Closed-Form Expressions for Moments of Two-Way Slabs under Concentrated Loads

Transcription

1 IOSR Jonl of Mchnicl nd Civil Engining IOSR-JMCE -ISSN: 7-6p-ISSN: 30-33X Vol Iss V. III Jn- F Closd-Fo Epssions fo Monts of Two-Wy Sls nd Concnttd Lods Mgdy Isl Sl Civil Engining Dpt. Fclty of Engining Kflshih Univsity Egypt Astct: Th nlysis of ctngl two-wy sls sjctd to concnttd lod nifoly distitd ov dfind ft dispsion down in th two dictions to th infocnt is pcticlly ipotnt cs. As psntd y Egyptin cod fo dsign nd constction of conct stcts ECCS th nlysis of this pol cn pfod y sing th lstic nlysis o y sing n ppoit thod which dpnds on th distition of th concnttd lod in th two dictions y th tio of th long lngth nd shot lngth of th ctngl sl. Significnt diffncs twn th slts of ths thods st t o ttntion. An ltntiv ppoit thod fo dtining th nding onts of th sl nd stdy in th two dictions is invstigtd in this pp sd on th lstic nlysis. Two closd-fo pssions w otind which dsci th ltion twn th nding onts nd ll fctos tht ffct it th spn tio th dinsions tio of th lodd nd th tio twn th shot spn to th plll lngth of th lodd. Copisons twn th stining ctions in th two dictions slting fo th psnt nlysis nd ths fo th isting ppoit thod of th ECCS with thos otind y igwd s thoy nd thos otind y finit lnt thod givn lso in this pp. Kywods: Sl two wys concnttd lod nding ont lstic nlysis. I. Intodction Whn sl is sppotd oth thn on two opposit sids only th pcis ont nd distition of th lod tn y ch sppot nd consqntly th gnitd of th nding onts on th sl not sily clcltd if ssptions sling pcticl conditions d. Thfo ppoit nlyss gnlly sd. Th thod pplicl in ny pticl cs dpnds on th shp of th pnl of sl th condition of stint t sppots nd th typ of lod. Two sic thods [-3] coonly sd to nlys sls spnning in two dictions. Ths th thoy of plts which is sd on lstic nlysis nd svic lods nd yild-lin thoy in which th hvio of th sl s collps ppochs is considd. A lss wll-nown ltntiv to th ltt is Hillog s stips thod [] []. Fo ctngl pnl cying nifo lod siply sppotd long ll fo dgs nd which no povision is d t th cons to pvnt th lifting o to sist tosion th Gshof nd Rnin thod [] is pplicl. Whn th cons of th sl pvntd fo lifting nd tosionl stint is povidd th sipl Gshof nd Rnin thod is inppopit. A o ct lstic nlysis ssing vl of oisson s tio is pfod nd th slting svic nding onts t id-spn is givn in tls fo siplicity s sd in ost intntionl cods [5-6]. Whn sl cis lod concnttd on pt only of th sl sch s whl lod on th dc of idg th contct of th lod is fist tndd y dispsion thogh th thicnss of th sl nd th flooing if ny. If th sl sppotd on two opposit sids only th width of sl cying th lod y ssd nd th totl concnttd lod is thn dividd y this width to giv th lod cid on nit width of sl fo ppos of clclting th nding onts. Fo sls spnning in two dictions cying lod nifoly distitd ov dfind on pt only of th sl igwd s thoy [] givs th nding onts on siply sppotd pnl long ll fo sids with stind cons y cvs t th Egyptin cod ECCS [7] psnts n ppoit thod which dpnds on th distition of th lod in th two dictions. In this pp o ttntion is fistly focssd to th significnt diffncs twn th slts of ths thods nd th copison twn ths slts nd th finit lnt thod. Th slts otind fo igwd s thoy gd with th slts otind y finit lnt thod with good cccy whil th diffnc twn th slts of th ECCS nd th finit lnt thod is not ccptl. Also in this pp closd-fo pssions otind sing thoticl nlysis sd on th thoy of plts to dtin th nding onts in th two dictions dictly s fnction of th vils. A copison of th slts with finit lnt thod [] is givn in this thsis. DOI: / g

2 / v / Closd-fo pssions fo onts of two-wy sls nd concnttd lods II. Modl nd Assptions Consid ctngl plt siply sppotd long ll fo dgs s shown in Fig. cying concnttd lod nifoly distitd ov dfind with th following ssptions - Th is no dfotion in th iddl pln of th plt. This pln ins ntl ding nding. - Th cons of th sl pvntd fo lifting nd tosionl stint is povidd. y Fig. : Siply Sppotd Rctngl Sl Sjctd to Concnttd Lod Unifoly Distitd Ov Dfind A III. Rviw Of Egyptin Cod ECCS And igwd s Thoy Egyptin Cod ECCS As intodcd in cls of th Egyptin cod fo dsign nd constction of conct stcts ECCS th concnttd lod is considd to disps down to ctngl with sids nd v which is dtind fo v t c t nd t c t Wh t nd t th contct of th concnttd lod in dictions ppndicl to nd plll to th in infocnt c is th flooing thicnss nd t is th sl thicnss. Th concnttd lod is distitd y n invs tio of th sl lngths ccoding to ths ltions nd To clclt th nding ont in diction is considd distitd on lngth A of th ffctiv spn nd in dth B in th diction ppndicl to diction. in th conty th nding ont in diction is considd distitd on lngth B of th ffctiv spn nd in dth A in th diction ppndicl to diction wh A 0. nd B v 0. [ ] 3 igwd s thoy Th concnttd lod is considd to disps down to ctngl with sids nd v which dtind fo v t c d nd t c d Th positiv nding onts on nit width of sl givn y ths pssions M nd M y y y 5 Wh vls of nd y otind y cvs fo th ppopit vl of th spns tio = / cosponding to th tios / nd v/. IV. Mthod Of Anlysis And Thoticl Eqtions Th nding onts of th sl spnning in two dictions cying concnttd lod nifoly distitd ov ctngl is otind in [] sing Nvi s thod y th following pssions: sin M ln 6 DOI: / g

3 Closd-fo pssions fo onts of two-wy sls nd concnttd lods DOI: / g sin ln M y 7 In which sin cosh sin cosh 3 tn tn tn tn v v nd In od to div sipl pssions to dtin nding onts w lt = / wh th i nding onts nd y so ngnts th nding ont in th shot diction M cn pt in th following fo ln M Wh tting wh cosh 9 And sing th nown sis 0 Th ts nd cn pssd s 3 3 / / Th ts nd wholly indpndnt of dinsions of nd v of lodd wh ths fnction in spns tio only. In th conty th vls of fctos nd dpnding only on th tio v/. Sstitting Eqs. - into Eq. 6 th ts nd cn pt in sipl pssions s tn tn nd 3 Sstitting Eq. 3 into Eq. nd ssing oisson s tio =0.0 which is condd y ECCS nd oth intntionl cods s BS 0 [6] th nding ont M cn pssd s M tn 0. tn ln

4 Closd-fo pssions fo onts of two-wy sls nd concnttd lods tting tn / tn 5 Eq. cn pssd in sipl pssion s follows M 0. tn.ln. 5 6 In siil nn w gt th nding ont in th long diction M y s follows M y.ln. tn 7 If t = t i.. =v = Eqs. 6 7 cn pssd in th sipl fos M M y 0.* ln * ln V. Illsttiv Epls To dsci th pol considd in this pp nd to clify pplying of th otind qtions in th psnt wo th nicl pls givn which solvd y th ppoit thod intodcd in ECCS igwd s thoy nd th psnt wo. Also th th pls solvd y Finit lnt thod [] to chc th cccy of ch on of th pvios thods Epl : A sq sl of 3.0 sid siply sppotd on ll fo sids sjctd to concnttd lod 00 N syticlly plcd t th cnt of th pnl. Contct ing 75 y 600. th sl thicnss is 00 d=75 nd th flooing is 50 thicnss. Using ECCS : th sids of th lodd ft th lod dispss down to th infocnt v : = =375 nd v = = 900 Th lods distitd in ch diction : =003/3+3=00 N nd =00 N is distitd on lngth A of th ffctiv spn nd in dth B in th diction ppndicl to diction wh A = =.575 nd B = [-3/3] =.0 Ths th svic nding onts p t width : M = 6.3 N./ M y = N./ Using igwd s thoy: Th sids of th lodd v = = 55 nd v = = 050 =3.0/3.0 =.0 Fo th ppopit cv =.0 nd cosponding to / = 0.55/3.0 = 0.75 nd v/ =.05/3=0.35 th cofficint nd y otind s = 0.6 nd y = 0.3 Ths M = = 37.0 N./ M y = = 3.0 N./ Using psnt wo: Th sids of th lodd v : = = 55 nd v = = 050 =3.0/3.0 =.0 = v/ =.05/.55 =.0 nd = / = 3.0/0.55 = 5.7 Sstitting in Eqs. 6 7 DOI: / g

5 M M y ln Closd-fo pssions fo onts of two-wy sls nd concnttd lods tn ln tn = 35. N./ 5 = 3.07 N./ Using F.E.M.: M = 36.6 N./ M y =3. N./ Epl : Consid ctngl sl which is 3.75 long y 3.0 wid siply sppotd on ll fo sids sjctd to th s lod nd ssing th s ssptions givn in th pvios pl. Using ECCS : = =375 nd v = = 900 = / =. N nd =.9 N A = =.575 nd B = [-3/3.75] =.3 Ths M = 6.6 N./ M y = 5. N./ Using igwd s thoy: = =55 v = =050 nd =3.75/3.0 =.5 Fo th ppopit cv =.5 nd cosponding to /=0.55/3.0 = 0.75 nd v/ =.05/3 = 0.35 th cofficint nd y otind s = 0.9 nd y =0. Ths M = =.0 N./ M y = = 3.60 N./ Using psnt wo: = 3.75/3.0 =.5 = v/ =.05/.55=.0 nd = / =3.0/0.55 = 5.7 Sstitting in Eqs.6-7 M = 0.3 N./ M y = 30.7 N./ Using F.E.M.: M = 0.7 N./ M y = 3. N./ Epl 3: Consid ctngl sl which is 5.6 long y.0 wid siply sppotd on ll fo sids sjctd to th concnttd lod 00 N syticlly plcd t th cnt of th pnl. Contct ing 300 y 300. th sl thicnss is 00 d=75 nd th flooing is 50 thicnss.. Using ECCS : = v = =600 = 005.6/+5.6 =7 N nd = 3 N A = =. nd B = [-/5.6] =.665 Ths M = 3.3 N./ M y = 0.63 N./ Using igwd s thoy: = v = =750 nd =5.6/.0 =. Fo th ppopit cv =. nd cosponding to /=0.75/.0 = 0. nd v/ =0.75/5.6 = 0.33 th cofficint nd y otind s = 0. nd y =0.55 Ths M = =.0 N./ M y = = 39.0 N./ Using psnt wo: = 5.6/.0 =. = v/ =.0 nd = / =.0/0.75 = 5.33 Sstitting in Eqs.-9.6*. M 0.*ln = 7.5 N./ DOI: / g

6 M y Closd-fo pssions fo onts of two-wy sls nd concnttd lods 0.*ln *. = 0.0 N./ Using F.E.M.: M = 6.7 N./ M y = 39.7 N./ VI. Copison Of Th Rslts Fo th slts of th pvios th pls tl shows th copison of th nding onts otind y Eqs. 6-7 of th psnt wo.w. th ppoit thod givn in cls of th ECCS nd igwd s thoy with th slts otind y lstic nlysis s pogd y finit lnt thod []. Tl. Copison of M nd M y otind y. W. ECCS nd igwd s thoy with F.E.M.[] Epl F.E.M.W. % Diff. ECCS. 03 % Diff. Epl =3.00 =3.00 =00 N Epl =3.75 =3.00 =00 N Epl 3 =5.6 =.00 =00 N igwd s thoy % Diff. M % % % M y % % % M % % % M y % % % M % % % M y % % % It cn noticd tht lthogh th psnt qtions sipl it gv o cct slts whn copd with th soltion y finit lnt thod. Thn th psnt qtions cn th sd y th dsign ngins. VII. Conclsion In this pp thoticl nlysis sd on th lstic nlysis is dvlopd to dtin th svic nding onts of th sl spnning in two dictions cying concnttd lod nifoly distitd ov dfind on pt only of th sl. Fo slts nd pls cid ot in this pp th following conclsions dwn. Significnt diffncs twn th slts of th ppoit thod intodcd in cls of th Egyptin cod fo dsign nd constction of conct stcts ECCS nd th lstic soltion.g. y finit lnt thod. Also significnt diffncs twn th slts of ECCS nd tht otind y oth thods s th igwd s thoy. Ths diffncs lg nogh to consid th ppoit thod intodcd in ECCS to nsf.. Th psnt nlysis dscis th ltion twn th nding onts nd ll th fctos tht ffct it which th spn tio th dinsions tio of th lodd nd th tio twn th shot spn to th plll lngth of th lodd psntd y fctos nd spctivly. Th gt dvntg of th psnt nlysis is th dtintion of th nding onts y sing hnd clcltions only withot th s of cvs o tls. Th nicl pls donsttd th s of th otind qtions fo sls nd stdy nd coping th slts with igwd s thoy [6] s wll s th finit lnt thod s pogd in SA000 []. Finlly th psnt nlysis povids ltivly sipl two pssions fo which th nding onts fo sls nd stdy cn sily clcltd. Ths qtions cn of gt hlp fo dsign pposs. Th copison of th slts with th finit lnt thod confis th cccy of sch qtions. Rfncs []. Tioshno And Woinowsy S. 97 Thoy Of lts And Shlls Mcgw-Hill Boo Copny London []. Szild R. 00 Thois And Applictions Of lt Anlysis: Clssicl Nicl And Engining Mthods John Wily & Sons Inc Hoon Nw Jsy. [3]. Richd J.Y. And Shngn N.E. 005 Hndoo Of Stctl Engining CRC ss. []. Rynolds C.E. Stdn J.C. And Thlfll A. J. 00 Rynolds's Rinfocd Conct Dsign's Hndoo Tylo & Fncis Gop London And Nw Yo Elvnth Edition. [5]. ACI Coitt Bilding Cod Rqints Fo Stctl Conct ACI 3-95 And Conty ACI 3 R- 95 Aicn Cod Institt Fington Hills Michign. [6]. BS0 95 Stctl Us Of Conct Bitish Stndd Instittion. [7]. Hosing And Bilding Rsch Cnt 007 Egyptin Cod Fo Dsign And Constction Of Conct Stcts Cod No. 03 Ministil Dc No Ministy Of Hosing Utilitis And Un Conitis. []. Sp000 Advncd Copt Softw Fo Sttic And Dynics Finit Elnt Anlysis Of Stcts Copt & Stcts Inc. Bly Clifoni U.S.A. DOI: / g