Seismic fragility analysis and index evaluation of concrete-filled steel tube column frame-core tube structures

ABSTRACT

To understand the effect of connection type on the seismic fragility of concrete-filled steel tube (CFST) column frame-core tube structures, typical structures with different numbers of storeys (15, 30, 45, and 60) and different types of beam-end connections (rigid connections, hinged connections, and rigid connections with outriggers) are designed. The probabilistic seismic demand models (PSDMs) of the structures are obtained using cloud analysis, and seismic fragility curves based on the mean period and the number of storeys are established. The results show that the differences in the seismic fragilities among the structures with the three types of connections can be effectively determined by the critical inter-storey drift ratio (IDR) and the storey ductility and that the determination method with the critical IDR is relatively simple. The use of beam-end hinges in the 15-storey structure results in an advantageous seismic performance, with a lower probability of exceeding the limit states, and the performance of the 60-storey structure with beam-end hinges is sensitive to spectral acceleration. The spectral acceleration and peak ground acceleration (PGA) can effectively describe the fragility curves based on the number of storeys and the mean period, respectively.

Graphical Abstract

KEYWORDS:

• Concrete-filled steel tube column frame-core tube structures
• seismic fragility
• connection type
• mean period
• number of storeys

1. Introduction

Concrete-filled steel tube (CFST) frame-core tube structures are widely used in super-high-rise buildings. However, in terms of structural design, the shear ratio of the outer frame often exceeds the code thresholds, necessitating an increase in the stiffness of the outer frame to meet the code requirements. Researchers have argued that this design problem can be circumvented by decreasing the shear ratio of the outer frame and increasing the size of the core tube, thereby improving the collapse resistance of the structure (Lu et al. 2019). On this basis, we previously proposed a design solution for a CFST column frame-core tube structure system with adjustable connections, in which all the traditional rigid beam ends of the outer frame are replaced with hinged beam ends so that seismic shear is entirely resisted by the core tube (Zeng, Huang, and Zhou 2021). The use of bolted connections (similar to hinged joints) for the outer frame improves the on-site construction efficiency and meets the needs of building industrialization but fundamentally changes the load path between the outer frame and the core tube, calling for an in-depth investigation of the effects of different connection types on the performance of the outer frame (Bai et al. 2021; Li and Xie 2021).

Seismic damage to high-rise structures concentrates in the structural joints, so connections have become a key issue in seismic design and post-earthquake retrofitting. Peng et al. (Peng, Jinping, and Mahin 2020) studied a form of beam-to-column connection consisting of a mechanical hinge combined with a pair of buckling-restrained steel plates, in which the mechanical hinge is used to resist shear and the buckling-restrained steel plates are employed to transfer the bending moment to dissipate the seismic energy through the deformation of the steel plates; this combination ensures that other structural members remain elastic. Garoosi et al. (Moradi Garoosi, Roudsari, and Hashemi 2018) proposed two types of replaceable reduced beam section (RBS) connections, which were compared with an ordinary bolted connection and an ordinary RBS connection through experimental investigation and numerical simulation; the results showed that the replaceable RBS connections effectively reduced the damage to the columns and panel zones. Psycharis and Mouzakis (Psycharis and Mouzakis 2012) conducted monotonic loading, cyclic loading, and pure shear tests on precast beam and column members with pinned connections to investigate the effects of various parameters on the performance of the connections. Ghayeb et al. (Ghayeb, Hashim Abdul, and Ramli Sulong 2017) carried out cyclic loading tests on precast concrete joints and monolithic concrete joints and compared the strength, ductility, and energy dissipation of the joints. The test results showed that the new precast concrete joint for frame structures results in a drift ratio up to 9% and a satisfactory resistance to seismic loads. Sharma et al. (Sharma et al. 2021) compared the characteristics of the seismic fragility curves for semi-rigid steel frames subjected to near-field and far-field earthquakes and found that near-field earthquakes with a high directivity ratio led to a higher probability of exceedance than far-field earthquakes.

There have been many studies, including the aforementioned reports, on the seismic capacity and post-earthquake retrofitting of connections, whereas efforts have rarely been devoted to examining the influences of different types of connections on the global seismic-resistance capacity of high-rise structures. The damage mechanism of high-rise structures changes with the connection type; damage indices can be determined to effectively reflect the differences in the structural stress and thus clarify the elastic plastic behaviour of high-rise structures during earthquakes. Fragility analysis can describe the damage probability of the structure, which helps to realize the difference in the connections in probabilistic terms and to explore the effectiveness of different control induces. This work provides a foundation for the performance-based seismic design method.

In this present study, seismic fragility analysis is conducted to investigate the seismic failure probability of CFST column frame-core tube structures at different performance levels. The types of connections of the outer frame are divided into traditional types (rigid beam-end connections and rigid beam-end connections with outriggers) and a new type (hinged beam-end connections). The seismic failure probability curves of structures with different types of connections are analysed to provide a reference for seismic design in the future.

Research results of fragility analysis for composite steel-concrete structures have been widely used for seismic damage assessment. Pnevmatikos et al (Pnevmatikos, Papagiannopoulos, and Papavasileiou 2019) compared the fragility curves of mixed concrete/steel moment-resisting framed structures with those of concrete or steel framed structures and found that mixed concrete/steel framed structures designed according to the current code are less vulnerable. Mohammadgholibeyki et al (Mohammadgholibeyki and Epackachi 2020) developed drift-based fragility functions by statistically analysing the test data of 60 steel-plate concrete composite shear walls and indicated that the aspect ratio and axial load ratio have a significant effect on the fragility function. Xu et al (Chengxiang et al. 2018) carried out a seismic fragility analysis of steel-reinforced concrete frame structures with OpenSees software and investigated the differences in the seismic fragility curves based on the peak inter-storey drift ratio (IDR) and two damage indices, demonstrating that the analysis based on the damage index of the members is more cost-effective, practical, and robust. Xiong et al (Xiong, Xinzheng, and Lin 2019) executed the fragility analysis of the frame-shear wall structure using the storey curvature as the engineering demand parameter (EDP) and determined and validated the damage limit state (LS) of the shear wall. Skalomenos et al (Skalomenos, Hatzigeorgiou, and Beskos 2015) researched the seismic performance of plane moment-resisting frames composed of steel I-beams and CFST columns and established fragility curves based on the peak ground acceleration (PGA) and spectral acceleration.

2. Structural models and parameter selection

2.1. Structures with different types of connections

In total, 12 structures are designed considering combinations of three different types of connections (hinged beam-end connections, rigid beam-end connections, and rigid beam-end connections with outriggers, whose corresponding structures are hereafter called hinged-connection structures, rigid-connection structures and outrigger structures) with four different structure heights (15, 30, 45, and 60 storeys). The ground motion intensity levels conforming to an extreme value type I distribution (shown in Section 2.3) with seven means, each with 120 PGA samples, are considered. The structural analysis software CANNY (Guobin et al. 2019) is used to construct the numerical structural models, which have been verified by testing (Zeng, Huang, and Zhou 2021), and to carry out a total of 10,080 non-linear dynamic time history analyses.

Figure 1. Diagram of structural models: (a) elevations of structures with different heights; (b) plan layout; (c) elevation of outrigger storey.

Figure 2. Uniaxial spring model of beams.

Figure 3. Column element model.

Figure 4. Rigid wall-beam model: (a) wall model with a rigid wall-beam; (b) nodal rotations equal to the wall rotation.

The overall dimensions of the structures refer to the model structures shown in Figure 1. According to technical specifications (DBJ15-92-2013 2013), the structure is designed for seismic conditions corresponding to a seismic precautionary intensity of 7, site class III, and second seismic design group. Grade C60 concrete (with a cube compressive strength of 60 MPa) and HRB400 reinforcing steel (with a standard yield strength of 400 MPa) are used in the structures. Tables 1–4 show the design parameters of various structural models. The outriggers of the outrigger structure are located in storeys 6–8 and 11–13 of the 15-storey model, storeys 14–16 and 21–23 of the 30-storey model, storeys 21–23 and 33–35 of the 45-storey model, and storeys 29–31 and 44–46 of the 60-storey model. For the hinged-connection structure, when the core tube thickness is appropriately increased, the column and beam section sizes of the outer frame are reduced accordingly.

Table 1. CFST columns and core tube sections of the rigid-connection, hinged-connection and outrigger structures.

Storey layout of cross sections for structures				Rigid-connection and outrigger structures (mm)		Hinged-connection structure (mm)
15 storeys	30 storeys	45 storeys	60 storeys	CFST column	Core tube thickness	CFST column	Core tube thickness
8 ~ 15	21 ~ 30	34 ~ 45	47 ~ 60	1900	1100	1000	1400
2 ~ 7	15 ~ 20	28 ~ 33	41 ~ 46	2200	1200	1300	1500
1	9 ~ 14	22 ~ 27	35 ~ 40	2300	1300	1400	1600
-	3 ~ 8	16 ~ 21	29 ~ 34	2300	1400	1400	1700
-	1 ~ 2	10 ~ 15	23 ~ 28	2400	1500	1500	1800
-	-	4 ~ 9	17 ~ 22	2500	1600	1600	1900
-	-	1 ~ 3	11 ~ 16	2600	1800	1700	2100
-	-	-	5 ~ 10	2700	2000	1800	2300
-	-	-	1 ~ 4	2800	2200	1900	2500

Table 2. Dimensions and reinforcement ratios of the steel beams.

Structure	Steel beam (mm)
Rigid-connection and outrigger structures	Inner I-beam: 400 × 1600 × 36 × 20 Outer I-beam: 400 × 1800 × 36 × 20 Box girder of outrigger storey: 800 × 1200 × 35
Hinged-connection structure	Inner I-beam: 300 × 850 × 25 × 14 Outer I-beam: 300 × 1000 × 25 × 14

Notes: I-beam dimensions given as flange width×height×flange thickness×web thickness, box girder dimensions given as width×height×thickness.

Table 3. Reinforcement ratio of the core tube (%).

Structure	15 storeys (${\rho }_{\mathrm{A}\mathrm{Z}},{\rho }_{\mathrm{A}\mathrm{Q}}$)		30 storeys (${\rho }_{\mathrm{A}\mathrm{Z}},{\rho }_{\mathrm{A}\mathrm{Q}}$)		45 storeys (${\rho }_{\mathrm{A}\mathrm{Z}},{\rho }_{\mathrm{A}\mathrm{Q}}$)		60 storeys (${\rho }_{\mathrm{A}\mathrm{Z}},{\rho }_{\mathrm{A}\mathrm{Q}}$)
	1 F ~ 7 F	8 F ~ 15 F	1 F ~ 8 F	9 F ~ 30 F	1 F ~ 9 F	10 F ~ 45 F	1 F ~ 10 F	11 F ~ 60 F
Hinged-connection structure	1.3, 0.40	1.3, 0.35	1.4, -	1.3, 0.35	1.4, -	1.3, 0.35	1.4, -	1.3, 0.35
Rigid-connection structure	1.1, 0.40	0.9, 0.35	1.2, 0.40	0.9, 0.35	1.2, -	0.9, 0.35	1.2, -	0.9, 0.30
Outrigger structure	1.1, 0.38	0.9, 0.30	1.2, 0.35	0.9, 0.30	1.2, -	0.9, 0.30	1.2, -	0.9, 0.30

Notes: ${\rho }_{\mathrm{A}\mathrm{Z}}$ is the flexural reinforcement ratio for component AZ of the walls; ${\rho }_{\mathrm{A}\mathrm{Q}}$ is the ratio of the horizontal distributing bar for component AQ of the walls and the corresponding ratio of the vertical distributing bar is 1.2%; “-” means no component AQ in the wall; components AZ and AQ are shown in Figure 5.

Table 4. Reinforcement of coupling beams for rigid-connection, hinged-connection and outrigger structures.

Storey layout of cross sections for structures				Rigid-connection and outrigger structures			Hinged-connection structure
15 storeys	30 storeys	45 storeys	60 storeys	Upper or lower reinforcement	Web bar	Stirrup	Upper or lower reinforcement	Web bar	Stirrup
8 ~ 15	21 ~ 30	34 ~ 45	47 ~ 60	17C25	C18@150	C10@100(8)	17C25	C18@150	C10@100(8)
2 ~ 7	15 ~ 20	28 ~ 33	41 ~ 46	18C25	C20@150	C10@100(8)	18C25	C20@150	C10@100(8)
1	9 ~ 14	22 ~ 27	35 ~ 40	19C25	C20@150	C10@100(8)	19C25	C20@150	C10@100(10)
-	3 ~ 8	16 ~ 21	29 ~ 34	21C25	C20@150	C10@100(8)	21C25	C20@150	C10@100(10)
-	1 ~ 2	10 ~ 15	23 ~ 28	22C25	C22@150	C10@100(8)	22C25	C22@150	C10@100(10)
-	-	4 ~ 9	17 ~ 22	24C25	C22@150	C10@100(10)	24C25	C22@150	C10@100(10)
-	-	1 ~ 3	11 ~ 16	27C25	C25@150	C10@100(10)	27C25	C25@150	C10@100(12)
-	-	-	5 ~ 10	30C25	C25@150	C10@100(12)	30C25	C25@150	C10@100(12)
-	-	-	1 ~ 4	33C25	C22@100	C10@100(12)	33C25	C22@100	C10@100(14)

The uniaxial spring model is adopted to simulate the forces in the steel beams and coupling beams, as shown in Figure 2. The CFST columns employ the multi-axial spring model in Figure 3 to consider the coupling effect of the bending moment and axial force and to consider the seismic bidirectional effect of bending and shear. The shear wall model allows in-plane bending and shear resistance. The rigid diaphragm assumption leads to the upper and lower ends of the shear wall model using rigid beams, and the side and middle columns of the model simulate the in-plane seismic performance, as shown in Figure 4.

As shown in Table 5, the periods in the horizontal directions are almost equal to the structural symmetry, and the fundamental periods T₁ of the 15-storey structure with different types of connections are similar. For structures with other numbers of storeys, the more storeys there are, the more pronounced the difference in T₁, which indicates the strong influence of this type of connection. During the structural design process, the stiffness of the outer frame in the hinged-connection structure is reduced while increasing the core tube stiffness, which possibly results in its fundamental period being slightly lower than that of the rigid-connection structure. It is useful to examine the differences among the three types of connections and make subsequent comparisons. According to the magnitude of the translational periods, the outrigger structure has the largest stiffness, followed by the rigid-connection structure, and the hinged-connection structure is most flexible, as expected. The torsional periods of the outrigger structure and rigid-connected structure are similar due to the lower influence, and the hinged-connection structure with the highest stiffness core tube has the smallest torsional period.

Table 5. Vibration modes of the finite element models in CANNY.

Structure	Hinged-connection structure				Rigid-connection structure				Outrigger structure
Storey	15	30	45	60	15	30	45	60	15	30	45	60
1st-order translation in the X-direction (s)	0.52	1.61	3.28	5.4	0.56	1.62	3.06	4.76	0.53	1.49	2.8	4.33
1st-order translation in the Y-direction (s)	0.52	1.61	3.28	5.4	0.56	1.62	3.06	4.76	0.53	1.47	2.78	4.3
1st-order torsion (s)	0.33	0.64	0.93	1.19	0.39	0.76	1.1	1.41	0.38	0.76	1.1	1.4

The Rayleigh damping corresponding to a 5% damping ratio is applied in the first and fifth modes in this paper, and the corresponding simulation result errors, verified in the test (Zeng, Huang, and Zhou 2021), are relatively stable, which indicates that the spurious damping force does not affect the results too much. When the structure enters the elastic-plastic phase, the damping generated by the non-structural members is difficult to reflect in the numerical model, and the presence of the spurious damping force can consider this part of the damping force to alleviate the deviations in the computed results. This damping is also frequently used in related papers, and its results are accepted (Trautner et al. 2019; Birzhandi and Halabian, 2018; Moniri 2017).

Figure 5. Typical wall and coupling beam details: (a) construction plan of the core tube; (b) section of the coupling beam.

2.2. Selection of the ground motion records

A total of 12 ground motion records from the Centre for Engineering Strong Motion Data are selected (CESMD Center Management Group n.d.). The basic information about these ground motion records is shown in Table 6. Based on the test phenomenon, ground motions with longer periods cause greater damage to the hinged-connection structures (Zeng, Huang, and Zhou 2021). In this paper, the structures are subjected to ground motions with different frequencies, evaluated by the mean periods $T_m$, and the results are analysed to understand the differences in the seismic responses.

Table 6. Ground motion characteristics.

No.	Ground motion name	Station name	Date	ML	Epicentral distance (km)	PGA (cm/s^2)	$T_m$(s)
1	Anza	Pinyon Flat	2001	5.1	19.6	132.4	0.119
2	Anza	Pinyon Flat	2005	5.6	18.2	309	0.140
3	Mammoth Lakes	Bishop-Paradise Lodge	1980	6.2	24.4	98.6	0.205
4	Guadalupe Victoria	El Centro Array 07	2006	5.4	63.7	24.4	0.313
5	Mammoth Lakes	Long Valley Dam	1980	6.1	15.6	191.6	0.352
6	Guadalupe Victoria	El Centro Array 10	2006	5.4	61.8	11	0.409
7	Mammoth Lakes	Convict Creek	1980	6.0	16.4	158.3	0.446
8	Livermore	Walnut Creek	1980	5.5	27.3	43.0	0.503
9	Coalinga	Parkfield-Stone Corral 1E	1983	6.7	50.5	121.9	0.562
10	Livermore	Walnut Creek	1980	5.5	27.3	133.9	0.601
11	Gulf of California	El Centro Array 10	2001	5.7	107.4	16.9	0.871
12	Imperial Valley	Meloland Overpass FF	1979	6.6	3.1	311.2	1.055

The principles of seismic selection require that ground motions be the effective motivation for the structural response, and the range of earthquake frequencies should be represented by the ground motions selected. Therefore, the bracketed duration (Bommer and MARTÍNEZ-PEREIRA 1999) (the threshold value is 5% of the PGA) of ground motions is generally 5 times the T₁ of the structures, and the values of $T_m$, an index calculated using Equation (1)(1) $T_m=\frac{\sum _i{C}_i^2\left(\frac{1}{f_i}\right)}{\sum _i{C}_i^2}$(1) (Rathje, Abrahamson, and Bray 1998) to describe the seismic frequency, conform to a uniform distribution between 0 and 1.2 s. Bidirectional seismic excitations are adopted, where the ratio of the PGAs in the primary and secondary directions is 1:0.85.

To investigate the effects of $T_m$, the ground motion records are divided into four groups. Group 1 contains record nos. 1–3 (the low-period group with 0 <$T_m$< 0.3 s), Group 2 includes record nos. 4–6 (the short- to medium-period group with 0.3 <$T_m$< 0.45 s), Group 3 consists of record nos. 7–9 (the medium- to long-period group with 0.45 <$T_m$< 0.6 s), and Group 4 contains record nos. 10–12 (the long-period group with 0.6 <$T_m$< 1.2 s).

(1) $T_m=\frac{\sum _i{C}_i^2\left(\frac{1}{f_i}\right)}{\sum _i{C}_i^2}$(1)

where $T_m$ is the mean period, C_i is the Fourier spectrum value corresponding to each frequency and f_i is a frequency between 0.25 Hz and 20 Hz.

A comparison between the long-period design spectrum (DBJ15-92-2013 2013) and the response spectra of the 12 ground motion records is shown in Figure 6, which describes the acceleration response spectra for every ground motion with a PGA equal to 0.035 g. Typically, to reflect the seismic frequencies of most natural seismic waves and study the influence of the seismic frequency on the structures, the whole waves selected do not have larger spectral values at periods greater than 1 s. The numbers of the ground motions increase from the smallest to the largest mean period, so that the No. 12 ground motion in Group 4 representing long-period waves has the largest spectral acceleration value at periods greater than 1 s. The input seismic wave is amplified to the desired PGA (shown in Section 2.3) to derive the fragility curves.

The T₁ values of structures with 15 to 60 storeys, representing a wide range of structures in terms of height, are scattered on the platform and descent stage of the design spectrum so that the effects of the spectral acceleration on the structures of different heights can be fully reflected. The response spectra for the ground motion records are not matched to consider the influence of $T_m$, which results in a uniform distribution. As a result, the results are representative and reveal the corresponding regularities.

Figure 6. Response spectra for the selected ground motion records.

2.3. Random variable sampling

The PGA is randomly selected using MATLAB according to the extreme value type I distribution with parameters $\alpha$ and $\mu$. The mean values of PGA were set to 0.05 g, 0.1 g, 0.3 g, 0.5 g, 0.7 g, 0.9 g, and 1.2 g, and 120 samples were generated at each level. According to the extreme value type I distribution used in the literature (Zeng, Huang, and Zhou 2019), the ratio of the mean to the standard value is set to 0.75, and the coefficient of variation (COV) is set to 0.73. Other parameters are shown in Table 7, and the fitting effect between the sampled PGAs and the data obtained theoretically according to the extreme value type I distribution is illustrated in Figure 7.

Table 7. Parameters of the extreme value type I distribution.

Parameter	m = 0.05 g	m = 0.1 g	m = 0.3 g	m = 0.5 g	m = 0.7 g	m = 0.9 g	m = 1.2 g
$\alpha$	3.51	1.76	0.59	0.35	0.25	0.2	0.15
$\mu$	0.34	0.67	2.01	3.36	4.7	6.04	8.06

An increase in the scale factor can lead to an overestimation in the structural response, so it is recommended that the maximum scale factor be taken as 2 (Dávalos and Miranda 2019). Nevertheless, it is difficult to avoid choosing a scale factor greater than 2 in the fragility analysis, which is limited by the natural seismic waves and the study objectives. In this paper, cloud analysis, used widely in fragility analysis, can ensure that the results are not at the maximum deviation by regression analysis of mass data. Meanwhile, most scale factors of the PGA samples, conforming to the extreme value type I distribution, are within a reasonable range, effectively controlling the magnitude of the deviation. Therefore, the calculation results can be used in structural design.

Figure 7. Comparison of extreme value type I distribution and samples.

2.4. Ground motion intensity measures (IMs) and the demand parameter

Luco and Cornell (Nicolas Luco and Cornell 2007) established the efficiency and sufficiency criteria for IM selection, and related studies have quantified the sufficiency of IMs, concluding that spectral acceleration is an efficient and sufficient IM (Dhulipala et al. 2018; Ebrahimian et al. 2015). However, some studies have indicated that PGA may be more suitable in the case of near-field earthquakes (Liu, Paolacci, and Da-Gang 2017). The ground motion IMs recommended by HAZUS (HAZUS-MH MR4 2003) include PGA and spectral acceleration S_a(T₁) (corresponding to a damping ratio of 5% and T₁). In this paper, PGA and S_a(T₁) are applied to perform the fragility analysis, and the effectiveness of each is evaluated.

The IDR is commonly regarded as an EDP for fragility analysis. To extend the scope of application and compensate for the deficiency in the evaluation using the IDR, the critical IDR and the storey ductility are selected as additional indices to describe the horizontal displacement of the structure. The storey ductility is applied widely to reflect the seismic response of the structure(Ghayoumian and Emami 2020; Emami and Halabian 2018), and the storey ductility of the ith storey, ${\mu }_{ei}$, is defined as the ith maximum storey drift divided by the ith yield storey drift. The critical IDR of the ith storey, ${\theta }_{iD}$, is defined according to the specifications (JGJ 3-2010 2010) as follows:

(2) ${\theta }_{iE}=\frac{d_i-d_{i-1}}{h_{i-1}}$(2)

(3) ${\theta }_{iD}={\theta }_{iE}-{\theta }_{(i-1)E}$(3)

where $d_i$ and $d_{i-1}$ are the displacements of the ith storey and the (i-1)th storey, $h_{i-1}$ is the height of the (i-1)th storey, and ${\theta }_{iE}$ and ${\theta }_{(i-1)E}$ are the IDRs of the ith storey and the (i-1)th storey, respectively.

3. Fragility function for seismic risk assessment

3.1. Probabilistic seismic demand model (PSDM)

The PSDM is a probability distribution model that describes the seismic demand capacity of a structure under uncertain earthquakes. This model is the basis for establishing the seismic fragility curve of the structure. According to a study by Cornell et al. (Cornell et al. 2002) on PSDMs, it can be generally assumed that EDP and IM follow a power function relationship:

(4) $m_{EDP}=a{\left(im\right)}^b$(4)

where $m_{EDP}$ is the predicted seismic demand of EDP for the structure, im is the value of IM for the ground motion, and a and b are fitting constants.

The deviation between the time history analysis result D_i and the fitted result $ln{m}_{EDP}$ is denoted by${\beta }_{EDP|IM}$ (Zanini and Hofer 2019), which reflects the uncertainty in PSDM:

(5) ${\beta }_{EDP|IM}=\sqrt{\frac{1}{N-2}\sum _{i=1}^N{\left(ln{m}_{EDP}-ln{D}_i\right)}^2}$(5)

where ${\beta }_{EDP|IM}$ is the standard deviation of the fitted log-linear curves and N is the total number of data points of the time history analysis results.

3.2. Method for fragility curve construction

The seismic fragility curve describes the relationship between IM and the structural failure probability, which is the probability of EDP exceeding the seismic capacity demand (symbolized by C) of the structure at the LS. Given the assumption of log-normal distributions for EDP and C, the structural failure probability ${\beta }_{EDP|IM}$ can be expressed by the following general equation

(6) $P_f\left(C\left.⟨EDP\right|im=x\right)=\varphi \left(\frac{lnx-ln{m}_R}{{\beta }_R}\right)$(6)

where the median value $P_f\left(C\left.⟨EDP\right|im=x\right)=\varphi \left(\frac{lnx-ln{m}_R}{{\beta }_R}\right)$ denotes the quantile of a log-normally distributed random variable with an exceedance probability of 50%, as shown in Equation (7)(7) $m_R={\left(\frac{m_C}{a}\right)}^{1/b}$(7) , ${\beta }_R$ represents the uncertainty in the fragility curve, and ${\beta }_R$ is a standard normal distribution function.

(7) $m_R={\left(\frac{m_C}{a}\right)}^{1/b}$(7)

Considering the various uncertainties, the equation for ${\beta }_R$ is given by

(8) ${\beta }_R=\frac{\sqrt{{\beta }_{EDP|IM}^2+{\beta }_C^2+{\beta }_M^2}}{b}$(8)

${\beta }_C$ represents the uncertainty in parameters such as the structural dimensions and material strength, and ${\beta }_M$ represents the uncertainty in the assumptions such as those regarding the equation and numerical model. Previous research by Wen et al. (Wen, Ellingwood, and Bracci 2004) has indicated that the value of ${\beta }_M$ does not have a significant effect on the fragility curve, so the accepted value of ${\beta }_M=0.2$ is adopted in the present study (Yu, Lu, and Bing 2017; Rajeev and Tesfamariam 2012). Moreover, ${\beta }_C=0.25$ and ${\beta }_C=0.47$ are employed for slight to moderate damage states and severe damage states, respectively, which are based on the recommendations of HAZUS (HAZUS-MH MR4 2003) and some scholars (Tavares, Padgett, and Paultre 2012; Liel et al. 2009).

Given the properties of the log-normal distribution, the median seismic capacity $m_C$ is obtained by Equation (9). ${\mu }_C$ and ${\delta }_C$ are the mean (Table 9) and COV, respectively, of C for different LSs.

Table 8. Mean value of the correlation coefficients.

Influence factor	IDR		Critical IDR		Storey ductility
	Sa(T1)	PGA	Sa(T1)	PGA	Sa(T1)	PGA
$T_m$	0.44	0.83	0.63	0.87	0.39	0.63
Storeys	0.95	0.62	0.86	0.75	0.74	0.50

Table 9. Correlation coefficients of the fitting curves for different $T_m$.

Control index	Outrigger structure	Rigid-connection structure	Hinged-connection structure
IDR	0.83	0.83	0.82
Critical IDR	0.84	0.89	0.89
Storey ductility	0.6	0.68	0.62

$m_C=\frac{{\mu }_C}{1+{\delta }_C^2}$, ${\beta }_C=\sqrt{ln\left(1+{\delta }_C^2\right)}$ (9)

4. Probabilistic seismic demand analysis (PSDA) and performance level

To investigate the seismic demands of structures with different connection types, it is necessary to evaluate the corresponding EDPs based on the number of storeys and ${\beta }_C=\sqrt{ln\left(1+{\delta }_C^2\right)}$ and to establish their PSDMs with cloud analysis (Lombardi and Flavia 2020). Numerous finite element time history analyses are performed, and the resulting data are subjected to log-linear regression analysis. Table 8 gives the fitted results based on different IMs employing PGA and S_a(T₁). These values demonstrate that the PSDM adopting the PGA as the IM for $T_m$ has higher correlation coefficients, while the PSDM for considering the number of storeys has lower correlation coefficients. As a result, $T_m$ and the numbers of storeys are analysed with PSDMs considering only the PGA and S_a(T₁), respectively.

A pulse earthquake with longer-period ground motion has a larger mean period value, and PGA has obvious advantages in describing the fragility curves with different mean periods, which effectively shows the changing regularity of the damage indices under ordinary and pulse earthquakes. S_a(T₁) is suitable to reflect the influence of the number of storeys on the damage indices because the approximate values of T₁ decrease the fitting errors of the fragility curves.

Tables 9 and 10 show the average values of the correlation coefficients corresponding to Figures 8, 9. The correlation coefficients of the IDR and critical IDR are close to 1, which indicates an appropriate fitting effect. However, the fitting curves based on storey ductility have lower correlation coefficients. Large errors generally appear in the elastic phase of the structure, but the fitting curves in the elastoplastic phase conform to the tendency of the data distribution, reflecting the ductility demand, as shown in Figures 8, 9. In the seismic design of the structure, the seismic performance of the structure in the elastic-plastic phase must meet engineering needs.

Table 10. Correlation coefficients of the fitting curves for different numbers of storeys.

Control index	Outrigger structure	Rigid-connection structure	Hinged-connection structure
IDR	0.95	0.94	0.94
Critical IDR	0.89	0.86	0.84
Storey ductility	0.78	0.74	0.71

4.1. PSDM based on the mean period

The maximum IDR ${\theta }_E$, maximum critical IDR ${\theta }_D$, and maximum storey ductility ${\mu }_E$ are accordingly employed as EDPs, and Figure 8 shows their PSDMs for the three structural models and displays the small error in the regression analysis.

The trend of the variation in ${\theta }_E$ is similar to that of ${\theta }_D$ for each $T_m$. The larger the value of $T_m$ is, the larger the values of ${\theta }_E$ and ${\theta }_D$. In comparison to $T_m$, the types of connections have little influence on ${\theta }_E$ and ${\theta }_D$. Under medium-period and long-period ground motions, the values of ${\theta }_E$ are more stable and less affected by $T_m$ than those of ${\theta }_E$.

During low-period ground motions with a PGA less than 1 g ($ln\mathrm{P}\mathrm{G}\mathrm{A}<0$), the structures are still in the elastic stage since the values of ${\mu }_E$ are less than 1, meaning that almost no plastic deformation occurs. However, under the impact of long-period ground motions, the demand on ${\mu }_E$ increases rapidly with increasing PGA. Therefore, when the structure is subjected to intense long-period ground motions, ${\mu }_E$ should be strictly controlled by seismic measures to ensure that the structural members have sufficient ductility.

Figure 8. Fitting curves between EDP and $ln\mathrm{P}\mathrm{G}\mathrm{A}$ for different $T_m$: (a) outrigger structure; (b) rigid-connection structure; (c) hinged-connection structure.

4.2. PSDM based on the number of storeys

The values of ${\theta }_E$ and ${\theta }_D$ clearly increase with an increasing number of storeys. To facilitate the discussion, structures with 15, 30–45, and 60 storeys are referred to as sub-high-rise structures, high-rise structures, and super-high-rise structures, respectively. In comparison with the results in the previous section, the trend of ${\theta }_D$ becomes markedly different from that of ${\theta }_E$ with an increasing number of storeys. As shown in Figure 9, the fitted curves of ${\theta }_E$ are basically parallel to each other, but those of ${\theta }_D$ for structures with different numbers of storeys intersect close to lnS_a(T₁) = 0 as S_a(T₁) increases. The fact that the seismic demand on the 30-storey structure is higher than that on the 45-storey structure beyond the intersection point indicates that when subjected to ground motions with a large S_a(T₁), structures with fewer storeys are susceptible to greater damage.

As seen from the variation in the seismic demand of ${\mu }_E$ shown in Figure 9, the more storeys there are, the higher the seismic demand on ${\mu }_E$, indicating that the ductility capacity of high-rise and super-high-rise structures should be improved. In addition, as expected, the super-high-rise hinged-connection structure has the highest ductility demand; thus, the ductility capacity of the structural members should be enhanced to guarantee a greater seismic performance than those for the rigid-connection structure and outrigger structure.

Figure 9. Fitting curves between EDP and $ln{\mathrm{S}}_{\mathrm{a}}\left(T_1\right)$ for different numbers of storeys: (a) outrigger structure; (b) rigid-connection structure; (c) hinged-connection structure.

4.3. Structural performance level

The division of the structural performance levels is determined by the thresholds of different damage LSs, and these thresholds are specific values of the quantitative measure C. Three damage LSs, namely, slight damage (LS₁), moderate damage (LS₂) and extensive damage (LS₃), are considered in this study. A unified criterion, using the pushover analysis method, is adopted to quantify the thresholds of ${\theta }_E$ and ${\theta }_E$. The threshold for LS₁ is defined as the elastic limit of the pushover curve, represented by the first turning point of the curve; the threshold for LS₂ is defined as the equivalent yield point of the structure, and the threshold for LS₃ is defined as the peak point of the pushover curve. The determination method of the equivalent yield point is shown in Figure 10 (Wang, Wang, and Zheng 2018). Figure 11 schematically shows the values of different LS points on the pushover curve. Figure 12 exhibits the distribution of LS points, which are basically located in three regions, and the average value for each region is used as the corresponding LS value, as shown in Table 11.

Table 11. EDPs corresponding to the LSs.

Damage state	IDR	Critical IDR	Storey ductility
Slight damage	1/620	1/5000	1.6
Moderate damage	1/140	1/820	3.6
Extensive damage	1/65	1/350	5.6

This lower limit of the member ductility factor estimated by Equation (10)(10) ${\mu }_{\varphi }=2\mu -1\hspace{0.17em}\left(T_1\ge T_c\right)$(10) from the Eurocode is selected as the corresponding LS value of the storey ductility factor listed in Table 11.

(10) ${\mu }_{\varphi }=2\mu -1\hspace{0.17em}\left(T_1\ge T_c\right)$(10)

where ${\mu }_{\varphi }=2\mu -1\hspace{0.17em}\left(T_1\ge T_c\right)$ is the global ductility factor of the structure; $T_c$ is the period at the upper limit of the constant acceleration region of the spectrum in the Eurocode; $T_c$ of all the structural models considered here are greater than $T_c$; $T_c$ is the ductility factor of the structural member, which is regarded as the LS value of storey ductility.

Figure 10. Definition of the equivalent yield point.

Figure 11. Definitions of LSs identified from the pushover curves (15 storeys as an example).

Figure 12. LS thresholds identified from the pushover curves.

The lateral force-resistant capacity of each structural model, described by the average base shear at the yield and ultimate points, is shown in Table 12. The outrigger structure has the greatest lateral force resistance, and according to the higher stiffness of the core tube, the hinged-connection structure has a slightly greater lateral force resistance than the rigid-connection structure. It can be assumed that the reasonable stiffness enhancement of the core tube accompanied by a reduction of the material of the outer frame may be a feasible way to increase the seismic performance.

Table 12. Average base shear of each structure in the pushover analysis ($\times {10}^5\mathrm{k}\mathrm{N}$).

Yield point			Ultimate point
Hinged-connection structure	Rigid-connection structure	Outrigger structure	Hinged-connection structure	Rigid-connection structure	Outrigger structure
286,118	275,883	310,116	345,937	327,332	365,626

5. Seismic fragility curves

5.1. Effect of the mean period of earthquake motion

5.1.1. Damage analysis of the maximum IDR

The fragility curves in Figure 13 show the patterns of damage (using $\times {10}^5\mathrm{k}\mathrm{N}$ as the index) to the hinged-connection structure, rigid-connection structure and outrigger structure at different performance levels. As shown in Figure 13, most of the fragility curves of the different models are highly consistent, especially under long-period ground motions, indicating insignificant differences among the performance levels of these structures. Therefore, the different mechanical behaviours among the three types of connections cannot be distinguished well by using ${\theta }_E$ as a damage index, which reflects the deformation in only the middle and upper storeys of the structures, as shown in Figure 14. However, the PSDM is fitted well as a function of ${\theta }_E$, which is therefore an appropriate index for measuring structural safety. Under LS₁ and LS₂, the structures subjected to long-period ground motions have a greater exceedance probability of ${\theta }_E$ than those subjected to low- and medium-period ground motions, as shown in the labels of Figure 13b,c

Figure 13. Damage fragility curves of ${\theta }_E$ for the ground motion groups: (a) slight damage; (b) moderate damage; (c) extensive damage.

Figure 14. Storey layout of ${\theta }_E$: (a) outrigger structure; (b) rigid-connection structure; (c) hinged-connection structure.

5.1.2. Damage analysis of the maximum critical IDR

Overall, the exceedance probability of ${\theta }_D$ is higher than that of ${\theta }_E$. As shown in Figure 15, ${\theta }_D$ basically occurs in the lower storeys of the structures, which corresponds with the damage phenomena observed in actual structures. This feature explains the inherent relation between ${\theta }_D$ and the lower storeys of structures.

More remarkable differences are displayed in the fragility curves of ${\theta }_D$ than those of ${\theta }_E$, especially under LS₁ and LS₂. As shown in Figure 16, the hinged-connection structure is generally associated with a lower exceedance probability than other structures, and the distinctions of the fragility curves among the structures reflect the different mechanical behaviours. According to ATC-63 (Applied Technology Council 2008), which states that “the probability of collapse under the maximum considered earthquake must be less than 10%,” a 14.8% difference in the exceedance probability at PGA = 0.4 g cannot be ignored, and ${\theta }_D$ becomes an effective index to distinguish the structures with the three types of connections, as shown in the labels of Figure 16 b,c. Hence, to compensate for the deficiency of ${\theta }_E$, ${\theta }_D$ should be used to control the exceedance probabilities of different structures more accurately.

Figure 15. Storey layout of ${\theta }_D$: (a) outrigger structure; (b) rigid-connection structure; (c) hinged-connection structure.

Figure 16. Damage fragility curves of ${\theta }_D$ for the ground motion groups: (a) slight damage; (b) moderate damage; (c) extensive damage.

5.1.3. Damage analysis of the maximum storey ductility

The hinged-connection structure usually has a higher ductility demand compared to the other structures, and as shown in Figure 17, it is associated with a higher exceedance probability for ductility, which is consistent with the forecast. The value of ${\mu }_E$ is an ideal control index, directly reflecting the seismic energy dissipation capacity of the structure and identifying the types of connections adopted in the structures, similar to ${\theta }_D$. In particular, ${\mu }_E$ has a better identification effect than ${\theta }_D$ in the slight damage state. However, the thresholds for the LSs of ${\mu }_E$ are difficult to determine, and the calculation of ${\mu }_E$ is very complicated. Because there is no uniform definition of yielding, the structural elastic-plastic strain is difficult to simulate realistically.

Figure 17. Damage fragility curves of ${\mu }_E$ for the ground motion groups: (a) slight damage; (b) moderate damage; (c) extensive damage.

5.2. Effect of the number of storeys

The number of storeys of a structure significantly affects the EDP. Through the establishment of seismic fragility curves as functions of the number of storeys, it is possible to estimate the structural damage and provide a basis for the structural performance index. Super-high-rise structures require stricter displacement demand under a specific ground motion, which is a basic structural response phenomenon. To decrease the displacement demand of super-high-rise structures, controlling T₁ is the key for aseismic design because of the strong effect of S_a(T₁) in the fragility assessment. The T₁ values of super-high-rise structures are generally greater than 4 s and are in the long-period descent stage of the design response spectrum. Figure 18,19 and 20 show that a slight difference in S_a(T₁) can cause a very large change in the exceedance probability, making the descent stage of the design spectrum an important basis for design. For sub-high-rise structures, the hinged-connection structure has the lowest probability of exceedance for both ${\mu }_E$ and ${\theta }_D$.

Figure 18. Damage fragility curves of ${\theta }_D$ for different numbers of storeys: (a) slight damage; (b) moderate damage; (c) extensive damage.

Figure 19. Damage fragility curves of ${\theta }_D$ for different numbers of storeys: (a) slight damage; (b) moderate damage; (c) extensive damage.

Figure 20. Damage fragility curves of ${\mu }_E$ for different numbers of storeys: (a) slight damage; (b) moderate damage; (c) extensive damage.

6. Conclusions

The effects of factors including the beam-end connections, number of storeys, PGA, S_a(T₁), and $T_m$ on structural seismic fragility were studied to investigate the exceeding probabilities of the control index, considering the IDR ${\theta }_E$, critical IDR ${\theta }_D$ and storey ductility ${\mu }_E$, and the conclusions are summarized as follows:

• Although ${\theta }_E$ is a common control index for the seismic design of structures, it is not very effective in judging the differences among the types of connections. The relevant control index should also be considered.

• The parameter ${\mu }_E$ is directly related to the structural energy dissipation capacity and has a clear physical meaning; thus, it can reflect the differences in connections. However, ${\mu }_E$ has some disadvantages involving complex calculations, discrete results and difficulty in deriving the LS values.

• The mechanical behaviours among structures with different types of beam-end connections are diverse; thus, ${\theta }_D$ should be used as an additional control index to reflect the effects of different connection types on the structural fragility more comprehensively. In comparison to the calculation results using ${\mu }_E$, those using ${\theta }_D$ have simpler calculation processes and smaller errors in the regression analysis.

• A structure with more storeys is more sensitive to S_a(T₁), and the spectral acceleration corresponding to the fundamental period T₁ of super-high-rise structures should be strictly controlled. The long-period segment of the acceleration spectrum needs to be carefully explored.

• For sub-high-rise structures, the hinged beam-end connections have the lowest exceedance probabilities of ${\theta }_E$ and ${\theta }_D$ and a moderate exceedance probability of ${\mu }_E$ among the three types of connections. Therefore, the hinged beam-end connections are suitable for sub-high-rise structures.

• The cumulative exceedance probabilities of the structures with the three tested connection modes differ remarkably for ground motions with large IMs. Using ${\theta }_D$ or ${\mu }_E$ as the control index is necessary to obtain more accurate exceedance probabilities.

• PGA is an appropriate IM for fragility curves based on mean periods. It can be used for damage assessment of structures under ordinary and pulsed earthquakes.

• For structures with different numbers of storeys, the use of S_a(T₁) can better reflect the log-linear relationship between the seismic demand and IM.

Acknowledgments

The authors gratefully acknowledge the financial support provided by the Natural Science Foundation of Guangdong Province (grant no. 2018A030313688) and the Natural Science Foundation of Hunan Province (grant no. 2020JJ4268).

Additional information

Funding

This work was supported by the Natural Science Foundation of Hunan Province [2020JJ4268]; the Natural Science Foundation of Guangdong Province, China [2018A030313688].

Notes on contributors

Fanliang Zeng

Fanliang Zeng is currently enrolled as a Ph.D. candidate in the Department of Civil Engineering at South China University of Technology, China. His main research interest is earthquake resistance of engineering structure.

Yansheng Huang

Yansheng Huang is a professor of Civil Engineering at South China University of Technology, China. His main research interest is earthquake resistance of engineering structure.

Jing Zhou

Jing Zhou is an associate professor of Civil Engineering at South China University of Technology, China. His main research interest is earthquake resistance of engineering structure.