Stability loss in thick orthotropic cylindrical shells under external pressure

The stability of equilibrium of a n orthotropic cylindrical shell subjcctcd t o external pressure is investigated. In this co~itext , the structure is considered a three dimensional body. T h c results show that the shell theory predictions can produce non-conservative results on the critical load of composite shells with moderately thick constructio~l. T h e solution provides a means of accurately assessing tlie limitations of shell theories in predicting stability loss. Introductioil A class of i11il)ortant structural applicatio~ls of fiberreinforccd composite materials involves the configuration of lanlinatcd shc~lls. Although thin platc co~istruction has been the tlir~lst of the initial applications, l i i ~ ~ l l attent,ion is now being p i d to configurations classified as moderately tllick sl1c.11 st,nlcturcs. S11c11 tlesigns call be used in co~nponents in the aircraft and autolriobile industries, as well as in the marine industry. Moreover. composite laminatf's have been considered in space vehicles in t h r form of circular cylindrical shells as a p r i~nary load carrying s truct l~re. In these light-1vl-c.ig11t shell structures. loss of stability is of p r i~nary concern. This snbject has been researched to-ilat,o through the application of the cylilidrical shcll t,lic~ory (c..g. Sirriitsrs, Shaw and S h e i n m x ~ , 1985). Honcl.sr, prcvious work (Pagano and \4%itliey. 1970; Pagimo. 1971 ) has sliowri that considerable care must be exc,rcisetl in applying thin shell theory formula t io~ls t o predict tllv response of composite rylinders. Besides tlic anisotropy, composite sliells h a w one other important t l is t i~lg~ishing fcat~lrc . namely extensionalto-shear 1llodl1111s ratio m ~ l r h largcr than that of their n ~ e t a l coulltcrpiirt s. 111 ort1c.r t o marc accurately account for the above rnrntioncd cffvcts. various niodificatio~ls in the classical theory of 1;lnlinated shells have generally been performed ( ~ I ' l ~ i t ~ i e ~ and Sun, 1974; Librescu, 1975; Reddy and Liu. 1965. scc also Noor and Burton, 1990 for a review of shear dcfor~nation theories). These higher order shell t11eoric.s can be applied to buckling problems with the potential of improved predictions for tlie critical load ( Aliastnsiadis, 1990). However, tlicsc, lias not yet been any effort to produce a n solutioli I~ascd on three-dimensional elasticity to the ~ x o b l s m of I)llrkling of composite shell structures, against wllicli results fro111 various shell theories could be compared. Towastls this ol>jcrtive, this work presents an elasticity solution to the problenl of buckling of composit? cylindrical ortliotropic shells subjected to external pressure. Colvright 0 1 9 9 2 113. the American Institute of Aeronautics and Astrouautics, Inc. All rights reserved, Formulatioil The equations of cquilibriu~n are taken ill t,erms of the second Piola-I posi t ic)~~. A perturbed posit,ion is de11otcd by where a is an infinitesimally small quantity. Here. C I U ~ ( T . 0. 2). 0 v I ( 7 . B . z ) . ( ~ ~ o l ( r . , 0 , 2) are the tlisplnccxmerits to which the points of tllv body must be s ~ ~ h j c r t c d to shift thenl from the initial position of eq~~i l ih r i~ l rn t o the new equilibrium position. The functions t ~ ~ ( r . 0 . z ) , c l ( r , 6 . z ) , w ~ ( r , 0 z ) are assilmed finite and cu is an infinitcsimally small ql~antity independent of r , 0. z . Following Kardo1ll;rtcas (1991), u7e obtairi the following 1)uc.kling cyl~atioris: ~, 111 t11v ~)rcviot~s equations, IT: and w" J are t , l ~ values of IT,, and d l at tllc initial cquilibrillrn position. i.e. for 11 = uo. 1 % = I." ant1 11% = q. and a:, and w(, arc: t,he values at tliv pr!rt~~rl)td position. i.e. for u = ul. 1 = 211 and 111 = 71.1 . The 1)01111<lary conditions associated with ( l a ) can ljr expressctl as: where f is the, tract icm vector on the surface wllicll hits outward unit norllii11 il = ( 1 ) m , 71 ) before any defor~natimi. Tlic tractioli v c ~ t o r tlcpcrids on the displaceiric.rlt fic.lc1 ? = ( 1 1 . 1.. 1 1 . ) . A~airl . following Iiardolnatcas (1991). U Y ~ o l ~ t a i ~ ~ and for thc lateral surfaces, i.e. for m = n = 0 and 1 = 1, c7:, + ~ : ~ d ; = O , ( 8 a ) Pre-bucklin,q State. The problem at hands is that of a hollow cylinder rigidly fixed at its ends and dcformed by uniformly distributed exter~ial pressure p (Fig. 1). The axially symmetric distribution of external forces produces stresses identical at all cross sections and dependent only on thc radial coordinate I-. In this manner the forces at the ends are distributed identically over both surfaces and rcduce to equal and opposite resultant forccs and moments. Let Rl be the internal and R2 the external radlus and set c = Rl /RE Lekhnitskii (1963) gave thr strcss ficld as follows: Fig. 1. Hollow cylintlvl 111lde1 ext(wla1 prcxssure We denote by r.,] the stlffne5s constants for the orthotropic constitut~vc behavior and by a,, the colnpli ance conc;tantc. i (. .