Karunes, B

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IACS Institutional Repository

STRESS DISTRIBUTION IN AN INFINITE PLATE WITH AN ELLIPTIC HOLE ACTED UPON BY A FORCE AND A  COUPLE A T AN INTERNAL POINT *54Bv B . K A R IW E SD epar tm ent  of A p p u e d  P h y s i c s , Un iv e r s it y  Colm-c'.k of S cifnck a m »T ec h NOLOf; v , C alcxITTa(Received for puhlicafinn, March 2,ABSTRACT. The Function theoretic method of ^rn^chelisvili fi<)3>'> for solvine two (h’mensionnt problems in elnsticity is employed to olitniii solutions of the problems of stress distribution in n lpr>?e plate containing an elliptic hole under the action of an isolated force or a couple at anv point inside the plate From these solutions particular cases of the force and couple acting on the boundary of the h(de aie derived and the results are compared with lh">se obtained by Rothman ^1^50), who used complex potentials as introduced by vSlevenson (1^45).I N T R O D n C r  T O NTn a re c e n t p a p er 'R o th m an  ( io 5 d ) lias obtain ed the solution o f the p ro b le m s o f s tre ss  d istrib u tio n  in an in fin ite p late  w ith  ellip tic  an d circu la r  h o les u n d e r th e a c tio n s  o f isolated fo rces on the b o u n d a ry  o f the hole. F o llo w in g  S te v e n s o n  1 1 9 4 5 ) ,  he h as in tro d u ced  tw o  fu n c tio n s  0(e) and  in  te rm s o f w h ic h  th e stresses can be e x p re sse d . Tn the p ie se n t p ap er the  so lu tio n  is  g iv e n  o f th e p ro b le m s o f stress d istrib u tio n  in an in finite p late  w ith  an  e llip tic  h o le  u n d e r th e actio n  of an i.solated fo rce or a co u p le  at a n y  p o in t in sid e  th e p la te . T h e  m eth o d  o f M u sc h e lisv ili ( 1 9 3 3 )  h as been  fo llo w e d  h e re . It  m a y  be m en tio n ed th a t th e co m ])lex stress fu n ctio n s  in tro d u c e d  b y  M u s c h e lis v ili are co n n ected  w ith  the co m p le x  p o ten tials  in tro d u c e d  b y  S te v e n s o n . O n e of th e fu n c tio n s  of Steven so n  is a c o n s ta n t  m u ltip le  o f on e o f th e fu n c tio n s  o f M u sch e lisv ili an d th ed e r iv a tiv e  o f th e  seco n d  function of S te v e n so n  is a co n stan t m u ltip le  o f th eseco n d  fu n c tio n  o f M u s c h e lis v ili. R o th m a n ’s p ro b lem s a p p ea r a s  p a rtic u la rcases of the p ro b le m s so lv e d  h ere, if  w e  m a k e  the p o in t w h e re  th e fo rce  isa p p lie d  te n d  to  th e b o u n d a ry  o f th e h o le. A n o th e r  p a rtic u la r  c a ^  is  th e  p ro b le m  o f an  in fin ite  p late  w ith  a  c irc u la r  ho le acted  u p o n  b y  a fo rce  at a n  in te r n a l p o in t. T h is  p ro b lem  h as b een  so lved  b y  S e n  (194.S) b y  a  m e th o d  w h ic h  he h a s  d e velo p ed  in th e case  o f a  fo rc e  a c tin g  a lo n g  a ra d iu so f  th e  h o le . „  , ..* Comnionirated by Prof P. C. Mahantia—iSjaP—9440 B . Kara tiesIt has been shown by Muschelisvili (1933) that in state of generalisedplane stress the stress combinations x x  +  yy  and yy  — x x  + j i  xy  of the average stresses are given byx.\ + y y  =  4 y real part of2 =  •••andy y  — XX +  2txy =  2[z<l>i''iz)+ ^i'{z)^ ... (2)where ^i(s) and ^i(r) are two analytic functions of z( =  x  + iy) and a bar over a function represents the complex conjugate of the function.The resultant traction across an arc A B  (which the material on the right exerts on the left) is given byI)(3)(4)(5)_ _ _ aX  + i Y — —il<l>i(z) + ztfti(.z)—\(>,{z)']Aand the moment of the traction across A B  about the origin is given by_ nMo =  real part of [ — 22^ /(z )— 2^1(2)+ ^3(2)! <The displacements arc obtained from the relation 2fi{u + iv) — Ktl>t{z)- 2 ^ /(2 )-^ '1(2 )where7v ='^1 ± 3M*v , •\ '  +  /xTo determine <p,(z) and ^i(z) when stresses are prescribed on the boundary, we have from (3)where0,(2) + 2^/(2)+ (z)—/ , + i/3 1^,(2) +20/(2) + (z)=/i — j/a Jf i  +  ifa — i J {Fi +  iF 2)d 8 +  constant.... (6)n.F,(S) dS  and F ii^ )  dS  being the x  and y  components of tractions across the arc dS.For an isolated force X  + iY  at the origin supposed to be in the interior of the plate, the parts of 0/2) and ^i(z) which give rise to the force areX + i Y  ,2 t ( X  +  l)(7)Stress Distribution in an Infinite Plate, etc. For a couple of moment M, we have them as441iM2n/... (S)When the force A '+ iY  acts at the point 2;«, the relevant parts of and (z) are^xiz)=  log U - r „ ) +2Jt ( K  + I) 2t (/\ 4- l)*(9)and when the couple acts at the point we have,i>iiz) =  oxP,(z) = iM( 1 0 )If 2 — transforms the region outside the hole in ihe 2 --plane conformally on the region outside the unit circle y in the  ^—plane, we have on y0((r) + <xUr)Y(o-)>(<r)+ yp(cr)=f, +if.ji>(fr)+ <P"{<r)+xp{<r)=^ft~iftw VfT}where0 (il =?>i{">(D} an d  =  a n d  i  =  <r g iv e s  a  p o in t  on  y, so th a t  cr<r =  1.I N F I N I T E  P L A T E  W I T H  A N  IC L  L  I P  T  I C H O L E  A C T  E  D U P O N  B Y  A N  I S O L A T E D  F O R C EI«et th e  b o u n d a ry  of th e  ho le  be g iv en  by=  IT h e  tra n s fo rm a tio no <  m <  1 , i? >  o... (12)... (1 3 )442 B . Karunestransforms the region outside the hole in the s-planc, to the region outside the circle 7, '  ^ | ~  i in the 1,-plane and the point at infinity in the two planes correspond. Substituting from (13) in (11) and replacing <r by i/o-, w'e get on y(t ( i  ~  m tr ) \  <r /  \  cr /J  l )  +  4 * ( 4 ) = / ,  -  ,7.\ cr / cr^ — m... (141If a force X  + iY  act at the point 2„ of the plate, we may take----(2-2o) -t2n-{/v + 1)27tU \ + J )  27r(K “i- i ) C — Cowhere 0i®(r) and are analytic outvside the ellii>lic hole (12),Using the transformation (13) we write... (15)log i L ; y ( f c - " L 4 ^ .(y2n-'/v-t-l) s^.02 ^ X 1+ 1). X  + iY  + )«2;r(K + r)(16'where and are analytic outside the unit circle y in the ^-plane.Since the boundary of the hole is free from tiactions, substituting in (14) /j ~ o , /2®=o, we gel on the unit circle yr w )  4  ( J . ) + n - ( i ) =  .+5 ': logo -(i—tMO"*) \ t r /  \  cr /  2 > r(/v + j) tri^ o— log (l ^  X - i Y  ioUr-^o)(cr{o-  m)2^Jk V j )^o'Ur) +  =  log\<r /  o-‘ ~ m  ' 2h(K + i ) ®— — i Y)  A' +  i Y  ^e(i — wtr)2n ( K + i )  "  U a --L ){< r} : . -m i '(1 7 )

Stress Distribution in an Infinite Plate with an Elliptic Hole Acted Upon by a Force and a Couple at an Internal Point

http://arxiv.iacs.res.in:8080/jspui/bitstream/10821/1771/1/Stress%20distribution%20in%20an%20infinite%20plate_B%20Karunes.pdf

Stress Distribution in an Infinite Plate with an Elliptic Hole Acted Upon by a Force and a Couple at an Internal Point

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