seismic shear forces on rc walls: review and bibliography

Bull Earthquake Eng (2013) 11:1727–1751DOI 10.1007/s10518-013-9464-1

ORIGINAL RESEARCH PAPER

Seismic shear forces on RC walls: reviewand bibliography

Avigdor Rutenberg

Received: 24 October 2012 / Accepted: 14 May 2013 / Published online: 30 May 2013© Springer Science+Business Media Dordrecht 2013

Abstract Effect of higher vibration modes on the seismic shear demand of reinforced con-crete cantilever walls has been studied since the 1970’s. The shear amplification becomesmore important with increasing fundamental period (tall buildings) and increasing ductilitydemand (R or q factors). Yet, studying the relevant recommendations of structural engineer-ing researchers and provisions of various seismic codes reveals that there is no consensusregarding the extent of shear amplification and of the inter-wall distribution of shear demandin structural systems comprising walls of different lengths. The paper presents the availableformulas for predicting shear amplification in ductile walls and dual systems (wall-frames).One effect that impacts the shear amplification is shear cracking mainly in the plastic hingezone of the wall near the base leading to appreciably lower shear amplification than previ-ously predicted. Post yield shear redistribution among interconnected unequal walls is alsoaddressed. Finally, an extensive bibliography is provided.

Keywords Earthquake engineering · Structural walls and wall frames · Higher modesshear amplification

1 Introduction

This paper reviews the literature on the seismic shear demand on RC cantilever walls. Theinterest in shear demand derives, in the words of Paulay and Priestley (1992), from the need“to ensure that shear will not inhibit the desired ductile behavior of wall systems, and thatshear effects will not significantly reduce energy dissipation during hysteretic response”. It ismainly concerned with shear amplification due to higher vibration modes, and it also reportson the post-yield redistribution of shear demand among the walls in systems comprisingseveral walls of different lengths. Amplification of wall shear due to higher vibration modes

Revised and updated version of COMPDYN 2011 paper.

A. Rutenberg (B)Technion—Israel Institute of Technology, 32000 Haifa, Israele-mail: [email protected]

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1728 Bull Earthquake Eng (2013) 11:1727–1751

in the elastic range was not widely recognized by structural engineers until the early 1970’s,although it was, or at least should have been, a known phenomenon to users of responsespectrum modal analysis for the seismic design of RC walls in tall buildings. However, thelarger shear magnification in yielding flexural walls due to higher modes was first observedby New Zealand consulting engineers, which led to the 1975 pioneering paper by Blakeley etal. (1975). The problem was also studied by researchers at the Portland Cement Association,and these resulted in a series of PCA reports and papers by Derecho et al. (1978, 1980, 1981),and Derecho and Corley (1984).

The amplification tables by Blakeley et al. (1975), the charts by Derecho et al. (1981), aswell as later amplification expressions or charts (e.g. Ghosh and Markevicius (1990); Ghosh1992; Filiatrault et al. 1994; Seneviratna and Krawinkler 1994; Seneviratna 1995; Priestley2003b,c; Priestley et al. 2007; Rutenberg and Nsieri 2006), and for wall-frames (e.g. Goodsiret al. 1983, 1984; Kabeyasawa and Ogata 1984; Aoyama 1987, 1993; Calvi and Sullivan2009; Priestley et al. 2007; Rutenberg and Nsieri 2010) were mainly based on results of non-linear response history parametric studies using suites of accelerograms. Obviously, the resultdepended on the choice of these suites. Theoretically oriented general formulation for theshear amplification was given by Eibl and Keinzel (1988), Keintzel (1990, 1992). Extensionswere proposed by Priestley (2003b,c), Priestley and Amaris (2003), Priestley et al. (2007); andfor wall-frames by Sullivan et al. (2008). Very recently Fischinger et al. (2010a,b) and Rejec etal. (2011) proposed corrections to Keintzel’s formula, and Pennucci et al. (2010) modificationthereof. The role played by shear flexibility in reducing shear amplification was demonstratedby Rad and Adebar (2006, 2008), Adebar and Rad (2007), Rad (2009) and Yathon (2011).

Shear amplification becomes more important with increasing fundamental period (tallbuildings) and increasing ductility demand (R or q factors). Yet, studying the relevant rec-ommendations of structural engineering researchers and provisions of various seismic codesreveals that there is no consensus regarding the extent of shear amplification, its height-wisevariations, and the inter-wall shear demand distribution when the structural system consistsof interconnected wall of different lengths.

This paper is concerned only with shear demand amplification on structural walls dueto the effects of higher modes, but not with moment magnification along the height, whichderives from the same source. Issues related to the shear capacity and post-yield shear flexureinteraction are beyond the scope of the present study.

The phenomenon is described first, then available procedures to predict the amplificationare presented, and relevant code provisions are briefly discussed. Shear amplification in dualor wall-frame structures and coupled walls is also included in the review, and post-elasticshear demand redistribution among the walls in systems with walls of unequal length isconsidered. The paper closes with a discussion including some unresolved issues.

The structural engineering communities in North America and in many other countrieshave been slow in recognizing the need to amplify shear demand, if at all. The objectionof many researchers and engineers to the rather large amplification in the nonlinear rangeobtained from numerical simulations may perhaps explain the plethora of publications asessays in persuasion, manifested by the quite long bibliography at the end of the paper onthis rather limited problem.

2 The shear amplification phenomenon

In the linear range the amplification of shear demand in flexural walls due to higher vibrationmodes has been routinely accounted for by means of response spectrum analysis. This matter

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Bull Earthquake Eng (2013) 11:1727–1751 1729

Fig. 1 Normalized base shear Vb/W for 3 values of ρ; for walls ρ = 0 (Chopra 2001)

is well covered in standard textbooks. Figure 1, taken from Chopra (2001), shows the baseshear Vb normalized by the total weight W in planar structures analyzed by the modal responsespectrum technique for a typical design spectrum. It can be seen that for a flexural cantileverwall (ρ = 0, ρ = beam to column stiffness ratio) the base shear increases appreciably withincreasing natural period due the effect of higher modes compared with the design spectrum.This fact explains the requirement in many seismic codes (e.g. CEN 2004) to permit use ofthe equivalent lateral force (ELF) procedure only for buildings with T1 < min (4Tc, 2.0 s),Tc = corner period to the velocity related spectral acceleration region. As can be seen, inframe structures (ρ � 0) the shear amplification due to higher modes is appreciably muchsmaller. The main reasons for this difference are: (1) the combined effective mass of thehigher modes in flexural walls is approximately 35 % of the total mass vs. circa 25 % inframes; (2) the ratios of the 1st mode period to those of the higher modes are much largerfor flexural walls (e.g. 1st/2nd ≈ 6 vs. ≈ 3 for ideal frames). Hence, when T1 of a flexuralwall is in the velocity spectral region, the higher modes—the spectral accelerations of whichare usually located at the high acceleration plateau—have a larger effect on the shear thanin frames. This effect is even more pronounced when T1 is longer, so that Sa(T2) and evenSa(T3) are in the velocity spectral region. Indeed, the main effect of the higher modes is dueto the 2nd mode (e.g. Keintzel 1990; Panagiotou and Restrepo 2007, 2011) and, to a muchlesser extent, to the 3rd mode. Evidently, the shape of the spectrum also strongly affects theextent of amplification. The reader is referred to Humar and Mahgoub (2003) for detailedconsideration of the phenomenon in the linear range. The way seismic codes consider thiseffect is described subsequently.

As already noted, the shear amplification due to higher modes in the nonlinear range hasattracted considerable interest since the Blakeley et al. (1975) paper. They showed, by meansof nonlinear response history analysis (NLRHA), that shear demand increases with naturalperiod and falling damping ratio. Similar results were obtained by Derecho et al. (1978);

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Fig. 2 Fixed base and hinged cantilever mode shapes (Wiebe and Christopoulos 2009)

Keintzel (1984) and Kabeyasawa and Ogata (1984), Kabeyasawa (1987, 1993, 1988), whoalso demonstrated that shear demand increases with ductility demand.

The higher mode shapes of hinged and fixed base cantilever are quite similar, as shown inFig. 2, suggesting that the formation of a plastic hinge at the wall base does not significantlyaffect the higher modes, and these increase proportionally with ground motion input inten-sity (e.g. Sangarayakul and Warnitchai 2004). Hence, while the plastic hinge constrains the1st mode shear response to the level commensurate with its flexural capacity, the responseof higher modes remains circa their elastic level. In other words, the shear response can beapproximated as a superposition of the 1st mode shear reduced by the force reduction factorR (q in Europe), while the higher modes’ response remains elastic, thereby amplifying theirshear demand relative to that due to the first mode. Note that due to the plastic hinge for-mation and unloading/reloading softening the periods of higher modes are also elongated,hence amplification is likely to be somewhat lowered. Similar conclusions were drawn fromthe response of a flexural wall with substantially reduced stiffness at its base—the plastic

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hinge (Panagiotou and Restrepo 2007). This approach forms the basis for several analysisprocedures developed to predict shear amplification, as described subsequently. As in the lin-ear range, the base shear amplification increases with increasing natural period since highermodes become more important, and as a result the shear resultant moves downwards. Also,as in the linear range, the spectral shape plays an important role.

It is interesting to note that base shear amplification in dual wall frame systems (WFs)is lower than in structural walls. As already observed, the period ratios in frames are muchlower, and their 1st mode mass participation is higher. Also, the shear demand in framesdepends mainly on moment demand. Indeed, shear amplification in WFs strongly dependson the wall-to-frame shear ratio. More details on WFs are given in Sect. 5.

3 Procedures to predict higher modes shear amplification

The work of Blakeley et al. (1975) brought about changes in the provisions of the New Zealandseismic code (NZS 1982, 1995, 2006) and in European model codes (CEB 1980, 1983, 1985)and (CEN 1988). However, the latter provisions explicitly restricted the magnifications to theELF procedure. This turned out to be unfortunate since the corollary was that amplificationwas understood not to be required when using modal analysis (as, e.g., in the Israeli seismiccode SI 413 (1998). It also led engineers to overlook the fact that a major part of the shearamplification due to higher modes takes place in the post-elastic range.

This attitude still persists also in the National Building Code of Canada (NBCC 1995,2005), which only accounts for linear multi-mode shear amplification (also Humar and Mah-goub 2003; Mitchell and Paultre 2006). However, the Canadian Concrete code (CSA 1994,2004) requires considering nonlinear shear amplification, yet gives no guidance to quantify-ing that effect (also Boivin and Paultre 2010), although the Commentary to the 1994 versionof that code suggested adopting the New Zealand shear amplification formulas. Note, thatin the majority of the provisions and the proposals described subsequently, it was tacitlyassumed that the plastic hinge could form only at the wall base, and that the prescribedmoment envelope took care of that.

The amplified shear Va is given in terms of the base shear Vd from analysis, as follows:

Va = ωvVd (1)

In the New Zealand Code (NZS 1982, 1995, 2006):

ωv ={

0.9 + n/10 n ≤ 61.3 + n/30 ≤ 1.8 n > 6

}(2a)

and in CEB (1980, 1983, 1985):

ωv ={

0.9 + n/10 n ≤ 51.2 + n/25 ≤ 1.8 n > 5

}(2b)

in which n = number of storeys. As can be seen, the NZ code is somewhat more conservativethan CEB for 5 < n < 15. Note that both the 1999 and 2009 SEAOC recommend adoptingthe NZ amplification values when Vd, is based on the ELF procedure while somewhat lowervalues are permitted for response spectrum analysis. However, these are not mandatory.

In spite of their widespread use there are several problems with these formulas. First, theonly governing parameter is the number of storeys, which is a crude proxy for the naturalperiod. Also, the upper bound amplification is arbitrarily set at 1.8 for a 15 storey building.However, perhaps a more serious limitation is their independence of the excitation level,

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Fig. 3 Base shear demandamplification for several valuesof rotational ductility (Derechoet al. 1981)

max

V

dyn

V T

FUNDAMENTAL PERIOD , T1 (sec) 0 1.0 1.5 2.0 2.5 3.0

1.0

5.5

5.0

4.5

4.0

3.5

3.0

2.5

2.0

1.5

My

H

VT

4

3

2

5

= 6 a

~ H23

r

0.5

μ μ

which affects the shear demand appreciably, and of the shape of the applicable responsespectrum. Note also that, by today’s standards, the Blakeley et al. (1975) study was based ona very limited data set: only 5 accelerograms, two of which were artificial, and apparentlyEqn. 2 was based on only two of these. Also, bilinear moment-curvature hysteresis wasassumed, while many modern studies use the modified Takeda model, considered to be morerealistic. On the other hand, the bilinear model is somewhat more conservative (Nsieri 2004).

Shear amplification design charts were presented by Derecho et al. (1980) and Derecho andCorley (1984). These are based on a parametric study on structural walls, 10–40 storey high,using the modified Takeda model for the NLRHA carried out for 6 earthquake records. Theypresented shear amplification relative to the base shear, as evaluated by the then governingstatic UBC provisions, vs. fundamental natural period for several values of rotational ductilitydemand μr , as shown in Fig. 3. Since the set of records was normalized for a given Housnerspectral intensity (SI = 1.5) a correction factor (practically proportional) was provided fordifferent SIs. One difficulty in applying this approach is its dependence on μr rather thanon the strength reduction factor R, which, in routine analysis, requires making assumptionsregarding the relation between these two parameters (R − μ − T formulas).

Based on parametric NLRHA several investigators proposed simple expressions to esti-mate the maximum base shear Va by decomposing it into the 1st mode (assumed triangular)and the higher modes responses. Kabeyasawa and Ogata (1984), Kabeyasawa (1987, 1988),and also Aoyama (1987, 1993) gave the following expression for wall frames:

Va = Vd + DmWAe (3)

in which Vd = base shear of an inverted triangular loading, W = total weight, Ae = peakground acceleration and Dm is a given factor increasing with the number of storeys (i.e., ∼period). Based on shake-table tests, Eberhard and Sozen (1989) proposed Dm=0.3 for 9 and

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10 storey wall frames, in agreement with Kabeyasawa’s results. The actual amplification inwall frames depends to a large extent on the base shear share carried by the wall. Hence thepredictive ability for wall frames of formulas in the form of Eq. 3 is rather limited.

A similar formula for isolated walls was presented by Ghosh and Markevicius (1990) andGhosh (1992), based on a parametric study similar to that of Derecho et al. (1981):

Va = My/0.67H + DmWAe (4)

in which My = the yield moment at base and H = wall height. The advantage of this form isthe explicit statement that the base shear is that at moment capacity level rather than at thedesign shear obtained from analysis, which is usually lower. For isolated walls the study bySeneviratna and Krawinkler (1994), Seneviratna (1995) described subsequently, confirmedthe equation format of Kabeyasawa and of Ghosh except for the short period range and forlow values of R.

In 1988 Eibl and Keintzel proposed a simple expression for base shear amplification byexplicitly assuming that the 1st mode shear is limited by the flexural capacity of the wall at thebase, and that the higher modes respond linearly. This formula was eventually incorporatedinto Eurocode 8 (CEN 1993, 2004):

ωv = q ·√(

γRd

q

MRd

MEd

)2

+ 0.1 ·(

Sa (Tc)

Sa (T1)

)2

≤ q (5)

ωv = 1st mode shear amplification factor – but see next paragraphq = R = Strength reduction (behaviour) factorMEd = Design bending moment at wall baseMRd = Design flexural resistance at wall baseγRd = Overstrength factor due to steel strain-hardening (suggested value 1.2)T1 = Fundamental periodTc = Higher period corner of the constant acceleration plateauSa (T) = Spectral ordinate

This is a very useful formula since it considers most of the parameters governing shearamplification in structural walls. It does not require exact evaluation of the 2nd mode naturalperiod, and is practically spectrum-shape independent, in contradistinction to parametricallyevaluated amplification formulas or plots that are not (e.g., Priestley 2003b,c; Rutenberg andNsieri 2006; Seneviratna and Krawinkler 1994; Seneviratna 1995). It also accounts for the twosources of overstrength: MRd/MEd and γRd. However, it is likely to be too conservative fortall buildings, i.e., those with T2 and even T3 in the velocity spectral region. It is also sensitiveto the calculated value of the 1st mode spectral acceleration Sa1, which for RC walls dependson design assumptions. Also, the formula is inherently somewhat conservative since codespectral ordinates represent peak values whereas it is unlikely that for a given earthquakeboth the 1st and 2nd modes would reach their peaks. It is believed that Sa at the naturalperiod based on the yield displacement should be the appropriate one (e.g. Panagiotou andRestrepo 2011). In view of the extensive use of this formula (EC8 is now applied in circa20 countries) some background information may be in order. Based on parametric studies,Keintzel assumed that the SRSS modal combination can also be applied in the post-elasticrange, and also that only the contributions of the first two modes are important, hence:

Va =√(

VEd,1)2 + (

qVEd,2)2 = qVEd,1

√(1

q

)2

+(

VEd,2

VEd,1

)(6)

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in which VEd,i are the 1st and 2nd mode shear forces from analysis. The 1st term in the paren-theses of Eq. 5 is obtained by accounting for the two sources of overstrength γRd MRd/MEd,and assuming that these affect only the 1st mode. Hence, the 1st term under the sqrt signin Eq. 5 replaces 1/q in Eq. 6. Since VEd,2/VEd,1 ≈ 0.32 Se(T2)/Se(T1), (0.32)2 ≈ 0.1 ≈the 2nd mode to 1st mode mass contribution ratio, and replacing Sa(T2) by Sa(Tc) lead tothe 2nd bracketed term in Eq. 5. This development shows, as was made clear by Fischingeret al. (2010a,b) and by Rejec et al. (2011) that the amplification ωv should factor only the1st mode shear VEd,1, i.e., Va = ωvVEd,1. However, in EC8 ωv (therein ε) amplifies VEd

as evaluated either by the ELF procedure or by modal analysis; hence, without making thedistinction, it evidently leads to different and more conservative results. Keintzel concludedthat ωv should be bounded by q(R), but this is too low considering that the elastic boundshould be related to the SRSS shear—not the 1st mode one—as was noted by Rejec et al.(2011). Also, the present EC8 restricts the application of Eq. 5 to walls designed for highductility (DC-H), while for medium ductility walls (DC-M) ωv = 1.5. This has been foundto be too low (e.g. Fischinger et al. 2010a,b; Rutenberg and Nsieri 2006), suggesting that Eq.5 should be applied also to DC-M walls, as indeed was the case in past versions of EC8.

Keintzel (1990) already observed that Eq. 5 is a simplification of an expression he consid-ered to be more accurate. A different correction, which is based on an extensive parametricstudy using the Modified Takeda model and assuming hinge formation only at wall base, wasproposed by Fischinger et al. (2010b) and Rejec et al. (2011). This correction affects the 2ndterm in Eq. (5) when MRd/MEd is large, hence is of limited practical use.

Priestley (e.g. 2003b; 2003c) and Priestley et al. (2007) extended the applicability ofKeintzel’s formula to the full height of the wall (rather than just to the base shear) throughrecasting it into the standard SRSS format. This also relieves the formula from the arbitraryreplacement of the 2nd mode period by the corner period (Tc):

Va =√(

V∗Ed,1

)2 + μ2{(

VEd,2)2 + (

VEd,3)2 + · · ·

}(7)

in which V∗Ed,1 is the overstrength factored 1st mode design shear and μ is the displacement

ductility demand. Priestley (2003b,c) reported satisfactory agreement with NLRHA resultsfor a wide range of wall heights. Note that for routine analysis it may be assumed that μ ≈ q.

Another modification of Keintzel’s formula was recently proposed by Pennucci et al.(2010). The basic assumption therein is that the total response can be obtained as the absolutesum of the 1st mode shear reduced by the strength reduction factor, denoted Rm, and of thehigher modes SRSS response, which is reduced by another factor (Rp). It has similar formatas Eqs. 3 and 4, and is given by:

Va = VEd,1

Rm+

√(Vpin,2

)2 + (Vpin,3

)2 · ··Rp

(8)

in which VEd,1 and Vpin,i are the modal shear forces at their elastic level, and the subscript pindenotes pin-based modal shear response. Based on parametric studies Pennucci et al. (2010)found that, as expected, the higher modes reduction factor Rp is related to Rm: large whenRm is low, and becoming smaller with increasing Rm, asymptotically reaching 1.0, i.e. linearbehaviour. Considerable improvement over Eq. 7, particularly for long periods and largestrength reduction factors, and very satisfactory agreement with NLRHA results along thebuilding height were reported. Hence, it appears that the approach can make better allowancefor the period elongation due to base hinging and for the lowered stiffness of the unload-ing/reloading hysteresis loop (Modified Takeda). Yet, since Rp is evaluated parametrically,it is spectrum shape dependent.

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Fig. 4 Base shear demandamplification for several valuesof displacement ductility(Seneviratna 1995)

A simple formula also in the format of Eqs. 3 and 4 was proposed by Panagiotou andRestrepo (2007). It is based on the observation that the amplification can practically beaccounted for by adding a factored 2nd mode shear envelope to that of the 1st mode shear atbase yield. The shear demand at storey i can then be approximated as:

Va = V∗Ed,1 + q ρ

12

⎛⎝ n∑

j=i

�2�j2

⎞⎠ mSa2 (9)

in which ρ12 is a factor to be determined, and �2�j2 = min[4.28hi/H, 0.6;−2.5hi/H + shape

1.85], �2 = 2nd mode participation factor,�i2 = modeordinate at floor i, hi = floor height

from base, H = total wall height, Sa2 = 2nd mode spectral acceleration. Good agreement withNLRHA results for near-fault records was reported. Note that the application of the formulato other cases depends on ρ12, which, however, was provided only for a small number ofpulse-type near-fault ground motions.

Base shear demand amplification charts based on 15 records for wall structures havingelastic–plastic moment-rotation hysteresis designed using SRSS load pattern were providedby Seneviratna (1995). Figure 4 presents the mean amplification, defined as the ratio of themaximum base shear of the multi-degree of freedom Vb (MDOF) to the single degree offreedom (SDOF) base shear Fy(μ), versus T1. These are displayed for several values ofμ(SDOF)—the ductility ratio of the SDOF system. It can be seen that the amplificationincreases with T1 and with μ (SDOF). As already observed, the amplification w.r.t the SRSSloading would be smaller since higher modes also amplify the shear in the linear range.

The amplification formula provided by Priestley (e.g., 2003b; 2003c) was based onNLRHA results for walls assuming Modified Takeda hysteresis. It has a very simple form,and is given in terms of the displacement ductility demand μ and T1:

ωv = 1 + μB(T)/�; B(T) = 0.067 + 0.4 (T1 − 0.5) ≤ 1.15 (10)

in which � is the flexural overstrength.A somewhat longer expression given in terms of q (or R) was presented by Rutenberg and

Nsieri (2006). It is given by:

Va = ωvVd = [0.75 + 0.22 (T1 + q + T1q)

]Vd (11)

in which Vd is the base shear evaluated by the ELF procedure. Equation 11 is shown in Fig. 5by the straight lines. Elastic-plastic response and 5 % tangent stiffness damping in the 1st and5th modes were assumed. Note that 5 % damping is now believed to be too high for tallerbuildings (e.g. Satake et al. 2003) for which major sources of damping are appreciably lower.

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q = 6

q = 5

q = 4

q = 3

q = 2

q = 1

0

1

2

3

4

5

6

7

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Fundamental period T (sec.)

From analysis

Eqn. 4

Bas

e sh

ear

amp

lific

atio

n f

acto

rω∗ v

Fig. 5 Equation 10—Base shear demand amplification for several values of strength reduction factor q (=R)(Rutenberg and Nsieri 2006)

A correction for modal response spectrum analysis derived base shear was provided in the2009 Amendment to the Israeli seismic code (SI 413 Amendment No. 3).

Another approximation also based on parametric response history results was proposedby Celep (2008) for which the Takeda degrading stiffness was assumed. It takes the followingform:

ωa = 1 + (0.281T1 + 0.394) (q − 1.5)−0.553 ≤ q (12)

A comparison of Eq. 11 with Eqs. 12 is given in Fig. 6. It is seen that the two studies lead tosimilar results even though they were based on a different hysteresis rule and different setsof records (larger Sa(T = 1.0s)/Sa(Tc)).

Note that in applying Eqs. 11 and 12, the flexural overstrength can be accounted forthrough dividing q by γRd MRd/MEd.

A summation formula resembling in form Eqs. 3, 4, 9, and conceptually closer to Eq. 8,was very recently proposed by Yathon (2011):

Va = VRSA/Rd + Hf Fpf VRSA (13)

VRSA = Response spectrum analysis base shearRd = Ductility related force modification factor per Canadian code (< R)

Hf = P – e1.6(1−Rd)

P ={

1.0 for Sa(0.2)/Sa(2.0) < 81.1 for Sa(0.2)/Sa(2.0) ≥ 8

}

Rpf ={

2T2; T2 < 0.25S

0.5; T2 ≥ 0.25S

}

T2 = Second period of the structure

The first term in Eq. (13) is the RSA based capacity component, Hf is the fraction of pinnedbase shear as based on NLRHA parametric results and Rpf is a correction for the pinned tofixed base shear ratio. Close agreement with NLRHA was reported for the Vancouver andMontreal spectra with Rd ranging from 2.0 to 5.0. Also, the formula is quite unique inpredicting amplification also for very long period structures (up to 70 storeys).

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Fig. 6 Comparison of shear amplification factors in Eqs. 11 and 12 (Celep 2008)

Most of the amplification formulas or charts reviewed above are parametrically based;hence they are only applicable to spectra having shapes that are similar to the mean spectraof the suite of records forming the data base for the NLRHA. Also, it is important to checkwhether in the structural model nonlinear elements were used above the anticipated plasticregion up the height of the wall. As noted subsequently, even minor yielding in the upperlevel is likely to reduce shear amplification, thereby resulting in overestimation. Moreover,the formulas are sensitive to the chosen damping ratios. Hence, they should be used withcaution.

4 Storey shear demand

Whereas, as noted, Eqs. 7, 8 and 9 are able to predict the shear demand for the full wallheight, this is not the case with the other design equations listed above that apply only tothe base shear. This is because in many cases the ωv normalized code based storey shearsoverestimate the expected shear values for large wall portions of long period structures, asshown in Fig. 7. However, Yathon (2011), based on extensive NLRHA, concluded that forvery long period low R (≤ 2.6) structures the base normalized SRSS results are adequate.

Apparently, this problem was first addressed by Iqbal and Derecho (1980) who provided awall height shear demand envelope that has not changed appreciably since. A design envelopeas function of the fundamental period T1 (Rutenberg and Nsieri 2006) is shown in Fig. 8, inwhich ξ is given by:

ξ = 1 − 0.3T1 ≥ 0.5 (14)

Note that Fig. 8 resembles Fig. 5.4 in EC8 (CEN 2004), but therein it is confined to wallsin dual WF systems. A similar envelope was proposed by Celep (2008).

A straight-line envelope with a wall-top shear of:

(0.9 − 0.3T1) Va ≥ 0.3Va (15)

was given by Priestley et al. (2007) and Model Code (Calvi and Sullivan 2009).

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0

1.0

0.8

0.6

0.4

0.2

0.0

Rel

ativ

e H

eigh

t

Vi / Vbase

Vdes, i / Vdes, base (UBC 94)Vdyn, i / Vdyn, base (Elas. Resp)Vdyn, i / Vdyn, base ( (SDOF)=4)Vdyn, i / Vdyn, base ( (SDOF)=4)

0.3 0.6 0.9 1.2 1.5

Fig. 7 40-storey wall shear envelopes: code loading design pattern, T = 2.05 s: mean for S1,a records, bilinear,α = 0 %, damping 5 % (Seneviratna and Krawinkler 1997)

Fig. 8 Shear force designenvelope (Rutenberg and Nsieri2006)

5 More on wall frame (WF) systems and coupled shear walls (CSWs)

Some studies on dynamic shear force amplification in WFs followed naturally from those onflexural walls, but not all. As already noted, wall shear amplification in WFs was studied byKabeyasawa (1987) and Aoyama (1987, 1993), see Eq. 3. Goodsir et al. (1983, 1984) observedthat, as in flexural walls, higher vibration modes amplified the shear force demand on thewalls of WFs, but to a lesser extent, and proposed a simple prediction formula accounting forthis response reduction. This extension of the shear amplification formula for flexural wallsbased on Blakeley et al. (1975) to walls of dual systems is also given by Paulay and Priestley(1992). This WF shear amplification factor ω∗

v is given by:

ω∗v = 1 + (ωv − 1) η (16)

in which ωv is the free standing wall shear amplification factor and η is the fraction of thebase shear carried by the walls. The format of this formula is appealing as it assumes thatthe contribution to shear amplification of the frame part is negligible. It was found, however,that its predictive capability appears to be limited (Rutenberg and Nsieri 2010).

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Alwely (2004) made ω∗v dependent on the non-dimensional frame-to-wall stiffness ratio

α H, as evaluated from the linear continuum approximation (e.g. Stafford Smith and Coull1991). Kappos and Antoniadis (2007) proposed a procedure to estimate more realisticallythe shear demand in the upper storeys and also a modified version of the (Goodsir et al.1983, 1984) formula (Eq. 16) to account for shear amplification in WFs with unequal walls.Very recently (Kappos and Antoniadis 2010, 2011) provided formulas to improve the sheardistribution match with NLRHA results all along the wall height.

The codified guidance regarding the amplification factor ω∗v is rather limited. The New

Zealand seismic code (NZS 3101-2006) refers only to Paulay and Priestley (1992). Note thattherein ωv is that given by Eq. 2a. It is clearly seen that when η ≤ 1.0 then 1 ≤ ω∗

v ≤ ωv.Equation 16 assumes that ω∗

v, as ωv, is independent of the expected ductility demand or q(R). Note also that Eq. 2 depends indirectly on T1 through its dependence on the number ofstoreys n.

EC8 (CEN 2004) considers WFs in which η > 0.5 as “wall–equivalent”, hence in suchcases wall shear in high ductility class (DC-H) systems should be amplified per Eq. 5.

Applicable US codes do not include dynamic shear amplification requirements for wallframes. However, in US practice, amplification is considered in tall buildings design, asdescribed subsequently. Priestley et al. (2007), in the draft displacement-based seismic designcode at the end of their book (see also Calvi and Sullivan 2009), proposed a dynamic base shearamplification expression for WF walls that requires predicting the displacement ductilitydemand μsys of the dual WF system. When 0.4 ≤ η ≤ 0.8 the wall base shear amplificationis given by:

ω∗v = 1 + μ

�C; C = 0.4 + 0.2(T1 − 0.5) ≤ 1.15 (17)

in which � is an overstrength factor (usually 1.25). The base shear at the wall top Va,top =0.4Va,base. Note that ω∗

v is assumed independent of η, hence cannot be in agreement withEq. 15.

Based on the deflected shapes observed in several time windows during NLRHA, (Sullivanet al. 2006, 2008) proposed a variant of Keintzel’s formula using so called “transitory inelasticmodes”. They concluded that the higher modes’ periods elongated, and suggested that goodagreement with NLRHA of the base shear per Keintzel could be obtained when the highermodes’ periods were evaluated based on the low post-yield secondary stiffness at wall base.This basically amounts to replacing in Eq. 7 the μ2 factored higher modes fixed base shears(at design level) by their pin-based counterparts.

A NLRHA based formula was very recently proposed by Rutenberg and Nsieri (2010):

Va = 0.4 + 0.2 (T1 + q + η) + 0.13q η T1 (18)

in which η is the elastic wall base shear to total base shear ratio. As all formulas based onNLRHA parametric studies it is applicable to design spectra similar to the mean spectrumof the suite of records used for their derivation. It was also found that the results are quitesensitive to the yield levels of the girders over the frame height.

Chaallal and Gauthier (2000), in a paper devoted solely to CSWs, observed that the shearresistance should be based on the shear capacity of the compression wall which has largeroverstrength than the tension wall. This implies that for CSWs located in Canada there is noneed to account for dynamic shear amplification.

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q = 3

q = 1

q = 6

q = 4

q = 5

q = 2

0

1

2

3

4

5

6

0.0 0.5 1.0 1.5 2.0 2.5

Fundamental period T (sec.)

Bas

e sh

ear

amp

lific

atio

n f

acto

r

Plastic hinge at base

Plastic hinges allowed at higher levels

q = 2, 3, 4, 5, 6

Fig. 9 Plastic hinge at base only versus plastification at higher levels—base shear flexural strength assumedover building height (Rutenberg and Nsieri 2006)

6 Effects on shear amplification: seismic isolation, multiple hinges, rocking, and sheardeformation

6.1 Seismic isolation, multiple hinges along height and rocking

It is often necessary to increase the strength over significant portions of the wall height ifyield is to be confined—per capacity design—to the plastic hinge zone at the base; whereasallowing yield to propagate upwards (with base level flexural strength) leads to some reductionin the shear demand, e.g., as shown in Fig. 9 (Rutenberg and Nsieri 2006). Adding anotherplastic hinge at wall mid-height, as proposed by Panagiotou and Restrepo (2009), leads tomore pronounced lowering of shear demand envelope mainly above that hinge. Furthermore,when the wall is designed such that plastic hinges can develop at several locations along theheight (Rad and Adebar 2008; Rad 2009), the shear demand falls appreciably, as illustratedin Figs. 10, 11. Evidently, the last approach requires special reinforcement detailing along allthe wall height. The recent study by Yathon (2011) supports the predicted lowering of shearamplification, mainly above the hinge, due to yielding in the upper levels of the wall.

Wiebe and Christopoulos (2009) proposed a design to allow rocking of several wallsegments with respect to each other in order to reduce the contribution of higher modes,using unbonded post-tensioning. Appreciable reductions in shear and bending moment werereported.

Twenty storey seismically isolated cantilever wall buildings designed with lead-plug rub-ber bearings were studied by Calugaru and Panagiotou (2010). Three variants were consid-ered: single isolator at base, isolators at base and at mid height, and isolators at base and at0.7 of height. As expected, these buildings demonstrated significant reduction in the impor-tant response parameters compared with standard fixed base, i.e. with a single plastic hinge,buildings. Dual isolation led to further reduction in shear demand at the upper 60 % of theheight.

6.2 Shear deformation

Already in 1988 Eibl and Keinzel (1988) observed that smaller shear rigidity (relative toflexural rigidity) lowers the higher modes dynamic amplification. In their study on the shear

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Fig. 10 Shear force envelopes with hinging permitted only at wall base, R = 5 (Rad and Adebar 2008)

Fig. 11 Shear force envelopes, flexural hinging at many locations over height, R = 5 (Rad and Adebar 2008)

distribution among wall of different lengths Rutenberg and Nsieri (2006) noted a substantiallowering in shear demand in cracked sections when the effective shear rigidity GAve wasevaluated based on the analogous truss model (Park and Paulay 1975), which is appreciablysmaller than the recommended values based on the gross section (GAvg) for both crackedand uncracked walls by ASCE 41-06 (2007), or 50 % thereof for cracked walls by EC8 (CEN2004).

Gérin and Adebar (2004) provided expressions to predict the shear deformation, includingshear cracking and yielding, observing that a typical effective shear rigidity of diagonallycracked concrete could approximately be taken as 10 % GAvg, somewhat lower than thevalue assumed by Rutenberg and Nsieri (2006). Rad and Adebar (2008), Rad (2009) usedlow effective shear stiffness as a simple way to demonstrate the significant influence ofcracking on shear amplification, as shown in Fig. 12. For effective shear stiffness at yieldPEER/ATC-72-1 (2010) recommends lager values, varying from 5 to 10 % GAvg dependingon shear strength of concrete. Wallace (2010) suggested modelling the post-cracking shearstiffness of walls with moderate shear stresses in the plastic hinge zone, considering shear-flexure interaction, as circa 2.5 % GAvg.

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Fig. 12 Influence of effective shear stiffness on average shear force envelopes, flexural hinging at manylocations along wall height, R = 5 (Rad and Adebar 2008)

It is interesting to note that a model incorporating shear deformation including shearcracking as well as flexural behaviour was implemented in ANSR-II computer program byKelly (2004). Kelly observed that although the wall was capacity designed to prevent brittlefailure, the dynamic amplification factor ωv per the NZ code (Eq. 2a) was found to be toolow for the particular wall configuration studied.

7 Notes on practice in the USA

Seismic design practice in the US, particularly in California, is highly regarded worldwideand often adopted; hence it may be useful to comment on the situation there with respect todynamic shear amplification.

It is interesting that although researchers, code-writers and practicing engineers wereexposed to the many publications on the post-yield shear demand magnification due to highermodes, this has not been manifested in US seismic codes (e.g. ASCE 7-05 2006), as wouldbe expected, particularly, since other major codes such as NZ3101 (1982, 1995, 2006) andEC8 (CEN 2004) included such provisions for many years. As noted, only the Commentaryto the SEAOC (1999, 2009) considered this issue, referring the reader to the New Zealandcode. Also, shear amplification is recognized, albeit to a very limited extent, by ASCE41-06 code (2007) for upgrading of existing buildings: the shear force at the wall base iscalculated assuming uniform lateral force distribution over the wall height, i.e., the shearresultant is located at wall mid-height (clause 6.7.2.4 therein) rather than at approximatelyits 2/3rd, thereby increasing the base shear by circa 33 %. It is only when NLRHA—one ofthe recommended procedures by US codes—is carried out that dynamic shear amplificationis accounted for. However, code based analysis per this procedure appears to be the exceptionrather than the rule.

The resurgence in high-rise construction in the west coast of the United States duringthe early 2000’s has led to the application of performance-based non-prescriptive designprocedures (CTBUH 2008; LATBSDC 2005, 2008; SEAONC 2007). For collapse preventionunder Maximum Considered Earthquake (MCE) level NLRHA-based design is required;

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hence, dynamic amplification is thereby considered. Indeed, researchers and practitionersin California were soon reporting on large higher modes magnifications (Klemencic et al.2007; Moehle 2006, 2008; Moehle et al. 2007; Zekioglu et al. 2007). For very tall buildings,circa 40 storeys, these studies predicted shear amplifications on the order of 4 relative tocode.

Apparently most of the reported amplification values were obtained using the commer-cially available nonlinear computer code PERFORM-3D (CSI 2006, 2007). It uses nonlinearfibre model for axial force and flexure and trilinear model for shear, but cannot model shear–flexure interaction (e.g. Oesterle et al. 1980). However, shear stiffness degradation was nottaken advantage of, and hence amplification could have been over-predicted. It is interestingto note that one designer (Klemencic 2008) observed, after designing over two dozen tallbuildings, that “design which follows the prescriptive provisions of the building code willlikely result in a shear capacity that falls well short of the likely demands the structure willexperience during a significant seismic event. As shear failure of a structural wall is typicallyviewed as non-ductile, undesirable response, this outcome raises serious concerns….”.

8 Post-elastic redistribution of shear demand among walls in systemswith interconnected walls of unequal length

The shear force distribution among the walls in regular multistorey structures laterally sup-ported by a number of reinforced concrete cantilever walls with markedly different lengthsis strongly affected by the different yield displacement levels of the walls. In the post-elasticrange the share of the shear force carried by the shorter walls is larger than their share in thebending moment.

In a single storey structure the shear distribution among the walls within the elastic rangeis proportional to their stiffness. Following yielding of the longer wall additional loads willbe carried by the shorter ones. It is evident that design for shear on the basis of relativestiffness underestimates the forces on the shorter walls since shear capacity should be madeproportional to flexural strength. However, the situation in a multistory building is differentas shown in Fig. 13, showing a simplistic model of a two storey system supported by twowalls. To fix ideas, assume Wall 2 is very much stiffer than Wall 1; hence, the horizontalforce on Wall 1 is negligible so long as Wall 2 has not yielded. With Wall 2 yielding at basethe force �H is distributed as shown in Fig. 13b, where for simplicity it was assumed thatthe floor diaphragms are pin-connected to the walls and are infinitely rigid in plane. It can beseen that displacement compatibility requires a shear force redistribution resulting in a verylarge shear force acting on Wall 1.

Rutenberg and coworkers (e.g. Rutenberg 2004; Rutenberg and Leibovich 2002a; Ruten-berg and Nsieri 2006) demonstrated that the shear demand on the flexible walls could bevery much larger than commensurate with their relative stiffness or even flexural strength.These studies were mainly carried out by means of cyclic pushover analysis on bilinear force-displacement hysteretic models. Formulas for simple hand calculation procedure were alsopresented to follow the 1st yield shear force distribution among the walls. However, sinceredistribution is mainly due to compatibility enforcement it is sensitive to modelling assump-tions. Hence further studies replaced the routine modelling (G = 0.4E, in-plane floor rigidity,bilinear hysteresis) with more realistic models. Indeed, the earlier results were shown (Beyer2005; Rutenberg and Nsieri 2006) to overestimate the base shear demand on the shorter wall.Beyer (2011) also observed that the greatest effect was due to “locking in” of compatibilityforces transmitted by the floor diaphragms. Indeed, based on Beyer’s 2005 work Priestley

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Wall 1 Wall 2(a) (b)

1.5ΔHΔHΔH

h

h

3ΔH

I1 (I1<<) I2

I1 (I1<<) I22.5ΔH 1.5ΔH

Fig. 13 Two-storey wall system: a properties and loading, b floor forces and deflected shapes (Rutenberg2004)

et al. (2007) concluded that these compatibility induced forces might normally be ignoredin design. Yet, it appears that the effect of this relaxation depends on the extent of yield-ing along the wall’s cross-section, which in turn also depends on whether the longitudinalreinforcement is mainly concentrated at the wall ends (flanges) or is uniformly distributed, aparameter that was not considered. Moreover, Beyer’s conclusions were based on a two-wallmodel with modest lengths ratio (1.5: 6.0 & 4.0 m respectively); larger stiffness ratios arelikely to increase the share of the short wall shear force. These expectations have recentlybeen supported by Simonini (2011). In view of the above, it is obvious that the distribu-tion formulas presented in, e.g., Rutenberg and Nsieri (2006) which are based on a lumpedplasticity model and ignore shear deformations, are in need of revision.

Beyer (2005) proposed using the pushover analysis (Rutenberg 2004) to test the propensityof the shorter wall for excessive shear demand. In that procedure the resultant lateral forceis located at the minimum dynamic effective height heq,dyn given by Eq. 19:

heq,dyn = heq,static/ωv (19)

in which heq,static is the height of the statically computed shear resultant, and ωv is thedynamic amplification factor. Alternatively, heq,dyn may be taken for the applicable spectrumfrom plots such as Fig. 2 in Rutenberg et al. (2004), if available. If the base shear on the shortwall sharply increases while the base shear of the long wall drops off after formation of theplastic hinge at the base of the long wall, coupling effects are expected to be significant. Notethat ωv in Eq. 19 is predicated on the assumption that the total base shear on a group of wallsis equal to that on an isolated wall whose strength and stiffness equal the respective sums ofthese properties of the individual walls. This was found to be justified for the record suitesused by Rutenberg and Nsieri (2006).

Adebar and Rad (2007), using pushover analysis, studied a two-wall system with markedlydifferent properties (a long flanged wall and a short rectangular one). They were mainlyinterested in the effect of shear deformation on the response, modelling both flexure andshear with tri-linear force displacement relations, i.e., a kink at cracking and a yielding plateau(very short for shear). As expected, cracking and yielding were found to affect the shear forceredistribution along the height, in some cases predicting shear failure. In particular, diagonalshear cracking led to significant shear force increase in the short wall at the 2nd floor level.Note, however, that the moderating effect of distributed plasticity was not considered.

Post-yield redistribution of shear demand has also been studied by Kappos and Antoniadis(2007).

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9 Discussion

In many countries dynamic shear amplification is now routinely accounted for in wall design,yet, as shown, there are large differences in the extent of amplification—from relatively lowωv in the New Zealand code to quite large ones in EC8. As already noted, the Canadian coderequires considering amplification only when the ELF procedure is carried out, and then onlythe linear multi-modal effect. Interestingly, the Romanian seismic code P100-1/2006 (2006),while generally adopting the EC8 provisions for RC structures, excluded Eq. 5, based mainlyon the peculiarities of the 1977 Vrancea earthquake record. Instead, it specifies ωv = 1.5for statically computed base shear or 1.2� (� = flexural overstrength), whichever is larger(Postelnicu 2011; Zamfirescu 2011).

The very large amplifications predicted by NLRHA were met in many instances with someskepticism. For a widely expressed attitude it is perhaps best to quote Paulay (2007), whowrote w. r. t. computed magnifications much larger than given in the New Zealand provisions:“I found at that time (early 1980’s, AR) that the duration of predicted extremely high sheardemands was typically 0.02–0.03s. So we pacified our conscience with the unansweredquestion: ‘is this time interval long enough to enable a wall region to fail in shear?”’. Onthe other hand, ignoring shear peaks with durations less than 0.03 s does not appear to leadto substantial reduction in shear demand. A preliminary study by Panagiotou and Calugaru(Panagiotou 2011) on records of three earthquakes showed that demand was lowered onlyby 6–15 %. In this connection, Linde and Bachmann (1994) noted, as is well known on thebasis of experimental results, that the material strength of concrete increased with high strainrate. Hence, design per peak shear value appears to be somewhat conservative.

It has often been observed that the maximum shear may not occur simultaneously withmaximum moment over the wall height; hence the large amplification factors could be ques-tioned (e.g. Leger 2010). Yet, simultaneous peaks at wall base level were observed in analysesby Rad (2009) and Tremblay et al. (2008) for several historical records; and the large ampli-fications were evaluated for the wall base. No doubt these two issues deserve study.

The main reason for the skepticism appears to be the feeling that standard nonlinearmodelling of RC walls, particularly in the plastic hinge zone, does not capture faithfullythe complexity of the response. As noted above, using the Takeda rather than the bilinearhysteresis lowers computed shear demand somewhat. A considerable reduction was reportedby Yathon (2011) using a particular trilinear model. Also, lowering shear stiffness to representshear cracking and yielding as well as shear-flexure interaction results in substantial fall inshear demand (Fig. 12). On the other hand, it has been demonstrated that the 5 % dampingroutinely assumed in NLRHA is nonconservative for tall buildings design, as damping ratioslower than 2 % were reported (e.g. Satake et al. 2003). The effect on shear amplification inwalls was considered by Boivin and Paultre (2010). Note also that overestimating the effect ofhigher modes, particularly the 2nd one, may result when using Rayleigh rather than constantdamping in NLRHA (Luu et al. 2011; Yathon 2011).

Another important issue is the experimental and historical evidence for nonlinear dynamicshear amplification. The studies by Eberhard and Sozen (1989, 1993) and Sozen and Eberhard(1992) on wall frames suggest that the formula proposed by Kabeyasawa and Ogata (1984)and Aoyama (1987, 1993) is in approximate agreement with the experimental results. Theexperimental study by Panagiotou and Restrepo (2007), Panagiotou et al. (2011) did notfocus on shear amplification per se, yet this phenomenon has been observed in their shaketable tests. Shear amplification was also observed by Tremblay, Leger and their co-workersin their shake-table testing program, but the large dynamic amplifications predicted by themany parametric studies listed in the bibliography and by Keintzel’s formula, or variants

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thereof, were questioned (Leger 2010). Yet, very recently the same research group (Luu etal. 2011) reported agreement with that formula (ωv = 2.2 experimental vs. 2.6 per Eq. 5) inone case.

On the whole it appears that the seismic performance of well-designed and constructedcantilever wall buildings has been satisfactory. Yet, damage to shear walls was reported afterseveral earthquakes. Blakeley et al. (1975) referred to reports on two cases of extensivediagonal tension cracking; however, it is not clear whether those or other such failures canbe attributed to dynamic shear amplification. Shear failures in walls were observed afterthe February 2010 Chile earthquake, but these were attributed mainly to discontinuities andsimilar effects such as stress concentrations near openings. Targeted studies are under way(Boroschek 2011).

The reader who chooses not to perform NLRHA with refined modelling—which providesthe best available response representation—may well ask which of the several formulasreviewed here he/she should adopt. In order to answer this question it is necessary to performextensive parametric studies. On the other hand it does not appear that the differences amongthe several variants of Keintzel’s formula (Eq. 5) are excessive. Yet, Eq. 7 (Priestley 2003b,c)extends that approach to the full height of the wall and avoids the ambiguity and conservatismnow present in evaluating ωv per EC8. Also, for large values of T1(T1 > 6Tc) it avoids theproblems inherent in using Tc in lieu of T2, and of ignoring the contribution of T3. Hence,its use should be encouraged. As noted, these design procedures do not account for multiplehinges when the vertical wall reinforcement follows approximately (in several steps) modalanalysis moment envelope, reducing shear demand up the wall. Similarly, parametricallyderived formulas could account for shear reduction resulting from upper level yielding onlyif in the analysis nonlinear wall elements were used above the wall base. Also, they mustbe compatible with the expected spectrum at the site. All these formulas do not considerwall shear cracking, as well as the inelastic shear deformations in the hinging region atthe wall base triggered by flexural yielding. Considering these effects could lower shearamplification, suggesting that Eq. 7 and even more so Eq. 5 are likely to yield safe side shearamplifications.

Acknowledgments The author is indebted to T. Postelnicu and D. Zamfirescu for explaining the relevantprovisions in the Romanian code P100-1/2006, to M. Panagiotou for providing peak shear plots for severalUS records and to K. Beyer for her comments on interconnected walls. Thanks are due to R. Adoni, A. Ainesand I. Elkayam for their help. Finally, the constructive criticism and the meticulous reading of the manuscriptby Reviewer #1 are gratefully acknowledged.

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