Probabilistic seismic vulnerability assessment of the structural deficiencies in Iranian in-filled RC frame structures

2.1. Characteristics of considered RC frame structures

Three subclasses of RC frame structures with three, five and eight stories and with the same typical plan were considered. Designing of these structures were carried out according to the third edition of the Iran earthquake code named standard 2800 [5] and also Iran concrete national code [6]. The structural system in both directions (two orthogonal directions) is assumed to be RC intermediate moment frame.

All considered structures are assumed located in relatively high-risk region with soil type III in accordance with Iran earthquake code [5]. The compressive strength of concrete used in RC frame members are assumed to be 210 kg/cm² and the yield stress of longitudinal and transverse bars are taken 4000 and 3000 kg/cm², respectively. Dead load of typical stories is taken 650 kg/m² and the live load of the roof and all other stories are taken as 150 and 200 kg/m², respectively. Considering the customary practical cases, the typical story height of 3.2 m and bay length of 5 m are assumed. The analytical models of considered structures are presented in Figs. 1 and 2.

Fig. 1Typical plan of all considered buildings

Fig. 23D OpenSEES models of 3, 5 and 8 story considered buildings

In order to considering the effect of upper vibration modes, beside static equivalent analysis, modal dynamic analysis are performed too. The resultants characteristics of the frame elements of the three considered building models are listed in Table 1.

2.2. Classifications of the considered buildings based on HAZUS-MH definitions

HAZUS-MH [7] divides the framed structures into three categories including Low-Rise, Mid-Rise and High-Rise structure. According to this classification, structures with three, five and eight stories in this article are considered to be Low-Rise, Mid-Rise and High-Rise, respectively. The definition regarding to the frame type are listed in Table 2.

2.3. Modeling and analyzing software

OpenSEES [8] is an object-oriented software framework that allows users to create finite element applications for simulating the response of structural and geotechnical systems subjected to earthquakes. The library of materials, elements and analysis powerful tools for numerical simulation of nonlinear systems, easy modeling, and high accuracy made it a powerful and useful analytical tool in earthquake engineering researches. For these reasons, in this article, the OpenSEES platform [8] is used for incremental time history dynamic analysis. The nonlinearity of materials and also p-delta effect are included in the 3 dimensional nonlinear dynamic analysis. A detailed information about the finite element considered models and their element behaviors can be found in Table 3.

2.4. Characteristics of considered materials

Reinforced concrete behavior is modeled using the Concrete 02 material provided in OpenSEES with tensile strength and linear tension softening. The behavior of this material is shown in Fig. 3. This material used the model proposed by Mander et al. [10] to define the monotonic stress-strain curves for confined and unconfined concrete. The effect of confinement is pronounced on the peak compressive stress and ultimate strain in the confined concrete stress-strain relationship. The numerical parameters which are used to model concrete nonlinear stress-strain relations in the OpenSEES are listed in Table 4.

Fig. 3Concrete 02 material: a) nonlinear behavior and b) typical hysteretic stress-strain relation [9]

a)

b)

Reinforcing steel is modeled using the Steel 02 material provided by OpenSEES which includes isotropic strain hardening. The behavior of this material is presented in Fig. 4 and numerical parameters which are used to model steel stress-strain relations in OpenSEES are listed in Table 5.

Fig. 4Steel 02 material; hysteretic behavior of model with isotropic hardening: a) in compression and b) in tension [9]

a)

b)

2.5. Modeling of the frame element sections in OpenSEES

Fiber sections are selected to define the cross section of the beams and columns in OpenSEES. Two type of concrete materials are defined, confined and unconfined for the beams and columns (Fig. 5). The cover concrete above the stirrups and the longitudinal bars has less strength. Since it is not confined with reinforcements, it will crack soon. Thus, unconfined concrete material are defined to consider the effect of this matter with the unique behavior presented by Mander et al. [10]. As in finite element problems meshing of the sections are essential to have reliable responses, in this study discretization and meshing are performed by patch quad command in OpenSEES platform. Accordingly, beam and column sections are discretized to fine sections as shown in Fig. 5.

Fig. 5Dissection of a rectangular shaped reinforced concrete cross-section [11]

Fig. 6Stress-strain model implemented for confined and unconfined concrete [10]

The confinement will increase concrete strength and ductility. The increased strength is defined as confinement coefficient ($K$) that depends on the dimension of the cross-section, the diameter and the number of stirrups. Confined concrete material is defined with equations introduced by Mander et al. [10]:

where $f_{yh}$ is the yield stress of steel in MPA, $b\text{'}\text{'}$ is the width of core concrete, $d\text{'}\text{'}$ is the height of core concrete, $A\text{'}\text{'}$ and $S$ are respectively area and distance between shearing bars.

Fig. 6 represents the Stress-Strain curves of the confined and unconfined concrete, where strain corresponding to nominal compressive strength $f_{C0}^{\text{'}}$ is expressed by ${\epsilon }_{c0}$, and ${\epsilon }_{sp}$ and ${\epsilon }_{cu}$ are defined as strain at the time of failure for unconfined and confined concrete, respectively and $f_{cc}^{\text{'}}$ is the compressive strength of concrete for confined concrete material, which is calculated according to Eq. (1).

2.6. Modeling the infill walls in OpenSEES

Rigid Frames are commonly used as the load carrying system in Iranian buildings. The beams and the columns are main elements of these kind of systems. Since no practical and generally recognized method has yet been developed to determine the nonstructural components such as architectural walls, and are not usually taken into consideration in the analyses, in this study the infill walls are modeled with frame system. In order to modeling the infill walls in external bay of the frame systems, equivalent diagonal strut model is used based on FEMA-306 [12] definitions. The length of these struts are equal to walls diagonal, their thickness are the same as the thickness of the walls and their equivalent width are calculated based on the following equations [12]:

where $A$ is equivalent width of the diagonal strut, $h_{col}$ is the center to center of the story height, $h_{inf}$ is the height of in-filled wall, $E_{fe}$ is the expected elasticity coefficient of frame material, $E_{me}$ is the expected elasticity coefficient of infill wall material, $I_{col}$ is the column’s moment of inertia, $r_{inf}$ is the diagonal length of in-filled wall and $\theta$ is the angle between strut and horizontal axis.

4.1. Uncertainty in material parameters

Treatment of uncertainty in seismic vulnerability assessment has been a subject of research for many years. Many parameters can cause the structural deficiency in structures but the most important parameter especially in RC frame structures is the deficiency created by the poor concrete specifications. The building classes considered in this study use concrete as the construction material and usually the cast-in-place concrete used in building construction in Iran has design strength of 210 kg/cm² at 28 days. Based on an extensive field investigation done by authors on Iranian as-built RC frame structures, in this study, reduced strength of concrete is considered as structural deficiency. According to the FEMA356 [13] definitions and guidelines for rehabilitation of existing building, based on the suitable combination of destructive and nondestructive concrete tests conducted by authors among 800 residual building with RC frame structure which have the material deficiencies typically built in north of Iran, the compressive strength of concrete is modeled in this probabilistic assessment using a normal distribution with mean value, ${\mu }_{fc}$ of 170 kg/cm² and standard deviation, ${\sigma }_{fc}$ of 30 kg/cm² and coefficient of variation, $\zeta =$ 0.18. Regarding to AIII class reinforcement steel bars, with minimum $F_y$ of 4000 kg/cm², they assumed to follows a normal distribution too with median value of ${\mu }_{fy}$ of 4450 kg/cm² and standard deviation, ${\sigma }_{fy}$ of 320 kg/cm² and coefficient of variation, $\zeta =$ 0.08. These parameters are adopted in the present study to model the distribution of the concrete strength and the yield strength of the reinforcing bars.

4.2. Uncertainty in earthquake prediction model

One of the most important step in Incremental Non-linear Dynamic Analysis (IDA) for probabilistic assessment of structures is selection of the ground motions, because the results obtained from analysis are depended widely on the ground motions used in the analysis. Therefore, selecting the suitable pairs of ground motions is extremely sensitive and will affect the proposed fragility functions. The similarity in site characteristics and also the number of required records for IDA analysis are important matters that should be addressed carefully. The more number of ground motion records used for analysis, the more reliable results will be obtained. However, the time needed for performing the analysis will be increased too. So Suitable number of ground motions are needed to cover aleatoric uncertainty of earthquake events for building fragility analysis.

According to recommendation of Shome and Cornell [14], 10 to 20 ground motion records provide acceptable accuracy for estimating the vulnerability of buildings. However, FEMA P695 [15] proposed 22 records of ground motions for building performance assessments. In this study, far-field ground motions recommended by FEMA P695 [15] are used with some minor modifications. The aforementioned 22 ground motions consist of 6 ground motions which are recorded in soil type C according to NEHRP [16]. This soil class is closely similar to soil type II of Iran earthquake code [5], and the other 16 ground motions are recorded in soil type D according to NEHRP [16] which are closely similar to soil type III of Iran earthquake code [5]. Since in this study the assumed soil type for the considered buildings is type III, in order to provide more compatibility between characteristics of ground motions and site characteristic of the considered structures, the aforementioned 6 ground motion records of soil type C in FEMA P695 [15] is replaced with 4 new ground motion records. (Records number 4, 7, 10 and 15 in Table 7). Finally, 20 modified ground motions listed in Table 7 are used in IDA analysis.