A General Equation for Stress Concentration in Countersunk Holes

Transcription

1 oprigh c 7 Tech Science Press M, vol., no., pp.7-9, 7 General Equaion for Sress oncenraion in ounersunk Holes Kunigal N. Shivakumar, nil Bhargava and Sameer Hamoush srac: deailed and accurae hreedimensional finie elemen sress analsis was conduced on counersunk rive holes in a plae sujeced o ension loading. The analsis included a wide range of counersunk dephs, plae hicknesses, counersunk angles and plae widhs. The sud confirmed some of he previous resuls, addressed heir differences, provided man new resuls, and invesigaed counersunk angle and widh effecs. Using he deailed FE resuls and he limiing condiions, a general equaion for sress concenraion was developed and verified. Keword: Sress concenraion facor, counersink, counersunk holes, ensile loading, sress concenraion equaion, finie elemen analsis Nomenclaure E FE h K r SF SS w x,, sraigh-shank deph counersink deph ) Young s modulus finie elemen analsis one-half heigh of plae maximum sress-concenraion facor along ore of hole under ension radius of sraigh-shank porion of hole sress concenraion facor sraigh-shank plae hickness plae half-widh aresian coordinae ssem orresponding auhor. Tel.: --7 x ; fax: address: kunigal@nca.edu. Deparmen of Mechanical and hemical Engineering, ener for omposie Maerials Research, Norh arolina & T Sae Universi, Greensoro, N 7. Infoss Technologies Limied, Tennson Parkwa Suie, Plano, TX 7. Deparmen of ivil and rchiecural Engineering, ener for omposie Maerials Research, Norh arolina & T Sae Universi, Greensoro, N 7. ν Poisson s raio counersink angle applied remoe ension sress Inroducion Riveing is a common mehod of joining srucural componens. Joining inroduces disconinuiies sress risers) in he form of holes, change in load pah, and addiional secondar loads such as rive earing and ending. Because of hese reasons, local sresses a he join are elevaed compared o srucural nominal sresses. Wherever he aero/hdro dnamic surfaces are required, counersunk rives are ofen used. ounersinking furher complicaes he sress flow and causes addiional elevaion of local sresses. These prolems require a hree dimensional -D) analsis. ccurae deerminaion of local sresses is essenial o predic join srengh and faigue life of he componen. Exhausive sudies on sress concenraion in holes and noches for wo-dimensional -D) odies sujeced o a wide varie of loadings have een repored in he lieraure and in handooks [Pilke 997), Savin 9)]. Three-dimensional -D) resuls have een repored for plaes wih circular sraigh-shank)holes sujeced o remoe ension [Green 98), Neuer 9), Sernerg and Sadowsk 99), Folias and Wang 99), Shivakumar and Newman 99, 99), Whale 9)]. ll hese soluions were limied o wide plaes. summar of hese are lised elow.. Sress concenraion is due o hoop sress along he ore of he hole.. Sress concenraion facor SF) depends on he dimensionless parameers /r) and /),where is he hickness co-ordinae direcion and is he plae hickness.

2 7 oprigh c 7 Tech Science Press M, vol., no., pp.7-9, 7. Sress concenraion is highes in he midhickness of he plae for hin o moderael hick plaes, u for hick plaes, he sress concenraion peaks near he free surface, a aou /). The sae-of-sress is nearl plane-srain in he cenral region, is nearl plane-sress a he free surface and is inermediae in he ransiion region. Thus, he sress flows ino he cenral region of he plae causing higher sress concenraion han a he plae surface.. The sress concenraion is aou o 7% higher han plane-sress value K =)andi increases wih plae hickness. Plane-sress value is recovered for ver hin plaes. The sress concenraion is aou % lower han plane-sress value a he free surface. Onl few papers have een pulished for counersunk rive holes ha direcl relae o sress concenraion. The firs was Whale 9), using irefringen coaing on aluminum plaes. He measured he sresses on he surface of he plae insead of measuring in he inerior of he hole. The oher experimenal work was heng 978); he used sress freeing echnique o oain sress hrough he hickness of plae wih a counersunk hole. He invesigaed a oal of configuraions wih differen counersink angles and dephs; seven specimens for ension loading and six for ending loading. heng s resuls showed conclusivel ha he highes sress concenraion is a he edge of he counersink. In earl 99, Shivakumar and Newman 99, 99) conduced a deailed -D FE analsis of counersunk holes in a wide plae sujeced o ension, ending and wedge loadings. Their sud included a wide range of counersink deph o hickness raio /), plae hickness o radius raio /r), and a counersunk angle ) of o. Numerical resuls were presened in he form of chars, ales along wih a simple FORTRN program including an inerpolaion mehod o deermine K for a wide plae and counersunk angle of o. In 99, Young and Lee 99) conduced an independen -D FE analsis of plaes wih counersunk holes sujeced o ension load and proposed a design equaion comining heir FE resuls and Briish erospace s -D design equaion. Young and Lee s soluions were ased on a ver coarse FE model and heir equaion even did no reduce o -D soluion in lieraure. Furhermore, Shivakumar and Newman s and Young and Lee s soluions differed widel for / =. Therefore, he presen sud is underaken o verif he wo resuls, provide new resuls, and hen o develop an accurae sress concenraion facor equaion ha saisfies he limiing condiions of he configuraions. n exhausive -D FE sud wih ver fine modeling will e conduced differen counersunk angle ), hickness o radius /r) raio, counersink deph o hickness /) raio and plae widh o radius w/r) raio. Based on hese resuls and limiing configuraional condiions, an equaion for SF in counersunk hole will e developed. Resuls and equaions oained from he preliminar sudies conduced he auhors have alread een presened in Shivakumar, Bhargava and Hamoush ). This paper presens an improved equaion compared o he previous equaion. Meshless mehods have een developed in references Li, Shen, Han, and luri ), and hen and hen ), o solve prolems in hree-dimensional elasici wih singulariies and maerial disconinuiies. onfiguraion and Maerial Figure shows a configuraion of a plae wih a counersunk rive hole sujeced o ension. ll he geomeric parameers used in his sud are defined in he figure. The aresian coordinaes x represen he reference coordinae ssem. The plae widh and he heigh are, respecivel, w and h, and he hickness is. The plae is sujeced o a remoe ensile sress,. The counersunk hole consiss of a sraigh-shank par hickness ) and a counersunk par hickness ). The oal deph of he hole, which is also he hickness of he plae, is relaed o and = +. The radius of he SS par of he hole is r and he counersunk angle is. common angle of counersink used in aircraf consrucion is o ;hek resuls evaluaed for his angle are

3 General Equaion for Sress oncenraion in ounersunk Holes 7 h x θ B c h θ c r φ w x a) ) w B x c) Figure : onfiguraion and Nomenclaure of a ounersunk Rive Hole: a) x Plane; ) Plane; c) x Plane; d) -D onfiguraion d) considered o e he aseline. The variaion of parameers, and will simulae all cases of hole configuraions from SS / = ) o knifeedge / =). The maerial is assumed o e homogeneous and isoropic wih E = 8.9 GPa and ν =.. n value of E is accepale, ecause in an isoropic prolem sujeced o force oundar condiions he resuling sresses are independen of E. Thus he SF is independen of E. However,hePoisson s raio ν ma influence he resuls u is effec is secondar and negleced. Definiion of Sress-oncenraion Facor The -D sress concenraion facor definiion is given in man classical ooks on heor of elasici and in sress concenraion handooks. For -D configuraions, he sress concenraion varies along he ore of he hole. Therefore, he sress concenraion ecomes a funcion of coordinae along he line formed he inersecion of = plane and he hole. The line connecing he poins,b, in Figure ) and c) defines he pah of ineres. The sress concenraion K ) is he raio of hoop sress σ along he line -B- and he remoe sress, K )=σ )/. The sress concenraion facor K is he maximum of K ). Finie-Elemen nalsis commercial FE analsis code, NSYS Version, was used for geomeric modeling, FE modeling and analsis of he prolem. NSYS Parameric Design Language PDL) programs were wrien o auomaicall generae geomeric and FE models, imposiion of loading and oundar condiions, and conducing he analsis in a ach mode.. Geomeric Models The various geomeric parameers considered in he sud have een presened in he previous secion. Because he geomer and he loading are smmeric Fig. ), onl he smmeric-quarer of he model was considered. Figure illusraes

4 7 oprigh c 7 Tech Science Press M, vol., no., pp.7-9, 7 a) ) c) Figure : Quarer Smmeric Geomeric and FE Models Showing he Effec of /: a) Sraigh-Shank Hole w/r = h/r =, / =, /r =); ) ounersunk Hole w/r = h/r =, / =., /r = ); c) Knife- Edge Hole w/r = h/r =, / =, /r = ) Figure : Quarer Smmeric Geomeric and FE Models of Narrow Widh w/r =, h/r =, / =., /r = )

5 General Equaion for Sress oncenraion in ounersunk Holes 7 he hree possile counersink rive hole configuraions, namel, SS /) = ), pical counersink, and knife-edge /) = ). Figure shows a narrow widh model used for conducing a sud on widh effec.. FE Models Three-dimensional hexahedron elemens solid) were generaed over he volumes using he iso-parameric mapping concep. are was aken in generaing he elemens so ha onl hexahedron elemens were generaed. The FE mesh idealiaion was finer in he high sress gradien regions and coarser a he low or no sress gradien regions. Three levels of mesh refinemens were used o conduc he convergence sud. Resuls of he convergence sud will e presened laer. The Figs. and illusrae he pical FE meshes used for SS, counersunk and knife-edge geomeries, and narrow widh plaes.. Boundar ondiions, Loading and nalsis Smmer oundar condiions were imposed on he model consraining u displacemen in - direcion on = plane and he u x displacemen in x-direcion on x = plane. To arres he rigid od moion in -direcion, he u displacemen was resrained a a node a x = = and = r. The plane = h was loaded wih a uniform sress =. ll oher oundar regions were allowed o deform freel. linear FE sress analsis was conduced. Deformaion, sress and srain field were examined. The hoop sresses along he nodal line -B-, see Fig. c)) were exraced from he PDL program. These sresses direcl gave he sress concenraion, K ) and he maximum value of K ) is he sress concenraion facor K.. Mesh onvergence Sud mesh convergence sud was conduced o evaluae he accurac of he K resuls. The geomeric model emploed for his sud was /r =,w/r =and /) =.7. The hree specific meshes were sudied: aseline, finer and fines. To provide a clear picure of he refinemen, he mesh regions around he counersunk hole are enlarged and shown in Fig.. The finer mesh was oained douling he numer of divisions in all hree direcions. The fines mesh was also oained douling he divisions u onl in he area of ineres. The mesh convergence approach used was similar o he pach es defined in Zienkiewic and Talor 989). Figure shows K for he hree meshes and noe ha he fines mesh resul has almos reached he asmpoic value. The aseline and he finer mesh K differed. and. percen, respecivel compared o he fines mesh. These errors are small and are assumed o e accepale. Therefore, he aseline mesh was used in all oher sudies. Deails of his sud are given Bhargava ). omparison of Presen FE Resuls wih Lieraure The K resuls from he presen finie elemen models are compared wih oher finie elemen and experimenal resuls in he lieraure.. omparison wih Oher FE Resuls The presen aseline FE model resuls were compared wih Shivakumar and Newman s 99 and 99) revised FE resuls which had some minor misakes) for differen hickness-o-hole radius raios /r) of.,, and wih w/r =and = o. The presen FE resuls and Shivakumar and Newman s resuls agreed ver well for all cases, he difference was less han %. Young and Lee s FE resuls are compared wih he presen FE resuls ploing he variaion of K as a funcion of /) for differen plae widhs of w/r =,,and,infig.. Thepresen FE are represened solid smols and joined solid lines while Young and Lee s FE resuls are represened hollow smols. From Fig. i is clear ha for all values of / >., Young and Lee s resuls are consisenl higher han he presen FE resuls. The difference is larges and is aou % for / = wih w/r =. The reason for his difference is e due o coarseness of he FE mesh used in Young and Lee s sud.

6 7 oprigh c 7 Tech Science Press M, vol., no., pp.7-9, 7 a) ) c) Figure : Mesh Sies Used for he onvergence Sud: a) Baseline; ) Finer; c) Fines.9 smpoic Value K Elemen Lengh Raio, Baseline Model urren Model Figure : Plo of K for he Three Mesh Refinemens 7 h/r = /r = ; = º K Presen Young & Lee w/r B / Figure : omparison of Presen and Young and Lee s FE K Resuls

7 General Equaion for Sress oncenraion in ounersunk Holes 77 Model # Tale : omparison of K from heng s Experimenal Resuls wih urren FE K / w/r h/r /r Percen Degrees heng s Exp. Presen FE Difference omparison wih Experimens Finie elemen models for all of heng s experimenal models were generaed and he sress concenraion facor was deermined. heng s experimenal configuraions, his resuls, and he presen FE resuls are lised in Tale. The percen differences for all seven models are also summaried. The percen difference is defined as K eq. K FE )/K FE. heng s experimenal resuls were 9% o % lower han he FE resuls. These differences are due o experimenal inaccuracies which are ecause of he numer of variales involved in he sress-freeing echnique, for example he cuing of he specimens ino slices, he susequen measuremens of he isochromaic fringes and he difficul in measuring he sresses a he counersunk edge. Shivakumar and Newman 99, 99) and Young and Lee 99) have also compared heir FE resuls wih heng s 978) experimenal daa. For model #7, Shivakumar and Newman s resul was % higher, and Young and Lee s resul was 7% lower han he presen FE resuls. Sress oncenraion Resuls from FE nalsis s previousl menioned, a deailed hreedimensional finie elemen analsis was conduced for a wide range of /), /r, and w/r parameers o evaluae heir effec on SF. Throughou he analsis, h/r = was used so ha he loading = ) could e considered remoe wih no or ver lile influence on he hole geomer. The values of /) were sraighshank hole),.,.,.7 and. knife-edge) and he /r values were.,, and. The plae widh-o-hole radius raio, w/r, was varied from. o and he counersunk angle was varied from o o o. The influence of on he SF was firs assessed performing he analsis for values ranging from o hrough 7 o in incremens of o for / =.and/r =.. ddiional analses were also conduced o evaluae he effec of in cominaion wih he counersunk deph raio /) and he plae hickness raio /r)onk. Secondl, he effec of / and /r were assessed for a wide plae w/r = ) wih = o performing analses varing / and /r. Finall, he effec of he plae widh-o-hole radius raio w/r was assessed for a counersink configuraion wih = o performing an analsis for w/r =. o. The resuls of he analses are presened in he following su-secions. lhough he SS hole configuraion is differen from ha of he counersunk hole, for he purpose of presenaion and discussion i is considered o e a special case of he counersunk hole wih / =. The sress concenraion K ) along he ore of he hole B of Fig. ) was examined. The maximum value of K ), hais K, was exraced for furher analsis.. Effec of counersunk angle ) Figure 7 shows he variaion of K ) along / for values ranging from o o o including he resuls for = o ) for h/r = w/r =, /r =and / =.. The K ) is maximum a he counersink B) and i decreases owards oh he free edges. The K ) decreases much more

8 78 oprigh c 7 Tech Science Press M, vol., no., pp.7-9, 7 h/r = w/r = /r = ; / =. B K ) σ B s, deg / Figure 7: Effec of counersink angle on K ) for a wide plae.. h/r = w/r = K K θ = /r = ; / θ c = 8 c [ ] s K.8. Slope m =.. θ = =. c B Figure 8: Variaion of K wih w/r = h/r = ; /) =.; /r =) / = K ) h/r = w/r = /r = ; = º B s / Figure 9: Effec of ounersunk Deph on Sress oncenraion Disriuion for a Wide Plae /r =)

9 General Equaion for Sress oncenraion in ounersunk Holes 79 rapidl owards he counersunk par B-) han owards he sraigh shank par B-). The K ) a is lower han ha a. The K ) a B increases wih. This rend is in agreemen wih Shivakumar and Newman s 99, 99) resuls while Young and Lee s 99) resuls showed an opposie rend. The difference in K eween = 9 o and o and eween = o and o is aou %; herefore, for a small deviaion ± o ) of from o, K can e assumed consan. Figure 8 presens he K for a larger range o o 8 o ) of,for /) =. and /r =.TheK for = is from he SS hole for /r =andhek value for of 8 o is exrapolaed from he following equaion for he loss of plae hickness equal o /. =8 o = = o ) [ /] The K has a nonlinear relaion wih ;however, for a range of o o, effec can e approximaed a linear relaion as illusraed he roken line in Fig. 8. The reason for he monoonic increase in K wih, is ecause of coninuous channeling of load owards he counersink edge and around he hole as increases [Bhargava )].. Effec of counersink deph /) The disriuion of K ) along / for varing counersink dephs is shown in Figure 9 o for /r =. For hin plaes /r ), he maximum of K ),haisk, occurs a he counersink edge. For hicker plaes /r =and)and /., K occurs no a he counersink edge u slighl awa from he edge % of ) and owards he SS porion of he hole. This rend coninues for all shallow counersink configuraions. The K values generaed for differen values of / and /r are lised in Tale and ploed in Fig.. These resuls show ha K increases monoonicall wih / unil / is equal o aou.8 and hen decreases for all plae hicknesses less han. For hick plaes,, K coninues o increase wih / including he knife-edge case), as also noed Shivakumar and Newman 99, 99). Figure also shows ha K increases monoonicall wih /r.. Effec of hickness o radius raio /r) Figures illusraes he effec of /r on K ) for /) =.. Values of /r =., and represen he pracical range of hole configuraions used in he aircraf indusr and /r isused for hick srucures such as hose found in marine applicaions. The K ) increases monoonicall wih /r on he SS porion of he hole excep near he free edge, while i decreases wih /r on he counersunk porion of he hole. Bu he K increases wih /r as illusraed in Figs. and. s also noed in he previous secion, K increases wih plae hickness /r) and counersunk deph /). The variaion of K is nonlinear and coupled wih oh / and /r.. Effec of widh o radius raio w/r) The rae of change of K ) and K wih w/r is higher for smaller w/r and is small for larger w/r. More deails for differen / are in Bhargava ). The K plo is shown in Fig.. Here also, K increases rapidl for w/r < and ecomes infini for w/r =, as prediced. For w/r >, K is nearl consan indicaing ha for wide plaes K is independen of w/r. The plo also includes Hewood s 9) -D soluion; he differences eween he wo are due o he -D effec, which is no much. Such a close agreemen indicaes ha Hewood s widh correcion equaion could e used for counersunk holek. The K versus w/r for differen /) values wih /r =and = o are shown in Fig. and he values of K are lised in Tale. s shown in he figure, he K increases rapidl as w/r ecomes smaller and i ecomes almos consan for w/r >. 7 -D Equaion for Sress oncenraion Facor K ) Resuls of FE resuls can e summaried as follows:. The K occurs a or near he counersink. a) K is a funcion of /r for SS holes.

10 8 oprigh c 7 Tech Science Press M, vol., no., pp.7-9, 7 h/r = w/r = = º /r =.. K.. -D SF B... /..8. Figure : Variaion of Maximum SF as a Funcion of /) /r.. h/r = w/r = / =.; = º K ).. B / Figure : Effec of /r on Sress oncenraion Disriuion for / =. h/r = w/r = = º /..7.. K. B /r Figure : Variaion of Maximum SF as a Funcion of /r

11 General Equaion for Sress oncenraion in ounersunk Holes 8 h/r = /r = ; / =.; = º K ) w/r = w / Figure : Effec of w/r on Sress oncenraion Disriuion / =.).. -D FE K... Smoohened FEM Hewood w + Y ) r w) K H w r = r w)... h/r = /r = ; / = D w/r w/r Figure : Effec of w/r on Sress oncenraion K for a Sraigh-Shank Hole... K.. w / = h/r = /r = ; = º w/r Figure : Effec of w/r on Sress oncenraion K for ounersunk Holes

12 8 oprigh c 7 Tech Science Press M, vol., no., pp.7-9, 7 / Tale : Sress oncenraion K w/r = h/r = ) from FE /r / Tale : The K Resuls for Finie Widh Plaes from FE w/r values K is a coupled nonlinear funcion of /) and /r, and he wo effecs canno e separaed.. K can e approximaed a linear funcion of over he range of angles o o wih as he aseline daa.. K increases as w/r decreases and i ecomes a consan value for large w/r. The widh w/r) effec can e represened Hewood s widh correcion equaion for - D prolems. Based on hese resuls and limiing soluions will e saed laer), a general K equaion is developed here so ha a designer can use i in heir preliminar designs. The K equaion ma e represened as a produc of four funcions, namel, K SS /r) for hickness effec in SS hole, K /), /r) for counersunk effec, K H w/r) for plae widh effec, and K θc ) o accoun for correcion. The general K equaion is K = K SS /r) K S /,/r) K H w/r) K θc ) ) The finie widh effec is inroduced hrough Hewood s -D sress concenraion equaion. The - D SF varies from for large w/r o for w/r =. The widh effec equaion is K H w/r)= + r/w) r/w) ) This equaion was verified laer a -D FE analsis in Bhargava ). Having seleced K H w/r), he remaining hree funcions ecome a correcion o -D SF due o he -D and counersink geomer effecs. The funcions K ss, K and K θc mus var from uni o a finie value depending on he hole geomer. The limiing condiions ha hese hree funcions have o saisf are: a) SS hole /)=) K = K θc = andk SS = for plane sress /r ) and plane srain /r ) condiions.

13 General Equaion for Sress oncenraion in ounersunk Holes 8 ) ounersunk hole / ) k θc = if = o and k θc if o. The K ss, K and K θc are deermined fiing equaions o he FE resuls generaed in he previous secion. The K ss is oained from SS daa, K from counersink daa for = o, and finall K θc from one counersunk hole geomer wih differen values. The FE resuls used for fiing he K ss and K equaions are lised in Tale. 7. The K ss Equaion Fi TheSShole /) = ) resuls in Tale are normalied -D SF) and used o perform he K ss equaion fi. The form of he funcion chosen is such ha i saisfies he limiing condiions ha K SS =for/r plane-sress) as well as /r plane srain). The equaion is wrien in he form K SS /r)= + a/r)m +/r) m ) The four consans a,, m and m are deermined he minimiaion of error eween he equaion and he FE resuls and he susequen reducion of he consans o a simple form. Noe ha consans a,, m and m are emporar variales chosen for curve fi and he do no have an meaning eond his secion. The resuling K ss equaion is K SS /r)= +./r). +/r). ) comparison of he Eq. ) wih he FE resuls is shown in Fig.. The K ss Eq. ) agrees wihin.% of he FE resuls and he correlaion facor of he fi was greaer han The K /,/r) Equaion Fi The FE K resuls for he counersink hole in Tale are normalied SS / = ) daa and hese resuls were used o perform a muli-parameer fi in /r and / o oain he K equaion. simple quadraing polnomial in /) muliplied he /r power funcion was chosen. K, ) = r +a ) r ) a ) ) ) r Through he muli-parameer fi, he consans a, a, and were deermined. These consans were furher simplified o arrive a simple values. The resuling equaion is K, ) = r +. ) r ).. ) ). 7) r Figure 7 compares he equaion K = K SS K wih he FE daa in Tale for differen values of /)and/r. The solid lines represen he equaion and he smols represen he FE daa. The maximum difference eween he equaion and he FE daa is aou % and he correlaion facor of he equaion wih he FE daa is greaer han.99. Thus, he K equaion is sufficienl accurae o accoun for he effec of counersunk deph and plae hickness. 7. The K θc Equaion Fi s illusraed in Figs. 7 and 8, he FE resuls show ha K is independen of he counersink angle for small variaions ± o ) from he aseline case of = o. However, for large variaions of, is a funcion of and over he range o o i can e represened a linear equaion see Fig. 8). Therefore, K θc is represened { K θc )= + m o } ) 8) = o where m is he slope of K versus plo and = o is he SF for = o. = o = K SS/r) K S /,/r) K H w/r) 9)

14 8 oprigh c 7 Tech Science Press M, vol., no., pp.7-9, 7.7. K SS / r ) =.. / r ) +. + / r ) K SS.... h/r = w/r = / = FEM F Daa.. 8 /r Figure : -D sress concenraion facor for a sraigh-shank hole wide plae) h/r = w/r = = º /r =.... K. B s *K SS *K ) FE /r =. /r =. /r =. /r =.... /..8. Figure 7: omparison of -D SF Equaion Thickness and ounersink Effec) wih FE Resuls 7.. h/r = Presen D-FE K H + ) r w) w r = r w) K.. ) =. 7.8 w / r) +.7 w / r). w / r) K YL K w/r ) r r r ) = r w w w w Isida. w x Young and Lee Hewood. w/r Figure 8: omparison of -D SF from Various Soluions for Finie Widh Plae

15 General Equaion for Sress oncenraion in ounersunk Holes 8 Tale : Summar of all m Values for Differen / and /r Values wih w/r = /r / The slope of he line m) is. over o o for / =.and/r =.seefig.8). alculaed slopes m) for differen /) and/r wide plaes w/r = h/r = ) are lised in Tale. The correlaion facor of he fi was greaer han.99. The plo of m versus /r for differen /) values is shown in Fig. he use of smols. Since m varies wih oh / and /r, apower law equaion in /r) was proposed in he form m /,/r)= /r) λ ) where and λ are funcions of /. Them reduces o ero for /) = SS hole). Through a muli-parameer fi o /)and/r), he funcions for and λ were deermined and he are /)= ) [. +.9 λ /)= ) [. ).9 ) + ) ] ) ] ) ) Finall he general K equaion for he counersunk rive hole of an /), /r, and w/r is given as K = K SS /r) K /,/r) K H w/r) K θc ) ) Each of hese funcions are K SS /r)= +./r). +/r). ) K, ) = r +. K H w/r)= and K θc )= ) r ).. + r/w) r/w) { + m o } ) = o ) ). ) r ) 7) For special cases, he aove equaion simplifies o: a) ounersink ngle ) is o K = = o = K SS /r) K /,/r) K H w/r) 8) ) Wide Plae and = o K = K SS /r) K S /,/r) 9) 8 Verificaion of K Equaion 8. omparison wih presen FE Resuls The K Eq. ) was verified comparing i wih he FE resuls used for fiing he equaion and he new daa generaed for his purpose. The comparisons were made firs for he wide plae w/r =) and = o, hen for he wide plae w/r =) for differen values, and finall for finie widh plaes. a) Wide plae w/r = h/r = ) and = o Figure 9 shows he comparison of he FE resuls and Eq. ) for differen /) and/r

16 8 oprigh c 7 Tech Science Press M, vol., no., pp.7-9, 7.. m =.)/r).9. Slope, m... / =..7.)/r).78.8)/r) )/r) /r Figure 9: Variaion of m wih /r for Differen / Values h/r = w/r = = º /r =.... K. B / Figure : omparison of Eq. ) wih FE for Wide Plae w/r = h/r = ) h/r = w/r = = º /r =.. K S.. S / Figure : omparison of Eq. ) wih FE for = o w/r = h/r = )

17 General Equaion for Sress oncenraion in ounersunk Holes 87 7 h/r = w/r = = º /r =... K S. s S / Figure : omparison of Eq. ) wih FE for = o w/r = h/r = ) K / =.7.. h/r = /r = ; = º B s 9 w/r Figure : omparison of Eq. ) wih FE for Finie Widh Plaes /r =;h/r = ) 7 / =... h/r = /r = ; = º K B / = w/r. Presen Eq. Young and Lee s Eq. Figure : omparison of Presen and Young and Lee s K Equaion Resuls as a Funcion of w/r /r =)

18 88 oprigh c 7 Tech Science Press M, vol., no., pp.7-9, 7 / Tale : Percen Difference Beween Eq. ) and FE for a Wide Plae w/r = h/r = %Di f f. = K eq ) K FE K FE /r Tale : Percen Difference Beween Eq. ) and FE for Finie Widh Plaes /r =;h/r = ) / %Di f f. = K eq ) K FE K FE w/r values. The percen difference eween he wo resuls is summaried in Tale. The maximum error in he equaion is aou % for /r =and / =., and i is less for all oher cases analed. ) Wide Plae and Differen Figures and compare K from Eq. ) wih FE resuls for = o and o for differen values of / and /r. The percen error eween he wo is aou % or less for oh exreme angles. c) Finie Widh Plae and = o Figure shows a comparison eween he FE resuls and Eq. ) for finie widh plaes for differen /. The percen difference eween he wo resuls is summaried in Tale. The difference is less han or equal o % for all cases excep for / =.7andw/r =, where he difference is aou 7%. The error is large for a ver narrow widh plae w/r =., which is no a pracical case. Based on he aove comparison i is concluded ha he Eq. ) is accurae for a wide range of /r, /, w/r and. The maximum error is aou % or less for mos of he cases, excep for a few configuraions where he error could e as high as 7%. 8. omparison wih Young and Lee s Design Equaion In 99, Young and Lee proposed a design equaion ased on heir FE analses and Briish erospace s experimenal design equaion for he widh effec. The K equaion is represened K YL and is given K YL =K S /)) YL K w w/r)) YL ) where K S /)) YL = /) K w w/r)) YL =. 7.8w/r)+.7w/r).w/r)

19 General Equaion for Sress oncenraion in ounersunk Holes 89 Noe ha K YL is no a funcion of /r and,and limied o w/r. This equaion was inended for aircraf join configuraions. The Eq. ) and Young and Lee s Eq. ) were compared for four plae hicknesses, namel, /r =.,,and,for a wide range of w/r, /) and = o.onl he comparison for /r = is presened in Figs. 8. From his comparison, i can e concluded ha he wo K equaions differ widel and his difference ma e as small as ero or as large as %. Because he presen K equaion is ased on an accurae FE model resuls and a deailed analsis, Eq. ) is elieved o e accurae. 8. omparison of FE wih heng s Experimen Because K Eq. ) was developed using configuraions differen from hose in heng s experimen, herefore he equaion is verified comparing i wih FE resuls for hese configuraions. heng s experimenal configuraions and he associaed FE resuls are lised in Tale. The difference eween FE resuls and he K equaion are lised in Tale 7. The difference is less han % excep for ver hick plae /r =.79), where he difference is aou.9%. Thus, K Eq. ) is accurae and can e used for a wide range of counersunk hole configuraions in a plae sujeced o ensile loading. 9 onclusions deailed hree-dimensional finie elemen sress analsis was conduced on counersunk rive hole in a plae sujeced o ensile loading. The analsis included a wide range of counersunk dephs / ), plae hicknesses. /r ), counersink angles o o ) and plae widhs. w/r ). The sress concenraion K ), along he edge formed he inersecion of = plane and he hole oundar was analed. The maximum value of he sress concenraion is he sress concenraion facor K, and is variaion wih hole geomeries and plae widh are assessed. The FE resuls show ha K is influenced he counersunk angle, he counersunk deph raio /), he hickness raio /r and he widh raio w/r. TheK occurs a or near he counersink. However, for hicker plaes /r ) and shallow counersink dephs /.) holes, K occurs slighl awa from he edge % of ),owardshe SS porion of he hole. The K increases monoonicall wih plae hickness /r)and / excep a / = for moderael hin plaes /r. Boh /r and / have a nonlinear coupled relaionship wih K. The counersunk angle has a ver small impac on K for small deviaions ± o ) from = o. However, for large variaions of, K increases wih and his variaion over he range o o can e approximaed a linear equaion. The K increases wih decreasing w/r and i ecomes infinie as w approaches r; whereas for w/r >, K is unaffeced w/r. The presen FE resuls also confirm he resuls and rends ha are oserved in he lieraure. The presen FE resuls for he sraigh shank hole in wide plae agree wih Sernerg and Sadowsk and Shivakumar and Newman and he also agree wih revised resuls of Shivakumar and Newman for counersink holes. Because of coarse FE mesh used in Young and Lee, he equaion grossl overesimaes K for man cases. Based on he FE resuls and he limiing condiions of configuraions, a general K equaion for SF was developed. The equaion is given K = K SS /r) K S /,/r) K H w/r) K θc ) K SS /r)= +./r). +/r). K H w/r)= and K θc )= + r/w) r/w) { + m o } ) = o The K equaion is accurae wihin % of he finie elemen daa for a wide range of widhs, counersunk dephs, plae hicknesses and, excep for a few cases where he error could e as large as 7%. Young and Lee s equaions varied consideral from he presen equaion. The difference

20 9 oprigh c 7 Tech Science Press M, vol., no., pp.7-9, 7 Model# Tale 7: Verificaion of Presen K Eq. ) for heng s Experimenal Models Degrees K Presen FE Eq. ) %Di f f. = K eq ) K FE K FE ma e as small as ero or as large as %. In summar, he sress concenraion equaion presened in his paper is general, accurae, and saisfies all limiing cases. cknowledgemen: The auhors wish o hank Dr. Rajapakse and he Office of Naval Research hrough he gran no. N -- and he Federal viaion dminisraion su-conrac hrough Iowa Sae) hrough he conrac no. DTF-98-D-8 for heir suppor. References NSYS., NSYS Inc., anonsurg, P Bhargava,. ): Three-Dimensional Tensile Sress and Srain oncenraion Equaions for a ounersunk Hole. Ph.D. Disseraion, Norh arolina griculural and Technical Sae Universi, Greensoro, N. hen, Wen-Hwa; hen, heng-hung ): On Three-Dimensional Fracure Mechanics nalsis an Enriched Meshless Mehod. MES: ompuer Modeling in Engineering & Sciences, vol. 8, no., pp heng, Y. F. 978): Sress-oncenraion Facors for a ounersunk Hole in a Fla Bar in Tension and Transverse Bending. J. ppl. Mech.,vol., no., pp Folias, E. S.; Wang, J. J. 99): On he Three- Dimensional Sress Field round a ircular Hole in a Plae of rirar Thickness. ompu. Mech., vol., no., pp Green,. E. 98): Three-Dimensional Sress Ssems in Isoropic Plaes. I. Trans. Roal Soc. London, ser., vol., pp Hewood, R. B. 9): Designing Phooelasici. hapman and Hall, London. Li, Q.; Shen, S.; Han, Z. D.; S. N. luri ): pplicaion of Meshless Local Perov- Galerkin MLPG) o Prolems wih Singulariies, and Maerial Disconinuiies, in -D Elasici. MES: ompuer Modeling in Engineering & Sciences, vol., no., pp Neuer, H. 9): Theor of Noch Sresses: Principles for Exac Sress alculaion. J.W.Edwards nn ror, Michigan). Pilke,W.D.997): Peerson s Sress oncenraion Facors, nd Ediion. John Wile & Sons, Inc. Savin, G. N. Eugene Gros, ransi.) 9): Sress oncenraion round Holes. Pergamon Press, Inc. Shivakumar, K. N.; Bhargava,.; Hamoush, S. ): n Equaion for Sress oncenraion Facor in ounersunk Holes. M: ompuers, Maerials & oninua, vol., no., pp Shivakumar, K. N.; Newman, J.. Jr. 99): Three-Dimensional Sress oncenraion Facor in Sraigh-Shank and ounersunk Rive Holes in Plaes Sujeced o Various Loading ondiions, NS TP 9. Shivakumar, K. N.; Newman, J.. Jr. 99): Three-Dimensional Sress oncenraion Equaions for Sraigh-Shank and ounersunk Rive Holes in Plaes Sujeced o Various Loading ondiions. Jl. pplied Mechanics, Trans. of SME, vol., pp Sernerg, E.; Sadowsk, M.. 99): Three-

21 General Equaion for Sress oncenraion in ounersunk Holes 9 Dimensional Soluion for he Sress oncenraion round a ircular Hole in a Plae of rirar Thickness. J. ppi. Mech., vol., no., pp Whale, R. E. 9): Sress-oncenraion Facors for ounersunk Holes. Exp. Mech., vol., no. 8, pp. 7-. Young, J. B.; Lee, K. K. 99) Sress concenraion facors in counersunk holes. eronauical J, pp Zienkiewic,. O.; Talor, L. R. 989): The Finie Elemen Mehod, Vol I: Basic formulaions and linear prolems. London: McGraw-Hill.

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