cycle counting method [8]. Dirlik [9] (ee lo [7]) developed n empiricl reltionhip for etimting the rinflow rnge PDF bed on the pectrl moment of the tre power pectrl denity (PSD). It i intereting to note tht Dirlik PDF only etimte the rin flow rnge mplitude, i.e. the etimte do not include men tree. β (11) 3

5 PROPOSED TWO DIMESIOAL STRESS & DAMAGE PDF The propoed method to include men tree in rndom liner or nonliner dmge eqution i: β p (, ) E[ D ] [ ] m = E P T d d (14) k, where p (, ) i the joint PDF of tre mplitude & men tre rinflow rnge nd k (, ) i function of mteril propertie nd the choen non-zero men tre ftigue eqution. The problem i tht there i no known cloed-form or empiricl reltionhip for p (, ). The joint probbility denity function p (, ) cn be etimted numericlly hitogrm with cycle counting lgorithm (ee for exmple the dtrfm function in WAFO [1]). WAFO Rin Flow Mtrix The D hitogrm of rinflow rnge clculted by the WAFO dtrfm function i clled rinflow mtrix (RFM). A RFM i convenient numericl etimtion of the joint PDF. The RFM h imum nd mximum tre xe, which cn eily be relted to the mplitude nd men tre dimenion propoed for the D rinflow PDF. Only hlf of the RFM i ued becue the imum tre cn never be greter thn the mximum tre. The WAFO group h choen to diply the RFM hown in Figure 1 with the imum tre on the x-xi nd mximum tre on the y-xi. The figure lo h line drwn long contnt vlue of the tre rtio R, which give inight into the ditribution of rnge. Contnt mplitude condition re found long digonl of poitive lope nd contnt men tre rnge re found long digonl of negtive lope. + Men = C R= -1 R = 1 Mx R= R= - Amp = C o Rin Flow Rnge + Figure 1 The WAFO convention for Rin Flow Mtrix (RFM), with pecil tre rnge ce. 4

6 Rndom Ftigue Dmge uing the RFM The propoed dicrete rndom ftigue dmge eqution bed on the RFM D hitogrm etimte of the joint PDF i: RFM (, mx ) RFMn(, mx ) D= = RF (15) mx f, mx mx f, mx where the normlized rinflow mtrix RFMn nd the number of rin flow cycle RF re defined by: RFM (, mx ) RFMn = RF = RFM (, mx ) (16) RF mx ote tht RFMn i relted to the D tre PDF by: RFMn, p, d d = 1 (17) mx mx The denotor of (15) i detered bed on choice of ftigue eqution. The etimte uing the lternting tre eqution () (with no men tre effect) i: RFM (, mx ) β Dr = = RF RFMn(, mx )[ A] (18), fr( ) The lternting tre ( ) mx mx = mx vrie over the RFM nd i clculted for ech hitogrm bin. The etimte uing the Morrow eqution (3) i: RFM (, ) fm(, ) The men tre = ( + ) D = = RFMn, A 1 mx M RF mx ' mx mx f mx lo vrie over the RFMn nd i clculted for ech hitogrm bin. An lternte form of the Morrow eqution (4) follow thi form with % ubtituted for f. The dmge etimte uing the Wlker eqution (5) i: fb ( ) γ RFM, mx mx 1 R DW = = RF RFMn(, mx ) () mx fw, mx A mx The R vlue i clculted for ech hitogrm bin uing eqution (6). The Smith- 1 Wton-Topper eqution (7) follow thi form with the exponent γ =. The dmge etimte uing the modified Wlker eqution nd the equivlent tre eqution (8) from [] i: D 5 RFM =, = RFMn, mx mx RF mx f 5, mx mx 1 ^ A1+ Alog A4 ( eq ) A et of function h been developed to etimte dmge given RFM nd choice of ftigue eqution (18) through (1). Thee will be compred with experimentl reult in the next ection. β β (19) (1) 5

7 APPLICATIO OF TWO DIMESIOAL STRESS & DAMAGE HISTOGRAMS WPAFB Experimentl Dt The reult here re from erie of brodbnd be excited (input -5 Hz) clmped-clmped bem experiment, conducted t WPAFB [11]. The lrge deflection repone of the bem reulted in ignificnt nonliner membrne tree, reulting in kewed urfce tre rnge PDF. The reult of numericl nlyi of thi repone dt (88 t 496 Hz mple rte for ech input) uing the WAFO RFM function nd the propoed dmge function re preented below. Figure how the normlized RFM nd RFD mtrice from the WPAFB dt t the lowet tet level of.5 g RMS input. The dt w normlized uing the mple men µ nd tndrd devition σ of the tre repone by: µ n = () σ The RFM how nerly ymmetric ditribution bout the R = -1 digonl. There re lo mny low mplitude cycle in the dt from the widebnd multi-modl repone of the continuou bem. () (b) Figure ormlized Totl Stre () nd Dmge (b) Rin Flow Mtrice: σ = Mp (.84 ki), µ =.15 Mp (.31 ki), E[Dm]= 6.9e-13, Input =.5 g The RFD mtrix w clculted uing the Morrow eqution (19) nd SAE 115 mteril propertie from [1]. ote tht mot of the dmge i reult of the lrge mplitude cycle tht occur between ~ nd 4 igm (ee Figure (b)); the lrge quntity of low mplitude cycle cue very little dmge. A the input level w increed, the tre repone of the bem becme much more nonliner, reulting in kewed RFM hown in Figure 3() nd Figure 4(). A with the low level ce, the mjority of the dmge occur between ~ nd 4 igm. A comprion of the dmge clculted uing different ftigue eqution i hown uing the 8 g input dt 6

8 in Figure 4(b) (Morrow eqution (19)), Figure 5() (Wlker eqution ()) nd Figure 5(b) (lternting tre eqution (18)). A expected, the lternting tre eqution etimte the mllet dmge (5.5e-6) while the Morrow nd Wlker etimte greter dmge (7.38e-6 nd 1.31e-5 repectively) due to the nonliner tenile men tree. () (b) Figure 3 ormlized Totl Stre () nd Dmge (b) Rin Flow Mtrice: σ = 14.7 Mp (.13 ki), µ =.978 Mp (.14 ki), E[Dm]=.77e-9, Input = g () (b) Figure 4 ormlized Totl Stre () nd Dmge (b) Rin Flow Mtrice: σ = 3.48 Mp (4.71 ki), µ = 9.4 Mp (1.36 ki), E[Dm]= 7.38e-6, Input = 8 g SUMMARY A method to include tenile men tree in ftigue dmge nlyi for nonliner rndom repone h been developed bed on everl commonly ccepted inuoidl ftigue eqution. Experimentl nonliner reult were ued to illutrte the 7

9 uefulne of the propoed method. Further exmple nd theoreticl dicuion of the finding will be publihed in the future. The primry uthor thnk WPAFB Structurl Mechnic Brnch for providing the experimentl dt nd ITT Indutrie for their eductionl upport t the ISVR. () (b) Figure 5 ormlized Dmge Rin Flow Mtrice uing Wlker Eqution() nd Alternting Stre Eqution (b): E[Dw]= 1.31e-5, E[D]= 5.5e-6, input = 8 g REFERECES [1]. E. Dowling, Men Stre Effect in Stre-Life nd Strin-Life Ftigue, SAE Pper # , in Proceeding of Ftigue 4: Second SAE Bril Interntionl Conference on Ftigue, São Pulo, Bril, June 4. [] DOT/FAA/AR-MMPDS-1, Metllic Mteril Propertie Development nd Stndrdiztion (replcement to MIL-HDBK-5), Jnury 3. [3] J. Morrow, Ftigue Propertie of Metl, in Proceeding of Diviion 4 of the SAE Iron nd Steel Technicl Committee, ov. 4, [4] K. Wlker, The Effect of Stre Rtio During Crck Propgtion nd Ftigue for 4-T3 nd 775-T6 Aluum, Effect of Environment nd Complex Lod Hitory on Ftigue Life, ASTM STP 46, Am. Soc. for Teting nd Mteril, Wet Conhohocken, PA, 197. [5] K.. Smith, P. Wton, nd T. H. Topper, A Stre-Strin Function for the Ftigue of Metl, in ASTM Journl of Mteril, Vol. 5, o. 4, December 197. [6] R. G. Lmbert, Anlyi of Ftigue Under Rndom Vibrtion, The Shock nd Vibrtion Bulletin o 46, vl Reerch Lbortory, Whington, DC, My [7] K. A. Sweitzer,. W. M. Bihop, V. L. Genberg, Efficient Computtion of Spectrl Moment for Detertion of Rndom Repone Sttitic, in Proceeding of Interntionl Conference on oie & Vibrtion Engineering, Leuven Belgium, September, 4. [8] M. Mtuihi, nd T. Endo, Ftigue of metl ubjected to vrying tre, Jpn Society of Mechnicl Engineer, Fukuok, Jpn, Mrch, [9] T. Dirlik, Appliction of Computer in Ftigue Anlyi, Univerity of Wrwick Thei, [1] WAFO Toolbox - Wve Anlyi for Ftigue nd Ocenogrphy, Verion.1., 7-Apr-4, Lund Int. of Tech., Lund Univerity, Sweden. [11] R. W. Gordon, J. J. Hollkmp, nd S. M. Spottwood, onliner Repone of Clmped - Clmped Bem to Rndom Be Excittion, in Proceeding of VIII Interntionl Conference on Recent Advnce in Structurl Dynmic Southmpton, UK, July 3. 8