New proposed drift limit states for box-type structural systems considering local and global damage indices

Abstract

The maximum displacement responses under the seismic motions are usually considered as an indicator for damage evaluation. It is obvious that appropriate selection of drifts corresponding to various damage levels plays an important role in safety and economy of a design project. Despite the extensive use of the box-type structural system in mass construction and housing industry, there is no special design requirement for this structural system. Due to three-dimensional behavior and interaction of intersecting walls and slabs, it is expected that this system presents different seismic performance in comparison to the conventional shear wall buildings. This study evaluates the overall and story failure mechanism as well as global and local damage indices in this structural system. Maximum allowable drift ratios of 0.45%, 0.65%, and 0.8% are suggested for the immediate occupancy, life safety, and collapse prevention levels, respectively. Moreover, a damage index based on the maximum relative inter-story drifts is proposed to assess the failure in the height domain. According to the assessments, the story and global failure occurring due to considerable damages in main load bearing elements reveals high importance of local damage indices in box-type structural system. Based on the results, it is concluded that the proposed maximum values for drifts in different standards and codes are not reliable. Considering the shear-control behavior and depending on the expected performance levels, the proposed local damage indices are considered as accurate control indicators for box-type structural system.

Keywords

box-type structural system damage index drift limit state performance level

Introduction

In earthquake engineering, over the past four decades, the maximum stories drift profile has been used to estimate damage in building structures subjected to the earthquake ground motions. Inter-story drift is a very important parameter for evaluating damage demand under the seismic loading. Hence, the correct and precise determination of these values and their allowable performance limits plays an important role in the analysis and design of building structures.

The new reinforced concrete (RC) box-type structural system, consisting of shear wall and slab, is considered as an industrialized construction technique. Performance of this structural system is based on the interactions of thinner shear wall and slab configurations as the common components of the box-type structural system. In other words, slabs and walls are used as vertical and lateral load bearing elements, which will be casted simultaneously. As a result of this, the seismic performance of the buildings improves considerably, and in addition, number of the cold joints reduces in comparison with the other conventional RC structural systems (Figure 1).

Figure 1.

Box-type formworks.

Although similarities exist between the box-type structural systems and lightly RC walls (Brunesi et al., 2016), the walls in box-type structures are completely filled. There are many differences between these two systems in terms of seismic weight and capabilities. Accordingly, in the technical literature, the selected structural system in this study introduces “tunnel-form” system as well.

High strength and efficiency of the box-type structural system were proved during two earthquakes Kocaeli (M_w = 7.4) and Düzce (M_w = 7.2) in 1999 in Turkey. Field evaluations showed that this structural system demonstrates much better seismic performance in comparison with the other conventional RC frame-type and dual systems (Balkaya and Kalkan, 2004).

In spite of the intensive applying box-type structural system within the construction projects, this technique is not considered as an independent system within the current codes. Most of the past studies regarding this system have been restricted to some relationships for estimating natural vibration period. These studies showed that available equations in seismic codes offer inaccurate results for determining the vibration period of box-type structural systems (Goel and Chopra, 1998; Lee et al., 2000).

Balkaya et al. (2012) examined the effect of soil–structure interaction on the dynamic characteristics of box-type buildings with totally different plans and heights. They developed a relationship, which estimates vibration period of these structures considering soil–structure interaction.

According to Tavafoghi and Eshghi’s (2008) investigation, fundamental period of box-type buildings in each direction relies directly upon structural height as well as aspect ratio. However, shear wall ratio does not have any impacts. The main characteristic of this structural system is shear dominant and brittle behavior. Shear punching occurs in the slab-wall connection area (Eshghi and Tavafoghi, 2012). Utilizing the method proposed by ATC-63, they concluded that response modification factor of 4 leads to reasonable results for this structural system (Tavafoghi, and Eshghi, 2013).

Balkaya and Kalkan (2003, 2004) suggested response reduction factors of 5 and 4 for low- and high-rise box-type buildings, respectively. They showed that the governing behavior of this structural system is membrane action and tensile–compressive performance. Therefore, shear walls play a key role in the load-bearing capability of the system.

Based on the experimental evaluation, Yuksel and Kalkan (2007) developed a numerical model for box building structures. They found that adding the concentrated longitudinal bars at the edge of shear walls prevents brittle performance even in the case of low reinforcement ratio (Kalkan and Yuksel, 2007).

İlerisoy and Tuna (2013) showed that errors created in determining the soil category of tunnel-form buildings could result in poor decisions throughout selecting the number range of floors and within the design of structural components and foundations of buildings.

Performance of tunnel-form buildings was compared to that of beam–column framed buildings by Sasidharan and Aslam (2015). They concluded that tunnel-form buildings experience lower story displacement, lower inter-story drift ratio, and higher base shear when compared to framed buildings. Opening locations also affect the performance of tunnel-form buildings under the dynamic loading.

Beheshti-aval et al. (2018) evaluated the seismic performance of tunnel-form system subjected to a set of near and far-field earthquake records. This study revealed that the forward directivity can influence the failure modes and reduce the design reliability.

Mohsenian et al. (2018) analyzed the seismic vulnerability of tunnel-form structures under the accidental eccentricities of mass and stiffness. They found that the structural responses are not influenced by the mentioned eccentricities and their configuration.

Mohsenian and Mortezaei (2018) evaluated the seismic behavior of box-type (tunnel-form) buildings under the accidental torsion caused by asymmetric mass distribution using incremental dynamic analysis (IDA). The results demonstrated high capacity and appropriate seismic behavior of box-type structural system, despite the asymmetric mass distribution. In other words, an eccentricity equal to 10% of the plan dimension does not change the building performance level. In a follow-up study, Mohsenian and Mortezaei (2019) proposed a replaceable steel link beam instead of concrete one in such a way that the damages could be optimally distributed in plan and height of tunnel-form buildings. They concluded that the coupling link beam can considerably affects the structural response of the box-type system.

Based on the studies that have been conducted in literature, there is very limited information about seismic performance of this structural system. Based on this fact, seismic performance evaluation of this type of structures is very substantial for developing seismic design code considering the effects of various parameters. In this study, failure consequence of the system in the nonlinear range is evaluated, and some allowable limit states are proposed for the story drift corresponding to different performance levels. Then, the suggested limit states in this study are compared with the corresponding ones in some modern seismic design codes. Proposing a simple damage index (DI) based on the relative inter-story drifts is the final evaluation step of this research work. It must be noted that the proposed DI is verified by means of incremental dynamic and push-over analyses as well as fragility curves.

Methodology

Properties of selected models

Plans of selected buildings which are used for this study are presented in Figure 2. It is evident that the selected plans are regular and symmetric in both directions.

Figure 2.

Plans of the selected box-type buildings: (a) preliminary evaluation, and (b) practical examination.

It is noteworthy that the plan (a) is used for preliminary evaluation of the failure mechanism, proposing allowable inter-story drift and finally suggesting a DI for box-type structural system.

In order to investigate the effects of asymmetry on the behavior of intersecting walls, three different combinations of wall length (L_w₁ and L_w₂) are utilized in this plan, while length of the link beam is kept constant. The detailed plan (b) is used as a practical example for verifying the proposed drift values and error estimation of the proposed DI in predicting damage extension of box buildings.

Dashed lines in the plan specify the location of the link beams over the opening whose length and depth are equal to 1 and 0.7 m, respectively. In order to study the effects of height on structural behavior in modeling, the maximum height of 30 m is considered. Buildings that located in the area with high seismicity are assumed to be residential with the story height of 3 m. It is also assumed that the site soil in is type II category ( $375 (m / s) \leq V_{s} \leq 750 (m / s)$ ) according to the Iranian code of practice for seismic resistant design of buildings (Permanent Committee for Revising the Standard 2800, 2014).

The models are designed according to the American provisions for design of concrete structures (ACI Committee 318-14, 2014). All requirements for design of box-type structural systems are also considered. It must be noted that the R-factor for these buildings are taken 5, which is commonly used by designers. In the structural modeling, the shell behavior is considered for walls and slabs (in-plane and out-of-plane deformations simultaneously) and the optimum finite element mesh is derived by try and error process. The thickness (t) of 20 cm was considered for all walls, and as illustrated in Figure 3, vertical and horizontal reinforcing bars ( $ϕ_{v}$ and $ϕ_{H}$ ) are placed in two layers. It should be noted that the diameter of all bars is equal to 8 mm, and the vertical bars in the first four stories of the taller building are of 12 mm in diameter.

Figure 3.

Schematic representation of detailing and arrangement of reinforcing bars in the walls and link beams.

With respect to the ratio of free length to height of link beams (less than 2) and also, following the dominancy of nonlinear shear behavior, to provide more ductility and increase the shear strength, the special transverse reinforcement ( $ϕ_{D}$ ), as well as the diagonal reinforcement ( $ϕ_{A}$ ), were designed in model (Figure 3). The slab’s thickness is 15 cm, which was modeled using “shell element.” The concrete and reinforcement properties in finite element model are shown in Table 1.

Table 1.

Material properties.

Mechanical property	Steel rebars	Concrete
Young’s modulus (GPa)	200	26.8
Poisson’s ratio	0.3	0.2
Yield stress (MPa)	400	–
Ultimate stress (MPa)	600	–
Tensile strength (MPa)	–	0
Compressive strength (MPa)	–	25

Modeling nonlinear behavior and deformation parameters

The PERFORM_3D (Computers and Structures Inc (CSI), 2016) is used for nonlinear modeling and analysis of the models. Both nonlinear shear and flexural behavior of the walls are modeled. Due to dimensional properties and geometry-dependent behaviors, link beams of the walls are only modeled as nonlinear shear-deformable elements (Paulay and Binney, 1974). In order to model nonlinear shear behavior of the members, nominal shear strength (V_n) is taken as the ultimate strength, according to the ASCE 41-13 (2014) provision. As the depth to length ratio is less than 2, the relationships of deep beams are used for determining the shear strength of link beams (Zhao et al., 2004).

In this study, the parameter V_n is determined from equation (1), in which V_c and V_s are shear capacities supplied by concrete and rebars, respectively. Equation (2) is used to estimate V_c, in which the axial force, N_u, can be neglected conservatively. In this equation, f_c, A_g, b_w, and d represent the characteristic compressive strength of concrete, the total cross-sectional area, the thickness, and effective depth of the element, respectively. For walls, the effective depth is usually considered as 80% of the horizontal wall length (d = 0.8 l_w).

In the walls, in order to calculate the shear capacity provided by the rebars, equation (3) is used, in which the effect of both vertical and horizontal reinforcements is considered simultaneously. In this equation, f_y is the yield stress of rebars. The parameters A_h and S_h represent the cross-sectional area of the horizontal rebars (φ_H in Figure 3) and the distance between them. A_v and S_v represent the cross-sectional area of vertical rebars (φ_V in Figure 3) and the distance between them.

For the link beams, the closed stirrups and diagonal bars (φ_D and φ_A in Figure 3) will supply V_s. In these elements, equation (4) can be used to calculate the shear capacity of reinforcement. In the aforementioned relationship A_a and α are, respectively, the cross section of diagonal rebars and their angles with respect to the horizontal line. A_d and s also represent the cross section of stirrups and their distance (see Figure 3)

Vn = Vc + Vs

(1)

Vc = [1 + \frac{N_{u}}{14 A_{g}}] \frac{1}{6} \sqrt{f_{c}} b_{w} \cdot d

(2)

Vs (Wall) = [\frac{A_{h}}{12 S_{h}} (1 + \frac{l_{w}}{d}) + \frac{A_{v}}{12 S_{v}} (11 - \frac{l_{w}}{d})] f_{y} \cdot d

(3)

Vs (Link Beam) = [\frac{A_{d} d}{s} + 2 A_{a} \sin (α)] f_{y}

(4)

The walls and link beams are modeled by “Shear Wall” elements in the software. According to Figure 4(a), other necessary parameters for modeling nonlinear behavior (F_u, F_y, F_r, D_u, D_l, D_r, and D_x) are derived from load–displacement and corresponding tables for shear-deformable members (Figure 4(b); ASCE 41-13, 2014). F_u, F_y, and F_r are ultimate, yield, and residual shear strength of the RC element, respectively. D_u and D_l are shear strain corresponding to starting and ending points of ultimate shear strength, and D_r and D_x are shear strain at the starting and ending points of the residual shear strength, respectively.

Figure 4.

Nonlinear shear behavior of walls and spandrels (a) adopted in the software and (b) proposed in ASCE 41-13 (2014) for the shear-control members.

For bending component in the walls, “fiber” elements are used in order to achieve more accurate nonlinear behavior. Based on the structural behavior, different criteria are used to evaluate the ductility of structural members. For the walls and link beams that are shear-deformable and ductility is provided by shear yielding, relative lateral displacements and chord rotation are selected as ductility criteria as shown in Figure 5. In Figure 5, $γ_{2}$ and $γ$ are relative lateral displacement of the wall and chord rotation of the link beam, respectively.

Figure 5.

Local damage criteria in the elements and the limit states corresponding to different performance levels (schematic).

Proposed limit states and DI

Damage measurement in RC buildings due to seismic ground motions has the highest importance. Seismic damage indices are extensively used to predict possible damage.

According to the term definition, damage indices are functions that define the damages of members, stories, and the whole structure based on quantitative values. In these dimensionless parameters, zero stands for non-damaged condition and 1 defines structural collapse. Values between 0 and 1 define various damage levels.

Damage indices can be classified into three broad categories:

Member-level (local) damage indices which define the damage intensity in a structural member. In this study, relative lateral displacement, plastic rotation of the walls, and chord rotation of the link beams are considered as the local damage indices.

Story-level damage indices which are used to express damage extension in a story or generally in a sub-structure.

Global damage indices which define the intensity of damages in the whole structure. As the overall damage of a structure is dependent on distribution and intensity of the local damages, the global DI can be determined by summation of the local damage indices. Note that since collapse of a story affects the serviceability of a structure, the story-level damage indices can be considered as global indices.

Nonlinear deformations are the main cause of damages in the structures (Cai et al., 2014). Therefore, it would be more desirable that the damage variable in some way represents deformation. Relative displacements of stories (drifts) are one of the main and simplest global damage criteria. Attempts have been made to relate damages to the maximum or residual drifts that occur during a ground motion. Despite some shortcomings, this index is vastly used due to its simplicity (Toussi and Yao, 1983).

Since four decades ago, the profiles of maximum story drifts (and the residual displacements) have been used to estimate the damages of buildings under the earthquake excitations. Currently, story drift ratio is also considered as an important parameter to evaluate damage demand in different structural systems under the seismic loadings (Liao and Goel, 2014). Based on the Ghobarah (2004) studies, corresponding drifts for various damage levels of elements and structural system are completely different.

In this study, in addition to defining a drift range corresponding to different damage levels of structural systems with different ductility, the available drift limits for ductile and non-ductile structural systems are also introduced.

It is obvious that accurate estimation of this parameter in the allowable performance level under the various types of ground motions would be very beneficial in the design and analysis of different structural systems. Palermo et al. (2017) performed extensive studies to evaluate and provide profiles of maximum drifts and inter-story velocities in the frame structures equipped with the viscose dampers under the seismic excitations. They proposed simple equations for prediction of these parameters.

Yang et al. (2010) systematically studied inter-story drifts of structures for different near-fault excitations. In empirical studies of Dai et al. (2017), the residual drift was discussed as a parameter for seismic performance evaluation of structures. Test results were used to estimate accuracy of this parameter in damage estimation and seismic performance evaluations. They concluded that the degree of uncertainty is high and should be considered as an integral part of this estimation.

In many standards such as FEMA 356 (2000), the residual drift is used as a criterion for defining the performance level of different lateral load bearing systems. Unfortunately, there is no damage criterion for the novel box-type structural system.

In this study, relative lateral displacement, plastic rotation of the walls, and chord rotation of the link beams are considered as the local damage indices and the maximum relative inter-story drift is taken as the global damage criterion. This criterion is evaluated for different performance levels as follows.

In plan (a) (Figure 2), different models are developed by assumptions of 15, 21, and 30 m height (5-, 7-, and 10-story buildings) and three different ratios of L_w₁/L_w₂ (k = 1, 2, and 5). Push-over analysis is performed for each model in the y-axis direction. The analysis results for the maximum inter-story drifts are recorded when the first wall and link beam reach the immediate occupancy (IO), life safety (LS), and collapse prevention (CP) performance levels. Averages of the recorded values have been presented in Table 2.

Table 2.

Averages of the maximum inter-story drifts when the first element reaches a certain performance level (%).

Performance levels	IO	LS	CP
Link beam (5-story)	0.22	0.377	0.57
Link beam (7-story)	0.29	0.405	0.565
Link beam (10-story)	0.35	0.45	0.56
Wall (5-story)	0.431	0.623	0.817
Wall (7-story)	0.451	0.648	0.801
Wall (10-story)	0.472	0.667	0.799

IO: immediate occupancy; LS: life safety; CP: collapse prevention.

It must be noted that the modal lateral load pattern is used for push-over analysis. This distribution is compatible with the effective modes of vibration in y-axis direction. The number of modes is selected such that at least 90% of the structural mass participates in the analysis.

Evaluation of the damages in the different configurations proved that shear damages occur before flexural ones in the shear walls. In fact, under the lateral displacements, the flexural demand is less than its capacity and the relative lateral displacements of the wall reach the values corresponding to different performance levels. Under the lateral displacements, variation in the wall relative displacements is considerably higher than the variation in the wall base-level plastic rotations. This can be explained and justified by considering the effects of intersecting walls and the standards minimum requirement for shear design of elements. Hence, from hereafter, only the parameter “ $γ_{2}$ ” is considered as the damage criteria of the walls.

There is an acceptable correlation between the analytical results and the results of experimental studies (Hidalgo et al., 2002) on squat shear walls. In the experimental studies (Hidalgo et al., 2002), the maximum drift corresponding to the maximum wall capacity has been estimated as 0.8%, which is consistent with the corresponding limit state (CP) of the walls in this study.

Moreover, according to Figure 6, it is evident that in the asymmetric case, damages in the element are also asymmetric and the longer wall experiences the performance levels sooner than the shorter one. This is due to the higher stiffness of the longer wall.

Figure 6.

Sample of damage initiation in the elements in both cases of symmetric and asymmetric wall configuration.

As the stiffness and strength of the walls adjacent to the link beam are different, by forming a shear joint in one of them, the lateral motion will only follow the other one.

Accordingly, due to the potential shear failure of one of the walls, the link beam does not fail. However, in some cases, the mechanism and instability of the story occur before reaching the higher levels of failure. This behavior would intensify by increasing the difference between the lengths of the two walls. It is noteworthy that the observations were almost the same for all 15, 21, and 30 m models. Based on the observations and quantification values presented in Table 1, it seems that the maximum allowable inter-story drifts equal to 0.45%, 0.65% and 0.8% are appropriate for the IO, LS, and CP performance levels, respectively. Therefore, these limit states are proposed for the box-type structural systems composed of intersecting walls and slabs.

As the nonlinear analysis of shell structures need considerable time and efforts, using a simple DI based on the maximum inter-story drifts would be highly helpful and efficient. In the following, according to the proposed values and the performed evaluations, an index is proposed to determine performance level of box-type structural system based on the maximum inter-story drift.

Each damage variable has an initial start value (threshold) and a critical value. If the damage variable is equal to (or smaller than) the threshold value, then the DI is 0 (or smaller than 0).This means that the structure does not experience remarkable damage. On the condition that the damage variable approaches the critical damage value, the DI approaches 1, which means total collapse. In order to compute quantitative values of the DI between the initial and critical values, a relationship is needed to connect the DI and the damage variable. A simple approach is a linear relationship according to equation (5), which is utilized in this study

DI = \frac{D_{m} - D_{t}}{D_{u} - D_{t}}

(5)

where D_m is the maximum inter-story drift which is recorded in the analysis and D_u is the critical value of the drift. As the story mechanism occurs by reaching the first wall to the CP performance level, the critical drift is selected as 0.8% (Table 2). D_t is the initial value of damage parameter. Quantitative value of this parameter is equal to the maximum inter-story drifts corresponding to the IO performance level of the link beams, which is equal to 0.2%, 0.3%, and 0.35% for 15, 21, and 30 m structures, respectively. As depicted in Figure 7, for the proposed index, several functional areas are considered as follows. It is obvious when the DI is less than 0 (DI < 0), the structural damages are negligible and the structural performance level is higher than the IO.

Figure 7.

Proposed damage index for box-type structural system.

In zone 1 $(0 \leq DI < 0.25)$ , the damage is limited to the link beams and the walls performance level is IO. In zone 2 $(0.25 \leq DI < 0.7)$ , the link beams experience higher damage levels. In this region, when the DI is around 0.5 or higher, most of the link beams are yielded and have no coupling effect or lateral load bearing; nevertheless, the walls perform in a performance level higher than LS (i.e. IO).

In zone 3 $(0.7 \leq DI < 1)$ , the only lateral load bearing elements are the walls that perform in a performance level higher than CP (i.e. LS). In zone 4 ( $1 \leq DI$ ), the structure fails due to story failure.

Verifying the proposed DI

In this section, as described previously, the 5- and 10-story box-type building structures with plan (b) (introduced in section “Methodology”) are evaluated by means of push-over, incremental dynamic and fragility analyses in order to verify the proposed DI. The dead and live loads in this modeling are same as those used for structural design. In the combination of lateral and gravitational loads, the upper limit of gravitational loads is used based on equation (6) (ASCE 41-13, 2014)

Q = 1.1 [Dead Load + Effective Live Load]

(6)

The percentage of shear walls in plan shows that stiffness and strength of the buildings in x-axis direction is more than y-axis direction. Therefore, behavior of the buildings is only evaluated in y-axis direction. Assessing the order of translational modes (Table 3) verifies this statement. As it is evident, in both models, the first modes are completely rotational, while the second and third modes are translational. For the 5-story model, the vibration period is less than 1 and the effective mass factors in both directions are more than 75%. Therefore, the triangular distribution assumption for the earthquake loads and utilizing the static methods for analysis and design seems appropriate. For the higher structure, spectrum or time history analysis is preferred.

Table 3.

Vibration period (seconds) and coefficient of effective translational mass (%) in the first three vibration modes.

Models	Mode no.	Parameters	Values
5-Story	1	Time period	0.2235
	1	Effective mass factor	0
	2	Time period	0.1397
	2	Effective mass factor	79.6
	3	Time period	0.1342
	3	Effective mass factor	76.05
10-Story	1	Time period	0.7485
	1	Effective mass factor	0
	2	Time period	0.4453
	2	Effective mass factor	75.41
	3	Time period	0.3187
	3	Effective mass factor	67.35

Push-over analysis

To perform the push-over analysis, the modal pattern is selected for the distribution of the lateral loads in y-axis direction. This pattern is consistent with the effective modes in the considered direction. The number of these modes is selected such that more than 90% of building mass participates in the analysis (Mortezaei and Ronagh, 2013). In the following, to evaluate the damages of the selected buildings under the push-over analysis, the roof drift (ratio of the roof displacement to the building height) is recorded when the first wall and link beam reach the subsequent performance levels: IO, LS, and CP.

In Figures 8 and 9, the drift values corresponding to different limit states and the capacity curves are demonstrated for the 5- and 10-story buildings with and without link beams. W and SP in these Figures 8 and 9 stand for wall and link beam, respectively.

Figure 8.

Capacity curve of 5-story building.

Figure 9.

Capacity curve of 10-story building.

Comparing the capacity curves of selected buildings with and without link beams shows that these elements have considerable impact on the lateral stiffness and strength of the structures. However, although the link beams act like the damage fuse, premature failure of these elements does not lead to the structural collapse. In fact, failure of the buildings occurs when the walls in one or more stories reaches damage levels. Damages initiate symmetrically from the elements along the axes 2 and 3 of the plan (Figure 2). This can be attributed to the higher percentage of the walls in these parts of the plan. As can be seen in the capacity curves in the selected structures, the distance between LS and CP limit states of the walls is negligible. By reaching the first wall to the LS performance level, a considerable drop is observed in the curve.

Based on the assessment results in this section, considering damages in the link beams and walls as local and initiation of global failures is correct. In fact, the global failure of the system occurs in the following local damages in the main lateral load bearing elements, that is, walls.

Developing average building drifts for damage limit states (0, 0.25, 0.7, and 1) shows that the proposed damage indices provide an acceptable evaluation of performance and damage level in the structures.

IDA

Lack of sufficient information about ground motions with different intensities and compatible with the site condition is one of the main challenges in the seismic performance evaluation of structures. Previous attempt to solve this problem leads to a novel approach which is called IDA. In this method, the well-known concept of scaling ground motions is used. It is developed to provide an appropriate means for estimating demand and capacity of structures in a wide range of elastic behavior till collapse occurs (Vamvatsikos and Cornell, 2002). In order to study the effects of variation in amplitude, frequency content, and duration of the ground motions on the response of the models, IDA is also performed. Appropriate selection of accelerograms, intensity measure (IM), and damage measure (DM) parameters is a prerequisite of IDA method.

Selected ground motion records

The first step of IDA is selecting the ground motions records which are compatible with the site condition. The selected ground motions must closely reflect the properties of the earthquake origin, fault mechanism, distance to fault rupture, ground motion intensity, and soil type. In an analytical study, in addition to the properties of records, number of utilized accelerograms is also important. Using more accelerograms would reduce uncertainty in the analysis. Relying on the past studies, using 10 to 20 ground motion records provides an acceptable accuracy in estimation of seismic demand by IDA (Shome and Cornell, 1999).

Therefore, in this study, 10 pair of far-fault ground motion records from the PEER Ground Motion Database (n.d.) database are used for IDA analysis (http://peer.berkeley.edu/peer_ground_motion_database). The soil type of these ground motions is consistent with the site (type C of ASCE 41-13 (2014) categorization, $360 (m / s) \leq V_{s} \leq 760 (m / s)$ ).

By determining the response spectrum of each pair of accelerograms and comparing them, the main component of the earthquake is then selected based on the greater spectral values in the range of structural frequencies. The characteristics of the selected accelerograms are listed in Table 4.

Table 4.

Selected ground motion records for IDA.

Record number	Earthquake and year	Station	R^a (km)	Component	M_w	PGA₀ (g)
R1	Cape Mendocino, 1992	Eureka—Myrtle and West	41.97	90	7.1	0.1782
R2	Northridge, 1994	Hollywood—Willoughby Ave	23.07	180	6.7	0.2455
R3	Northridge, 1994	Lake Hughes #4B—Camp Mend	31.69	90	6.7	0.0629
R4	Cape Mendocino, 1992	Fortuna—Fortuna Blvd	19.95	0	7.1	0.1161
R5	Northridge, 1994	Big Tujunga, Angeles Nat F	19.74	352	6.7	0.2451
R6	Landers, 1992	Barstow	34.86	90	7.4	0.1352
R7	San Fernando, 1971	Pasadena—CIT Athenaeum	25.47	90	6.6	0.1103
R8	Hector Mine, 1999	Hector	11.66	90	7.1	0.3368
R9	Chi Chi (Taiwan), 1999	TCU045	77.5	90	7.6	0.5120
R10	Friuli (Italy), 1976	Tolmezzo	15.82	0	6.5	0.4169

IDA: Incremental dynamic analysis.

Closest distance to fault rupture.

Selecting intensity and DMs and discussion

Intensity of the ground motions which increases during the analysis is nominated by IM, and the output of the analysis due to the applied excitation is designated by DM. The IDA curves are in fact graphical representation of variation of DM with respect to the IM (Vamvatsikos and Cornell, 2002). In this study, the maximum ground acceleration (PGA(g)) is selected as IM and the maximum relative lateral displacement $(γ_{2})$ and chord rotation $(γ)$ are selected as DM for the walls and link beams, respectively (Figure 5). In order to increase the accuracy, the incremental step of intensity parameter is selected as 0.05g $(Δ PGA = 0.05 g)$ . Obviously, the maximum earthquake acceleration in step n is determined based on equation (7).

Thus, taking into account (PGA₀) as the maximum initial acceleration in the main component (Table 4), it is evident that in step n, the scale factor for record SF is determined in such a way that the maximum acceleration corresponds to the desired step PGA_n (see equation (8))

PG A_{n} = 0.05 n (g)

(7)

S F_{n} = \frac{PG A_{n}}{PG A_{0}}

(8)

Figures 10 and 11 demonstrate the developed curves from IDA and the limit states of the elements (local damage criteria).

Figure 10.

IDA curves (5-story building).

Figure 11.

IDA curves (10-story building).

In the following, the maximum acceleration corresponding to the damage limit states of the walls and link beams is derived from the IDA curves. According to the values in Table 5, it is evident that the average of maximum acceleration which is needed for structural members (specially the walls) to reach various performance levels is much higher than the maximum acceleration of design earthquake (0.35g). This shows the high strength and acceptable seismic performance of box-type structural system in the areas with high seismicity hazard.

Table 5.

Average of maximum ground acceleration (g) to reach different limit states.

Performance levels	Immediate occupancy	Life safety	Collapse prevention
Wall (5-story)	0.97	1.17	1.25
Link beam (5-story)	0.81	0.97	1.14
Wall (10-story)	1.05	1.35	1.40
Link beam (10-story)	0.77	0.96	1.07

These can be attributed to the high redundancy and over-strength of the box-type structural systems. In the experimental studies on sandwich panel structures (Palermo et al., 2014), similar to those in this study, similar results were observed.

The link beams are damage fuse and as it is evident, the necessary intensity needed to reach various performance levels is less than the walls. This was predictable based on the higher seismic demand of these elements in comparison with the walls. By scaling each record to the intensity corresponding with a damage limit state, the maximum inter-story drifts of each analysis are recorded. As depicted in Figure 12, the average values are used for evaluation. As can be seen, in both models, the inter-story drift is less than 1%. As the drift of higher stories is very limited, these stories usually remain undamaged or experience very limited damage at the structural failure time. The minimum requirement of the walls and rebar in these stories is too strict and results in waste of materials.

Figure 12.

Average of maximum drifts corresponding to different performance levels of the elements and proposed limit states.

As shown in Figure 12, selecting allowable drifts equal to 0.45%, 0.65%, and 0.8% for the IO, LS, and CP performance levels, respectively, seems reasonable. Despite the considerable stiffness of the system, drift values are substantially less than the allowable drifts of the Iranian code of practice for seismic-resistant design of buildings (Permanent Committee for Revising the Standard 2800, 2014; Uniform Building Code (UBC), 1997), that is, 2% for the structures higher than 5-story and 2.5% for the lower ones. So, these values should not be used for analysis and determining the performance level of this structural system. As it is shown, the system fails by experiencing much lower drifts.

In the European Seismic Code (European Committee for Standardization (CEN), 2003), the allowable drift values are proposed based on the attached non-structural components and their participation in lateral load bearing. For the condition that non-structural components do not exist or non-structural components do not participate in the lateral load bearing (the condition of selected buildings), this standard proposes an allowable drift of 1%. The New Zealand Standard (Fenwick and MacRae, 2009) also proposes the same value (1%) for the structures in the high seismic hazard zones, which is not appropriate for the box-type structural system.

Evaluations show that the proposed allowable drifts by IBC are also not appropriate for the selected buildings. In International Building Code (IBC, 2006), depending on the condition, the maximum allowable drifts of 1%, 1.5%, and 2% are proposed.

In Japanese seismic code for buildings, the allowable drifts are limited to 0.5%. Observation shows that this value is enough for performance of the structure in a level higher than the LS. Chilean Standard NCh433.Of 96: Seismic Design of Buildings (Instituto Nacional De Normalizacion (INN), 1996) also proposes 0.2% for the allowable drift, which is very strict.

In FEMA 356 (2000), the maximum allowable drifts for the IO, LS, and CP performance levels of common concrete shear walls are 0.5%, 1%, and 2%, respectively. However, 0.5% is sufficient to keep the structure in the LS performance level; however, 1% and 2% for the other performance levels are not appropriate.

Based on the Ghobarah (2004) studies regarding squat and ductile walls, the limit values of drifts have been listed in Table 6 for various damage states. The results reported for squat walls and the described damage states are in good agreement with the obtained results in this study.

Table 6.

Limit drift values corresponding with different damage levels of the shear walls (%) (Ghobarah, 2004).

Types	Structural damage states
	No damage	Repairable damage		Irreparable damage	Severe damage	Collapse
	No damage	Light	Moderate	Irreparable damage	Severe damage	Collapse
Ductile walls	0.2<	0.4	0.8<	0.8>	1.5	2.5>
Squat walls	0.1<	0.2	0.4<	0.4>	0.7	0.8>

Risk Management Solutions Inc (1997) predicted the drift limits for shear wall system in four different damage states, namely, slight damage (SD), moderate damage (MD), extensive damage (ED), and complete damage (CD). For the selected structures, these values have been presented in Table 7. As can be seen, the suggested values by this standard for extensive and CDs in both 5- and 10-story structures are not applicable. Based on the analysis results, the suggested values are appropriate only for the SD level and result in damages in the link beams. For the values given at the MD level, although the main load carrying elements perform in the IO level, they do not experience the level of LS. Due to the fact that in these situations the main elements of load bearing system have reached to the performance levels and the structure is not far from collapsing, using the term “moderate damage” might not be appropriate for the box-type structural system.

Table 7.

Limit drift values corresponding with different damage levels for the selected structures based on HAZUS.

Models	Structural damage states
	Slight	Moderate	Extensive	Complete
5-Story	0.27	0.53	1.34	3.34
10-Story	0.20	0.40	1.00	2.50

The evaluations in this section proved that the proposed allowable maximum drifts in many of the standards and codes are not appropriate for box-type structural system and using those values leads to incorrect results.

Fragility analysis

In this section, to evaluate accuracy and efficiency of the suggested DI in various performance levels for box-type structural system, fragility analysis is performed based on the IDA results.

If it is assumed that R is a parameter which represents the structural response, and LS_i is the performance level or limit state corresponding to this response, the fragility function can be expressed in the following mathematical form

Fragility = P [R > L S_{i} | IM = S]

(9)

In fact, fragility curves demonstrate the cumulative distribution of the damage probability (Cimellaro et al., 2006).

According to Figures 10 and 11 and considering the maximum relative lateral displacement and chord rotation as the response parameter for the walls and link beams, respectively, the defined performance levels by ASCE 41-13 (2014) are utilized (Figure 5). For different hazard levels, the probability of exceeding the hazard level is calculated and the resulted fragility curves are depicted in Figure 13. Noted that in order to compare the results of fragility curves with the predictions of suggested DI, only three hazard levels with the maximum accelerations of 0.35g, 0.55g, and 0.75g are used. First, the fragility curves are derived for different performance levels in the link beams and walls. Then, the listed ground motions in Table 4 are scaled to the mentioned maximum accelerations (0.35g, 0.55g, and 0.75g). Next, the resulted accelerograms are applied to the models to compute their maximum inter-story drifts. Finally, the DI is computed using developed values and previously proposed relationships (Table 8).

Figure 13.

Fragility curves for various performance levels of walls and link beams: (a) 5-story building and (b) 10-story building.

Table 8.

Average of the maximum inter-story drifts in different intensity levels and the corresponding damage indexes.

PGA	0.35g		0.55g		0.75g
Parameters	Drift (%)	DI	Drift (%)	DI	Drift (%)	DI
5-Story	0.058	<0	0.176	<0	0.335	0.225
10-Story	0.148	<0	0.285	<0	0.440	0.2

DI: damage index.

According to Table 8, it is expected that in the hazard levels with the maximum accelerations equal to 0.35g and 0.55g, in which the computed DI is less than zero, the structures perform in a performance level higher than the IO. For the hazard level with maximum acceleration of 0.75g, it is expected that the damages be limited to the link beams and the wall still performs in a level higher than the IO level.

Based on the fragility curves of 5-story structure (Figure 13(a)), it is evident that the probability of exceeding the IO performance levels under the earthquake with maximum acceleration of 0.35g is almost zero. In the case of maximum ground acceleration of 0.55g, the probability that the link beams reach the mentioned performance level is less than 6%, and this quantity is less than 1.5% for the walls. The probabilities of exceeding the IO level in the earthquake with maximum acceleration of 0.75g are less than 43% and 16.5% for the link beams and walls, respectively.

In the 10-story structure, probabilities of exceedance in the case of IO level under the earthquake with maximum acceleration of 0.35g are less than 2.5% and 0.5% for the link beams and walls, respectively. The limit state values under the earthquake with maximum acceleration of 0.55g are 23% and 4.5%, respectively. Finally, in the earthquake with maximum acceleration of 0.75g, the probabilities that the walls and link beams exceed the IO performance level are almost 20.5% and 55%, respectively (Figure 13(b)).

The results of fragility curves in this section are again in good agreement with the predictions of the proposed DI.

Conclusion

This study evaluated the overall and story failure mechanism as well as global and local damage indices in box-type structural system. The results, which are limited to the developed model and adopted assumptions, are as follows:

The box-type structural system has high strength and acceptable seismic performance. In the selected structures which were designed based on the minimum requirements, the necessary ground motion intensity needed for the structure to reach the high damage level is higher than the intensity of design earthquake.

At the point of structural collapse, in higher stories, the main lateral load carrying elements, that is, the walls, perform in the high-performance levels. This observation necessitates a review of the proposed design requirement for this structural system in order to provide the minimum percentage of the walls in each direction of the plan.

The link beams have considerable effect on the lateral strength and initial elastic stiffness of the box-type structural system and plays the rule of damage fuse in the structure. Based on the observations, removing the link beams from the system results in unacceptable outcomes.

Failure of the link beams does not lead to significant drop in the capacity curve of the structures. However, the lateral load bearing capacity of the structure falls significantly when the first wall reaches the LS performance level. In fact, the overall failure of the system is the result of the moderate local damage levels in the main lateral load bearing elements.

Because of the high stiffness of the system, the story drifts are less than the allowable drifts proposed by many of standards and design codes. Therefore, when the structure collapse, the story drifts are considerably lower than the proposed values in the standards. It becomes evident that these suggested values are not applicable for the box-type structural system.

Considering the relative lateral displacements and the chord rotation as the local damage criteria for the walls and link beams, respectively, the maximum drifts of 0.45%, 0.65%, and 0.8% are proposed as the damage limits for the IO, LS, and CP performance levels, respectively.

The proposed DI based on the maximum inter-story drift provides an acceptable estimation of damage development and performance level of the system. The mentioned index can be updated including the effects of some parameters, such as percentage of the walls in the plan, overall height of the structure, and plan irregularities in the future research works.

Based on the limit state values of the proposed DI and the analysis results, the selected structures perform in the IO performance level under the design hazard level.

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.

ORCID iD

Alireza Mortezaei

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