8 TWO-WAY SLABS. Figure 1: Two way slabs

Transcription

1 1 8 TWO-WAY SLABS To-ay Sab: When the ratio (L/ S) i e than 2.0, it i caed to-ay ab, hon in Figure 1. Bending i take pace in the to direction in a dih-ike form. Accordingy, main reinforcement i required in the to direction. Figure 1: To ay ab Type A- Fat Pate: A fat pate foor i a to-ay ab ith no upporting beam, ony coumn. The ab ha mooth under urface ithout coumn capita or drop pane, hon in Figure 2.a. Thi type of to-ay ab i uited for ight oad aociated ith apartment contruction and can be economicay ued for pan up to 6.0 meter. Negative moment reinforcement i paced near the top of the pate coe to the coumn hie the poitive moment reinforcement i paced near the bottom urface in the midde of the pate in both direction.

2 Ue of fat pate i imited by their punching hear capacitie here arge thickne i required in cae of heavy oad or arge pan, thu rendering the ytem uneconomica. 2 Figure 2.a: Fat pate B- Fat Sab: A to-ay ab ith coumn capita or drop pane, or both, hon in Figure 2.b. Thi ytem i appropriate for heavier oad encountered in office or indutria buiding. The ytem i ued for pan up to 8.0 meter, due to the increae in ab thickne around the coumn. Figure 2.b: Fat Sab C- To-ay Edge-upported Soid Sab: In thi ytem, hon in Figure 2.c, beam provide moment interaction ith the coumn epeciay hen moment reiting frame are ued to reit atera oad. The ab overa thickne i greater becaue beam project donard, thu the ytem become infexibe

3 in term of mechanica ayout. The ytem can be ued economicay for pan up to 7.0 meter. 3 Figure 2.c:To-ay edge upported oid ab D- To-ay Ribbed Sab: 1. Waffe Sab: The affe ab i capabe of providing the arget pan of the conventiona concrete foor ytem and can be economicay ued for pan up to 12.0 meter. Thi i attributed to the coniderabe reduction in dead oad a compared to other foor ytem. Waffe ab contruction conit of orthogona et of rib ith oid part at the coumn, a hon in Figure 2.d. The rib are formed ith fiberga or meta dome form. The rib are uuay 0.60 to 0.90 meter on center. Shear i tranferred to the coumn by uing beam or hear head. Figure 2.d:Waffe ab

4 2. To-ay Edge-upported Ribbed Sab: Thi ytem, hon in Figure 2.e, can be economicay ued for pan up to 7.0 meter. It i imiar to the affe ab but the void beteen rib are fied ith hoo bock. Hidden or drop beam can be ued ith thi ytem depending on their pan. Figure 2.e:To-ay edge upported ribbed ab Deign Method In the ACI Code , to empirica method are given for anayi and deign of toay ab, regarde of their type. The firt of thee method i the Direct Deign Method hich ha to meet certain imiting condition. Thi method i be eaboratey deat ith ater on in thi chapter. The econd method i caed the Equivaent Frame Method, hich ha no retriction on it ue. Thi method i rather compex and i not be covered in thi book. Detaied coverage of the method can be found in the ACI Code For deign of edge upported ab, impified deign method i be ued. Thee deign method are baicay imiar in nature to the method provided in the ACI Code and oder verion. Thee method are baed on oad ditribution in to principa direction parae to the hort and ong ide of edge upported pane, depending on ong ide to hort ide ratio and upport condition at ab edge. Simpified Deign Method The oad carried to each upport and the bending moment in to-ay ab are not eay to cacuate if a the condition are to be atified. Therefore, empirica formuae and

5 5 approximate theorie have been formuated hich give bending moment in ab upported on a four edge. Bending moment can be obtained by one of the fooing method: Grahoff method: it i ued for oad ditribution hen the ab i impy upported on a four edge and the corner are not hed don. Marcu method: it i ued for oad ditribution hen the ab i impy upported on a four edge and the corner are hed don. Sab buit into thin a or monoithic ith the beam come into thi category. Egyptian Code method: it i ued for variou upport condition aociated ith continuou ab. Grahoff Method Conider a ab pane of ength L and idth S upported on a four edge and ubjected to uniformy ditributed oad, a hon in Figure 3.a. Thi method i baed on ditribution of the oad to to oad, one in the hort direction, here, and the other in the ong direction = + (1) Conider to trip of unit idth in the direction of L and S at the center of the ab. The defection of each trip at the center i be the ame. The centra defection of the trip in L direction, hon in Figure 3.b, i given a 5 L = (2) 38 EI The centra defection of the trip in S direction, hon in Figure 3.c, i given a 5 S = (3) 38 EI

6 6 (a) (b) (c) Figure 3: (a) Strip in the hort and ong direction; (b) defection in the ong direction; (c) defection in the hort direction Equating Eq. (2) and (3) give 5 L 38 EI 5 S = () 38 EI L or = (5) S Eq. (5) can be ritten a or + L = L + S (6) Since = +, L = (7) L + S Dividing the right hand ide of Eq. (7) by S and etting r = L/S,

7 7 r 1 + r = = α (8) The oad tranferred in the ong direction i evauated by a 1 = β r (9) 1 + = The oad coefficient α and β are given in Tabe 1. Tabe 1: Grahoff oad coefficient r α β The maximum bending moment occur aong horter pan and depth of the ab i deigned for thi bending moment. The reinforcement of uch ab i provided in to direction paced one above the other. The reinforcement in the horter direction i paced coet to the concrete urface o a to provide greater effective depth to reit arger moment aociated ith thi horter direction. Marcu Method In Grahoff method, effect of retraint againt ifting of the corner and effect of torion in the ab ere negected. Marcu ha given an approximate method for determining bending moment in ab, impy upported on four edge ith corner prevented from being ifted and conidering torion in the ab. The bending moment at midpan of uch a ab are the midpan moment a cacuated by Grahoff mutipied by a modification factor hich i e than unity a the poitive bending moment in the ab are reduced due to corner retraint and torion. Thi modification factor i given by r r Load ditribution coefficient according to Marcu are given in Tabe 2.. Tabe 2: Marcu oad coefficient r α β

8 8 Egyptian Code Method Grahoff and Marcu method have baicay deat ith anayi of impy upported ab pane. The Egyptian Code method ha extended thee method for anayi of continuou ab here individua pane have different upporting condition. To account for the variou upporting condition, the rectanguarity ratio r =L/S i modified by mutipying the nominator and the dominator of thi expreion by the ratio of ditance beteen infection point to the ength beteen upport for onger and horter pan, repectivey, or r L m S m m m 1 = = r. 2 For impy upported pan, m = 1.0, for pan continuou from one ide, for pan continuou from both ide, m = m = 0.87, and The oad ditribution factor for edge upported to ay ab according to the Egyptian Code of Practice are given in Tabe 3. Tabe 3: Egyptian Code oad coefficient r α β Beam for To-ay Sab Deigned by Approximate Method The oad upported by each beam i conidered to be the entire oad on tributary area bounded by 5-degree ine dran from the corner of the Figure.a and Figure.b. Trianguar or trapezoida ditribution are poibe. If avaiabe anayi oftare can not cope ith uch oad ditribution, a hon in Figure 9..c it i poibe to repace thee oad ditribution by equivaent uniform oad that give ame maximum hear and moment a the origina oad do. For trianguar oad ditribution, the equivaent hear force coefficient and the equivaent bending moment coefficient ditribution, C and C b are given in Tabe. C i equa to 0.50, C b i equa to 2/3. For trapezoida oad

9 9 (a) (b) Beam B3-B (c) Beam B1-B2 Figure 9.: (a) Tributary area for ab oad on beam; (b) tributary area for oad on continuou beam; (c) oad on beam B3-B and B1-B2

10 10 In Tabe, Tabe : Shear and moment equivaent oad coefficient for trapezoida oad ditribution r C C b C = 1, and 2 r C b 1 = r The tota oad on a given beam conit of it on eight, eight of partition, if any, and the oad reuting from the ab, a expained earier.