Fatigue safety factor general formula proposition for the prestressed components subjected to arbitrary CA stress cycling process

Transcription

1 tigue fety fctor generl forul opoition for the etreed coponent ubjected to rbitrry C tre cycling oce Dir Jelk Univerity of Split-ESB R. Bošković b.b., Split, Croti Tel: x: E-il: [email protected] btrct: Known Goodn concept of deterining the ftigue fety fctor i derogted in thi pper. It i deontrted tht, in the eence of ttic etre, the ftigue trength plitude clculted fter Goodn' criterion i le thn the rel one. The itke inight i on the "fe ide" in ftigue deign coputtion, but till wrong. Thi iperfection hd been perceived in oe book nd other pper, but the unique, generl forul for deterining the ftigue fety fctor in the eence of ttic etre h not been offered. In thi pper, the unique forul for deterining the ftigue trength plitude nd ftigue fety fctor of coponent ubjected to contnt plitude (C) tre cycling oce in the eence of ttic etre i derived. The extreely iple wy for deterining the HC life of the etreed coponent ubjected to n rbitrry C tre cycling oce for known S-N curve nd en nd plitude tree, on the bi of Goodn line in High digr, i uggeted, well. Key word: High digr, en tre, ttic etre, lod line, ftigue trength plitude. 1. Introduction In nowdy of trongly developed obbilitic poche to ftigue eent, deterinitic poch i however till in ue nd therefore till iportnt, epecilly in the deign phe of the chine t, tructurl coponent nd joint. Such coponent re frequently ubjected to the high cycle ftigue (HC) tre cycling oce nd therefore the tre poch to ftigue deign i uitble. Thi poch h been ore thn hundred yer bed on the concept of Goodn (tright) line [1] in High digr nd correponding ftigue fety fctor. Goodn line i bed on huge nuber of teting nd it i ore or le unquetionble nd generly ccepted in deign counity. It i locu of ftigue frcture tte, i. e. locu of liit vlue of plitude tree for the certin en tre or converely. It connect the end point (0, ) nd (, 0) of High digr, where i the endurnce liit of oberved coponent t tre rtio r = in / x = 1 nd i oe ttic operty of teril trength (originly by Goodn [1] ultite trength), ee ig. 1. Conequently, the fety fctor for ny point of thi line equl unity. or ny en tre x nd correponding plitude tre y, it eqution i y x. (1)

2 oble rie when deterining point of Goodn line which give correct liit vlue of the ftigue trength plitude. fter Goodn, thi oble i byped: one h not to deterine the ftigue trength plitude in order to get the fety fctor. The reult i bd: ftigue fety fctor obtined i oetie correct, nd oetie incorrect! The entioned iperfection hd been perceived nd the concept of the lod line h been introduced in ftigue clcultion, e.g. [2, 3, 4, 5, 6]. fter thi concept, the ftigue trength, it i the liiting vlue of the plitude tre, i deterined with interection point of the lod line nd the Goodn line. ollowing thee concept, one h to think bout how to chrt lod line to get the ftigue trength. The reder re intructed to derive the ftigue trength theelve. So, thee bicly correct poche didn't reult with the n unique nlyticl exeion for deterining the correct vlue of the ftigue trength. The reon i iple: the uthor didn't perceive tht the reulting tre hitory of oe coponent i not only iple u of the en nd lternting tree, but it i lwy u of ttic etre nd the ource contnt plitude (C) tre cycling oce which generlly h it own en tre! The lod incree doen't ffect the ttic etre. It ke only the plitude nd the en tree of the ource tre cycling oce to incree long the pth of the lod line which origin i therefore oved long the bci of the High digr for the vlue of the ttic etre, ee ig. 2. or ticulr ource tre cycling oce the lod line i ingle one. In word, ll iperfection in deterining the ftigue fety fctor rie fro no ditinguihing ong en tre nd ttic etre. ll of tht hd been perceived nd the correct exeion for the fety fctor in the ce of etreed bolt hd been offered e.g. by Shigley nd Michke [7], nd correct generl exeion for deterintion the ftigue trength in the ce of ttic etreing hd been obtined by uthor [8, 9, 10], but there were not enough reverbertion in ofeionl bience. Tht i reon for thi pper. In the next ection the correct exeion for ftigue fety fctor re derived, coed with tht fter Goodn, nd dicued. 2. tigue trength nd fety fctor in the bence of ttic etre In iple ce where the chine or tructurl (unnotched) coponent, joint or pecien re ubjected to the C tre cycling oce with en tre nd plitude tre, without ttic etreing, the lod tright line i deterined with origin point (0, 0) of High digr nd it lope /, ee ig. 1. The tree nd vry long the lod line (which i lo line of contnt tre rtio r). It eqution i y x. (2)

3 = lod line ( = cont ) Goodn line M = ig. 1: Deterining the ftigue trength when no ttic etre i eent The liiting vlue nd M of nd re the plitude nd en tre of the ftigue trength repectively, which re deterined with interection point of the lod line (2) nd the Goodn line (1). Solving Eq. (1) nd (2) yield 1 1 y r r 1 (3) x r 1 M 1 r 1. (4) where r in x i tre rtio (or tre cycle yetry fctor). Obviouly, nd M do not depend on nd, but do depend on their rtio, it i on the tre rtio r. The ftigue fety fctor f here i equl to the rtio of nd, but lo, only in thi iple ce, to the rtio of + M nd x = + nd to the rtio of M nd f M r M x 1 (5) where r i the ftigue trength t r tre rtio, exeed in ter of xiu tre. Obviouly, the ftigue fety fctor equl the Goodn fety fctor, which en tht Goodn' forul i vlid for uch ce of treing.

4 = 3. tigue trength nd fety fctor in the ce of tticl etreing bove eented Goodn ftigue fety fctor i generly ccepted for ny en tre regrdle of it nture, nd here i hidden itke. Nely, when coponent i tticly etreed nd fter tht ubjected to the ource (working) tre cycling oce of the certin tre rtio r, the ttic etre doen't ticipte in lod nd tre increing. Thu, ttic etre σ ty e nd only the working en tre nd the plitude tre of the ource tre cycling oce incree, of coure, long the lod line pth which therefore h the e lope the ource tre cycling oce h. So, the origin of the lod line i oved for the vlue of en tre long the bci (in High digr), ig. 2, nd in the point (σ, σ ) long the yetrle of Sith digr, ig. 2b. Sith digr h poibility to eent the tre tte lo in tie-tre digr, it i ore poite in order to deontrte the tte of ll tree nd it chnge. Obviouly, the lod line in High digr pe the point (σ, 0) t the lope /. It i defined by eqution y x. (6) ) x r b) = cont ( = 0) lod line ( = cont) Goodn line = cont ( = 0) lod line ( = cont) Goodn line M = ig. 2: Deterining the ftigue trength in the eence of ttic etre ) in High digr b) in Sith digr 45 + M The vlue of the nd incree long the lod line nd it liiting vlue nd M, it i the ftigue trength, re deterined (in High digr) with interection point of the lod line (6) nd the Goodn line (1). So, olving Eq. (1) nd (6) yield:

5 r 1 1 r. (7) The ftigue fety fctor i the rtio of liiting vlue of tre plitude nd the tre plitude itelf. It i obtined: f 1. (8) fter Goodn, the liiting vlue,g nd M,G of nd, it i the ftigue plitude liit, i deterined, bove, with interection point of the Goodn line (1) nd the tright line ping the origin of the High digr nd the point ( +, ), ee ig. 3. In tht figure, the ltter tright line h been ned Goodn lod line becue it correpond to Goodn fety fctor. But, it cnnot be lod line, becue it i not the pth long which the tree incree! The ttic etre cnnot ticipte in lod nd tre incree, it ty e! nyhow, in ccordnce with Goodn, it i obtined:,g M,G. (9),G lod line Goodn lod line (wrong) Goodn line M + ig. 3: Deontrtion of wrong deterintion of ftigue trength fter Goodn for tticly etreed coponent Conequently, the known ftigue fety fctor fter Goodn i obtined once gin: M,G f,g,g 1. (10)

6 lod line ( = ) Obviouly, in the eence of the ttic etre, the ftigue trength plitude,g fter Goodn i lwy le thn the rel one, for the e rtio Goodn ftigue fety fctor i le thn the rel one. n extreely gret itke rie when the ource tre oce i of r = 1 tre rtio, ig. 4. In uch ce, the rel ftigue trength plitude i deterined gin by the interection point of the Goodn line (1) nd the lod line =. It i obtined: The rel ftigue fety fctor i: fter Goodn, the ftigue fety fctor i then. (11) 1 1. (12) f f,g (13) The rtio f / f,g = /,G becoe 1 1 f f,g (14) nd obviouly, it i uch greter thn one.,g Goodn lod line (wrong) M,G ig. 4: Coion of Goodn' nd rel ftigue trength plitude t the eence of ttic etre nd t r = 1 tre rtio of the ource tre cycling

7 It i poite to notice tht there i liiting vlue,b of ttic etre for which the ftigue trength plitude equl the tre plitude, ig. 5. or ny >,b, becoe le thn nd ftigue fety fctor le thn one. It i not difficult to obtin,b :,b 1. (15) Thu, the rtio,b / could be lo tken ftigue fety fctor, epeccily in oe pecil, very rer ce, when plitude tre ty contnt by increing the lod. Goodn line lod line,b M ig. 5: Liiting vlue of ttic etre The correct forule (7) nd (8), jut like (3) nd (5), for deterining the ftigue plitude trength nd ftigue fety fctor cn be pplied lo to the finit life of coponent. It i necery only to chnge there the endurnce liit with finite life ftigue trength N where ltter i deterined fter Woehler (or Bquin): 1,N 1 f N N (16) where N f i the ftigue life exeed in the nuber of cycle nd N nd re the ftigue life t the knee nd the lope of the Woehler curve, repectively. 4. tigue life etition The obletic delth with bove i relted to HC nd could be pplied in low cycle ftigue (LC) if deling with true tree. In both ce it doen't ipct ftigue life eent, becue ll the tree re then ituted on the Goodn line nd one h not to tke into ccount lod line, it i h not to ditinguih ong en tre nd ttic etre. Thu, Mnon, Morrow nd other forule for totl trin, bicly derived fro Goodn line ty e lo in the ce of ttic etreing nd cn be ued for deterining the ftigue life in the zone of LC. However, in the zone of HC, the ftigue life cn be deterined in n extreely iple wy. Nely, for given plitude nd en tree, which lie on Goodn line becue they re liiting tree in the e tie, regrdle the en tre coehend the ttic

8 etre or not, the Goodn line in High digr i deterined with point (, ) nd (, 0), ig. 7. It eqution i: y x. (17),N Goodn line ( ) Goodn line f ( f = cont) 8 ig. 7: Deterining finit life Goodn line for the certin, nd The finite life ftigue trength N reeent the vlue of ordinte for zero bci:,n 1. (18) The ftigue life i now obtined fro the Woehler curve eqution: 1 1 Nf N N 1 1N. (19) If deling in LC zone, iilr ocedure, but with true tree, cn lo be pplied for the etition of the ftigue life. Since LC zone i lo zone of eltic-pltic trin, the ttention ut then be pid to deterining the ftigue tre concentrtion fctor which differ fro tht in HC zone which i otly zone of only eltic trin. 5. Concluding rerk ro the reding-piece eented herein, the following concluding rerk cn be derived: It i explined tht well-known Goodn forul for deterining the ftigue fety fctor for certin C tre cycling oce i not correct if the coponent i eviouly ubjected to ttic etreing. The correct forul i derived nd it ue i uggeted. The entioned itke doen't ffect the ethod of LC ftigue eent (Morrow, Mnon, Berkovitz) bed on Goodn line in High digr. The liiting vlue of ttic etre i derived which reult with criticl ftigue fety fctor.

9 The iple forul for etiting the ftigue life of coponent for the certin C tre cycling oce nd the certin ttic etre i derived nd uggeted. ppliction of the forule obtined led to ore robut deign, or for the e deign to increing the currcy of eent. eented, Goodn' iperfection in deterining the ftigue fety fctor in the eence of ttic etre, ried fro equlizing the en tre nd ttic etre, i "on the fe ide". Tht i why it ppliction in ftigue eent of tructure, coponent nd joint couldn't ke ny hr nd tht i why thi iperfection h not been yet derrogted. By uthor' opinion, thi urplu of fety i one of the reon for chieving o gret ucce in the ervice life olonging of tructure, coponent nd joint by the tool of rcture Mechnic in lt few decde.. The e poch to ftigue fety fctor deterintion cn be lo pplied to the known Gerber' bol, Soderberg' or ny other criterion of ftigue filure which tke into ccount pecific teril, tte of treing or ervice condition. The forule derived cn be lo ued for the ftigue eent of coponent ubjected to vrible plitude or rndo loding: it i necery only to reduce it tre hitorie to the C one. Siilrly, in the ce of ultixil loding, it i necery to reduce the tree to the equivlent norl one. Reference [1] Goodn J, Mechnic pplied to Engineering. London: Longn; [2] Bnntine J, Coer J.J. nd Hndrock J.L. undentl of etl ftigue nlye (chpter 2). N Jerey: Prentice-Hll; [3] Deign of Mchine Eleent, Lecture 16, Worceter Polytechnic Intitute, Mech. Eng. Dept., vilble online: [4] Peteron RE. Stre concentrtion fctor. John Wiley & Son, N York, London, Sydney, Toronto; [5] tigue Sfety ctor. (MIT 2009). vilble online: [6] Wikipedi. vilble online: Theoreti885/0888-tigue_888 [7] Shigley JE, Michke CR. Mechnicl Engineering Deign. New York et l.: McGrow-Hill; [8] Jelk D. Generl forul opoition for coponent ftigue trength evlution (in Crotin). Strojrtvo 1990; 32:

10 [9] Jelk D, Podrug S. Etition of ftigue trength t opertionl lod. Proc. Int. Syp. tigue Deign, Vol. 2. Mrqui G, Solin J. (Editor). Epoo: Julkiij-Utgivre-Publiher; 1998; p [10] Jelk D. Opertionl Strength of Stedy Preloded Prt. Mterilove Inziniertvo 1999; 17: 60.