Stress Analysis of Infinite Plate with Elliptical Hole

Transcription

1 Sess Analysis of Infinie Plae ih Ellipical Hole Mohansing R Padeshi*, D. P. K. Shaa* * ( P.G.Suden, Depaen of Mechanical Engg, NRI s Insiue of Infoaion Science & Technology, Bhopal, India) * ( Head of, Depaen of Mechanical Engg, NRI s Insiue of Infoaion Science & Technology, Bhopal, India) ohan.padeshi@gail.co*, dpkshaa@gail.co* ), Absac In his pape he elecosaic poble fo hoogeneous isoopic infinie plae ih he loading a infiniy (in and ) fo ellipical hole is solved.fo his e have used he Schaa s alenaing echnique given by Sokolnicoff and Ukadgaonka. Afe ha he analyical soluion is used fo geneaion of code in C++ o find ou he sess coponens a a given paicula poin. Keyod: Sess analysis, Infinie plae, Ellipical hole. I.INTRODUCTION The pesence of ellipical hole in plaes and achined coponens hich ae subjeced o loading inoduces sess concenaion in he viciniy of holes. Sudden changes in coss secion lead o sess concenaion a a localied poin. I is ipossible o esiae he ise of sesses in neighbohood of hole using eleenay heoy of sengh of aeials. The sess concenaion depends on geoey of coponens, sie of hole, and loading condiion. A classical exaple is plae loaded ih ension hich conains cenally locaed hole is consideed hee. The soluions in fo of siple analyical equaions fo he ehod of coplex vaiable appoach can give us he check agains he effec of change in he paaee in design pocess. -Infinie Plae (neglecing edge effec) -Majo Axis: a, Mino Axis: b -Bounday Condiion: x y xy P; ; ; Sol: The above poble can be solved by vaious ehods. The elaso plasic poble fo hoogeneous isoopic plae ih loading a infiniy fo cicula, ellipical, iangula, ecangula holes and cacks ae solved using a novel ehod called Scha s Alenaing Mehod given by Sokolnikoff and Ukadgaonka ih successive appoxiaions. Iniially a plae ihou any hole is consideed. The bounday condiion on he hole bounday is found ou. To nullify his, he negaive of his bounday condiion is applied on he hole.the soluion of his poble is supeiposed on he fis soluion hich give he equied closed fo soluion.[,] By he ehod e can supeposiion he o pobles o ge he final soluion. II. THEORETICAL SOLUTION: A. Poble Definiion:- P Fis Poble Second Poble

2 To obain he bounday condiion fo fis poble can be found ou as given by he foula f () Fo second poble bounday condiion can be iposed as = f() f() Fis poble: Re Puing he given bounday condiion and inegaing e ( is bounday poin lie on hole). (. =R) B.Second Poble: ge he () P /; No, i We ge, () P /; No, e use he confoal apping fo apping he egion ouside of ellipical o egion ouside he uni cicle. Z Plane Plane f() = R ( + /) f ()=R(-/ ) f ()=R(- ) R=a+b/; M= (a-b) / (a + b) f f() () () () () f() PR( /)/R( /)/(R( )*PR( )/PR(/ )/ PR [ ]/ f() = f() f( ) d i PR [ ] i ( ) ( ) PR [ ] Solving Cauchy s inegal, PR [ ] fd () ( ) i PR( ) ( ) PR( ) [ ] i ( ) PR( ) [ ] ( ) Finally e ge () () () PR PR ( ) ( ) PR [ ] () () () ( )

3 PR PR ( ) ( ) [ ] PR ( )( ) [ ] ( ) Finding sesses: Re Re{ } f ( ) P [ ] Sepaaing eal and iaginay pa and conveing in pola fo e ge AA B ( A G) AB D C F H AC AE BB BD AD E B C D C F H A B C 6 ( cos ) L cos M () i i e x ( *cos ) ( *sin ) ( sin cos ) D E F G H J K 6 Sepaaing eal and iaginay pa Whee, N * D N * D P D D P N * D N * D D D N AAABcos ACsin ADcosAEsin N BBBCsin BDcos ADsinAEcos D ( JK)cosLcosM D ( JK)sinLsin y ( *sin ) ( *cos ) ( sin cos ) () xy ( )*cos *sin ) (cos sin ) () C Poble Definiion:-

4 -Infinie Plae (neglecing edge effec) - Majo Axis:a, Mino Axis:b -Bounday Condiion: x y xy ; P; ; Sol: By he ehod e can supeposiion he o pobles o ge he final soluion. No, i We ge, () P /; No, e use he confoal apping fo apping he egion ouside of ellipical o egion ouside he uni cicle. Z Plane Plane + f() = R ( + /) f ()=R(-/ ) f ()=R(- ) Fis Poble Second Poble To obain he bounday condiion fo fis poble can be found ou as given by he foula f () Fo second poble bounday condiion can be iposed as = f() f() Fis poble: Re [] R=a+b/; M= (a-b) / (a + b) f f() () () () () f() PR( /)/ R( /)/(R( )*PR( )/ PR(/ )/ PR [ ]/ =P/ + P/ + P/ ( is bounday poin lie on hole) =P/+ PR/ (Z) (. =R) Puing he given bounday condiion and inegaing e ge he () P /;

5 D. Second Poble: f() = f() Finding sesses: Re P PR Sepaaing eal and iaginay pa and conveing in pola fo e ge R P cos i i e f ( ) d i R Pd P d i i Solving Cauchy s inegal, = -PR/() f( ) d i P d PR d [ ] i ( ) PR Finally e ge R ( ) () () () P PR () () () P PR R ( ) Sepaaing eal and iaginay pa PR R R P ( ) c o s P R R s in Solving equaion and e ge P R R ( ) ( ) c o s P R R R ( ) ( ) c o s P R R s in [,, 5] III. COMPUTER CODE FOR THE ANALTICAL SOLUTION: 5

6 The above analyical expessions ae used o find ou sess filed aound he hole. Also i geneaes he ex esul file of all he nodes (in he egion hich is fou ies he hole adius) infoaion (all sess coponens). IV. CONCLUSION By adoping he copue code e can find ou he sess a any given paicula poin in he neighbohood of cicula hole. The esul file so as geneaed can be used fo visual display of esul so obain such as eshing and hen conou plo fo he sess coponens. V. REFERENCES:. Muskhelishvili N. I., Soe Basic Pobles of he Maheaical Theoy of Elasiciy, P. Noodoff Ld., (96).. Ukadgaonka V. G. and P. J. Aasae; A Novel Mehod of Sess Analysis of an Infinie Plae ih Ellipical Hole ih Unifo Tensile Sess, IE (I) Jounal- ME, Vol. 7, pp Jess Coe, Julie Bannanine; Fundaenals of eal faigue analysis.. A.E.H Love I. S., Maheaical Theoy of Elasiciy, Cabidge univesiy pess. 5. Coplex Analysis, Coplex Vaiables, Advance Maheaics 6. Mohaed Aeen Copuaional Elasiciy Naosa Pub. 6