Probabilistic assessment of plan-asymmetric structures under the near-fault pulse-like events considering soil

Abstract

This study aims to evaluate the torsional effects and soil–structure interaction simultaneously under near-fault pulse-like earthquakes in a probabilistic framework. Incremental dynamic analysis and fragility curves are employed for this goal. An eight-story R/C dual lateral load-resistant building consisting of shear walls and moment resisting frames is used. The median incremental dynamic analysis curves reported the maximum capacity for the symmetric structure in each foundation conditions. In addition, the capacity of structure will be increased when more shear wave velocity is assumed. Therefore, from this view, neglecting the soil–structure interaction will not be in the safe side. Fragility curves (using intensity measure directly) show that for different cases (except for very low shear wave velocity), more value of eccentricity leads to more probability of collapse. Moreover, the fragility curves show that (for each eccentricity), soil–structure interaction effect is significant only for the flexible base structure with the very low shear wave velocity (100 m/s) and more eccentricity value leads to less soil–structure interaction effects. Results show that the significant eccentricity value may lead to reduce the soil–structure interaction effect in the shear-wall structures under the near-fault events.

Keywords

asymmetric structures fragility curves incremental dynamic analysis near-fault events soil–structure interaction

Introduction

Real structures are not usually plan-symmetric and fixed base; therefore, the torsion and the soil–structure interaction (SSI) are two inevitable phenomena that must be considered in the seismic evaluation of structures. In the past studies, these two effects were investigated separately, but in this article, a probabilistic assessment of torsional effects and SSI is targeted simultaneously.

Effects of torsion have been studied in the recent five decades. In some newest researches, the effects of torsion on the moment frame (Stathopoulos and Anagnostopoulos, 2005) and the inelastic torsional response of two-way asymmetric RC dual structures (Halabian and Birzhandi, 2014) were evaluated. In addition, the seismic demand of low-rise multi-story structures with general asymmetry was investigated (Dutta and Roy, 2012). A comprehensive review of the torsional response of asymmetric structures was carried out by the Anagnostopoulos et al. (2015). Sometimes deterministic studies about the torsional effects lead to contradictory conclusions and do not have clear and efficient outcomes because of the record-to-record variability. This fact actuates the researches to the probabilistic approach in seismic response of asymmetric structures. Using fragility curves and introducing a new damage index for the plan-asymmetric structures, Jeong and Elnashai (2006, 2007) showed that using this new damage index leads to a higher probability of collapse than the Park–Ang damage index (Park and Ang, 1985) especially for severe earthquakes. This new damage index decomposes a three-dimensional (3D) frame to several two-dimensional (2D) frames, and considering the local damage of every frame, calculates the global damage index. They concluded that using traditional damage indices for the asymmetric structures is not in the safe side. Aziminejad and Moghadam (2010) used the probabilistic assessment and fragility curves as a tool for optimum design of asymmetric one-story shear-wall structures. This study confirms an old idea about the effect of torsion (Erdik, 1975; Kan and Chopra, 1981), claiming that by increasing the intensity of excitation (Peak Ground Acceleration (PGA)), differences between the symmetric and asymmetric system especially for the edge drifts have been decreased. In other words, by entrance to the inelastic range, the effects of torsion will be reduced. Manie et al. (2015), by investigation of the three- and six-story RC concrete moment frames under the far-field earthquakes, concluded that increase in the mass eccentricity leads to a brief increase in the probability of collapse. Karimiyan et al. (2013) investigate the seismic progressive collapse of three-story RC moment resisting buildings with different levels of eccentricity in plan. Their results show that an increase in the level of asymmetry in structure leads to an increase in potential of the progressive collapse at both sides of plan in the asymmetric structure. Birzhandi and Halabian (2017) presented an approximate method for the incremental dynamic analysis (IDA) and fragility curves of plan-asymmetric structures. Also Soleimani et al. (2018), using a bidirectional energy-based pushover, proposed an approximate two-component IDA. Lucchini et al. (2009, 2011) followed the Base shear and torque surface idea (De La Llera and Chopra, 1995) for the one-way and two-way asymmetric structures and compared the results with those in the IDA.

Effects of SSI on the seismic response of structures are other major phenomena that involved the many different researches. In the recent studies such as Pioldi and Rizzi (2018), two different approaches of time domain and frequency domain are still used for the problem with heavy damping such as SSI. Rodriguez and Montes (2000), using a simple model, concluded that if the roof displacement or hysteretic energy is considered as damage index, it is acceptable to use the fixed base system with the modified period. Mofid and Ghannad (2008) investigated the SSI effect on the Park–Ang damage index of a bilinear single degree of freedom and stated that this effect increases the damage. In other words, neglecting the SSI and considering the fixed base structure are not in the safe side especially for the short-period structures on soft soils. Halabian and Emami (2013) concluded that SSI effects could change the height distribution of ductility demand (as a damage measure (DM)) especially for the short and medium structures. They also investigated the SSI effects on the damage index distributions of RC dual lateral load-resistant multi-story structures under bidirectional earthquakes (Emami and Halabian, 2018). Seismic response of reinforced concrete tall structures to the near-fault earthquakes considering the SSI has been investigated by Galal and Naimi (2008). Recently, the inelastic displacement ratios for soil–structure systems with embedded foundation have been investigated by Bararnia et al. (2018). They found that using the inelastic displacement ratios of fixed base structure evaluates an underestimated demands for soil–structure systems.

Limited studies had probabilistic approaches to the SSI problem and its effects. Considering the inelastic dynamic SSI in the fragility curves, it is demonstrated that nonlinear behavior for the soil usually leads to a decrease in the seismic demands (Saez et al., 2011). A research on the seismic reliability of multi-story systems considering dynamic SSI states that reliability functions are not significantly different for the fixed and flexible support structures (Bárcena and Esteva, 2007). Using IDA technique, the SSI effects were investigated, and they showed that fragility curves for the flexible base RC shear wall show a less probability of damage than the fixed base (Tang and Zhang, 2011).

Considering the SSI and the torsional effects simultaneously is only carried out in a limited number of parametric studies in the linear range. Using modal analysis, Wu et al. (2001) investigated the SSI effect on the elastic response of asymmetric buildings in the frequency domain. In addition, Shakib and Fuladgar (2004) proposed a method for the linear solution of the three-dimensional dynamic SSI of eccentric structures that idealize the structure as a single-story system and solve it in the time domain. In the following study (Shakib, 2004), an evaluation of dynamic eccentricity of linear system, considering the SSI effects, was proposed. As seen, all of these researches are limited to considering torsional effects and SSI simultaneously in the linear range. In addition, the investigation of these two phenomena simultaneously is attractive for the researchers to design the active multiple-tuned mass dampers for the asymmetric structure with SSI effect (Li, 2012). Recently, Birzhandi and Halabian (2018) presented a simplified method to consider the plan-asymmetry and SSI effects simultaneously in the nonlinear solution of seismically excited structures.

As mentioned above, the nonlinear behavior of asymmetric structures, considering the SSI effects, was neglected in the deterministic and probabilistic past studies. Therefore, this study aims to investigate the torsional effects and SSI simultaneously in a probabilistic framework. In this regard, for the selected structure and excitations, the detailed IDA curves and median IDA curves of the fixed and flexible base conditions with different values of eccentricity will be discussed. The Spectral Pseudo-Acceleration of the first mode of the structure with 5% modal damping ( $S_{a} (T_{1}, 5 %)$ ) is selected as intensity measure (IM) and the overall drift of structure is selected as damage measure (DM). Then, the fragility curves of collapse limit state will be presented and the effects of eccentricity and SSI on these curves will be evaluated. To extract the fragility curves, the IM-based approach will be used. It means that the calculated fragility functions are independent from the DM. This investigation may lead to practical recommendation in the modeling of plan eccentricity and base conditions. Finally, it will attempt to answer the question that torsional behavior will increase or decrease the SSI effect.

The effects of near-fault events on inelastic seismic responses of steel frame structures compared to their responses to far-fault ground motions were studied by the Alavi and Krawinkler (2001). The effects of near-fault ground motions are further studied by several researchers. Employing fragility curves, Aziminejad and Moghadam (2010) investigated the differences of near-fault and far-fault records on the seismic response of single-story asymmetric buildings. Emami and Halabian (2015) concluded that directionality of the near-fault events can change the ductility demand and damage distribution in the structures. Therefore, when it comes to record-to-record variability, the present study aimed to focus on the near-fault events due to the shortage of researches compared to the far-field events. In this study, similar to Ji et al. (2009) and Emami and Halabian (2017), 10 ground motion records are chosen as the seismic scenario.

Asymmetric structure model

The SSI is more significant for the short-period structures (Wolf, 1985). Among the building structures, reinforced concrete shear-wall structures have more stiffness and, for a constant elevation and mass, they have the minimum period of vibration. Therefore, an eight-story R/C dual lateral load-resistant building that consists of shear walls and moment resisting frames in both principal directions of the structure (Halabian and Birzhandi, 2014) is used to evaluate the effects of SSI and torsion simultaneously. The design procedure of this structure and its properties can be found in the previous study (Halabian and Birzhandi, 2014). To review this procedure briefly, it must be mentioned that in all stories, the story height is assumed to be the same and equal to 3.0 m. The plan of this asymmetric structure is shown in Figure 1. All stories are assumed to be rigid diaphragm in plane. In the design of moment frame, the generic frame technique (Alavi and Krawinkler, 2001) was used that leads to a uniform elastic drift profile along the structure height under the Square Root of the Sum of the Squares (SRSS) load pattern of Uniform Building Code (UBC) (International Conference of Building Officials (ICBO), 1997) design spectrum and avoids the local mechanism in different stories. Then to produce a dual structure, shear walls were added to the generic frames so that the stiffness ratio of shear walls equal to 75% of the total lateral structural stiffness is produced. Considering the conclusion of previous studies such as Fernández-Dávila and Cruz (2006), that the number of lateral load resisting planes (more than three planes) in the direction of excitation has no significant effect on the torsional response, three shear walls have been arranged in three separate planes parallel to the direction of excitation. In the seismic design of the structure, the base shear strength factor (C) was assumed to be equal to 0.10. The frame sections (beam and column), wall sections, and longitudinal reinforcement are presented in Table 1. It must be noted that in the generic frame technique (Alavi and Krawinkler, 2001), all frame sections of one story (beams and columns) must have the same dimensions. All of the seismic provisions of ACI 318M-11:2011 (2011) code such as demand-to-capacity ratio (D/C) of shear-wall boundary elements were satisfied.

Figure 1.

Plan of the eight-story concrete dual asymmetric structure (Halabian and Birzhandi, 2014).

Table 1.

Frame sections (beam and column), wall sections, and longitudinal reinforcement.

Story level	Frame sections (mm)	Wall thickness (mm)	Column reinforcement percentage
Story level	Frame sections (mm)	Wall thickness (mm)	Shear-wall column	Frame column
Story 8	296 × 296	223	1.0	3.9
Story 7	346 × 346	223	1.0	2.1
Story 6	368 × 368	223	1.2	1.6
Story 5	383 × 383	223	2.2	1.1
Story 4	392 × 392	223	3.3	1.0
Story 3	404 × 404	223	4.0	1.1
Story 2	405 × 405	223	4.0	1.9
Story 1	405 × 405	223	4.0	2.8

The building eccentricity is caused by the mass eccentricity. The stiffness of the structure is assumed symmetric. Because the eccentricity is assumed in one direction (one-way asymmetric structure) and the excitation is applied in the x-direction, if the shear walls of the orthogonal direction (y-direction) are present in the model, they will remain elastic and impose unrealistic torsional stiffness. Therefore, the shear walls of the orthogonal direction are eliminated to avoid the unrealistic torsional stiffness that they will produce. In the previous studies such as Wong and Tso (1994), Correnza et al. (1994), Riddell and Santa-Maria (1999), Humar and Kumar (1999), and Halabian and Birzhandi (2014), it has been concluded that for structures with the shear-wall resisting system, one can consider one-directional excitation if the shear walls of the orthogonal direction are removed from the model to avoid the additional torsional stiffness that they will produce. In other words, if the orthogonal excitation is applied simultaneously, the orthogonal shear walls enter the inelastic region and their stiffness will reduce significantly. Therefore, one can remove the orthogonal excitation and orthogonal shear walls simultaneously.

The periods of the second and third (translational and rotational) mode shapes of the structure for different values of eccentricity and different types of support conditions are given in Table 2 (the first mode of structure is translational in the y-direction and not excited). Dominant motion of the second mode is x-translation (x) and that of the third mode is z-rotation (r). As can be seen in Table 2, the structure is basically a short-period structure, and therefore the SSI might have significant effects on its seismic response (Wolf, 1985). It must be noted that SSI effect is only considered for the x-translation, y-rocking, and z-rotation. The damping matrix of the superstructure is assumed as Rayleigh’s stiffness and mass proportional damping ( $C = α M + β K$ ), with $α = 0.7239$ and $β = 0.0022$ , so that the damping ratio of the first and fifth modes was taken equal to 5%.

Table 2.

Structural periods for the second and third mode shapes of the structure (SSI effect only considered for the x-translation, y-rocking, and z-rotation).

Support	Fixed base		Vs = 200 m/s		Vs = 100 m/s
Mode number	2	3	2	3	2	3
Dominant motion	X	r	X	r	X	r
e = 0.00	0.673	0.565	0.801	0.592	0.954	0.634
e = 0.05	0.688	0.558	0.822	0.576	0.972	0.636
e = 0.10	0.727	0.544	0.847	0.561	0.991	0.639
e = 0.15	0.781	0.53	0.872	0.545	1.016	0.642
e = 0.20	0.846	0.519	0.902	0.53	1.041	0.645
e = 0.25	0.918	0.51	0.931	0.521	1.074	0.649

SSI: soil–structure interaction.

To include SSI using the substructure method, the stiffness and damping of the foundation should be available. The DYNA5 (Novak et al., 1993) program was used to calculate the foundation impedance functions (Table 3). To calculate these functions in this semi-analytical method, the Poisson’s ratio of the soil medium was assumed to be 1/3, and the density of the soil medium was taken to be 1900 kg/m³. To evaluate the effect of foundation flexibility on seismic response of plan-asymmetric structures resting on soft soils, two values for the soil shear wave velocity ( $V_{s}$ ) are taken (100 and 200 m/s).

Table 3.

Stiffness and damping of the supporting foundation.

	$K_{T} (N / m)$	$K_{Z} (N / m)$	$K_{r} (N / m)$	$C_{T} (N . s / m)$	$C_{Z} (N / m . s)$	$C_{r} (N / m . s)$
Vs = 100 m/s	1.16E+09	2.52E+11	1.60E+11	1.31E+08	6.85E+09	7.11E+09
Vs = 200 m/s	4.70E+09	1.11E+12	7.52E+11	2.91E+08	1.52E+10	2.01E+10

The nonlinear response history analysis (NRHA) is carried out in the 3D nonlinear static/dynamic structural analysis computer program called Canny (Li, 2002). The nonlinear modeling of the superstructure was explained in the previous study (Halabian and Birzhandi, 2014). Additional substructure system is modeled using impedance functions (Table 3) of foundation. Because of the eccentricity in the structure plan, the NRHA is a 3D analysis with a 3D model. It must be noted briefly that nonlinear modeling of superstructure includes different types of plastic hinge for different types of elements (Halabian and Birzhandi, 2014).

For the beams, the moment–curvature model is used at the ends to model the plasticity of beams. The moment–curvature model $(M - ϕ)$ , because of its simplicity and low computational cost, is the most practical approach used to express the nonlinear behavior of beam plastic hinge. So, this method was used to model nonlinear behavior of beam elements. The moment–curvature relationship for a section at a constant axial load was calculated using material stress–strain relationships. For the plastic hinge length in the RC beams, Paulay and Priestley (1992) proposed the following relationship that L is the distance between the support and zero moment point of the beam. The $d_{b}$ is the flexural reinforcement diameter and $f_{y}$ is its yield stress

L_{p} = 0.08 L + 0.022 d_{b} f_{y}

(1)

For the traditional dimension and reinforcement, they proposed the half of the beam section depth for the plastic hinge length as a simple estimation.

For the columns, because of the presence of axial forces and biaxial bending moments, the moment–curvature model does not have enough accuracy. Therefore, the fiber model (multi-spring model) that incorporates the axial forces and biaxial bending moments employed for the nonlinear behavior of columns. The shear walls are modeled using panel element combined with fiber model for the axial-bending behavior and nonlinear spring for the in-plane shear behavior. The fiber-based multi-spring model is a useful technique to simulate the columns and shear walls in the similar studies such as Tang and Zhang (2011), Emami and Halabian (2017, 2018), Brunesi et al. (2016), and Brunesi and Nascimbene (2017). The out-of-plane behavior is assumed elastic. For the plastic hinge length of the RC shear walls, the Paulay and Priestley (1992) recommendation (following relationship) was used

L_{p} = 0.2 D_{W} + 0.044 h_{e}

(2)

where $D_{W}$ is the wall section depth and $h_{e}$ is the effective wall height.

A multi-linear backbone curve adjusted to the stress–strain model proposed by Mander et al. (1988) was used for the confined concrete of shear walls and CS3 hysteretic model (Li, 2002) with stiffness degradation used for hysteretic behavior. This model was verified by Archila (2011). The A706 steel material specifications with a trilinear backbone curve is used for stress–strain relationship of reinforcing bar employing adjusted post-yield stiffness and cyclic deterioration proposed by Orakcal and Wallace (2006). Elastic-perfect-plastic model was established for in-plane shear behavior of shear walls using ACI 318M-11:2011 (2011) code. The nonlinear modeling of structure can model the extensive range of nonlinear behavior from life safety (LS) to collapse prevention (CP) and global instability of structure.

Earthquake excitations

The pulse-like ground motions include strong velocity pulses, such as those imposed by near-fault directivity. In these motions, $T_{p}$ , the period of the largest velocity pulse, can be extracted using velocity spectrum (Alavi and Krawinkler, 2001) or wavelet analysis (Baker, 2007). The research conducted by Champion and Liel (2010), whereas IDA curves and fragility assessment were employed, showed that under the near-fault pulse-like events, the collapse capacity of the structure depends on $T_{p} / T_{1}$ ratio, whereas T₁ is the fundamental natural period of the structure. It must be noted that by entering the structure into the nonlinear region, the stiffness of the structure is reduced and subsequently the fundamental natural period of the structure is increased. Therefore, the records with the pulse period larger than the fundamental natural period of the structure should also be taken into account in the analysis procedure. But the results presented by Champion and Liel (2010) showed that for $T_{p} / T_{1}$ ratio greater than two, an increase in T_p does not significantly affect the probability of structural collapse. Considering this fact and the periods of structures (listed in Table 1), 10 records have been selected (listed in Table 4). Other criteria are listed as follows:

The R_rup (the closest distance to co-seismic rupture) is less than 13 km.

Moment magnitude (M_w) is greater than 6.5, according to the Federal Emergency Management Agency (FEMA, 2009) P695 recommendation.

To simulate the more realistic conditions of the soft soil for SSI effects, the site shear wave velocity ( $V_{S, 30}$ ) of the selected ground motions is limited to the 350 m/s (the $V_{S, 30}$ is the average shear wave velocity over a subsurface depth of 30 m).

Table 4.

Earthquake excitations.

	Event	Station	Year	T_p (s)	PGA (g)	Vs (m/s)	M_w	R_rup
1	Dezce-Turkey	Bolu	1999	0.88	0.739	293	7.1	12.04
2	Kobe	KJMA	1995	1.09	0.821	312	6.9	0.96
3	Northridge	Pardee—SCE	1994	1.23	0.557	325	6.69	7.46
4	Northridge	Rinaldi Receiving	1994	1.246	0.825	282	6.69	6.5
5	Tottori, Japan	TTR008	2000	1.54	0.391	139	6.88	6.61
6	Kobe	Takatori	1995	1.55	0.618	256	6.9	1.47
7	Loma Prieta	Gilroy Array #2	1989	1.729	0.370	270	6.93	11.07
8	Montenegro, Yugoslavia	Ulcinj—Hotel Olimpic	1979	1.97	0.293	318	7.1	5.76
9	Imperial Valley-06	Agrarias	1979	2.34	0.287	242	6.53	0.65
10	Superstition Hills-02	Parachute Test Site	1987	2.39	0.432	349	6.54	0.95

PGA: Peak Ground Acceleration.

The elastic pseudo-acceleration spectrum and pseudo-velocity spectrum of the selected near-field records (5% damping assumed) are presented in Figure 2.

Figure 2.

Elastic pseudo-acceleration spectrum and pseudo-velocity spectrum of the selected records (5% damping assumed).

Detailed and median IDA curves

In recent years, the extensive tendency of researchers is oriented to the IDA. Now, the IDA is a traditional tool for the seismic assessment of different structures (Emami and Halabian, 2017; Sadraddin et al., 2016). The IDA is a parametric analysis method to determine the seismic dynamic capacity of structures in different levels of performance. The main reason is the significant development in the nonlinear dynamic analysis software and capability of the new generation of computers. The result of IDA is the variation of DM as a function of IM. In previous studies, different damages and IMs were used. According to the definition, a monotonic scalable ground motion IM (or simply IM) of a scaled accelerogram is a nonnegative scalar that constitutes a function that depends on the unscaled accelerogram and is monotonically increasing with the scale factor (Vamvatsikos and Cornell, 2002). Traditional types of scalable IMs are the PGA, Peak Ground Velocity, and the Spectral Pseudo-Acceleration of the first mode of the structure with assumed modal damping usually 5% ( $S_{a} (T_{1}, 5 %)$ ). The last is more efficient because $T_{1}$ (the first period of the structure) can represent the fundamental properties of structure such as mass and stiffness. Therefore, its contribution in the IM can improve the judgment accuracy compared to the IM such as PGA that only depends on the excitation. Different cases of base conditions and eccentricity will be investigated in this study that leads to different structural periods (Table 2). These structural differences must be considered in the selected IM. Therefore, $S_{a} (T_{1}, 5 %)$ is an appropriate IM that can be used.

Moreover, according to the definition, DM is a nonnegative scalar that characterizes the additional response of the structural model due to a prescribed seismic loading (Vamvatsikos and Cornell, 2002). Traditional examples are the maximum base shear, maximum story ductility, maximum roof drift, the floor maximum inter-story drift angles, or their maximum and some proposed damage indices such as Park–Ang damage index. Usually, the comparison of drift of asymmetric structures with the symmetric one is a traditional method to investigate the effects of torsion. Therefore, the most popular DM extensively used in the asymmetric structure studies is drift in the form of maximum inter-story or overall drift because the example structure studied in this article was dual lateral load-resistant R/C asymmetric building and the cantilever shear walls impose their cantilever behavior on the seismic behavior of the structure. Furthermore, the moment frame is designed using generic frame technique that avoids the local mechanism in different stories (Alavi and Krawinkler, 2001). Therefore, the overall drift can be used as an efficient DM. A comparison between the use of maximum inter-story and overall drift as DM (Figure 3) shows that local mechanism in different stories is not seen and using overall drift or maximum inter-story drift leads to the same pattern of the IDA curve. Because the structure is asymmetric, for different points of plan of each story, different values of drift will be calculated. Usually, maximum drift occurs in the flexible edge of the plan. Therefore, it is expected that the maximum overall drift occurs at the plan edge. Thus, overall drift for all of the plan joints is recorded and their maximum is assumed as DM. It must be noted that in Figure 3, the PGA is used as IM for an initial investigation on the DM in a clearer condition. In the continuing IDA curves, the $S_{a} (T_{1}, 5 %)$ will be used as IM.

Figure 3.

IDA curves of symmetric flexible base structure (Vs = 100 m/s) with different DM: (a) maximum inter-story drift ratio and (b) overall drift ratio.

Seismic design references for reinforced concrete structures such as Paulay and Priestley (1992) emphasize that main desirable mechanism of energy dissipation in a laterally loaded cantilever wall must be the plastic behavior of the flexural reinforcement in the plastic hinge (at the base of the wall). Therefore, the failure modes such as instability of principal compression reinforcement or of the thin walled sections, diagonal tension, or compression caused by shear, shear or bond failure along lapped splices, or anchorages and sliding shear along construction joints must be prevented. Considering these principles of seismic design, in this study it is assumed that the above-mentioned undesirable failure modes are prevented with the complete detailing and only the allowable plastic behavior (yielding of the flexural reinforcement in the plastic hinge) will be occurred in the high level of excitation. But to monitor any possible global shear failure, in-plane shear behavior was modeled as elastic-perfect-plastic with nominal strength computed based on the ACI 318M-11:2011 (2011) code. Observations show that lateral collapse of this structure usually occurs due to failure of flexural reinforcement of shear wall in the plastic hinge (at the first story), and in few cases, the in-plane shear failure of shear wall (at the first story) leads to the lateral collapse. Therefore, the selected DM (overall drift) can properly represent the CP limit state because of the cantilever behavior that is affected by the plastic hinge at the base. Furthermore, the structural damage can be computed for a section, member, story, or structure. But for RC shear-wall structures, the story and global damage indexes (such as story ductility demand or drift and global ductility demand or drift) are traditional damage indexes in the previous studies (Emami and Halabian, 2017, 2018; Halabian and Birzhandi, 2014; Tang and Zhang, 2011). It must be noted that for dual structures that consist of shear walls and moment frames, use of local damage indexes of a shear-wall section (such as plastic hinge rotation) may not be a good representative for CP damage index because of the moment frame (secondary system) action.

In this section, the detailed IDA curves of the fixed and flexible base structure with different values of eccentricity will be presented in Figures 4 to 6 using $S_{a} (T_{1}, 5 %) (g)$ as IM and overall drift as DM. It must be noted that for the flexible base models, the foundation deflections are eliminated from the displacement to determine the relative displacement of superstructure from the base as a DM. To investigate the effects of foundation flexibility on seismic response, three cases are considered: fixed base structure, and soil shear wave velocity ( $V_{s}$ ) equal to 100 and 200 m/s. In addition, to evaluate the effect of torsion, different mass eccentricities are included: 0, 0.05, 0.10, 0.15, 0.20, and 0.25. Because of the multiplicity of the curves, detailed IDA curves are presented for some limited cases. According to Figures 4 to 6, using $S_{a} (T_{1}, 5 %)$ shows significant differences between different support conditions because of the SSI effects in some of the excitations. To extract the median IDA curves, the values of DM from the different excitations for a constant IM are sorted in an ascending order. Then, among these 10 values, the average of the fifth and the sixth data is calculated as the data median value.

Figure 4.

IDA curves of symmetric structures using S_a(T₁, g) and overall drift ratio: (a) fixed base and (b) flexible base (Vs = 100 m/s).

Figure 5.

IDA curves of asymmetric structures (e = 0.10) using S_a(T₁, g) and overall drift ratio: (a) fixed base and (b) flexible base (Vs = 100 m/s).

Figure 6.

IDA curves of asymmetric structures (e = 0.25) using S_a(T₁, g) and overall drift ratio: (a) fixed base and (b) flexible base (Vs = 100 m/s).

If the median IDA curves of different support conditions and different values of eccentricity are summarized in Figure 7, it will be seen that median IDA curves (as dynamic capacity curves of structure) of symmetric structure report the maximum capacity for each support condition and for all limit states (immediate occupancy (IO), LS, and CP). Because of the constant damage (drift), the symmetric structure (e = 0.00) reports the maximum intensity (S_a). In other words, for the constant intensity, it experienced the minimum damage. As shown in Figure 7, an increase in the mass eccentricity leads to a reduction in the structure capacity. In addition, the capacity of structure will be increased when more shear wave velocity is assumed. Therefore, if the SSI effects are neglected and the structure is modeled fixed base, the dynamic capacity of structure will be calculated more than the realistic dynamic capacity. Therefore, from this view, neglecting the foundation flexibility will not be in the safe side.

Figure 7.

Median IDA curves using S_a(T₁, g) and overall drift ratio: (a) fixed base, (b) flexible base (Vs = 200 m/s), and (c) flexible base (Vs = 100 m/s).

For the lower limit states, FEMA (2000a) 356 and ASCE/SEI 41-06:2009 (2009) recommend 0.5% and 1.0% inter-story drift ratio (IDR) for IO and LS performance level. Also Kircher et al. (1997) proposed IDR of 0.4%, 0.8%, and 2.3% for the slight, moderate, and extensive and complete damage states for low-rise concrete shear walls designed to the UBC (ICBO, 1997) requirements. Considering these range of IDR and Figure 3 (that shows the relation between the IDR and overall drift), it is concluded from Figure 7 that eccentricity value is less effective in the lower limit states (IO and LS) especially for the fixed base structure and the structure with the $V_{s}$ equal to 200 m/s. But the above-mentioned general trends for the effects of eccentricity and SSI are seen in these limit states; because when eccentricity is decreased and more shear wave velocity is assumed, more structural capacity will be reported.

As seen in all of the detailed and median (summarized) IDA curves, the effects of eccentricity are very different from one record to another. For example, in Figure 8, the IDA curves of asymmetric fixed base structure are shown for two different excitations (Northridge—Rinaldi and Loma Prieta—Gilroy Array #2). It is seen that for the Northridge earthquake, unlike the Loma Prieta earthquake, no significant differences are seen in different values of eccentricity (especially for the CP). It is an example of the important record to record uncertainty that encourages this study to use probabilistic framework.

Figure 8.

IDA curves of fixed base asymmetric structures using S_a(T₁, g) and overall drift ratio under the (a) Northridge (Rinaldi) and (b) Loma Prieta (Gilroy Array #2) excitations.

Probabilistic studies (Fragility curves)

The probabilistic analysis of structures under the earthquake excitations has recently become an attracted approach, and generalized probability density evolution method (Li and Chen, 2006) has been welcomed to use in the seismic reliability assessment of structures (Pang et al., 2018a) and stochastic seismic performance assessment (Pang et al., 2018b).

The fragility curves are the traditional tool of probabilistic assessment in earthquake engineering, and translation error has also been used in the latest researches such as Pang et al. (2018c). To extract the fragility curves, definition of the limit state is the main step. To define a limit state, one can use the Engineering Demand Parameter (EDP) or directly the IM. The first method is more practical and its criteria can be found in references such as FEMA (2000a) 356 and ASCE/SEI 41-06:2009 (2009). However, when the CP limit state is considered and a nonlinear modeling with appropriate accuracy is available, the second method (use of IM directly) is more accurate. Furthermore, if the EDP such as drift is used to extract the fragility curves of asymmetric structures, usually the peak value of drift is recorded in the flexible edge. This fragility curve may be improperly interpreted because in the CP, usually the flexible edge drift of asymmetric structure is more than the symmetric structure. However, it does not mean that the capacity of asymmetric structure is more than the symmetric one. Therefore, if the IM is used directly, this misunderstanding will be avoided. Thus, for asymmetric structures, using IM directly (IM-based approach) is more accurate.

In the IM-based approach, the collapse IM ( $I M_{C}$ ) must be defined exactly. The FEMA (2000b) 350 recommends the 20% tangent slope approach as an IM-based rule. It is based on the fact that the slope of IDA curve can represent the stiffness of the structure. This approach considers the $I M_{C}$ be corresponding to the last point where the tangent slope of the IDA curve (as an index of structure stiffness) is 20% of the elastic slope. This definition must be used carefully because it will be possible that after a significant reduction in the slope of the curve, a hardening may happen and the structure can stand the higher level of IM before the collapse. Therefore, the solution must be continued for the higher level of IM to be sure that this curve slope reduction leads to the collapse. Considering this phenomenon, the FEMA (2000b) 350 emphasizes that the last point with the tangent slope equal to 20% of the elastic slope is defined as capacity point.

ASCE/SEI 41-06:2009 (2009) proposed the 2.0% IDR for the CP limit state of concrete wall structures that related to extensive concrete crushing and buckling of reinforcement. But previous studies such as Kircher et al. (1997), Mwafy (2010), and Emami and Halabian (2017) emphasize that the code recommended drift limits tend to be on the conservative side and do not necessarily imply actual collapse. For example, Kircher et al. (1997) proposed IDR of 6.0% for the complete damage state for low-rise concrete shear walls designed to the UBC (ICBO, 1997) requirements. It is concluded in the recent researches such as Emami and Halabian (2017) that the actual collapse sometimes occurs in the higher level of drifts that excessive lateral displacements lead to lateral dynamic instability resulting in flattening of IDA curve (global instability of structure).

Therefore, in this study, three criteria are considered to recognize the CP IM:

The last point where the tangent slope of the IDA curve is 20% of the elastic slope.

Maximum inter-story drift is equal to 10%.

Global instability of structure.

To predict the probabilistic function for the collapse of structures, the lognormal distribution is the best selection in earthquake engineering studies. This function can be expressed in equation (3)

P = LogCDF (x; η, β)

(3)

where CDF stands for the cumulative distribution function, x is the value of random variable, $η$ is the mean, and $β$ is the standard division. The robust fitting of fitted functions was examined in terms of a correlation coefficient denoted as $R_{s}$

R_{S} = 1 - \frac{Residual Sum of the Squares}{Corrected Sum of the Squares}

(4)

The parameters of lognormal concentrated distribution function for collapse of different cases are listed in Table 5. In addition, the fragility curves are shown in Figure 9, and they include the exact values and predicted values. This figure shows that an increase in the eccentricity value leads to a reduction in the effects of support conditions. It may occur due to the changes in the period of vibration of structures with different eccentricities because an increase in the eccentricity value makes an increase in the first period of vibration. On the other hand, less shear wave velocity increases the first period of vibration, too. Therefore, if the mass eccentricity increases the first period of vibration, the increase due to the support conditions will be ineffective. In other words, the stiffness of structure will be reduced and the structure does not have the pervious rigidity to take effect from the SSI. Thus, the effects of SSI on it will be declined.

Table 5.

Parameters of lognormal concentrated distribution function for collapse of different cases.

Eccentricity	Fixed base			Vs = 200 m/s			Vs = 100 m/s
Eccentricity	η	β	R_s	η	β	R_s	η	β	R_s
e = 0.00	3.501	0.602	0.973	3.504	0.561	0.982	3.059	0.486	0.961
e = 0.05	3.543	0.561	0.955	3.582	0.496	0.958	3.062	0.476	0.977
e = 0.10	3.64	0.505	0.948	3.504	0.477	0.962	3.08	0.463	0.971
e = 0.15	3.532	0.483	0.964	3.463	0.47	0.969	3.222	0.437	0.954
e = 0.20	3.511	0.479	0.981	3.364	0.461	0.987	3.317	0.413	0.982
e = 0.25	3.425	0.461	0.966	3.254	0.436	0.956	3.404	0.393	0.978

Figure 9.

Fragility curves (probability of collapse) for different values of eccentricity using S_a(T₁, g) and lognormal CDF: (a) e = 0.00, (b) e = 0.05, (c) e = 0.10, (d) e = 0.15, (e) e = 0.20, and (f) e = 0.25.

The other fact seen in Figure 9 shows that for each eccentricity, the fragility curve of structure with the shear wave velocity equal to 200 m/s is closer to the fragility curve of fixed base structure. In other words, SSI effect is significant only for the flexible base structure with the shear wave velocity equal to 100 m/s.

The fragility curves are categorized from another view and are arranged according to the support conditions in Figure 10. In each case, different values of eccentricity are considered. It is seen that for the fixed base structure and flexible base structure with Vs = 200 m/s, increase in the eccentricity leads to an increase in the probability of collapse especially for the high-intensity earthquakes. This observation is predictable and compatible with the nature of torsion. However, in the flexible base structure with Vs = 100 m/s, the probability of collapse of asymmetric structure is equal to or less than the symmetric one. It may be due to this structure’s condition and may not be a general observation. Although, similarly in the old studies about the torsion (Erdik, 1975; Kan and Chopra, 1981), it was explained that torsional effects in the inelastic range are less than the elastic range. This idea is emphasized in the newer researches (Aziminejad and Moghadam, 2010) that increase in the intensity level will reduce the torsional effects (differences between the drift of symmetric and asymmetric structures).

Figure 10.

Fragility curves (probability of collapse) for different support conditions using S_a(T₁, g) and lognormal CDF: (a) fixed base, (b) flexible base (Vs = 200 m/s), and (c) flexible base (Vs = 100 m/s).

As seen in Table 5, for high eccentricity value, the effect of base condition on the mean value of fragility function ( $η$ ) will decrease. It means that plan asymmetry can lead to reduce the SSI effect. This phenomenon may be due to the fact that in the eccentric plan structure with the significant eccentricity, the contribution of torsional response in the lateral displacement will increase and the contribution of translational response in the lateral displacement will decrease. Therefore, the SSI phenomenon may be less effective for the structure with the significant eccentricity.

Also for this structure, the standard division ( $β$ ) of fragility function is decreased when the eccentricity value is increased. It means that the median IDA curves are more valid for the high eccentricity value, or in the worst case, an increase in the plan eccentricity dose not decrease the validity of median IDA curves.

These observations give the idea that in the practical engineering aims, for the asymmetric plan shear-wall structures with the high eccentricity, one can use the fixed base condition (instead of flexible base) and median IDA curves (instead of fragility curves).

Conclusion

In the deterministic and probabilistic past researches, considering the SSI effects in the inelastic asymmetric structures is neglected. Therefore, this study aims to evaluate the torsional effects and SSI simultaneously under the near-fault pulse-like events in a probabilistic framework. For this purpose, an eight-story R/C dual lateral load-resistant building that consists of shear walls and moment resisting frames is used. To evaluate the effects of SSI on seismic response, three cases are included: fixed base structure, and soil shear wave velocities equal to 100 and 200 m/s. In addition, to investigate the effect of torsion, different mass eccentricities are considered: 0, 0.05, 0.10, 0.15, 0.20, and 0.25. The IDA curves of structures were presented using $S_{a} (T_{1}, 5 %) (g)$ as IM and maximum overall drift as DM.

If the median IDA curves are assumed as dynamic capacity curve of structure, the symmetric structure reports the maximum capacity for each support condition as predicted. To overcome record-to-record uncertainty, fragility curves using IM directly are employed. They show that as expected, for the fixed base structure and flexible base structure with Vs = 200 m/s, more value of eccentricity leads to more probability of collapse especially for the high-intensity earthquakes. However, in the flexible base structure with Vs = 100 m/s, the asymmetric structures are safer than the symmetric ones (from the probabilistic view). Thus, just for this case of support conditions, IDA curves and fragility curves do not have the same manner to predict the effects of torsion.

Using IDA curves shows that if the SSI effects are neglected and the structure is modeled fixed base, the dynamic capacity of structure will be evaluated more than the realistic dynamic capacity. In addition, using fragility curves shows that for each eccentricity, SSI effect is significant only for the flexible base structure with the shear wave velocity equal to 100 m/s, and more eccentricity value leads to less SSI effects. Considering the plan-asymmetry and SSI effect simultaneously shows that significant eccentricity value may lead to reduce the SSI effect in the shear-wall structures under the near-fault events.

Despite efforts made to extract approximate IDA and fragility curves of symmetric structures, the proper method has not been developed for the plan-asymmetric structures. The development of approximate IDA and fragility curves for flexible base asymmetric structures to evaluate their probabilistic manner faster is the aim of future works.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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Probabilistic assessment of plan-asymmetric structures under the near-fault pulse-like events considering soil–structure interaction

Abstract

Keywords

Introduction

Asymmetric structure model

Earthquake excitations

Detailed and median IDA curves

Probabilistic studies (Fragility curves)

Conclusion

Footnotes

Declaration of Conflicting Interests

Funding

References