.b. BEHAVUR F CYlNDRCAL SHELLS BULT F HLLW CLAY BLCKS K NAG B. c.:; (En g), D..C., pi),d. _. C. En.g., 'C;; "JStruc:t. E., FP.!':, '_7F,'(T),." ')('';' SR DAVES 8. 5(;., P7),D., C. E'lg., F.. C. E. DepY'tm3nt f' Civ'il EnrTneeJng an'.: Bui Z'7irt;J 8cienC9 Un.iVel"sl:ty f Edi'lbuY'gl, > 'l:din.'x-t! gj., Cren Eritain BEHAVUR " CYLNDRCAL SHELLS BULT " HLWY CLAY BWCKS VERYALTEN VN ZYLNDRSCHEN SCHALEN AUS HHLZECELN The authrs have previusly develped expressins fr the slutin f plates built f hllw c:lay blc:ks and in this papel" the expressins are further develped t inc:lude shezls. Theretic:al results btained by substitutin f finite differenc:e apprximatins and then slving the resulting equatins by means f a c:mputer are c:mpared with thse btaineâ experimentally using a c:ylindpic:az shell built f hlzw c:lay blc:ks. v,:e Autrn haben schn f Mel' Rec:henrrr8thdEn fúr die Berec:hnung vn Platten aue Hhlzieg;!ln entwic:keu. n diesem Benc:ht werden nun die FY'!Teln weiterntwic:ke U und errrdglic:hen nun auc:h die Berchnung vn Sc:halen. Theretisc:h ermittelte Ergebnisse, die durc:h Urrwandlungen vn Naherungs fy'!te n erzie Zt wvyden und die dann mittezs Cmputer errec:hnet wurden, werden mit Versuc:hsergebnissen ve rg li c:hen, bei denen zy Zindnsche ZiegeZsc:haZen aus Hh lziege Z verwandt wvyden. LE C!>fPRTEMENT DES PARS CYLNDR';JUES CNSTRUTES EN BWCS CREUX EN TER_'?E CUTE HET GEDRAG VAN CYNDRSCHE UT HLLE BAKSTENEN SCHALEN Les auteurs nt djjvelppé auparavant de", frmules purle c:alc:ul des Danneaux cnstru;' ta en b lc:s creux en terre cuite. Dana c:ette cmmunic:atin c:es frmule s snt déve lppéea jusqu 'à l 'inc:lusin des pal'is. Des résultats thériques btenus par la méthde d'apprximatin d 'é léments finis avec: la rés lutin de c:es équatins d 'apprximatin à l 'aide d ' un rdinateur snt c:mparés avec: l slutin expérimentale truvée pur une pyi JlindPique en blca c:reux en terre c:ui te. De auteurs hebben vraf de frrrru les ntulikkeld vr de berekening van baksteenplaten uit hlle steen, en in de ze mededeling ntwikkelen ze deze verder vr sc:halen. Theretisc:he resultaten bekmen d' de methde van benadering met finite elementen met plesing van deze benaderingsekwaties die d.m.v. cmpu"ers wrden vergeleken met de experimentele plssing gevnden vr een cylindrisc:he sc:/mal uit hle blkken.
. b. -. NTRUCTN Hllw c lay blcks have been used in the Eurpean cuntries, espacially i n taly, fr ver frty years, as structural cmpn e nts fr the cnstructin f shells f varius shapes. Hwever, there seems t be n reasnable design me t hds available fr the design f a cylindrical shell bui lt f hllw clay blacks which takes bth membrane and bending actins int cnsideratin. The authrs have therefre deelped expressins applicable t such structures which takes bth membrane and bending actin int accunt. These expressins are slved by using finite difference apprximatins and a cmputer. Teses were carried ut n a cylindrical shal l built f hh mdel b lcks. The experimental results ""ere in clse agreement wi th the theretical results. n practice hllw clay b l ck s may be f different crcss-sectins and t simpli fy the analysis, the crss-sectins f the shells cnsidered n this paper are represented by f i gu r es ' and. The methd can hwaver be mdified t allw fr ther types f crssse ctin.. EVELPMENT DF THERY The thery was develped frm first principes using tha usual assumptins f elasticity and setting ut relatinships cnnecting : a) Middle surface strai ns and changes f curvatura, lth the displacement f the middle surface b) Parallel surface strains with middle surface st rains and changes f curvat ure c) Nrmal and shear stresses with parallel surface strains d) Frces and mments with nmal and shear stresses e) Frces and mmen t s i n the frm f the equilibrium eq uatins Three partial differential equatins in terms f U, V and vj cnnecting the di sp lacements f t he mi ddle s urface and externa frces are btained by successlve substitutin f each relatinship int the next. Tha rasulting partial differential aquatins are : d ' () + J (+lj )l } () dx l dxdy (.!:!.) R + P x... () { ( - lj )l} () + { ( + lj )l } ( ) d,,( _3.) óx ÓXdy dy d " - (R ) ( 3W) + ay P Y () =............. (lj a ) () d"a ( av) ()w - k ' () (R) - R óx ay R óx" P z ( 3) ó"w - ( - ) K " () + K..... ax 3y óy" \lhere, d ' k ' a = /K 0, d " x K K, k" x, x and y are the extensinal rigidities and defined by : E( - ljf dz s E( - lj)f dz + Ef (") dz x s r = E( - lj) f s dz + Ef () y r bj K, K and K are the bending rigidites and defned x y by : K = E( - lj)f Z dz s K = E( - lj)f Z dz + Ef ( " ) x s r... dz Z dz K = E ( - lj ) J Z LJz + Ef ( ' ) Z dz Y s r j where f s integratin ver slab f r intsgréltin ver rlb An eighth rder ditfarantial equat0n in terms f dsplacement 'w' is btai ned frm the thrse displacement equatins by successive dlfferentii:lt.in and eliminatin f U and V terms. 3. ANAL YTCAL S LUTmJ The three gverning equati::>ns in terms f U, V and W f which the first tw are secnd erder ",nd the third furth rder are slved LJsin g the finit'l di fterence apprximatins. The differe:tial equacins are re duced t a finite number f linear simultaneus equatins and thes8 squatills dre then scl'led by cmputer. nce the values f U, V and vj are btained, the values f frces and mments at,my pint can 5a5ily be celculated (ÓU) óv W N + 0( - - -) x x dx dy R N ( - ) + lj(l Y Y óy R d Q U óv r N ( - \) (- + - ) xy yx ay óx M - { Kx () a \<J + pk( x ) óx ay 'W M - { Ky ( ) + ljk( _d_ ] Y dy dx } M M - K( - lj ) (ó w ) xy yx dxóy
. b. - ' FGURE RB < "- ::: C) -'-i." '-'l '"! y r b b"- \ rl l ' i \ ""' FGURE FXED.-. _. x! _. - - - - - r- (7 rrrn ) 30" J -T, 3...- - 0-8 7 Bttm LDCATDN DF STRAN GAUGES FGURE 3D (7rrrn).. 9 f- -r Li r!30 "(7 mm)j DMENS NS DF CLAY-BLDCK SHELL FG URE 3 A-A R = 30" ( 7rrrn) y L-7?7E 7 5 9 3 /3 3 3."".r c:....." " " S... N J c:.... li "" "'; ' '" THEDRET CAL EXPERMENTAL L- -=E 9 5 LAD = / 5 Lb/sq. i n. (0. x 0-3 N/mm) 8 0 8 3 35 5 93 7 5 933 3 0 8 30 FGURE CDMPARSDN DF EFLECTDNS - W SHELL BULT DF HDLLW CLAY BLDC<.S
.b.-3 5. EXPRMENTAL WRK 7. ACKíi<JLEDGEMENT A cyl tr"jricc'l "nall flxed d:: t ile r<dges and free at anc-a".hds vjas juilt cf " "G8 hljw L:lay blcks. Ha!f-5!e l0cks WEr US5 tu stagger the lngitudinal júi'lts. leinfdrc:e''snt, bcth at tp and bttm faess, was l ' vldd nly alrlg the trnsv8rs8 directin. T ensura a09quatr edge fity tha fixed adges f the shaj,8'8 enledded in i'lrge L:ncret8 cuge bsams. TMe aetaile af tn8 mddsl sell ensidered were as fnllws (Figurs 3: 30 in. (75 mm) 30 in. (7 mml The authrs wish t thank h8 British Structural Ceramic P,ssciatilln anel tlle Edj.nburgh University fr financiõlg the prjct anel Pr P. A. \ J. H8nCiry fr his assistancs and encljragemgnt. 8. REFE"':NCES Due ta imitatin Df SCG it is nt pesible t incl ude a cmplete list Df rrfe;-enl:8s. rl\,ever, il Lst Df references Ciln be fund in "An Analytical anel Experimental nvestigatin Df Stresses in Plates And Snells Df Hllw Crss-Sectins by K. Nae, Ph.D. Thesis, Edinbrgh Universjty, May, '958, Raclius 30 in. ( 7 iln) PRNCPAL NTATNS MS5h lengths a '. A75 in. [7, G mml b ' 3. 75 ill. (95. mrnl Ladd int:gnsity b/in (usad ú= bass fr anc:.l ysii J EA. 57 X 0 5 lb ( l h =.BrJ) f:a x EA Y E x E :/.57x[l'lb.7 X 0 5 b.5 7 X 0' lb / is 7. x 0" b/jns Ncte : Th8se vaues wir8 clet8l'rr.ins,j by ex:jsrimerlt Th :nrter used fr Ct:Fí;ínti:g t!-ll blcks tgether 'Nas in t h fllning prncrticgs: :n gms 700 gms /5 CG x,y, z U, V, W P,P, P x y z N, N x Y NXy, N yx M,M x Y M,M xy yx R L, B c-ardinata system Displac8ment cmpnents in the x, y and z di;-etins respsctiv8y E.dernal lads ill the x,y anj z dir8ctinns l'esp'jcti'j)ly Nrmal FDrc8s Sheôr F[cs Twistin Mments PiS5cn 's Rat.i Radius Df Cur'vature Areas Df [rss ssctlcns Mament cf nsrtia Df crss-sectin Length and Breadth reseativ8y C?:;/t;'í:: US8rJ t-j"=s Ferr cement and sand irias Lt0r Huz3rd N. 5.B. T;;;8': PrnCE'Gll'e a, b, a,b r'8sh 8fi;;t:lS K, K,K x Y Bending Fiiuities A uni tcnnly distributed lad was anplied in steps af!b5. 7 KN up t a mbium Df 30 bs (. KN by uig Jead wights and baga filleu with stne chips. Ds+!ectlns were m8asureu by dial gauges the lcatins bing sh\'ln i n Figures t, and 5.. crjnclusns A ccmpdrisan between the experimental and theretical values f tha deflectin 5rB :hqw in figures G and 7. Cemparisn Df the va luas f dsf8ctiuns btaine by analytj8 calculbtins ad by 8X8 ' lrnts shw clse agreern8llt. Tre ciscrepancics êlnl hj_gh8r near the bufldari3'j... lhich,tlby 08 d\ja ".0 th8 ilycrductin f the fictitius Js in the analysis. n addit j. n these rni,tn; 'Jc; 0.8 t \'ii'kmanship, S,'pCJrt C'Jfl'Ü tins, rn8thc!jf ld:''s :3n,j p.j :3!:d.bls.:hai'.g8 in t8mperôture dur:lng r."jtlns. Crót.l'? í J t vjllld hv'e!j0'3f rth!"est..i!!g t(.l ct: rre.ra thl results )t:áinf:d frgrr. tu SC é-l8 t8st. bljt the :tuthürs l...jers ln.bl. t. Find f.lny pctjl isr,ed :('Lsult3 fr cr:.r:-ipfjrisn.!-.- L-.-L._'-_ - L-.-,-,,,-J - r f- i t L -e-t-,.-,_l -.- - - L-, ' ' ' - - 0 í' l - - - - l f-. r', --,Q 5 f..- -, f - j, -r._ - f fj f..- i_. _. - _.- " 'b LCAT N r,f ral C'.UGf.S FGURE S
. b. CYLNRCAL SHELL BU LT DF HDL LDW CLAY BLCKS: FXE AT EGES AN FREE AT EN-ARCHES CDMPARSN DF EXPERMETAL AN THEDRETCA L RES ULTS TABLE EFLECTDNS (W ia l Gauge N. Lcat irl f ef eculls in 0 in. Remar ks ial Gauge eviat in frm Experi.mental Thereti cal (Mesn Pint Ther. resu l ts 3.75. 8. 5. +3. 0% + 5. 3% 3 5.0. 8 +. % 3. 0. 8 + 7.% 5 9. 0. - 5. 7% 7 8 9 0 7 0 5 30 -. 0. 5. 0.8 3..8. 0.. 9. 8-5. 7% +. % +0. 7% - 5.7% +. % ;( La d ntenshy (Un i frmly ist ribut ed ( /5 b/in THEDRETCAL EXP ER MENT AL L-.-.-.- =E 7 0 3..... <::>.-., s:: li."........,; <::> '".-., s::." ','.., tj. :::\ L -..- r-e 5 8 3 L,_rE 30 33 35 3 FGURE 7 CDMPARSN DF DEFLECTNS SHELL BULT DF HDLLW CLAY BLDCKS LAD = / 5 lb/in ( 0. X 0-3N/ f )