Modication of PBSD Procedure for Moderate Shear Walls Subjected to Near-Field Excitation

In the near-eld areas, VCE effectively exists which is so sensitive depending on the distance from a source and the earthquake magnitude. This has remarkable effects when shear wall is to be in moderate size which causes it to face wide diversity modes of failure. So, a modied procedure is conceived to ensure the response of the shear wall subjected to biaxial excitation. Thereby, there is an evaluation process that is based on the intersection of capacity and demand curves commonly called the performance point. The modication factors are developed based on the validation of performance points from which are derived demand and capacity curves. al., 2010). Thus, the present study is formed in such a way to develop modication factor in the nonlinear static procedure to achieve a reliable response. The modication factors are developed based on the validation of a pseudo performance point and the differentiation of pseudo and actual performance points which are derived from demand and capacity curves. The modication factors are formed based on distance from a source and steel ratio of moderate shear wall. Four earthquakes are considered in the following text such as Northridge, Lomaperita, Kobe, and Landers in 3 distances as 0, 15, 30 kilometers. Finally, verication procedures with experimental data have presented that the modication factors can effectively provide reliable outcomes. results of the present study show that the modication of the PBD method for moderate shear wall should be accounted for in a wide range of steel ratios and distances from a source earthquake if the collapse-prevention performance level has been adopted. However, in life-safety performance level, this modication should be accounted for in a narrow range of low steel ratio intermingled with a small distance from a source of an earthquake. The variations show that the general modication factors, which are presented in the present study in the last part for moderate shear wall, can be utilized for different sizes of moderate shear wall with relatively high condence (88.6% captured from the maximum case of average variation).


Introduction
The main scope of the performance-based seismic design (PBSD) method is conceptually formed based on the development of structural and component performances with the utilization of the nonlinear static analysis instead of nonlinear dynamic analysis which its goal is to capture a reliable result by reduction of the computational effort. Meanwhile, a rational procedure which can provide a reliable result in an area that a structure is subjected to synchronous vertical (VCE) and horizontal (HCE) earthquake excitations (biaxial excitation) has not been clari ed yet ((H. J. JIANG et al., 2011, He and Agrawal, 2008, Ellys Lim and Chouw, 2018, Laila Elhifnawy et al., 2017, Collier and Elanashai, 2001, Naeim, 1998, S., 1997, Huy T. Tran et al., 2016. Researchers indicated that in the near-eld area due to intensities of VCE and HCE and change of their characteristics, the interaction between them and distance from a source, performances of components and structures may be considerably affected (Nikolaos Simos et al., 2018, Sinan Akkar et al., 2018, Jalal Kiani et al., 2018, Yang Xiang et al., 2017, Heng Li and Chen, 2017, Laila Elhifnawy et al., 2017, He and Agrawal, 2008, Bozorgnia, 2004, Naeim, 1998, S., 1997, Aki, 1980, Bommer, 2001, Ghatee et al., 2007, Haibo Yang, 2014, Aruna Rawat. et al., 2017. Since VCE effectively exists in near-eld area sensitively depending on the distance from a source and the earthquake magnitude (Collier andElanashai, 2001, Asgarian Behrouz et al., 2012); There are a few methods that the VCEs are commonly transformed to an additional weight which is vertically distributed in different levels of a structure. Correspondingly, the effect of vertical component in demand spectral acceleration response of a single degree of freedom is shown to be signi cant by a study on the Hyogo-ken Nanbu earthquake (Loh, 1997). And also, Bousias et al. (2002) have shown the effect of near-eld on a capacity curve of the column (Bousias, 2002). Since biaxial excitation may also produce different inelastic (damage and/or plastic) surfaces in tensorial material matrices space can produce different elastic/inelastic loadde ection manner (Su andWong, 2007, Sima et al., 2008); From a different point of view, the performance of components which are governed by biaxial stress in a material matrix such as shear walls and structures consist of them can be impacted signi cantly by biaxial excitation. The exibility of shear wall structure is lesser than the other structural systems (actually lesser inter-story drift) but the inter-story drifts can be affected by vertical and horizontal excitations simultaneously. These effects may be ampli ed when shear wall is to be in moderate size; Walls are considered moderate if their aspect ratio is between 1.5 to 3 (FEMA-273, 1996), hence, they face wide diversity modes of failure (Sheu and Huang, 1992). The capacity-demand evaluation process that is based on the intersection of capacity and demand curves could be affected by considering the static analysis procedure in a near-eld area where it has different excitation characteristics corresponding to distance from a source. So, a modi ed procedure is conceived to ensure the response of the shear wall subjected to biaxial excitation. In spite of all above mentioned, a study by analytical and experimental procedures shows that the nonlinear static analysis may be applicable for the near-eld area with the decent response of structure (Wuchuan Pu et al., 2018, Basu, 2007, Ghatee et al., 2009a, Ghatee et al., 2009b, Ghatee et al., 2010. Thus, the present study is formed in such a way to develop modi cation factor in the nonlinear static procedure to achieve a reliable response. The modi cation factors are developed based on the validation of a pseudo performance point and the differentiation of pseudo and actual performance points which are derived from demand and capacity curves. The modi cation factors are formed based on distance from a source and steel ratio of moderate shear wall. Four earthquakes are considered in the following text such as Northridge, Lomaperita, Kobe, and Landers in 3 distances as 0, 15, 30 kilometers. Finally, veri cation procedures with experimental data have presented that the modi cation factors can effectively provide reliable outcomes.

Earthquake loading
The selection of a suitable set from the collected earthquake events plays an important role in developing general results. As the nonlinear response of a structural system strongly depends on characteristics of motion such as frequency content, magnitude, strong motion duration, and pulse sequencing; therefore, several ground motions representing a range of amplitudes and frequencies should be utilized to anticipate the upper and lower bounds of the nonlinear response. Considering the characteristics of the earthquakes, which are tabulated below, show that the four earthquake records of Northridge, 1994, Loma Prieta, 1989and Landers, 1992(Iwan and Gates, 1997 Kobe, 1995 are able to generalize the results. Besides the target drift straightly depends on the performance level. So, there should be a reliable statistical study, for instance, the SEAOC has presented the values of earthquake loadings and the corresponding drift targets for the structures, i.e., in Life Safety (LS), the drift is 1.5% while in Collapse Prevention (CP) is 2.5%. Although all these values are not accurate, it has no signi cant impact in comparison with the actual performance point and the pseudo performance point that is the overall procedure of the present study. It should be noted that for shortening of the sentences in the graphs and tables, the probability of exceedance 2% and 10% per 50 years are demonstrated by 2-50 or 10-50, respectively.
It is obvious due to the table that the effect of near-eld characteristics on the response of structures almost vanishes after 30 km. Therefore, the performance points are captured in 0, 15, and 30 Km distances. Besides, it is di cult to record the VCE from the exact distances from a source. Hence, the vertical records are scaled based on the statistical values of V/H ratios which are developed based on the 104 worldwide records by Ambraseys et.al. In 1996(Ambraseys, 1996.

Material model
When static horizontal and vertical cyclic loadings are imposed on a structure, nonlinear characteristics, and damage aspects of a material matrix within each component may produce different performances (Su and Wong, 2007). In most mathematical models that represent the behavior of concrete, the stress-strain relationship is considered linear in plastic surfaces before damage criteria. Although this de nition of concrete behavior can be a good estimation, it can be totally different for the structures and elements subjected to biaxial stress, vertical and horizontal earthquakes, damage state, and damage mode. In other words, a pair of the same component which is affected by different protocols of bidirectional cyclic loadings may have disparate performances due to differences in decreasing stiffness, hardening and/or softening in the material matrix and diminishing of energy absorption capacities because of different cyclic loading protocols (Mo andChan, 1996, Loh, 1997). All the above mentioned are shown schematically in Figure 1.
The performance of components, which are governed by biaxial stress in the material matrix, can be impacted signi cantly by biaxial excitation. Since biaxial excitation may also produce different inelastic (damage and/or plastic) surfaces in tensorial material matrices space afterward produce different elastic/inelastic load-de ection manner (Su andWong, 2007, Sima et al., 2008).
The nonlinear behavior of the moderate aspect ratio of shear wall is more complex and involves different modes of failure in comparison with the other aspect ratios. This leads to different aspects of damage, behavior, and performance of the shear wall which is correlated with the different aspects of reinforced concrete damage. Hence, an appropriate material model that is able to consider different aspects of damages and failure modes should be utilized in a numerical simulation to produce acceptable performance points. Therefore, a material model has been developed by authors (Ghatee et al., (Ghatee et al., 2018, in press)) which is capable of handling the characteristics of near-eld earthquakes (the partial unloading/reloading effects), their effects on concrete material matrices for the multitude of damage modes, and realistic stress-strain relationships of concrete under crack closure-reopening phenomenon, is adopted for this study. This model has been developed based on the damage formulations which has covered all cyclic loading conditions including partial and complete unloading/reloading ones; the program of the model has been generated via ABAQUS which has been published separately (Ghatee et al., 2017, Ghatee et al., 2018

Development of performance points
As discussed before, VCE has a signi cant impact on the behavior of the structure. In order to develop the modi cation factor this effect should be taken into account by considering it as an additional mass for a preliminary NDA analysis. The design is done with respect to VCE and HCE, distance from a source, and certain reinforcements of moderate shear wall to reach drift target of the top of the wall to a limitation value corresponding to performance level (Botta and Mezzi, 2008). Within the iteration procedure, the mass has changed, from 8950 kg to 9250, which makes the drift change of 0.08m to 0.1m to capture the actual performance point. In this way, to reach the actual performance point, the wall was allowed to reach the drift target which is 1.5% in LS. After the mass causes the drift limitation, the mass and the actual performance point is captured. This procedure is shown in Figure 2. Hereafter the mass is transformed into a force and applied to the shear wall to reach the displacement under a cyclic procedure to displacement target which results in pseudo performance point.
For generating pseudo performance points, the nonlinear static analysis is adopted in such a way that the increasing horizontal displacement loading history in presence of constant vertical load in compression state is applied to the top of the shear wall until the drift of the wall reached limitation values (drift targets) in different performance levels. This procedure is shown in Figure 3. With an evaluation of actual and pseudo performance point, the modi cation factor is developed.
Veri cation procedure Considering the time interval between peak vertical and horizontal accelerations effects on the response of the structure, the rst part of the records cannot be dropped. Hence, in the present study, the end part of the records is only dropped out based on the 95 percent of delivered seismic energy.
Due to this veri cation procedure, ve experimental results of shear walls (SW21 to SW25) with moderate aspect ratio and different concrete and steel properties are adopted. These shear walls have been monotonically tested by Lefas et.al. (Lefas . L et al., 1990) and the corresponding results have been used also in the veri cation procedure of the other material models by Vecchio (Vecchio, 1992) andWu et.al.(Wu et al., 2006). The experiment setup, boundary conditions, material properties, which have been used in the experiments, and nite element models are shown in Figure 5, Tables 2 and 3.
The nite element modeling and mesh discretization, which is adopted in the present study, is illustrated in Figure 6. The present material constitutive law was adopted due to simulation. The maximum top displacements of the walls, which have been reached due to experiments procedure in CP are adopted as targets of displacement loadings which are applied to the models via nite element analyses. The target displacements are presented in Table 4.
Considering Table 4. All analytical simulations, which have been done via the other or present studies, are developed based on the target displacements with small variations but are not developed based on the exact values. It is because of the fact that the target displacements, which have been developed by experiments, are located in the collapse limit of the shear walls. These targets may cause some numerical instability in nite element simulation based on constitutive law capability, material properties (such as damage indices and steel ratio), convergence rates, number of iterations, and time intervals. The variation of imposed displacement on top of the walls is also presented in Table 4.

Results And Discussion
In this study, the modi cation factors are presented in such a way that can be directly applied in capacity curves of moderate shear wall in a modi ed PBSD procedure. Since the modi cation factors are developed based on the evaluation procedure of performance points, there is a capacity-demand evaluation process that is based on the intersection of capacity and demand curves commonly called the performance point. Besides the actual performance points can be captured from drift demand curves, if the displacement of the top of the shear wall reaches displacement objectives. While the pseudo performance point can be captured from the push of hysteresis behavior of the shear wall which is illustrated in Figures 7 to 9. The nal results are presented in tables 5, 6, and 7.
The base shear value of the pseudo performance point should reach the actual one that has been presented in Figure 10.
Considering above mentioned, the modi ed shear capacity of the moderate shear wall is adopted which can be developed based on the following formulation: where;V cm is modi ed shear capacity, V c is provided shear capacity and M f is modi cation factor for moderate shear wall subjected to performance level, steel ratio, and distance from a source, respectively.
The PBSD method for moderate shear walls is conceptually modi ed to enhance its accuracy and reliability subjected to near-eld excitation. This modi cation is done by application of some general modi cation factors which may be applied in the capacity of a moderate shear wall. The modi cation factor's values corresponding with different sizes of moderate shear walls are different. Nevertheless, the modi cation factors, which may be developed by other sizes of moderate shear walls, should be located in a narrow band of variations. This is the most important issue for the achievement of reliability on the estimation of moderate shear wall performance with different size subjected to near-eld excitation.
The modi cation factors can also be generated by a trend-surface, which passes through the modi cation factor points in three-dimensional spaces. The Cartesian axes X, Y, and Z of the three-dimensional spaces are steel ratio, distance from a source, and the modi cation factors, respectively. The trendsurfaces are calculated by MATLAB software. The generalized modi cation factors can be achieved by averaging the previously developed modi cation factors. It is not recommended that the modi cation factors greater than one to be utilized in the practical case of shear wall design. Because the modi ed performance points will be located in the greater safety margin limit if the maximum values of the modi cation factors are adapted equally to one. This issue is just a recommendation to keep a greater safety limit in practical cases.
The surfaces of modi cation factors generally show that the effects of near-eld characteristics on the moderate shear walls are reduced if the steel ratios of the moderate shear wall increase and vice versa. It is deduced from the effect of the partial unloading-reloading state of plain concrete on the overall response of the moderate shear wall which is decreased by overcoming the behavior of the steel layer. Moreover, the axial demand loads subjected to one of the distances are commonly decreased, if the steel ratio increases. Since the self-weight of the shear walls are participated in their inelastic responses, while the axial demand forces are calculated based on lumped masses only. Also, the total demand masses subjected to one of the distances are commonly increased, if the steel ratio increases; because the axial force demand of a shear wall with a high steel ratio should be greater than a shear wall with a low steel ratio.
Considering the capacity curves, in a condition that the steel ratio of the wall is not small, hardening of steel reinforcement can play an effective role in diminution or even enhance shear wall performance. Considering the terminology of PBSD, the demand force of a component is de ned as a minimum requirement of component force resistance, which the component remains in the performance level. When distance goes to a large extent, total demand masses on the top of the moderate shear wall commonly increases, and the axial demand forces of moderate shear walls commonly decrease. It is because, rst due to updating of the shear wall design in the iteration procedure, the steel ratios increase. Hence, the weights of shear walls increase; and the axial force demand in close distance to a source should be greater than the far distance from the source.

Conclusion
The modi cation of the PBSD method subjected to near-eld excitation is successfully developed by directly applying the modi cation factors in the capacity curves. All of the modi cation factors are developed based on different sizes (3*3m, 3*8.4m, 5*10m), aspect ratios (2, 2.8), and the same conditions (Northridge earthquake, CP 2-50, ρ=4.4%, distance from a source 15km) can be compared as presented in Table 8. Considering Table 8, there are two comparison procedures between the modi cation factors for different sizes. The modi cation factors can be compared, rstly, by the modi cation factors which have been produced by similar earthquake imposed to the models, secondly, by the general modi cation factors which have been developed by the response's average of three different earthquakes loading for generalization. The variations of modi cation factors in the second comparison procedure are greater than the rst comparison procedure (Table 8). Because the comparison procedure should be done in such a way that all boundary conditions to be similar. The second comparison procedure has been presented for a demonstration of the fact that the variations of modi cation factors in the second comparison procedure are even small. Considering modi cation factor ratios within performance level, which is adapted to be, low. I mean that the capacity of moderate shear wall in LS may be increased from provided capacity depending on the distance from a source and steel ratio. In case of a greater modi cation factor than 1, an increase of the distance from a source may consequently increase the effect of near-eld characteristics, when a large amount of steel ratio and the low-performance level such as LS is selected. The effects of neareld characteristics on moderate shear wall are almost reduced when the distance from a source is increased. Clearly, the modi cation factors should be different, if the size of moderate shear walls is changed. However, based on the fact that the PBSD method is an estimation method, the proposed modi cation factors are acceptable, if the difference of the values or ratios of the modi cation factors produced by other moderate aspect ratios below. The results of the present study show that the modi cation of the PBD method for moderate shear wall should be accounted for in a wide range of steel ratios and distances from a source earthquake if the collapse-prevention performance level has been adopted. However, in life-safety performance level, this modi cation should be accounted for in a narrow range of low steel ratio intermingled with a small distance from a source of an earthquake. The variations show that the general modi cation factors, which are presented in the present study in the last part for moderate shear wall, can be utilized for different sizes of moderate shear wall with relatively high con dence (88.6% captured from the maximum case of average variation). Tables   Table 1. Collection of horizontal components of near eld earthquakes and their major characteristics and their Scaled vertical earthquakes records Horizontal earthquakes loading collection (Iwan, 1997, SAC, 1997