Seismic response and fragility analysis of BRB-retrofitted highway pile-slab structures with an improved load-transmission path

Abstract

Pile–deck structures are emerging alternatives to conventional highway-widening methods, which have been increasingly applied in highway reconstruction projects under strict land-use constraints. However, the monolithic pile–deck connection renders such structures vulnerable to extreme disasters such as earthquakes due to the increased seismic force demands. Considering this, the current study proposes a new seismic resistant system for the highway Pile–deck structures, mainly comprising sliding mechanism between the deck and the piles, diagonally installed buckling restrained braces (BRBs) to generate an improved load-transmission path. Based on the established numerical models of the structures, nonlinear time history analysis is conducted to investigate the seismic responses of the proposed structures and their influential parameters. Furthermore, seismic fragility analysis is performed to demonstrate seismic vulnerability of the proposed Pile–deck structures. The results of the analyses indicate that by altering the transmission path of the inertia force from the deck to the piles, the seismic responses and probability of damage of the piles either above or under the ground can be significantly reduced. Compared to the prototype structure, the proposed structure experiences reductions of 98% and 55% in terms of the peak pile curvature and the peak deck displacement responses, respectively.

Introduction

With the rapid development of national transportation infrastructure and networks, the reconstruction and widening of existing highways have become increasingly necessary to accommodate growing traffic demands^1,2. However, due to strict land-use constraints and limited material resources, conventional embankment widening methods are often difficult to implement. These traditional approaches typically require extensive excavation and backfilling, leading to significant environmental impacts, increased construction costs, and longer project durations. To address these challenges, innovative widening techniques, such as the pile-slab structures, have been proposed and gradually applied in highway reconstruction projects with severe land-use restrictions. The pile–slab structures are characterized by using prefabricated reinforced concrete slabs and prestressing high-strength concrete pipe piles, which are monolithically connected, as illustrated in Fig. 1. These prefabricated structural components can be easily manufactured, assembled, and constructed, which facilitates a rapid highway widening construction.

Fig. 1

Typical pile-slab structure for highway reconstruction and widening.

As pile–slab structures are increasingly applied in highway and embankment engineering, it is essential to investigate their performance under various loading conditions, including earthquakes. Extensive studies have been carried out on the static and dynamic behaviors of pile-supported embankment systems. Deng et al.³ experimentally and numerically examined the load-bearing mechanisms of pile-supported embankments in loess, revealing the key characteristics associated with the soil arching effect and tensile membrane effect within the embankment. Brown and Shie⁴ developed three-dimensional finite element models for laterally loaded piles, considering the nonlinear behavior of surrounding soils, thereby providing an effective numerical approach to assess the seismic response of pile-slab systems. Xu et al.⁵ investigated the load transfer mechanisms of floating pile-supported embankments under cyclic loading and demonstrated that the pile end conditions exert a significant influence on the performance of both the floating piles and their supported embankments. Zhang et al.⁶ evaluated the effectiveness of cement–fly ash–gravel (CFG) pile-slab-supported embankments in the Beijing–Shanghai high-speed railway, highlighting the important role of the slab in load redistribution and settlement control.

However, most of the aforementioned studies focus on pile-supported embankments used primarily for foundation reinforcement, whose mechanical behavior differs from that of the pile–slab structures considered in the present study. The seismic performance of pile–slab structures employed in highway widening projects has received limited attention. Chen et al.⁷ investigated the seismic behavior of such structures incorporating a novel elastoplastic column–deck joint, and the results demonstrated that the proposed joint can act as a seismic fuse, effectively dissipating energy and mitigating damage to the prefabricated piles.

In fact, implementing energy dissipation braces like buckling restrained braces (BRBs)^8,9,10,11 in the pile–slab structures is also a promising approach to enhance the structural seismic performance. A comparison study of steel structures equipped with seesaw braces and BRBs has been conducted by Katsimpini et al.¹², which highlighted the effectiveness of BRBs in reducing the residual interstory drift of steel structures. This study proposes a new earthquake-resistant system for the pile–slab structures by adopting combined sliding mechanism and BRBs. In the proposed system, the conventional monolithic connection between the slab and the piles is replaced by a sliding mechanism (e.g., using sliding bearings) to release the rigidity of the connection. Moreover, BRBs are diagonally installed between the piles and the slab to provide lateral stiffness and energy dissipation for the structure. Such a new system alters the load-transmission path of the structure, enabling inertia forces of the slab transmitted through BRBs to the lower part of the piles, thereby mitigating the bending moment and curvature responses of the piles. Nonlinear time-history analyses, parametric studies, and probabilistic seismic fragility assessments are performed to comprehensively demonstrate the seismic effectiveness of the proposed system.

Overview of proposed pile–slab structural seismic system

In conventional pile–slab structures with monolithic pile–slab connections, as illustrated in Fig. 2, the inertial forces from the slab are directly transmitted through the above-ground piles to the foundation. Consequently, significant axial forces, shear forces, and bending moments develop at the pile bases, leading to a complex stress state in the embedded portions of the piles. Moreover, the above-ground sections of the piles are also subjected to considerable force demands.

In contrast, the proposed pile–slab structural system modifies the load-transfer mechanism of the structure, allowing the inertia forces of the slab to be transmitted to the base through the diagonal BRBs, as indicated in Fig. 2. This differs from previous studies in which BRBs were directly installed on the prototype structure without isolation. Although such systems can provide additional stiffness and energy dissipation on top of conventional structures, the lack of isolation at the connection between the deck and the piles allows seismic forces to be transmitted to the above-ground piles. As a result, bending moments still develop at the pile bases, similar to those observed in conventional structures. This alternative load path bypasses the above-ground piles via the sliding bearings installed between the piles and the slab, thereby effectively isolating the piles above the ground from seismic forces. Moreover, since bending moments at the pile base are largely eliminated, both the force demands and potential damage in the underground piles can be substantially reduced.

The BRB is adopted in this system due to its superior mechanical characteristics. Its intrinsic behavior is well recognized in the literature: it can achieve full-section yielding under both tension and compression, effectively eliminating buckling in conventional braces and providing excellent energy dissipation capacity. More importantly, BRBs deliver only axial force. In the proposed structural system, this allows the axial force to be transferred directly to the pile bases, thereby enabling the intended mechanism—isolating the piles above the ground from seismic forces and preventing bending moments from being transmitted into the piles below the ground. In addition, the yield strength of the BRB core can be precisely tuned by adjusting its cross-sectional dimensions, allowing designers to intentionally define the force level at which inelastic action occurs. In both new construction and seismic retrofitting, these advantages enhance seismic performance, reduce post-earthquake downtime, and contribute to improved lifecycle economy.

However, the failure of the sliding mechanism of bearings should be considered. Sometimes, the sliding mechanism of support structures may undergo failure under specific conditions. The first well-recognized failure mode is stick–slip behavior^13,14, an unstable motion characterized by alternating “sticking” and “sliding” phases due to non-constant frictional resistance. This phenomenon typically occurs when the static friction coefficient exceeds the dynamic friction coefficient. During the sticking phase, the sliding interface remains stationary, allowing shear stress to accumulate. Once the accumulated stress surpasses the static friction threshold, rapid slip occurs along the interface, resulting in abrupt stress release. In this study, stick–slip may also be triggered by surface roughness of the sliding interface or plastic deformation of the material. The second failure mechanism involves temperature effects^15,16, where temperature changes alter the physical-chemical properties of the contact surfaces. In the PTFE-metal interface, stick-slip motion, particularly the high-speed sliding during the transition from stick to slip, may generate sufficient frictional heating to soften the interfacial material, thereby reducing the friction coefficient. It should be noted that such temperature effects can also occur under high-load conditions independently of stick-slip. Another relevant mechanism is velocity dependence^17,18, which is often coupled with the two conditions above. The primary concern of the failure of the sliding mechanism is velocity-weakening, a decrease in the friction coefficient with increasing sliding velocity. Both stick-slip behavior and temperature effects can induce or exacerbate velocity-weakening, leading to the destabilization or functional failure of the friction mechanism.

Fig. 2

Load-transmission mechanisms of prototype and proposed pile–slab structures.

Prototype structure and numerical modelling

The prototype structure is a three-span pile–slab system, as illustrated in Fig. 3a. Each reinforced concrete (RC) slab measures 6.0 m × 6.0 m in plan and has a thickness of 50 cm. The piles are fabricated from high-strength prestressed concrete and feature a hollow tubular cross-section with an outer diameter of 60 cm and a wall thickness of 7 cm. The concrete used for the piles has a design compressive strength of 50.2 MPa. Each pile is reinforced with 16 longitudinal steel bars (32 mm in diameter, 400 MPa tensile strength) and 16 prestressing tendons (10.7 mm in diameter, 1395 MPa design prestress). The total pile length is 20 m, with 5 m above ground and 15 m embedded below ground. The embedded portion is surrounded by dense sand, characterized by an effective unit weight of γ = 20 kN/m³ and an internal friction angle of φ = 35 °. In the proposed structures, the rigid pile–slab connection is replaced by the flat sliding bearings, and simultaneously, diagonal BRBs are installed between the slab and the piles, as shown in Fig. 3b. The BRBs employed in this study have an overall length of 5.83 m, with a core cross-sectional area of 638 mm². The yield strength of the BRB core material is 235 MPa, and the Young’s modulus is 2.1 × 10⁵ MPa. The post-yield stiffness ratio of BRB core is taken as 0.01.

Fig. 3

Description of prototype and proposed pile–slab structures and their numerical modelling.

Finite element models of the prototype and proposed pile–slab structures are established in OpenSees for dynamic analysis, as shown Fig. 3. The slab is modelled using elastic plate/shell elements assuming it does not experience plastic responses during earthquakes. The tubular piles are simulated by nonlinear fiber-section-based beam-column elements. The mesh size for the pier elements is set as 1.0 m. The fiber sections of the piles are discretized into multiple fibers representing the high-strength concrete, longitudinal reinforcement and prestressing tendons. Specifically, the pile section is discretized into 2142 fibers, including 63 wedges and 34 rings for high-strength concrete, 16 fibers for reinforcement, and 16 fibers for prestressing tendons. The stress–strain constitutive behavior of the high-strength concrete is modelled using the Concrete01 material in OpenSees, with the values of the peak compressive strength (50.2 MPa) and its corresponding strain (0.002), and the ultimate compressive strain (0.005) specified in the material model. The nonlinear behaviors of the longitudinal reinforcement and prestressing tendons are modelled by the Steel01 material model by specifying the values of modulus of elasticity, yield strength, and the strain hardening ratio. These values are 200 GPa, 400 MPa, and 0.005 for normal reinforcement, and 195 GPa, 1860 MPa, and 0.005 for prestressing tendons. Particularly for prestressing tendons, the initial stress applied in the tendons is set to be 0.75 times of the yield strength.

In the proposed pile–slab structures, the sliding bearings between the slab and the piles are simulated using a Flat Slider element in OpenSees. The coefficient of friction is specified to be 0.06 for the element. The BRBs are modelled by the truss elements with the nonlinear axial force-deformation relationship characterized by the Bouc-Wen material in OpenSees.

The soil action on piles was simulated through nonlinear zero length p-y, t-z and q-z springs, which were capable of accounting for effects of soil–pile gap formation and closure as well as the radiation damping effects in the soil. Specifically, the TzSimple1 and QzSimple1 materials in OpenSees are used to model the vertical friction and pile tip, respectively. The PySimple1 material was chosen to represent the lateral resistance. The load-deformation (p_soil-y) relationship of the soil is determined using Eq. (1), originally proposed by Parker and Reese¹⁹ and adopted by API²⁰ guidelines:

$$\:{p}_{{soil}}=A{p}_{u}{tan}h\left(\frac{{n}_{h}z}{A{p}_{u}}y\right)$$

(1)

where A = loading factor of 0.9 for cyclic loads; p_u= ultimate resistance of soil surface; z = depth below the soil surface; and n_h = rate of increase of the modulus of the horizontal subgrade reaction.

Previous study has conducted a 1:2.5 scaled prefabricated PHC pipe pile was used for quasi-static tests to obtain the hysteretic response of the component, which was then compared with the finite element analysis results²¹. The failure mode of the specimen is shown in Fig. 4, and the comparison of numerical simulation results is presented in Fig. 5. Although the numerical results slightly overestimate the experimental measurements since the differences in the steel constitutive behavior, the hysteresis curves obtained from the numerical model still agree reasonably well with the experimental results, and the accuracy is sufficient for subsequent seismic behavior analyses. For the soil springs, Brandenberg et al. have conducted study for the feasibility of the p-y plasticity model used for piles in the soil²². In addition, the TzSimple1 and QzSimple1materials have been validated can be useful for the vertical friction and the vertical resistance at pile tip, respectively^23,24.

Fig. 4

Cyclic loading test and final damage pattern.

Fig. 5

Numerical model validation.

Input ground motions

A set of 20 near-fault earthquake ground motions was selected from the PEER Strong Ground Motion Database, with detailed information provided in Table 1. To evaluate the seismic response of the pile–slab structure under design-level earthquakes, these ground motions were adjusted using a wavelet-based algorithm to match the design response spectrum specified in the Chinese design code. It is important to note that the current study focuses solely on the transverse seismic responses of the structure.

For seismic fragility analysis, the 20 original ground motions were scaled using six scale factors, 0.5, 1.0, 1.5, 2.0, 2.5, and 3.0, based on references from previous studies^25,26,27,28. This scaling procedure generated a total of 120 ground motions, effectively covering a broad spectrum of earthquake intensities.

Table 1 Characteristics of adopted ground motions in the current study.

Results of nonlinear time history analysis

Incremental Dynamic Analysis (IDA) was performed to evaluate the seismic performance evolution of the bridges under increasing earthquake intensities. Figures 6 and 7 present the IDA curves in terms of the slab displacement and pile curvature under varying earthquake intensities. The responses are presented as the mean values of the 20 design earthquake-compatible real ground motions. An earthquake intensity scale factor (SF) of 1.0 corresponds to the design-level earthquake (DE), while lower and higher intensities were obtained by linearly scaling the design ground motions.

Regarding the maximum pile curvature, as shown in Fig. 6a and b, the proposed pile–slab structure exhibits substantially smaller seismic demands compared with the conventional structure, both above and under the ground. This improvement is primarily attributed to the modified load-transfer path. Specifically, for the above-ground portion, the maximum curvature is reduced by 93.3% at 1.0DE and up to 98.1% at 2.0DE, as shown in Fig. 6a. It is noteworthy that when the seismic intensity factor (SF) exceeds 1.7, the above-ground pile in the prototype structure yields, leading to a sharp increase in curvature with increasing seismic intensity. In contrast, the piles in the BRBs-equipped system do not yield, exhibiting consistently smaller and more stable seismic responses. For the under-the-ground portion, the corresponding reductions are 71.1% and 81.5% due to the decreased overall system stiffness, as indicated in Fig. 6b.

For the maximum slab displacement responses (Fig. 7), the inclusion of BRBs and pile–slab sliding mechanism modifies the load path by directly linking the slab to the ground-level piles, thereby mitigating displacement demands of the slab. The reductions of 54.8% and 28.2% are observed at 1.0DE and 2.0DE, respectively.

Figure 8 shows the curvature and moment envelopes of the piles at 1.0DE, clearly indicating that the proposed pile–slab structure shifts the locations of critical responses due to the altered force-transfer mechanism and path. Nevertheless, the responses at all critical positions remain consistently smaller than those of the conventional structure, implying a reduced likelihood of structural damage.

To further demonstrate the response mitigation effect achieved by the proposed structural system, Fig. 9 presents representative time histories of deck displacement and pile curvature at 1.0DE and 2.0DE. The results demonstrate that the proposed system is particularly effective in reducing the pile curvature response, while also contributing to a noticeable reduction in the slab displacement.

Fig. 6

IDA curves in terms of pile curvature responses: a above ground and b under ground.

Fig. 7

IDA curves in terms of deck displacement response.

Fig. 8

Comparison of maximum seismic responses along the elevation of piles under design-level earthquake for a bending moment and b section curvature.

Fig. 9

Sample response time histories in terms of deck displacement and pile curvature above and under the ground under different earthquake intensities.

Results of parametric analysis

To further investigate the influence of BRBs on the seismic performance of the proposed Pile–deck structure, a parametric analysis was conducted. Two critical parameters were considered: the installation angle and the design strength of BRBs. The considered variation of the BRB installation angles (α) are listed in Table 2. and Fig. 10 illustrates the three types of the installation of the BRBs. Variations in BRB inclination were introduced to examine their effect on the lateral load-resisting mechanism and the force redistribution between the slab and piles, while variations in BRB strength were intended to evaluate their role in controlling the displacement demands and enhancing the energy dissipation of the pile–slab structures.

Figure 11a and b present the pile curvature envelopes for two scenarios: (i) varying BRB installation angles with constant strength, and (ii) varying BRB strength with a fixed inclination. These responses are derived from the mean values of 20 spectrum-matched ground motions. The results indicate that the installation angle of the BRBs plays a critical role in the seismic performance of the system, with smaller angles being more effective in reducing the pile curvature.

Furthermore, when the installation angle is fixed, increasing the BRB strength leads to greater curvature demands in the piles. Figure 12 illustrates the trend of pile curvature variation with increasing BRB strength for three different installation angles. It is observed that, for smaller inclination angles, changes in BRB strength have a minimal effect on above-ground pile curvature. In all three inclination cases, above-ground curvature generally increases with BRB strength. However, when the BRB strength becomes excessively large, the brace yielding may not occur, limiting energy dissipation. For example, in Cases No. 2 and No. 3, the maximum curvature response occurs at a BRB strength of 200 kN, beyond which curvature demand decreases due to the absence of BRB yielding.

Table 2 Variation of the installation angles of BRBs in parametric analysis.

Fig. 10

BRBs installation angle (α) of a No.1, b No.2 and c No.3.

Fig. 11

Effect on the mean curvature distribution along the pile elevation of a BRB installation angle and b BRB design strength (kN).

Fig. 12

Effect of BRB strength on the maximum curvature response of different installation angle a above and b under the ground.

Seismic fragility analysis

To perform a probabilistic performance assessment of the prototype structure and the proposed pile–slab structures, seismic fragility analysis was conducted by generating fragility curves for key structural components. In this study, the earthquake demand parameter (EDP) is taken as the maximum pile curvature ductility (MPCD), which is used to develop the probabilistic seismic demand models (PSDMs) for the bridge.

The PSDMs describe the relationship between engineering demand parameters (EDPs) and intensity measures (IMs). Peak ground velocity (PGV) is adopted as the intensity measure (IM) for developing the PSDMs of the prototype structure subjected to near-fault ground motions. PGV is chosen because it inherently reflects the velocity-dominated nature of near-fault earthquakes. Numerous studies^26,27,29,30 comparing alternative IMs for bridges under near-fault seismic actions have consistently demonstrated that PGV provides superior performance relative to commonly used IMs such as PGA. In terms of efficiency, practicality, proficiency, sufficiency, and hazard computability, PGV has been shown to be the most robust and reliable choice. It is assumed that the median of the EDPs follows a lognormal distribution, and the logarithmic relationship between the median EDP and the selected IM, here, the peak ground velocity (PGV), is expressed as:

$$\:{ln}(EDP)={ln}(a)+b{ln}(IM)$$

(1)

where a and b = constants that are obtained from the regression analysis on the calculated seismic responses versus the input earthquake intensities.

To generate sufficient data for developing the probabilistic seismic demand models (PSDMs), a combined scaling and cloud method was employed. Six scaling factors (0.5, 1.0, 1.5, 2.0, 2.5, and 3.0) were applied to the 20 original ground motion records, resulting in an expanded dataset of 120 scaled ground motions.

By comparing the seismic demand and capacity models, the basic fragility function can be expressed as^30,31:

$$P\left[ {LS|IM} \right]=\Phi \left[ {\frac{{\ln (EDP) - \ln (S)}}{{\sqrt {{\beta _d}^{2}+\beta _{c}^{2}} }}} \right]$$

(2)

in which EDP is calculated using the established PSDMs, S is the mean capacity value, β_d and β_c are the standard deviations of the demand and the capacity, respectively; $\Phi \left[ {} \right]$ = standard normal cumulative distribution function.

Table 3 Definition of damage States in terms of maximum pile curvature ductility (MPCD).

Figure 13 presents the maximum pile curvature predicted by the PSDM for both systems at different locations. A fitted regression line is overlaid, with regression coefficients a and b used to establish the PSDM. The calculated R² values for all considered systems range from 0.66 to 0.68, indicating that the linear fit using PGV as the intensity measure (IM) is generally reasonable and yields acceptable accuracy. Since piles play a critical role in the seismic fragility of pile–deck structures, this study adopts the curvature ductility factor of piles as the criterion for assessing their damage states. Table 3 presents the recommended values of the curvature ductility factor corresponding to four defined damage states^32,33.

Figure 14  illustrates the seismic fragility curves for the above-ground portions of the piles under four defined damage states. The comparison between the prototype and BRB-retrofitted systems clearly demonstrates that the incorporation of BRBs greatly reduces the probability of damage across all intensity levels. At the design-level earthquake (PGV ≈ 60 cm/s), the probabilities of exceeding slight and moderate damage states decrease from 0.81 to 0.63 in the prototype system to 0.42 and 0.25 in the retrofitted system, corresponding to reductions of approximately 48% and 60%, respectively. For extensive and complete damage states, the exceedance probabilities are reduced from 0.37 to 0.19 to only 0.09 and 0.04, indicating an improvement of over 75%.

Fig. 13

Established PSDMs in terms of MPCD for different scenarios.

The marked fragility reduction is consistent with the curvature response results, where the above-ground pile curvature decreased by 93.3% at the design earthquake and up to 98.1% at twice the design intensity. This demonstrates that the BRBs effectively modify the lateral load path, allowing seismic inertia forces from the deck to be transferred through the braces rather than concentrating at the pile head. Consequently, the above-ground pile segments experience smaller curvature demands and enhanced deformation capacity, substantially lowering the probability of severe or complete damage and ensuring improved reparability after strong earthquakes.

Fig. 14

Seismic fragility curves associated with the damage of piles above the ground for a slight, b moderate, c extensive and (d) complete damages.

Figure 15 presents the seismic fragility curves for the below-ground portions of the piles under different damage states. Compared with the prototype structure, the BRB-retrofitted system exhibits a substantial reduction in the probability of damage across all performance levels. At the design-level ground motion (PGV ≈ 60 cm/s), the probability of exceeding the slight and moderate damage states decreases from 0.72 to 0.46 in the prototype system to 0.38 and 0.21 in the retrofitted system, respectively. Similarly, for extensive and complete damage, the probabilities drop from 0.25 to 0.12 to 0.08 and 0.03, representing reductions of approximately 68–75%.

This improvement is attributed to the modified load-transmission path and the supplemental energy dissipation provided by the BRBs. The braces absorb a portion of the horizontal seismic energy, thereby alleviating bending and curvature demands on the piles. Specifically, as shown in the time-history analyses, the maximum below-ground pile curvature is reduced by 71.1% and 81.5% under DE and 2.0DE excitations, respectively. These quantitative results demonstrate that the inclusion of BRBs effectively mitigates the potential for plastic hinging and nonlinear soil–pile interaction, significantly lowering the likelihood of hidden damage in the embedded pile segments and improving the overall seismic resilience of the pile–deck structure.

Fig. 15

Seismic fragility curves associated with the damage of piles under the ground for a slight, b moderate, c extensive and d complete damages.

Conclusion

This study investigates the effectiveness of a new pile–slab structure incorporating buckling-restrained braces (BRBs) and pile–slab sliding mechanisms in mitigating the structural seismic responses by providing an improve load-transmission path. The main conclusions are summarized as follows:

• (1) In the conventional pile–deck structure, the rigid pile-to-deck connection induces large pile curvatures under seismic loading, increasing the likelihood of pile damage compared with the proposed BRB-retrofitted system.

• (2) The proposed system with BRBs significantly reduces seismic responses in both the above-ground portion, which is easier to repair, and the below-ground portion, which is difficult to access.

• (3) The parametric analysis indicates that the installation angle and strength of the BRBs are critical factors affecting pile response. With their controllable and tunable properties, BRBs are well suited for pile–deck structures and can be effectively designed to enhance seismic performance.

• (4) Seismic fragility analysis confirms the effectiveness of the BRB-retrofitted system. The results show that pile fragility is substantially lower than in the prototype structure, demonstrating the superior seismic resilience of the new configuration.

• (5) The seismic response and vulnerability analysis show that the proposed system sustains less damage than traditional structures, particularly in hard-to-repair underground sections, resulting in lower maintenance costs and improved economic efficiency.

Overall, this study compared the proposed BRBs-equipped system with the prototype structure and demonstrated that modifying the load-transmission path through the combined use of BRBs and sliding bearings can effectively improve the seismic performance of Pile–deck systems. However, this study focuses only on the transverse seismic vulnerability of a conventional Pile–deck structure, without considering the effects of bi-directional or tri-directional earthquake excitations, nor the influence of modeling uncertainties on data scatter. Future research will develop more comprehensive models to broaden the scope of analysis, incorporate parameter-based sensitivity studies to quantify sources of dispersion, and further account for practical engineering effects.