Theoretical Evaluation on Effects of Opening on Ultimate Load-carrying Capacity of Square Slabs

Transcription

1 Eectrnic Jurna f Structura Engineering (8 008 Theretica Evauatin n Effects f Opening n Utimate ad-carrying Capacity f Square Sabs Chee Khn Ng Universiti Maaysia Sarawak, Maaysia Timthy Juius Edward Universiti Maaysia Sarawak, Maaysia Danie Kim Tee ee Universiti Maaysia Sarawak, Maaysia, Emai: ckii96@gmai.cm ABSTRACT: Reinfrced cncrete sabs are the mst cmmn structura eements fund in buiding cnstructin. Despite the arge number f sabs designed and buit, the detais f the eastic and pastic behaviur f sabs are nt aways appreciated r taken int accunt. Studies n sabs with penings are as scarce in the research n reinfrced cncrete eements. Therefre, a study n simpy-supprted and fixed-end, square sabs with pening at utimate imit state using the yied ine methd was carried ut and the resuts are presented herein. Fr simpy-supprted sabs, the anaytica study n the utimate ad capacity f the sab shws that the utimate tta ad decreases with the size f the pening. Hwever, when the utimate tta ad is cnverted t utimate area ad, the resuts shw therwise. In the study f fixed-end sabs, the resuts shw that the pening has insignificant effect n the utimate area ad capacity fr a sma pening size f up t 0.3 times the sab dimensin. Fr pening size f mre than 0.5 times the sab dimensin, the utimate area ad capacity increases drasticay. The utimate tta ad f a fixed-end sab with pening up t 0.3 times the sab dimensin is as nt affected by the pening. Hwever, the utimate tta ad increases drasticay fr pening size f 0.5 times r mre f the sab dimensin. A the resuts in this study are ptted in nmgraphs frm and can be used as guideines fr the design f sabs with pening particuary in determining the utimate ad capacity. KEYWORDS: Utimate ad, sab, sab pening, capacity 1 INTRODUCTION Reinfrced cncrete sabs are the mst cmmn eements amngst structura eements in the cnstructin f buidings. In these buidings, many pipes and ducts are necessary t accmmdate essentia services such as eectricity, teephne, cmputer netwrk, water suppy, sewerage and aircnditining. Due t the need fr instaing these pipes and ducts, sabs f buidings may need t prvide penings fr them t be intercnnected. Mansur & Tan (1999 had prpsed anaysis and design prcedure fr beams with circuar and rectanguar penings. The anaytica mde prpsed is abe t hande cmbined bending, shear and trsin in beams with penings, and subsequenty design the reinfrcements required fr this cmbined actin. Hwever, the prpsed anaysis and design prcedure are nt appicabe t reinfrced cncrete sabs. Park & Gambe (000 cnducted a review n anaysis f reinfrced cncrete sabs with penings and reprted that an pening in a simpy-supprted square sab with dimensin f 0. t 0.3 times f the sab dimensin wud cause a reductin f 11% in terms f utimate ad per unit area. arger pening with dimensin f 0.5 r mre times the sab dimensin wud nt resut in reductin f utimate ad per unit area. E-Saakawy et a. (1999 tested six fu-scae reinfrced cncrete sabs, f which five were sabs with varius arrangements f penings in the vicinity f the cumn. The penings in the prttypes were square with the sides parae t the sides f the cumn; and there were tw pening sizes, ne which is the same size as the cumn and the ther is 60% f the cumn size. It was reprted that the arger and smaer pening sizes ed t reductin in utimate strengths f 30% and 1% respectivey. Anther fu-scae testing n reinfrced cncrete sabs 1

2 Eectrnic Jurna f Structura Engineering (8 008 with penings was carried ut by Teng et a. (004. In this study, 0 sabs with penings supprted n cumns f varius sizes were tested. This study distinguishes itsef frm the study carried ut by E- Saakawy et a. (1999 by the arrangements f cumns and penings. The sabs tested by Teng et a. (004 had cumn supprt in the midde f the sabs whereas sabs tested by E-Saakawy et a. (1999 were having cumn supprt in the midde f the nger edges f the sab. It was reprted by Teng et a. (004 that penings reduce punching shear strength f sabs cnsideraby, and the recmmended catins fr penings in sabs are ang the nger side f a cumn. In a separate study carried ut by Tan & Zha (004, six prttype ne-way reinfrced cncrete sabs with penings were tested. Five sabs were strengthened with externay bnded carbn fibrereinfrced pymer (CFRP systems and the resuts were cmpared t the sab with pening withut strengthening. It was fund that the CFRP systems are effective in enhancing the ad-carrying capacity and stiffness f the reinfrced cncrete sabs with penings. Frm the studies as discussed abve, pening in a sab reduces the ad-carrying capacity. Therefre, a theretica study n square sabs with pening was carried ut t investigate the effect f penings n the utimate ad-carrying capacity. The anaytica cnsideratins and resuts are shwn in the impending discussin. YIED INE ANAYSIS.1 Simpy-supprted sab Figure 1(a shws a simpy-supprted square sab with an pening at the centre. The sab is assumed t have simiar amunt f reinfrcement in bth the ngitudina and transverse directins. A δ 1 B (b Defrmed shape acrss Sectin A-A VOID Pastic hinge δ 1 A Yied ine B (a Yied ines (c Defrmed shape acrss Sectin B-B Figure 1. Yied ines and defrmatins f a simpy-supprted square sab with an pening at the centre. i At utimate imit state, each yied ine was assumed t stretch frm the crner f the pening t the crner f the sab. When a virtua dispacement defined by a unit vaue under unifrm area ad f w is appied t the crner f the pening as shwn in Figure 1(b, the externa wrk dne by the ad is W e ( 1 w 4 1 ( 1 + w ( 3 ( w ( + 3 ( ( + w 3 (1 in which dimensin f the square sab; and dimensin f the pening. 13

3 Eectrnic Jurna f Structura Engineering (8 008 Defining mment m as the mment f resistance per unit ength ang the yied ines, the tta interna wrk dne by the mment f resistance ang the yied ines is W 4 m ( i i where ength f the yied ines; and i rtatin ang the yied ines given as [see Figure 1(c] 1 1 i + (3 Substituting Equatin (3 int Equatin ( and equating the externa wrk dne in Equatin (1 t the interna wrk dne in Equatin (, 8m w ( ( + 3 4m ( ( + (4 Fr a sab which has simiar amunt f reinfrcement prvided in bth x-axis (ngitudina and y-axis (transverse directin, the mment f resistance per unit ength ang the x-axis m x is the same as the mment f resistance per unit ength ang the y-axis m y. This type f sab is knwn as istrpicay reinfrced sab (Nisn 1997 in which the unit mment m ang the yied ines is the same as m x r m y. Defining w u as the utimate area ad fr a sab withut an pening, that is, 0, Equatin (4 gives 4m w u (5 Hence, the rati f utimate area ad fr a sab with an pening t the utimate area ad fr a sab withut an pening can be btained by dividing Equatin (4 by Equatin (5, that is, w w (6 ( u ( + Fr the study f tta ad, P is defined as the tta utimate ad n the sab with an pening given as 4m( P w ( ( ( + 4m( + ( + (7 Simiary, fr the tta utimate ad n the sab withut an pening P u, it is given by Pu wu 4m (8 Hence, the rati f tta utimate ad fr a sab with an pening t the tta utimate ad fr a sab withut an pening can be btained by dividing Equatin (7 by Equatin (8, that is, P P + (9 u +. Fixed-end sab Figure shws a fixed-end sab with an pening at the centre. The externa wrk dne by a unifrm area ad f w is the same as the ne in simpysupprted sab given in Equatin (1. Defining n as the rati f the mment f resistance per unit ength at the fixed end t the mment f resistance per unit ength ang the yied ine extending frm the crner f the sab t the crner f the pening, the tta interna wrk dne by the mment f resistance ang the yied ines is (see Figure W 4 m + nm (10 i i 4 in which and i are the same as in Equatin (3; and rtatin at the fixed-end given as [see Figure (b] 1 (11 Substituting Equatin (3 and Equatin (11 int Equatin (10 gives W i 8m + 8nm (1 + n 8m 14

4 Eectrnic Jurna f Structura Engineering (8 008 Pastic hinges C δ 1 D (b Defrmed shape acrss Sectin C-C VOID Pastic hinges δ 1 C Yied ine D (a Yied ines (c Defrmed shape acrss Sectin D-D Figure. Yied ines and defrmatins f a fixed-end square sab with an pening at the centre. i Equating the externa wrk dne, W e, in Equatin (1 and the interna wrk dne, W i, in Equatin (1, and sve fr w, 4m( + n w (13 ( ( + Fr a fixed-end sab withut pening, that is, 0, the utimate area ad is given as w u w w 4m( + n 3 4m(1 + n Hence, the rati f w t w u is given as (14 ( + n (15 ( ( + (1 n u + It is nted that Equatin (15 reduces t Equatin (6 fr the case f simpy-supprted sab, that is, when n 0. In rder t btain the rati f P t P u, simiar prcedure fr the simpy-supprted sab can be used. Hwever, it can as be cnvenienty assessed using Equatin (15, which gives P w ( ( + n( ( + P w ( ( + (1 + n u u ( + ( + n ( ( + (1 + n (16 It is nted that Equatin (16 reduces t Equatin (9 fr n 0, that is, a simpy-supprted sab. 3 RESUTS AND DISCUSSION 3.1 Simpy-supprted sab Figure 3 shws the resuts f the effect f size f pening n the utimate area ad capacity n the sabs. It is ptted using Equatin (6 as derived in the preceding sectin. There are three sab sizes cnsidered in the anaysis, that is, 3 m, 4 m, and 5 m. It is hwever shwn in Figure 3 that the sab size has n effect n the resuts btained in which a three curves cincide with each ther. Therefre, there is n size effect when the parameters are nrmaised and hence this effect is nt discussed further in the impending sectins. 15

5 Eectrnic Jurna f Structura Engineering ( Nrmaised utimate area ad, w /w u m m 3 m Rati f pening dimensin t sab dimensin, / Figure 3. Effect f size f pening n utimate area ad f simpy-supprted square sabs. It is indicated in Figure 3 that the utimate area ad capacity decreases with the size f pening fr penings in between 0 t 0.5 times the sab dimensin. The maximum decrease is at penings f 0. and 0.3 times the sab dimensin, which is 11%, in agreement with the findings presented in the review by Park & Gambe (000. When the pening size is increased beynd 0.5 times the sab dimensin, there is an expnentia increase in utimate area ad capacity in the sabs, with the maximum increase f 57% at an pening equa t 0.9 times the sab dimensin. The decrease in utimate area ad capacity with the increase in pening size up t 0.5 times the sab dimensin is due t the fact that the ad is sti distributed near t the centre f the sab. Under this circumstance, the externa wrk dne by the ad is sti quite significant with ne unit defectin impsed upn the sab at the edges f the pening. Therefre, a smaer area ad is required t d the wrk in rder t baance ut the interna wrk dne by the resisting mment ang the yied ines. As the pening size increases beynd 0.5 times the sab dimensin, the ad is cncentrated twards the supprting edges f the sab. Under this circumstance, the externa wrk dne by the ad is quite minima under a unit defectin impsed at the edges f the pening, resuting in mre area ad required t d the wrk in rder t achieve equiibrium with the interna wrk dne by the resisting mment ang the yied ines. Figure 4 shws the resuts f the effect f pening in simpy-supprted sab n the utimate tta ad ptted using Equatin (9. Simiar t Figure 3, there is n effect f sab size as a three curves cincide with each ther. Therefre, this effect is nt discussed further. 16

6 Eectrnic Jurna f Structura Engineering ( Nrmaised utimate tta ad, P /P u m 4 m m Rati f pening dimensin t sab dimensin, / Figure 4. Effect f size f pening n utimate tta ad f simpy-supprted square sabs. It can be seen in Figure 4 that the utimate tta ad capacity f the sab is decreasing with the increase in pening size. At an pening f 0.5 times the sab dimensin, the reductin in utimate tta ad is 5%. There is n increase in ad carrying capacity as seen in Figure 3 fr arger pening size, but instead, there is a cntinuus reductin in utimate tta ad fr arger pening with up t 3% reductin fr an pening f 0.9 times the sab dimensin. It is nted that the pening f a sab in practice is usuay 0.3 times the sab dimensin r ess. Therefre, mst sabs with pening wi be subjected t reductin in utimate area ad f up t 11% and a reductin in utimate tta ad f 19%. 3. Fixed-end sab Figure 5 shws the effect f pening size n the utimate area ad f fixed-end square sab. Each rati f mment f resistance at fixed-end yied ine t mment f resistance at yied ine at crner f sab t crner f pening, n, is ptted separatey. In Figure 5, it can be seen that when the pening size is ess that 0.3 times the sab dimensin fr the case f n vaue f 0.5, there is a decrease in utimate area ad capacity with the maximum decrease f 4% ccurring at the rati f pening size t sab dimensin, /, f 0.1. Fr sabs with n vaue f 1, the maximum decrease in utimate area ad capacity f % as ccurs at / f 0.1, and there is n decrease in utimate area ad capacity when the / vaue is mre than 0.. Fr sabs with higher n vaues, there is a decrease in the utimate area ad capacity fr / f ess than 0., where the maximum decrease is abut 1% at / f 0.1. Beynd the demarcating pints where the utimate area ad capacity decreases, that is, / f 0.3 fr n vaue r 0.5 and / f 0. fr higher n vaues, the utimate area ad capacity increases expnentiay with the increase in rati f pening size t sab dimensin, /, particuary fr / f mre than 0.5. This is due t the fact that fr arger rati f pening size t sab dimensin, the ad is mre cncentrated at the area f the sab in which the defectin is smaer, therefre there is an increase in area ad t create enugh externa wrk t baance the interna wrk dne by the yied ines extending frm the crner f sab t crner f the pening as we as thse yied ines at the fixed ends. There is a drastic increase in nrmaised utimate area ad, w /w u, at / f 0.8 t 0.9, with the maximum recrded w /w u f 5, fr n f and / f 0.9, shwing an increase f 400%. This arge pening eaves a sma area near the fixed ends fr ading, eading t a very high stiffness f the sab. As a resut, the ad-carrying capacity f the sab becmes very high, even mre s fr higher n vaue. 17

7 Eectrnic Jurna f Structura Engineering (8 008 Nrmaised utimate area ad, w /w u 30 n 0.5 n 1 5 n 1.5 n Rati f pening dimensin t sab dimensin, / Figure 5. Effect f size f pening n utimate area ad f fixed-end square sabs. Figure 6 shws the effect f pening size n the utimate tta ad f fixed-end square sab with different rati f mment f resistance at fixed-end yied ine t mment f resistance at yied ine at crner f sab t crner f pening, n. Fr the sab with n vaue f 0.5, the nrmaised utimate tta ad, P /P u, decreases fr pening size f 0.5 times f sab dimensin r ess, with the maximum decrease f 7% at / f 0. and 0.3. As fr the sab with n vaue f 1, P /P u decreases fr / f 0. r ess, with maximum decrease f 4% at / f 0.1. In the sabs f higher n vaues, there is n significant decrease in nrmaised utimate tta ad fr pening size f 0.3 times f sab dimensin r ess. Nrmaised utimate tta ad, P /P u n 0.5 n 1 n 1.5 n Rati f pening dimensin t sab dimensin, / Figure 6. Effect f size f pening n utimate tta ad f fixed-end square sabs. Except fr the sab with n vaue f 0.5, a the ther sabs with higher n vaue shw increase in utimate tta ad ver the sab withut an pening fr / f mre than 0.4, that is, the P /P u vaues are mre than 1. The P /P u vaues increase drasticay fr / f mre than 0.6, with the maximum recrded P /P u vaue f 4.75 fr n f and / f 0.9, shwing an increase f 375%. This is then again due t the arge pening resuting in ad cncentratin 18

8 Eectrnic Jurna f Structura Engineering (8 008 near the supprt, which eads t very high stiffness f the sab. On the nte that mst penings in sabs are generay sma, that is, / is nrmay 0.3 r ess, the resuts in Figures 5 and 6 may be summurised as that pening in a fixed-end square sab reduces the utimate ad capacity. The reductin in utimate area ad capacity is up t 4% and the reductin in utimate tta ad capacity is up t 7%. 4 CONCUSIONS A study n the effect f pening n the ad carrying capacity f simpy-supprted and fixed-end sabs was presented and discussed. In simpy-supprted sabs, it was fund that the utimate area ad capacity decreases with the increase in size f pening up t 0.5 times the sab dimensin. After which the utimate area ad capacity increases with the pening size and registered a maximum increase f 57% at an pening size f 0.9 times the sab dimensin. Hwever, when the utimate area ad capacity is cnverted t utimate tta ad capacity, there is a cntinuus decrease in utimate tta ad capacity with the increase in pening size. The maximum decrease is 3% with an pening size f 0.9 times the sab dimensin in this study. The resuts f the fixed-end sabs shw that the pening reduces the utimate area ad capacity fr a sma pening f 0.1 times f the sab dimensin, with the maximum reductin f 4% in the sab with the rati f mment f resistance at fixed-end yied ine t mment f resistance at yied ine at crner f sab t crner f pening f 0.5. When the pening size is mre than 0.5 times the sab dimensin, the utimate area ad capacity increases expnentiay with the maximum recrded increase f 400% at an pening size f 0.9 times the sab dimensin fr a sab where the rati f mment f resistance at fixed-end yied ine t mment f resistance at yied ine at crner f sab t crner f pening f. The utimate tta ad f a fixed-end sab decreases with sma pening size f up t 0.3 times the sab dimensin. The maximum reductin is 7% at pening sizes f 0. and 0.3 times the sab dimensin in the sab with the rati f mment f resistance at fixedend yied ine t mment f resistance at yied ine at crner f sab t crner f pening f 0.5. Hwever, the utimate tta ad increases drasticay fr pening size f 0.5 times r mre f the sab dimensin, with the maximum recrded increase f 375% at an pening size f 0.9 times the sab dimensin fr a sab where the rati f mment f resistance at fixed-end yied ine t mment f resistance at yied ine at crner f sab t crner f pening f. Mst sabs with pening have sma pening size f up t 0.3 times the sab dimensin. Therefre, a simpy-supprted sab may have a reductin in utimate area ad f up t 11% and a reductin f utimate tta ad f up t 19%. A fixed-end sab has ess significant reductin in bth utimate area ad and utimate tta ad capacities f 4% and 7%, respectivey, fr a sma pening f up t 0.3 times the sab dimensin. The charts were presented in nrmaised ad capacity and pening size. Therefre, they can be used as guideines fr predicting the ad capacity f simpy-supprted and fixed-end sabs with penings. ACKNOWEDGEMENT The authrs wud ike t express their heartiest appreciatin t Universiti Maaysia Sarawak fr the supprt in this research prject with Fundamenta Research Grant N. 0(70551/05(50. REFERENCES BSI, BS 8110: Structura Use f Cncrete Part 1. Cde f Practice fr Design and Cnstructin, ndn: British Standards Institutin, E-Saakawy, E. F., Pak, M. A., and Siman, M. H., Reinfrced Cncrete Sab-Cumn Edge Cnnectins with Openings, ACI Structura Jurna, V. 96, N. 1, January- February 1999, pp Mansur, M. A., and Tan, K. H., Cncrete Beams with Openings: Anaysis and Design, Bca Ratn: CRC Press, Nisn, A. H., Design f Cncrete Structures, 1 th editin, New Yrk: McGraw-Hi, Park, R., and Gambe, W.., Reinfrced Cncrete Sabs, nd editin, New Yrk: Jhn Wiey & Sns, 000. Tan, K. H., and Zha, H., Strengthening f Openings in Oneway Reinfrced-cncrete Sabs Using Carbn Fiberreinfrced Pymer Systems, Jurna f Cmpsite fr Cnstructin - ASCE, V. 8, N. 5, Octber 004, pp Teng, S., Cheng, H. K., Kuang K.., and Geng, J. Z., Punching Shear Strength f Sabs with Openings and Supprted n Rectanguar Cumns, ACI Structura Jurna, V. 101, N. 5, September-Octber 004, pp