Impact of including a slab effect into a 2D RC frame on the seismic fragility assessment: A comparative study

1 Introduction

Assessing the seismic fragility of a structure to seismic excitation is a crucial endeavour in earthquake-prone areas [1,2]. Such assessment is especially crucial in developing countries that lack the necessary infrastructure to withstand the impact of seismic forces [3]. Analysts widely employ two main methods to achieve this task: nonlinear static analysis (NSA), such as pushover analysis (POA), and nonlinear dynamic analysis (NDA), including incremental dynamic analysis (IDA) [4,5]. However, the former possesses certain limitations and shortcomings that hinder its range of applications and increase uncertainties in its efficiency in estimating structural seismic demands [6,7]. For instance, in cases where higher modes dominate the structural behaviour or when a structure is highly pushed into its nonlinear post-yield range, the prediction of the deformation can be significantly inaccurate. Furthermore, given that it is a static method, it only considers the deformation of materials and ignores other forms of energy dissipation associated with dynamic response, such as viscous damping, kinetic energy, and duration impacts [8]. Therefore, the NSA method lacks many important features and cannot consistently substitute for NDA, which is the most accurate means for structural seismic analysis [9,10]. In this context, analysts usually resort to applying NDA to obtain precise results. However, NDA is costly and time consuming, particularly in complicated 3D models [11,12]. As a result, they resort to simplifying the structure by using an equivalent “SDOF” system that represents 3D models [13]. Some analysts have adopted idealised framing models, which extract seismic fragility for internal and external 2D frames from the 3D models in one or two principal directions [11,12,14]. This approach, in turn, allows for a reduction in computational efforts. However, these simplifications also have shortcomings. 2D modelling simplifications fail to capture the relative displacement effects of transverse and longitudinal members. This issue can cause the induction of biaxial stresses, which leads to changes in the structural responses. In addition, these simplifications may overlook important aspects of structural behaviour, such as torsional effects, shear wall interactions, irregularities in the third-dimension diaphragm action, and floor-to-floor interaction, which are important in the seismic response of RC structures [15]. Involving slab effects in a 2D framing model can provide a more accurate depiction of structural behaviour and enhance the reliability of seismic fragility assessments, specifically for buildings with shear walls [14]. Ignoring this aspect could potentially create a different failure mechanism than the 3D model and might result in the transfer of plastic hinges from the columns to the beams. This allocation is due to the absence of the slab contribution to resist seismic-induced shear forces and the imposed force by shear walls. This study examines and compares the seismic behaviour of three scenarios: (1) a 3D model (3D), (2) a 2D traditional frame (2D-F), and (3) a 2D frame with a slab (2D-S). The investigation focused on two RC frame buildings with shear walls: one with five stories, and the other with ten stories. These buildings were assumed to be situated in the state of Penang, Malaysia. Initially, the two structures were created as fully 3D models and designed to withstand the forces of gravity and wind, which adhered to the specifications outlined in EC2 and MS1553 (2002). Subsequently, an internal frame oriented in the weaker direction was extracted. The internal frame was assessed as a 2D frame without the slab effect. The same internal frame with the slab effect was also analysed. An IDA study was conducted on the three cases involving six buildings. Lastly, analytical fragility curves based on the IDA results were developed for all models. Interesting findings are presented in the following sections.

2 Finite element models

The 3D finite element models of the 5- and 10-story buildings were created using the ETABS 2022 platform, as shown in Figure 1.

Figure 1

3D finite element models of models 3D-5 (left) and 3D-10 (right).

Both buildings were designed to resist the forces of gravity and wind, which adhered to the specifications outlined in EC2 and MS1553 (2002). The deformation of the slabs was carefully considered during the design process to prevent serviceability failure. Furthermore, the structural elements were not designed with any specific seismic detailing because the seismic load was disregarded. In both models, the concrete compressive strength was 30 MPa for beams and slabs, while the strength was 30 MPa for 5-story and 35 MPa for 10-story in the vertical elements (column and shear walls). According to IEMP projects [16], the yield and ultimate tensile strengths of the used reinforcing bars were 500 and 540 MPa, respectively. Table 1 lists the sizes of the beams, columns, and shear walls. Additional details can be found in Arada et al. [2].

The nonlinear behaviour of the members was considered by simulating plastic hinges at both ends, where the maximum bending moment occurs, of each member. The moment–rotation relationships of plastic hinges were determined based on the proposed model in ASCE/SEI 41-13 (2014) considering the lack of seismic detailing using smaller rotational capacities, as stated by Ahmad et al. The fibre-based distributed plasticity model simulated the nonlinear behaviour of concrete slabs and shear walls. Table 2 presents the considered threshold for strains in concrete and reinforcing bars of the shear walls at different limit states.

3 Modal analysis of 3D models

Modal analysis is a method used to derive modal parameters, such as modal periods, damping loss factors, natural frequencies, and mode shape deformations [1]. These parameters are then utilised to construct a mathematical formula that describes the dynamic behaviour of a system. In the seismic engineering field, mode shapes or vibration periods are critical. A mode is a depiction of the deformation of a structure that corresponds to each natural frequency. According to the seismic codes’ guidelines, the analysis must consider a sufficient number of modes to cover at least 90% of the total modal mass of the structure. This work conducted 3D finite element analysis to determine the initial mode shapes, which encompassed over 90% of the overall mass participation ratio in the X and Y directions (ΣUX and ΣUY). Table 3 presents the first 10 mode shapes in each direction and the corresponding mass participation ratio.

4 Earthquake demand parameter and damage level relationship

An essential step in developing analytical fragility curves is the establishment of engineering demand parameters (EDPs) [17]. EDPs are measurements that represent the structural response to seismic load. Typically, EDP values are related to various predefined performance levels, which indicate specific levels of damage. Roof displacement (RD) and inter-story drift ratio (ISDR%) are widely used as EDP parameters in most seismic fragility assessment works [18]. Codes and standards frequently utilise three performance levels, which are denoted as damage states. These damage states are immediate occupancy (IO), life safety (LS), and collapse prevention (CP). Each indicates a certain level of damage, which ranges from slight damage to collapse. This study used RD in the static analysis and ISDR% in the dynamic analysis. Moreover, the values of the damaged states were determined based on the analysis results and the formation of the plastic hinges.

5 NSA and selection of the critical 2D frame

As reported in this section, nonlinear static POA was performed. Based on its outcomes, the most vulnerable frame was selected as a critical frame to be studied and compared with those of the 3D models.

5.1 POA of 3D models for selecting the critical 2D frame

An initial POA was conducted in the weaker direction of the 3D model. Two distinct lateral load patterns were employed in POA. The first pattern corresponded to the shape of the first mode, while the second was proportional to the mass of each level. The form of the plastic hinges propagation was detected, as shown in Figure 2, which represents the failure mechanism. The capacity curve for each model was also plotted, and the three damage states were determined based on the formation of the plastic hinges, as shown in Figures 3 and 4. The POA showed that the plastic hinges in the three specified limit states tended to develop initially in the middle internal frames and subsequently propagate to other frames. Figure 2 illustrates the first CP plastic hinge, which formed in both 3D models. It was located in the column of the first story at the middle internal frames. Figures 3 and 4 show that the 3D-5 model hit the CP limit state when the RD was approximately 186 mm under both load patterns. Furthermore, the 3D-10 model reached the CP limit state when the RD values were 393 and 408 mm under the first and second load patterns, respectively.

Figure 2

3D-10 (top) and 3D-5 (bottom) models presenting the propagation of the plastic hinges and demonstrating the location of the first CP.

Figure 3

Capacity curves of models 3D-5.

Figure 4

Capacity curves of models 3D-10.

5.2 Design of 2D frames and modal analysis of 2D models

The critical frame was isolated (following the same design) and separately studied as a 2D model to compare the findings with the other 2D model. It was constructed with a solid slab in one instance, while it was built with only the frame in another. The first scenario requires simulating the slabs in the 2D framing models and support from the boundaries. Thus, the slabs are supported at their boundaries by adding springs with stiffnesses in the vertical direction. These vertical stiffnesses were determined based on the vertical elastic stiffness of the slabs in the 3D models, as shown in Figure 5. However, Figure 6 shows the second scenario, which is a normal 2D-frame models. Notably, the selection of the slab width is based on the principle of load transfer from the slab to the supporting beam. Specifically, the loads exerted by the slab are distributed across the sides of the beam. This process is influenced by the geometric dimensions of the slab and the orientation of its reinforcement. Certain assumptions were made to streamline the analysis and ensure a simplified but effective design, including the maintenance of the rectangular shape of the slab. Consequently, for the purposes of load distribution, half of the slab on each side of the beam – commonly referred to as partial loading – was considered. This approach allows for the symmetrical distribution of loads, which simplifies the calculations involved in beam design.

Figure 5

Capacity curves of models 3D-10.

Figure 6

Capacity curves of models 3D-10.

The building configuration (3D or 2D) can considerably affect the mode shapes, which in turn affect the dynamic response of the building. Thus, this study compared the mode shapes of 2D models together and with the corresponding 3D models (Section 3) to highlight the impact of building configuration on vibration periods. Table 4 presents the first 10 mode shapes for 2D models. The dominating vibration mode period values in 2D-S models provided a better representation of the 3D models than the traditional 2D-F. The periods for the A1 and B1 models were 0.473 and 1.246 s. These values demonstrate strong consistency with the values of the 3D models, which were 0.479 and 1.109 s. The traditional 2D-F models had period values of 0.351 and 0.869 s for the A1-F and B1-F models. These values poorly reflect the values of the 3D models compared with those of the 2D-S models.

Table 4

Modal information of the 2D 5- and 10-story models

	2D 5-story models				2D 10-story models
Model	2D-S	Y	2D-F	Y	2D-S	Y	2D-F	Y
	Period	Mass P.	Period	Mass P.	Period	Mass P.	Period	Mass P.
1	0.479	0.7284	0.351	0.7054	1.109	0.7228	0.869	0.7016
2	0.104	0.1781	0.073	0.1844	0.288	0.1471	0.213	0.1562
3	0.042	0.056	0.032	0.0553	0.126	0.0573	0.09	0.0602
4	0.024	0.0198	0.022	0.0172	0.07	0.0298	0.051	0.0308
5	0.018	0.0047	0.02	0.0046	0.045	0.0167	0.034	0.017
6	0.0002	0	0.013	0	0.032	0.0097	0.027	0.0093
7	0.0002	0	0.01	0.0003	0.025	0.0056	0.023	0.0049
8	0.0002	0	0.01	0.0013	0.02	0.003	0.021	0.0026
9	0.0001	0	0.008	0.0016	0.018	0.0013	0.02	0.0012
10	0.0001	0.0000	0.007	0.0014	0.016	0.0003	0.01	0.0007
Sum		0.987		0.9742		0.9933		0.9846

Period in second, 2D-S: 2D with a slab, 2D-F: traditional 2D frame, mass P.: mass participation.

5.3 POA for 2D models

Similar to the 3D model scenarios, a POA was conducted for the selected 2D models. The formation of the plastic hinges and the failure mechanisms were observed, as shown in Figures 7 and 8.

Figure 7

5-story (right) and 10-story (left) 2D-with-slab models presenting the propagation of the plastic hinges and demonstrating the location of the first CP.

Figure 8

5-story (right) and 10-story (left) 2D-frame models presenting the propagation of the plastic hinges and demonstrating the location of the first CP.

Figures 7 and 8 demonstrate two distinct failure mechanisms, each indicated by the presence of plastic hinges at different locations. In the first scenario, the plastic hinges were mostly formed in the columns of both 2D-S (A1 and B1) models; such a case corresponds to the 3D model, including the location of the first CP hinge, as shown in Figure 7. On the contrary, the plastic hinges were only formed in the beams in the second scenario 2D-F (A1-F and B1-F), which is a different failure mechanism than that in the 3D models, as shown in Figure 8. The capacity curves for all models are plotted in Figures 9–12. From the capacity curves shown in Figures 9 and 10, A1 model crossed the CP limit state when the RD was approximately 190 mm under both load patterns. Moreover, the RD values for the same state in the B1 model were 369 and 409 mm under the first and second load patterns, respectively. The results align closely with those observed in 3D models. In the meantime, the A1-F model hit the CP limit state when the RD was approximately 216 mm under both load patterns, as exhibited in Figure 11. The RD values for the same state in the B1-F model were 410 and 413 mm under the first and second load patterns, respectively, as shown in Figure 12. The resulting building capacities were slightly higher than those obtained from the 3D models.

Figure 9

Capacity curves of model A1.

Figure 10

Capacity curves of model B1.

Figure 11

Capacity curves of model A1-F.

Figure 12

Capacity curves of model B1-F.

6 NDA for 3D and 2D models

As presented in this section, IDA was performed using eight far-field records, and the IDA curves were developed. The performance thresholds were determined based on the IDA results and the formation of the plastic hinges.

6.2 GM selection and scaling

The appropriate selection of GM records is an important task that requires careful consideration in any IDA investigations because it significantly impacts the findings [1]. Several criteria, including magnitude range (Mw), PGA, distance from the site to the source, and soil types, warrant attention during the selection process. The GM records used in this project comply with the specifications outlined in Eurocode 8 (EC8). These specifications are in accordance with the prevailing design principles, which recommend employing a minimum of three base records and adopting the highest response value. However, applying for at least seven ground records is required to adopt the mean value of responses. To meet these requirements, eight far-failed records with specified parameters were selected from the PEER Ground Motion Database. Therefore, these seismic records had Mw values between 5 and 8, which align with the predominant intensities observed in the earthquakes in Malaysia. Unscaled PGA values between 0.125 and 0.684 g were also considered, as reported by Kassem et al. [21]. The selected GMs are listed in Table 5. The soil type and seismicity zone are crucial elements in the seismic GM recordings, and they lead to substantial variations in the structural seismic response [22]. Previous researchers have indicated that unscaled ground recordings need to be scaled to meet site specifications for obtaining accurate findings from linear and nonlinear analyses. Therefore, the spectral matching approach was utilised to incorporate site-specific effects into the seismic prediction process using SeismoMatch software. The GM records were scaled to match the target spectrum applied to the building design according to the Malaysian NA code with ag = 0.05 g and soil type D. Figure 13 shows the input spectrum curves compared with the spectral acceleration curve recommended by the Malaysian NA code.

Table 5

Selected far-field GM records for IDA (PEER)

No	Event	Station	Year	Distance (km)	Magnitude (Mw)
1	Hector Mine	Baker Fire Station	1999	64	7.1
2	Hector Mine	Fort Irwin	1999	66	7.1
3	Landers	Barstow	1992	34.8	7.28
4	Landers	Fort Irwin	1992	62.98	7.28
5	Northridge	Elizabeth Lake	1994	36.5	6.7
6	Northridge	City Terrace	1994	35	6.7
7	San Fernando	Pearblossom Pump	1971	38.9	6.61
8	San Fernando	Antonio Dam	1971	61.7	6.61

Figure 13

Matching of GMs to the target spectrum in the Malaysian NA code.

6.3 Describing the propagation of plastic hinges and defining the damage state thresholds

This study adopted three damage states: IO, LS, and CP. These damage states are widely recommended in seismic codes and standards. Codes and standards, such as ASCE/SEI 41-13 (2014), define certain inter-story drift capacities for these states as 0.5%, 1%, and 2% for IO, LS, and CP, respectively. These values are mainly valid for buildings designed to resist seismic loads; however, they are conservative values. Therefore, these values were established similarly in this study by observing the first plastic hinge that reached the maximum specified rotation after performing IDA. Firstly, the propagation on plastic hinges was detected. Then, the damage state values were computed. The propagation of plastic hinges followed the same pattern as detected in the POA, with the critical frame being the same middle internal frame, as shown in Figure 2. The values of the derived damage state are shown in Table 6 and Figures 14 and 15 16.

Table 6

Derived damage state in terms of ISDR%

Number of stories	Modeling type	Damage state in terms of ISDR%
		IO%	LS%	CP%
5-story	3D	0.67%	1.06%	1.45%
10-story	3D	1.01%	1.28%	1.54%
5-story	A1	1.00%	1.20%	1.38%
10-story	B1	1.05%	1.35%	1.58%
5-story	A1-F	0.80%	1.20%	1.26%
10-story	B1-F	0.83%	1.47%	1.49%

Figure 14

IDA curves for 3D-5 (left) and 3D-10 (right) presenting the three damage states.

Figure 15

IDA curves for A1 (left) and B1 (right) presenting the three damage states.

Figure 16

IDA curves for A1-F (left) and B1-F (right) presenting the three damage states.

The abovementioned observation suggests that the codes may potentially overestimate or underestimate the values of damage states. For example, all the models had a CP value below 2%, as denoted by the codes. The reason is that these values are proposed mainly for buildings under seismic loads. The IDA demonstrated that neglecting the slab effect in the 2D frame can substantially impact the configuration of the failure mechanism in such building types. In the 3D and 2D-S models, the first CP was observed to form in the columns. However, in the 2D-frame models, the first CP formed in the beam, which is similar to the PAO findings. Furthermore, the scatter in the IDA results for the 2D models was more than that in the 3D models. Among the model sets, the highest scatter for the ISDR% was observed in the traditional 2D frames. The median IDA curves of the 2D models and their corresponding 3D models were plotted, as shown in Figures 17 and 18, to compare the demand and capacity of 3D and 2D models. These figures illustrate that modelling the slabs with the 2D models can better reflect the correct demands and capacities of the 3D models. The 5-story buildings required seismic demands of 0.5 and 0.6 g to hit CP levels of 1.45% and 1.38%, respectively, in the 3D and 2D-S scenarios. However, the 5-story 2D-F needed a seismic demand of 0.8 g to reach a CP level of 1.47%. The 10-story buildings required seismic demands of 0.7 and 0.7 g to hit CP levels of 1.54 and 1.58%, respectively, in the 3D and 2D-S scenarios. However, the 10-story 2D frame needed a seismic demand of 0.85 g to reach a CP level of 1.5%.

Figure 17

Median IDA curves for A1 (left) and B1 (right) and their corresponding 3D models.

Figure 18

Median IDA curves for A1-F (left) and B1-F (right) and their corresponding 3D models.

7 Analytical derivation of overall fragility curves

The IDA findings, as outlined in Section 7, were used to generate analytical fragility curves for each model. These curves were then used to compare the probability of damage for the 3D and 2D models. A fragility curve is a valuable tool in contemporary earthquake engineering for evaluating the susceptibility of structures to seismic activity. This statement explains the likelihood of an EDP reaching or exceeding a predetermined performance threshold for a specific IM. Seismic fragility is quantified using the lognormal cumulative distribution function [23], which is written as follows:

From equation (1), the fragility curve consists of two parameters: μ, which represents the median, and β, which refers to the lognormal standard or dispersion. Notably, the analytical approach, which focuses on modelling physical parameters with known structural properties, serves as a more precise and consistent means to develop fragility curves, compared with empirical assessments based on observed damage [24]. In this context, analytical fragility curves using Equation (1) were established based on the damage states identified from the IDA results for each model. Figures 20–22 show the developed curves and the determination of the earthquake intensity at which a 50% probability of collapse exists, also known as the CP level.

Figures 19–21 show that the 3D models of the 5- and 10-story buildings required seismic demands of 0.5 and 0.7 g, respectively, to achieve a 50% probability of collapse. Furthermore, 2D models with the slab effect (referred to as 2D-S) were preferred instead of the conventional 2D-F. The 2D-S models exhibited comparable performance to the 3D models. To achieve a 50% probability of collapse, A1 and B1 buildings required seismic demands of 0.625 and 0.7 g, respectively. Conversely, the traditional 2D-F exhibited a 50% probability of collapse for A1-F and B1-F structures when subjected to seismic accelerations of 0.775 and 0.825 g, respectively.

Figure 19

IDA curves for 3D-5 (left) and 3D-10 (right) presenting the three damage states.

Figure 20

IDA curves for A1 (left) and B1 (right) presenting the three damage states.

Figure 21

IDA curves for A1-F (left) and B1-F (right) presenting the three damage states.

Furthermore, the fragility curves for the 2D-S models were more consistent with the 3D models. The 2D-S fragility curves showed a slight variation in the bending shape (or the slop) of the 3D fragility curves compared with the traditional 2D-F models, which bend differently, as shown in Figure 22.

Figure 22

Fragility curves for 3D-5 (in the left) and 3D-10 (in the right) and their corresponding 2D models.

8 Conclusion

This study compared 2D and 3D models of ‘RC frames with shear walls’ in terms of the structural seismic capacity and required seismic demand to achieve predefined performance levels. The comparison was performed using (1) modal analysis, (2) POA with two different load patterns, and (3) incremental dynamic analysis (IDA) with eight far-field GM records. The fragility curve results were also compared. The objective was to examine the extent to which 2D models accurately represent 3D models. The slab effect was considered in some instances during the modelling process to simulate more reliable models and compared with 3D and 2D traditional frames. A total of 6 models comprising 3D, 2D-S, and 2D-F models were compared. A modal analysis was performed, and the results of the dominant vibration periods were compared. POA was conducted by applying two different load patterns: the first pattern corresponded to the shape of the first mode, while the second pattern was proportional to the mass of each level. A total of 480 NDAs were also performed with eight far-field records to develop IDA curves and then derive the fragility curves. RD and ISDR% were used as engineering demand parameters in POA and IDA, respectively. The observations and findings can be summarised as follows:

• The values of the damage states introduced in the codes and standards are mainly valid for buildings designed to resist seismic loads; however, they are conservative values.

• The use of 2D simplifications offers significant advantages in terms of computational efficiency. Compared with 3D models, 2D models reduce computational time by 3–5 factors, depending on the GM duration and intensity. In addition, 2D models demand lower computational resources and require less processing power and advanced hardware specifications. Thus, they are a more cost-effective option for large-scale analyses.

• The 2D-S models show better representations of the 3D models in terms of the dominant vibration period.

• The scatter in ISDR% is generally more in 2D models than in 3D models, and it is more apparent and observed in the traditional 2D-F.

• Although the 2D-S models slightly overestimate the structural seismic capacity and the required seismic demand to achieve predefined performance levels, it exhibits comparable performance to the 3D models. On the contrary, the traditional 2D frame tends to significantly overestimate the results with respect to the 3D analysis.

• The 2D-S models also show a comparable formation of plastic hinges to the 3D models, which results in a comparable failure mechanism. By contrast, the traditional 2D-F fails in a different manner.

• The shapes of the analytical 2D-S model curves (POA, IDA, and fragility) are compatible with their equivalent 3D model curves. However, this compatibility is often absent while analysing the conventional 2D frame.

In general, the findings indicate that the 2D representation of 3D models requires careful modelling. The dominant periods of 2D and 3D models should be closer to obtain a proper representation. Simulating the slab with the 2D models can provide a closer period to the 3D model. Considering the slab effect in the 2D modelling allows a proper depiction in terms of plastic hinge formation, which in turn leads to a close failure mechanism for the 3D models. Furthermore, the slab effect in the 2D models permits obtaining compatible outcomes in terms of seismic capacity and seismic demand. All these advantages are attributed to the additional slab mass and stiffness, which are ignored in the traditional 2D design.