Performance-based global reliability assessment of a high-rise frame-core tube structure subjected to multi-dimensional stochastic earthquakes

Liu Zhangjun, Ruan Xinxin and Liu Zixin

1. School of Civil Engineering and Architecture, Wuhan Institute of Technology, Wuhan 430074, China

2. Key Laboratory of Building Collapse Mechanism and Disaster Prevention, Institute of Disaster Prevention, China Earthquake Administration, Sanhe 065201, China

Abstract: When evaluating the seismic safety and reliability of complex engineering structures, it is a critical problem to reasonably consider the randomness and multi-dimensional nature of ground motions. To this end, a proposed modeling strategy of multi-dimensional stochastic earthquakes is addressed in this study. This improved seismic model has several merits that enable it to better provide seismic analyses of structures. Specifically, at first, the ground motion model is compatible with the design response spectrum. Secondly, the evolutionary power spectrum involved in the model and the design response spectrum are constructed accordingly with sufficient consideration of the correlation between different seismic components. Thirdly, the random function-based dimension-reduction representation is applied, by which seismic modeling is established, with three elementary random variables. Numerical simulations of multi-dimensional stochastic ground motions in a specific design scenario indicate the effectiveness of the proposed modeling strategy. Moreover, the multi-dimensional seismic response and the global reliability of a high-rise frame-core tube structure is discussed in detail to further illustrate the engineering applicability of the proposed method. The analytical investigations demonstrate that the suggested stochastic model of multi-dimensional ground motion is available for accurate seismic response analysis and dynamic reliability assessment of complex engineering structures for performance-based seismic resistance design.

Keywords: multi-dimensional stochastic ground motion; dimension-reduction representation; frame-core tube structure; global dynamic reliability; performance-based seismic design

1 Introduction

Studies have shown that the traditional force-based seismic design approach usually underestimates the deformation of injured structural components since excessive deformation or displacement could often lead to severe damage to building structural members (Ghobarah, 2001; Liuet al., 2016a). In this light, a novel method called performance-based seismic design, with an emphasis on changing from “strength” to “performance” has received extensive attention and multi-faceted research done by seismology and engineering scholars worldwide since the method was first proposed by the Pacific Earthquake Engineering Research Center (PEER) of the United States in the early 1990s. In essence, the purpose of performance-based seismic design is to find effective ways to investigate and describe the failure state of a structure subjected to various seismic excitations and conduct determination and quantification of a structural performance level with an appropriate critical index, which is of significant importance for the analysis and calculation of structures, earthquake damage prediction and structural seismic loss assessment. To date, a series of performance goals have emerged in ATC-40 (1996), FEMA-273 (1997), and Vision 2000 (1995). In 2000, the Japanese Building Code officially adopted the capability spectrum method, based on the performance design concept (Kotani, 2000). It is worth mentioning that in 1997 the Joint Committee on Structural Safety (JCSS) developed a model code for full probabilistic design, which can directly employ structural reliability. The targets are by and large compatible with observed failure rates and with outcomes of cost-benefit analyses (Vrouwenvelder, 2002). Actively following the developmental trend, the Chinese Code for Seismic Design of Buildings (GB 50011-2010, 2016) also suggests a reference method to realize the goal of performance-based seismic design, including the reference indicators of bearing capacity and inter-story drift angle response for structural members, and so forth. In addition to the revision of the code, researchers have conducted extensive research on the methods of performance-based seismic design. Allahvirdizadeh and Mohammad (2016) presented a directly equivalent static method for different damage states. Afterwards, Allahvirdizadehet al.(2017) did a probabilistic comparative investigation on introduced performance-based seismic design and assessment criteria for reinforced concrete (RC) frames. Arroyoet al.(2018) proposed a design procedure based on eigenfrequency optimization for RC frames. Kalapodiset al.(2018) developed a performance-based seismic design method for eccentrically braced frames and buckling restrained braced frames. Subsequently, this method has been successfully extended by Muhoet al.(2019) to the seismic design of bare RC moment resisting frames. Recently, Muhoet al.(2020) proposed a performance-based seismic design method for wallframe dual systems and infilled-moment resisting frames, and constructed the modal behaviour factors for the first few modes and four performance levels (immediate occupancy, damage control, life safety and collapse prevention). In general, the idea of performance-based seismic design has changed the onefold design method based on the principle of life safety found in previous seismic codes, enabling the proprietors to maintain the best balance between structural seismic performance and post-earthquake repair costs (Sullivanet al., 2014). Therefore, in this paper, the analysis and research on the dynamic response and reliability of the structure under earthquake action is also based on the goal of attaining a satisfying seismic performance.

Meanwhile, presently, high-rise buildings are springing up in major cities due to the booming populations of cities, the lack of land resources, and progress in science and technology (Lu and Jiang, 2014), among which the frame-core tube system has undoubtedly become the most preferred structural form, given that the frame and core tube can effectively reduce lateral displacement of structures subjected to small or medium earthquakes (Luet al., 2016; Liet al., 2018). Therefore, the seismic performance, behaviour and collapse-resistant design of high-rise framecore tube structures have been increasingly explored by researchers who adopted both the numerical and experimental methods in recent years (Huanget al., 2017; Shen and Qian, 2019). However, most studies only focus on the seismic effect of the one-dimensional strong ground motion record. A large number of strong earthquake observation records indicate that the movement of a seismic wave when passing through the ground is extremely complicated, and three translational components, including two horizontal directions and one vertical direction as well as three rotational components, can be observed in each part of the ground (Penzien and Watabe, 1975; Hernández and López, 2003). Accordingly, the structural response during earthquakes is also multi-dimensional, especially for high-rise buildings which commonly have relatively complicated structural forms, of which the difference between the nonlinear seismic responses under a one-dimensional and multi-dimensional rare occurrence earthquake is remarkably obvious. Hence, the influence in terms of an earthquake′s multi-dimensional characteristics is an indispensable consideration for minimizing the earthquake damage for such structural systems (Liet al., 2004). For that, Miuet al. (2009) implemented a simulation of complicated seismic resistant behaviour on an actual project of a hybrid steel reinforced concrete (SRC), high-rise frame-core tube structure induced by two horizontal earthquake components with different intensities. Chenget al. (2018) conducted a nonlinear time-history analysis and studied the typical seismic behaviour of a RC, high-rise frame-core tube structure with a fundamental period of 2.64 s subjected to multidimensional long-period earthquake records, employing the analytical method of numerical simulation.

However, the seismic performance and behaviour of high-rise frame-core tube structures suffering from multi-dimensional ground motion and corresponding multi-dimensional earthquake-resistant design methods, as well as the associated computation, have been inadequately studied. This set of circumstances also pertains to the relevant performance-based design applied in engineering practices. In addition, several improvable aspects summarized from existing research are specified as follows: I) The above research all employed the deterministic dynamic response analysis methodology which chose certain strong motion records as seismic excitations for engineering structures, without sufficiently considering the randomness of earthquakes; Ⅱ) The correlation between different components of a multi-dimensional earthquake, which can be obtained from statistical analysis results of abundant motion records, was unintentionally neglected; Ⅲ) Though the particular seismic behaviour analysis was presented in the aforementioned studies, the global reliability assessment of frame-core tube structures under multi-dimensional earthquake actions with regard to the current seismic design code for reference still remains a great challenge. Therefore, the purpose of this study is motivated by addressing the above-listed bottlenecks, of which includes modelling of stochastic, multi-dimensional, fully non-stationary ground motion and conducting a reliability evaluation of a high-rise frame-core tube structure corresponding to different performance-based requirements, in accordance with current seismic design codes.

Due to the extremely limited records of the rotational earthquake components and the complexity between different rotational components, this study only concerns the three translational seismic components. Commonly, research associated with the correlation of a multidimensional earthquake usually involves two aspects: the correlation upon the seismic parameters between different components and the correlation upon the crosspower spectrum density (PSD) of different components in the PSD matrix. Specifically, the former studies mainly focus on peak value, sometimes referred to as the response spectrum. A series of representative research results have been achieved by Ohsakiet al. (1980), Hu (1988), Li and Li (1992), Xueet al. (2004), and Li and Liu (2011). These studies have already been widely used in engineering applications for their simplicity and practicality. On the other hand, although the latter studies could provide a relatively comprehensive description of the correlation upon the cross-PSD of a multi-dimensional earthquake, its complexity is too difficult for engineers to accept, and at present, there is no unified statistical outcome for accurately defining the cross-PSD, which leads to the impracticality of employing engineering applications. In addition, a recent study (Das and Hazra, 2018) showed that the correlation between different components of multi-dimensional ground motion is frequency-dependent. Bearing the above situation in mind, this paper adopts a simplified method to model stochastic multi-dimensional fully nonstationary ground motion from the local point of view, such as the definition of modulation function, power spectrum and response spectrum of the ground motion acceleration process. In the meantime, the fitting of the multi-dimensional design response spectrum is taken as the control objective, and then a brief frequencydependent principal component analysis of the simulated multi-component ground motions is also carried out to illustrate their correlation characteristics.

As for the refined reliability assessment of engineering structures, an efficacious technique named the probability density evolution method (PDEM) (Li and Chen, 2008, 2009; Liet al, 2009), developed by Li and Chen during the past decade, provides an alternative approach and has been diffusely accepted in engineering applications. It is particularly noteworthy that the equivalent extreme-value event approach involved in the PDEM can readily tackle the structural exact dynamic reliability evaluation, regardless of whether the safety event is a single limit state function or a combination of multiple ones (Chen and Li, 2007; Li and Chen, 2007). Generally, the most versatile method for simulating the stochastic processes as excitation inputs in structural dynamic time-history analysis appears to be the spectral representation method, due to its characteristics of complete theory, simple calculation and easy implementation, which in recent decades have been vigorously developed and increasingly utilized for generating non-stationary earthquake ground-motion processes with evolutionary power spectrum (Shinozuka and Deodatis, 1991; Deodatis, 1996; Huanget al., 2013; Konget al., 2016; Wuet al., 2017). Nevertheless, when the conventional spectral representation belonging to the family of Monte Carlo methods is employed to proceed with the stochastic ground motion simulation, a large number of samplings for a series of high-dimensional random variables is always required to acquire an acceptable level of simulation accuracy, which not only tremendously increases the calculation cost, but also immensely limits the application of this method in the accurate analysis of dynamic response and seismic reliability for randomly-excited complex structures. For the purpose of solving the above issues, Chenet al.(2013, 2017) successfully addressed the stochastic harmonic function representation for expressing both the stationary and non-stationary stochastic processes by means of a small number of random harmonic components. In recent years, a random function-based dimension-reduction methodology was proposed by Liuet al.for simulating the stationary and non-stationary stochastic processes (Liuet al., 2016b, 2018b). The investigations demonstrate that the dimension-reduction strategy is of fair accuracy and efficiency, with the computational efforts of a case involving millions of random variables reduced to the level of that involving merely one or several random variables. As a result, each generated representative sample has an assigned probability and all the representative samples constitute a complete probability set, so that it can be used as the stochastic excitation source in the PDEM to implement the structural refined dynamic response analysis and dynamic reliability assessment.

However, to date, the dimension-reduction method has only been used to simulate one-dimensional ground motions, and has not been applied to consider multi-dimensional ground motions. Consequently, the dimension-reduction approach is extended in this work to obtain input representative samples of a multidimensional earthquake for dynamic time-history analysis. It is also worth mentioning that the long-period range of the target response spectrum in the seismic design code is artificially increased to ensure structural safety. Therefore, the structural reliability assessment in this study is essentially some kind of conditional reliability, since the earthquake input is compatible to the design response spectrum. On the basis of such an engineering background, this study first decomposes the ground motion vector process into three mutually uncorrelated and perpendicular components, of which the intensity modulation function, time-varying power spectrum and design response spectrum are established respectively by adequately considering the correlation upon seismic parameters between different components (see Section 2). Next, the dimension-reduction representation and the response spectrum fitting scheme are both employed herein to realize the simulation of the representative time-history set, which is compatible with the design response spectrum (see Section 3). Further, the non-linear finite element model of a high-rise framecore tube structure with a height of 98.1 m is established via using Midas Building software (see Section 4). Following that step, the seismic response analysis and the accurate global reliability assessment of the framecore tube structure that is subjected to multi-dimensional severe earthquake events are investigated (see Section 5). In this section, the performance-based seismic requirements are introduced. Since the deformationbased performance design method has the merits of a relatively simple concept, commendable operability, facile acceptance by structural engineers, and a natural integration with current specifications, the performancebased seismic requirements that take the inter-story drift angle response as the critical index, in accordance with the GB 50011-2010 (2016), are presented in detail. In general, there are five performance levels: Level 1 is equivalent to immediate occupancy; Levels 2 and 3 are equivalent to damage control; and Levels 4 and 5 are equivalent to life safety and collapse prevention, respectively (detailed description can be found in Section 5.2). Finally, some concluding remarks are provided (see Section 6).

2 Stochastic model of multi-dimensional fully non-stationary ground motion

2.1 Multi-dimensional seismic design response spectrum

The research in the reference (Li and Liu, 2011) indicates that different seismic components vary distinctly in spectrum, intensity and non-stationarity of frequency content. Therefore, it cannot be simply considered that the response spectra of the two horizontal ground motions are completely the same, nor can the response spectrum of the vertical earthquake easily be regarded as a proportional coefficient of the response spectrum upon the horizontal component. Indeed, the response spectra of different seismic components show differences in the maximum peak value and the characteristic period, as well as, simultaneously, the curve index of the descending section. For the sake of simplicity, the maximum value of the seismic impact coefficient of the vertical ground motion component is taken as two-thirds of the average value upon the two horizontal components, and the characteristic period of the vertical component is shorter than the horizontal component by about 0.10.

According to the GB 50011-2010 (2016), the expressions of the seismic acceleration response spectrum in the case of the structural damping ratio taken as 0.05 (for RC structures) are defined as follows:

wheremaxαdenotes the maximum value of the seismic impact coefficient,0Tdenotes the natural vibration period of structures,gTdenotes the characteristic period of site soil,γdenotes the attenuation index of the descending section of the curve,ηdenotes the adjustment coefficient of the descending slope in the straight descending section, and g denotes the gravitational acceleration valued by g=1000 cm/s2, respectively.

As for the multi-dimensional seismic design response spectrum, the suggested parameter values are concretely provided in the reference (Li and Liu, 2011). For instance, the specific values of the four parametersmaxα,gT,γandηfor estimating the multi-dimensional seismic design response spectrum corresponding to an earthquake of 8-degree and 9-degree seismic precautionary intensity and site classification II are listed in Tables 1–3. Here, thex-component and they-component represent two mutually orthogonal horizontal directions, respectively, and thez-component represents the vertical direction. It should be noted that the concepts of the design scenario are contained in GB 50011-2010 (2016). Usually, a design scenario refers to a set of earthquakes and site characteristics such as earthquake magnitude, source-tosite distance, and soil conditions of the site. In GB 50011-2010 (2016), these three characteristics are considered in terms of seismic precautionary intensity, classification of design earthquake and site classification.

Table 1 Parameters of the maximum value of the seismic impact coefficient for the multi-dimensional seismic design response spectrum

Table 2 Parameters of the characteristic period of site soil for the multi-dimensional seismic design response spectrum

Table 3 Parameters of the attenuation index and the adjustment coefficient for the multi-dimensional seismic design response spectrum

Table 4 Parameters of the time-varying power spectrum for site classification Ⅱ in the rare earthquake case

The multi-dimensional seismic design response spectrum of the rare earthquake corresponding to an 8-degree seismic precautionary intensity, site classification II, and the first group is typically depicted in Fig. 1, which conforms to the parameters presented above.

Fig. 1 The multi-dimensional seismic design response spectrum

Fig. 2 The multi-dimensional seismic intensity modulation function

2.2 Multi-dimensional fully non-stationary seismic evolutionary power spectral model

Studies have shown that multi-dimensional ground motion at a particular point is a vector process. However, in seismic measurement and structural dynamic analysis, it is commonly of great necessity to decompose the vector process of ground motion into three mutually perpendicular components. In 1975, Penzien and Watabe (1975) defined the orthogonal set of principal axes for seismic ground motions along which the component variances have maximum, minimum and intermediate values, and the covariance equals to zero. Meanwhile, it was also concluded that artificially generated components of an earthquake are not required to be correlated statistically provided they are directed along a set of principal axes. Consequently, in the present study, coordinate axesx,yandzare treated as the three principal axes of ground motions, wherexandyrepresent the two horizontal directions andzrepresents the vertical direction, respectively.

For non-stationary ground motion, four kinds of evolutionary power spectrum models are suggested in the reference (Liuet al., 2019b), and the third, which could comprehensively describe the earthquake nonstationarities upon both time and frequency, is employed in this paper, given by:

whereGv(ω,t) denotes the evolutionary power spectrum of seismicv-component,Sv(ω,t) denotes the timevarying power spectrum of the seismicv-component,andfv(t) denotes the intensity modulation function of the seismicv-component, respectively. Herein,vrepresentsx-component,y-component, andz-component, respectively.

2.2.1 Time-varying power spectrum

In order to adequately reflect the frequency of nonstationary characteristics of ground motion, the Clough-Penzien time-varying power spectrum (Deodatis, 1996; Liuet al., 2018a) is employed in Eq. (2), followed by:

whereωg,v(t) andζg,v(t) denote the time-varying dominant circular frequency and the damping ratio of site soil, respectively.ωf,v(t) andζf,v(t) denote the timevarying dominant circular frequency and the damping ratio of bedrock, respectively. The four parametersaandbcan be determined according to the site soil classification, andTis the seismic duration,respectively.

In Eq. (3), the spectral intensityS0,v(t) is defined as follows:

wheremaxAsignals the mean of peak ground acceleration (PGA), andvrpresents the equivalent peak factor of the seismicv-component, respectively.

In this study, for the two horizontal components of the multi-dimensional ground motion, say thex-component andy-component, the site parameters of the time-varying power spectrum can take the same value. While for the verticalz-component, the parameters correlating to the time-varying power spectrum between the horizontal direction and the vertical direction possess the following relationship (Xueet al., 2004):

Meanwhile, for the equivalent peak factorvrof different seismic components, the relationship between the two horizontal components and that between the horizontal component and the vertical component are defined on the basis of the multi-dimensional seismic design response spectrum in this study, such that:

whereαmax,x,αmax,yandαmax,zdenote the maximum value of the seismic impact coefficient upon thex-component, they-component and thez-component, respectively. The equivalent peak factorvrand the vertical correction factorλmust be determined by means of the optimal fitting of the design response spectrum. In this study, the parameter values ofvrandλcorresponding to site classification Ⅱ and the first seismic design group are suggested as:xr=2.75, 1.1λ= for a moderate earthquake, andxr=2.65, 1.1λ= for a rare earthquake.

2.2.2 Multi-dimensional seismic intensity modulation function

In the families of intensity modulation function for describing one-dimensional horizontal ground motion, Amin and Ang suggested using the three-segment envelope function, for example, the Amin-Ang model (Amin and Ang, 1968), which for decades has been widely applied in engineering practices. The main merit of this model is its simple and intuitive form, and it can effectively reflect the rising stage, stationary stage and attenuation stage of ground motion acceleration timehistories. However, for the multi-dimensional seismic components, their rising rate, stationary duration and attenuation speed are completely different, especially the significant difference between vertical component and horizontal component. In order to reflect the intensity non-stationarity of multi-dimensional seismic components, on the basis of the Amin-Ang model, Li and Liu (2011) proposed an improved four-segment continuous intensity modulation function model:

wheret0andt1denote the starting time and ending time of the ascending segment, respectively.t2denotes the beginning time of the descent segment.cdenotes the attenuation index of the descent segment.T1=t1–t0denotes the duration of the ascending segment.Ts=t2–t1denotes the duration of the stationary segment.

According to the statistical data of a large number of earthquake records, the reference (Li and Liu, 2011) conducted an in-depth study on the correlation between the components in different directions of multi-dimensional ground motions, and addressed the parameter values of the four-segment intensity modulation function of nearfield and far-field earthquakes with seismic precautionary intensities of 7-degree, 8-degree and 9-degree. In order to be consistent with the current GB 50011-2010 (2016), the near-field and far-field earthquakes in this paper are correspondingly adjusted as the first and third seismic design groups, while the second seismic design group is directly taken as the average of the first and third groups. Table 5 shows the parameter values of the four-segment intensity modulation function for multidimensional ground motion belonging to 8-degree and 9-degree seismic intensity of the first design group for site classification Ⅱ.

Table 5 Parameters of the four-segment intensity modulation function

Figure 2 presents the four-segment intensity modulation function upon two horizontal components and one vertical component for multi-dimensional ground motion corresponding to that of 8-degree seismic intensity, site classification Ⅱ and the first seismic design group, with a duration of 30 s. As shown in the figure, remarkable differences between the peak arrival time of intensity and the duration of the plateau for seismic components in different directions can be intuitively observed, especially the vertical component, whose intensity peak arrives earlier and lasts longer. Accordingly, the employed four-segment intensity modulation function can effectively describe the differences in the intensity of seismic components in different directions.

3 Dimension-reduction representation of multi-dimensional non-stationary ground motion process

3.1 Dimension-reduction representation

where ()Nεdenotes the maximum value of the mean square relative errors associated with the three seismic components, and usually the limit value for earthquake engineering is defined asε(N) ≤10%.Tdenotes the duration of the ground motion process. The truncation frequencyuNωω=∆, thus the number of expansion termsN, can be determined through the truncation frequencyuωand the discrete step intervalω∆ .

Obviously, the number of random variables in the original spectral representation, i.e., Eq. (12), is dramatically reduced from 6Nto 3 by means of the random functions that are defined in Eq. (14), thus overcoming the limitation caused by millions of random variables required for the conventional Monte Carlo simulation method, thereby providing a solid foundation for accurate dynamic response analysis and dynamic reliability evaluation of randomly-excited complex engineering structures. Additionally, another attractive feature of this method is employing three mutually independent elementary random variables for representing the corresponding three seismic components, by which the uncorrelated characteristics between different components can thus be commendably presented. This effectively shows the validity of the proposed dimension-reduction approach.

3.2 Spectrum-compatible realization

3.3 Simulation procedure

Ⅲ) Generate the representative sample set of the multi-dimensional ground motion, compatible with the design response spectrum. According to the representative sample set generated in the second step, the average response spectrum can be calculated and should be compared with the target response spectrum. Note that if the tolerance requirements defined by Eqs. (16) and (17) are satisfied, the generated representative sample set can directly act as the multi-dimensional ground motion excitation of stochastic structural dynamic analysis. Otherwise, the iteration scheme needs to be performed using Eq. (18) until the two allowable relative errors defined in Eqs. (16) and (17) are both satisfactory.

3.4 Numerical investigation

In Fig. 3, the mean and standard derivation (Std. D) of the generated multi-dimensional ground motion, compatible with the design response spectrum, is presented. As pictured in the figure, the mean and Std. D of the simulated multi-dimensional ground motion fit commendably well with the corresponding target values for all the three components. The mean is almost zero for the duration of the entire time-history, which fully reveals that the ground motion process is a stochastic one, with zero mean. On the other hand, the Std. D timehistory has significant non-stationary characteristics, which verifies that the ground motion process is a nonstationary one. At the same time the shape of the Std. D curve in different directions is basically consistent with the applied intensity modulation function, as shown in Fig. 2.

Figure 4 depicts the earthquake acceleration representative time-histories for different directional components without iteration (Fig. 4(a)) and after three iterations (Fig. 4(b)). As distinctly observed in the figures, the waveforms of different components of the multi-dimensional ground motion are notably different, which is mainly due to the differences between different earthquake components, which are considered across the board in the various parameter values in terms of the evolutionary power spectrum. Specifically, the amplitude of the seismicx-component is the largest, slightly larger than that of they-component, while the amplitude of the seismic component in the vertical direction (z-component) is the smallest. At the same time, the peak of the vertical component arrives much earlier, and the duration of the plateau is longer, which can be attributed to the application of the four-stage intensity modulation function. Hence, the uncorrelated characteristics among different seismic components and the feasibility of the evolutionary power spectrum model applied in the present paper are adequately revealed. Moreover, by inspecting the simulated results at none iteration and the third iteration, it can be determined that the wave patterns of the acceleration diagrams at the long period part are evidently adjusted by implementing the iterative scheme.

The comparisons for the multi-dimensional seismic response spectrum between the simulated spectrum and the design spectrum at none iteration and the third iteration are shown in Fig. 5. It can be clearly seen from the figures that the difference between the simulated response spectrum for all the three seismic components and the design response spectrum is of large variation before performing the iteration scheme, while the simulated result after three iterations is ideally fitted to the normative response spectrum, which can fully satisfy the requirements of the structural seismic design. The validity of the iterative method is thus sufficiently proved.

In Fig. 6, the evolutionary power spectra of the seismicx-component at none iteration and the third iteration are lucidly apparent. Obviously, it can be seen that the evolutionary power spectrum amplitude of the low-frequency part is significantly increased after three iterations, which indicates that the iteration method has a large adjustment to the low-frequency part, which corresponds to the improvement of the amplitude of the long-period part in the response spectrum simulation. The same results can be obtained for the other two seismic components, which again validates the availability of the iterative scheme.

Fig. 3 Mean and standard derivation of multi-dimensional ground motion at the third iteration

Fig. 4 Representative time-histories of multi-dimensional ground motion components at none and the third iteration

Fig. 5 Comparisons of the multi-dimensional seismic response spectrum between the simulated and design spectrum at none and the third iteration

Fig. 6 Evolutionary power spectrum of the seismic x-component at none and the third iteration

In order to show frequency-dependent correlation characteristics between different components of generated multi-dimensional ground motion, the frequency decomposition of each componentUv(t) is taken as (Basu and Gupta, 1998; Das and Hazra, 2018)

By using Eqs. (19) and (20), the representative time-histories of the three seismic components (i.e., thex-component,y-component, andz-component) are decomposed, and 32 × 3nselsub-components time-histories can be obtained. To show the presence of frequency-dependent correlation structure, the correlation coefficients are computed between the decomposed sub-components time-histories of the three seismic components. Figure 7 shows the frequency-dependent correlation coefficient between the decomposed sub-components time-histories of Sample 100. As described in the figure, the correlation coefficient varies with frequency (represented by a vertical blue line). Meanwhile, the correlation coefficients without considering the frequency-dependent (i.e. considering the seismic components themselves) are also given (represented by a horizontal red line). It is obvious from Fig. 7 that the correlation between the decomposition motions is not uniform. Although the correlation between composite motions is low, some decomposition motions are highly correlated. This finding is consistent with the reference (Das and Hazra, 2018). Figure 8 shows the mean (at the level of sample functions) of the frequency-dependent correlation coefficient between the decomposed sub-components timehistories (represented by a vertical blue line). Also, the mean of correlation coefficients without considering the frequency-dependent are given in Fig. 8 (represented by a horizontal red line). As shown in the figures, although the correlation coefficient of individual samples may be relatively large, the mean value of the correlation coefficient of all representative samples is less than 0.7. The three correlation coefficients have some similarity to the decomposition level, but they are not entirely the same, which may be caused by the same evolutionary power spectrum model and different parameters.

Fig. 7 Frequency-dependent correlation coefficient between the decomposed ground motion components of sample 100 (the thick horizontal red line represents the correlation between the two simulated motions at the composite level)

Fig. 8 Mean of frequency-dependent correlation coefficient between the decomposed ground motion components (the thick horizontal red line represents the mean of correlation between the two ground motion components)

4 Background description of the high-rise frame-core tube structure model

4.1 Engineering information

A high-rise frame-core tube structure used for an office building with 27 stories is investigated herein as an engineering case study. The structure adopts the boxshape closed section steel reinforced concrete (SRC) frame-reinforced concrete (RC) core tube structural system, which has a total height of 98.1 m, involving a 1st-story height of 4.5 m and a 2nd–27th-story height of 3.6 m. The structure has three spans in both the longitudinal and transverse directions, with dimensions of 24 m by 24 m. The plane layout is drawn in Fig. 9(a). The example structure used for case study herein can be considered as a bench mark structure.

Assuming that the seismic precautionary intensity of the project area is the 8-degree, the design earthquake group is the first group and the site category is classified as class II. Correspondingly, according to the GB 50011-2010 (2016), the site design characteristic period is 0.35 s, the design basic seismic acceleration value is 0.2 g, and the structural seismic fortification category is classification A.

With respect to the structural materials, a concrete strength grade of C40 is employed for floor slabs, frame beams and SRC frame columns. The steel grade of HRB400 is used for shear-walls, floor slabs and frame beams, and the shape steel grade of Q345 is adopted for SRC frame column members. Moreover, all the frame beams are set as 350 mm wide by 750 mm high, and all the floor slabs have the same thickness of 180 mm. The concrete strength grade of SRC frame columns and shear-walls, the section of SRC frame columns, and the size of box-shape steel are provided in Table 6. In addition, the uniform dead load of 6.0 kN/m2and the uniform live load of 2.0 kN/m2are loaded on the floors, including the roof of the frames.

Table 6 Size and materials of structural members

4.2 Nonlinear finite element model

In this study, the widely used Midas Building software, which has an excellent nonlinear computational capacity, is utilized to conduct the establishment of the finite element model of the high-rise frame-core tube structure and to acquire the structural nonlinear randomly-excited dynamic response. A total of 324 frame column members, 540 frame beam members, 108 shear-wall members and 243 floor slab members are produced for the structural finite element model, which is displayed in Fig. 9(b). The self-possessed weight of structural components is automatically loaded by the software calculation program. In this model, the dead load, live load and self-possessed weight are generally considered in terms of the vertical load, while seismic action is taken into account as the exclusive induced horizontal load for the target model. Furthermore, it is assumed that the bottom of the structure model is rigidly connected to the ground, and the presence of the floor slab is considered by assuming a rigid diaphragm for all stories.

Fig. 9 Plane layout and finite element model of the high-rise frame-core tube structure

The structural nonlinear characteristics are reflected in the following ways:

I) Material nonlinearity. In terms of the nonconfined concrete, the corresponding constitutive relationship adopts the uniaxial compressive stress-strain curve of concrete proposed in the Chinese Code for Design of Concrete Structures (GB 50010-2010, 2015). As for the reinforcement bar, the corresponding constitutive relationship employs the simplified bilinear model, assuming that the tension and compression are symmetrical and the hardening of the reinforcement bar is also considered, of which the hardening coefficient is taken herein as 0.01 s.

Ⅱ) The bending moment-rotation angle relationship between the beam and the column end is considered as a nonlinear relationship. Specifically, the hysteretic rule of the beam-column plastic hinge adopts the modified Takeda trilinear model, as shown in Fig. 10, in which the parameters are valued by: initial stiffness06KEIL= , and the first and second stiffness reductions are10.5α= and20.1α= , respectively. With respect to the RC beams, it is assumed that the internal forces of the plastic hinge of the beam end are not related to each other. While being associated with the SRC columns, the internal force relationship of the plastic hinge of the column end is assumed to be the bending moment-axial force (P-M) behavior type, of which the axial force is respectively related toyMandzM(yMandzMare not related to each other, and the three internal forces are related when the initial yield surface is calculated by means of the initial internal force). It should be noted that the plastic hinges of beams and columns are modeled as concentrated hinges and are located at both ends. Figure 11 shows the hysteretic relationship of the beam plastic hinge subjected to a rare earthquake, for which the bending moment-rotation angle relationship represents significant nonlinearity.

Fig. 10 The modified Takeda trilinear hysteretic model for moment M versus rotation θ

Fig. 11 The non-linear hysteretic relationship of the beam plastic hinge subjected to a rare earthquake

Ⅲ) Nonlinear wall element. The shear-wall adopts the nonlinear shear wall element on the basis of the fiber model provided by the Midas building software, and each wall element is again divided into a certain number of vertical and horizontal fibers. The plastic hinge is assumed to be distributed in the upper and lower parts of the fiber. Moreover, the ideal elasticplastic constitutive model is adopted for the backbone curve in terms of the shear characteristics of the wall elements. For the backbone curve upon the tensile and compressive properties of the wall elements, the concrete and reinforcement are considered respectively, and the models described in I) are adopted.

Ⅳ) Geometric nonlinearity. Commonly, for the column element groups, especially in the case of a large slenderness ratio, the actual internal force and displacement of the component should be calculated by sufficiently considering the P-delta effect, which could allow the structural design to be more reasonable. However, the P-delta effect could be ignored for the beam element groups. In the specific operation, the element groups that are required to consider the P-delta effect are calculated by selecting “YES” in the Midas Building software during the analysis series of Analysis Phase.

Rayleigh damping is extensively applied in engineering structural dynamic analysis for its simplicity and convenience, which is also employed herein for the sake of simulating structural damping. It is generally defined as a function of mass and stiffness: for example, the structural Rayleigh damping matrixCgiven byC=αM+βK, whereMdenotes the mass matrix,Ksignifies the initial elastic stiffness matrix,αis a mass damping constant, andβrepresents a stiffness damping constant, respectively. The values of the two constantsαandβcan be determined from the structural fundamental period of the first two modes, for example,Ts1=Ts2=1.848 s , and the damping ratio corresponding to the structure assumed to be 5%(considered as the Rayleigh damping of RC structures in most cases); thus, the target values areα= 0.170 andβ= 0.0147, respectively. Accordingly, through the above methodologies, the nonlinear finite element model of the high-rise frame-core tube structure is established effectively and reasonably, which provides a powerful basis for, in the future, performing the structural dynamic response analysis and dynamic reliability assessment using numerical simulation approach.

5 Seismic response analysis and reliability assessment

Within this section, the representative time-history set generated by the updated dimension-reduction representation for multi-dimensional fully nonstationary ground motion in Sections 2 and 3 which is simultaneously compatible with the design response spectrum, is adopted as the earthquake excitation for the high-rise frame-core tube structure. For the purpose of systematically and meticulously researching the differences in dynamic response of the structure when subjected to multi-dimensional seismic components, four kinds of earthquake input conditions are considered, namely,X-excitation,X+Y- excitation,X+Z-excitation, andX+Y+Z-excitation (X,YandZindicate the loading seismic direction, respectively). Furthermore, the interstory drift angle is selected as the evaluation standard of structural seismic performance, and the reliability of the structural component and the global structural reliability are readily obtained and carefully analyzed, combined with the valid PDEM.

5.1 Seismic response analysis

Figure 12 distinctly presents the mean and Std. D time-histories regarding the inter-story drift angle of the first story, along theX-direction of the high-rise framecore tube structure that is subjected toX-excitation,X+Y-excitation,X+Z-excitation, andX+Y+Z- excitation.

As expressly depicted in Fig. 12(a), the variation trend of the mean time-history of the structural dynamic response underX-excitation andX+Z-excitation remains almost consistent. Meanwhile, the mean time-history curves produced by theX+Y-direction and theX+Y+Zexcitation coincide perfectly well from the beginning to around 18 s, and sustain a similar changing trend regardless of different amplitudes in the remaining duration. On the whole, compared with the dynamic response time-histories induced byX-excitation andX+Z-excitation, the ones under theX+Y-excitation andX+Y+Z-excitation change remarkably, and the amplitudes of the latter two become significantly larger, which indicates that the horizontalY-excitation might possess an amplification effect on the structural dynamic response.

In Fig. 12(b), one can readily observe that the Std. D time-histories of the structural inter-story drift angle subjected toX-excitation andX+Z-excitation are almost the same. Analogously, the Std. D time-histories induced byX+Y-excitation andX+Y+Z-excitation are also close to coinciding. As a whole, in the first five seconds, the Std. D time-histories of the structural dynamic response driven by the four earthquake input conditions line up over one another exactly. However, after the first five seconds, the Std. D time-histories underX+Y-excitation andX+Y+Z-excitation become observably larger than those ofX-excitation andX+Z-excitation, and the maximum value of the four excitations appears at around 10 s, which precisely corresponds to the peak arrival time modulated by the four segment intensity modulation function displayed in Fig. 2, verifying the accuracy of the numerical analysis operation.

Through further observation of Fig. 12, it can be found that the peak values of the mean response are only around 1/20 of the peak values of Std. D, which implies that the mean value is essentially approximate to zero. This is consistent with the property by which the mean of the seismic input is zero, which is described in Fig. 3. Moreover, it can be apparently concluded from the figure that the amplitudes of the mean and Std. D time-histories of the inter-story drift angle show a rapid increase when adequately considering the bidirectional horizontal earthquake action (X+Y-excitation orX+Y+Z-excitation), whereas the vertical seismic loading (Z-excitation) has little effect on the mean and Std. D values of the interstory drift angle. In addition, with regard to the mean and Std. D time-histories upon the inter-story drift angle of all the other stories of the investigated structure, the analogous law can be obtained, which matches other research conclusions (Miuet al., 2009), thereby convincingly verifying the correctness of the numerical simulation results.

Fig. 12 Mean and Std. D time-histories upon the inter-story drift angle along the X-direction of the first story, subjected to the four earthquake input conditions

Fig. 13 The PDFs upon the extreme-value of inter-story drift angle for the global structure

5.2 Performance-based seismic reliability assessment

The theory and techniques in terms of the seismic performance-based structural design, for which the basic idea is to enable the designed engineering structures to satisfy various predetermined performance objectives during their service periods, has become a research focus for the past few decades. In general, the specific performance requirements of a structure are usually determined according to the function and importance of the engineering project to be built. At present, performance-based structural design puts more emphasis on overall structural performance, resulting in finding ways to improve the structural global reliability has become a key consideration for engineers. Commonly, the corresponding design theory mainly includes performance design indicators, performance levels and other contents.

Specifically, performance design indicators mainly include deformation-based, energy-based, and damage-based indicators. Compared with energybased or damage-based seismic performance design, the deformation-based performance design method has the merits of a relatively simple concept, commendable operability, facile acceptance by structural engineers, and natural integration with current specifications, thus becoming one of the main ways to achieve the goal of performance-based design in practical engineering applications.

In the present study, the deformation-based failure criterion is adopted as the failure criterion of the investigated high-rise frame-core tube structure that is subjected to rare earthquakes. The core idea is to compare the structural maximum inter-story drift angle induced by rare earthquakes with the allowable deformation failure index, in order to judge whether the whole structure is damaged. Herein, the extreme-value of the inter-story drift angle of the whole structure can be expressed as follows:

wheremaxΦ denotes the extreme-value of the interstory drift angle of the whole structure, andBφdenotes the allowable (or threshold of) inter-story drift angle index, respectively. Regarding frame-core tube structures, according to the GB 50011-2010 (2016), the corresponding allowable elastic inter-story drift angle index is 1/800, and the corresponding allowable elasticplastic inter-story drift angle index is 1/100, respectively.

In light of the idea involved in the equivalent extreme-value event (Chen and Li, 2007; Li and Chen, 2007), the extreme-value of the inter-story drift angle of the whole structure can be obtained by calculating the maximum value for each story, such that:

whererNdenotes the story number of the structure.Φj,maxrepresents the extreme-value of the inter-story drift angle of thej-th story, which can be obtained by acquiring the maximum value of the inter-story drift angle time-histories of thej-th story along theX-direction andY-direction, following that:

Two remarks can be drawn from the above table, as follows:

I) The seismic component reliabilities for the 1st–5th stories are all above 95% under the aforementioned four excitation cases. Moreover, the component reliabilities associated with the four excitation cases all follow the same trend along the height of the story, in that they all decrease in the first place and, as a whole, rise with an increase in height. In the meantime, it can be observed that the minimum value of the component reliability appears at the 21st story, which may be attributed to the reduction in the cross-sectional size of the vertical structural members of the 21st story (as can be seen in Table 6).

Table 7 Structural storey reliability under 8-degree rare multi-dimensional earthquake

Ⅱ) Comparing the component reliabilities under the various working conditions, it can be found that the component reliabilities subjected toX+Z-excitation are 0.05%–0.14% lower than that ofX-excitation, except that the component reliabilities of each loading case on the 1st–2nd stories are 100%. However, not all the component reliabilities induced byX+Y+Z-excitation are less than that ofX+Y-excitation; the former component reliabilities are 0.03%–0.49% higher than the latter on the 5th–10th stories. In addition, the component reliabilities of each story underX+Y-excitation are 0.45%–5.54% lower than that ofX-excitation, and the component reliabilities of each story subjected toX+Y+Zexcitation are below those upon anX+Z-excitation of 0.41%–5.44%. As a result, these findings adequately indicate that structural component reliability under multi-dimensional earthquakes is lower than that excited by unidirectional seismic action. Especially in the case of bidirectional horizontal earthquakes, the reliability significantly declines, while the vertical seismic action is of limited effect on reliability change.

Figure 13 describes the PDFs of the extreme-value with respect to the inter-story drift angle of the global structure induced by the four earthquake excitations. As demonstrated in the figure, it is obvious that the PDF curve ofX-excitation basically coincides with that ofX+Z-excitation. A similar phenomenon can be seen between the PDF curves ofX+Y-excitation andX+Y+Z-excitation. Additionally, compared with the PDF curves ofX-excitation andX+Z-excitation, that ofX+Yexcitation andX+Y+Z-excitation shift significantly to the right and the peak value becomes smaller, indicating that there exists a larger mean and Std. D involved in the extreme-value of the inter-story drift angle ofX+Yexcitation andX+Y+Z-excitation.

Performance-based seismic design that emphasizes multi-level and multi-target fortifications can purposefully strengthen the key and weak parts of the structure during structural design. Therefore, structures designed according to this concept can preferably achieve the level of seismic performance requirements in the event of future earthquake disasters. Hence, it effectively provides a reliable basis for the construction of safe, economical and applicable buildings. Presently, the theory and technology of performance-based seismic design are developing rapidly. The theory is being extensively studied and the technology applied by structural engineers worldwide.

Specifically, the performance level of an engineering structure indicates the maximum extent to which the structure could be destroyed by a particular seismic precautionary intensity. Since the structural deformation can reflect the overall performance or failure of the structure and can readily be obtained by calculation, it thus becomes the most convenient and practical methodology to divide the structural performance level based on displacement (deformation). For the sake of realizing structural performance-based seismic design, the specific quantitative index for the structural performance level regarding inter-story drift angle, when subjected to rare earthquakes, is provided in the GB 50011-2010 (2016), as shown in Table 8. In general, Level 1 is equivalent to immediate occupancy; Levels 2 and 3 are equivalent to damage control; and Levels 4 and 5 are equivalent to life safety and collapse prevention. In this paper, the following performance levels are adopted for the global reliability assessment of the investigated high-rise frame-core tube structure that would encounter severe earthquake excitation.

Moreover, in combination with the aforementioned division criteria for the structural performance level, the threshold values of the inter-story drift angle corresponding to the above-listed five performance levels of frame-core tube structures are presented in Table 9. Simultaneously, according to the PDFs upon the extreme-valuemaxΦ of the inter-story drift angle for the global structure (shown in Fig. 13) and the threshold values under different performance levels, the overall failure probabilities of the investigated frame-core tube structure for the five performance levels are scrutinized in considerable detail, as shown in Table 10. Specifically, for instance, the overall failure probability of the framecore tube structure at performance level 5, subjected to an 8-degree seismicX-excitation, can be seen in the area below the corresponding PDF curve to the righthand side of the line titled “Inter-story drift angle = 0.01” pictured in Fig. 13, which is calculated as 9.73% (shown in Table 10). Conversely, the corresponding global reliability of the frame-core tube structure is 90.27%, which can be obtained by subtracting the failure probability from 1. Moreover, for comparison, the global failure probabilities of the structure suffering from the 9-degree rare earthquakes at performance level 5, are also addressed in Table 10.

The following comments can be concluded from Table 10:

Table 8 Detailed description of the structural performance level for a rare earthquake

Table 9 Threshold values of inter-story drift angle for different performance levels

Table 10 Structural global failure probability, with elastic-plastic inter-story drift angle as a failure index

I) With the improvement of structural performance requirements (from level 5 to level 1), the failure probability increases gradually. To be specific, the failure probability of each seismic condition corresponding to performance levels 1 and 2 is more than 95%, which indicates that the structure can barely satisfy the requirements of performance levels 1 and 2 under 8-degree rare earthquake actions. Further, in the case that the allowable failure probability is 10%, the structure can achieve the performance-based demand only when being subjected toX-excitation andX+Z-excitation of an 8-degree event for performance level 5. However, the structure can fulfill the performance request under all four earthquake excitations for performance level 5 andX-excitation andX+Z-excitation for performance level 4 of 8-degree severe seismic excitation if the allowable global failure probability increases to 20%.

Ⅱ) In the global failure probability of the structure corresponding to each performance level under the excitation of an 8-degree earthquake, the difference between the global failure probabilities ofX+Z-excitation andX-excitation is within the range of –0.04%–0.14%, and the difference between the global failure probabilities ofX+Y+Z-excitation andX+Y-excitation is –0.09%–0.14%. From the above, the absolute values of the difference between the two is less than 0.15%, demonstrating that the vertical seismic component has little influence on structural global reliability, with the inter-story drift angle being treated as the performancebased-level index. On the other hand, when comparing the global failure probabilities of the structure betweenX+Y-excitation andX-excitation, the differences at performance levels 3, 4, and 5 are approximately 20%, 10%, and 5%, respectively, and the analogical results can also be detected betweenX+Y+Z-excitation andX+Z-excitation. Consequently, it can be inferred that compared with the horizontal unidirectional seismic action, simultaneously considering the bidirectional horizontal seismic components would noticeably enlarge the global failure probability of the frame-core tube structure. That is, the more rigorous the structural performance requirements, the greater the extent of the increase.

Ⅲ) In the case of resisting 9-degree rare earthquakes, the difference of the structural global failure probabilities betweenX+Z-excitation andX-excitation is 0.39%, and the difference betweenX+Y+Z-excitation andX+Yexcitation is 0.20%. The difference in terms of the global failure probability is less than 0.40% in both the above two cases, which is similar to the condition of 8-degree earthquake action. In addition, the difference of the structural global failure probabilities betweenX+Yexcitation andX-excitation and that betweenX+Y+Zexcitation andX+Z-excitation both approach 20% at performance level 5, which is already equivalent to the case corresponding to performance level 3 of 8-degree rare occurrence earthquake. Therefore, as the seismic intensity increases, the structural performance level requirements are correspondingly improved, and in the meantime time, multi-dimensional seismic component action requires to be sufficiently taken into account when structures are exposed in high-intensity earthquake disasters for the performance-based seismic design.

6 Concluding remarks

In this study, a framework of random function-based dimension reduction for simulating multi-dimensional fully non-stationary ground motion that satisfies the provision imposed by certain seismic specifications is elaborated. The simplified method, which originates from a local point of view, establishes the corresponding intensity modulation function, time-varying power spectrum and design response spectrum for multidimensional fully non-stationary ground motion by means of adequately taking into account the correlation regarding seismic parameters between different components. Further, aiming at the average response spectrum of the generated representative time-history set can be compatible with the design response spectrum, the iterative scheme is adopted by considering the fitting of the two. As a result, the simulated multi-dimensional seismic representative time-histories possess typical earthquake characteristics, including non-stationarities in the time and frequency domains, differences in intensity among the three seismic components, and so forth. Meanwhile, the statistical average response spectrum following iteration is also in agreement with the target value, which verifies that the proposed multidimensional stochastic ground motion model is of significant effectiveness and applicability in engineering practice.

Furthermore, the global reliability assessment of a high-rise frame-core tube structure with a height of 98.1 m subjected to multi-dimensional fully nonstationary ground motion that corresponds to 8-degree and 9-degree rare earthquakes is discussed in detail. The establishment of a nonlinear finite element model and the acquisition of the structural seismic response are numerically implemented by applying Midas Building software, which could comprehensively consider material and geometric nonlinearity. In the meantime, with the performance-based seismic design in mind, allowable values of the inter-story drift angle for five levels with regard to performance-based requirements that resist severe earthquake excitations, referred to as GB 50011-2010 (2016), are addressed. Next, with the help of the PDEM, refined structural global reliability under the four earthquake input conditions for various performance levels are thus readily obtained. The numerical calculation results thoroughly examined the influence of different ground motion components on the global reliability of a structure adapting to different seismic performance requirements. Some findings and suggestions are as follows:

● The middle and upper part of the frame core tube structure are weak parts, and the structure′s crosssectional size should not be excessively reduced during design process.

● The vertical seismic component has little effect on the inter-story drift angle. Thus, it is suggested that a more reasonable index needs to be further determined for evaluating global structural reliability to consider the influence of the vertical seismic component.

● Multi-dimensional seismic component action should be sufficiently considered when structures are exposed in high-intensity earthquake disasters for a performance-based seismic design.

In summary, the proposed strategy for simulating multi-dimensional stochastic earthquakes is kept simple, practical and feasible all at once, providing a strong basis for the performance-based seismic design of earthquakeexcited complex engineering structures.

Acknowledgement

This work was supported by the National Natural Science Foundation of China (Grant Nos. 51978543, 52108444, and 51778343), the Plan of Outstanding Young and Middle-aged Scientific and Technological Innovation Team in the Universities of Hubei Province (Project No. T2020010), and the Natural Science Foundation of Hebei Province (Grant No. E2021512001). The above organizations are highly appreciated.