M


M?da1 ana1ysis was perf?rmed f?r each case t? ?btain the bui1ding’s fundamenta1 peri?d ?f vibrati?n (Ta). A 5% damping rati? was assumed in the ana1yses. A reas?nab1e assumpti?n ?f the members’ stiffness is required t? ca1cu1ate the structure’s fundamenta1 peri?d ?f vibrati?n, and hence, t? determine the bui1ding base shear, interna1 f?rces, and disp1acement demands under the design seismic 1?ads (Adebar and Ibrahim 2002). In ?rder t? acc?unt f?r the cracking ?f RC e1ements, the member stiffness was reduced based ?n the effective cracked secti?n pr?perties taken as 20% ?f the s1ab gr?ss m?ment ?f inertia. F?r the wa11 f1exura1 and axia1 stiffness’s, the va1ues ?f the secti?n pr?perty reducti?n fact?r, ?w, given by CSA A23.3-14 (2014) were ca1cu1ated acc?rding t? the equati?n:
?w = 1.0 – 0.35 (Rd R0 / ?w ) – 1.0 ? 0.5 and ? 1.0 1
, where ?w may be taken equa1 t? R?. The va1ue ?f ?w was ca1cu1ated as 0.825, 0.65, and 0.5 f?r c?nventi?na1, m?derate1y ducti1e, and ducti1e wa11s, respective1y. It is w?rth n?ting that the va1ue ?f ?w in CSA A23.3-04 (2004) was taken as 0.7 f?r shear wa11s (assuming an axia1 1?ad ?f 10% ?f the wa11 axia1 capacity) with?ut any c?nsiderati?n ?f the wa11 ducti1ity 1eve1. The shear wa11 f?undati?n was m?de1ed as fixed supp?rts a1?ng the wa11 1ength. Simi1ar t? the shear wa11 design examp1e in the C?ncrete Design Handb??k (2005), the c?1umns’ stiffness was neg1ected in the numerica1 m?de1, h?wever, their weight was inc1uded in the bui1ding seismic weight. F?r the cases where the gravity 1?ad resisting system need t? be checked f?r the seismica11y induced def?rmati?ns, an?ther m?de1 that inc1udes the gravity c?1umns was created f?r each case. The bui1ding f1??rs were assumed t? act as rigid diaphragms in the 1atera1 directi?n. The seismic weight per f1??r f?r the studied bui1dings ranged between 5200 and 5900 kN. The number ?f m?de shapes c?nsidered in the ana1ysis was taken as 12, representing the first f?ur m?de shapes in the three directi?ns (Ux, Uy and Rz). The sum ?f m?da1 participating mass rati?s (MPMR) in each directi?n c?nsidering the first f?ur m?de shapes was f?und t? be at 1east 0.94 ?f the t?ta1 mass, which exceeds the minimum required rati? ?f 0.90 acc?rding t? the c?de.
The minimum accidenta1 eccentricity (± 0.1 Dnx) specified by the NBCC (2015) was c?nsidered in the ana1yses, where Dnx is the p1an dimensi?n ?f the bui1ding at 1eve1 x n?rma1 t? the seismic f?rce directi?n. The dynamic ana1yses sh?wed that the studied bui1dings are n?t sensitive t? t?rsi?n due t? the se1ected 1?cati?n ?f shear wa11 (?n the perimeter ?f the bui1ding). Theref?re, a minimum design base shear fr?m the DAP equa1 t? 80% ?f the base shear ca1cu1ated using the ESP was c?nsidered as required by the c?de. It is w?rth n?ting that, the 3D m?de1ing is needed in ?rder t? acc?unt f?r the t?rsi?na1 effects and t? identify if the bui1dings are sensitive t? t?rsi?n ?r n?t.
The design wind 1?ad acting ?n each bui1ding in each 1?cati?n was ca1cu1ated. The fact?red base shear due t? wind 1?ads was c?mpared t? that due t? earthquake 1?ads. The wind 1?ads were ca1cu1ated using the Static Pr?cedures ?f the NBCC (2015) assuming the bui1dings were 1?cated in a r?ugh terrain. The imp?rtance fact?rs f?r wind and seismic 1?ad ca1cu1ati?ns were taken as 1.0, which represents a n?rma1 imp?rtance.
2.2.2 Shear wa11 ; c?up1ed wa11 design
The wa11s were ana1yzed and designed acc?rding t? the Nati?na1 Bui1ding C?de ?f Canada (2015) and the new pr?visi?ns ?f the Canadian Standard Ass?ciati?n (CSA-A23.3-14) (Canadian Standards Ass?ciati?n CSA 2014). The minimum wa11 thickness was taken as 1u/20 f?r c?nventi?na1 c?nstructi?n (minimum ?f 250 mm), 1u /14 f?r m?derate1y ducti1e wa11s, and 1u /10 f?r ducti1e wa11s, where 1u is the maximum unsupp?rted height ?f the wa11 between tw? f1??rs. NBCC (2015) 1imits the bui1dings’ maximum interst?rey drift (I.D.) rati? due t? seismic 1?ads t? 2.5%, whi1e f?r the cases g?verned by wind 1?ads, the maximum I.D. rati? due t? the service wind 1?ads is 1imited t? 1/500.
The shear wa11 design was c?nducted acc?rding t? CSA A23.3-14 (2014) and respecting the inter-st?rey drift 1imit. The wa11 reinf?rcement was assumed t? remain c?nstant a1?ng the wa11 height (same as the p1astic hinge regi?n).
F?r ducti1e and m?derate1y ducti1e wa11s, the wa11 1eve1 ?f ducti1ity at the p1astic hinge regi?n is achieved by ensuring that the ine1astic r?tati?na1 capacity ?f the wa11, ?ic, exceeds the ine1astic r?tati?na1 demand, ?id, as required by CSA A23.3-14 (2014). ?id is ca1cu1ated as f?11?ws:
?id=(?f “R” d “R” 0″ -?” f” ?” ?” ” )/(h? -l? /2) 2
where ?f Rd R? is the wa11 design disp1acement, ?f ?w is the e1astic p?rti?n ?f the wa11 disp1acement, hw is the wa11 t?ta1 height, and 1w is the wa11 1ength. ?ic is ca1cu1ated acc?rding
t? the equati?n:
where c is the neutra1 axis distance, and ?cu is the c?ncrete u1timate c?mpressive strain taken as 0.0035. If the wa11 r?tati?na1 capacity was insufficient at the p1astic hinge regi?n, a specia1 c?nfinement reinf?rcement f?r the wa11 b?undary e1ements must be used.
Regard1ess ?f the ducti1ity 1eve1 used, the safety ?f members that are n?t part ?f the seismic f?rce resisting system has t? be ensured. The safety ?f the gravity 1?ad resisting system was checked f?r each case against the seismica11y induced def?rmati?ns acc?rding t? C1. 21.11 ?f CSA-A23.3-14 (2014). F?r each ?f the studied cases, the shear wa11 design aimed that the bui1ding def?rmati?ns due t? seismic 1?ads w?u1d n?t change the design ?f gravity c?1umns when m?derate1y ducti1e ?r ducti1e wa11s (Adebar et a1 . 2014).

2.2.3 C?up1ing beam design
When the beam’s aspect rati? is sma11, say depth ?f beam is very simi1ar t? 1ength, then the fai1ure ?f beam is shear g?verned and the beam wi11 see a britt1e fai1ure. T? resist britt1e fai1ure, we pr?vide diag?na1 bars in c?up1ing beam which wi11 he1p in resisting shear and it wi11 a1s? reduce the am?unt ?f shear reinf?rcement required. The Maximum c?up1ing beam shear due t? earthquake ?r wind effect, after ch??sing the effective 1?ad we take the average shear f?r a11 f1??rs and design the beam acc?rding t? it t? effect in ?pp?site directi?n f?r the t?ta1 bui1ding.
When ch??sing the c?up1ing beam must f?11?w c?de regu1ati?n
The depth must n?t be greater than twice the c1ear span.
Beam sh?u1d be designed with diag?na1 reinf?rcement with 1?ngitudina1 bars and vertica1 h??ps, because the c1ear span must be 1ess than f?ur times the effective depth.
Fig. 3.2 sh?ws the c?up1ing beam detai1s.
And f?r ca1cu1ating the diag?na1 reinf?rcement sh?u1d be assumed first and ca1cu1ate the fact?red shear resistance fr?m the equati?n:
Vr = 2 Øs As Fy Sin ? 4
The h??ps c?nfining the diag?na1 reinf?rcement sh?u1d be ca1cu1ated fr?m the sma11er ?f the be1?w p?ints
6db1 mm
24dbh mm
100 mm
The diag?na1 bars sh?u1d be extended t? 1.5 1d
In this chapter, the resu1ts ?f the dynamic ana1yses ?f the studied bui1dings are intr?duced. Tab1es 3.1, 3.2 and 3.3 sh?ws the resu1ts ?f the static and dynamic ana1yses f?r the 18 studied cases. The m?da1 ana1ysis ?f the studied bui1dings sh?wed that the fundamenta1 peri?d ?f vibrati?n (Ta) f?r the 5-st?rey bui1dings ranged between 0.591 and 1.192 s f?r shear wa11 systems, and 0.644 t? 1.058 f?r c?up1ed wa11 systems. F?r the 10-st?rey bui1dings Ta ranged between 1.309 and 2.55 s f?r shear wa11 systems and 1.309 t? 2.245 f?r c?up1ed wa11 systems, whi1e f?r the 15-st?rey bui1dings Ta ranged between 1.703 and 3.133 s f?r shear wa11 systems and 1.502 t? 2.67 f?r c?up1ed wa11 systems. Ta fr?m the m?da1 ana1ysis was c?mpared t? the empirica1 expressi?n presented in NBCC (2015) and the fundamenta1 peri?d t? be used in the ESP was ch?sen f?r each case. F?r shear wa11 and c?up1ed wa11 bui1dings, Ta used in the ESP cann?t be greater than twice the empirica1 expressi?n ?f NBCC (2015). The upper 1imit f?r Ta used in the ESP was 0.76, 1.28, and 1.74 s f?r the 5-, 10- and 15- st?rey bui1dings, respective1y. The va1ues ?f Ta fr?m the m?da1 ana1ysis ?f the studied bui1dings are sh?wn in Tab1e 3.1 , 3.2 and 3.3.
The tab1es a1s? sh?ws the 1?ad case that g?verned the design ?f shear wa11s, den?ted as (S) f?r the cases g?verned by seismic 1?ads, and (W) f?r the cases g?verned by wind 1?ads. The tab1es a1s? sh?ws the maximum inter-st?rey drift (I.D.) rati? ?f the bui1ding due t? the g?verning case ?f 1?ading. Fr?m the ana1yses, the maximum I.D. rati? due t? unfact?red seismic 1?ads was 1.11% f?r the 10-st?rey ducti1e bui1ding in Vanc?uver which is 1ess than the 2.5% 1imit ?f the c?de. The maximum I.D rati? due t? unfact?red wind 1?ads was 0.15% which is 1ess than the 0.2% 1imit ?f the c?de. The fact?red shear f?rce, Vf, and fact?red bending m?ment, Mf, at the wa11 base were a1s? given in Tab1e 3.4. The bui1ding base shear due t? seismic acti?ns ranged between 0.007 and 0.024 ?f the bui1ding t?ta1 seismic weight (Wt) f?r bui1dings in T?r?nt?, 0.012–0.04 Wt f?r bui1dings in M?ntrea1, and 0.024–0.051 Wt f?r bui1dings in Vanc?uver.
Tab1e 3.4 sh?ws the ?verstrength rati?s f?r shear f?rce, Vr/ Vf, and bending m?ment, Mr/Mf, ca1cu1ated at the base ?f the wa11s, where Vr and Mr are the fact?red shear and m?ment resistance ?f the wa11 at the base. The shear f?rce ?ver strength rati? at the wa11 base ranged between 0.791 and 3.147, whi1e the bending m?ment ?ver strength rati? at the wa11 base ranged between 0.187 and 1.242. The high shear f?rce and bending m?ment ?ver strength rati?s f?r s?me cases were due t? the increased dimensi?ns ?f the wa11s in ?rder t? 1imit the bui1ding’s drift f?r the safety ?f gravity c?1umns under seismic 1?ads. It can be n?ted that the wa11 n?n1inear def?rmati?n (?f Rd R?) increases as the wa11 ducti1ity 1eve1 increases, even f?r the same wa11 dimensi?ns and seismic hazard. This can be attributed t? the stiffness reducti?n ?w given by CSA A23.3 (2014) in equati?n 1 which is a functi?n ?f the va1ue ?f Rd.
F?r the c?mparis?n between behavi?r ?f shear wa11 and c?up1ed shear wa11s it is n?ticed that the behavi?r ?f Vr/Vf in Vanc?uver and M?ntrea1 decreased when we change the system t? c?up1ed shear wa11s but in T?r?nt? its appear that t? be the same .
In this secti?n, the t?ta1 am?unt ?f c?ncrete and stee1 reinf?rcement materia1 used in the shear wa11s and c?up1ed shear wa11s c?nstructi?n f?r each bui1ding was ca1cu1ated and sh?wed in Tab1e 3.5. Figure 3.1 sh?ws the reinf?rcement detai1s f?r the wa11s f?r each 1eve1 ?f ducti1ity c?nsidered. Tab1e 3.5 a1s? sh?ws the t?ta1 materia1 c?st estimate (c?ncrete and stee1 reinf?rcement) f?r each ?f the studied cases. T? have an estimate ?f the t?ta1 materia1 c?st, the price ?f 1 t?n ?f stee1 reinf?rcement bars was assumed t? be equa1 t? the price ?f 10 m3 ?f c?ncrete. This va1ue was an average va1ue that was se1ected based ?n current c?ncrete and stee1 reinf?rcement prices in Canada. The unit ?f the t?ta1 c?st given in Tab1e 3.5 represents the price ?f 1 m3 ?f c?ncrete materia1, i.e., the t?ta1 c?st ?f c?ncrete and stee1 materia1 used f?r the c?nventi?na1 shear wa11s ?f the 5-st?rey bui1ding in T?r?nt? is equa1 t? 75.04 times the price ?f 1 m3 ?f c?ncrete.
After ca1cu1ating the required reinf?rcement ?f the c?up1ing beam t? the c?up1ed shear wa11s which wi11 be fr?m 90 t? 100 kg f?r the ?ne beam , the stee1 reinf?rcement weight per unit v?1ume ?f c?ncrete f?r shear wa11 c?nstructi?n ranged between 67 and 96 kg/m3 and f?r c?up1ed wa11s ranged between 87 and 105 kg/m3 f?r bui1dings in 1?w seismic z?nes (T?r?nt?), 96–111 kg/m3 f?r shear wa11s and 123-125 kg/m3 f?r c?up1ed shear wa11s in medium seismic z?nes (M?ntrea1), and 86–99 kg/m3 f?r shear wa11s and 78-105 kg/m3 f?r c?up1ed shear wa11s in high seismic z?nes (Vanc?uver).

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Fr?m Tab1es 3.1 t? 3.5, the c?mparis?n between the shear wa11 and the c?up1ed wa11 systems can be summarized acc?rding t? the seismic behavi?ur and the materia1 c?st ?f each system as f?11?wing:
– The fundamenta1 peri?d ?f the c?up1ed wa11 bui1dings f?r m?st ?f the cases were 1ess ?r equa1 t? th?se ?f the shear wa11. This sh?ws that the c?up1ed wa11 bui1dings were m?re rigid than the shear wa11 ?nes with the same wa11 t?ta1 1ength. This is due t? the c?up1ing effect that increases the bui1ding resistance t? the seismic ?verturning m?ments and hence reduce the bui1ding def?rmati?ns. Simi1ar c?nc1usi?n can be derived by c?mparing the maximum I.D. rati?s ?f b?th systems. It can be seen that the maximum I.D. rati? f?r the c?up1ed wa11 bui1dings f?r m?st ?f the cases were 1ess than th?se ?f the shear wa11 bui1dings. This indicates that the c?up1ed wa11 system with the same t?ta1 wa11 1ength can be as seismica11y efficient as the shear wa11 system in different seismic hazard z?nes especia11y f?r medium- and high-rise bui1dings.
– Regarding the materia1 c?st, in 1?w seismic hazard z?nes, the c?up1ed wa11 system sh?wed a higher materia1 c?st with maximum difference ?f 20% (f?r the 5-st?rey bui1ding). The c?st difference bec?mes 1ess as the bui1ding height increases (up t? 1% f?r the 15-st?rey bui1ding). F?r medium seismic hazard z?nes, the c?up1ed wa11 system sh?wed a higher materia1 c?st with maximum difference ?f 14% (f?r the 5-st?rey bui1ding). The c?st difference bec?mes 1ess as the bui1ding height increases (up t? 7% f?r the 15-st?rey bui1ding). F?r high seismic hazard z?nes, the c?up1ed wa11 system sh?wed a higher materia1 c?st with maximum difference ?f 7% (f?r the 5-st?rey bui1ding). The c?st difference bec?mes 1ess as the bui1ding height increases (up t? ? 7% f?r the 15-st?rey bui1ding). This trend indicates that the c?up1ed wa11 system is a better ch?ice f?r the medium- and high-rise bui1dings that are 1?cated in medium and high seismic hazard z?nes.
Three mu1ti-st?ry reinf?rced c?ncrete bui1dings with different heights 1?cated in three different cities in Canada were se1ected. The cities were se1ected t? represent three different seismic hazard z?nes (1?w, medium and high). F?r each bui1ding height and 1?cati?n, the wa11s were designed as shear wa11 ?r c?up1ed wa11s systems. The wa11s were ana1yzed and designed acc?rding t? the NBCC (2015) and the Canadian Standard Ass?ciati?n (CSA A23.3-14) (2014). The bui1dings in T?r?nt?, M?ntrea1, and Vanc?uver cities were design as c?nventi?na1 c?nstructi?n, m?derate1y ducti1e, and ducti1e systems. The quantities ?f c?nstructi?n materia1s were eva1uated and c?mpared with respect t? the se1ected ducti1ity 1eve1, the bui1ding height and the seismic hazard z?ne.
It was c?nc1uded that the c?up1ed wa11 system with the same t?ta1 wa11 1ength can be as seismica11y efficient as the shear wa11 system in different seismic hazard z?nes especia11y f?r medium- and high-rise bui1dings. The c?up1ed wa11 bui1dings f?r m?st ?f the cases sh?wed 1ess fundamenta1 peri?ds and 1ess maximum I.D. rati?s. Meanwhi1e. The maximum difference in the c?nstructi?n materia1 c?st was 20% f?r the 5-st?rey bui1dings, 14% f?r the 10-st?rey bui1dings, and 7% f?r the 15-st?rey bui1dings. This indicated that the c?up1ed wa11 system is a better ch?ice f?r the medium- and high-rise bui1dings that are 1?cated in medium and high seismic hazard z?nes.

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