# Assessment of the seismic capacity of tall wall buildings using nonlinear finite element modeling

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## Abstract

Two existing RC shear wall buildings of 17 and 26 stories were analyzed using fully nonlinear finite element models, i.e., models that include nonlinear material behavior and geometric nonlinearities. The buildings are located in Santiago, Chile and are representative of Chilean residential buildings in the sense that they have a large number of shear walls. The buildings withstood undamaged the 2010 Chile earthquake even though they were subjected to demands much larger than the code-specified demand. The approach to model the RC shear walls was validated through comparisons with results experimentally obtained from cyclic static tests conducted on isolated wall specimens. Several pushover analyses were performed to assess the global response of the buildings under seismic actions and to evaluate the influence of several modeling issues. Response history analyses were performed considering a ground motion recorded in Santiago during the 2010 Chile earthquake. In general, results (in terms of both global and local response quantities) are consistent with results given by pushover analysis and with the empirically observed lack of damage, a consistency that was not found in a previous study that considered linearly elastic models. The tangential inter-story drift deformation was found to correlate much better with the lack of observable damage than the total inter-story drift deformation typically considered in practice. The analysis also revealed that foundation uplift is possible but does not seem to significantly influence the response. Other modeling issues that were found to deserve further research are the shear stiffness of the walls and the influence of the slabs.

## Keywords

Shear walls Shear wall buildings Shear wall models Seismic response 2010 Chile earthquake## 1 Introduction

The M_{w} 8.8 2010 Chile earthquake (one of the strongest ever recorded) caused significant damage to only 2% of the affected inventory of buildings having 9 or more stories (Massone et al. 2012). This observation is somewhat surprising because, as it will be shown later, spectral ordinates of several recorded ground motions are in many cases (Naeim et al. 2011) much larger than the design spectrum specified in the Chilean seismic design code NCh433 (INN 1996), which is the code the great majority of the building inventory in 2010 was designed for. Hence, the 2010 Chile earthquake should have caused damage to a significant amount of multistory buildings, but such scenario did not occur (Naeim et al. 2011).

It is clear then that the actual seismic capacity of typical Chilean RC shear wall buildings is (much) larger than the strength required by the seismic design code, but the reasons for such larger-than-expected capacity have not been thoroughly evaluated. While several studies analyzed the response of multistory buildings damaged by the 2010 Chile earthquake (Carpenter et al. 2011; Rojas et al. 2011; Wallace 2011; Bonelli et al. 2012; Hube et al. 2012; Telleen et al. 2012; Wallace et al. 2012; Song et al. 2012; Westenenk et al. 2013; Parra and Moehle 2014; Alarcon et al. 2015; Junemann et al. 2016), apparently only one study analyzed undamaged buildings (Junemann et al. 2015) and its sole finding is that the characteristics of damaged buildings are essentially not different from those of undamaged buildings. A research program was then initiated by the authors of this paper to obtain more insight into why undamaged buildings were indeed not damaged even though they were supposed to. It is emphasized that the objective of this research program is different from the objective of almost all post-earthquake studies, which focus on *damaged* buildings rather than on *undamaged* structures. In a strict sense the latter should be referred to as “buildings in which no damage was observed”, but the expression “undamaged buildings” is nevertheless used just for the sake of brevity.

In the case of RC structures an important question related to the analysis of *undamaged* buildings is whether a linearly elastic model is appropriate. On the one hand, it can be argued that the amount of inelastic deformations must be very low (if any at all) in undamaged structures, and linearly elastic models are also attractive because they do not require neither sophisticated modeling techniques nor large computational efforts. On the other hand, however, it can be argued that the stress–strain relationship of concrete is intrinsically nonlinear both in tension (i.e., cracking) and in compression (i.e., nonlinear behavior even at low deformation levels), and such nonlinearity might significantly affect the overall structural response even in the absence of observable damage. Since no information was found in the literature on this issue (as mentioned before the literature focuses on damaged buildings rather than on undamaged structures) it was decided to consider both types of models (i.e., linear and nonlinear).

In former studies (Ugalde and Lopez-Garcia 2017a, b), three actual and representative residential RC shear wall buildings of 5, 17 and 26 stories that withstood undamaged the 2010 Chile earthquake were analyzed using linearly elastic models. In a first series of analyses it was found that the elastic capacity/demand ratio (denoted *elastic overstrength* in Ugalde and Lopez-Garcia (2017a)) is large enough to explain the lack of damage in the 5-story building but not in the other two buildings. These findings were ratified by results obtained in a second series of analysis (response history analyses, Ugalde and Lopez-Garcia 2017b), which also revealed that foundation uplift might be the reason for the lack of damage in the 17- and 26-story buildings. However, since very little evidence of foundation uplift was observed after the 2010 Chile earthquake, the results reported in Ugalde and Lopez-Garcia (2017b), i.e., obtained considering linearly elastic models, were deemed somewhat inconclusive.

In this paper the seismic response of the same 17- and 26-story buildings analyzed in Ugalde and Lopez-Garcia (2017a, b) is re-evaluated considering a fully nonlinear finite element model, i.e., a model in which a nonlinear concrete material model is adopted and other nonlinear features (such as the nonlinear behavior of reinforcement bars and PΔ effects) are also accounted for. In doing so the specific objectives of this study are: (1) to realistically evaluate the strength of typical Chilean residential RC shear wall buildings; (2) to identify the most relevant modeling issues for the analysis of RC shear wall buildings that did not suffer observable damage; and (3) to provide insight into why most Chilean residential RC shear wall buildings did not suffer observable damage during the severe 2010 Chile earthquake. Full 3D finite element models of the buildings were developed in PERFORM-3D (CSI 2016a), a commercial computer program that was deemed to provide an adequate balance between modeling capabilities and computational cost. The approach to model RC shear walls is presented and validated through comparisons with experimental results. The building structures are described in detail. The structural models are subjected to a series of pushover analysis to evaluate the impact of several nonlinear modeling considerations and to assess the seismic capacity of the structures. Finally, response history analyses were performed considering a ground motion recorded close to the location of the buildings. Particular attention is paid to examination of foundation uplift.

## 2 Description of the building structures

Direction-dependent building properties

Building | Direction | Period (s) | Code reduction factor | Effective reduction factor | Design base shear coefficient | Wall density |
---|---|---|---|---|---|---|

R* | R | |||||

17 stories | Long | 0.43 | 7.48 | 7.48 | 0.061 | 0.026 |

Short | 1.11 | 9.46 | 3.88 | 0.050 | 0.024 | |

26 stories | Long | 1.49 | 9.89 | 3.68 | 0.050 | 0.025 |

Short | 1.73 | 10.13 | 3.17 | 0.050 | 0.024 |

Wall thickness varies from 150 to 170 mm in the superstructure, while thicker walls (up to 250 mm) are found at the basement levels, either for shear or for soil retention. Typical of Chilean RC wall buildings, all nonstructural partitions are light partitions of negligible stiffness. The buildings were designed in accordance with the Chilean code NCh433, which refers to ACI 318 provisions for concrete design. However, motivated by the good performance of buildings without confined boundary elements in past earthquakes, the special boundary requirements were explicitly excluded when Chile formally adopted the ACI-318 in 1993. Consequently, none of the RC walls in any of the selected buildings has confinement detailing. Typical detailing at wall boundaries is shown in Fig. 1c. It is important to mention that both buildings satisfy by a wide margin the inter-story drift requirement indicated in NCh433 (Ugalde and Lopez-Garcia 2017a).

_{eff}) shown in Table 1. The design spectra for these structures, both elastic and reduced by R

_{eff}, are presented in Fig. 2, along with the elastic response spectra of the ground motions recorded in the Greater Santiago Area on soil during the 2010 Chile earthquake (both components). Nominal fundamental periods along each direction are highlighted with vertical lines. As already reported by Naeim et al. (2011), it can be observed that spectral ordinates of the (reduced) design spectra are generally much smaller than those of the recorded ground motion, but even so no damage was reported. Further details of the buildings analyzed in this study can be found in Ugalde and Lopez-Garcia (2017a, b).

## 3 Shear wall modeling and validation

Building models were developed using the commercial software PERFORM-3D (CSI 2016a), a structural analysis tool oriented to Performance Based Design (PBD) and currently the most commonly used for this purpose by practitioners. The software offers two types of elements to model shear walls: the Shear Wall element and the General Wall element. Both are 4-node finite elements specifically formulated to model RC walls, in which separate layers account for vertical axial-bending interaction and in-plane horizontal concrete shear. Weak-axis shear and out-of-plane bending are elastically modeled. The General Wall element has one additional layer to model horizontal axial-bending interaction and two additional layers to model diagonal shear related to strut-and-tie action. The General Wall element is then more sophisticated, but is computationally much more demanding and its parameters are also much more difficult to calibrate. Therefore, the General Wall element is recommended only in cases where horizontal axial-bending interaction or diagonal shear are relevant (e.g., to model squat walls or horizontal wall segments). On the other hand, the Shear Wall element offers reasonable accuracy in most cases, making it the preferred element in most building projects. This element was adopted for the analysis described in this paper.

In the Shear Wall element the vertical axial-bending behavior is accounted for using fiber cross sections, i.e., the wall is manually meshed into several Shear Wall elements, each with a cross section composed of concrete and steel fibers. The latter are given in terms of a vertical steel ratio (Auto Size cross-section) or in terms of area and coordinates of each bar (Fixed Size cross-section). Alternatively, bar groups can be explicitly modeled using Steel Bar/Tie/Strut elements (Kolozvari et al. 2017). The latter approach is advantageous to model boundary steel of walls that are not confined, such as the ones in the buildings analyzed in this study.

In the models, the vertical mesh is such that there are two vertical elements per story at locations where nonlinear response is expected, namely the underground stories and a few stories above grade level (2 and 5 stories in the 17- and 26-story buildings, respectively). Horizontal meshing was intended to set square-like elements at these stories. One vertical element per story was considered sufficient at all other stories, as recommended by the software documentation (CSI 2016b).

- a.
Concrete elastic modulus defined per ACI 318-14 (ACI 2014) as E

_{c}= 4700 \(\sqrt {{\text{f}}_{\text{c}}^{ '} }\) (MPa units). - b.Concrete compression stress–strain relationship follows a YURLX envelope (trilinear with strength loss, Fig. 3) whose parameters are summarized in Table 2. The quantity \(\varepsilon_{cmax}\) is given by Eq. 1 (Paulay and Priestley 1992). Variables ρ
_{h}, f_{ym}and*ε*_{hm}are, respectively, the volumetric ratio, the yielding strength and the strain at maximum strength of the confining reinforcement. Confined concrete compressive strength (f_{cc}^{′}) was computed according to Mander et al. (1988).$$\varepsilon_{cmax} = 0.004 + 1.4\rho_{h} f_{yh} \varepsilon_{hm} /f_{cc}^{\prime}$$(1)Table 2Concrete compressive stress–strain parameters according to Lowes et al. (2016)

Concrete

FY

FU

FR/FU

DU(mm)

DL (mm)

DR (mm)

Unconfined

0.75 \(f_{c}^{\prime}\)

\(f_{c}^{\prime}\)

0.00

0.002

0.00202

0.01

Confined

0.75 \(f_{cc}^{\prime}\)

\(f_{cc}^{\prime}\)

0.20

0.004

0.00404

\(\varepsilon_{cmax}\)

- c.
Steel stress–strain relationship follows a YURLX envelope adjusted to measured data (Fig. 3).

- d.
Shear stiffness equal to 0.1 G

_{c}A_{cv}, where G_{c}= 0.4 E_{c}as recommended by ATC (2010) and A_{cv}is the gross wall area. - e.
- f.
Steel energy dissipation factor equal to 0.75 and steel unloading/reloading stiffness factor equal to 0.5.

- g.Regularization of unconfined concrete envelope by reducing the compression strain at residual stress to the value given by Eq. 2, where unconfined crushing energy G
_{fc}= 0.0876 kN/mm and L_{elem}is defined by the height of the element.$$DR = \varepsilon_{u} = DU - \frac{{f_{c}^{\prime} }}{{E_{c} }} + 2 \frac{{\left( {\frac{{G_{fc} }}{{L_{elem} }}} \right)}}{{f_{c}^{\prime} }}$$(2) - h.Regularization of confined concrete envelope by reducing the compression strain at residual stress to the value given by Eq. 3, where confined crushing energy G
_{fcc}is given by Eq. 4 (“cc” subindex refers to confined concrete)$$DR_{cc} = \varepsilon_{ucc} = DU_{cc} - \frac{{0.8f_{cc}^{\prime} }}{{E_{c} }} + \frac{5}{3}\frac{{\left( {\frac{{G_{fcc} }}{{L_{elem} }}} \right)}}{{f_{c}^{\prime} }}$$(3)$$G_{fcc} = 0.0876\frac{kN}{mm} < 0.4378\frac{kN}{mm} \left( {\frac{{f_{cc}^{\prime} }}{{f_{c}^{\prime} }} - 0.85} \right) < 0.2189\frac{kN}{mm}$$(4) - i.
Simplified model for steel buckling, where the steel tensile stress is reduced to its residual value when the strain at the residual stress of concrete (given by Eqs. 2 and 3) is reached.

Shear force–deformation relationship was modeled as elastic, perfectly plastic. Shear strength was computed with Eq. 5. Since most of the walls are slender with aspect ratios greater than 2.0 parameter α_{c} was set equal to 0.17 (MPa units). Equation 5 was adapted from Eq. 18.10.4.1 of ACI 318-14 but increased by 50% to approach ultimate conditions (Değer et al. 2015). Variables ρ_{t} and f_{y} are the volumetric steel ratio and the yielding stress of transverse steel, respectively.

Model parameters of specimens RW2 and TW2 and of the walls of the buildings

RW2 | TW2 | 17-story building | 26-story building | |
---|---|---|---|---|

Wall width t (mm) | 101.6 | 101.6 | 150 to 200 | 150 to 300 |

Concrete strength f | 43.5 | 41.7 | 26 | 32.5 |

Concrete elastic modulus E (MPa) | 31,174 | 30,541 | 24,116 | 26,962 |

Mander confined ratio K | 1.095 | 1.3 | 1 | 1 |

Shear modulus G (Mpa) | 12,470 | 1222 | 9646 | 10,785 |

Steel yielding stress f | 427.5 | 427.5 | 5250 | 5250 |

Steel ultimate stress f | 620.5 | 620.5 | 6840 | 6840 |

Steel ultimate strain DU (mm/mm) | 0.071 | 0.071 | 0.11 | 0.11 |

Steel loss strain DL (mm/mm) | 0.094 | 0.094 | 0.16 | 0.16 |

Longitudinal steel ratio ρ | 0.33 | 0.45 | 0.3 to 5.1 | 0.3 to 4.5 |

Transverse steel ratio ρ | 0.33 | 0.45 | 0.3 to 0.7 | 0.3 to 0.8 |

In general, the global response of the walls is reasonably well predicted by the models (Fig. 4 center). Notably, the strength of specimen RW2 is slightly overestimated at low drift ratios but accurately predicted at larger displacements. Further, the strength of specimen TW2 is also very well predicted by the model when the flange is in compression. However, when the flange is in tension the model overestimates the strength, by 24% at peak strength. Analytical models of specimens RW2 and TW2 developed in PERFORM-3D have already been presented in ATC-72 (ATC 2010), but they were calibrated differently and results are less accurate (e.g., in ATC-72 the strength of specimen TW2 is overestimated by 53% when the flange is in tension). Even so, such results are deemed reasonable in ATC-72 given the wall geometries, loading histories, and particularly the simplified material models implemented in PERFORM-3D. More sophisticated models of specimens RW2 and TW2 developed with research-oriented software (Orakcal and Wallace 2006) give results that are only slightly more accurate than those shown in Fig. 4 (e.g., the strength of specimen TW2 is overestimated by 19% when the flange is in tension). Larger differences, on the other hand, are observed in local response quantities, e.g., concrete strains at the base of the walls (Fig. 4, right) measured with Deformation Gage elements at the base nodes. While the models predict well the experimental measurements at the center of the web, strains at the boundaries are overestimated in tension (34% on average) and underestimated in compression (54% on average). Again, the largest differences are observed in specimen TW2 when the flange is in tension. It must be noticed that the models assume constant strains between the end nodes of the Deformation Gage elements (located in this case between the base and mid-height of the first story), while actual strain gages implemented in the tests have much shorter spans.

Despite the limitations described in the former paragraph and despite other limitations as well (i.e., the lack of shear-flexure interaction, Kolozvari and Wallace 2016), results shown in Fig. 4 indicate that the Shear Wall element of PERFORM-3D is accurate enough for the purposes of this study. It is important to note that the “PERFORM-3D + Shear Wall element” modeling approach is possibly the most used of the few options currently available to perform nonlinear response history analyses of full 3D models of RC shear wall buildings.

## 4 Model implementation

The buildings presented in Sect. 2 were modeled in PERFORM-3D using the previously described model for the shear walls. This approach required identification of the dimensions and steel reinforcement of all walls in order to define the Shear Wall compound sections (it is recalled that this element is composed of one layer for vertical axial-bending and another layer for shear). In actual buildings like the ones considered in this study, the number of combinations of thickness, longitudinal steel ratio and transverse steel ratio is very large, which lead to numerous different Shear Wall compound sections whose definition and assignation to the model is cumbersome. Thus, longitudinal (web only) and transverse steel ratios were rounded to the third decimal number, e.g., two different walls having longitudinal web steel ratios (ρ_{l} in Fig. 4) equal to 0.0028 and 0.0032 were both modeled with ρ_{l} = 0.003. Results given by a separate sensitivity analysis (not shown here for brevity) confirmed that the effects of such rounding are negligible. It is emphasized that only the web steel ratios (i.e., longitudinal and transverse) were rounded, areas of boundary steel were not. Even with this simplification, 53 different Shear Wall compound sections were required for the 17-story building, and 77 for the 26-story building. Once regularization of concrete material in compression was included, the number of Shear Wall compound sections increased to 62 and 91, respectively.

_{p}/2 from each end, where L

_{p}is the length assigned to the hinge (Fig. 6a). Concrete columns, whose behavior is affected by axial load variation, were modeled using elastic frame elements with plasticity distributed along a distance L

_{p}from each end, to which a fiber cross-section was assigned (Fig. 6b). For both beams and columns the length of the plastic hinge (L

_{p}) was computed according to Priestley and Park (1987). Slabs were modeled using linear elastic area elements whose moment of inertia was reduced by 75% to account for stiffness reduction due to concrete cracking (ACI 2014). Rigid diaphragm constraints were included to reduce the number of degrees of freedom and, consequently, to reduce the computational cost.

Gravity loads and mass properties were modeled as concentrated loads and lumped nodal masses, respectively, and their values were set based on tributary areas. Following recommendations for PBD (LATBSDC 2015), the gravity loads include 100% of dead load and 25% of live load. On the other hand, seismic weight includes only the mass from dead load, i.e. live load was excluded from mass calculation (LATBSDC 2015; ASCE 2017). The models include PΔ effect in walls and columns. In the models, some features of the actual floor layouts were slightly modified in order to simplify the structural models and computation of member capacities (for instance, walls having slightly separated parallel axes were assumed coaxial). Modified floor layouts are shown later in Fig. 10.

For concrete material, expected compressive strength was taken as 1.3 times the nominal strength (LATBSDC 2015) whereas the steel backbone curve was adjusted to match experimental test data of typical Chilean reinforcing bars available elsewhere (Alarcon 2013), which indicate yielding stresses 25% larger than the nominal value. These and other relevant material properties are listed in Table 4. Ground supports were modeled using elastic nonlinear compression-only springs (denoted Nonlinear Elastic Gap-Hook Bar elements in PERFORM-3D) in order to capture possible foundation uplift, as the shallow RC footings of the buildings were not designed to carry tension forces. Compression stiffness of these nonlinear springs was set equal to the foundation area times the modulus of subgrade reaction.

Modal damping was set equal to 2% for all modes. Besides, following recommendations given by the software documentation (CSI 2016c) a small amount of Rayleigh damping was provided in order to prevent undamped higher mode displacements. Thus Rayleigh damping was set equal to 0.1% at 0.2 T_{1} and at 1.5 T_{1} where T_{1} is the first (elastic) mode period. These period values are recommended by Deierlein et al. (2010). With this damping model total damping remains below 2.3% in all the computed modes, i.e. below the maximum value of 2.5% stablished by ASCE (2017).

## 5 Incremental static (pushover) analysis

_{max}) over the design base shear (V

_{d}), computed using the effective reduced spectrum of NCh433 (Eq. 6). The overstrength is a measure of the capacity of a structure beyond that required by code specifications, and thus it is a quantitative descriptor of the structural behavior that can be realistically expected under a given earthquake scenario.

Identification of modeling issues not included in the modified models

Modelling feature not included | Model ID | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

M1 | M2 | M3 | M4 | M5 | M6 | M7 | M8 | M9 | M10 | |

Nonlinear soil structure interaction | X | |||||||||

Soil flexibility | X | |||||||||

PΔ effect | X | |||||||||

Bar buckling | X | |||||||||

Nonlinearity in frame elements | X | |||||||||

Shear capacity | X | |||||||||

Effective shear stiffness | X | |||||||||

Material regularization | X | |||||||||

Slab elements | X |

Figure 7 shows that in its short direction the 17-story building reaches a maximum strength equal to approximately 15% of the seismic weight, leading to an overstrength value of 3.11. This value is close to the 3.5 value reported for a similar 15-story building in Chile (Restrepo et al. 2017). On the other hand, the response of the 17-story building in the long direction does not exhibit strength degradation, and maximum strength is reached at a very large lateral displacement. A sharp stiffness degradation, however, is observed at a strength level roughly equal to 40% of the seismic weight. This large strength value is then assumed equal to the maximum strength and is used to compute the overstrength (= 6.34), which turned out to be much larger than that in the short direction because of the significant difference between horizontal dimensions (and therefore between lateral stiffness as well). In the case of the 26-story building, whose short and long directions are quite similar to each other, the response in both directions reaches a maximum strength of approximately 10% of the seismic weight, leading to similar overstrength values (2.03 and 2.12, respectively). The trends observed in these results, and also in those corresponding to the Chilean building analyzed by Restrepo et al. (2017), are consistent with those reported in several previous studies in the sense that overstrength decreases with increasing height (Jain and Navin 1995) and with increasing fundamental period (Humar and Rahgozar 1996). The 17- and 26-story buildings reach their maximum strength at a roof drift ratio that varies between 0.6 and 1.0%, and at that point the tangent stiffness is substantially smaller than the initial stiffness.

It was found that slabs between closely spaced walls are subjected to relatively large bending moment demands, which in turn is indicative of outrigger action (further evidence will be shown later, when describing results given by a model in which slabs are not included). Upon examination it was found that the bending moment demands on slabs exceed the capacity only at a very few locations (at less than 1% of the slab elements in each model).

*undamaged*buildings, the analysis presented below focuses on the response up to maximum strength. The modified models and their corresponding modeling issues are:

Overstrength values for the reference model (M1) and the modified models (M2–M10)

Building | Direction | Overstrength (Ω) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

M1 | M2 | M3 | M4 | M5 | M6 | M7 | M8 | M9 | M10 | ||

17-story | Short | 3.11 | 3.66 | 3.58 | 3.22 | 3.11 | 3.22 | 3.12 | 3.23 | 3.31 | 1.84 |

Long | 6.34 | 6.69 | 6.31 | 6.53 | 6.35 | 7.06 | 6.68 | 5.27 | 6.48 | 2.57 | |

26-story | Short | 2.03 | 2.03 | 2.00 | 2.18 | 2.04 | 2.07 | 2.01 | 2.31 | 2.11 | 1.04 |

Long | 2.12 | 2.11 | 2.00 | 2.23 | 2.12 | 2.16 | 2.20 | 2.21 | 2.20 | 1.32 |

*M2*Nonlinear soil-structure interaction: in these models the elastic nonlinear, compression-only support springs are replaced by linearly elastic springs (denoted Support Spring elements in PERFORM-3D) that do take tensile forces. In the case of the 17-story building this model leads to a strength along the short direction that is 18% larger than that of the reference model. In the other cases, however, no significant changes are observed, most likely because of the large dimension of the 17-story building along its long direction and because of the podium of the 26-story building, which increases overturning stability and reduces possible foundation uplift.*M3*Soil flexibility: in these models the elastic nonlinear, compression-only support springs are replaced by fixed supports that restrain displacements and rotations in all directions. Differences with respect to the reference model are quite similar to those corresponding to model M2, i.e., there is an increase in the strength of the 17-story building (15%) along the short direction and very small variations in the other cases, suggesting that overturning stability reduces the relevance of soil-structure interaction.*M4*PΔ effect: in these models the nonlinear geometric transformations for small deformations, i.e., the PΔ effect, is omitted. Thus all the elements use linear geometric transformations, and nonlinear behavior is due only to material nonlinearity. The analyses show that PΔ effect slightly reduces the strength, up to 4% in the 17-story building, and up to 7% in the 26-story building (Fig. 5). The influence of PΔ effect is even less at displacement levels smaller than that at maximum strength, suggesting that geometric nonlinearity has a very modest effect in stiff buildings such as the ones considered in this study.*M5*Bar buckling: in these models the compression strength of steel was not reduced to account for bar buckling. Instead, the steel constitutive relationship is symmetric and based on rebar tensile tests. The pushover curves of these models are virtually identical to those of the reference models up to large displacements levels, indicating that bar buckling (at least in the context of the simplified approach used herein) has a negligible effect on the stiffness and strength of the buildings.*M6*Nonlinearity in frame elements: in these models the inelastic features in beams and columns (Fig. 6) were removed and these members were modeled using linearly elastic frame elements. This simplification turned out to be relevant only along the long direction of the 17-story building, in which case leads to a significant increase in stiffness at strength levels greater than 20% of the seismic weight. In the other three cases it is relevant only after maximum strength is attained. (it must be noticed that in these three latter cases the maximum strength is less than 20% of the seismic weight). The lack of relevance of inelastic deformations in beams and columns at relatively low strength levels was somehow expected because most of the lateral forces are carried by the walls.*M7*Shear capacity: in these models the elastic-perfectly plastic constitutive relationship for shear in walls was replaced by a linearly elastic relationship whose stiffness is equal to the initial (elastic) stiffness of the nonlinear relationship. In both buildings these models result in a slight increase of strength (around 5%) in the long direction but do not cause appreciable differences (with respect to the reference models) in the short direction. These observations suggest that the seismic response of the walls of the buildings under study is dominated by axial-flexure rather than by shear.*M8*Effective shear stiffness: in these models the shear stiffness is equal to 0.4 E_{c}A_{cv}(i.e., it is not reduced) rather than equal to just 10% of 0.4 E_{c}A_{cv}as in the reference models M1 (Sect. 3). The reduction of the shear stiffness adopted in the reference models is recommended by ATC (2010) on the basis that the shear stiffness computed assuming isotropic conditions greatly overestimates the actual stiffness of cracked RC walls. Similar recommendations for stiffness reductions can be found in the literature (Gerin and Adebar 2004; Powell 2007; Değer et al. 2015) but they are often omitted in practice. Results indicate that unreduced shear stiffness leads to a significant overestimation of global stiffness. It must be noted, however, that the actual shear stiffness most likely degrades progressively as cracks develop, evolving from a value similar to that based on the gross cross-section properties at low deformation levels to a relatively small (effective) value at maximum strength. In other words the actual pushover curve probably evolves progressively from that of model M8 to that of model M1 as lateral displacements increase. In any case these results clearly indicate that shear stiffness is definitely relevant even in buildings like the ones considered in this study whose walls are slender and the contribution of shear deformations is expected to be small, and even when the displacement demand is smaller than that at maximum strength (i.e., even when there is no significant incursion into the inelastic range).*M9*Material Regularization: in these models the compressive constitutive relationship of concrete was not regularized to account for mesh sensitivity (regularization is not common in practice). Results indicate that the maximum strength is overestimated by 2–6% when regularization is not included, while the lateral stiffness is not affected. Larger differences are observed in the descending branch of the pushover curves. This result was somehow expected because regularization affects only the descending branch of the compressive constitutive relationship of concrete.*M10*Slab elements: in these models the elastic slab elements were removed, leaving only a rigid diaphragm constraint to account for in-plane slab stiffness (slab self-weight was modeled as nodal loads). This is a common approach in practice because explicit modeling of slabs highly increases the computational cost. As it can be observed in Fig. 7 this modification has by far the largest impact on the pushover curves as stiffness and strength are drastically reduced when the slabs are removed. It is recalled that the flexural stiffness of the slabs in the M1 reference models is relatively very low to account for cracking (equal to just 25% of the gross cross-section stiffness), hence even so the slabs greatly influence the global response, most likely by coupling the walls to each other. Qualitatively, the relevance of the slabs in the type of buildings analyzed in this study (i.e., typical Chilean multistory residential buildings having a seismic force-resisting system made up exclusively of RC shear walls) has already been mentioned in Zhang et al. (2017) and in Restrepo et al. (2017). The analyses performed in this study provide a clear quantitative measure of such relevance, which turned out to be much larger than expected.

## 6 Response history analysis

Properties of the ground motion used in RHA

Station name | Santiago Centro |
---|---|

Soil type | II |

PGA North–South | 0.21 g |

PGA East–West | 0.31 g |

Record duration | 140 s |

Distance to 17-story building | 4.8 km |

Distance to 26-story building | 9.1 km |

## 7 Evaluation of the effect of stiffness degradation on the seismic performance

In the former section it was found that results given by RHA of fully nonlinear models are essentially consistent with the empirically observed lack of damage. When stiffness degradation (even at relatively low levels of roof drift ratio) is taken into account to determine the dynamic properties of the buildings, more qualitative insight into the reasons for the lack of observable damage can be obtained by comparing seismic demands and capacities in terms of Acceleration-Displacement Response Spectra (ADRS).

*S*

_{a}= pseudo-acceleration,

*V*= base shear, α

_{1}= first mode mass coefficient,

*S*

_{d}= spectral displacement,

*u*

_{roof}= lateral roof displacement, Γ

_{1}= first mode participation factor and ϕ

_{1,roof}= first mode ordinate at the roof level. The resulting capacity spectra in the short direction (critical, particularly in the 17-story building) are shown in Fig. 14 along with the same elastic response spectra already shown in Fig. 2 (recorded ground motions). The response spectrum of the short-direction component of the record used in RHA (Sect. 6) is highlighted in red color. Strictly speaking, the point at the intersection of the capacity spectrum and the response spectrum is not really the performance point because at such point the capacity spectrum is not strictly linear any more. However, such point is located well before the point at maximum strength of the capacity spectrum, which means that the level of ductility at the intersection point must be very low (if any at all). Therefore, the intersection point is assumed as a very good approximation of the actual performance point.

Consistent with results given by RHA, Fig. 14 clearly shows that at the performance point the demand imposed by the Santiago Centro record is less than the capacity (i.e., peak strength) of the buildings. Further, the same can be said about the demands imposed by the other records. At the performance point of the Santiago Centro record the effective period (i.e., the one based on secant stiffness) is equal to 1.85 s (17-story building) and 2.49 s (26-story building). These periods roughly coincide with the periods than can be inferred from the roof displacement responses (short direction) shown in Fig. 8. The effective periods are substantially larger than the nominal periods reported in Table 1 and shown in Fig. 2. The nominal periods are the ones that were considered in the actual design of the buildings (i.e., the ones with which the design seismic loads were determined), and were calculated following the standard practice in Chile. They were obtained from linearly elastic 3D models in which shear and bending stiffness of all members are based on gross cross-section properties, walls and slabs were modeled with shell elements, and diaphragm constraints were incorporated at each floor level. Such practice is believed to be conservative, and is widely adopted because of simplicity and also because effective stiffness properties are not considered in the local seismic design codes. Pseudo-acceleration ordinates at the *nominal* periods are equal to 0.29 g (17-story buildings) and 0.18 g (26-story building), but at the effective periods are significantly smaller, 0.11 g and 0.12 g, respectively. Hence, the significant amount of stiffness degradation even at a displacement level smaller than that at peak strength leads to significant period elongation, which in turn leads to demands that are substantially smaller than those at the nominal periods.

## 8 Concluding discussion

Two existing RC shear wall buildings of 17 and 26 stories representative of Chilean residential buildings were analyzed using fully nonlinear finite element models, i.e., models in which a nonlinear concrete material model is adopted and other nonlinear features (such as the nonlinear behavior of reinforcement bars and PΔ effects) are also accounted for. The buildings are located in Santiago, Chile and have a large number of walls to carry seismic and gravity loads. Although the ground accelerations recorded during the 2010 Chile Earthquake widely exceeded the code-specified design spectra, the buildings did not suffer observable damage, suggesting (at least in principle) that their response was essentially linear. However, in former studies that considered linear models of the buildings (Ugalde and Lopez-Garcia 2017a, b) the seismic demands were found larger than the capacity. Only when foundation uplift was accounted for using a nonlinear soil-structure interaction model, the seismic demands did not exceed the capacity. However, no evidence of foundation uplift was observed in the buildings analyzed in this study after the 2010 Chile earthquake, and very little evidence was observed elsewhere after this earthquake. This inconsistencies between analytical results and empirical evidence were the main motivation for the study described in this paper.

The buildings were modeled using the structural analysis computer program PERFORM-3D. Shear walls were modeled using Shear Wall and Steel Bar/Strut/Tie elements calibrated according to recommendations available in the literature. The modeling approach adopted in this study was validated through comparisons with results experimentally obtained from a cyclic static test conducted on isolated wall specimens. A good match was found in terms of force–displacement hysteretic cycles but strains were predicted with a lesser degree of accuracy.

Incremental static (pushover) analyses were performed first. The overstrength of the 17-story building was found roughly equal to 3 and 6 in the short and long directions, respectively, and the overstrength of the 26-story building was found roughly equal to 2 in both directions. In all cases the maximum strength is attained at a roof drift ratio that varies between 0.6 and 1.0%. The influence of soil-structure interaction was found relevant only in the short direction of the 17-story building. In this case the pushover curve of the model that accounts for nonlinear soil-structure interaction deviates from that of the model that considers fixed support conditions at a roof drift ratio roughly equal to 0.2%, and deviates from that of the model that considers linear soil-structure interaction at a roof drift ratio roughly equal to 0.4%. In all other cases the influence of soil-structure interaction was found negligible at roof drift ratios that are smaller than that at maximum strength.

Response History Analysis (RHA) was performed considering a ground motion recorded during the 2010 Chile earthquake in Santiago. In terms of global response, peak roof drift ratios were found in all cases less than 0.4%, well below the values at maximum strength obtained from pushover analysis. Also consistent with pushover analysis, the roof drift ratio response was found essentially insensitive to soil-structure interaction, even though the model that considers nonlinear soil-structure interaction indicates considerable uplift levels (up to 60 mm). In terms of local response, the *total* inter-story drift deformation was found to take considerable values at some walls, especially at the upper stories where the limits established by ASCE 41-13 and LATBSDC were in some cases exceeded. This is not consistent neither with the lack of observable damage nor with the expected location of peak demands on walls in multi-story buildings (typically the first few stories above the grade level). This seems to indicate that the *total* inter-story drift deformation is not really an appropriate parameter to evaluate the seismic performance of slender walls. Values of *tangential* inter-story drift deformation, on the other hand, were found to take values that are smaller than the established limits, which is indeed consistent with the lack of observable damage in the buildings analyzed in this paper. Further, the *tangential* inter-story drift deformation was indeed found to take its maximum value at the grade level. These results indicate that in fact the *tangential* inter-story drift deformation relates much better with seismic damage (or lack of) in slender walls, and therefore it should be considered for accurate assessments of the seismic demand on tall RC wall buildings. While shear demands were found smaller than the capacity, axial-flexural demands and strains at wall boundaries do take values slightly greater than that at onset of damage at a few locations. Such demands, however, seem to affect only the boundary of a few walls and are deemed unlikely to produce visible damage. Hence, in general it is concluded that results given by the nonlinear models considered in this study are consistent with the observed lack of damage, hence nonlinear concrete material models are indeed necessary for the analysis of RC shear wall buildings even in the absence of observable damage. It was also found that the strength of the buildings considered in this study is such that they might withstand earthquakes somewhat greater than the 2010 Chile earthquake without significant damage. It is noticed, though, that the models also reveal that foundation uplift is indeed possible, and even though uplift does not seem to significantly affect the seismic response it nevertheless deserves further study.

The pushover curves were converted into capacity spectra. The performance point consistent with the response to the 2010 Chile earthquake was assumed equal to the point at the intersection of the capacity spectrum and the response spectrum of the same recorded ground motion used in RHA (both plotted in ADRS format). The effective periods (i.e., based on secant stiffness) at the performance point turned out to be significantly longer than the nominal periods considered for design, and at such long periods it was found that the strength of the buildings (including overstrength) is in fact larger than the demand imposed by the 2010 Chile earthquake. Hence the main motivation for this study (i.e., lack of observable damage despite demands larger than capacities) turned out to be not really true at the effective periods consistent with the response to the 2010 Chile earthquake. Effective periods longer than the nominal were certainly expected, but not as long as those found in this study. Future research on this issue should obviously include data from building instrumentation to quantify this finding with greater precision.

Pushover analyses were also performed to analyze the influence of several modeling issues. It was found that PΔ effect, bar buckling, inelastic deformations in beams and columns, nonlinear constitutive law for shear (elastic perfectly plastic for this study) and regularization of the concrete material model do not seem to be relevant to the seismic analysis of *undamaged* RC shear wall buildings having a large number of walls such as the ones considered in this study. However, two modeling issues were indeed found to have a considerable influence on the analysis (even in the absence of damage) and are deemed worthy of further research. The first issue is the shear stiffness of the walls, which was found to be relevant even when the walls are slender and subjected to demands that are well below the full strength. The value of shear stiffness adopted in this study (10% of the gross cross-section stiffness) is the value recommended in the literature for the analysis of *damaged* structures. Further research is needed to accurately determine the actual effective shear stiffness at relatively low deformation levels. The second issue is the influence of the slabs. The response of the models in which the slabs are explicitly modeled (even with an effective flexural stiffness that is 25% of that of the gross cross-section) is vastly different from those in which the slabs are not explicitly modeled and only a rigid diaphragm constraint is imposed. The quantitative differences shown in this study are much larger than the differences qualitatively suggested in former studies. For the class of buildings analyzed in this study (i.e., lateral force-resisting system made up of a large number of RC shear walls having intricate C, L, and T shapes) further research on this issue was found essential.

## Notes

### Acknowledgements

The authors are grateful to VMB Structural Engineering (Santiago, Chile) for providing information on the buildings analyzed in this paper and valuable comments on the Chilean structural engineering practice. The first author’s doctoral studies were financially supported by the Chilean National Commission for Scientific and Technological Research (CONICYT) through the CONICYT-PCHA/Doctorado Nacional/2015-21151184 scholarship. Further support was provided by the National Research Center for Integrated Natural Disaster Management (CIGIDEN) CONICYT FONDAP 15110017. This support is gratefully acknowledged.

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