MECHANICS OF ADVANCED MATERIALS AND STRUCTURES 2016,VOL.0,NO.0,1 20 http://dx.do.org/10.1080/15376494.2016.1194505 ORIGINAL ARTICLE Strtchng and bndng dformatons du to normal and shar tractons of doubly curvd shlls usng thrd-ordr shar and normal dformabl thory P. H. Shah and R. C. Batra Dpartmnt of Bomdcal Engnrng and Mchancs, Vrgna Polytchnc Insttut and Stat Unvrsty, Blacksburg, Vrgna, USA ABSTRACT W analyz statc nfntsmal dformatons of doubly curvd shlls usng a thrd-ordr shar and normal dformabl thory (TSNDT) and dlnat ffcts of th curvlnar lngth/thcknss rato, R/a, a/h, radus of curvatur/curvlnar lngth, and th rato of th two prncpal rad on through-th-thcknss strsss, stran nrgs of th n-plan and th transvrs shar and normal dformatons, and stran nrgs of strtchng and bndng dformatons for loads that nclud unform normal tractons on a major surfac and qual and oppost tangntal tractons on th two major surfacs. In th TSNDT th thr dsplacmnt componnts at a pont ar rprsntd as complt polynomals of dgr thr n th thcknss coordnat. Advantags of th TSNDT nclud not ndng a shar corrcton factor, allowng strsss for monolthc shlls to b computd from th consttutv rlaton and th shll thory dsplacmnts, and consdrng gnral tractons on boundng surfacs. For lamnatd shlls w us an quvalnt sngl layr TSNDT and fnd th n-plan strsss from th consttutv rlatons and th transvrs strsss wth a onstp strss rcovry schm. Th n-hous dvlopd fnt lmnt softwar s frst vrfd by comparng dsplacmnts and strsss n th shll computd from t wth thos from thr analytcal or numrcal solutons of th corrspondng 3D problms. Th stran nrgy of a sphrcal shll s found to approach that of a plat whn R/a xcds 10. For a thck clampd shll of aspct rato 5 subjctd to unform normal tracton on th outr surfac, th n-plan and th transvrs dformatons contrbut qually to th total stran nrgy for R/a gratr than 5. Howvr, for a cantlvr shll of aspct rato 5 subjctd to qual and oppost unform tangntal tractons on th two major surfacs, th stran nrgy of n-plan dformatons quals 95 98% of th total stran nrgy. Numrcal rsults prsntd hrn for svral problms provd nsghts nto dffrnt dformaton mods, hlp dsgnrs dcd whn to consdr ffcts of transvrs dformatons, and us th TSNDT for optmzng doubly curvd shlls. ARTICLE HISTORY Rcvd 30 Dcmbr 2015 Accptd 12 May 2016 EYWORDS Bndng and strtchng dformatons; doubly curvd lamnatd shlls; stran nrgy; TSNDT 1. Introducton Lamnatd compost shlls ar wdly usd n arcraft, arospac, marn, automobl, powr gnraton, and chmcal ndustrs. A shll s calld shallow (dp) f ts rs quals at most (at last) on-ffth of ts smallst planform dmnson [1]. A shll s trmd thck, modratly thck, and thn f th rato of th smallst curvlnar lngth and/or th smallst radus of curvatur to th thcknss s, rspctvly, at most 10, btwn 10 and 20, and at last 20 [1]. For thn shallow shlls, classcal thors, such as Lov s frst approxmaton thory (LFAT) [2], Donnll s[3], Sandrs [4], and Flügg s [5] shll thors, among othrs, accuratly prdct thr mchancal dformatons. Th LFAT s basd on th followng postulats: () th shll s thn and shallow, () dsplacmnts and dsplacmnt gradnts ar nfntsmal, () th transvrs normal strss s nglgbl, (v) a normal to th mdsurfac stays normal to t durng th dformaton procss, and (v) th shll thcknss, h, dos not chang. It s assumd that h/r << 1and,hnc,trmsz/R 1 and z/r 2 ar <<1, whr z s th thcknss coordnat wth th orgn at th shll s md-surfac, and R 1 and R 2 ar th prncpal rad of curvatur. Postulats (v)and(v)mplythatthtransvrssharandthtransvrs normal strans vansh. Donnll s [3], Sandrs [4], and Flügg s [5] shll thors ar also basd on th abov fv hypothss. Howvr, thy us dffrnt approxmatons for strans and th shll curvaturs. Donnll s thory [3] nglcts gradnts of n-plan dsplacmnts n xprssons for th shll curvaturs. Ths maks th thory nconsstnt wth rspct to rgd body motons. For xampl, raus [6] showd thatwhn a rgdbody rotaton s appld to a nonsphrcal curvd shll or an unsymmtrcally loadd shll of rvoluton, torson s nducd n th shll. In Sandr s thory [4], xprssons for shll curvaturs ar modfdtonsurthatargdbodymotondosnotnduc strans n th shll. Flügg [5] xpandd th trms 1/(1 + z/r ) ( = 1, 2) apparng n xprssons of strans as bnomal srs and rtand trms up to scond ordr n z/r. For thck and modratly thck shlls, transvrs dformatons may bcom sgnfcant, n whch cas classcal thors wll not provd accurat rsponss. Rssnr [7] pontd out n 1947 that for sandwch shlls wth (te f )/(he c ) >> 1both transvrs shar and transvrs normal dformatons should b consdrd. Hr, t and E f (h and E c )qual,rspctvly,th thcknss and th longtudnal modulus of th fac sht (cor). CONTACT R. C. Batra rbatra@vt.du Dpartmnt of Bomdcal Engnrng and Mchancs, Vrgna Polytchnc Insttut and Stat Unvrsty, 333E Norrs Hall, M/C 0219, Blacksburg, VA 24061. Color vrsons of on or mor of th fgurs n ths artcl ar avalabl onln at www.tandfonln.com/umcm. 2016 Taylor & Francs Group, LLC
2 P. H. SHAH AND R. C. BATRA Rssnr dvlopd a thory wth ach fac sht modld as a mmbran and th cor as a 3-dmnsonal (3D) contnuum. For lamnats wth a snusodal prssur of ampltud q appld on th top surfac, Vl and Batra [8] found that th avrag transvrs normal stran nar a tracton fr dg s of th ordr of q/e, whr E quals Young s modulus n th fbr drcton. Th frst-ordr shar dformaton thory (FSDT) assums constant transvrs shar strans across th shll thcknss, and an approprat shar corrcton factor s usd to account for nonunform through-th-thcknss shar strans. Hghr-ordr shar dformaton thors [9 13] do not rqur a shar corrcton factor. Rddy and Lu [11], LwandLm[12], and Xao-Png [13] dvlopd thrd-ordr shar dformaton thors (TSDTs) for analyzng shlls dformatons n whch th nplan dsplacmnts ar xprssd as complt polynomals of dgr thr and th transvrs normal dsplacmnt s assumd to b constant across th shll thcknss. Ths lads to a parabolc dstrbuton of transvrs shar strans through th thcknss and zro transvrs normal stran. Th numbr of ndpndnt unknowns nvolvd n ths thors s rducd to that for th FSDT by rqurng that transvrs shar strsss vansh on th major surfacs of th shll. Xo-Png [13] ncorporatd Lov s frst-ordr gomtrc approxmaton and Donnll s smplfcaton. Shll thors accountng for transvrs shar and normal dformatons hav bn dvlopd, among othrs, by Hldbrand t al. [14], Rssnr[15], and Whtny and Sun [16, 17]. Brt [18], Lssa[19], Qatu[1, 20], andlwtal.[21], among othrs,havrvwdshllthors.foracompltlstofpaprs on shlls, th radr should s th wbst www.shllbucklng. com dvlopd and rgularly updatd by Dr. Davd Bushnll. For analyzng dformatons of lamnatd shlls, thr an quvalnt sngl layr (ESL) or a layr-ws (LW) shll thory s oftn mployd. In th ESL (LW) thory th numbr of unknownssthsamas(numbroflayrstms)thatfora monolthc shll. Thus, a LW thory s computatonally mor xpnsv than an ESL thory. Howvr, n an ESL thory th transvrs normal and th transvrs shar strsss obtand from consttutv rlatons and th shll thory dsplacmnts ar gnrally not accurat and may not satsfy tracton contnuty condtons across ntrfacs btwn adjonng layrs and tracton boundary condtons on major surfacs. Ths strsss computd usng a strss rcovry schm (SRS), whch usually nvolvs ntgraton of th 3D qulbrum quatons along th thcknss drcton by startng from a major surfac of th shll, ar rasonably accurat. Dffrnt ESL and LW thn and thck plat/shll thors hav bn rvwd by Carrra [22], Ambartsuman [23, 24], Rddy[25], Rddy and Arcnga [26], and apana [27]. Th SRS dscrbd abov s a on-stp mthod [28] and has bn mployd by Pagano [29] and Rohwr [30], among othrs. It s possbl that tractons on th top surfac computd usng ths schm startng from th bottom surfac do not satsfy th appld surfac tractons thr. Ths dffrnc can b usd to masur rror n th numrcal soluton. Tornabn t al. [31] mployd th Murakam functon for satsfyng th tracton boundary condton on th top surfac. Th two-stp mthods, such as thos mployd by Noor t al. [32, 33] and Malk and Noor [34], us tratv tchnqus to comput th transvrs strsss and satsfy tracton boundary condtons at both major surfacs. Rohwr t al. [28] and ant and Swamnathan [35] hav rvwd dffrnt mthods to stmat ntrlamnar transvrs strsss. Th stat-spac approach usd by Tarn and Wang [36] s a on-stp mthod and smultanously ntgrats along th thcknss drcton th thr dsplacmnt componntsandththrtransvrsstrsss. Vdol and Batra [37] and Batra and Vdol [38] dducd a varabl ordr plat thory for pzolctrc and orthotropc lnar lastc plats n whch th ordr up to whch trms n th thcknss coordnat, z, ar kpt s a varabl. Thy calld th plat thory mxd ( compatbl ) whn th consttutv rlatons for th plat ar drvd from th Hllngr Prang Rssnr prncpl (from th plat thory dsplacmnt fld and th consttutv rlaton of th 3D lnar lastcty thory [LET]). Th mxd thory xactly satsfs tracton boundary condtons spcfd on th two major surfacs of th plat. Batra and Vdol [38] showd that for a thck bam rsults from th mxd plat thory ar n bttr agrmnt wth th LET solutons than thos from th compatbl thory of th sam ordr. Qan t al. [39] and Batra and Amman [40] havusdthcompatbl(mxd) thory of dffrnt ordrs to show that n-plan mods of fr vbraton n a thck plat ar wll capturd whn = 3. Qan and Batra [41] hav xtndd th compatbl thory to analyz thrmo-lastc dformatons. Batra and Xao [42] hav shown that a LW compatbl thrd-ordr shar and normal dformaton thory (TSNDT) gvs strsss n curvd lamnatd bams that agr wll wth thos obtand by usng th LET. Usng th TSNDT, Shah and Batra [43] found that for a monolthc clampd squar plat loadd wth a unform normal tracton on a major surfac, th stran nrgy of transvrs shar dformatonscanbabout7%(20%)ofthtotalstrannrgyofdformatons for aspct rato (.., lngth/thcknss) of 20 (10), and can b as hgh as 50% for an aspct rato of 5. In ordr to optmally dsgn a doubly curvd shll for a gvn applcaton, t s mportant to undrstand how gomtrc paramtrs (.g., rato of two prncpal rad, rato of th maxmum prncpal radus of curvatur to th curvlnar lngth, curvlnar lngth/thcknss) affct ts dformatons. Sh t al. [44] nvstgatd th ffct of th rato of two prncpal rad of curvatur on frquncs of fr vbraton of monolthc sotropc doubly curvd shlls usng th LFAT. Thy found that natural frquncs ncras wth an ncras n th R 2 /R 1 rato for a gvn R 2 /l 2 rato and dcras wth an ncras n th R 2 /l 2 rato for a gvn R 2 /R 1 rato, whr l 1 and l 2 ar th two planform lngths. Fan and Zhang [45],Wu t al.[46],and Huang t al.[47] computd analytcal solutons for statc dformatons of lamnatddoublycurvdshllsandrportdthcntrodaldsplacmnt of shlls for dffrnt curvaturs. Thy found that th dsplacmnt dcrass wth an ncras n th curvatur for a fxd aspctrato.thformrtwostudsusdth3dletandthlattr usd a hghr-ordr shar dformaton thory (HSDT). For a monolthc sotropc cylndrcal shll, Calladn [48] drvd an xprsson rlatng th gomtrc paramtrs of th shll and th rato of strtchng and bndng stffnsss. alnns [49] studd fr vbratons of monolthc sotropc sphrcal shlls usng classcal bndng thory and computd stran nrgs du to bndng and strtchng for ach mod. H dntfd th vbraton mod as bndng (strtchng) f th stran nrgy
MECHANICS OF ADVANCED MATERIALS AND STRUCTURES 3 du to bndng (strtchng) s largr than that du to strtchng (bndng). H found that frquncs of strtchng mods ar ndpndnt of th shll thcknss but thos of bndng mods ncras wth th thcknss, and th frquncy of th frst strtchng mod s rlatvly hgh compard to that of th frst bndng mod. W could not fnd works dscrbng th ffct of gomtrc paramtrs on stran nrgs of varous dformaton mods for doubly curvd shlls dformd undr xtrnal loads. Ths knowldg can hlp structural dsgnrs dntfy sgnfcant dformaton mods for gvn loadng condtons and optmz gomtrc paramtrs of th shll. Hr, w study nfntsmal dformatons of doubly curvd shlls dformd statcally wth arbtrary surfac tractons appld on thr major surfacs usng th compatbl TSNDT andthfntlmntmthod(fem)tonumrcallysolvsvral problms usng n-hous dvlopd softwar. For lamnatd shlls, w us an ESL and on-stp SRS to comput strsss. Frst, w study for varous valus of lngth/thcknss and (radus of curvatur)/lngth ratos dformatons of doubly curvd lamnatd shlls and compar rsults from th TSNDT wth th corrspondng analytcal 3D LET solutons avalabl n th ltratur. Nxt, w study thr xampl problms for a monolthc doubly curvd shll subjctd to () a unform normal tracton on on of th major surfacs, () combnd normal and tangntal tractons on a major surfac, and () qual and oppost tangntal tractons on th two major surfacs. For most of ths problms w dlnat ffcts of gomtrc paramtrs on strsss, and stran nrgs of bndng and strtchng dformatons and of n-plan and transvrs dformatons, and compar prdctons from th TSNDT wth thos from th solutons of th 3D LET quatons. 2. Formulaton of th problm W analyz statc nfntsmal dformatons of a lamnatd doubly curvd shll schmatcally shown n Fgur 1. Thshlls composd of N layrs of not ncssarly qual thcknss. Each layr s mad of a homognous, orthotropc, and lnar lastc matral wth adjacnt layrs prfctly bondd to ach othr. Th orthogonal curvlnar coordnats (y 1, y 2, y 3 ) ar such that y 1 = constant and y 2 = constant ar curvs of prncpal curvatur on th md-surfac, y 3 = 0. Th poston vctors of a pont wth rspct to th fxd rctangular Cartsan coordnat axs (X 1, X 2, X 3 ) and (x 1, x 2, x 3 ) wth th X 3 -andthx 3 -axs paralll to th y 3 -axs ar dnotd by X and x n th rfrnc and th currnt confguratons, rspctvly. W dnot th total thcknss and th constant prncpal rad of curvatur of th mdsurfac of th shll by h, R 1m,andR 2m, rspctvly, th arc lngths of th shll md-surfac n th y 1 -andthy 2 -drctons by a and b, rspctvly, and th corrspondng planform lngths by l 1 and l 2,rspctvly. Componnts G j ofthmtrctnsornthrfrncconfguraton ar gvn by: G j = A A j, A = X ( = 1, 2, 3), (1) y whr A A j quals th nnr product btwn vctors A and A j.wnotthatforthorthogonalcurvlnarcoordnat systm, G j s nonzro only whn = j. Th unt bas vctors (ē 1, ē 2, ē 3 ) assocatd wth th curvlnar coordnat axs (y 1, y 2, y 3 ) ar ē = A () H () (no sum on ), whr H 1 = (1+ y 3 R 1 ), H 2 = (1+ y 3 R 2 ), H 3 = 1, and R 1 and R 2 ar rad of curvatur at th pont (y 1, y 2, y 3 ). Th dsplacmnt u of a pont s gvn by u = x X. Th physcal componnts of th nfntsmal stran tnsor n th curvlnar coordnat systm ar gvn by [50]: 11 = 1 ( u1 + u ) 3, 22 = 1 ( u2 + u ) 3, H 1 y 1 R 1 H 2 y 2 R 2 2 12 = 1 u 2 + 1 u 1, H 1 y 1 H 2 y 2 2 13 = 2 23 = 1 H 1 ( u3 y 1 u 1 R 1 1 H 2 ( u3 y 2 u 2 R 2 ) + u 1, y 3 ) + u 2 y 3, 33 = u 3 y 3, (2) whr ( = 1, 2, 3) s th normal stran along th y -drcton, 12 sthn-plansharstran,and 13 and 23 ar th transvrs shar strans. InthTSNDT,thdsplacmnt,u, at a pont s xprssd as a complt polynomal of dgr 3 n th thcknss coordnat, y 3 : d (y 1, y 2, y 3 ) = (y 3 ) d (y 1, y 2 ), ( = 0, 1, 2, 3). (3) Unlss statd othrws, a rpatd ndx mpls summaton ovr th rang of th ndx. In Eq. (3): d = [u 1 u 2 u 3 ] T and d = [u 1 u 2 u 3 ] T ( = 0, 1, 2, 3), (3.1) whr u 0 = u (y 1, y 2, 0) ( = 1, 2, 3). On ntrprtaton of varabls s: u 1 = u y, 2u 2 = 2 u 3 y3 =0 (y 3 ) 2, y 3 =0 6u 3 = 3 u (y 3 ) 3 ( = 1, 2, 3). (3.2) y 3 =0 Th 12D vctor d = [d 0, d 1, d 2, d 3 ] s rfrrd to as th vctor of gnralzd dsplacmnts at a pont on th shll s mdsurfac. W substtut for u from Eq. (3)ntoEq.(2)toobtan: whr =Z (y 3 ) Ld (y 1, y 2 ) ( = 0, 1, 2, 3), (4) =[ 11 22 33 2 23 2 13 2 12 ] T, (4.1) and matrcs Z ( = 0, 1, 2, 3) and th oprator matrx L ar dfnd n Appndx A, rspctvly, by Eqs. (A.1) and(a.2). Smlar to componnts of th stran tnsor, w wrt componnts of th Cauchy strss tnsor as a 6D vctor: σ = [σ 11 σ 22 σ 33 σ 23 σ 13 σ 12 ] T. (5) Th consttutv rlaton (Hook s law) for a lnar lastc matral s: σ j = C jmn mn, C jmn = C mnj = C jmn, (, j, m, n = 1, 2, 3), (6)
4 P. H. SHAH AND R. C. BATRA Fgur 1. Gomtry and coordnat systm of a doubly curvd lamnatd shll. whr C s th fourth-ordr lastcty tnsor havng 21 ndpndnt componnts for a gnral ansotropc matral. For an orthotropc, a transvrsly sotropc, and an sotropc matral, th ndpndnt componnts of C rduc, rspctvly, to 9, 5, and 2. Wth rspct to th matral prncpal axs, Eq. (6) foran orthotropc matral of layr k bcoms: σ k 11 C1111 k C1122 k C k 1133 0 0 0 σ22 k C1122 k C2222 k C k 2233 0 0 0 σ33 k C1133 k C2233 k C3333 k 0 0 0 = σ23 k 0 0 0 C k 2323 0 0 σ13 k 0 0 0 0 C1313 k 0 σ12 k 0 0 0 0 0 C1212 k k 11 k 22 k 33, (7) 2 k 23 2 k 13 2 k 12 whr quantts for thkth layr ar ndcatd by th suprscrpt k. In th global coordnat axs (y 1, y 2, y 3 ) C jmn ar computd by usng th tnsor transformaton ruls for th strss and th stran tnsors, and th 6 6 matrx may b fully populatd;.g., s [26]. For th TSNDT, lastc constants n Eq. (7)arthsam asthosusdnthlet,..,thyarnotmodfdtosatsfy σ 33 = 0assoftndonnthLFAT. W us th prncpl of mnmum potntal nrgy to drv quatons govrnng statc dformatons of th shll: δ = δ(w W f ) = 0, (8) whr δ s th varatonal oprator, s th potntal nrgy of th shll, and W and W f ar, rspctvly, th stran nrgy of dformaton and th work don by th xtrnal forc n th absnc of body forcs gvn by: W = 1 2 W f = Ā k ( k ) T σ k d k, (8.1) d T f dā. (8.2) Hr k s th rgon occupd by th kth layr n th rfrnc confguraton, and Ā s th part of th boundng surfac of th shll on whch surfac tracton, f, s spcfd. Ponts on th rmandr of th boundary of th doman,, occupdby th shll hav thr null tractons (.., ar on a fr surfac) or hav dsplacmnts prscrbd on thm. Th work don by racton forcs at ponts of th boundary whr dsplacmnts ar prscrbd s not ncludd n Eq. (8) bcaus varatons n thprscrbddsplacmntsarnull. W substtut n Eq. (8.1)forσ k n trms of k from Eq. (7), andsubsttutfor k n trms of th gnralzd dsplacmnts dfnd on th md-surfac of th shll from Eq. (4). Also, w substtut n Eq. (8.2) ford n trms of d ( = 0, 1, 2, 3) from Eq. (3). In th rsultng xprsson for δ, wntgratwth rspct to y 3 ovrthshllthcknsstoobtananntgralquaton for δ wth th ntgrand dfnd on th md-surfac. Thmd-surfacofthshllsdscrtzdntoafntlmnt (FE) msh of dsjont 8-nod so-paramtrc quadrlatral lmnts. Thus δ quals th sum of ntgrals ovr ach lmnt. Followng th standard procdur (.g., s [43] or Appndx B) w obtan quatons govrnng statc dformatons of th shll as U = F. (9)
MECHANICS OF ADVANCED MATERIALS AND STRUCTURES 5 Tabl 1. Nomnclatur for boundary condtons spcfd at y 1 = 0ora. Notaton Typ BCs n th 3D LET BCs n th TSNDT C Clampd u 1 = 0, u 2 = 0, u 3 = 0 u 1 = 0, u 2 = 0, u 3 = 0 S Smply supportd σ 11 = 0, u 2 = 0, u 3 = 0 M 11 = 0, u 2 = 0, u 3 = 0 F Tracton fr σ 11 = 0, σ 12 = 0, σ 13 = 0 M 11 = 0, M 12 = 0, M 13 = 0 In Eq. (9) s th global stffnss matrx, U th global vctor of gnralzd nodal dsplacmnts, and F th global load vctor; thr xprssons ar gvn by Eq. (B.10) n Appndx B. Th vctor F of gnralzd forcs at nods s work quvalnt to surfac tractons appld on th boundng surfacs of th shll. For a FE msh of N nod nods, bfor applyng ssntal boundary condtons, th lngth of vctor U quals (12 N nod )sncanodhas 12 dgrs of frdom. W consdr thr typs of boundary condtons (BCs) spcfd at a pont on a shll dg. For xampl, at th dgs y 1 = 0 and a, dfntons of ths BCs n th 3D LET and thr quvalnc n trms of varabls of th TSNDT ar lstd n Tabl 1. In Tabl 1, th ndx taksvalus0,1,2,and3,and h/2 M 1n = (y 3 ) σ 1n dy 3, (n = 1, 2, 3). (10) h/2 Th dsplacmnt (or ssntal) boundary condtons appld at ponts on a shll dg ar satsfd whl solvng algbrac quatons (9). 3. Exampl problms Whn prsntng and dscussng blow rsults for svral problms,wusthmorcommonnotatonandrplac(x 1,X 2,X 3 ) and (u 1,u 2,u 3 ) by (x, y, z) and (u, v, w), rspctvly. W compar numrcal rsults computd usng th TSNDT wth th corrspondng 3D LET solutons obtand by thr analytcal mthods or th FEM wth th commrcal softwar, ABAQUS. Unlss mntond othrws, th 3D FE rsults ar obtand usng a unform msh of 8-nod brck lmnts wth 100 lmnts along th x- and th y-drctons and 10 lmnts n th thcknss drcton, whch corrsponds to 336,633 nodal dgrs of frdom (DoF). W comput strsss from th consttutv rlaton and th shllthorydsplacmnts.forlamnatdshllswfndnplan strsss (σ xx, σ yy, σ xy ) from th consttutv rlatons and th shll thory dsplacmnts, and comput transvrs strsss (σ xz, σ yz,σ zz ) by usng th on-stp SRS; ths ar ndcatdblowbylabls C and SRS, rspctvly.wnotthat thn-planstrsssnafearvaluatdat3 3quadratur ponts and by fttng to thm a complt quadratc polynomal by th last squars mthod, strsss and thr drvatvs wth rspct to x and y at any pont n th FE ar computd. 3.1. Convrgnc of th soluton wth FE msh rfnmnt W consdr a 0 /90 /0 orthotropc lamnatd sphrcal shll (R 1m = R 2m = R) wth a = b, R/a = 2, h = 10 mm, and a/h = 10, only th outr surfac subjctd to a unform normal tnsl tracton, q 0, and assgn followng valus to matral paramtrs: E 1 = 172.4 GPa, E 1 /E 2 = 25, E 3 = E 2, G 12 = G 13 = 0.5E 2, G 23 = 0.2E 2,ν 12 = ν 13 = ν 23 = 0.25. Hr, E 1 dnots Young s modulus along th Z 1 axs, and G 12 (ν 12 ) th shar modulus (Posson s rato) n th Z 1 Z 2 -plan whr th Z 1 -, th Z 2 -, and th Z 3 -arthmatralprncpal axs. Valus of lastc constants wth rspct to global coordnat axs ar obtand by usng th tnsor transformaton ruls. Th lamnaton schm, α 1 /α 2 /.../α N, for a lamnat havng N layrs wth layrs 1 and N bng th nnr and th outr layrs, rspctvly, mpls that fbrs n th kth layr ar orntd at angl α k masurd countr-clockws from th x-axs. Unlss statd othrws, ach layr of a lamnatd shll s assumd to b of qual thcknss. In Tabl 2,w hav lstd for th shll wth all dgs thr clampd or smply supportd, valus of th nondmnsonal dflcton, w (0.5a, 0.5b, 0), at th cntrod of th md-surfac and th axal strss, σ xx (0.5a, 0.5b, 0.5h), at th cntrod of th top surfac for sx unform n mfemshs wth n and m qualng th numbr of lmnts along th x- and th y-axs, rspctvly. Unlss mntond othrws, th dsplacmnt and th strsss ar nondmnsonalzd as: w = w (h 3 /b 4 )E 2 /q 0 and σ = σ/q 0.Wnotthatnumbrsnthcolumn Dff. dnot th chang n th valu from that obtand wth th mmdat prvous FE msh. Rsults rportd n Tabl 2 mply that wth th msh rfnmnt th dflcton convrgs fastr than th axal strss and th convrgnc rat dos not dpnd much upon th BCs. For smply supportd dgs th axal strss at th pont (0.5a, 0.5b, 0.5h) s narly thr tms that for th clampd dgs, and th dflcton at th pont (0.5a, 0.5b, 0) for smply supportd dgs s fv tms that for th fxd dgs. For both problms th 23 23 FE Tabl 2. Convrgnc of solutons for a 0 /90 /0 lamnatd sphrcal shll (a = b,r/a = 2, a/h = 10). Clampd dgs Smply supportd dgs Msh w 10 3 (0.5a,0.5b,0) %Dff. σ xx (0.5a,0.5b,0.5h) %Dff. w 10 3 (0.5a,0.5b,0) %Dff. σ xx (0.5a,0.5b,0.5h) %Dff. 15 15 1.829 25.081 9.178 72.788 17 17 1.829 0.0000 25.079 0.008 9.178 0.000 72.768 0.027 19 19 1.829 0.0000 25.077 0.008 9.178 0.000 72.753 0.021 21 21 1.829 0.0000 25.075 0.008 9.178 0.000 72.743 0.014 23 23 1.829 0.0000 25.074 0.004 9.178 0.000 72.735 0.011 25 25 1.829 0.0000 25.074 0.000 9.178 0.000 72.729 0.008
6 P. H. SHAH AND R. C. BATRA Tabl 3. Nondmnsonal dflcton, w (0.5a, 0.5b, 0.5h) x10 3, at th cntrod of th outr-surfac of th 0 /90 sphrcal shll. R/a a/h = 100 a/h = 10 3D LET [45] TSNDT %Dff. 3D LET [45] TSNDT %Dff. 1 0.0725 0.0726 0.14 6.6628 6.8453 2.74 2 0.2869 0.2870 0.03 13.2075 13.3982 1.44 3 0.6461 0.6465 0.06 16.1084 16.1938 0.53 4 1.1440 1.1447 0.06 17.4280 17.4427 0.08 5 1.7569 1.7580 0.06 18.1020 18.0743 0.15 msh gvs convrgd valus of th two quantts. Unlss statd othrws, w us th 25 25 unform FE msh corrspondng to 23,712 DoF. 3.2. Comparson of TSNDT rsults wth thos from th 3D LET For dffrnt valus of th lngth/thcknss and (radus of curvatur)/lngth ratos of th lamnatd doubly curvd shll, w compar th TSNDT prdctons wth thos from th 3D LET. Valus of matral paramtrs ar lstd n Scton 3.1. 3.2.1. Sphrcal shll W study dformatons of a smply supportd 0 /90 cross-ply lamnatd sphrcal shll wth a unform normal tnsl tracton, q 0, appld only on th outr surfac. In Tabl 3,whav compard nondmnsonal dflcton, w (0.5a, 0.5b, 0.5h), at th cntrod of th top surfac of th shll for a = b, a/h = 100 and 10, and dffrnt valus of R/a found usng th TSNDT wth that gvn by th analytcal 3D LET soluton of Fan and Zhang [45]. Thr s at most 2.74% dffrnc btwn th two sts of rsults for R/a = 1anda/h = 10, and ths dffrnc dcrass wth an ncras n R/a. Th shll dflcton ncrass wth an ncras n th rato R/a. 3.2.2. Cylndrcal shll For smply supportd [0 /90 /0 / ] lamnatd cylndrcal shlls (1/R 2m = 0) subjctd only on th outr surfac to th snusodal dstrbutd normal tnsl tracton: q (x, y) = q 0 sn(πx/a) sn(πy/b), (11) w hav compard n Tabl 4 valus of th nondmnsonal dflcton at th cntrod of thr md-surfacs wth R 1m /a = 4, b/a = 3, and a/h = 10 and 50 computd wth th two thors. Th two sts of rsults ar n xcllnt agrmnt wth ach othr Tabl 4. Nondmnsonal dflcton, w (0.5a, 0.5b, 0) x10 2, at th cntrod of md-surfac of a lamnatd cylndrcal shll of N layrs wth R 1m /a = 4andb/a = 3. N 2 3 4 5 10 a/h = 50 3-D LET [46] 2.139 0.5129 1.043 0.6059 0.9113 TSNDT 2.132 0.5169 1.050 0.6101 0.9189 %Dff. 0.33 0.78 0.67 0.69 0.83 a/h = 10 3-D LET [46] 2.783 0.9396 1.609 1.020 1.381 TSNDT 2.787 0.8981 1.501 0.9524 1.314 %Dff. 0.14 4.42 6.71 6.63 4.85 Tabl 5. Nondmnsonal strsss n th 90 /0 doubly curvd shll wth a = b, a/h = 10, R 1m /a = 5, and R 2m /R 1m = 2. z/h 3-D LET [47] TSNDT %Dff. σ xx (0.5a, 0.5b, z) 0.5 72.5015 72.2213 0.39 0+ 55.7425 54.2168 2.74 0 1.5182 1.4397 5.17 0.5 8.2074 8.3104 1.25 σ yy (0.5a, 0.5b, z) 0.5 8.8503 8.8137 0.41 0+ 2.0775 2.0171 2.91 0 62.9176 62.4791 0.70 0.5 69.0028 69.0674 0.09 σ xz (0, 0.5b, z) 0.25 3.2347 3.1925 1.30 0 1.3965 1.3937 0.20 0.25 0.9362 0.9414 0.56 σ yz (0.5a, 0, z) 0.25 0.727 0.7050 3.03 0 0.9964 0.9680 2.85 0.25 3.1326 3.1665 1.08 w (0.5a, 0.5b, z) 0.5 11.9190 11.8748 0.37 0 11.9581 11.9150 0.36 0.5 11.8910 11.8468 0.37 for thn shlls. Howvr, for thck shlls, thy dffr at most by 6.71%. 3.2.3. Doubly curvd shlls wth dffrnt rad of curvatur W now consdr a smply supportd 90 /0 cross-ply lamnatd doubly curvd shll wth a = b, a/h = 10, R 1m /a = 5, h = 10 mm, R 2m /R 1m = 2, and subjctd only on th outr surfac to snusodal dstrbutd tnsl tracton gvn by Eq. (11). W hav lstd n Tabl 5 th nondmnsonal dflcton and strsss at varous locatons through th shll thcknss obtand by th two thors. For th TSNDT, th transvrs strsss ar computd usng th SRS. Th two sts of rsults dffr by at most 5% mplyng that for ths problm also th TSNDT wth th SRS gvs through-th-thcknss strsss clos to thos obtand from th 3D LET [47]. 3.3. Clampd sotropc shll subjctd to unform normal tracton on th outr surfac 3.3.1. Comparson wth th 3D LET soluton 3.3.1.1. Sphrcal shll. W study dformatons of a clampd sphrcal shll mad of an sotropc matral wth h = 10 mm, a/h = 10, a = b, R/a = 5, E = 210 GPa, ν = 0.3, and a unformly dstrbutd tnsl normal tracton, q 0 = 10 MPa, appld only on th outr surfac. In Fgur 2, whavplottd th dsplacmnt, w (x, b/2, 0), of ponts on th md-surfac of th shll along th ln y = b/2 obtand by usng th TSNDT and th 3D LET soluton computd wth th FEM; th nst n th fgursaschmatcofthproblmstudd.rsultsfromthtwo thors agr wll wth ach othr havng at most 0.3% dffrnc btwn thm. Strsss σ xx (0.5a,0.5b, 0.5h) and σ xz (0.055a, 0.5b, 0) n th 3D LET solutons wth unform 100 100 10 and 80 80 10 mshs dffrd by only 0.07 and 0.6%, rspctvly. In Fgur3a, w hav xhbtd through-th-thcknss dstrbutons of th axal strss, σ xx, and th transvrs normal strss, σ zz, along th transvrs normal passng through th cntrod of th md-surfac. Th axal strsss computd from th two thors dffr by 0.82 and 0.42% at z = h/2and h/2,rspctvly. Bcaus of th shll curvatur, magntuds of th axal
MECHANICS OF ADVANCED MATERIALS AND STRUCTURES 7 Fgur 2. For a clampd sphrcal shll mad of an sotropc matral and subjctd to a unform normal tnsl tracton on th outr surfac, varaton of th z-dsplacmnt along th curvlnar ln y = b/2 on th shll md-surfac. tnsl and th axal comprssv strsss at corrspondng ponts on th outr and th nnr surfacs of th shll ar dffrnt. Th TSNDT transvrs normal strss obtand drctly from th consttutv rlaton dos not xhbt th boundary layr phnomnon nar major surfacs as prdctd by th 3D LET and dffrs from th appld normal tracton at th outr surfac by 20%. Howvr, ths rror s rducd to 2.7% and th boundary layr ffct s capturd whn σ zz s computd usng th SRS. In Fgur 3b, w hav dpctd through-th-thcknss varatons of th transvrs shar strss, σ xz, along th transvrs Fgur 4. For a clampd doubly curvd shll mad of an sotropc matral and subjctd to a unform normal tnsl tracton q 0 on th outr surfac, through-ththcknss dstrbutons of σ xx (0.5a, 0.5b, z) and σ xz (0.055a,0.5b, z). Th strsss ar normalzd by q 0. normal nar and away from th dg x = 0, passng through th ponts (0.055a, 0.5b, 0) and (0.25a, 0.5b, 0), rspctvly.it s vdnt that for both cass th TSNDT SRS strsss dffr from th 3D LET strsss by at most 2.8% at z = 0. Th transvrs shar strss at th pont (0.055a, 0.5b, 0) nar th dg quals about 2.5 tms that at th ntror pont (0.25a, 0.5b, 0). W not that th zro tangntal tracton boundary condtons at ponts (0.055a, 0.5b, ±h/2) nar th clampd dg x = 0 ar wll satsfd whn σ xz s computd wth th TSNDT and th SRS. Howvr, ths boundary condtons ar wll satsfd at ponts (0.25a, 0.5b, ±h/2) away from th dg x = 0 whthr or not th SRS s usd. 3.3.1.2. Doubly curvd shll wth two dffrnt rad of curvatur. In Fgur 4, w hav portrayd through-ththcknss dstrbutons of σ xx and σ xz along th transvrs normal passng through ponts (0.5a, 0.5b, 0) and (0.055a, 0.5b, 0), rspctvly, for th problm studd n Subscton 3.3.1.1 but now wth R 1 = 3a and R 2 = 2R 1.Thdstrbutonsofσ xx obtand from th TSNDT and th 3D LET agr wll wth ach othr havng th maxmum dffrnc of 1.5% at z = h/2. Howvr, at ponts (0.055a, 0.5b, ±h/2) nar th dgs on th two major surfacs of th shll, th tracton boundary condton s wll satsfd only f σ xz s computd usng th SRS. Th maxmum dffrnc btwn th two valus of σ xz at ponts (0.055a, 0.5b, z) obtand usng th SRS and th 3D LET s 1.37%. Fgur 3. For a clampd sphrcal shll mad of an sotropc matral and loadd wth a unform normal tnsl tracton q 0 on th outr surfac, through-ththcknss dstrbutons of (a) σ xx (0.5a, 0.5b, z) and σ zz (0.5a, 0.5b, z),and(b) σ xz at ponts (0.055a,0.5b,z)and(0.25a,0.5b, z). Th strsss ar normalzd by q 0. 3.3.1.3. Cylndrcal shll. W now study dformatons of a cylndrcalclampdshll(1/r m2 = 0) composd of an sotropc matral havng h = 10 mm, b/a = 2, a/h = 10, and R m1 /a = 1, E = 210 GPa, ν = 0.3, and subjctd to a comprssv vrtcal tracton (.., paralll to th Cartsan X 3 -axs) of magntud q 0 = 10 MPa on th outr surfac only. Thus, n th curvlnar (xyz) coordnat systm ths corrsponds to combnd normal (along th z-axs) and tangntal (along th x-axs) nonunform surfac tractons. W hav dpctd n Fgur 5 th z- and th x-dsplacmnts along th ln y = b/2 on th md-surfac of th shll computd usng th TSNDT and th 3D LET; th nst shows a schmatc of th problm studd. It should b clar that th two sts of rsults agr wll wth ach othr havng th maxmum dffrnc of 0.15% n w at x/a = 0.5 and 1.67% n u at x/a = 0.21 and 0.79. W hav portrayd n Fgur 6a through-th-thcknss dstrbutons of th n-plan axal strsss (σ xx, σ yy )andnfgur 6b
8 P. H. SHAH AND R. C. BATRA th transvrs strsss (σ xz, σ zz ) along th transvrs normal passng through ponts (0.5a, 0.5b, 0) and (0.25a, 0.5b, 0), rspctvly. Th dffrncs n th magntud of σ xx at th cntrodofthoutrandthnnrsurfacscomputdfromth TSNDT and th 3D LET ar 1.86 and 0.81%, rspctvly, and th corrspondng dffrncs for σ yy ar 0.25 and 2.52%. Th transvrs strsss computd usng th TSNDT and th SRS satsfy thtractonboundarycondtonswthonly0.57and1.3%rror n th normal and th tangntal tracton at th pont (0.25a, 0.5b, 0.5h) on th outr surfac. Fgur 5. For a clampd cylndrcal shll mad of an sotropc matral and subjctd to a comprssv load along th X 3 -drcton on th outr surfac only, x- and z-dsplacmnts along th ln y = b/2 on th md-surfac of th shll. 3.3.2. Effct of shll s curvatur on strss dstrbutons In Fgurs 7a and 7b, w hav dpctd through-th-thcknss dstrbutons of σ xx along th transvrs normal passng through th cntrod of th md-surfac and σ xz computd usng th SRS along ponts (0.055a, 0.5b,z)forclampdsphrcalshllsstudd n subscton 3.3.1.1. but R/a = 1, 2, 3, 10, 50. Th nst n Fgur 7a dpcts curvs of ntrscton of shll s md-surfacs Fgur 6. For a clampd cylndrcal shll mad of an sotropc matral and subjctd to a comprssv load along th X 3 -drcton on th outr surfac only, through-th-thcknss dstrbutons of (a) σ xx and σ yy at (0.5a,0.5b,z)and(b)σ xz and σ zz at (0.25a,0.5b, z). Th strsss ar normalzd by q 0. Fgur 7. For a clampd sphrcal shll composd of an sotropc matral and subjctd to unform normal tnsl tracton q 0 on th outr surfac, through-ththcknss dstrbutons of (a) σ xx (0.5a, 0.5b, z) and (b) σ xz at (0.055a,0.5b,z)for dffrnt valus of R/a. Th strsss ar normalzd by q 0.
MECHANICS OF ADVANCED MATERIALS AND STRUCTURES 9 wth a plan X 2 = constant. Ths rsults rval that wth an ncras n th shll curvatur, th plan of zro axal strss shfts towards th nnr surfac of th shll; thus, th porton of th shllnaxaltnsonncrassascompardtothatnaxalcomprsson. For R/a = 1, th axal strss at all ponts on th transvrs normal passng through th pont (0.5a, 0.5b,0)s tnsl. Th magntud of th rato of th axal strsss at th top and th bottom surfacs ncrass from 1.06 for R/a = 50 to 10.94 for R/a = 1. It s found that th magntud of th transvrs shar strss ncrass wth a dcras n th shll curvatur. Howvr, thratofncrasnthmagntudofσ xz slows down wth an ncras n th valu of R/a. Thmagntudofσ xz at z = 0for R/a = 50 s 2.54 and 1.02 tms th corrspondng valu for R/a = 1 and 10, rspctvly. 3.3.3. Effct of th loadng on strss dstrbutons For th problm studd n Subscton 3.3.1.3, w now consdr four dffrnt appld loads,.., a unform tracton of magntud10mpaonthrthoutrorthnnrsurfaconlypontng thr radally (along th ngatv z-axs) or vrtcally (along th ngatv X 3 -axs) downwards; ths loads ar dnotd by q r and q v, rspctvly. For tractons along th X 3 -axs, both normal and tangntal tractons act on th shll surfac. In Fgurs 8a and 8b, w hav dpctd through-th-thcknss dstrbutons of th axal strss, σ xx, and th transvrs shar strss, σ xz,nth plan y = b/2 along th sctons x = 0.5a and 0.25a, rspctvly. It s found that whn th shll s subjctd to th radal (vrtcal) load on th outr surfac, th plan of zro axal strss ntrscts th transvrs normal passng through th cntrod of th md-surfac at z = 0.46h ( 0.385h). Thus, a largr porton of th shll s n axal tnson for th vrtcal tracton as compard to that for th radal tracton. Th magntuds of th maxmum tnsl and comprssv axal strsss at th cntrods of th nnr and th outr surfacs for th vrtcal tracton qual, rspctvly, 2.73 and 0.89 tms that for th radal tracton. Furthrmor, du to th shll curvatur, whn th radal or th vrtcal tracton s appld on th nnr surfac, th axal strss nducdnthshllsdffrntfromthatwhnthsamtracton s appld on th outr surfac. Whn th nnr surfac s subjctd to th radal tracton, magntuds of σ xx at cntrods of th nnr and th outr surfacs, rspctvly, ar 2.21 and 0.84 tms that for th sam tracton appld on th outr surfac, and th corrspondng ratos for th vrtcal tracton ar 1.56 and 0.97. Smlarly, dstrbutons of th transvrs shar strss for th four appld tractons ar dffrnt. Whn th tracton s appld on th outr (nnr) surfac, th rato of th magntud of σ xz at z = 0forthvrtcalloadtothatforthradalloads 1.59 (0.86). Morovr, whn th outr surfac s subjctd to th radal (vrtcal) load th magntud of σ xz at z = 0 s 1.06 (1.96) tms that whn th nnr surfac s subjctd to th samload.althoughnotshownhrforthsakofbrvty, t s found that th axal strss changs only th sgn wth a nglgbl chang n th magntud whn only drcton of th appld load s changd for ach of th four tracton boundary condtons. Fgur 8. For a clampd cylndrcal shll mad of an sotropc matral and subjctd to four dffrnt loadngs, through-th-thcknss dstrbutons of (a) σ xx (0.5a, 0.5b, z) and (b) σ xz (0.25a,0.5b, z). Strsss ar normalzd by th magntud of th appld tracton. 3.3.4. Effct of gomtrc paramtrs on th stran nrgy of dformaton Th total stran nrgy, W, of dformaton s gvn by: whr W 1 = 1 2 W 3 = W = 6 W j, (12) j=1 k σ k xx k xx d k, W 2 = 1 2 k σ k xy k xy d k, k σ k yy k yy d k,
10 P. H. SHAH AND R. C. BATRA W 4 = W 6 = 1 2 k σ k xz k xz d k, W 5 = k σ k yz k yz d k, k σ k zz k zz d k. (12.1) Altrnatvly, W can b wrttn as th sum of stran nrgs ofthn-plan(w ), th transvrs shar (W s ), and th transvrs normal (W n ) dformatons,.., W = W + W s + W n, (13) whr W = W 1 + W 2 + W 3,W s = W 4 + W 5,andW n = W 6. W substtut for u from Eq. (3)ntoEq.(2)towrtphyscal componnts of th nfntsmal stran tnsor as: whr b αβ = ( b αβ 2H (β) + b 11 = u 1b y 1 αβ = (y 3 ) b b αβ, (14) b βα 2H (α) ), + u 3b R 1, b 12 = u 1b y 2, b 13 = D jbu 1j, b 21 = u 2b, b 22 y = u 2b + u 3b, b 23 1 y 2 R = D jbu 2j, 2 b 31 = u 3b y 1 u 1b, b 32 R = u 3b u 2b, b 33 1 y 2 R = D jbu 3j, 2 D 10 = 1, D 21 = 2, D 32 = 3, D j = 0 ( j + 1). (14.1) In th abov xprssons, ndcs b,, and j tak valus 0, 1, 2, and 3, and othr ndcs tak valus 1, 2, and 3. Th rpatd ndx n th parnthss s not summd. W substtut for αβ from Eq. (14) ntoeq.(6) toobtancomponntsofthcauchy strss tnsor as: whr σ αβ = (y 3 ) b σ b αβ, (15) σ b αβ = C αβmn b mn, (15.1) n whch, ndcs α, β, m,andntakvalus1,2,and3,andb taksvalus0,1,2,and3. Thus, Eq. (12) rprsntng th total stran nrgy of dformatoncanbwrttnas: W = W strtch-n + W bnd-n + W strtch-tr + W bnd-tr + W coupl, (16) whr W strtch-n and W bnd-n ar stran nrgs of strtchng and bndng dformatons, rspctvly, corrspondng to th n-plan strans, W strtch-tr and W bnd-tr, ar thos corrspondng to th transvrs strans, and W coupl sthstrannrgydu to couplng btwn strtchng and bndng dformatons; thr xprssons ar gvn blow: W strtch-n = 1 2 k [ (y3 ) m (y 3 ) n (σ m 11 n 11 +2σ m 12 n 12 + σ m 22 n 22 )] d k (m, n = 0, 2) Tabl 6. Stran nrgy, W, of dformaton and work don, W, by th xtrnal forc for shlls wth a = b = 10 cm. a/h R 1m /a R 2m /R 1m W (mj) W (mj) %Error 10 1 5 734 730 0.54 2 651 648 0.46 1 494 494 0.00 5 1 5 232 238 2.59 2 223 230 3.14 1 198 204 3.03 W bnd-n = 1 2 W strtch-tr = 1 2 W bnd-tr = 1 2 W coupl = 1 2 k [ (y3 ) m (y 3 ) n (σ m 11 n 11 +2σ m 12 n 12 + σ m 22 n 22 )] d k (m, n = 1, 3) k [ (y3 ) m (y 3 ) n (2σ m 13 n 13 +2σ m 23 n 23 + σ m 33 n 33 )] d k (m, n = 1, 3) k [ (y3 ) m (y 3 ) n (2σ m 13 n 13 +2σ m 23 n 23 + σ m 33 n 33 )] d k (m, n = 0, 2) k [ (y3 ) p (y 3 ) q σ p j q j +(y 3) r (y 3 ) s σ r j d k (, j = 1, 2, 3; p, s = 0, 2; q, r = 1, 3) (16.1) W not that for a plat W coupl quals zro. W consdr a clampd doubly curvd shll wth a = b = 10 cm, mad of an sotropc matral wth E = 210 GPa, ν = 0.3, and subjctd to a unform normal tnsl tracton, q 0 = 10 MPa, on ts outr surfac only, and nvstgat ffcts of th aspct rato, a/h, th normalzd radus of curvatur, R 1m /a, and th curvatur rato, R 2m /R 1m,ofthshllonstrannrgs of dffrnt dformatons. For statc lastc dformatons of th shll, th total stran nrgy of dformaton, W, quals th work don by xtrnal forcs, W.Itshouldbclarfromvaluslstd n Tabl 6 that W and W dffr by at most 3.14%. Ths furthr vrfs th accuracy of th computd rsults. 3.3.4.1. Stran nrgs of n-plan and transvrs dformatons. 3.3.4.1.1. Effct of aspct rato. W frst assum R 1m = R 2m = R and llustrat n Fgur 9a th ffct of th aspct rato, a/h, for thr valus of th radus of curvatur, R/a = 1, 10 and (plat), on stran nrgs of n-plan and transvrs dformatons. Unlss mntond othrws, stran nrgs of dffrnt dformaton mods plottd n th followng svral fgurs ar normalzd wth rspct to th total stran nrgy of dformaton, W, for ach confguraton of th shll. Valus of W for shlls wth R/a = 1 and 10 and dffrnt aspct ratos lstd n Tabl 7 ndcat that for th sphrcal shll wth gvn a and R, th total stran nrgy of dformaton ncrass wth an ncras n th aspct rato. It s clar from plots of Fgur 9a that, for a gvn curvatur, th stran nrgy of n-plan dformatons ncrass s j ]
MECHANICS OF ADVANCED MATERIALS AND STRUCTURES 11 Tabl 8. Total stran nrgy of dformaton, W (mj), for clampd sphrcal shlls wth dffrnt valus of th radus of curvatur; a = b = 10 cm. R/a 1 2 3 4 5 6 7 8 9 10 a/h = 5 204 258 268 270 271 270 270 270 269 269 a/h = 10 494 905 1074 1148 1184 1204 1216 1224 1229 1233 Fgur 9. Varaton wth (a) th aspct rato, a/h, and (b) th normalzd radus of curvatur, R/a, of normalzd valus of stran nrgs of dformaton du to n-plan, transvrs shar, and transvrs normal dformatons for a monolthc clampd sphrcal shll mad of an sotropc matral subjctd to a unform normal tnsl tracton on th outr surfac. Plottd nrgs ar normalzd wth rspct to W (lstd n Tabls 7 and 8) for ach confguraton of th shll. and that of transvrs shar and transvrs normal dformatons dcrass wth an ncras n th aspct rato. W not that th stran nrgs, W 4,W 5,andW 6, of transvrs dformatons hav bn computd wth strsss found usng th SRS and th corrspondng strans by pr-multplyng th strss tnsor wth thcomplancmatrx.forr/a = 1anda/h = 5, th contrbutons to th normalzd total stran nrgy of dformaton from n-plan and transvrs shar dformatons ar 73.2 and 23.8%, rspctvly. For a/h > 20, stran nrgs du to transvrs shar dformatons bcom nglgbl and thos du to n- plan dformatons domnat. Ths agrs wth rsults of [51] n that for a/h > 20 prdctons from shar dformaton thors convrg to thos from th classcal thory, whch nglcts transvrs shar dformatons. Th stran nrgs of th shll wth R/a = 10 narly qual thos for th plat. W not that rsults for th plat ncludd n Fgur 9a ar from our arlr work [43]. 3.3.4.1.2. Effct of th shll curvatur. W hav dmonstratd n Fgur 9b for th clampd sphrcal shll th ffct of th normalzd radus of curvatur, R/a, fortwovalusofthaspct rato, a/h = 5 and 10, on two componnts of th stran nrgy of dformaton. Wth an ncras n R/a (.., a dcras n th curvatur), th stran nrgy of n-plan dformatons dcrass and that of transvrs shar and transvrs normal dformatons ncrass. Howvr, th rat of dcras and ncras n W and W s, rspctvly, slows down wth an ncras n th magntud of R/a. For xampl, for th shll wth a/h = 10 th contrbuton to th normalzd total stran nrgy of dformaton from nplan dformatons dcrass from 94.3% for R/a = 1 to 80.7 and 79.5% for R/a = 5and10,rspctvly.Forallshllconfguratons studd, th stran nrgy du to transvrs normal dformatons s found to b lss than 3% of th total stran nrgy. In Tabl 8,whavrportdforsphrcalshllswtha/h = 5and 10 valus of W for dffrnt R/a ratos. 3.3.4.1.3. Effct of th curvatur rato. W now consdr th shll wth R 1m /a = 1, and dffrnt valus of R 2m > R 1m.In Tabl 9,w hav lstd valus of W for ths shlls.for a thck shll wth a/h = 5, valus of R 2 /R 1 btwn 2 and 5 hav a nglgbl ffct on W. Howvr, for a shll wth a/h = 10, th valu of W ncrass from 648 to 730 mj wth an ncras n R 2 /R 1 from 2to5.InFgur 10, w hav portrayd th ffct of th curvatur rato, R 2m /R 1m, on stran nrgs of th n-plan and th transvrs dformatons for a/h = 5 and 10. It s found that th stran nrgy of th n-plan dformatons dcrass and that of th transvrs shar and th transvrs normal dformatons ncrasswthanncrasnthcurvaturrato.thnormalzd valu of W b dcrass by 4.8 and 12.6% n gong from R 2m /R 1m = 1to5fora/h = 10 and 5, rspctvly, whl th corrspondng ncras n th normalzd valus of W ar 40.4 and 28.7%. For a/h = 5andR 2m /R 1m = 5, th stran nrgy of transvrs shar dformatons accounts for 35.8% of th total stran nrgy of dformaton. Tabl 7. Total stran nrgy of dformaton, W (mj), for sphrcal shlls wth dffrntvalusofthaspctrato;(a= b = 10 cm). a/h 5 6 7 8 9 10 15 20 25 30 40 50 R/a = 1 204 253 308 367 429 494 837 1190 1547 1909 2643 3388 R/a = 10 269 382 530 718 951 1233 3555 7778 14318 23668 49917 86984 Tabl 9. Total stran nrgy of dformaton, W (mj), for doubly curvd shlls wth dffrnt ratos of th rad of curvatur; (a = b = 10 cm). R 2 /R 1 2 3 4 5 a/h = 5 230 235 237 238 a/h = 10 648 696 718 730
12 P. H. SHAH AND R. C. BATRA Fgur 10. Varaton wth th curvatur rato, R 2m /R 1m, of th normalzd stran nrgs of dformaton du to n-plan, transvrs shar, and transvrs normal dformatons for a monolthc clampd doubly curvd shll mad of an sotropc matral subjctd to a unform normal tnsl tracton on th outr surfac. Plottd nrgs ar normalzd wth rspct to W (lstd n Tabl 9) for confguraton of th shll. 3.3.4.2. Stran nrgs of strtchng and bndng dformatons. 3.3.4.2.1. Effct of th shll curvatur. For a sphrcal shll wth a/h = 10 and 5 w hav plottd n Fgurs 11a and 11b, rspctvly, th ffct of th rato R/a on stran nrgs of strtchng and bndng dformatons. W not that W strtch = W strtch-n + W strtch-tr and W bnd = W bnd-n + W bnd-tr. It s found that for a gvn aspct rato, th stran nrgy of bndng (strtchng) dformatons ncrass (dcrass) wth an ncras n R/a. Howvr,thratofncras(dcras) n W bnd (W strtch ) slows down wth an ncras n R/a. For xampl, for th shll wth a/h = 5, th normalzd valu of W bnd (W strtch ) s 50.5% (51.2%) whn R/a = 1andtncrass (dcrass) to 95.6% (4.5%) and 98.6% (1.5%) whn R/a = 5and 10,rspctvly.For th shll wth a/h = 10, contrbutons from W bnd and W strtch to th total stran nrgy of dformatons ar 29.5 and 70.8%, rspctvly, for R/a = 1 and th corrspondng valus for R/a = 10 ar 98 and 2%. It s obsrvd that for th shll wth a/h = 10, th contrbuton to W bnd from n-plan dformatons s hghr than that from transvrs dformatons. Howvr, for th thck shll (a/h = 5), n-plan and transvrs dformatons contrbut about qually to th total stran nrgy of bndng dformatons. For xampl, whn R/a = 10 bndng dformatons du to n-plan and transvrs strans account for 77.5 and 20.5%, rspctvly, of th total stran nrgy of dformaton for a/h = 10 and th corrspondng valus for a/h = 5 ar 48.3 and 50.2%. Th stran nrgy of strtchng dformatons du to transvrs strans, W strtch-tr, s at most 2.2% and th magntud of W coupl rprsntng th ntracton btwn strtchng and bndng dformatons s at most 1.7%, whch s nglgbl as compard to othr componnts of th stran nrgy; ths maxmum valus corrspond to th shll wth R/a = 1anda/h = 5. 3.3.4.2.2. Effct of th aspct rato. W hav lucdatd n Fgurs 12a and 12b th ffct of th aspct rato, a/h, on stran nrgs of strtchng and bndng dformatons for a sphrcal shll wth R/a = 10 and 1, rspctvly. It s clar that th stran nrgy of strtchng (bndng) dformatons ncrass (dcrass) wth an ncras n th aspct rato. For th shll Fgur 11. Varaton wth th normalzd radus of curvatur, R/a,of th normalzd valus of stran nrgs of dformaton du to strtchng and bndng dformatons for a monolthc clampd sphrcal shll mad of an sotropc matral wth (a) a/h = 10, (b) a/h = 5, and subjctd to a unform normal tnsl tracton on th outr surfac. Plottd nrgs ar normalzd wth rspct to W (lstd n Tabl 8) for ach confguraton of th shll. wth R/a = 10, (W bnd,w strtch ) = (98.5%, 1.5%) of W whn a/h = 5 and quals (69.6%, 30.4%) of W whn a/h = 50. Not that th ffct of transvrs dformatons has bn ncludd n ths two nrgs. For th dp shll (R/a = 1), th rat of ncras (dcras) of W strtch (W bnd ) slows down wth an ncras n th magntud of a/h. For xampl, W strtch ncrass from 0.512W for a/h = 5 to 0.9W and 0.945W for a/h = 25 and 50, rspctvly. For an sotropc cylndrcal shll (R 2 = R, 1/R 1 = 0), Calladn [48] computd th rato of th strtchng and th bndng stffnsss as ξ = 12 a 4 (1 v 2 ) [1 + (a/b) 2 ] 4.Thus,forashll π 4 R 2 h 2 wth ν = 0.3, R/a = 1, and a = b, th strtchng stffnss wll xcd 10 tms th bndng stffnss whn a/h > 37. Th corrspondng aspct rato for th sphrcal shll for th stran nrgy of strtchng dformatons to xcd that of bndng dformatons s found to b about 30 from th plots of Fgur 12b.Furthrmor, th xprsson for ξ suggsts that for fxd valus of a and b, th strtchng stffnss domnats ovr th bndng stffnss as thr th aspct rato ncrass for a gvn curvatur or th curvatur ncrass for a gvn aspct rato. Th stran nrgy of strtchng dformatons s manly du to n-plan strans wth nglgbl contrbuton from that du to
MECHANICS OF ADVANCED MATERIALS AND STRUCTURES 13 Fgur 12. Varaton wth th aspct rato, a/h, of th normalzd valus of stran nrgs of dformaton du to strtchng and bndng dformatons for a monolthc clampd sphrcal shll mad of an sotropc matral wth (a) R/a = 10, (b) R/a = 1, and subjctd to a unform normal tnsl tracton on th outr surfac. Plottd nrgs ar normalzd wth rspct to W (lstd n Tabl 7) for ach confguraton of th shll. Fgur 13. For a cantlvr shll mad of an sotropc matral and subjctd to qual and oppost unform tangntal tractons on th outr and th nnr surfacs: (a) x- and z-componnts of dsplacmnt along th ln y = b/2 on th md-surfac of th shll and (b) dformd shap ofth cross-scton y= b/2 wth dsplacmnts magnfd by a factor of 10. transvrs strans. Th contrbuton to th stran nrgy of bndng dformatons from th transvrs strans dcrass monotoncallywthanncrasnthaspctrato.forxampl, W bnd-tr /W for R/a = 10 and 1, rspctvly, quals 50.2 and 24.3% whn a/h = 5anddcrassto9.9and1%whna/h = 15. Howvr, wth an ncras n th aspct rato, th stran nrgy of bndng dformatons du to n-plan strans ncrass for a/h β and dcrass for a/h >β,whrβ quals 25 and 7 for R/a = 10 and 1, rspctvly. 3.4. Cantlvr sotropc shll subjctd to qual and oppost unformly dstrbutd tangntal tractons on th outr and th nnr surfacs Ths problms ar motvatd by th obsrvaton that durng ntry of a shp or a boat hull nto watr both tangntal and normal tractons act on th hull/watr ntrfac. Smlarly, surfacs of a fsh movng n watr ar subjctd to both normal and tangntal tractons. For lnar problms, th prncpl of suprpostonnablsontostudysparatlyproblmsfor normal and tangntal tractons. 3.4.1. Comparson wth th 3D LET soluton W analyz dformatons of a sphrcal shll clampd at th dg x = 0, wth th rmanng thr dgs tracton fr, havng h = 10 mm, a/h = 10, a = b, R/a = 5, mad of an sotropc matral wth E = 210 GPa, ν = 0.3, and qual and oppost unform tangntal tractons of magntud q 0 = 10 MPa appld on th outr and th nnr surfacs. W hav dpctd n Fgur 13a th z- and th x-dsplacmnts along th ln y = b/2 on th mdsurfac of th shll wth thr scals on th lft and th rght vrtcal axs, rspctvly, and n Fgur 13b th dformd shap of th cross scton at y = b/2; th nst s th schmatc of th problm studd. Th rsults from th TSNDT and th 3D LET ar n xcllnt agrmnt wth ach othr havng at most 0.5% dffrnc btwn thm for th two dsplacmnt componnts. In Fgurs 14a and 14b, w hav portrayd through-ththcknss dstrbutons of th axal strss, σ xx, and th transvrs shar strss, σ xz, along dffrnt transvrs normal sctons nar and away from th dgs x = 0andx= a. Th axal strss along th transvrs normal passng through ponts (0.055a, 0.5b, 0), (0.5a, 0.5b, 0), and (0.945a, 0.5b, 0), rspctvly, computd usng th TSNDT dffrs from th corrspondng 3D LET soluton at most by 3.1, 1.4, and 2.6%; ths maxmum dffrncs
14 P. H. SHAH AND R. C. BATRA Fgur 15. For a cantlvr shll mad of an sotropc matral and subjctd to qual and oppost unform tangntal tractons on th outr and th nnr surfacs, varaton of σ xx (x, 0.5b, 0.45h) along th lngth n th x-drcton of th shll. Fgur 14. For a cantlvr shll mad of an sotropc matral and subjctd to qual and oppost unform tangntal tractons q 0 on th outr and th nnr surfacs, through-th-thcknss dstrbutons of (a) σ xx (x, 0.5b, z) for x/a = 0.055, 0.5, and 0.945, and σ xz (x, 0.5b, z) for x/a = 0.055 and 0.945. Th strsss ar normalzd by q 0. occur at z/h = 0.15, 0.5, and 0.15, rspctvly. Th magntuds of σ xx at corrspondng ponts on th outr and th nnr surfacsofthshllardffrntdutothcurvaturofthshll. Furthrmor, th magntud of th axal strss at ponts (x, b/2, h/2) locatd along th ln at th ntrscton of th outr surfac and th cross scton y = b/2, for x = 0.055a s about 2 tms and 19 tms that for x = 0.5a and 0.945a, rspctvly. Th transvrs shar strss nar th clampd dg obtand from th consttutv rlaton dffrs from th appld tangntal tracton at ponts (0.055a, 0.5b, 0.5h) and (0.055a, 0.5b, 0.5h) on th outr and th nnr surfacs by 2.78 and 14.23%, rspctvly, and th corrspondng dffrncs nar th tracton fr dg, x = a, at ponts (0.945a, 0.5b, 0.5h) and (0.945a, 0.5b, 0.5h) ar 2.2 and 3.56%, rspctvly. Howvr, ths dffrncs at (0.055a, 0.5b, 0.5h) and (0.945a, 0.5b, 0.5h) rduc to 0.69 and 0.5% whn σ xz s computd usng th SRS. W not that at th nnr surfac of th shll σ xz computd usng th SRS xactly satsfs th tangntal tracton boundary condton, snc th appld tracton thr s usd as an ntal condton durng th ntgraton of th qulbrum quatons. Th transvrs shar strss computd drctly from th consttutv rlaton dffrs from th corrspondng 3D LET soluton at most by 12.56 and 9.6% for x = 0.055a and 0.945a, rspctvly. Howvr, ths dffrncs rduc to 8.04 and 7.9% whn σ xz s computd usng th SRS. W hav not chckd f rfnng th FE msh nar th dgs (.., usng a gradd FE msh) wll rduc ths dffrncs. Th maxmum valu of σ xz n th plan y = b/2 along th transvrs normal scton at x = 0.055a s about 3.75 tms that for x = 0.945a. In Fgur 15, w hav xhbtd th varaton of σ xx along th lngth n th x-drcton of th shll at y = b/2 and z = 0.45h computd usng th TSNDT and th 3D LET. Th two sts of rsults agr wll wth ach othr. At th clampd dg, th dffrnc btwn valus of th axal strss computd from th two thors s 1.44%; ths dffrnc rducs to 0.95% at x/a = 0.055 and 0.92% at x/a = 0.5. At th fr dg, x = a, σ xx accuratly satsfs th zro tracton boundary condton. Th axal strss s maxmum at a pont locatd on th clampd dg, x = 0, and dcrass wth an ncras n th dstanc of th pont from th clampd dg. 3.4.2. Effct of th curvatur on strss dstrbutons In Fgurs 16a and 16b, whavlucdatdthffctofth curvatur on through-th-thcknss dstrbutons of σ xx and σ xz along th transvrs normal passng through th cntrod of th md-surfac and th pont (0.055a, 0.5b, 0) nar th clampd dg x = 0, rspctvly. It s found that th magntud of σ xx ncrass, that of σ xz dcrass, and th plan of th zro axal strss shfts towards th md-surfac of th shll wth an ncras n th R/a rato.howvr,thratofncrasanddcrasn magntuds of σ xx and σ xz slows down wth an ncras n th valu of R/a. Forxampl,thnormalzdmagntudofσ xx at th pont (0.5a, 0.5b, 0.5h) on th outr surfac of th shll ncrass from 18.1 for R/a = 1 to 30.4 for R/a = 5andto30.6for R/a = 10. Th normalzd magntud of σ xz at th pont (0.5a, 0.5b, 0) on th md-surfac of th shll dcrass from 6 for R/a = 1 to 1.9 for R/a = 5 and furthr to 1.6 for R/a = 10. 3.4.3. Effct of gomtrc paramtrs on th stran nrgy of dformaton W nvstgat th ffct of th aspct rato and th shll curvatur on stran nrgs of dffrnt dformatons for th problm studd n Subscton 3.4.1 wth a = b = 10 cm. 3.4.3.1. Stran nrgs of n-plan and transvrs dformatons. 3.4.3.1.1. Effct of th aspct rato. W hav dmonstratd n Fgur 17a for a sphrcal shll th ffct of th aspct rato, a/h, for R/a = 1 and 10, on stran nrgs of n-plan and transvrs