Nonlinear dynamic seismic analysis of a modern concrete core wall building in Los Angeles using the BTM‐shell methodology

This paper discusses the nonlinear dynamic seismic response of a 14‐story reinforced concrete core wall building designed with the prescriptive requirements of the 2019 California Building Code for a near‐fault site in Los Angeles, California, simulated with a detailed nonlinear analysis of the complete structure and subjected to an ensemble of three‐component ground motion input. The study follows up on a recently published work by the authors, which examined the response of the specific building to static lateral loads. The nonlinear model of the building was developed in the software FE‐Multiphys using a shell‐element implementation of the beam–truss model (BTM). The BTM has been extensively validated through prior studies for planar, flanged, and coupled walls under static and dynamic loading. The building model was subjected to an ensemble of ground motions scaled at the design and risk‐targeted maximum considered earthquake levels. The vertical ground motion component and the out‐of‐plane shear degradation of wall piers were included in the simulations. The analyses elucidate the profound effect of nonlinear flexure–shear interaction and of the multidirectional behavior of the flanged coupled core wall on the evolution and localization of damage and on residual deformations. The coupling behavior between the core wall, the slabs, and the columns is also analyzed, and the importance of modeling the nonlinear behavior of all slabs for the residual damage evaluation is demonstrated. The aim of this study is to improve the understanding of issues arising from prescriptive design requirements as well as of issues associated with limitations of standard nonlinear seismic analysis.


NOVELTY
• Nonlinear dynamic analysis of a modern RC coupled flanged core wall building designed according to the current California Building Code.• Explicit modeling of nonlinear flexure-shear interaction in a coupled flanged wall and its effect on damage and failure evolution.• Modeling and quantification of the effect of out-of-plane shear strength degradation of walls.
• Evaluation of wall shear stress limits of ACI code and LATBSDC PBSD guidelines.
• Evaluation of slab design.
• Modeling and quantification of complex interaction between wall piers, slabs, and columns.
• Understanding issues arising from prescriptive design requirements as well as issues associated with limitations of standard nonlinear seismic analysis.

INTRODUCTION
In a recently published study, 1 the authors developed an improved beam-truss model (BTM) for nonlinear static analysis of reinforced concrete (RC) core wall buildings.This enhanced BTM considers the nonlinear behavior of all structural elements, including post-tensioned floor slabs, and accounts for the salient stiffness and strength degradation mechanisms.The authors noted that the enhanced BTM overcomes several significant limitations of current nonlinear analysis practices, which are discussed in more detail in Mavros et al. 1 and summarized below.
The current nonlinear analysis practice primarily relies on four-node wall panel element models with a fiber crosssectional representation, 2 also known as fiber models (FMs).These models do not consider the nonlinear flexure-shear interaction (NLFSI) and, thus, fail to capture the walls′ stiffness and strength degradation mechanisms accurately.This shortcoming is critical, 1,3 as both code-prescriptive 4 and performance-based seismic design (PBSD) approaches for core wall buildings, 5,6 aim for wall configurations with minimal thickness.Consequently, the design shear stresses for structural walls tend to be high, approaching the allowable limit values. 7Furthermore, FM is incapable of precisely representing the damage localization (see Figure 1) exhibited by RC structural walls.Additionally, the accuracy of FM analysis results concerning maximum vertical (longitudinal) strain calculations is heavily influenced by mesh refinement.This limitation is apparent in PBSD guidelines, 5,6 where the selection of an ad hoc plastic hinge region length for modeling wall piers directly impacts the length over which maximum tensile strains and minimum compressive strains are expected to remain constant.FM, as specifically used in standard practice, assumes a linear out-of-plane behavior, neglecting to model local inelastic buckling and fracture due to low cycle fatigue of reinforcing steel, despite the existence of material models capable of doing so 8 while they do not also account for global out-of-plane buckling of plastic hinge regions.
In current PBSD practice, the nonlinear response of floor slabs is not modeled for spans exceeding 6 m.However, Mavros et al. 1 showed that due to the significant vertical elongation of the core walls, floor slabs could undergo substantial F I G U R E 1 Crack patterns and failure modes of experimentally tested ductile planar and barbell RC walls. 21RC, reinforced concrete.
inelastic deformations even for 9-m-long spans.Another limitation of existing PBSD practice is the disregard of the vertical component of ground motions.
The seismic behavior of ductile RC structural walls (RCSWs) is governed by a variety of mechanisms and damage patterns, such as flexural and shear cracking (Figure 1A), vertical bar buckling and rupture (Figure 1A), horizontal bar rupture, web diagonal tension failure or crushing (Figure 1B,C), vertical crushing of boundary elements (Figure 1D), and sliding shear (Figure 1E) as well as out-of-plane global inelastic buckling.These damage patterns have been observed in experimental testing 7,[9][10][11][12][13][14][15][16][17] and earthquake-damaged buildings. 18Numerous experimental studies have investigated the hysteretic seismic response and damage of RCSW over the last 40 years (e.g., [9][10][11]13,14,19,20 ). These studis have been primarily focused on rectangular and barbell-shaped walls under in-plane loading.There is a lack of experimental testing data for flanged and coupled core walls designed to develop shear stresses near the current limit values established in PBSD guidelines 1 and subjected to multidirectional loading.Multiaxial dynamic, large-scale experimental testing is the most thorough approach for understanding the actual behavior of such systems.Still, it entails several technical challenges that may render it cost-prohibitive.This necessitates the availability of accurate, reliable, and efficient computational seismic analysis methods.
3][24][25][26][27] While an in-depth discussion of this review falls outside the scope of this paper, it is crucial to highlight that none of the examined methods have been validated for all the damage and failure modes discussed earlier.Moreover, none of these methods have been applied to investigate the nonlinear dynamic response of contemporary core-wall buildings, such as those explored in this study.These buildings experience substantial inelastic deformations and story drifts, accompanied by significant vertical elongation of the walls, which impose considerable demands on floor slabs in conjunction with gravity and vertical inertia.
A method specifically tailored to the analysis of RCSW components and systems is the BTM, initially developed by Refs.28, 29 The BTM allows the efficient and accurate simulation of rectangular, flanged, and coupled walls and slabs while explicitly accounting for the effects of NLFSI and various sources of strength degradation.9][30][31][32] It has been used in simulations of complete multistory building systems under dynamic loads, 33 including a collapse simulation of the Alto Rio building. 34It is worth adding that the BTM was used by the winners of a recent blind prediction competition focused on two U-shaped walls tested under cyclic loads. 35his paper employs the BTM for the nonlinear dynamic analysis of a 14-story RC core wall building, designed according to the current California Building Code 36 and satisfying the minimum requirements of the ASCE 7−16 4 and ACI 318-14 37 for a near-fault site in Los Angeles.The response of the specific building to static lateral loads was considered in a recently published study by the authors. 1 The specific earlier study considered the nonlinear out-of-plane behavior in shear of walls in a simplified fashion, that is, without considering shear strength degradation.The present paper eliminates this simplification by introducing and calibrating a law to account for out-of-plane shear (OOPS) strength degradation of planar wall segments.
The nonlinear time history analysis (NLTHA) in this study utilizes a collection of triaxial ground motion records, which are scaled at both the Design and Maximum Considered Earthquake Response (MCE R ) levels.It is important to note that NLTHA is the only and most accurate method used in PBSD practice for design validation, and cannot be replaced by nonlinear static analysis (monotonic or cyclic) such as the one used in Mavros et al. 1 The present study provides quantitative specificity to several qualitative conclusions of the work in Mavros et al. 1 within the PBSD context, taking into account the effects of multiaxial ground motion on the nonlinear dynamic behavior.The simulation results indicate the critical effect of NLFSI on the evolution and localization of core wall damage and residual deformations.The complex multidirectional behavior of the core wall and its critical coupling behavior with the slabs and the columns are presented.The effect of the vertical component of ground motions and the out-of-plane wall shear degradation is quantified.The study provides important observations regarding the adequacy of prescriptive code requirements such as story force demands and offers recommendations for improving the design and nonlinear analysis procedures.Additionally, the paper presents parametric sensitivity studies on the effect of numerical viscous damping and mesh size.

OVERVIEW OF THE ANALYSIS SCHEME
The analyses presented herein employ the BTM, which relies on the representation of a RCSW component with horizontal, vertical, and inclined line elements, as shown in Figure 2A.The analyses are conducted in the finite element program Overview of the BTM and uniaxial constitutive material stress-strain relationships.BTM, beam-truss model. 1,30,38,39that includes an implementation of the BTM as a rectangular shell macroelement, termed BTM shell.Each BTM shell is an assemblage of six internal elements, that is, two horizontal, two vertical, and two inclined line elements, as shown in Figure 2B.The vertical and horizontal elements use a beam formulation with a sectional fiber law, having multiple fibers along the thickness to capture the nonlinear out-of-plane flexural resistance of the segment.The inclined truss elements (also called diagonal elements) represent the in-plane inclined compression field in concrete panels.

FE-MultiPhys
The reinforcing steel material is described by the uniaxial constitutive stress-strain relationships described in Kim and Koutromanos, 8 which can phenomenologically account for the effect of buckling via the unsupported bar length to bar diameter ratio parameter (L / d b ), and rupture via a damage parameter (D cr ), which quantifies the accumulated inelastic work under tensile stresses. 8Figure 2C shows the uniaxial stress-strain behavior of steel for the expected steel properties used in the analysis of the building, as described in the next section.The concrete material is represented through a uniaxial constitutive model formulated in Lu and Panagiotou. 3 For the diagonal truss elements, the concrete material accounts for the effect of transverse tension on the compressive resistance.Specifically, the compressive axial stresses for these elements are multiplied by a reduction coefficient, β, which is a decreasing function of the transverse tensile strain, ε n , in accordance with the law presented in Figure 2D.The softening branch of the concrete stress-strain law and the reduction law for transverse tension are regularized, as described in Panagiotou et al., 40 to prevent the occurrence of spurious mesh size effects associated with strain localization.The regularization relies on adjusting the constitutive laws based on the ratio of the actual element size over a constant reference length value.The reference length in the horizontal and vertical line elements is set equal to 600 and 450 mm for unconfined and confined concrete, respectively.In contrast, the diagonal elements use a reference length value of 800 mm. Figure 2D shows the regularized and unregularized stressstrain behavior for confined concrete adopted herein.It is worth noting that the regularized laws give a more abrupt strength degradation in the softening regime because the actual size of the elements in the building model exceeds the corresponding reference length.

BUILDING AND MODEL DESCRIPTION
The present study is focused on a hypothetical 14-story building located in Downtown Los Angeles and designed 41 in accordance with the pertinent minimum code requirements. 4,36,37The plan configuration of the building is presented in Figure 3A.The building was detailed for a design base shear coefficient C s = 0.13 in both principal directions.The concrete specified compressive strength,  ′  , is 48.3 MPa (7 ksi) for the walls and columns and 34.5 MPa (5 ksi) for the slabs.Grade 60, ASTM 615 reinforcing steel with a specified yield strength, f y , of 414 MPa (60 ksi) is used throughout.The floors consist of 203 mm (8 in.) thick post-tensioned RC flat slabs, having a square plan configuration with a dimension of 28 m (92 ft).All stories have a height of 3.66 m (12 ft), and the total building height H is 51.2 m (168 ft).The total seismic weight, W, of the building, consisting of the self-weight and 1.92 kPa (40 psf) of superimposed dead loads, is 93.5 MN (21,020 kip).The core wall consists of two C-shaped segments coupled along the X-direction with 1016 mm (40 in.) deep coupling beams.
The modeling scheme of the core wall considered three regions along the height of the building.The first region consisted of the bottom story, the second region was comprised of stories 2 through 5, and the third region included stories 6 through 14.The wall in the first two regions had a thickness of 610 mm and a longitudinal steel ratio, ρ l , of 1.54 and 1.49% for the wall piers parallel to the X-and Y-directions, respectively.The wall in the third region had a thickness of 508 mm and a longitudinal steel ratio ρ l = 1.4%.For all stories, the horizontal reinforcement ratio provided in the X-and Y-directions was ρ h = 0.37 and 0.26%, respectively.The reinforcing details of the wall sections at the bottom story are shown in Figure 3B.Special boundary elements, per the requirements of ACI 318-14, 37 were designed in the corners and edges of the wall section, as shown in Figure 3B.The confining volumetric ratio of the special boundary elements was 2.4% for the first region, 0.7% for the second region, and 0.9% for the third region.The square columns (600 by 600 mm) had ρ l = 1.8% and a confining reinforcement volumetric ratio of 1.74%.Each floor slab has a longitudinal steel ratio of 0.66% in both directions (0.33% at the top and 0.33% at the bottom) and is post-tensioned with 0.5 in.unbonded strands spaced at 304 mm (1 ft) in both directions, with an effective post-tensioning force of 120 kN.Finally, each coupling beam was provided with diagonal reinforcement (20 #11 diagonal bars in the bottom five stories and 20 #10 in the remaining stories) and was fully confined along its length.
An overview of the computational model for the building is depicted in Figure 3C.The mesh scheme of the core wall for a typical story, which is shown in Figure 3D, uses six different parts.Parts 1-3 represent confined regions at the corners or ends of the core wall, parts 4 and 5 represent the unconfined regions in the walls, and part 6 represents the coupling beams.The analyses use expected values for the material strength parameters.The expected concrete compressive strength was 63 MPa (9.1 ksi) for the core walls and columns and 45 MPa (6.5 ksi) for the floor slabs.The expected yield stress f y in the steel material law equals 478 MPa, the ultimate stress f u is 769 MPa, and the modulus of elasticity is E s = 200 GPa.At the confined regions of the core wall, the overlap of the horizontal and confinement reinforcement was modeled in the assigned horizontal steel properties.Table 1 summarizes the volumetric reinforcement ratios, confined strengths, and The equations of motion are solved through a Hilber-Hughes-Taylor (HHT) time-stepping scheme, 42 combined with a Newton-Raphson iterative strategy.Rayleigh proportional viscous damping, based on the initial stiffness of the structure, is used.The values of the mass-and stiffness-proportional coefficients are determined so that a viscous damping ratio of 0.5% is anchored at periods equal to 0.5 and 5 s.It should be noted that the value of the viscous damping ratio used is lower than the value of 2.5% recommended by the LATBSDC guidelines.This is for two reasons.First, large-scale shake table testing of RC and masonry wall buildings 43,44 has shown that the magnitude of viscous damping that best reproduces the test results by detailed nonlinear finite element analysis for highly inelastic response is of the order of 0.5%.Second, the value of 2.5% is meant to indirectly account for hysteretic structural energy dissipation, which is not explicitly incorporated in standard engineering practice.Since the computational model of the building in the present study explicitly accounts for all sources of hysteretic energy dissipation, a much lower value of the viscous damping ratio is justified.Additionally, previous studies have reported the overestimation of numerical viscous damping for nonlinear response history analysis of steel frames with initial stiffness Rayleigh damping formulation. 45For RC wall buildings, this spurious damping can be even larger due to the larger initial stiffness of RC walls compared to steel frames studied in Chopra and Mckenna. 45It is for this reason that software solutions used in PBSD practice such as PERFORM3D 46 reduce the stiffness contribution of concrete fibers area in the composition of the initial stiffness damping matrix by 85%.

MODELING OUT-OF-PLANE SHEAR STRENGTH DEGRADATION
An important capability that was not included in the original BTM implementation pertains to accounting for OOPS degradation in wall webs and flanges.This type of degradation has been determined to be the leading cause of the failure in a planar wall of the Grand Chancellor hotel 47 during the 2011 Christchurch earthquake.Such failure patterns have also been observed in experimental tests of walls with different sectional shapes under cyclic static or dynamic loading. 44,48,49he study by Alvarez et al. 41 using the BTM for simulations of core walls employed springs at the base of the wall model to capture this degradation.Recent work by Mavros et al. 1 used BTM shell elements that enforced an upper bound (cap) to the value of OOPS that can be carried by wall components.This upper bound had a constant value throughout the simulation and did not account for the degradation of OOPS resistance.To address the need for accurately capturing potential OOPS failures, the present study enhances the BTM shell element with an OOPS degradation law, calibrated to phenomenologically account for the impact of axial force and shear deformations on the corresponding shear capacity.The specific law is implemented in the shell element as a cap to the OOPS resistance, similar to the approach in Mavros et al. 1 A numerical parametric study was first conducted to enable the calibration of the OOPS strength degradation law in a BTM model.An assemblage of BTM shell elements was established to investigate the through-thickness shear resistance mechanism of a planar wall, as shown in Figure 4A.The model was subjected to monotonically increasing lateral deformations under a constant axial force until the occurrence of strength degradation.Two different cases of concrete material behavior were considered, one representing unconfined concrete and the other confined concrete.The reinforcing steel ratio in the vertical and horizontal (through-thickness) directions was equal to 1.49 and 0.13%, respectively.Analyses were conducted for different values of axial load ratio (ALR), that is, 0, −0.05, −0.10, −0.15, −0.20, −0.30, −0.40, and −0.80 (negative values of ALR implying compression).Figure 4B shows an example deformed mesh and failure mode for the case of unconfined concrete with an ALR of −0.15. Figure 5A,B presents the curves giving the OOPS force ratio (V out /l) versus the lateral drift ratio, γ, for the unconfined and confined models, respectively.For low values of ALR, the OOPS force-deformation behavior is relatively ductile.Increasing the ALR value leads to a decrease in the OOPS inelastic deformability.The following equation is established to provide the OOPS strength, V r , of a BTM shell element, as a function of the corresponding element deformational (i.e., after removal or rigid-body rotation effects) drift ratio, γ: where the V r0 is a reference, user-specified, shear strength value, and F is a dimensionless reduction factor that depends on γ.
The relation between F and γ is presented in Figure 6A.If γ is less than a value γ d , the value of F is equal to 1. Exceedance of the γ d value leads to a degradation of the F factor up to a residual value of 0.1, which is attained for a value of γ equal to 1.1γ d .The value of V r0 is calibrated using the results shown in Figure 5, resulting in values of V r0 /l equal to 3.28 and 1.55 MN/m for the confined and unconfined regions, respectively.The value of γ d is considered to be a function of the ALR, as shown in Figure 6B.Since the corner region of the core wall is being modeled with two orthogonal BTM shells that each have a tributary area of 50% of the corner area, the V r0 /l value assigned to these elements is 50% reduced (1.64 MN/m).
Although further investigation is necessary to validate the OOPS modeling approach presented in this paper, such validation is beyond the scope of the present study.However, we recognize that further validation should be based on experimental data, which is currently limited, and should also include more detailed and refined finite element models that accurately capture diagonal tension failures without suffering from shear-locking effects. 44,50,51

ELASTIC MODAL ANALYSIS
An elastic model is developed utilizing RC stiffness modifiers recommended by LATBSDC 5 to calculate key modal properties of the building under study.The concrete modulus (Ec = 57, 000 √  ′ ce ) is multiplied by a stiffness modifier equal to 0.35 for walls, 0.17 for coupling beams, 0.5 for slabs, and 0.7 for columns.The periods for the first mode shapes in translation in the Y-and X-directions, as well as rotation about the Z-axis, were determined to be 1.52, 1.45, and 1.23 s, respectively.Similarly, the corresponding values for the second mode shapes in each direction are calculated to be 0.32, 0.38, and 0.41 s, respectively.The lowest mode dominated by vertical oscillations of the slabs had a period of 0.41 s.The modal analysis relied on linearly elastic, quadrilateral shell elements rather than BTM shells.This decision is made since the use of BTM shells leads to an overestimation of stiffness as a result of the overlap of diagonal truss elements with vertical and horizontal beam elements.This feature renders BTM-based models rather unsuitable for calculating effective modal properties that correspond to MCE R response, where cracking-induced softening occurs over a significant portion of the structural components.

GROUND MOTION ENSEMBLE
A set of 11 triaxial ground motion records, sourced from seven earthquakes with moment magnitudes ranging between Mw 6.2 and Mw 7.6, was used in the dynamic analysis of the building model.The ground motion characteristics are summarized in Table 2.3][54] To this end, this study includes records of pulse-type near-fault ground motions, with the closest proximity to the rupture plane (R RUP ) ranging between 0 and 6.5 km.While common PBSD uses approximately half of the 11 ground motions as pulse-like near-fault and the other half as far-field ground motions to represent a Mw 8 earthquake scenario of the San Andreas fault, the authors decided to use a record set consisting exclusively of near-fault pulse-like motions, to examine an upper bound scenario.The selected ground motions are derived from earthquakes and fault ruptures with characteristics that closely resemble those of the seismic hazard in downtown Los Angeles.
The mean nonlinear seismic response is primarily controlled by the mean spectrum.To that end, the two horizontal acceleration time history components of each motion were modified, that is, scaled, using the continuous wavelet transformation as described in Montejo, 55 so that the mean of the maximum-direction (RotD100) spectrum matches the risk-targeted maximum considered earthquake (MCE R ) response spectrum for a site class D in downtown Los Angeles with spectrum parameters S MS = 1.97 g and S M1 = 1.19 g.The mean spectra of the unscaled and scaled ground motions exhibit, on average for the period range of 0-8 s, a deviation of only 17 and 3% from the target MCE R spectrum, respectively.This is particularly important because the scaling procedure involved minor amplitude and frequency scaling, resulting in scaled motions that preserve a large part of the original motions′ character.As shown in Figure 7A,B, the mean RotD100 pseudo-spectral acceleration (PSA) and spectral displacement (SD) of the eleven spectrally matched motions closely match the corresponding MCE R design spectra.The specific figures also provide the mean spectra in the two principal horizontal directions, X and Y.The peak ground velocity (PGV) of the unscaled and scaled motions ranged between 0.87 and 2.65 m/s and 0.82 and 2.30 m/s, respectively.The shape of the mean unscaled spectrum of the vertical component of the ground motion records closely resembles that of the MCE R design spectrum.As a result, no spectral matching was performed on the vertical component and only a scaling factor of 1.65 was applied to match the mean spectral acceleration with the

RESULTS
The distribution along the building height for five key response quantities, obtained from the analysis of the building model for the DE-and MCE R -level motions is given in Figure 8.The response quantities include the interstory drift ratio (IDR), residual IDR, floor acceleration, story shear force ratio (V/W), and story moment ratio (M/WH).Separate distributions are calculated for lateral loads and deformations along the X-and Y-axes of the building.The plots in the figure provide the mean quantities for the entire ground motion ensemble, as well as the envelopes (i.e., the maximum value of the quantity recorded among all analyses).The mean IDR for the DE reaches 1.04%, which is significantly lower than the prescriptive IDR limit in the CBC of 2%.The mean IDR for the MCE R does not exceed 1.79%, which is also lower than the 3% limit that is typically adopted in PBSD guidelines for the MCE R level.Despite the relatively low IDR values, the analysis gave significant damage and residual deformations for five ground motions at the MCE R level, as it will be described below.
At the DE and MCE R , the IDR envelope reached 1.57 and 2.84%, respectively.The mean residual IDR reached 0.11% for the DE and 0.30% for the MCE R .The corresponding values of the residual IDR envelope profiles were 0.24 and 0.96%.The MCE R 's maximum and residual IDRs occurred at the bottom levels of the building.This is due to the localization of damage and inelastic flexural and shear deformations in the plastic hinge region.The mean and envelope residual IDRs at the MCE R are less than the limits of 1 and 1.5%, respectively, recommended in the LATBSDC-20. 5The mean floor acceleration profiles correspond to values equal to the PGA for the bottom 80% of the building's height.For the upper 20% of the height, the total horizontal accelerations exceed PGA, with the maximum acceleration attained at the roof equal 1.43 and 1.48 times the PGA for the DE and MCE R , respectively.The design acceleration for diaphragms, C px , calculated using the Alternative Design Provisions of ASCE 7-16 and plotted in Figure 8C, closely matches the mean acceleration recorded in the analyses in the bottom 80% of the building and it systematically exceeds the peak floor accelerations at higher levels.However, C px values closely match the maximum accelerations recorded in the roof for the eleven motions.
The distributions of mean story shear forces were similar for the two principal directions of the building.The story shear obtained in the X-direction slightly exceeded that in the Y-direction, and the base shear ratio, V b /W (i.e., the ratio of the base shear over the seismic weight), reached values of 0.27 and 0.35 for the DE and MCE R , respectively.These story shear values greatly exceed the design base shear ratio, V u,b /W, which was equal to 0.13.This discrepancy is due to the fact that the ASCE 7-16 4 and ACI 318-14 37 requirements do not consider the effect of overstrength and higher modes of response on the base shear force.As deduced from Figure 8, using the shear design equations of ACI 318-19, 56 which account for the effects mentioned above, would give a much higher value of design base shear, equal to 1.28 times the value obtained for the DE ground motions and 0.98 times the value obtained for the MCE R ground motions.
The mean base story moment about the Y axis (i.e., for lateral deformations along the X-direction) was up to 11% greater than the corresponding value of moment about the X axis, with the maximum base moment ratio, M b /W, being equal 0.13 for the DE and 0.15 for the MCE R .These values are 1.68 and 1.94 times the design base moment ratio, M u,b /(WH), which was equal to 0.078.
The significant overstrength developed by the building system considered herein for the MCE R is also evident in Figure 9A,B, showing peak base shear and peak base moment ratios, respectively.The values are provided for each of the individual ground motions considered, for the X-and Y-directions, as well as for the direction at an angle of 45 o with respect to each of the two principal directions.The specific figures also include horizontal dashed lines, providing the mean value obtained for each of the three directions.Clearly, the analytically mean base shears and moments greatly exceeded the corresponding design values, which are also marked in the figures.
Nonlinear time history analyses have highlighted the role of the higher modes in the dynamic response of structural systems. 57Higher modes lead to an increase in shear forces, bending moments, and horizontal floor accelerations.A quantity that can provide insights into the higher mode effect is the instantaneous effective height, defined as the ratio of the base overturning moment divided by the concurrent base shear force at a given time during the analysis.Figure 10A plots the minimum height H e,min , for each ground motion and each direction, over all instances where the magnitude of the base overturning moment for the specific ground motion and the specific direction was at least 85% of the maximum overturning moment.Figure 10B plots the effective height, H e , corresponding to the instant of peak base shear force for each analysis.While the values in Figure 10B are much lower than those in Figure 10A, the latter figure is deemed more significant because plastic hinge regions of RCSW components are more vulnerable to combinations of large inelastic flexural deformations (dominated by displacement and hence base moment response) and large shear stresses.The mean effective height values for Figure 10A,B are equal to 0.44H and 0.29H, respectively.Both these values are significantly smaller than the effective height of the first mode of response or of the equivalent lateral force procedure in ASCE 7-16. 4 Figure 11 depicts the distribution of mean wall longitudinal (vertical) tensile and compressive strain profiles of the bottom three stories of the core wall for the DE and MCE R .The distributions are given for the middle of the thickness of the wall depth and the extreme (outer) fibers.For the DE, the mean tensile and compressive vertical strain profiles have a peak value equal to 0.37 and −0.16%, respectively, indicating that the core wall response was essentially elastic.For the MCE R intensity, the mean values of the tensile and compressive strains of the middle fiber are 0.84 and −0.97%, respectively, and occur at the base of the core wall.The corresponding strains of the extreme fibers are very large and equal to 2.69 and −3.41%, respectively, due to the effect of simultaneous out-of-plane flexure.Figure 12 shows the peak longitudinal wall tensile and compressive strains at the base story for the (a) middle and (b) extreme fiber computed from the responses to the eleven MCE R input ground motions.These strains imply concrete crushing in unconfined and confined regions for five ground motions.The peak strains localize at the base of the core wall and are much larger for the extreme fiber because of out-of-plane nonlinear behavior in bending moment and shear.As discussed below, the bottom corner BTM shell elements are the first to experience OOPS degradation for 45% of the MCE R ground motions.
Figure 13A depicts the peak roof drift ratios computed in the X-, Y-, and 45 o -directions for the MCE R ground motions.The corresponding peak-bottom story drift ratios are depicted in Figure 13B.The peak roof drift values are systematically greater than their bottom story counterparts.Figure 14A provides the residual horizontal displacement, vertical elongation, and vertical shortening displacements of the bottom story of the core wall.Figure 14B shows the peak out-of-plane drift ratio recorded in the bottom corner BTM elements.The ground motions that are found to be the most demanding in terms of drift ratios, vertical strains, and residual deformations are TCU068, TCU065, TAK, GDLC, and SCS.These were the motions with the greatest spectral demands for a period of 2 s, implying a more pronounced 1st mode inelastic displacement response.Similar conclusions can be drawn by examining the damage patterns observed for each MCE R level motion, provided in Table 3.Four out of 11 ground motions did not give any serious damage modes listed in the table.For the five most demanding motions listed above, the core wall experienced longitudinal and diagonal concrete crushing in one corner at its base and OOPS failure.For the TCU065 and TCU068 ground motions, corner crushing was accompanied by rupture of the longitudinal reinforcement.Horizontal reinforcement rupture was not observed for any of the ground motions.
Figure 15 presents the deformed shape and vertical strain contour plots obtained at the instant of maximum vertical strain for the motions TCU068, TAK, and GDLC.The plots indicate the occurrence of large localized compressive strains in a corner toe region of the core wall.For the first two motions, the compressive strains correspond to softening, (i.e., compressive strength degradation due to crushing).Significant OOPS distortion is also observed in the response to the TCU068 motion.The tensile strains on the opposite diagonal corner of the core wall are fairly distributed over the height of the bottom three stories, with their peak value ranging between 1.1 and 2%. Figure 16A plots the maximum values of vertical compressive force, C z,corner , at the corners of the bottom story, normalized by the total weight, W. The corner region considered, shown in Figure 16B, has a concrete area of 1.36 m 2 (0.37-m 2 confined concrete, 0.99 m 2 of unconfined concrete), which corresponds to 16% of the total cross-sectional area of each Cshaped wall pier.The figure also provides the normalized values of C z,corner obtained at the instant of maximum base shear force along the X-and Y-directions.On average, the maximum corner vertical compression force is 11% higher than the weight of the structure and far exceeds the value due to gravity loads, also marked in Figure 17A.For many input ground motions, the corner compression force at the instant of peak base shear force in both the X-and Y-directions is significantly smaller than the maximum compression force.Another important observation is that there is no strong correlation between the magnitude of the peak corner compressive force and damage, since some of the input ground motions (e.g., TAK) that resulted in corner crushing did not result in one of the five largest compressive forces.
To quantify the effect of NLFSI, Figure 17 plots the ratio of the shear deformation Δ s over the total deformation Δ t of the base story along the X-and Y-directions, corresponding to the instant of the peak base story drift ratio.The shear deformation is calculated as the difference of the flexural deformation, Δ f , from Δ t .The value of flexural deformation is calculated by integrating the wall curvature along the bottom story height.For each motion, the Δ s /Δ f ratio corresponds to the wall pier that resists the majority of the shear force.The 11-motion mean shear deformation ratio for the MCE R input ground motions in the X-, and Y-direction is 0.58 and 0.79, respectively, indicating that nonlinear shear deformations significantly affect the overall plastic hinge deformations.It is noted that if the corresponding ratios are calculated by focusing on the effective height of the building (and not the bottom story), the values are lower and within the range of corresponding values observed in previous experimental tests. 60n average, at the instant of peak base shear V b,max along the X-direction for the MCE R intensity, 67% of V b,max is found to be resisted as in-plane shear force, 18% as an OOPS force in the compression flanges, 10% in the gravity columns, and 5% as OOPS in tension flanges of the core wall.The corresponding values in the Y-direction were 72, 15, 10, and 3%,  respectively.These values are generally consistent with the base shear resistance distributions obtained by Mavros et al. 1 for static analyses with two different lateral force distributions along the height.Figure 18 plots the maximum shear stress ratio values of the six segments (four webs and two flanges) of the core wall for the DE and MCE R intensities.The shear stress ratio is defined as ∕(  √  ′ ce ) (MPa), where V and A g are the shear force and the gross area of each wall pier, respectively.The plots include the values for the individual motions and the corresponding mean values for the entire motion ensemble.The shear stress ratio limit of 0.62 √  ′ ce MPa (0.71 √  ′ ce psi), corresponding to a mean longitudinal tensile strain >0.01 per the guidelines of the LATBSDC, 5 is also marked in the plots.The web peak shear stress ratio occurs at the base story and reaches a mean value of 0.71 √  ′ ce MPa (8.55 √  ′ ce psi).The corresponding peak value for the flanges occurs in the second story and is equal to 0.44 √  ′ ce MPa (5.3 √  ′ ce psi).The shear stress ratios for the individual MCE R ground motions are also provided in Table 4.The specific table also reports the peak diagonal compression and horizontal strains of the base story as quantitative indicators of the extent of shear damage in the walls.The data provided in Tables 3 and 4 indicate no direct correlation between the peak shear stress ratio of the wall piers and the extent of the damage.For example, the peak shear stress ratio obtained for the Tabas motion is 1.10 √  ′ ce MPa, well in excess of the limit value, but this motion still did not give substantial damage.One important observation pertains to the significant values of peak and residual vertical elongation of the wall, as deduced from Figure 19A.This behavior, attributed to the accumulation of crack opening and inelastic tensile strains in the reinforced steel, is typical for cyclically loaded RC components having high longitudinal steel ratios and/or low axial compression load ratios. 12,20On average, the peak core wall elongation at the roof is 163 mm, and the corresponding residual value is 84 mm.Contrary to the core wall, the columns experienced much lower peak and residual elongations.The differential vertical elongation of the core wall with respect to the columns, combined with the rotational deformations of these components impose significant deformation demands in the floor slabs.These deformation demands, together with the vertical inertial and gravity loads, lead to significant inelastic and residual deformations of the slabs and are an important aspect of the response, which is not considered in standard PBSD practice.
Figure 19B provides the values of peak slab-to-core wall (SCW) vertical drift ratio for the 11 ground motions.The specific ratio is calculated by dividing the relative vertical displacement between the center of the slab and the corners of the core wall over the horizontal distance of these points.The SCW drift ratio due to gravity loads is also shown in Figure 19B (black continuous line), and is only 0.6% (40 mm relative vertical displacement of the slab) while the 11-motion-mean SCW drift ratio reached a value of 4.3% at the roof.The mean residual SCW drift ratios are higher in the upper stories wherein the core wall rotation is larger (e.g., 2.6, 3.3, 3.8, and 4.2% for 1st, 4th, 8th, and 14th story, respectively).The peak and residual vertical displacement contour plots for the TAK motion are shown in Figure 20A,B, respectively, indicating a significant settlement of the slabs, especially at the midspan locations between the core wall corners and the columns.This type of system behavior and slab damage can decisively impact the repairability and functional recovery of this type of buildings, as it crucially affects the performance of drift-sensitive structural and nonstructural components.The present study's remarks related to the inelastic response of the slabs should be evaluated with knowledge of the various simplifying modeling assumptions adopted herein, such as the representation of the post-tensioning tendons as straight lines.

MESH-SIZE OBJECTIVITY ANALYSIS
As mentioned in previous sections, the concrete constitutive equations involving softening that is properly regularized to eliminate spurious mesh-size effects.Given that it was important to verify the objectivity of the softening laws and also that the discretization employed in the analyses ensured a convergence in the finite element approximation an additional analysis was conducted, this time using a finer mesh.As shown in Figure 21, the smaller element size is only used for the bottom story of the wall, as this was the region that experienced the most significant inelastic strains and damage.The element size in the refined (fine) discretization corresponds to half the size of the original (coarse) analyses presented in previous sections.The peak values of five key response quantities along the building height, obtained from the original analysis (i.e., with a "coarse" mesh) and the additional analysis (with the "fine" mesh), are compared in Figure 22.It can be seen that the discrepancies in the values obtained in the two analyses are practically negligible.Additionally, the interstory drift time histories, vertical residual slab displacements, wall elongations, and peak shear stresses between the coarse and fine analyses were compared and found to be practically identical for the two analyses.Finally, the two analyses gave identical failure modes.The results of the additional analysis verify that the discretization scheme employed for the building is bound to give accurate values for all quantities of interest.

PARAMETRIC ANALYSIS OF RAYLEIGH DAMPING MODEL
A parametric study was conducted to assess the impact of Rayleigh viscous damping on the numerical results.Three distinct sets of analyses were employed, each with varying damping ratios and target frequencies, as summarized in Table 5.
Case D1 in the table constitutes the default viscous damping model utilized in the analyses discussed in preceding sections.Case D2 signifies the LATBSDC recommended damping for MCE-level dynamic analysis, with target damping ratios of 2.5% at target period values of 0.2T = 0.3 s and 2T = 3 s.Lastly, Case D3 represents a scenario where damping for short periods (i.e., the stiffness-proportional damping) aligns with the damping of Case D1, while damping for long periods (i.e., the mass-proportional damping) closely matches the damping of Case D2. Figure 23A illustrates the Rayleigh damping ratio versus period employed in the three analysis cases for a period range of 0-8 s, while Figure 23A provides a closer look at the range of 0-0.2 s (short periods).The plot indicates that Case D2 results in higher damping ratios than Case D1 across the entire period range, whereas Case D3 exhibits comparable damping ratios with Case D1 for short periods and higher damping ratio values for longer periods.
A comparison of the results for various quantities along the building's height for the three analysis cases is presented in Figure 24.In Case D2, the maximum peak and residual interstory drifts are reduced by up to 12 and 34%, respectively, in comparison to the corresponding values of Case D1.The analogous comparison between Case D3 and D1 exhibits a decline of 5 and 19% in maximum and residual interstory drifts.The peak SCW drift ratio reduction in Cases D2 and D3 (compared to Case D1) is approximately 15 and 2.5%, respectively.Comparable reductions were observed for the residual SCW drift ratios.Finally, the failure mechanisms observed in Cases D1 and D2 were identical.However, only three out of 11 motions in the D3 case displayed diagonal crushing, vertical crushing, and OOPS failure modes.
This parametric study corroborates that the employment of high damping ratios for short periods results in erroneous excessive numerical damping, which subsequently leads to an artificial reduction in displacement responses, as documented in other studies. 61Our computational model explicitly accounts for all structural elements′ hysteretic energy dissipation, and viscous damping should only be incorporated to circumvent potential convergence issues.The application of high damping ratios has a substantial impact and should be justified in PBSD when all sources of hysteretic energy 4 Comparison of mean results for analysis using three different damping ratio models.dissipation are already encompassed in the nonlinear models.The damping ratios proposed by PBSD guidelines should take into account the various nonlinear modeling approaches.Furthermore, a specific nonlinear modeling approach that uses the proposed PBSD guidelines damping ratio values should be able to reproduce data from system-level tests. 43

DISCUSSION
The building examined in this paper experienced, on average, lower IDR than the design limits for both the DE and MCE R input ground motions.However, the mean peak base shear and base overturning moments recorded during the analyses greatly exceeded the corresponding design values.These observations indicate that the building had substantial design overstrength (up to 2.9 in shear and 2.3 in overturning moment).The base shear overstrength is attributed to the fact that the ACI 318-14 37 requirements do not consider the effect of overstrength and higher modes of response on the shear force demands of walls.The improved approach in ACI 318-19 56 results in higher values of design base shear.The underestimation of the design story shear forces for the DE did not result in significant damage because the design procedure involved a strength reduction factor φ = 0.6 and nominal material properties.The combined effect of factor φ and expected material properties for reinforcing steel, which are 17% larger than their nominal counterparts, resulted in an overstrength factor of 1.94.The significant flexural overstrength is attributed to the code-prescribed flexural design of the coupled wall piers, which relies on a linear sectional analysis and considers uncoupled wall segments subjected to the axial forces due to the gravity loads.This design procedure does not account for the significant impact of the coupling beam action 12 on the overall core wall moment strength.Mavros et al. 1 showed that for the specific building and loading in the X-direction, 80%-85% of the core wall base shear was resisted by the compression piers.Similar observations have been made in the analytical studies of Refs.12, 28, 31 The moment and shear overstrength have critical implications for the design of the foundations, which are not accounted for, in the code prescriptive requirements of ACI 318-19.This is a subject worthy of future studies for improving the seismic design of core wall buildings.Alternatively, any unnecessary moment overstrength for controlling drifts can be eliminated by iterative design procedures using nonlinear analysis.
F I G U R E 2 5 Snapshots of vertical strain distributions at the base of the core wall obtained for the analysis using the MCE R -scaled TAK motion.
The building's response indicated the significant effects of NLFSI in core wall buildings that are not captured by standard PBSD practice.The localized crushing of the corner of the core wall, observed in five out of the 11 MCE R input ground motions is attributed to the combination of inelastic deformation demands of the bottom story and the significant multidirectional loading of the corners of the core wall, including OOPS degradation.To elucidate this behavior, Figure 25 plots the distributions of vertical strains obtained at different instants of the MCE R TAK ground motion.Figure 25A shows the bottom story drift ratio time-histories along the X-and Y-directions, and Figure 25B plots the vertical strain distributions corresponding to three characteristic instants (also marked in Figure 25A), that is: (i) t = 7.25 s, when the peak value of compression force C z,corner is obtained in the corner region; (ii) t = 7.65 s, when the peak vertical strain value is obtained; and (iii) t = 12.45 s, when the peak base IDR is obtained.Apart from the vertical strain distributions for the specific instants, Figure 25B, also marks the neutral axis depth and reports the normalized base moment of the core wall in the two principal directions.At the instant of peak corner compressive force, the base moment vector is at an angle equal to 28 o relative to the Y-axis.The skew loading leads to exhaustion of the compressive capacity in the confined corner area, leading to softening associated with crushing and the development of large compressive vertical strains.The loss of vertical compressive capacity in the confined region entails an increase in the vertical compressive force in the adjacent, unconfined regions, which in turn leads to the propagation of crushing damage in the latter regions.The occurrence in crushing is evident in the plot corresponding to t = 7.65 s, wherein the peak compressive strain equals 0.024, exceeding the value of 0.021, which corresponds to the onset of compressive strength degradation in confined concrete due to crushing.The occurrence of localized compressive crushing in the corner region is irreversible, as deduced from the strain distribution corresponding to t = 12.45 s.Although the base moment vector at this instant is nearly aligned with the Y-axis, the orientation of the neutral axis depth remains skewed with respect to the Y-axis.
The extent and spread of plasticity is affected by the NLFSI and tension shift. 62Figure 11B demonstrates this phenomenon, where positive strain (tension) remains almost constant along the first three stories while negative strain (compression) is concentrated at the base.This mechanical behavior of the plastic hinge regions affects the extent of plastic hinge detailing required in terms of antibuckling ties of the vertical reinforcement, as well as the design of the splices.This is an important issue that is not captured in standard nonlinear seismic analysis practice, which uses beam-based FMs that enforce the plane sections remain plane assumption.
The motions that resulted in the vertical crushing of the wall, in general, correspond to larger flexural deformation ratios in the bottom story.Furthermore, the shear deformation ratios in the Y-direction are generally larger than those obtained in the X-direction.This is attributed to two reasons: first, the shear reinforcement ratio in the Y-direction is smaller than that in the X-direction; second, given that the shear deformation ratio is calculated for the pier that resists the largest shear force, the values of shear deformation ratio in the X-direction correspond to the wall pier that experiences large axial compression due to the coupling action of the beams, and these piers experienced smaller shear deformation ratios than the piers experiencing lower axial compression.This behavior is illustrated in Figure 26, which shows the maximum principal strains of the bottom three stories of the core wall for the TAK ground motion at the instant of peak base story drift ratio in the X-and Y-direction.In both cases, the piers with the largest axial compression experienced the smallest shear deformation ratio at that instance.
The effect and significance of the OOPS degradation were quantitatively examined by repeating the set of analyses for the MCE R ground motions without modeling the OOPS degradation of the wall piers.All mean response quantities discussed in the previous sections (e.g., vertical strains, maximum and residual drifts, base reactions, etc.) were found to decrease when the impact of OOPS was neglected.The response parameters that experienced the largest reductions (in % in parenthesis) by not including the OOPS degradation were the residual first story shortening (30%), the peak compression strain (28%), the peak vertical tensile strain (20%), the residual first story horizontal displacement (16%), and the peak out-of-plane elemental drift ratio (12%).It is worth noting that the OOPS modeling approach presented in this paper requires further validation, not addressed in the current study, using experimental data and enhanced finite element models.
For some motions, the peak shear stress ratio of individual wall piers significantly exceeded the recommended PBSD limits without experiencing damage, indicating a weak correlation between the shear stress level and the anticipated wall damage.This observation demonstrates the additional impact of inelastic deformations in the plastic hinge regions on the propensity for significant shear damage under high shear stresses.It should also be noted that using the current PBSD guidelines 5 for the specific structure would require an increase in the shear capacity of the bottom three-story piers by a factor of 1.72.This is because the 11-motion-mean wall shear stresses at the MCE R intensity, when multiplied by 1.5 according to PBSD practice, would surpass the corresponding limit of 0.62 √  ′ ce MPa limit requiring an increase in wall thickness and/or concrete compressive strength.Employing FB models for PBSD-oriented analyses of this building would probably require even greater increases in shear capacity, given that such models cannot accurately capture the effect of in-plane shear nonlinearity and, consequently, they would have been expected to give larger shear stress ratios than those reported in Table 4.
Furthermore, substantial damage and residual deformations were observed in the slabs due to the elongation and rotation deformation experienced by the core wall, especially in the upper stories.In standard nonlinear analyses using PBSD, the nonlinear behavior of slabs with spans exceeding 6.10 m (20-ft) is neglected.Such damage and residual deformations can be a decisive factor for the functional recovery and repairability of these types of buildings.However, it is worth noting that in this case study, which pertains to a regular core wall structure, no collapse (total or partial) was observed in any of the MCE R motions.Thus, at least for these types of buildings, the current code design objective of preventing collapse is satisfied.It is important to note that the code does not consider the influence of overstrength on the behavior and design of foundations and basements, which are excluded from this model.
A set of analyses using the MCE R -level ground motions without their vertical acceleration component was conducted to examine its effect on the peak and residual slab deformation.The results showed that the peak and residual SCW vertical drift ratio decreased by 23 and 19%, respectively.The response parameters that undergo the maximum reduction due to the absence of the vertical component of the ground motions were the mean peak axial compression force of the core wall (30%) and the mean peak axial compression force of an individual column (24%).All other response parameters were reduced by less than 10%.
In all of the ground motions, the columns did not experience crushing of the confined concrete or rupture of the reinforcing steel.The mean tension and compression strains did not exceed 0.008 and −0.004, respectively, while the maximum tension and minimum compression strains recorded in all analyses were 0.037and −0.017.
Finally, the present study supports the conclusions of the previous study, 1 The similarities between the two studies include the predominance of NLFSI in failure patterns, significant system overstrength, an approximately 20% contribution to base shear resistance from OOPS resistance of wall segments in compression, and slab damages due to vertical elongation and rotation of the core wall.The present study offers quantitative specificity to several qualitative conclusions from the research in Mavros et al. 1 In contrast, the study by Mavros et al. 1 revealed differences in the behavior of the core wall under certain loading conditions.Specifically, when a load pattern dominated by the first mode was employed for the pushover analyses, the results exhibited a relatively ductile behavior, with computed shear stresses on the core wall substantially lower than the average shear stresses computed in this study (as depicted in Figure 18).Additionally, none of the analyses in Mavros et al., 1 where loads were applied parallel to the main X-and Y-directions, demonstrated crushing of the core wall corner.

CONCLUSIONS
This paper presents an analytical study of the nonlinear seismic response of a 14-story RC core wall building designed according to CBC 2019 for a site in Los Angeles, using the previously validated BTM.The BTM was enhanced to consider the nonlinear OOPS degradation of wall piers.Analyses were conducted for a collection of 11 triaxial ground motion records, scaled at both the DE and MCE R seismic hazard levels.The computational model considered the nonlinear behavior of all structural elements, including the floor slabs.
The study found that for the DE seismic hazard level, the building response was essentially elastic due to its substantial moment and shear overstrength.The prescriptive design resulted in a stiff structure with significant flexural overstrength, which in turn resulted in relatively high shear forces.
At the MCE R seismic hazard level, the building experienced crushing in the wall corner regions for five of the motion records.This localized damage mode was affected by the multidirectional dynamic response of the core wall, the significant effect of NLFSI, as well as from the OOPS degradation of wall piers.NLFSI resulted in significant shear deformations at the bottom story of the core wall, which corresponded to more than 50% of the base story drift.This behavior is not captured in standard fiber-section, beam-based models typically used in PBSD practice.The flexure-shear interaction (specifically, the tension shift effect) also affected the spread of inelastic strains of the vertical reinforcement of the core wall.
The study also found that even though the shear stress values of the wall piers exceeded the corresponding limit values of PBSD guidelines for all MCE R ground motions, significant damage, and residual deformations developed only for the five ground motions with the greatest base story drift demands.This indicates that shear stress values alone (without consideration of inelastic deformation demands) may not be adequate for deciding a wall's propensity to shear damage and failure.The predominant damage mode observed was the crushing of the confined concrete at the corner regions.
The dynamic increase of axial compression in the corner regions of the core wall compared to the corresponding compression under gravity loads was significant.On average, the maximum compression load in a single corner of the core wall exceeded the entire weight of the structure.This is attributed to the skewed dynamic horizontal loads to which the structure was subjected during the motion, as well as the significant coupling of the two C-shape segments of the core wall via the coupling beams.
These findings may indicate a need to increase the confinement reinforcement and the extent of confinement to fully confine at least the bottom story of the core wall.Additionally, it is recommended to account for the significant inelastic deformations and residual deformations in the floor slabs because of the significant peak and residual vertical elongation of the core wall.These inelastic deformations can significantly affect the repairability and functional recovery of core wall buildings.Furthermore, a relevant design enhancement might be the vertical post-tensioning of the core wall to reduce the residual core wall elongation.It is important to note that these recommendations are based on a single case study and further studies, including both analytical and experimental investigations, are needed to support or refute the findings of this study.

F I G U R E 3
Building configuration and overview of the model.(A) Plan configuration; (B) core wall section; (C) overview of the model; (D) core wall mesh.

F I G U R E 4
Model used for the parametric study of the out-of-plane behavior of the core wall.(A) Through wall thickness discretization; (B) failure mode of unconfined element for axial load ratio = −0.15.F I G U R E 5 Relation between out-of-plane shear (OOPS) resistance and OOPS drift ratio, γ, obtained from analyses of (A) unconfined and (B) confined concrete.F I G U R E 6 (A) Out-of-plane shear degradation law; (B) dependence of drift value γ d on axial load ratio.

F I G U R E 7
Mean scaled spectra and original spectra of individual ground motions compared to the target ASCE 7-16 MCE R spectrum.(A) Pseudo-spectral acceleration; (B) spectral displacement; (C) vertical acceleration.F I G U R E 8 Profiles of five response parameters: (A) interstory drift ratio (IDR); (B) residual IDR; (C) floor accelerations; (D) story shear force ratio; (E) story moment ratio for the DE and MCER analyses.MCE R design spectrum, as demonstrated in Figure 7C.For the calculation of the Design Earthquake acceleration time histories, a uniform scale factor of 2/3 is applied to the corresponding MCE R -level scaled motions.The five scaled motions with the larger PSA values at T = 2 s were the most damaging motions as discussed in the next section.

F I G U R E 9
Peak (A) base shear (V b /W) ratio and (B) base moment (M b /WH) ratio recorded in X-, Y-, and 45 o -directions for the 11 MCE R motions.

F I G U R E 1 0
Normalized H e,min for (A) the instances with base moment larger than 85% of the peak base moment at each direction (X, Y, and 45 o ) during the 11 MCE R motions and (B) the instance of peak base shear.F I G U R E 1 1 Mean vertical strain profiles at the bottom three stories of the core wall for: (A) DE level motions and (B) MCE R level motions.

F I G U R E 1 2
Peak tensile and compressive vertical strains of the base story for the MCE R motions.(A) Middle fiber; (B) extreme fiber.F I G U R E 1 3 Peak drift ratios recorded in X-, Y-, and 45 o -directions for the MCE R ground motions.(A) Roof drift ratio; (B) 1st story drift ratio.

FF I G U R E 1 5
I G U R E 1 4 (A) Residual displacement of the bottom-most level of the core wall for the 11 MCE R ground motions; (B) peak out-of-plane BTM element drift ratio for the eleven MCE R ground motions.BTM, beam-truss model.TA B L E 3 Damage patterns recorded in the eleven MCE R motions.The regularized concrete crushing stains are 0.004 (unconfined) and 0.021 (confined) and the out-of-plane failure drift ratio is 1.5%.Deformed shape and vertical strain contour plot in the core wall at the instant of the peak vertical strain, for motions involving compressive failures at the bottom corner of the section.(Deformation scale factor = 3).

F I G U R E 1 6
Maximum vertical compressive force (C Z,corner ) recorded at the core wall corner region to W ratio for the eleven MCE R motions.The motions with observed vertical damage are denoted with *.F I G U R E 1 7 Shear deformation relative to the total deformation of the piers in the X-and Y-direction at the instance of peak drift for each of the 11 ground motions at: (A) DE and (B) MCE R .

F I G U R E 1 8
Shear stress ratios in the individual wall segments.TA B L E 4 Peak shear stress ratios of wall piers for the MCE R -level motions.Direction Response quantity REHS RRS SCS SYLOV TAK LGPC GDLC Tabas TCU065 TCU067 TCU068

F I G U R E 1 9
Peak and residual values for: (A) core wall vertical elongation; (B) slab-to-core wall (SCW) vertical drift ratio.

F I G U R E 2 0 3 F I G U R E 2 3
Vertical relative displacement contours: (A) at peak vertical core wall elongation and (B) at the end of the analysis.FI G U R E 2 1Refined discretization scheme for the bottom story used in mesh-size effect study.F I G U R E 2 2Comparison of peak responses for analysis using coarse and fine mesh.TA B L E 5 Summary of cases considered for parametric study on the effect of Rayleigh damping.Rayleigh damping.

F I G U R E 2 6
Maximum principal contour plot at the instance of the peak 1st story drift in X-and Y-direction for the TAK motion.

TA B L E 1
Material parameters for the confined concrete regions of the core wall.
TA B L E 2