An easy way to analyse octagonal slab

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1 4 th Aual Paper eet ad 1 t Cvl Egeerg Cogre, December 22-24, 2011, Dhaka, Bagladeh ISBN: Noor, Am, Bhuya, Chowdhury ad Kakoled A eay way to aalye octagoal lab Sharm Reza Chowdhury Departmet of Cvl Egeerg, Ahaullah Uverty of Scece ad Techology, Dhaka, Bagladeh ABSTRACT: Th paper decrbe the aalye of octagoal lab wth a few elected upport codto. Baed o the avalablty of mddle trp, aalye decrbed th paper have bee dvded to two et. I frt et, where mddle trp ext, mddle trp momet ad lab cetre deflecto have bee compared wth that of rectagular lab order to detfy the rage of parameter whch would allow treatg a octagoal lab a a rectagular lab. I the ecod et, where mddle trp doe ot ext, aalye have bee performed to tudy octagoal lab behavor. Fally reult of aalye have bee utlzed to formulate coeffcet table to compute the momet ad deflecto of octagoal lab. Smplfed method ha bee decrbed where octagoal lab ca be aalyzed a a rectagular lab. 1 INTRODUCTION I degg the rectagular lab there are coveet deg method avalable to uer to obta the lab trp momet ACI, 1963 ad method are alo avalable to compute the deflecto of uch lab Ahmed ad Chowdhury, 1999a, b. Such method are ot avalable for aaly ad deg of octagoal lab. From the upport codto rectagular lab are clafed to e categore ACI, Th umber depedet o the umber of upportg edge of the lab. Coderg the ame Octagoal lab ca be clafed to a large umber of categore. Itead of gog through all the cae the paper am to detfy a few umber of elected cae that are mot lkely to occur buldg tructure. I addto to that t of mportace to detfy cae where a octagoal lab ca be mplfed a a rectagular lab ad whe t mut be aalyed a a octagoal lab. The preet paper detfe uch cae ad alo preet a tabular form of coeffcet to compute the deg momet ad deflecto of octagoal lab wth few elected upport codto. 2 SELECTION OF PARAETERS FOR ANALYSES To tudy the behavor of octagoal lab addto to the pa rato a/b ee fgure ; followg have bee codered: Support codto a1/a ad b1/b rato ee fgure 1 for defto of a1, a, b1 ad b 2.1 Support codto It kow that there are e poble upport codto for rectagular lab. Number of upport cae creae rapdly wth creag umber of lab edge. Thu t became mpoble to cover up the full et of poble cae the preet reearch cheme. Oly four upport cae are codered the preet paper Cae A mea all edge mply upported; Cae B mea all edge fxed; Cae C mea oly oe edge fxed whle other mply upported ad Cae D mea oly oe edge mply upported whle other fxed a how Fgure. 235

2 Fgure 1: elected upport codto 2.2 a1/a ad b1/b rato A metoed the troducto two et of aalye have bee elected. To coduct each et of aalye a certa rage of parameter ha bee elected a how Table 1. Table 1. Selected a1/a ad b1/b rato for et-1 ad et-2 Spa rato a/b Set-1 Set-2 a 1 /a b 1 /b a 1 /a b 1 /b For octagoal lab belogg to et-1 mddle trp momet ad lab cetre deflecto to be compared wth that of the rectagular lab. Ug reult of et-1 ad that of et-2 coeffcet ca be obtaed o that momet ad deflecto ca be calculated by ug thee coeffcet ug mplfed equato. 3 COPUTATION OF DESIGN OENTS FRO RESULTS To compute the mddle trp momet of octagoal lab belogg to the frt et et- ug reult method decrbed by Ahmed ad Chowdhury 1999a, b have bee followed. To calculate momet of octagoal lab belogg to the ecod et et- from the reult of EF model, ce mddle trp a/2 or b/ ot avalable, dfferet approach eetal that decrbed below. 236

3 Fgure 2: Calculato of momet the cled porto of octagoal lab cotued Fgure 2: Calculato of momet the cled porto of octagoal lab Fgure 2 how the procedure to calculate momet at cled porto of octagoal lab. omet -, -,, +, + ad ' -, ' -, ', ' +, ' + fgure 2a, 2b, 2 c, 2d are calculated ug the method to covert of odal momet to lab teral momet how Ahmed ad Chowdhury 1999a,b. From fgure 2 a, egatve upport momet hort drecto a - ca be obtaed by averagg them over the horzotal porto l b of the cled part of octagoal lab, a the reult are per ut wdth of elemet. Thu approxmately, 237

4 238 b a l Smlarly, a how Fgure 2 b, upport momet log drecto b - ca be obtaed by averagg them over the horzotal porto l a of the cled part of octagoal lab. Thu, approxmately, a b l Smlarly, from fgure 2 c, b a l 3 from fgure 2 d, a b l 4 For traght porto of octagoal lab, potve momet hort drecto ca be obtaed by averagg them over the traght porto b 1. Thu, from fgure 3 a, 1 b a 5 From fgure 3b, 1 a b 6 Smlar method appled for evaluato of the edge momet of traght porto. Fgure 3: Calculato of momet of traght porto of octagoal lab

5 Itead of computg momet, t derable to compute momet coeffcet. It poble to compute the coeffcet for dfferetal cae ad pa rato, a follow: 2 C Wl 7 a, a, ea C a, a, 8 2 Wl ea Where lea a1 2 l2a 2 l3a a ee fgure 4, ad W = Total uformly dtrbuted load 6 Smlarly, C b -, C a +, C b + ca be obtaed. For traght porto of octagoal lab, a, 2 C Wa 9 a, a, C 10 a, 2 Wa where a = hort pa of lab Smlarly, C b +, C a -, C b - ca be computed. Fgure 4: Calculato of equvalet legth for calculatg the momet of cled porto of octagoal lab 4 RESULT OF ANALYSES 4.1 Verfcato of model wth elected cae A exteve lterature o octagoal lab ot avalable, t ot poble drectly verfy reult of model for uch lab. But for the purpoe of checkg the accuracy of troducg tragular elemet, yet ome verfcato eetal. Th coducted by comparg reult of rectagular lab from Chowdhury 2000, dcated by cae a, wth reult of meh decrbed above for cae a1/a = b1/b = 0.00 dcated by cae b ad a1/a = b1/b = 0.75, dcated by cae c, for varou a/b rato, a thee a1/a ad b1/b the two later cae cloely repreet or early repreet rectagular lab ytem. Thu thee hould produce ame or very cloe dplacemet at the cetre whe ubjected to ame load. Sce deflecto to be ame, the coeffcet of deflecto hould alo be ame. So comparo of lab cetre deflecto coeffcet hould be ueful for verfcato of the ew model whch how fgure 5. Bede th the momet hould be alo ame or very cloe ad the momet coeffcet ca alo be ued for the purpoe of verfcato reult compared well decrbed Chowdhury

6 Fgure 5: Comparo of deflecto of mehe obtaed from dfferet cae for upport cae D 4.2 Effect of a1/a ad b1/b rato o mddle trp momet Fgure 6 how the varato of potve momet coeffcet rato hort ad log drecto wth b1/b rato for dfferet a1/a rato for pa rato a/b omet coeffcet rato for dfferet elected upport cae ad a/b rato ha bee obtaed by dvdg the mddle trp momet coeffcet potve or egatve hort or log drecto of octagoal lab of gve a1/a, b1/b rato by the ame of rectagular lab the repectve drecto; both havg ame a/b rato ad upport codto. It ha bee oberved from Fgure 6 ad the reult of aalye Chowdhury 2000 that the momet coeffcet rato potve ad egatve ether cotat or creae wth a1/a ad b1/b rato for all upport codto. It dcate that mddle trp momet of octagoal lab ether equal or creae wth a1/a ad b1/b rato tha that of rectagular lab for ay upport codto. Th happeg becaue load dtrbuto ether cotat or creae wth a1/a ad b1/b rato. From fgure 7, It ca alo be ee that the rato of potve momet coeffcet log drecto ether cotat or decreae wth a1/a ad b1/b rato for all upport codto a load dtrbuto ether cotat or decreae. From fgure 6, 7 ad other reult of aalye Chowdhury2000, It ca be cocluded that maxmum creae log drecto potve ad egatve momet 30% ad 15% repectvely ad maxmum decreae log ad hort drecto egatve momet 60% ad 25% repectvely, wth repect to rectagular lab. Fgure 6: Varato of C a + rato wth b 1 /b rato for a/b rato

7 Fgure 7: Varato of C b + rato wth b 1 /b rato for a/b rato Effect of a1/a ad b1/b rato o momet of traght porto of the octagoal lab Aalye coducted by Chowdhury 2000 how that varato of Cb+ for dfferet pa rato how that Cb + creae gradually whe b1/b rato vare from 0.00 to 0.25 but after that t doe ot chage coderg all the elected upport cae ad a/b rato. Varato of Ca + for dfferet pa rato how that Ca + almot cotat wth b1/b rato. Varato of Ca -,Cb - for dfferet pa rato, how that Ca -,Cb - creae harply wth b1/b rato coderg all the elected upport cae. Varato of Ca +, Cb +, Ca -, ad Cb - wth a1/a rato creae. Th occurred due to the fact that all the momet coeffcet creae a load dtrbuto of the traght porto creae wth a1/a ad b1/b rato. It ot poble to obta the varato of Ca +, Ca - for cae A, B, C ad D of b1/b rato = 0.00 wth a1/a rato a the traght porto of octagoal lab ot avalable. For the ame reao the varato of Cb +, Cb - for cae A, B, C ad D of a1/a = 0.0 wth b1/b rato ot poble to obta. Reult preeted by Chowdhury [4] ca be utled for computg the momet of traght porto of octagoal lab by the help of equato Effect of a1/a ad b1/b rato o momet of cled porto of the octagoal lab Varato of Ca +, Ca -, Cb +, Cb - for dfferet pa rato how that Ca +, Ca -, Cb + decreae wth a1/a rato wherea coeffcet of egatve momet of cled porto of octagoal lab Cb - log drecto creae wth a1/a rato. Reult preeted by Chowdhury 2000 ca be utled for computg the momet of cled porto of octagoal lab by the help of equato Effect of a1/a ad b1/b rato o deflecto Fgure 8 how the varato of deflecto rato wth b1/b rato for dfferet a1/a rato, elected upport cae ad 0.50 pa rato. Here deflecto rato ha bee defed for dfferet elected upport cae ad pa rato, by dvdg the lab cetre deflecto of octagoal lab of gve a1/a, b1/b rato by the ame of rectagular lab coderg ame pa rato, upport cae. Coderg deflecto of umercal aaly et-1, lab ca be aalyed a rectagular lab ce maxmum 10% varato of deflecto have bee oberved. Slab cetre deflecto of octagoal lab et-1 ca be drectly computed kowg the lab cetre deflecto of rectagular lab ug reult preeted by Chowdhury From reult preeted by Chowdhury 2000 for et-1 ad et-2 t ca be ee that deflecto rato creae wth a1/a ad b1/b rato a load dtrbuto creae wth a1/a ad b1/b rato. 241

8 Fgure 8: Varato of deflecto rato wth b 1 /b rato for a/b rato DESIGN CONSIDERATIONS Table 2a ad 2b lt the rage of parameter whe the potve ad egatve momet log drecto are gfcatly hgher tha that of rectagular lab, a obtaed from the aalye. Table 2a Rage of parameter whe potve momet log drecto are gfcatly greater tha that of rectagular lab Support cae a 1 /a b 1 /b Spa rato a/b A to B to to 0.75 D 0.50 to to to 0.75 Table 2 b Rage of parameter whe egatve momet log drecto are gfcatly greater tha that of rectagular lab Support cae a 1 /a b 1 /b Spa rato a/b C to to 0.75 Table 2c ad 2d lt the rage of parameter whe egatve momet log ad hort drecto are gfcatly lower tha that of rectagular lab repectvely. Table 2 c Rage of parameter whe egatve momet log drecto are gfcatly lower tha that of rectagular lab Support cae a 1 /a b 1 /b Spa rato a/b 0.50 to to to 0.75 B to to to D to to

9 Table 2d Rage of parameter whe egatve momet hort drecto are gfcatly lower tha that of rectagular lab Support cae a 1 /a b 1 /b Spa rato a/b B to to 1.0 D to to 1.0 So octagoal lab wth the rage of parameter how Table 2a, 2b, 2c ad 2d mut be aalyed a octagoal lab. Table 3 lt the rage of parameter whe octagoal lab ca be aalyed a rectagular lab wth maxmum varato momet wth a rage of 10% wth repect to rectagular lab. Table 3 Rage of parameter whe octagoal lab ca be mplfed a rectagular lab Support cae a 1 /a b 1 /b Spa rato a/b to A 0.75 to to to to to 1.0 B 0.75 to to to to 0.75 C 0.75 to to to to to D 0.75 to to to 1.0 ddle trp momet of uch octagoal lab ca be drectly computed kowg the mddle trp momet of rectagular lab, f the lab are wth the above lmt. Fally ug reult of aaly, momet coeffcet table have bee formed Chowdhury 2000 o that mddle trp momet ad deflecto ca be computed drectly. 6 CONCLUSIONS The paper decrbed the fte elemet aaly of octagoal lab wth four upport cae are codered all edge mply upported; all edge fxed; oly oe edge fxed whle other mply upported; oly oe edge mply upported whle other fxed. Equato have bee formulated to obta trp momet for uch lab from ANSYS reult that are term of momet at a ode per elemet. From the reult of aaly t ha bee poble to detfy whe t poble to aalyze a lab a a rectagular lab. Bede th a et of coeffcet table have bee prepared preeted Chowdhury 2000 ug reult of aalye, that ca be ued to aalye oly octagoal lab wth metoed upport codto. RERENCES Buldg Code Requremet for reforced Cocrete, ACI Publcato , Amerca Cocrete Ittute, Detrot. Ahmed B. ad Chowdhury S. R.,Smplfed calculato method for ervceablty deflecto of edge upported lab; Part 1: odellg Slab Repoe, Joural of the Cvl Egeerg Dvo, The Ittuto of Egeer, Bagladeh, Vol. CE 27, No. 1, p , Jue 1999, Ahmed B. ad Chowdhury S. R.,Smplfed calculato method for ervceablty deflecto of edge upported lab; Part 2: Coeffcet method, Joural of the Cvl Egeerg Dvo, The Ittuto of Egeer, Bagladeh, Vol. CE 27, No. 1, p Jue1999, Chowdhury S. R.,Numercal modellg of edge upported lab repoe,.sc. the, Departmet of Cvl Egeerg, BUET, Dhaka-1000, Bagladeh, ANSYS 5.4, Operato Gude, SASIP Ic, 201 Joho Road, Houto, PA