Incremental dynamic analysis method application in the seismic vulnerability of infilled wall frame structures

3.1. Frame structure infilled walls-based analysis model and interstory displacement angles performance index limits

A building is comprised of both structural and non-structural components, and understanding their performance collectively is crucial in assessing the overall performance of the building. However, the existing research on performance indicators for non-structural frame structures is still lacking in comprehensiveness, which hinders the determination of overall performance indicators for building structures. To address this, research is undertaken from the perspective of non-structural components, specifically focusing on infill walls, to analyze the overall seismic performance of frame structures that incorporate infill walls. To begin, a corresponding nonlinear analysis model is established, utilizing the fiber model within Perform-3D software. The frame's beam-column elements are represented by fiber elements, which are divided into 7 and 5×5 sections respectively, employing a concentrated plastic hinge model. For the analysis of infill walls, the FEMA equivalent slant support model is employed. This is achieved through the construction of the Concrete Structure unit within Perform-3D software [14-16]. The calculation formula for the equivalent width of the slant support is presented in Eq. (1):

where, ${\lambda }_1$ represents the parameter, $h_{col}$ represents the frame column height, and the diagonal length of the infill wall is expressed as $r_{inf}$. The calculation formula for ${\lambda }_1$ is shown in Eq. (2):

where, the height, length, and thickness of the infill wall are expressed as $h_{inf}$, $L_{inf}$, and $t_{inf}$, the elastic modulus of the infill wall material is set as $E_{me}$, the moment of inertia of the column is expressed as $I_{col}$, and $\theta$ is the inter-layer displacement angle. Infilled walls usually have openings. For infilled walls with openings, when using the FEMA equivalent slant support model, the reduction coefficient method is used to simplify the impact of openings on frames’ performance. The Strength reduction coefficient $(R_1{)}_i$ is shown in Eq. (3):

where, the area of the infill wall and the area of the opening are $A_{panel}$ and $A_{open}$. When $A_{open}$ is 0.6 times greater than $A_{panel}$, the effect of infill walls can be ignored. The stiffness reduction coefficient $(R_1{)}_i$ outside the plane is shown in Eq. (4):

In the constitutive equation of unconstrained concrete, the relevant formula for the reference value $a_c$ of the descending segment of the concrete uniaxial stress-strain curve is shown in Eq. (5):

where, $n\mathrm{\text{'}}$, $x$, and ${\rho }_c$ all represent variables, and $d_c$ represents the concrete damage evolution parameters under uniaxial compression. The Mander model is used for the constitutive model of constrained concrete, and the peak strain ${\epsilon }_{cc}$ of constrained concrete is shown in Eq. (6):

where, the peak stress of unconstrained concrete is set to $f_{c0}\mathrm{\text{'}}$, and its peak strain is set to ${\epsilon }_{c0}$; The peak stress of the confined concrete is expressed as $f_{cc}\mathrm{\text{'}}$, with a peak strain of ${\epsilon }_{cc}$ and an ultimate peak strain of ${\epsilon }_{cu}$. The constitutive relationship of concrete materials is defined through Perform-3D software, as shown in Fig. 1.

Fig. 1F-D relationship of the five-line model

In Fig. 1, F represents strength; D represents deformation in the strength state; FY, FU, and FR represent material ultimate strength, yield strength, and residual strength. The first yield point is Y, the strength limit point is set to U, the strength degradation point is set to L, and the residual strength point is R. The deformation limit point is represented as X. The deformations corresponding to these key points are set as DU, DL, DR, and DX. In the constitutive model of steel reinforcement, the Non-Buckling model is used, and the strength of the infill wall material is the average axial compressive strength of the masonry. According to the performance indicators of ASCE 41-06, the limits for infill wall components are given in Table 1.

According to the performance index limits in Table 1, the performance of infill walls can be analyzed. The damage indicators of components are quantified and the overall performance indicators of the structure are extracted. When extracting indicators, the relevant steps are shown in Fig. 2.

Fig. 2Steps for extracting overall performance indicators of the structure

In Fig. 2, the structural components are first adjusted to their intact state. There are sporadic non-structural components with slight damage, and this time point is used as a benchmark. The relevant seismic motion recording time is adjusted in increments of 0.1 seconds before and after the benchmark, and the component failure rate is checked. When slight damage is found to structural components or the failure rate of non-structural components reaches 5.0 %, relevant adjustment work shall be stopped and this time point shall be recorded. $\theta$ is derived from the structure at each time point, and the maximum value ${\theta }_{max}$ is used as the limit value for the basic integrity of the structure. Repeating the above operation can obtain the limit values for different failure levels based on the component failure rate.

3.2. IDA method application in seismic vulnerability analysis

A comprehensive understanding of the seismic resistance of building structures is crucial for effective disaster prevention and mitigation efforts. In light of this, this study focuses on the infill wall frame structure as a representative example and employs the IDA method to analyze the seismic resistance of the building structure [17]. Specifically, the structural performance parameter chosen for evaluation is the maximum interlayer displacement angle. The steps involved in the IDA method are illustrated in Fig. 3.

Fig. 3The process of IDA method for seismic analysis of building structures

Fig. 3 illustrates the construction of a nonlinear analysis model for the structural examples. The selection of the target ground motion follows the ATC-63 screening principle, which determines the characterization parameters for the ground motion intensity measure (IM). In this study, Ground Peak Acceleration (GPA) is chosen as the IM characterization parameter for $IM$. A specific seismic motion is selected, and relevant principles of amplitude modulation are referred to in order to modulate the target seismic motion. The selected amplitude modulation method employs both equal step size and unequal step size. Following the amplitude modulation process, multiple ground motion records with the same spectral characteristics and duration but varying peak values are obtained from each original ground motion record. Nonlinear time-history analysis is then conducted to obtain the corresponding Engineering Demand Parameter (EDP) values under the relevant seismic amplitudes. The steps of seismic motion selection and amplitude modulation are repeated, and each seismic motion record undergoes analysis using the IDA method. This iterative process results in an IDA curve cluster for the structure, providing a comprehensive representation of its seismic response characteristics. By utilizing these techniques, this study aims to achieve a thorough understanding of the structural behavior under various seismic conditions, as depicted in the IDA curve cluster.

Based on the structural type, its limit state $LS$ is defined and $LS$ represented by $EDP$ parameters is quantified. Given the probability $L{S}_i$ of $EDP$ exceeding a certain limit state under different $IM$, i.e. $P\left(L{S}_i\left|IM=im\right.\right)$. $im$ represents the corresponding $IM$ value when $EDP$ exceeds a certain limit state. $L{S}_i$ is set as $ed{p}_i$ through the quantification of the $EDP$ parameter, and the probability of $EDP$ exceeding $ed{p}_i$ when $IM=im$ is shown in Eq. (7):

where, the conditional probability distribution conforms to the logarithmic normal distribution. With $IM$ and $P\left(L{S}_i\left|IM=im\right.\right)$ as the horizontal and vertical axes, the curve of exceedance probability under different $IM$ is fitted and drawn through the logarithmic normal distribution function. On the basis of the obtained exceedance probability curve, the exceedance probability of each $LS$ is determined under the fortification target and different $IM$ in the structure, and the seismic performance is evaluated. In vulnerability analysis, the nonlinear behavior of structures under different $IM$ conditions is generally predicted by drawing IDA curves of different ground motions. However, due to the inaccuracy of ground motion, the IDA curve plotted has a certain degree of nonlinearity. Thus, it is necessary to statistically analyze the curve data, using the median to represent the average level of the curve cluster, and using the 16 % and 84 % percentile curves to characterize the discrete type, as shown in Eq. (8):

where, $\mathrm{\Phi }(\cdot )$ represents the standard normal cumulative distribution function, the inter-layer displacement angle is expressed as $\theta$, ${\theta }_{50\mathrm{\%}}$, ${\theta }_{16\mathrm{\%}}$, and ${\theta }_{84\mathrm{\%}}$ represent the quantile curve relationship. The logarithmic mean of $\theta$ is set as ${\lambda }_D$, the logarithmic standard deviation of $\theta$ is set as ${\zeta }_D$, and ${\zeta }_D$ can represent the degree of dispersion of $\theta$. The mathematical expression of ${\lambda }_D$ is shown in Eq. (9):

where, $\mu$ represents the variable and $\delta$ represents the variable. The mathematical expression of ${\zeta }_D$ is shown in Eq. (10):

According to Eqs. (8), (9), and (10), the relevant mathematical expressions for ${\theta }_{50\mathrm{\%}}$, ${\theta }_{16\mathrm{\%}}$, and ${\theta }_{84\mathrm{\%}}$ can be obtained as shown in Eqs. (11):

where, $\mathrm{e}\mathrm{x}\mathrm{p}$ represents the exponential function based on the natural constant $e$.

3.3. Case models and seismic vulnerability construction

After explaining the design and establishment of the IDA calculation model, a city in China was selected to calculate the construction age of 500 frame structures. Over time, a larger proportion of structures were designed according to the new standards. To evaluate the seismic performance and understand the situation of newly designed structures, particularly in relation to important non-structural components like infill walls, a 6-story frame structure office in Tianjin was chosen as a representative prototype. During the selection process for the case structure prototype, the office building in Tianjin was redesigned using PKPM2010 software to represent many current frame structures. The seismic performance analysis was conducted using Perform-3D software. Specifically, the selected representative frame structure is a 6-story RC frame office building with a floor height of 3.6 m. The building has 5 spans in the $X$ direction and 3 spans in the $Y$ direction, each with a span of 7.5 m, the structural plan is shown in Fig. 4.

Fig. 4Structural plan

In Fig. 4, in terms of design parameters, the structure falls under Class III in the design site category, with a seismic intensity of 7 degrees (0.15 g). The exterior wall consists of clay brick-filled walls, and windows are installed along the A to D axis and 1 to 6 axis, with a hole opening rate of 40 %. The structure adopts C30 concrete material, Grade III steel for longitudinal reinforcement, and Grade I steel for stirrups. The infill walls have outer and inner wall thicknesses of 240 mm and 180 mm, respectively. MU10 standard clay bricks and M2 mixed mortar are used for masonry. Additionally, tie bars extend 500 mm beyond the column edge and are embedded inside column 2A6@500. The floor load is designed with a constant load of 6.25 kN/m², while the roof load is designed with a constant load of 7.5 kN/m². The live load reduction coefficient is taken as 0.5, and the basic wind pressure is designed with a value of 0.5 kN/m².

The reinforcement design of the frame structure is carried out using PKPM2010 and is verified through simplified arrangements based on the principle of equivalent reinforcement area. Subsequently, fiber models are defined based on these arrangements. The analysis model is constructed, and geometric modeling is established using SAP200. The obtained frame node coordinates, loads, and other information are then imported into Perform-3D software. To facilitate rapid modeling and analysis, material properties and load effects are redefined. Verify the rationality of the Perform-3D model using SAP200 basic modal calculations. The relevant situation of the model is shown in Fig. 5.

Fig. 5 (a) shows the three-dimensional model of the PKPM framework structure, Fig. 5(b) shows the three-dimensional model of the SAP2000 framework structure, and Fig. 5(c) shows the infill wall framework structure. Determine model damping. During an earthquake, when the input seismic energy dissipates, the building eventually stops shaking. This energy dissipation occurs through two main mechanisms: elastic-plastic energy dissipation and other forms of energy dissipation. However, accurately calculating the latter form of energy dissipation is challenging. Therefore, it is commonly represented by setting a damping coefficient. Due to the complexity of estimating damping, the damping ratio can serve as a reference value. The influence of damping on the structure can be analyzed through modal damping and Rayleigh damping. The expression for modal damping is shown in Eq. (12):

where, $C$ is the damping matrix, the number of damping modes is expressed as $N$, $n$ is vibration mode, the period of $n$ is $T_n$, the damping ratio of $n$ is ${\xi }_n$, the mass matrix is expressed as $M$, and the shape of $n$ is ${\varphi }_n$. In Rayleigh damping, $C$ is set as the linear combination of structural mass matrix and stiffness matrix. When $\beta =0$, $\beta$ represent variables, the relevant formula can be obtained as shown in Eq. (13):

where, $\alpha$ represents the variable, the mode is set to $i$ with a period $T_i$. During the calculation, 5 % modal damping is applied to the periodic interval, and high-frequency vibration is eliminated by adding 0.1 % Rayleigh damping. The values of Rayleigh damping are shown in Fig. 6.

Fig. 53D model of structure

a) 3D model diagram of PKPM framework structure

b) 3D model of SAP2000 frame structure

c) Infilled wall frame structure

Fig. 6 provides the values of Rayleigh damping that are used to match the model structural damping with the actual structural damping, ensuring the energy balance of the seismic input structure. To accurately consider the effect of infilled walls, the study simulates them using equivalent diagonal strut elements, which serve as simulation elements. In order to ensure the correctness of the nonlinear analysis, a method of deleting elements under gravity load is adopted. This means assuming that the internal force of such elements under gravity load is zero, and then activating these elements in subsequent analyses to incorporate their contributions.

Fig. 6Setting of Rayleigh damping

The deformation of the frame structure with infill walls, when subjected to horizontal forces, results in a transition from shear type to bending shear type, which introduces a gravity effect $P$-$\mathrm{\Delta }$. Therefore, this effect must be taken into account in the calculations. The selected performance indicators in the example structure include beam components, frame column sections, and infill walls. Table 2 presents the performance indicators specifically for beam components and infill walls.

Table 2 contains various performance indicators with different values corresponding to different indicator rating levels. These values serve as reference points for conducting correlation analysis. The study also involves the selection of seismic motion and amplitude modulation. Following the ground motion selection principle outlined in ATC-63, a total of fifteen far field motions and one El Centro motion have been selected. Table 3 presents some specific details about these selected motions.

In Table 2, the seismic magnitude selected is approximately 7.0, resulting in variations in the corresponding seismic acceleration time history curve and response spectrum curve. The study selected three equal step sizes of 0.05 g, 0.10 g, and 0.20 g for unequal step amplitude modulation. After 13 amplitude modulation and 208 nonlinear time history analyses, the required structural performance parameters were obtained. The seismic peak accelerations after 13 amplitudes of modulation are: (0.05 g, 0.10 g, 0.15 g, 0.20 g, 0.25 g, 0.30 g, 0.35 g) +(0.40 g, 0.50 g, 0.60 g) + (0.80 g, 1.00 g, 1.20 g), a total of 13 amplitudes. For the infill wall frame structure, overall performance indicators are derived, and the average inter-layer displacement angles are obtained under the selected seismic motion, which are then used as performance indicators. The overall failure rate is utilized to represent the failure of the infill wall and the subsequent repair requirements. During vulnerability analysis using the IDA method, the obtained IDA data undergoes regression analysis to establish a linear regression analysis function, as shown in Eq. (14). This regression analysis function aids in understanding the relationship between various parameters and provides valuable insights for assessing the structural vulnerability:

where, $a\mathrm{\text{'}}$ and $b\mathrm{\text{'}}$ represent parameters. Substituting $a\mathrm{\text{'}}$ and $b\mathrm{\text{'}}$ into Eq. (15), the conditional probability $P_f$ of exceeding a limit state under different $PGA$ can be obtained:

where, ${\theta }_c$ represents the structural capacity parameter, and $\sqrt{{\beta }_S^2+{\beta }_R^2}$ takes a value of 0.4. The relevant exceedance probability curve and the relevant vulnerability matrix can be obtained. Thus, the seismic vulnerability of infill wall frame structures can be analyzed.