Probabilistic IM-Based Assessment of Critical Engineering Demand Parameters to Control the Seismic Structural Pounding Consequences in Multistory RC Buildings

Abstract

This research aims to assess and quantify the significance of incorporating the seismic performance of global and local engineering demand parameters (EDPs) within probabilistic frameworks when structural pounding of adjacent buildings occurs. For this purpose, the seismic performance of six-story and twelve-story reinforced concrete (RC) frames subjected to floor–floor pounding is assessed. The pounding is caused by an adjacent shorter and stiffer structure with the top contact point at the middle of the tall building’s total height. Displacement-based and ductility-based EDPs are evaluated at different performance levels (PLs) and at different separation distances (d_g). The seismic performance of the RC frames without considering pounding is also evaluated. Incremental dynamic analyses (IDAs) are performed, and probabilistic seismic demand models (PSDMs) are developed to establish the fragility curves of the examined RC frames. The probability of earthquake-induced pounding between adjacent structures is properly involved with the median value of S_a,T1 that corresponds to an acceptable capacity level (acceptable PL) of an EDP. The results of this study indicate that excluding structural pounding consequences from the probabilistic frameworks related to the seismic risk of colliding buildings leads to unsafe seismic assessment or design provisions.

1. Introduction

An important issue in earthquake engineering that several researchers have focused on is the interaction of buildings in proximity. It is well known that insufficient separation distances between adjacent structures can alter free vibration modes, leading to collisions that result in high impact forces with short-duration acceleration spikes on both structures. Although neighboring buildings may exhibit either in-phase or out-of-phase vibrations, it is acknowledged that interaction impacts become more crucial when an out-of-phase vibration mode is observed. Out-of-phase vibrations of the adjacent structures are identified when significant differences in their dynamic characteristics are noted, such as in the pounding cases between a tall RC frame and a shorter stiffer structure.

Several studies have been reported in the literature to capture the structural seismic response during pounding impacts primarily based on the results of nonlinear time history analyses (deterministic assessments) e.g., [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23]. In these studies, various key parameters have been addressed to verify structural performance, such as the type of structural pounding, buildings height, buildings arrangement, separation gap distance, buildings material (i.e., RC or steel), soil–structure interaction, setbacks, masonry infills, height difference in the colliding structures, the beam–column joint damage effect, mitigation strategies, and characteristics of the seismic excitations, among others. Additionally, it has been demonstrated that the seismic performance of buildings that experience collisions depends on both the seismic hazard of the location and the performance criteria established for seismic demands [21]. Comprehensive reviews of the multiple aspects of structural pounding can be found in Miari et al. [24], Brown and Elshaer [25].

On the other hand, advanced concepts in seismic structural engineering incorporate performance-based approaches for the design and assessment of infrastructure [26]. A key issue in these methods is the definition of an accurate analytical fragility curve that identifies the vulnerability of structures to seismic damage. In recent years, the literature has also provided results based on fragility analyses that have been performed for probabilistic evaluation of the structural pounding phenomenon [27,28,29,30,31,32,33,34,35,36,37,38]. Tubaldi et al. [27] evaluated the seismic pounding risk by developing fragility curves for single-degree-of-freedom (SDOF) and multi-degree-of-freedom (MDOF) systems. The fragility assessment of multistory RC or/and steel structures subjected to pounding has also been studied [28,29,30,31,32,33,34,35,36,37,38]. Nazri et al. [28] developed fragility curves for different performance levels in terms of maximum interstory drift to study the effect of gaps between RC frame structures and the influence of regularity/irregularity on structural behavior under moderate seismic actions. The structural performance of the adjacent buildings was based on the results of the IDA method as a function of the peak ground acceleration (PGA). The local inelastic demands of the critical structural members in the fragility assessment of buildings subjected to pounding were introduced by Flenga and Favvata [29,30,31,32]. Both floor-to-floor and inter-story (floor-to-column) structural pounding were assessed. The results highlighted the necessity of coupling the probability of pounding with the pounding consequences. Additionally, a fragility-based methodology was proposed to evaluate the minimum separation gap distance required at different limit states [30]. These results indicated that beyond the relative displacements of the adjacent structures, global and local EDPs also needed to be considered. Other key issues related to the probabilistic assessment of the structural pounding effect that have been addressed include the investigation of (a) the discrepancies between the methodologies used to develop accurate structural seismic fragility curves [20], (b) the efficiency and sufficiency of the intensity measures (IMs) [31], and (c) the effect of the magnitude M_w and R_rup [32]. Nevertheless, in these studies [29,30,31,32], the verification of the proposed methods and the assessment of the pounding effects have been evaluated only for the case of an eight-story RC frame.

Mohamed and Romão [33] studied the structural pounding effect on the fragility assessment of non-seismically designed RC structures, taking into account both floor-to-floor and floor-to-column pounding conditions. Kazemi et al. [34] presented fragility curves to evaluate the impact of floor-to-floor structural pounding at different performance levels on the seismic performance of adjacent RC and steel frames. In 2022, Miari and Jankowski [35] studied the effect of soil type on the fragility assessment of colliding buildings. Sinha and Rao [36] investigated displacement-based fragility for floor-to-floor and floor-to-column collision within an eight-story non-ductile RC frame adjacent to a three-story rigid RC frame. Bantilas et al. [37] evaluated the seismic performance of an eight-story RC frame structure subjected to floor-to-floor pounding by an adjacent rigid structure. They developed fragility curves of the maximum interstory drift ratio at different limit states. In 2024, Kazemi et al. [38] investigated the seismic performance of steel structures in collision with RC structures under floor-to-floor pounding conditions. IDAs were performed and fragility curves in terms of IDR were developed based on M-IDA curves as a function of S_a,T1.

This review underlines the importance of conducting additional research on the seismic performance-based assessment and design of colliding buildings. The concept of including the structural pounding consequences in the probabilistic frameworks related to the seismic risk of colliding buildings should be further enhanced. Assessing the fragility results should extend beyond verifying the severity of the structural pounding phenomenon that causes damage or even collapse. It is essential to identify solutions that ultimately help to define adequate separation gaps and improve risk mitigation strategies. The most popular approach against structural pounding is the evaluation of a sufficiently large separation distance between the adjacent structures. However, while for new buildings it is possible (but not always applicable) to design sufficient gap distances, in the case of existing structures, this option cannot be ensured. Mitigation methods to control damage when an insufficient separation distance exists between adjacent buildings are also limited. The consequences of structural pounding on the damage level of the structure can only be quantified if the performances of critical global and/or local EDPs are properly involved in the probabilistic methods. However, the sole EDP that is usually adopted to evaluate separation distances is associated with displacements that are developed in the contact area of the colliding buildings. The same approach is used by current seismic design building codes, although several studies have proved that these codes’ provisions are either inadequate or conservative. Therefore, the code requirement cannot always be applied.

So, the aim of this study is to quantify the importance of several EDPs to control the seismic structural pounding consequences in multistory RC buildings, considering different separation distances (d_g) and building heights. In this context, the compounded fragility-based method proposed by Flenga and Favvata [30] has been adopted. The main advantage of the proposed probabilistic framework, compared to existing methods, is that it can be used to ensure structural safety against the seismic pounding phenomenon. This framework represents a pioneering approach that directly links the probability of pounding with the pounding consequences, specifically the likelihood of reaching or exceeding pre-defined performance levels. This innovative coupling effectively mitigates pounding risks and provides a strategic framework for determining and adjusting the minimum separation distance between adjacent structures, ultimately enhancing safety and resilience. This probabilistic framework is herein implemented to couple the probability of earthquake-induced pounding between adjacent structures with the median value of S_a,T1 that corresponds to an acceptable capacity level (acceptable PL) of an EDP.

Considering that pounding risk is amplified when the dynamic characteristics of the colliding buildings are significantly different, the seismic fragility of six-story and twelve-story RC frame structures subjected to floor-to-floor pounding caused by an adjacent shorter and stiffer structure is examined. Displacement-based and ductility-based EDPs are evaluated at different performance levels and at different separation distances d_g. A total of 1456 nonlinear dynamic analyses are performed based on the IDA method and linear and bilinear PSDMs are developed to establish the fragility curves of the examined RC frames. The results of this study highlight the critical impact of the ductility demands on the seismic performance of the structures subjected to floor-to-floor pounding, the vulnerability of the examined RC buildings, and the need to reconsider EC8 provisions regarding the structural pounding phenomenon.

2. Design of Multistory RC Buildings

In this study, six-story and twelve-story RC frame structures were designed according to Eurocode 2 [39] and Eurocode 8 [40]. Both structures had Ductility Capacity Medium (DCM) and a behavior factor q equal to 3.75. The material properties used were C20/25 for concrete and S400 for steel. The design base shear force was equal to V = (0.3 g/q)M, where M is the mass equal to M = (G + 0.3 Q)/g (G: gravity loads; Q: live loads). Moment of inertia of the gross concrete section (I_g) was reduced to take into account the flexural cracking [40]. Specifically, the effective moment of inertia (I_eff) was 0.5 I_g for beams and 0.9 I_g for columns [41]. The dimensions of the columns were primarily determined by the code’s axial load ratio limitation, while in a few cases (upper floor levels) by the code requirements for minimum dimensions. Figure 1a,b illustrates the geometry and reinforcement details of both RC frames.

3. Structural Modelling Assumptions

Using the computer program Drain-2DX [42], the structural members of the RC frames were modelled with two different types of one-dimensional beam–column elements, as follows: the common lumped plasticity model for beams and the special-purpose distributed plasticity element for columns. The column element is a fiber-based element that accounts for the spread of inelastic behavior both over the cross-sections and along the deformable region of the member length (Figure 1c). The deformable part of the column element is divided into several segments. Cross-section analysis at the control (centre) section of each segment is performed based on the fibre model, while axial–moment (N–M) interaction is also rationally accounted for.

Regarding the pounding phenomenon, the top contact point of the neighboring structures was modelled using a compression/tension link element between the adjacent nodes (Figure 1c). The response model of this link element is illustrated in Figure 1d. This idealization aligns with the structural modelling assumptions and is sufficient for assessing the pounding effects on the seismic performance of RC structures [1,21].

4. Fragility Curve Based on Linear or Bilinear PSDMs

A PSDM is a mathematical relationship between the median structural demand $\widehat{\mathrm{E}\mathrm{D}\mathrm{P}}$ and the seismic intensity measure (IM). To develop a PSDM, nonlinear dynamic analyses such as IDA are first performed to generate the EDP-IM pairs, and then a statistical process synthesizes the PSDM. The PSDM is expressed as:

$$\widehat{\mathrm{E}\mathrm{D}\mathrm{P}}|\mathsf{{\rm I}}\mathsf{{\rm M}}=\mathrm{a}{\mathrm{I}\mathrm{M}}^{\mathrm{b}}$$

(1)

In log-log space, the PSDM is formulated as

$$\ln \widehat{\mathrm{E}\mathrm{D}\mathrm{P}}|\mathsf{{\rm I}}\mathsf{{\rm M}}=\mathrm{b}\ln \mathrm{I}\mathrm{M}+\ln \mathrm{a}+\mathsf{\epsilon }|\mathsf{{\rm I}}\mathsf{{\rm M}}$$

(2)

where ε|ΙΜ is the random error with mean zero and standard deviation $\mathsf{\beta }$. The coefficients a and b are calculated through linear regression analysis.

On the other hand, local inelastic demands of structural members are adequately addressed when bilinear PSDMs are utilized [30]. In these cases, the mathematical relationship between $\widehat{\mathrm{E}\mathrm{D}\mathrm{P}}$ and IM is expressed as:

$$\ln \widehat{\mathrm{E}\mathrm{D}\mathrm{P}}|\mathrm{I}\mathrm{M}=({\mathrm{a}}_1+{\mathrm{b}}_1\ln \mathrm{I}\mathrm{M})(1-{\mathrm{H}}_1)+({\mathrm{a}}_2+{\mathrm{b}}_2\ln \mathrm{I}\mathrm{M}){\mathrm{H}}_1+\mathsf{\epsilon }|\mathsf{{\rm I}}\mathsf{{\rm M}}$$

(3)

where ${\mathrm{a}}_1$, ${\mathrm{b}}_1$, ${\mathrm{a}}_2$, and ${\mathrm{b}}_2$ control the intercepts and the slopes of the first and the second segment of the bilinear regression model. H₁ is a dummy variable that depicts the first segment of the bilinear PSDM when H₁ = 0.0, while the second segment of the bilinear model is described when H₁ is set equal to 1.

Having developed the PSDM, fragility curves are calculated as follows:

$${\mathrm{G}}_{\mathrm{E}\mathrm{D}\mathrm{P}|\mathrm{I}\mathrm{M}}(\mathrm{C}|\mathrm{I}\mathrm{M})=\mathrm{P}[\mathrm{E}\mathrm{D}\mathrm{P}|\mathsf{{\rm I}}\mathsf{{\rm M}}\ge \mathrm{C}|\mathrm{I}\mathrm{M}]=\mathsf{\Phi }\left({\displaystyle \frac{\ln \widehat{\mathrm{E}\mathrm{D}\mathrm{P}}|\mathsf{{\rm I}}\mathsf{{\rm M}}-\ln \widehat{\mathrm{C}}}{{\mathsf{\beta }}_{\mathrm{E}\mathrm{D}\mathrm{P}|\mathrm{I}\mathrm{M}}}}\right)$$

(4)

where Φ(.) denotes the standard normal cumulative function, $\widehat{\mathrm{C}}$ the median value of the capacity, and β_EDP|IM the logarithm standard deviation calculated as:

$${\mathsf{\beta }}_{\mathrm{E}\mathrm{D}\mathrm{P}|\mathrm{I}\mathrm{M}}=\sqrt{{\displaystyle \frac{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\left(\ln {\mathrm{E}\mathrm{D}\mathrm{P}}_{\mathrm{i}}|\mathsf{{\rm I}}\mathsf{{\rm M}}-\ln \widehat{\mathrm{E}\mathrm{D}\mathrm{P}}|\mathsf{{\rm I}}\mathsf{{\rm M}}\right)}^2}{\mathrm{n}-2}}}$$

(5)

where n is the number of the nonlinear dynamic analyses. It is noted that in the case of utilizing a bilinear PSDM, two standard deviations are considered, namely β₁ for the first segment and β₂ for the second segment. In this way, relaxation of the homoscedasticity assumption is achieved (see [30,31,32]).

5. Examined Parameters

In this study, the floor-to-floor structural pounding is addressed. The potential top contact point is located at the middle floor level of each RC frame structure, meaning at the 3rd floor level in the case of the six-story RC frame, and at the 6th floor level for the twelve-story RC frame. The probabilistic assessment of the pounding consequences on the seismic performance of the multistory RC frames is accomplished for three different separation gap distances (d_g): (a) ${{\mathrm{d}}_{\mathrm{g}}=\mathrm{d}}_{\mathrm{g},\mathrm{E}\mathrm{C}8}=\mathrm{q}·{\mathsf{\delta }}_{\mathrm{e}\mathrm{l}}$ (Eurocode’s requirement), (b) ${\mathrm{d}}_{\mathrm{g} }= 0.5{\mathrm{d}}_{\mathrm{g},\mathrm{EC}8}$, and (c) ${\mathrm{d}}_{\mathrm{g} }= 0.0 \mathrm{cm}$. The case without the pounding effect is also included.

Typically, the fragility analysis is based on the performance of the most critical component, while the effect of different EDPs on the seismic performance of the buildings is usually neglected [30,43]. Wang et al. [43] also highlight the importance of incorporating different EDPs at different limit states when structural fragility analysis is utilized. In this direction, multiple failure models of different structural components were considered to generate accurate explicit connectivity BNs for the structural reliability assessment [44]. Additionally, Wang et al. [45] proposed a probabilistic method to enhance the accuracy of predicting demand responses in structural health monitoring measurements.

The knowledge acquired over these years highlights that structural pounding yields a significant increase in the maximum interstory drift ratio (IDR_max), ductility demands, shear forces, impact forces, and maximum floor acceleration [1,8,9,10,24]. The pounding impact on the seismic performance of the structures depends on the type of collisions that occur (e.g., floor-to-floor or floor-to-column). Especially in the floor-to-floor pounding cases between RC structures (like the ones studied herein), critical EDPs are the IDR_max and the maximum ductility demands ${(\mathsf{\mu }}_{\mathsf{\phi },\mathrm{m}\mathrm{a}\mathrm{x}})$ of the external columns at the pounding side of the adjacent buildings [1,8,24,30]. For the assessment of the probability of pounding, the maximum displacement δ_max at the top contact level is considered. Therefore, three EDPs are evaluated, namely, (a) maximum displacement δ_max at the top of the contact point; (b) ratio of maximum interstory drift to the story height, h_st (IDR_max-%h_st); and (c) maximum curvature ductility demands (μ_φ,max) of the most critical external column of the multistory RC frame. Different performance levels for each EDP are considered. In the case of δ_max, the probability of pounding is estimated considering the examined values of separation gap distance d_g as capacity levels. The seismic fragility of the multistory RC frames in terms of IDR_max (%h_st) is assessed for two performance levels [46]: (a) Immediate Occupancy (IO)—IDR_max,capacity = 1 (%h_st), and (b) Collapse Prevention (CP)—IDR_max,capacity = 2.5 (%h_st). Two capacity levels are also considered when assessing performance in terms of μ_φ,max: (a) yield capacity state of performance (Y)—μ_φ,max = 1.0, and (b) ultimate capacity state of performance (U)$-{\mathsf{\mu }}_{\mathsf{\phi },\mathrm{m}\mathrm{a}\mathrm{x}}={\displaystyle \raisebox{1ex}{${\mathsf{\phi }}_{\mathrm{u}}$}\!\left/ \!\raisebox{-1ex}{${\mathsf{\phi }}_{\mathrm{y}}$}\right.}$ (φ_u = ultimate curvature and φ_y = yield curvature—based on RC column sectional analysis, taking into account a compressive axial load of N = G + 0.3 Q). Figure 2 shows the sectional moment–curvature responses of the critical RC columns of the examined structures.

For the seismic intensity measure, the S_a,T1 (spectral acceleration at the fundamental period T₁ of the structure) is used. The T₁ of the examined RC frames are: T_1,6-story = 0.823 s and T_1,12-story = 1.295 s. Finally, 1456 nonlinear time history analyses are performed based on the IDA method [47]. For this purpose, seven two-component seismic excitations were properly selected from the PEER’s database [48]. These seismic excitations cover a range of magnitudes (M_w) between 6.2 and 7.9 and closet distances-to-rupture area (R) between 9.6 km and 116.2 km. The soil average shear wave velocity in the upper 30 m of soil (v_s,30) is up to 813.5 m/s and corresponds to soil class A. The main characteristics of the selected seismic excitations are presented in

Table 1. Main characteristics of seismic excitations.

Seismic Excitations	Duration (s)	Maximum Acceleration αmax (m/s 2)		Mw 3	R
		Component FN 1	Component FP 2		(km)
Italy Arienzo, 1980 (EQ283)	24	0.268	0.405	6.9	52.9
Italy Auletta, 1980 (EQ284)	34	0.615	0.655	6.9	9.6
Chi-Chi Taiwan-06, 1999 (EQ3479)	42	0.073	0.070	6.3	83.4
Denali-Alaska, 2002 (EQ2107)	60	0.869	0.975	7.9	50.9
Loma Prieta, 1989 (EQ804)	25	1.090	0.509	6.9	63.1
Chi-Chi Taiwan-04, 1999 (EQ2805)	60	0.096	0.075	6.2	116.2
San Fernando, 1971 (EQ59)	14	0.153	0.181	6.6	89.7

¹ Fault Normal component, ² Fault Parallel component, ³ Moment Magnitude.

[29,30].

6. Results

The probabilistic seismic performance of multistory RC frame structures due to the structural pounding effect is herein evaluated. For this purpose, the PSDMs of δ_max|S_a,T1, IDR_max|S_a,T1, and μ_φ,max|S_a,T1 are initially developed using linear and bilinear regression models, as discussed in Section 4 and shown in

Table 2. PSDMs of δ_max|S_a,T1, IDR_max|S_a,T1 and μ_φ,max|S_a,T1 for the multistory RC frame structures for all the examined cases.

Pounding Case	RC Frame Structure	EDP = aΙΜb		βEDP|IM
without pounding	6-story	δmax = 0.096 Sa,T10.980		0.316
	12-story	δmax = 0.218 Sa,T10.913		0.291
without pounding	6-story	IDRmax = 2.354 Sa,T10.978		0.286
	12-story	IDRmax = 2.939 Sa,T10.819		0.275
dg = 0.0 cm	6-story	IDRmax = 2.835 Sa,T10.964		0.314
	12-story	IDRmax = 4.349 Sa,T10.845		0.266
dg = 0.5·dg,EC8	6-story	IDRmax = 2.601 Sa,T11.005		0.322
	12-story	IDRmax = 3.633 Sa,T10.883		0.267
dg = dg,EC8	6-story	IDRmax = 2.467 Sa,T10.994		0.310
	12-story	IDRmax = 3.133 Sa,T10.838		0.284
without pounding	6-story	μφ,max − C16 = 1.439 Sa,T10.836	Sa,T1 ≤ 0.647 g	0.222
		μφ,max − C16 = 1.632 Sa,T11.125	Sa,T1 ≥ 0.647 g	0.577
	12-story	μφ,max − C28 = 1.050 Sa,T10.728	Sa,T1 ≤ 0.960 g	0.175
		μφ,max − C28 = 1.075 Sa,T11.308	Sa,T1 ≥ 0.960 g	0.325
dg = 0.0 cm	6-story	μφ,max − C16 = 2.465 Sa,T10.914	Sa,T1 ≤ 0.373 g	0.289
		μφ,max − C16 = 7.000 Sa,T11.973	Sa,T1 ≥ 0.373 g	0.729
	12-story	μφ,max − C28 = 2.430 Sa,T10.787	Sa,T1 ≤ 0.325 g	0.205
		μφ,max − C28 = 13.929 Sa,T12.34	Sa,T1 ≥ 0.325 g	0.684
dg = 0.5·dg,EC8	6-story	μφ,max − C16 = 2.173 Sa,T10.952	Sa,T1 ≤ 0.443 g	0.168
		μφ,max − C16 = 5.658 Sa,T12.128	Sa,T1 ≥ 0.443 g	0.866
	12-story	μφ,max − C28 = 1.925 Sa,T10.882	Sa,T1 ≤ 0.478 g	0.387
		μφ,max − C28 = 28.646 Sa,T13.355	Sa,T1 ≥ 0.478 g	1.004
dg = dg,EC8	6-story	μφ,max − C16 = 1.831 Sa,T10.907	Sa,T1 ≤ 0.514 g	0.215
		μφ,max − C16 = 5.083 Sa,T12.441	Sa,T1 ≥ 0.514 g	0.970
	12-story	μφ,max − C28 = 1.426 Sa,T10.809	Sa,T1 ≤ 0.660 g	0.290
		μφ,max − C28 = 6.639 Sa,T14.513	Sa,T1 ≥ 0.660 g	1.206

. It is noted that, for the development of the bilinear PSDMs of μ_φ,max|S_a,T1, the breakpoint is defined at the IM^* where μ_φ,max = 1.0 to represent the transition from linear (μ_φ < 1.0) to nonlinear behavior (μ_φ > 1.0).

Thereafter, the fragility curves of each multistory RC frame structure at different performance levels and different separation distances d_g (including the case without pounding) are defined (see Figure 3, Figure 4 and Figure 5). In Figure 3, the probability of pounding considering different separation gaps (d_g) is assessed based on the maximum displacements (δ_max) at the top contact point of the examined RC frames.

These results indicate that when the separation distance d_g is increased, the probability of pounding decreases for the same value of S_a,T1. Also, while the probability of pounding is 100% when the structures are in contact from the beginning, in the case of free vibration, the corresponding probability is zero.

On the other hand, the consequences of the structural pounding on the damage level of the structure can be quantified only if the performances of critical global and/or local EDPs are properly involved in the probabilistic methods. So, in this study, the vulnerability of the structures in terms of IDR_max (global EDP) and μ_φ,max (local EDP) is assessed and is presented in Figure 4 and Figure 5.

Figure 4 and Figure 5 demonstrate that the fragility curves resulting from structural pounding configurations are shifted to lower values of S_a,T1 compared to the corresponding fragilities without pounding (free vibration of the structure). Also, as expected, the impact of the pounding on the vulnerability of the structures increases as the initial gap distance d_g between the adjacent structures decreases. Evaluating the seismic performance of the RC frames at different capacity levels, it can be stated that as the performance level becomes more demanding, the fragility curves shift towards higher values of S_a,T1. Of course, these observations align with existing knowledge on the seismic performance of structures under pounding conditions using probabilistic procedures.

To further quantify the consequences of the structural pounding, the median values of spectral acceleration at the fundamental period T₁ of the structures (S_a,T1) are estimated based on the PSDMs of IDR_max|S_a,T1 and μ_φ,max|S_a,T1. These values depict the S_a,T1 at which the EDPs of IDR_max and μ_φ,max reach a specific performance (capacity) level. Hereafter, the median values of S_a,T1 are referred to as S_a,wp when pounding conditions are precluded and as S_a,dg in pounding conditions.

So, Figure 6 shows the S_a,wp of the RC frames. It can be observed (Figure 6a) that the S_a,wp at which the six-story RC frame reaches the capacity of IDR_max at any PL is greater than the corresponding values of S_a,wp for the twelve-story RC frame. The six-story frame reaches the capacity of IDR_max at the performance level of IO (IDR_max–IO) with a magnitude of S_a,wp about 1.5 times greater than that of the twelve-story frame. In the case of IDR_max–CP, this increment is 1.3.

Now, evaluating the performance based on the demands of μ_φ,max, it can be observed (Figure 6b) that the twelve-story RC frame is less vulnerable than the six-story frame, as the examined column reaches its capacity level (Y or U) at higher values of S_a,wp. Nevertheless, the crucial seismic demand parameter of both structures is the IDR_max, as lower values of S_a,wp are required to reach the PLs compared to μ_φ,max, even at the CP state. This condition is compatible with the seismic design philosophy. In all cases, more demanding performance levels of IDR_max or μ_φ,max lead to increased values of S_a,wp.

Afterwards, in Figure 7, the median values of S_a,dg at which the corresponding EDPs (IDR_max or μ_φ,max) reach their ultimate capacity level are presented for the examined structural pounding cases. It can be observed that the structures reveal similar performance characteristics compared to the case without pounding when the assessment is based on IDR_max. Indeed, the six-story RC frame is again less vulnerable in IDR_max than the twelve-story RC structure, as it reaches the capacity level of CP at greater values of S_a,dg. The same results hold for the capacity level of IDR_max–IO, and when the assessment is based on either μ_φ,max-Y or μ_φ,max-U, the tall building (twelve-story RC frame) is more vulnerable than the six-story frame.

The notable information herein is that these results (Figure 7) indicate that to ensure the safe performance of the structural systems under pounding conditions, the local demands of the critical columns should be considered. As shown in Figure 7, the structures reach the PL of μ_φ,max at a value of S_a,dg that is lower than the S_a,dg required to reach the examined PL of IDR_max. This reduction is more pronounced in the case of the six-story RC frame and decreases as the building height increases. So, structural pounding significantly alters the seismic performance of structures compared to the case without pounding, making the ductility-based EDP, rather than the displacement-based EDP, the critical demand parameter for the evaluated RC frames.

Having evaluated the S_a,dg, the corresponding probability of pounding PL (referred hereafter as P_pounding) can also be defined through the fragilities of δ_max. In this way, the probability of pounding when a specific EDP-PL is achieved is directly expressed. These results are presented in Figure 8, where it can be observed that in most cases, the probability of pounding at the examined S_a,dg-PL is greater than 50%, even when the gap distance provided by EC8 is applied (Figure 8). This means that the use of the δ_max (P_pounding) as a sole criterion to design and assess the structural pounding phenomenon may yield conservative solutions.

For example, in the case of the six-story RC frame, if we choose 50% as an acceptable/targeting value of P_pounding when (i) d_g = 0.5 d_g,EC8 and (ii) d_g = d_g,EC8, then the corresponding acceptable values of S_a,T1 will be 0.26 g and 0.54 g, respectively (see Figure 7). However, these values are less than the one deduced based on the performance level of either IDR_max or μ_φ,max at the collapse limit states. As shown in Figure 7a, the minimum acceptable S_a,dg among IDR_max–CP and μ_φ,max–U is 0.61 g when (i) d_g = 0.5 d_g,EC8 and 0.68 g when (ii) d_g = d_g,EC8, respectively. The same results are noted for the twelve-story RC frame (see Figure 7b).

Finally, to further quantify the pounding impact on the values of S_a,wp the ratio of ${\displaystyle \frac{S_{{a,d}_g}}{S_{a,wp}}}$ is examined. These results are presented in Figure 9 as a function of the pre-defined separation gap distances, d_g, for both displacement-based and ductility-based performance levels.

It can be observed that structures are expected to reach the examined capacity level (PL) at an earthquake intensity level less than the expected one in the case of no pounding, since ${\displaystyle \frac{S_{{a,d}_g}}{S_{a,wp}}}<1.0$ in all of the examined cases in this study. The observed reduction is more intense when the μ_φ,max-PLs of the columns are assessed.

Also, an increase in stories from 6 to 12 decreases the ratio of ${\displaystyle \frac{S_{{a,d}_g}}{S_{a,wp}}}$ Therefore, the most critical pounding cases are posed on the tall building (twelve-story RC frame). This observation holds whether the assessment is based on the IDR_max-PLs or the μ_φ,max-PLs.

7. Conclusions

The main objective of this study is to provide insight into the importance of including critical EDPs to predict and control the severe damage caused by seismic pounding between adjacent multistory RC buildings. Displacement-based and ductility-based EDPs are evaluated at different performance levels and at different separation distances d_g.

The seismic performance of six-story and twelve-story RC frame structures is assessed based on probabilistic methods. The structures are subjected to floor–floor pounding caused by an adjacent shorter and stiffer structure, with the top contact point located at the middle of the building’s total height. Nonlinear dynamic analyses are carried out (IDA method) and PSDMs in terms of δ_max|S_a,T1, IDR_max|S_a,T1, and μ_φ,max|S_a,T1 are developed. Then fragility curves of the examined RC frames at different performance levels are defined, considering (a) ${{\mathrm{d}}_{\mathrm{g} }= \mathrm{d}}_{\mathrm{g},\mathrm{EC}8}= \mathrm{q}·{\delta }_{el}$ (Eurocode’s requirement), (b) ${\mathrm{d}}_{\mathrm{g} }= 0.5{\mathrm{d}}_{\mathrm{g},\mathrm{EC}8}$, (c) ${\mathrm{d}}_{\mathrm{g}} = 0.0 \mathrm{cm}$ (structures are in contact from the beginning), and (d) free vibration of the structures (case without structural pounding effect). Then, the median value of S_a,T1 that corresponds to an acceptable capacity level (acceptable PL) of an EDP is evaluated, and the probability of earthquake-induced pounding between adjacent structures is properly involved.

The results of this study deduce the following remarks:

• The first observations align with existing knowledge on the seismic performance of structures under pounding conditions using probabilistic procedures. Indeed, fragility curves due to structural pounding configurations are shifted to lower values of S_a,T1 compared to the corresponding fragilities without pounding (free vibration of the structure). Also, as expected, the impact of the pounding on the vulnerability of the structures is increased as the initial gap distance d_g between the adjacent structures is decreased. Evaluating the seismic performance of RC frames at different capacity levels, it can be stated that as the performance level becomes more demanding, the fragility curves shift towards higher values of S_a,T1.
• Structural pounding significantly alters the seismic performance of structures compared to the case without pounding, making the ductility-based EDP, rather than the displacement-based EDP, the critical demand parameter for the evaluated RC frames.
• The use of the δ_max (P_pounding) as a sole criterion to design and assess the structural pounding phenomenon yields conservative solutions.
• The structures reach their capacity level (examined PL) at an earthquake intensity level less than the expected one in the case of no pounding $(\mathrm{in} \mathrm{all} \mathrm{the} \mathrm{examined} \mathrm{cases}\frac{{\mathrm{S}}_{{\mathrm{a},\mathrm{d}}_{\mathrm{g}}}}{{\mathrm{S}}_{\mathrm{a},\mathrm{wp}}}<1.0)$.
• An increase in stories from 6 to 12 decreases the ratio of ${\displaystyle \frac{{\mathrm{S}}_{{\mathrm{a},\mathrm{d}}_{\mathrm{g}}}}{{\mathrm{S}}_{\mathrm{a},\mathrm{w}\mathrm{p}}}}$. Therefore, the most critical pounding cases are posed on the twelve-story RC frames.
• The probability of pounding cannot be set to zero even when the gap distance provided by EC8 is applied.
• Excluding structural pounding consequences from the probabilistic frameworks may lead to unsafe performance-based seismic assessment or design provisions regarding the seismic risk of colliding buildings.