Solving Equation [5.61], the helical fiber thickness required to contain the internal pressure is:

Transcription

1 5.4.3 eng Analyss of Cylndrcal Pressure Vessels S. T. Peers 001 Ths sofware s provded free for your use wh no guaranee as o s effecveness. I s copyrghed by Process-Research and may no be duplcaed, gven away or raded wh anyone. If you have problems wh we would apprecae some nofcaon. O WARRATIES. Process-Research dsclams all warranes eher express or mpled, ncludng bu no lmed o mpled warranes as o merchanably or fness for a parcular purpose wh respec o he SOFTWARE or he accompanyng wren maerals. Ths ncludes O LIABILITY FOR COSEQUETIAL DAMAGES. In no even shall Process-Research be lable for any damages whasoever (ncludng, whou lmaon damages for loss of busness profs busness nerrupon loss of busness nformaon or oher pecunary loss) arsng ou of he use of or nably o use hs produc, even f Process-Research has been advsed of he possbly of such damages. The followng equaons and he aached programs are descrbed n Flamen Wndng, Compose Srucure Fabrcaon pp , nd Ed., 1990, SAMPE Publshers eng analyss s a smple procedure for predcng sresses n a fber-renforced compose by neglecng he conrbuon of he resn sysem. The procedure has nuve appeal, and ganed accepance early n he developmen of compose srucures. The echnque apples sac equlbrum prncples wh no consderaon gven o sran compably. Even wh such heorecal flaws, neng analyss connues o survve, and s somemes useful for szng and for predcng falure n smple srucures. However he echnque should no be exended o oher han smple pressure vessels and he desgn of any compose srucure mus be based on a horough undersandng of lamnae plae heory As an example applcaon, consder a flamen wound cylnder of radus R pressurzed wh an nernal pressure p If he vessel s wound wh only helcal (±θ) fbers, wh an allowable fber sress σ deermne he helcal fber hckness ( ) and he wnd angle θ. Fgure 5.11 shows forces acng on a wo-ply helcal layer wh a cu of un wdh (a) n he axal plane, and (b) n he crcumferenal plane. Summng forces n he axal (m) drecon n Fgure 5-11a shows: σ θ θ cos θ m f f [5.61] Solvng Equaon [5.61], he helcal fber hckness requred o conan he nernal pressure s: σ cos θ [5.6] Summng forces n he crcumferenal (h) drecon n Fgure 5.1b, produces: h σf θ sn θ [5.63] 1

2 Usng from Equaon [5.6] n [5.63] shows ha an θ, or θ± 54.7 degrees. Ths s he wnd angle requred for a pressurzed cylnder wh helcal wndngs only. If he vessel s wound wh boh helcal (±θ) fbers and hoop (θ 90 o ) fbers, deermne he helcal and hoop fber hckness and f90 respecvely. Fgure 5-1 shows forces acng on a wo-ply helcal layer and a hoop ply wh a cu of un wdh, (a) n he axal plane, and (b) n he crcumferenal plane. Summng forces n he axal drecon (Fgure 5-1a) produces he helcal fber hckness requred o conan he nernal pressure, gven by Equaon [5.6]. Usng hs value for when summng forces n he hoop drecon (Fgure 5-1a) produces: ( an θ) [5.64] σ f 90 Where σ f90 s he fber sress n he hoop fbers Conversely, when he CSA value s no known (many fber manufacurers may no nclude CSA n her produc descrpons), can be deermned from he followng [5.65a] from he densy and yeld of he fber. 1 densy yeld CSA [5.64a] In he wndng process, spools are used o form a wndng band of wdh W. Each spool has a cross seconal area (CSA), values of whch are lsed n Table 5.3 for several rovngs. The fber hckness for each ply as wound s: CSA ( ) fp W [5.65] The fber sress a falure n a compose lamnae s less han ha for a undreconal coupon es or a srand ensle es. Ths reducon s called he ranslaon effcency; ypcal values are 70 o 80% of he srand ensle value. For example, he allowable sress for Kevlar 49 fbers s 3800 MPa (550 ks) n a srand ensle es (Table -). A hgh value of fber sresses for Kevlar 49 n a pressure vessel applcaon s 750 MPa (400 ks), a ranslaon effcency

3 of 73%. Translaon effcency s affeced by fber damage durng wnd, vods, complex sress saes, ec., whch are presen n he lamnae, and no n a srand ensle specmen. In a baxal enson feld, lke a pressure vessel, Kevlar 49 ypcally exhbs lower ranslaon effcences han glass or graphe. Kevlar, wh s fbrous mcrosrucure, s easly spl longudnally by marx cracks, f he marx s ghly bonded o he fber. If he fber s coaed wh a release agen before wndng, marx crackng can bypass he fbers, mprovng effcency. Usually, fbers where nerlamnar shear srengh s mporan would no be released. Reflecng hs, n a cylndrcal pressure vessel he hoop fber would be released, whle he helcals, wh dsconnues a he dome cylnder juncon and a he polar boss, would no be released. Translaon effcences of he hoop fbers would ypcally be ncreased from 70 o 85% by releasng. Releasng glass and graphe fbers has been found o be neffecve n mprovng ranslaon effcences for hese fbers. In a cylndrcal pressure vessel, ranslaon effcences are usually lower for helcal fbers han for hoop fbers. Ths s because he helcal paerns nclude more overlaps and crossovers, as well as dsconnues a he dome cylnder juncon and n he polar boss regon. For hese reasons, he helcal-o-hoop fber sress rao used n desgn s usually beween 75 and 100%. In wndng, he number of ples s seleced so ha he ply hckness n Equaon [5.65] dvdes equally no he oal fber hckness requred, Equaons [5.6] and [5.64]. Band wdhs are seleced n conjuncon wh he number of spools so ha a band hckness (ncludng resn) of abou 100 o 50 µ (0.004 o nches) s produced. Ths provdes a band ha s neher oo hck nor oo hn. For he helcal wndngs, he band wdh W θ s seleced o cover he cylnder wh no overlap wh a whole number of bands wound a he angle θ. The helcal ply hckness (ncludng resn) s hen: 3

4 Cθ ( CSA) WV θ [5.66] Where V s he fber volume fracon for he helcal wndngs. For he hoop wndngs, he band wdh W 90 s seleced so ha he hoop wndngs do no devae apprecably from he 90 o wnd angle. The hoop ply hckness (ncludng resn) s hen: C90 90( CSA) WV 90 f 90 [5.67] Where V f90 s he fber volume fracon for he hoop ples. Table 5-4 provdes ypcal ranges of fber volume fracons aaned wh dfferen fber sysems for helcal and hoop ples. Acual fber volume fracon depends on several process and geomerc consderaons ncludng resn vscosy, mandrel dameer, wndng enson, wnd angle, processng me, B-sage emperaure, and exernal pressure durng cure. The fber volume fracon s deermned by usng n-processng hckness measuremens. Good process conrol produces repeaable dmensons. umercal Example The above equaons are used o provde prelmnary hckness for he sandard ASTM D- 58 pressure vessel (Fgure 5-13) desgned for a 45.5 MPa (6600 ps) burs pressure and an nsde dameer of 146 mm (5.75 nch). If he wndng machne s se up for a 5.8 mm (0.3 nch) hoop band wdh and 5.1 mm (0. nch) helcal band wdh, he specfed helcal wnd angle θ n he cylnder s 1 o. The vessel s o be wound wh Kevlar 49, 4-end aerograde rovng (Table -). Desgn allowables are chosen as.93 GPa (45 ks) hoop fber sress, and.1 GPa (30 ks) helcal fber sress. Ths uses ranslaon effcences of 77% for he hoops, and 58% n he helcals, wh a 75% rao of helcal o hoop fber sress. The requred helcal fber hckness Equaon [5.6] s: σ θ cos cos mm( nch) Assumng 3 helcal layers (6 helcal ples), he calculaed number of helcal spools Equaon [5.65] s: θ f θwθ ( CSA) 079. /

5 The desgn would use θ, he closes whole number. I may be desrable o repea he calculaons wh a slghly modfed band wdh so ha he calculaed number of spools s closer o a whole number: The requred hoop fber hckness Equaon [5.64] s: ( an θ) σ f an cm( n) Assumng 9 hoop ples, he number of hoop spools Equaon [5.66] s: 90 f 90W / CSA Assumng a fber volume fracon of 0.6 for he helcal, and 0.65 for he hoops (Table 5-5), he oal compose hckness s Equaons [5.66 plus 5.67]: c 30. mm( 01. nch)