experimental seismic shear force amplification in scaled

ORIGINAL RESEARCH PAPER

Experimental seismic shear force amplification in scaledRC cantilever shear walls with base irregularities

Francisco J. Jimenez1 • Leonardo M. Massone1

Received: 10 October 2017 / Accepted: 28 February 2018 / Published online: 3 March 2018� Springer Science+Business Media B.V., part of Springer Nature 2018

Abstract RC structural slender walls under large seismic excitation are expected to reach

base moment capacity mainly affected by the first vibration mode. However, the base shear

could be affected by higher modes once yielding in flexure has occurred, which might

result in base shear underestimation in linear design. In this work, an experimental program

is carried out on five RC rectangular walls 1:10 scaled. All five specimens considered

irregularities at base, common in construction and one specimen did not consider shear

reinforcement or boundary detailing. Tests are carried on a unidirectional shaking table and

excitation is based on two Chile earthquake records with different intensities. Damage is

concentrated at the wall base for all specimens; primary due to flexure with some par-

ticipation of shear. For one of the records an average amplification of 1.3 is obtained, and a

decrease in height of the resultant equivalent lateral force closes to 0.4 hw. By increasing

the intensity of the input record, amplification grows to an average of 1.7, while it

decreases drastically when subjected to input records with low frequency content. No

significant difference is observed in shear amplification in specimens with a base central

opening, nor with setback, even though the cracking and failure mode was different for

such specimens. Ductility demand shows no correlation when two different earthquakes

are considered, whereas the frequency content and Arias intensity (Ia) of the input record

directly affected the shear amplification.

Keywords Seismic response � Shear wall � Base irregularities � Highermodes shear amplification � Experimental test

& Leonardo M. [email protected]

Francisco J. [email protected]

1 Department of Civil Engineering, University of Chile, Blanco Encalada 2002, Santiago, Chile

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Bull Earthquake Eng (2018) 16:4735–4760https://doi.org/10.1007/s10518-018-0343-7

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1 Introduction

On February 27, 2010 Chile was affected by an Mw 8.8 earthquake in the south-central

area. The damage observed in reinforced concrete (RC) structures was due to mainly base

moment and axial load, which resulted in concrete crushing and buckling of steel bars in

boundary elements of walls with poor detailing (Wallace et al. 2012). Although damage

was primary caused due to poor detailing, large axial loads and frequent irregularities also

affected the structural behaviour. A typical discontinuity in Chilean construction is the

result of the architectural requirements, where the lower floors (typically for parking) are

shorter in length than the upper floors (setback or flag-walls); and walls with openings,

primarily by gateways. The seismic event resulted in the modification of the reinforced

concrete design code, but the focus was on the flexural and axial damage, so that irreg-

ularities were not considered (Massone 2013). Furthermore, by providing a better detailing,

and therefore flexural ductility, the walls are more susceptible to brittle shear failure, if the

dynamic response is not well understood. Indeed, in the 22th February Christchurch

earthquake of 2011, some walls designed with modern code (post-1970s) were detailed for

flexural yielding, but failed in brittle shear compression (Kam et al. 2011). In the case of

flag-walls, there is little information in the literature about its general behaviour. Recent

work (Massone et al. 2017) has shown that the presence of such discontinuity affects the

damage distribution for slender cantilever walls given that plastic hinge tends to con-

centrate within the discontinuity region, which accelerates the damage process.

One of the phenomena associated with shear is the dynamic amplification. Shear

amplification takes place when base shear increases due to higher modes, especially when

the wall reaches the nonlinear range. This plasticization commonly develops at the wall

base, which is affected largely by the first mode, leaving higher modes nearly intact, and

increasing their participation after flexural yielding, causing a resultant equivalent lateral

force to be applied at a lower lever with a higher magnitude (base shear) since the

maximum base moment remains almost constant (see Fig. 1).

Shear amplification has been studied since the 1970s, when Blakeley et al. (1975)

proposed an amplification formulation for the New Zealand code (NZS 1982) that

increases the base shear with the number of floors. Derecho and Corley (1984) and others

(Keintzel 1984; Kabeyasawa and Ogata 1984) demonstrated that base shear increases with

ductility demand. Those mentioned author, as same as many researches, have reached

conclusions based on nonlinear time-history models using suites of records, whereas a

theoretical formulation was given by Eibl and Keinzel (1988), and Keintzel (1990), which

Fig. 1 Equivalent lateral forces and roof displacement in cantilever wall for the first two modes accordingto a flexural model with lumped masses

4736 Bull Earthquake Eng (2018) 16:4735–4760

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have been extended and modified by Pennucci et al. (2010) and Fischinger et al. (2010), as

summarized by Rutemberg (2013). Besides, dynamic shear amplification has been

observed in experimental tests as well. Panagiotou and Restrepo (2007) observed a

moment base over-strength, defined as the ratio of maximum measured base moment and

the design base moment, of 2.71; while shear base over-strength factor observed was 4.20,

yielding in an increase of 1.5 times for shear. Also, Ghorbanirenani et al. (2012) recorded a

shear amplification factor of 2.2 in their test. Both test programs have been reproduced

numerically (Martinelli and Filippou 2009; Luu et al. 2011). Recently Isakovic et al.

(2016), in a parametric non-linear study of coupled walls with one row of openings,

conclude that in this case it is conservative to use expressions for cantilever walls.

Different shear amplification factors are used in design codes. Amplification in the New

Zealand code depends on the number of floors; while Eurocode 8 (CEN 2004) formulation

requires moment resistance, period, ductility and response spectrum shape. On the other

hand, it is absent in the ACI 318-14 (ACI 2014), Chilean code and other current codes.

In order to investigate the impact of dynamic shear amplification an experimental

program that consists of five RC walls specimens at 1:10 scale is performed. Testing takes

place in a unidirectional shake table, in the structural dynamic laboratory at the University

of Chile. Specimens are constructed as cantilever elements, with masses lumped at five

levels.

2 Experimental program

2.1 Prototype design and scale factors

The prototype wall corresponds to a wall designed according to Chilean code (MINVU

2011; INN 2012) and ACI 318-14 code. It is located according to Chilean code in seismic

zone 3 and soil type C, which corresponds to a PGA = 0.4 g and a soil with shear velocity

between 350 and 500 m/s. The thickness is 40 cm, and the length (lw) 1.5 m is defined for

a minimum length to cross-section ratio of 3, which is similar to analytical studies by Rejec

et al. (2012) that include wall sections of lw = 2 m and bw = 30 cm with satisfactory shear

amplification results. Total height (hw) is 21.5 m (hw/lw * 14) distributed in 8 levels, with

a 2.7 m story height (Fig. 2a). The total mass is 100 ton, which considers a tributary area of

12.5 m2 per floor (common values range between 10 and 25 m2) and a total gravitational

load of 1 ton/m2. Nominal compressive concrete strength is fc0 = 25 MPa, resulting in an

axial force of 0.07Agfc0. Wall special boundary detailing consistent with ACI 318-14 were

fulfilled, providing enough transverse reinforcement at the boundary (Fig. 2b) as confining

length and reinforcement. Fundamental period, considering the uncracked section, is

1.12 s; higher than common 8-story buildings in Chile (Lagos et al. 2012), however

observable in other places. Incorporation of discontinuities considered: lengthening (set-

back) of 20% above the wall base—flag wall, which has been analytically studied by

Massone et al. (2017), a central opening at base with a short span of 10% of total length,

similar to the numerical research by Isakovic et al. (2016), and a central opening with a

span of 30% of total length.

The length scale factor used is k = 10 and the scale method used is the described by

Carvalho (1998) in which simultaneously fulfil the Froude and Cauchy laws for dynamic

analysis. On one hand, the Froude similitude is adequate for phenomena in which the

gravity forces are important. On the other hand, the Cauchy similitude is associated to

Bull Earthquake Eng (2018) 16:4735–4760 4737

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phenomena in which restoring forces are derived from stress–strain constitutive relation-

ships. Among other parameters to scale, there is a time scale kt =ffiffiffi

kp

= 3.16, while

acceleration, strain and stress are being equal to a factor of 1.

2.2 Design and properties of the test specimens

The relevant scale factor (10) requires careful selection of materials and dimensions for a

correct specimen characteristic scaling. One aspect is ductility, for which steel properties

need to be maintained (e.g., steel ratio, stress–strain curve), and are selected to closely

represent the prototype. In the case of concrete, the aggregate size is also relevant, and a

small size was selected. The selected concrete mix corresponds to a dry pre-mixed with a

nominal maximum aggregate size of 2.5 mm (prototype considers a 20 mm maximum

aggregate size). Water/cement (W/C) ratio of 0.26 is used, and a plasticizer is used in a

weight ratio of 0.78% to avoid concrete vibration. The average strength at the end of the

tests is 42 MPa (39.4–46.3 MPa). Longitudinal steel bars have a local steel classification

called A440-280H, which are bars of 6 mm diameter (not ribbed, but well anchored to

reach expected flexural strength) with a tested yield stress of fy = 387 MPa and initiation

of hardening at a strain 0.0043 and tested ultimate strength of fu = 556 MPa with a fracture

strain of 0.20. Steel for shear reinforcement bars and hoops is classified as AT56-50H with

a bar diameter of 4 mm, a tested yield stress of fy = 487 MPa and a tested ultimate strength

of fu = 685 MPa.

Final design of test walls (scaled) is presented in Fig. 3. All walls are 4 cm thick and

2.15 m high. Walls M1, M3, M4 and M5 are 15 cm long, while M2 is 15 cm at base (first

10 cm in height) and 18 cm long in the remaining height, i.e. 20% lengthened. The cross-

Fig. 2 Prototype dimensions. a Elevation and b detailing


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section aspect ratio is 0.27, which is considered a wall according to ACI 318 (2014). Wall

M3 has an opening at base of 10 cm in height and 1.5 cm in length (10% of total length),

and wall M4 has an opening of the same height and 4.5 cm in length (30% of total length).

Wall M5 is equal to wall M1, but without shear reinforcement and transversal boundary

reinforcement. All walls have two longitudinal bars at each boundary, that are bonded in

height and anchored within the foundation by a 90� hook extended 15 cm inside it. Also

two longitudinal debonded bars (greased and covered) are disposed at the inside side of the

boundary wall, which do not work in flexure, but help installing hoops for boundary

detailing (except with wall M5 which has only bonded bars at each end because hoops are

not provided). Transverse boundary reinforcement in walls M1, M2, M3 and M4 within the

first story corresponds to 4 mm diameter hoops spaced at 40 mm (* 6 times the longi-

tudinal bar diameter). The boundary detailing was placed mainly to prevent bar buckling

(Massone and Moroder 2009; Massone and Lopez 2014), which combined with a low axial

load (Wallace et al. 2008), are enough to provide displacement capacity required to

observed the dynamic response in the nonlinear range. The same rebar diameter is used for

shear reinforcement spaced at 80 mm (* d/2). Due to scale limitations, a single layer of

horizontal distributed reinforcement was placed.

Fig. 3 Wall and steel reinforcement detailing (dimensions in mm—section not to scale)


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Each wall has two holes placed horizontally and spaced at 10 cm throughout five levels,

where bars are placed to set metal plates that provide the required mass (inertial force) and

axial load. The foundation of each specimen is 20 cm height, 30 cm wide and 45 cm long,

these dimensions are considered to obtain minimum deformation and included 10

removable anchors attached to the base to prevent unwanted pedestal displacement or

lifting. Also, foundations are heavily reinforced to avoid unwanted cracking (/12 mm

spaced at 80 mm).

The design of the specimen tried to maintain most of the static and dynamic features

invariants for different specimens. Considering tested material properties, moment

capacity at the wall base is Mn = 3.48 kN m, base yield moment is My = 3.33 kN m and

base yield curvature is /y = 0.021 1/m, which are equal for all specimens under study.

Moment capacity is calculated according ACI318-14 code and is designed to be reached

while test is performed. Yield properties are calculated with a nonlinear fibre model,

defined with usual material properties (Kent and Park 1971; Mander et al. 1984). The shear

strength for walls M1 and M2 is 12.3 kN, for M3 and M4 is 10.4 kN, and for M5 is 6.1 kN

according to ACI318-14. The nominal fundamental and second mode periods are

T1 = 0.37 s and T2 = 0.06 s, for walls M1, M3, M4 and M5, whereas for wall M2 these are

T1 = 0.30 s and T2 = 0.05 s.

2.3 Construction, test set up and instrumentation

Concrete is placed in specimens entirely on the same day in two groups, first group

includes M2, M3 and M5; and the second group includes M1 and M4; making sure that the

composition and procedures are maintained. They were casted horizontally, because of the

difficulty of pouring concrete in a narrow element. Small size aggregate (maximum size

2.5 mm) and a plasticizer are used for the same reason. Prior to test, the specimens are

stored in a closed laboratory and cured for a few days. For the test setup, 50 steel plates are

disposed in order to add inertial and axial forces to the specimens during testing, resulting

in a total mass in each wall of 1017 kg, which is consistent with an axial compression of

0.07Agfc0 for nominal concrete strength (0.043Agfc

0 for actual material properties for M1).

A steel structure supports laterally the walls (Fig. 4b), preventing out of plane movement.

Instrumentation includes six unidirectional 24-bits accelerometers EPISensor ES-U2

placed at each level (including the base) aligned to the movement of the shake table to

capture inertial forces, which has an excellent match in similar test (Ghorbanirenani et al.

2012) with the force measured by means of a load cell. Acceleration is measured with a

sampling rate of 200 Hz and filtered with a Butterworth filter (order 4). Also, two linear

variable differential transformers (LVDT) are installed vertically (over an average length

of about 100 mm and separated 150 mm), one at each edge of the wall to measure average

curvature at wall base. In addition, the test setup includes a vertical accelerometer at each

end of the foundation, and an accelerometer measuring foundation movement, in order to

capture any possible unwanted base displacement or lifting. An additional accelerometer

also measures out-of-plan acceleration on the fourth floor. Instrumentation is schematically

shown in Fig. 4a. An example of actual test setup for specimen M1 is shown in Fig. 4b.

2.4 Input ground acceleration

Two input ground motion records are considered. One is a synthetic record based on

Constitucion record from the 2010 Chile earthquake. For this record, time is scaled at

kt = 3.16 to be consistent with the considered scale laws (Carvalho 1998). This time


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scaling results in a pronounced peak in acceleration spectrum (Sa) at T = 0.12 s, and

variation of natural period within different walls could result in different results. Thus, a

synthetic record is manipulated to smooth this behaviour following the method described

by Al Atik and Abrahamson (2010), preserving the nonstationary character of the ground

motion, in which the original time-history acceleration record and the desired acceleration

spectra corresponds to the input functions. Also, Llolleo record from the 1985 Chile

earthquake is used without any time scale (providing more energy to the first mode, which

would be consistent with a softer soil or stiffer structure), yielding an acceleration slightly

lower than the original record. Despite of this, small scatter within different runs was

obtained that resulted in good repeatability. Displacement spectra (Sd) and acceleration

spectra obtained from tests are shown in Fig. 5 (acceleration is scaled to compare input

with different PGA). The specimens are subjected to forcing actions, scaled in acceleration,

in the following sequence: synthetic constitucion (C) record scale to 10% (C010), 100%

(C100), 130% (C130), 150% (C150) and 200% (C200), and when possible to the Llolleo

record scaled to 100% (L100) and 150% (L150). The peak ground acceleration (PGA) and

Arias intensity (Ia ¼ p2g

R t

0a2g tð Þdt, with ag input acceleration) for the entire input record

values for each record are presented in Table 1.

Constitucion record was originally measured in a subduction earthquake, and it is rich in

high frequency content as is observed both in acceleration spectra (Fig. 5a, red lines) in

which high values are concentrated between T = 0.1 and 0.3 s and Fourier Transform of

ground acceleration (Fig. 5c, red lines), where energy is concentrated between 3 and

10 Hz. In the case of Llolleo record, the original ground motion was also obtained in a

subduction earthquake, but as it was not time-scaled, the energy provided for frequencies

of 4 Hz or higher is limited (Fig. 5c, blue lines). Also, time scaling that results in length

scaling for Constitucion record, produced reduced displacement demand compare to

Llolleo displacement demand (Fig. 5b). The acceleration–spectra for different tests vary

Fig. 4 Test setup. a General view and instrumentation and b specimen M1


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respect to the mean values in about 5% or less for most of the cases. Some of these

differences are observable in Fig. 5a, b (envelop and average responses are shown) and

they are typically due to limited performance of the shake table.

3 Test response

3.1 Failure patterns

Figure 6 shows the final stage (after the latest record) of all specimens. Specimens M3 and

M4 failed (severe damage that made impossible applying further records) after applying

the L150 record, whereas specimen M5 failed with C200 that resulted in a brittle failure. In

the case of wall M1 all records were applied without collapse, whereas records for M2

were only applied up to L100 without collapse, but the test was stop given a potential risk

Fig. 5 Actual input record. a Acceleration spectra, b displacement spectra and c Fourier transform ofground acceleration (*scaled in acceleration to compare with 100% scaled input acceleration)


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Table 1 Main tests characteristics

Wall/test PGA (g) Ia (g s) Droof (mm) D1 (mm) Vb (kN) Mb (kN m) T1 (s) T2 (s)

M1

C010 0.06 0.0031 2.6 0.4 0.26 0.3 0.56 0.09

C100 0.65 0.57 55 11.7 3.73 3.34 0.87 0.12

C130 0.84 0.94 60 10.2 3.87 2.93 0.91 0.13

C150 0.98 1.25 63 11 4.21 3.24 0.94 0.14

C200 1.13 2.07 76 12.4 4.97 3.32 0.99 0.13

L100 0.48 0.65 129 56.8 2.6 3.75 1.14 0.14

L150 0.62 1.35 200 104.5 3.1 3.8 1.41 0.14

M2

C010 0.06 0.0022 1.9 0.6 0.24 0.29 0.46 0.07

C100 0.74 0.62 45 19.8 3.17 3.29 0.85 0.09

C130 0.94 0.99 62 25.9 3.96 2.77 0.99 0.1

C150 1.03 1.3 83 29.1 3.44 3.09 1.06 0.1

C200 1.24 2.18 115 64.9 5.04 3.15 1.34 0.1

L100 0.47 0.65 209 124.1 2.36 3.63 1.67 0.11

M3

C010 0.05 0.0017 2 0.3 0.23 0.24 0.5 0.09

C100 0.72 0.57 54 8.7 4.29 3.48 0.82 0.12

C130 0.87 0.96 62 11.2 4.85 3.33 0.87 0.12

C150 1.23 1.23 63 15.2 5.1 3.29 0.9 0.13

C200 1.26 2.13 72 15.4 5.36 3.37 0.95 0.13

L100 0.47 0.67 131 61.9 2.88 4.09 1.08 0.14

L150a 1.09 0.99 182 115.1 3.98 5.7 1.2 0.14

M4

C010 0.01 0.0001 0.8 0.5 0.07 0.08 0.48 0.08

C100 0.62 0.5 54 11.2 3.79 3.26 0.87 0.12

C130 0.83 0.85 55 11.5 3.53 2.84 0.93 0.13

C150 0.95 1.11 58 12.1 4.08 3.24 0.95 0.13

C200 1.1 1.84 73 15.5 5.16 3.59 1 0.14

L100 0.51 0.62 119 45 2.48 3.73 1.26 0.13

L150a 0.8 0.61 191 84.4 4 4.97 1.2 0.14

M5

C010 0.05 0.0016 1.7 0.2 0.21 0.19 0.54 0.08

C100 0.74 0.62 53 9.5 3.55 3.27 0.8 0.11

C130 0.93 1.04 60 10.7 4.3 3 0.88 0.12

C150 1.07 1.33 58 10.9 4.79 2.93 0.92 0.12

C200a 1.1 1.07 61 61.3 4.17 2.86 0.96 0.13

aSpecimen collapse

Droof, D1, Vb, Mb correspond to peak values


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of collapse and the large roof displacement observed. M1 presented horizontal flexural

cracks; all of them of minor dimensions except for the one at the wall-foundation interface.

Wall M2 presents a similar behaviour as M1, except that the main crack changes to the

location at the upper part of the discontinuity region. Walls M3 and M4 presented a

horizontal crack at wall-foundation interface that resulted in flexural yielding, also both

developed a diagonal crack in the upper part of the opening and presented loss of concrete

cover at boundaries, near the wall base. For Wall M4, longitudinal rebar in the south

boundary slipped from concrete. Finally, wall M5 presented a brittle failure with a large

diagonal crack near the wall base.

3.2 Time-history (TH), peak and frequency–space response of selected walls

To provide details of the response observed in the specimens, a series of figures are

included that incorporate: (a) TH base shear, (b) TH base moment, (c) TH roof drift,

(d) TH spectrogram of base shear, (e) story acceleration at the time of maximum base

shear, moment and roof drift, and (f) spectral-ground acceleration measured with observed

experimental period. TH base shear and moment are obtained by means of inertial forces

derived from acceleration measurement at all levels that are provided with mass, while TH

roof drift from integrating measured acceleration. Spectrogram is calculated with a window

of 5 s, showing the period (T) in the vertical axis to compare with spectral acceleration

(darker colours indicate period prevalence). Profile accelerations for maximum values also

display two arrows that indicate resultant height of forces (Mb/Vb) for maximum base shear

and moment, and the influence of modes 1 and 2 for these peaks. Influence of each mode is

obtained by means of minimum squares of a combination with the linear–elastic mode

shapes. Spectral acceleration is generated with a linear elastic model with 5% damping.

Data from selected walls and records, such as base wall M1, set-back wall M2 and 30%

Fig. 6 Final state of test specimens at the base (M5b: Picture just before collapse)


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central-base opening M4, with C100, C200 and L100 records are shown (Figs. 7, 8, 9, 10,

11, 12, 13, 14, 15).

According to TH shear response, the peak base shear is observed at t & 10 s for walls

M1, M2 and M4 in test C100 and C200 (see Figs. 7a, 8a, 10a, 11a, 13a, 14a), and the

profile of acceleration at maximum Vb have similar shape, except for M2C100 (flag-wall,

Fig. 10e) in which the shape is in the opposite direction. In addition, walls have almost the

Fig. 7 M1 C100 response. TH (peaks values highlighted): a Vb, b Mb and c Droof; d Vb spectrogram,e acceleration profile at time of peak values and f Sa with measured natural periods (dashed line for M1C010values)



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same participation of the second mode of * 1.1 g�/2 and * 2.1 g�/2 for C100 and C200,

respectively (see Figs. 7e, 8e, 10e, 11e, 13e, 14e) at peak shear response. Peak value for

L100 tests is produced at t * 30 s (see Figs. 9a, 12a, 15a), with a modal participation in

acceleration profile of 0.29 g�/1 ? 0.21 g�/2 (mean value) for walls M1 and M2 (see

Figs. 9e, 12e), and 0.26 g�/1 ? 0.30 g�/2 in wall M4 (see Fig. 15e), where in all cases the

acceleration direction is the same. Base moment analysis shows similar values for walls

Fig. 9 M1 L100 response. TH (peaks values highlighted): a Vb, b Mb and c Droof; d Vb spectrogram,e acceleration profile at time of peak values and f Sa with measured natural periods (dashed line for M1C010values)



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M1, M2 and M4 in C100 test at almost the same time t * 7 s (see Figs. 7b, 10b, 13b), and

in addition, for walls M1 and M4 high base moment values are as well observed at

t * 21 s and the shape of acceleration profile are similar (see Figs. 7e, 13e), while wall

M2 neither presents this moment increase at t * 21 s nor the same profile pattern

(Fig. 10e). In C200 test, the maximum Mb is observed at t * 28 s in walls M1 and M4




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(see Figs. 8b, 14b), also low participation of mode 2 for both walls is reached (see Figs. 8e,

14e), whereas, for wall M2 base moment has an asymmetric behaviour and peak value is

reached at 8.7 s as is observed in Fig. 11b, with much higher participation of second mode

(see Fig. 11e). Finally, maximum Droof occurs at different times for walls M1, M2 and M4

in tests C100, C200 and L100, and high values are observed through the tests (see Figs. 7c,




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8c, 9c, 10c, 11c, 12c, 13c, 14c, 15c), without matching when the maximum moment Mb is

reached for most tests, except with M1C100, M1C200 and M4C200.

For modal analysis, a spectrogram and Sa with measured values are included. Periods of

darker colours in spectrograms, that indicates a prevalent response, are closely linked to

apparent measured period, as is appreciated when sub figures (d) and (f) for Figs. 7, 8, 9,

10, 11, 12, 13, 14 and 15 are compared. Also, the spectrogram response related to T1, in the

beginning of some of the tests, is located corresponding with T1 measured in C010 (i.e. in

the linear range), as is observable in the range from 5 to 7.5 s for C100 and from 10 to

12.5 s for L100 in the figures previously referred. This period change is associated with a

stiffness decrease, and is one of the main sources of shear amplification (other important

factors are over-strength and higher mode contribution by no-linear response) as there is a

large first mode shift, typically diminishing the acceleration spectrum value, while second

mode remains with little variation.

3.3 Fourier analysis: fundamental periods and energy

From the acceleration measures at each level, the apparent natural period during the test is

obtained. These periods are obtained by applying Fourier Transform to acceleration

response at each level. Also these frequency-space values are normalized by the Fourier

transform of the base acceleration to pick the apparent natural frequencies. The measures

are plotted for each test in Fig. 16a, b for modes 1 and 2, respectively. The elastic natural

periods in walls are T1 = 0.37 s and T2 = 0.06 s (T1 = 0.30 s, and T2 = 0.05 for wall M2)

according to the lineal model. However, for C010 that keeps the walls in the lineal range,

the periods observed are T1 = 0.53 ± 0.04 s (the ± indicates the maximum variation

between specimens) and T2 = 0.086 ± 0.003 s (0.46 and 0.071 for wall M2), resulting in

a * 40% higher value than the lineal model. In order to incorporate this stiffness decrease

in the lineal model, the effective flexural inertia is reduced homogeneously along the wall



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height to reach the periods observed in C010. The stiffness keeps decreasing in the fol-

lowing runs as intensity (and ductility) increases, especially for wall M2 due to the

development of an important crack in the setback that impacts the wall stiffness. However,

the lineal model is analysed according to the experimental response obtained for C010

only. In this context, natural fundamental period measured in C100–C200 (except for wall

M2) increases to 0.93 ± 0.13 s, i.e. 82% more than in C010, but between the same ground

motions T2 increases only to 0.13 ± 0.01, which is an increase of * 50% (except for wall

M2). This difference is more obvious when wall M2 or Llolleo records are included, and it

is consistent to considering damage (or stiffness decrease) concentrated at wall base. As

can be seen, for setback wall M2 first mode stiffness decreases 17% more than the other

walls for L100 (see Fig. 16a) due to concentration of damage at wall base, while mode two

stiffness decreases 16% less for M2 for the same input because of the additional moment

resistance in height (3 cm more in length above the discontinuity with the same longitu-

dinal reinforcement).

Fourier transform also allows to measure energy by mode using Rayleigh’s theorem.

This theorem refers to energy that could be measured by integrating a time function y(t) or

the Fourier transform of this time function Y(f), that is, energy = $-???|y(t)|2 dt = $-?

??|Y(f)|2

df. In this research, the input function (in the time–space) is the acceleration, and its

corresponding Fourier transform is integrated from 0 to 1.9 Hz to get the mode 1 energy

and from 5.0 to 12.6 Hz to get mode 2 energy (for tests C010 these values are 0–3.4 and

7.6–17.4 Hz, respectively). The ratio of energy in modes 1 and 2 to total energy are plotted

in Fig. 16c, d, respectively. These modes concentrate 83 ± 8% of the total energy (except

for test where the wall collapsed, reaching a minimum of 64%) and modal participation at

Fig. 16 Measured natural periods. a T1 and b T2 and energy in records for c mode 1 and d mode 2


123

Vmax is 87 ± 12% in tests. Besides, energy in mode 1 shows a constant decrease when

intensity increases for Constitucion record from 37 ± 7 to 14 ± 3% (test M5 with C200 is

not included because of collapse) of the total energy, but for Llolleo record, that is not rich

in high frequencies content, this value is much higher: 66 ± 2% for L100 and 55 ± 5% for

L150. On the other hand, the energy associated to mode 2 increases as intensity does for

constitucion record going from 51 ± 11 for C010 to 68 ± 2% for C200. Meanwhile, the

energy in Llolleo record related to mode 2 is only 14 ± 4% according to the nature of a

low frequency content signal. These experimental results are the first evidence of the

impact of higher modes in the structure, and therefore, of shear amplification. In addition,

modal energy is correlated to the intensity and frequency content of the earthquake,

although structural characteristics are not included at this point (e.g., irregularities).

3.4 Ductility demand

Maximum roof relative displacement (Droof) and peak relative displacement of the first

story (D1) for all tests are described in Table 1, and shown in Fig. 17a, b, respectively. For

C010 record, the roof displacement keeps under 2.6 mm = 0.0012 hw, while in the first

story its value is 0.6 mm = 0.0014 h1. For tests C100–C200 the peak roof displacement

increases moderately, which means that there is a small increase in ductility demand,

despite the larger increment of earthquake intensity. These values are

54 ± 1 mm = 0.025 ± 0.001 hw (45 mm = 0.021 hw for wall M2) for C100 and

67 ± 8 mm = 0.032 ± 0.004 hw (115 mm = 0.053 hw for wall M2) for C200. Also, when

focused on the first story, the peak response is 0.024 ± 0.004 h1 in C100 and

0.032 ± 0.04 h1 in C200, similar to roof value ratios, even though peak displacements are

not simultaneous in time. The similar magnitude indicates that most deformations con-

centrate within the first story (plastic hinge). On the other hand, L100 and L150 present

larger displacement demands whose values are 129 ± 6 mm = 0.058 ± 0.003 hw(209 mm = 0.055 hw for M2) for L100 record and 191 ± 9 mm = 0.089 ± 0.004 hw for

L150. These measurements are consistent with the ductility demand that is shown in the

displacement spectrum (Fig. 5b). In addition, first floor displacement results for Llolleo

records are 53 ± 9 mm = 0.126 ± 0.020 h1 (124.1 mm = 0.292 h1 for M2) for L100 and

99.8 ± 15.4 mm = 0.234 ± 0.036 h1 for L150.

Fig. 17 Wall relative maximum lateral deformation—a roof displacement and b 1st story displacement


123

3.5 Flexural demand

The peak base moment for the applied records C100–C200 is maintained in a range of

3.17 ± 0.32 kN m (Fig. 18a), which is close to the moment capacity of Mn = 3.48 kN m

and yield moment of My = 3.33 kN m. This result indicates that a plastic hinge has been

formed, and this flexural limitation produces a Mb peak between 5 and 10 s for C100 (see

Figs. 7b, 10b, 13b) and between 25 and 30 s for C200 correlated to larger Droof (Figs. 8b,

c, 14b, c), except for M2C200, which is at 8.7 s, influenced by second mode participation

with high frequency content and large reduction of Sa (T1) (Fig. 11b, d, e, f). The larger

displacement (ductility) demand of L100 and L150 records results in an increase of the

base moment of about 15% in average (M3 with L150 and M4 with L150 are excluded),

which can be attributed to material (steel) hardening (Fig. 18a) and no main differences in

modal participation are observed between setback wall M2 and base wall M1 (Figs. 9e,

12e).

Higher modes have an important participation in the moment distribution over the

height in walls. In Fig. 18b the envelopes of the bending moment distribution grouped in

C100–C200 and L100–L150 records for all walls are plotted. For Llolleo records, which do

not have important high frequency content, the envelopes are similar among them and have

the typical pattern that is consistent with an equivalent horizontal force profile as an

inverted triangle with a smooth and regular decrease of moment in upper levels. In con-

trast, for Constitucion records, bending moment envelopes have an inflexion point in level

2 that produces an average bending moment in level 3 of 0.55 Mb (Mb being the base

moment) for all constitucion record envelopes, while this value is 0.30 Mb for Llolleo

records. This flexural amplification in height is considered in some design codes such as

Eurocode (CEN 2004) and Canadian Code (NRCC 2005), and according to the results do

not depend on base openings as is observed in Figs. 13e, 14e, 19e and 20e, where both

height of resultant forces (Mb/Vb) and modal participation for walls M1 and M4 (with a

central opening) are almost the same at maximum Mb, while influence of setbacks is

understood as a stiffness matter.

As is observed in the moment-roof displacement hysteresis of M1 for C200 test in

Fig. 19a, the bending moment is well correlated to top relative displacement. Such

behaviour means that the base moment demand is linked to mode 1 response. This happens

because acceleration in the upper part of the wall (top stories) controls the moment

resultant at wall base (larger moment arm) and larger accelerations are observed in upper

Fig. 18 Maximum moment—a base moment and b vertical moment distribution


123

stories in the first mode. Moreover, wide loops in the hysteresis are explained by inelastic

behaviour. In the case of shear (Fig. 19b), there is no direct correlation to roof displace-

ment, since in this case acceleration in lower levels are relevant for base shear and are not

controlled by the first mode, but by higher modes.

3.6 Shear demand

Unlike maximum base moment, which is maintained near a fixed value (yielding), peak

base shear keeps growing for records C010–C200, whereas for records L100 and L150 it

decreases (see Fig. 20a). Maximum base shear for Constitucion record increases from

3.76 ± 0.56 in C100 to 5.17 ± 0.20 kN in C200 (M5 for C200 not considered because the

wall collapsed). All these peak shear values for constitucion have important scatter, which

could be related to the irregularities. To clarify this dispersion, it is considered both

maximum Vb (1 peak, Vb being the base shear) and the average of the largest 10 peak

values for base shear, in order to observe whether trends are maintained and are not just

isolated peak points. The results of the average of 10 peaks base shear are plotted in

Fig. 20b where values have less scatter and with similar values independent of irregu-

larities, proving that the sustained increment of base shear for records C010–C200 is

related to the dynamics of the problem.

Fig. 19 Overall response of M1 for C200—a base moment versus roof displacement, and b base shearversus roof displacement

Fig. 20 Maximum base shear—a peak shear value, and b average of 10 largest peaks


123

Maximum Vb is located between 8.8 and 11 s for constitucion record in all walls, i.e. in

the range with larger high frequencies influence as is observable in Figs. 7a, d, 8a, d, 10a,

d, 11a, d, 13a, d and 14a, d no matter the type of discontinuities. Flag-wall type (M2)

present slightly different results of base shear due to stiffness differences, since experi-

mental T2 is slightly lower (Fig. 16b) and Sa is sensitive to this parameter (see Fig. 5a), but

not because of higher mode participation as is explained later. Moreover, base shear

notably deceases for Llolleo records since the records are not rich in high frequency

contents according to both shear estimate criteria (1 and 10 peaks).

In tests where constitucion records were considered, base shear versus roof displace-

ment hysteresis present an erratic behaviour (see Fig. 19b for wall M1 for C200) as a result

of higher modes participation, which increases as intensity does. Furthermore, other

authors (Panagiotou and Restrepo 2007; Ghorbanirenani et al. 2012) have observed weak

or definitely no correlation between base shear and displacement when higher modes have

an important participation in shear response. Figure 19 explains this phenomenon: when

higher modes are participating, base shear could reach a peak value, while roof dis-

placement is underneath its peak (Fig. 19b), whereas for base moment, peak moment tends

to be correlated to peak roof value (Fig. 19a). Besides, when looking into the envelope

shear distribution over the height of wall M1 (Fig. 21a), there is an important increase of

shear in the lower levels for the records C100–C200, expected when a higher mode is

active, and is not observable for other records (Llolleo) where shear increases modestly,

and shear amplification is not highly expected due to the energy content. In this last case,

the shear distribution is more consistent with an equivalent horizontal force profile from an

inverted triangle as it has been seen before (Priestley 2003).

The location of the resultant of the equivalent horizontal force profile allows under-

standing the impact of higher modes. Thus, the base moment to base shear is determined and

normalized by the wall height in Fig. 21b when the peak base shear is being reached. In the

case of C010 it reaches 0.46 ± 0.06 of the total height (hw). For records C100–C200 there is

a progressive increase in the maximum base shear, which corresponds to a lower resultant

lateral force, starting at 0.38 ± 0.02 hw (0.31 hw for M2) and going as low as

0.18 ± 0.02 hw (0.12 hw for M4), with no important difference among walls with irregu-

larities even for setbackwallM2 (that have a lower peakVb). Also, influence of higher modes

at maximum Vb is almost the same (see Figs. 7e, 8e, 10e, 11e, 13e, 14e) despite of walls

differences.While for L100 record, height of resultant force reaches 0.61 ± 0.07 hw, similar

Fig. 21 Force distribution—a shear envelope distribution over the height for wall M1, and b force resultantheight versus base shear for all walls when reaching the maximum base shear


123

to consider an equivalent horizontal force profile as an inverted triangle. This result is due to

low influence of higher modes, and is not influenced by setbacks or central openings, even if

modal participation differs slightly among walls (Figs. 9e, 12e, 15e). Equivalent forces

applied at low height, as seen between C100 and C200 indicates involvement of higher

modes, since the base of the wall has reached its moment capacity amplifying the base shear.

3.7 Shear amplification results

In order to determine the shear amplification factors, a linear elastic model was considered

for the normalization, instead of using C010, because measurement and shake table control

errors are relatively larger for the low intensity runs. The lineal model (see Fig. 1) is a

flexural model with lumped masses distributed in five stories that considers only transla-

tional degrees of freedom and a fix condition at the wall base. In addition, the stiffness is

reduced according to the first two natural period values measure in C010 (where the walls

were kept in the elastic range) for each wall. The stiffness considered are 0.42, 0.38, 0.46,

0.55 and 0.43EIg (product of elastic modulus and gross inertia) for walls M1, M2, M3, M4

and M5, respectively. On the other hand, the dynamic amplification factor (xV) is defined

as the ratio between the experimental base shear to base moment ratio (VbEXP/MbEXP) to

the same ratio determined for the linear models (VbL/MbL), capturing the amplification due

to flexural yielding, as xV ¼ VbEXP

MbEXP= VbL

MbL. This definition is not dependent on a specific code,

but compares experimental and lineal results.

As was analysed previously, experimental maximum base moment is maintained in a

bounded range near yielding. Then xV is the ratio of shear over-strength and flexural over-

strength. Furthermore, this amplification factor could use any lineal model. In this case, it

is used an equivalent lateral force (ELF) model defined as the design response of the

fundamental mode of the structure with 100% of the mass, as early done by other authors

(Eibl and Keinzel 1988), and a modal spectral analysis (MSA) with the square-root-sum-

of-squares modal combination and a damping ratio of 5%, as many design codes use

nowadays. Both lineal analyses use actual input (particular measured record) for spectrum

and these amplification factors are identified as xV and xV*, for ELF or MSA, respectively.

Relevant differences are expected between both approaches, and are tabulated in Table 2.

In the case of normalization by MSA, the lineal model already includes the combination of

several modes in the linear range. Thus, the shear amplification indicates how the shear

increases when going from the lineal to the nonlinear range. In the case of normalization by

ELF, only the first mode is relevant, which implies that the shear amplification includes

two effects: (a) mode contribution or combination and (b) nonlinear response. Thus, ELF

normalization would result in larger values for the dynamic shear amplification, which

means that typical xV values yield more conservative results if MSA procedure is per-

formed for design as Rejec et al. (2012) reported.

The dynamic shear amplification reached in all tests is shown in Fig. 22. As was

expected since the specimens were maintained in the linear range, amplification for record

C010 is close to 1 (i.e. not amplified) in MSA, while a factor near to 1.5 was reached for

ELF due to the limited influence of higher modes in the model. Amplification gradually

increases from C100 to C200, presenting mean amplification of 1.30 in C100 and 1.69 in

C200, for results normalized by MSA. L100 and L150 present amplification with values

close to 1.10 and 1.28, respectively. In despite of ductility demand increment, shear

amplification decreases compared to constitucion record. This is explained because of

among the main factors that impact the shear magnification (over-strength, fundamental


123

Table 2 Maximum base shear according to experimental response and linear analysis

Wall/test EXP ELFb MSAb

Vb (kN) Mb (kN m) Vb (kN) Mb (kN m) Vb (kN) Mb (kN m)

M1

C010 0.3 0.3 0.2 0.4 0.3 0.3

C100 3.7 3.3 3.8 6.4 3.9 4.6

C130 3.9 2.9 5.0 8.3 5.6 6.1

C150 4.2 3.2 5.8 9.7 6.5 7.1

C200 5.0 3.3 7.8 13.0 8.3 9.4

L100 2.6 3.7 9.8 16.3 6.8 11.2

L150 3.1 3.8 13.8 23.0 9.6 15.8

M2

C010 0.2 0.3 0.2 0.4 0.3 0.3

C100 3.2 3.3 3.9 6.3 3.7 4.7

C130 4.0 2.8 5.1 8.3 4.9 6.2

C150 3.4 3.1 5.9 9.7 5.7 7.2

C200 5.0 3.1 7.9 12.9 8.0 9.6

L100 2.4 3.6 10.4 16.9 7.8 12.4

M3

C010 0.2 0.2 0.2 0.3 0.3 0.3

C100 4.3 3.5 3.7 6.0 4.1 4.5

C130 4.9 3.3 4.9 8.1 5.4 6.0

C150 5.1 3.3 5.8 9.6 5.9 7.0

C200 5.4 3.4 7.7 12.7 8.3 9.3

L100 2.9 4.1 9.2 15.2 6.8 10.8

L150a 4.0 5.7 12.9 21.3 9.9 15.2

M4

C010 0.1 0.1 0.1 0.1 0.1 0.1

C100 3.8 3.3 3.6 6.0 3.8 4.4

C130 3.5 2.8 4.8 7.9 5.0 5.8

C150 4.1 3.2 5.6 9.2 6.0 6.8

C200 5.2 3.6 7.5 12.4 7.8 9.1

L100 2.5 3.7 9.5 15.7 6.9 11.2

L150a 4.0 5.0 12.3 20.4 9.1 14.5

M5

C010 0.2 0.2 0.2 0.3 0.2 0.2

C100 3.5 3.3 4.0 6.7 4.1 4.8

C130 4.3 3.0 5.3 8.9 5.1 6.3

C150 4.8 2.9 6.1 10.2 6.5 7.4

C200a 4.2 2.9 5.8 9.8 7.2 7.3

aSpecimen collapsebLinear model considers spectrum of particular test and stiffness according to C010 measures


123

period shift and influence of higher modes), only the over-strength is completely repre-

sented, but as is observable in Fig. 9f neither the fundamental period shift (which use to

diminish spectrum demand in acceleration) nor second mode participation (Fig. 9e) have

an important participation in the shear response.

In order to compare shear amplification factor for different parameters, Fig. 23a shows

the amplification versus the top displacement. Although there is correlation between those

variables for a specific record, they do not correlate for different records (Constitucion and

Llolleo records are grouped by a dotted line). Wall M2 presents a different behaviour due

to its different period. The differences between walls, however, are less relevant than the

impact of increasing the record intensity. In the other hand, a good correlation between the

amplification and Arias intensity (Ia) is observed in Fig. 23b.

For this study, the experimental shear amplification results are compared with the

proposals of two previous works: Blakeley et al. (1975), included in the New Zealand code

in 1982, and Eibl and Keinzel (1988), included in the European code in 1993. The values

obtained from these formulations can be compared directly with the experimental results,

since the amplification factor has no scale.

Blakeley et al. (1975) proposes,

xV ¼ 1:3þ n=30� 1:8 if n[ 6 ð1Þ

where n = 8 (number of stories), then xV = 1.57. In particular, this study compares the

results of the nonlinear model with the shear design from a static model, including the

effect of nonlinear response and modal combination.

Fig. 22 Shear dynamic amplification—a based on ELF, and b MSA

Fig. 23 Dynamic shear amplification versus—a roof displacement and b Arias intensity


123

In the case of Eibl and Keinzel (1988), their proposal expression is,

xV ¼ Kc

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

My

KMI

� �2

þ 0:1max Sad

Sa T1ð Þ

� �2s

�K ð2Þ

where K is the response modification factor, considered as K = MME,1/My with MME,1 the

moment for the first mode response of MSA with 100% of the translational mass, MI is the

design moment at the base (MI = MME,1/K for this case), My = 3.3 kN m is the yield

moment, T1 the fundamental period and max Sad the acceleration on the spectrum plateau,

which for this study is considered as max Sad = Sa(T2). The periods T1 and T2 are the

measured values obtained during C010 record of each wall. c = 1 according to the authors

recommendations.

Figure 24 shows the comparison between the observed dynamic shear amplification

(xv, based on ELF for consistency with the analytical expressions) in the tests and the

predictions by Blakeley et al. (1975) and Eibl and Keinzel (1988). The simplified for-

mulation offered by Blakeley et al. (1975) mainly includes the effect of higher modes

observed as modal combination observed in taller structures, avoiding some of the most

representative parameters such as over-strength and the contribution of higher modes in the

nonlinear response, yielding an unconservative prediction in several cases. On the other

hand, the proposal by Eibl and Keinzel (1988) provides a better correlation with the

experimental values predicting, by means of a simple formulation of the dynamic theory

that incorporates two characteristic modes, in average (linear trend) a conservative esti-

mation of shear dynamic amplification. Other current formulations, such as the one by

Priestley (2003) would provide similar results, but incorporates a larger number of modes.

4 Conclusions

A series of 5 1:10 scaled cantilever walls with different base irregularities were tested in a

shake table. The design and detailing of specimens followed common design practice as

ACI-318. As for the input, two signals were used, one based on constitucion 2010 record

scaled in time properly, and another record, Llolleo 1985 not scaled in time, both with a

good repeatability.

y = 0.74xR² = 0.35

1

1.5

2

2.5

3

3.5

1 1.5 2 2.5 3 3.5 4ω

vexpe

rimen

t

ωvmodel

Eibl and KeintzelBlakeley et al.

Fig. 24 Comparison of dynamicshear amplification


123

Final damage pattern for each wall was analysed, obtaining different failure modes in

each specimen. Base wall M1 developed horizontal cracks mainly at the wall base, flag-

wall type wall M2 accumulated damage in the upper part of the discontinuity as a hori-

zontal crack, walls M3 and M4, which have a central opening at the base, yielded in flexure

with an important horizontal crack at the wall-foundation interface, but also both devel-

oped a diagonal crack over the opening, and finally wall M5 that lacks of confinement and

shear reinforcement showed a brittle collapsed due to a large diagonal crack. In addition,

natural frequencies, modal energy, overturning moment, horizontal base shear, and relative

roof displacement were analysed, obtaining several evidences of higher modes participa-

tion, which resulted in base shear amplification, while base moment reached its capacity.

The dynamic amplification factor of shear was obtained by comparing two linear

models: Equivalent lateral forces (ELF) and modal spectral analysis (MSA). The first one

is useful for observation of mode combination and nonlinear effects, while the second one

only shows the nonlinear effect, which is more appropriate for comparison with current

code design. Mean amplification values for all walls according to MSA are * 1 (i.e. no

amplification) in C010 where walls behave elastic, 1.3 in C100 and 1.69 in C200, while

low amplification (* 1.1) was observed in Llolleo record that is not rich in high frequency

content. Results are consistent with a decrease of the height of the resultant equivalent

horizontal forces. Furthermore, these amplification values vary with the type of earthquake

(frequency content and intensity) and the ductility demand of the structure. The frequency

content and Arias intensity (Ia) of the record directly affect the amplification, but ductility

shows no direct correlation with amplification when looking into two different records.

Despite of different failure modes, no significant influence was observed in shear ampli-

fication. In the case of flag-wall type discontinuity, changing stiffness (or period) seems to

be more relevant regarding the shear amplification, which is captured by the linear model.

The experimental response is compared with two formulations from the literature used

in previous design codes: Blakeley et al. (1975) and Eibl and Keinzel (1988). The first one

achieves a poor representation of the experimental results and underestimates the results in

several cases, given that the expression is only related to the building height. Whereas, the

formulation by Eibl and Keinzel (1988), similar to other modern shear amplification

methods that include several of the parameters that affect the shear amplification, achieves

a reasonable correlation with the observed data.

Acknowledgements This study was financially supported by Chile’s National Commission on Scientificand Technological Research (CONICYT) under the project Fondecyt 2013, Regular Research FundingCompetition under Grant No. 1130219. The contribution by prof. Fabian Rojas with the synthetic recordgeneration is also acknowledged.

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