Investigating the Influence of Dynamic Interaction Between Primary and Secondary Structures on Elastic Floor Response Spectra During Near- and Far-Field Earthquakes: A Comparative Analysis with EC8 Formulation

Analyzing secondary structures (SSs) during earthquakes is vital due to their vulnerability and potential impact on building functionality and occupant safety. Understanding the seismic performance of SSs requires analyzing the Floor Response Spectra (FRS). This research investigates how the dynamic interaction between the primary structure (PS) and SSs affects the FRS under near-field (NF) and far-field (FF) earthquake conditions. Both the elastic PS and SS are modeled as single-degree-of-freedom (SDOF) systems. The governing equations of motion of the PS and SS are derived and numerically solved using the RK4 method.  The study examines the influence of the PS vibration period (T_p), tuning ratio (T_r), mass ratio , and SS damping ratio on FRS. Twenty horizontal ground motion excitations were selected for the study. Time-history analysis results indicate that the dynamic interaction is negligible at a lower mass ratio . For values of 0.01, 0.1, and 0.5, the peak acceleration response of the SS under near-field (NF) excitation decreased by 15.7%, 68.3%, and 79.1%, respectively, and by 15.2%, 68.9%, and 78.8% under far-field (FF) excitation, compared to the uncoupled case. The spectral acceleration response of the SS is significantly influenced by dynamic interaction within dynamic interaction has no effect on FRS across all considered  values. Parametric analysis showed that NF earthquakes induce larger FRS peaks compared to FF events. In conclusion, a comparison between simulated FRS and those predicted by Eurocode 8 shows discrepancies, with the code-based formulation often either underestimating or overestimating the FRS magnitude.

Dynamic interaction; Far-field earthquakes; Floor response spectra; Near-field earthquakes; Non-structural component

Certain building components are unable to bear loads and are classified as secondary structures (SSs). The ground motion of an earthquake can be intensified by a structure, resulting in floor accelerations exceeding those of the ground. If secondary structures (SSs) post-earthquake is crucial for maintaining emergency services and public safety and mitigating financial losses. Despite their designation, secondary structures play a significant role and can sometimes exceed the primary structure (PS) in cost (Murty et al., 2012; Taghavi and Miranda, 2004). Recent decades have highlighted the vulnerability of secondary structures to earthquakes (Faisal et al., 2023; Challagulla, Parimi and Noroozinejad Farsangi, 2022; Kamble et al., 2022; Partono et al., 2022; Kamble, Bharti and Shrimali, 2021; Wang, Shang and Li, 2021; Challagulla, Parimi, and Anmala, 2020; Challagulla, Parimi and Thiruvikraman, 2020; Sullivan, 2020; Filiatrault et al., 2018). For instance, during the 1994 Northridge earthquake in Los Angeles, several major hospitals had to evacuate due to failures in critical secondary systems (Villaverde, 2009). Considering the critical need to protect SSs during earthquakes, additional research is essential to develop dependable design standards based on performance.

Extensive research has been conducted over the years to understand the behavior of secondary structures (SSs) during earthquakes, aiming to protect public safety and mitigate financial losses from resulting damage. The floor response spectrum (FRS) method is a widely used technique for assessing earthquake forces on secondary structures. It is common practice to utilise the FRS technique to determine the input force of a SS (Bhavani and Challagulla, 2023; Challagulla et al., 2023a; 2023b; Pesaralanka et al., 2023; Vyshnavi et al., 2023; Landge and Ingle, 2021; Pramono et al., 2020; Berto et al., 2020; Surana, Singh and Lang, 2018a). Engineers frequently employ this technique in the design of secondary structures. A key assumption of this method is that the secondary structure does not interact with the primary structure, meaning its presence has no influence on the dynamic response of the primary system, and vice versa.

In cases where the secondary structure (SS) carries significant weight, the assumption of independent behavior between the primary structure (PS) and SS may not hold. This interdependence necessitates a comprehensive consideration of the entire structural system to accurately assess seismic performance. As highlighted by (Annamdasu et al., 2024; Salman, Tran and Kim, 2020; Lim and Chouw, 2018; Kelly and Sackman, 1978), the dynamic interaction between the PS and SS can substantially influence the overall response of the structure during seismic events. Neglecting this interaction typically leads to an overestimation of SS demands, resulting in overly conservative designs that may not be cost-effective or efficient. Smith-Pardo et al. (2015) emphasize that this overestimation can lead to unnecessary material use and increased construction costs without a corresponding increase in safety or performance. Consequently, understanding and incorporating the dynamic interaction between PS and SS is crucial for accurate seismic performance evaluation and optimized structural design. Research into the seismic performance of secondary structures should, therefore, take into account their interaction with the primary structure. This approach ensures a more realistic and holistic assessment of structural behavior under seismic loads, leading to safe and economical designs. By integrating this dynamic interaction into the analysis, engineers can better predict the actual demands on both primary and secondary structures, leading to more effective mitigation strategies and improved overall structural resilience during earthquakes. The authors have performed numerical analysis to investigate the behavior of the Floor Response Spectrum (FRS). The primary aim of this research is to thoroughly investigate and understand the seismic performance of secondary structures considering their dynamic interaction with primary structures.

Several studies have explored the dynamic properties and interaction effects of integrated systems with a combined oscillator-structure model (Singh and Suarez, 1987; Suarez and Singh, 1987; Igusa and Der Kiureghian, 1985; Sackman and Kelly, 1979; Kelly and Sackman, 1978). However, prior studies have neglected to consider the effect of dynamic characteristics of both primary and secondary structures on the seismic response of secondary structures. While earlier research has explored the seismic response of primary and secondary structures under typical ground motions, there is a gap in understanding the seismic performance of secondary structures under near and far field earthquake excitations. Near-field and far-field ground motions significantly impact structural responses differently than typical ground motions. Near-field ground motions often induce higher acceleration peaks and demand on structures due to their proximity to the fault, causing rapid energy release. In contrast, far-field motions, originating further from the fault, generally produce lower-frequency content and extended shaking duration, leading to different dynamic responses in structures (Mehta and Bhandari, 2023; Lin, 2022; Salimbahrami and Gholhaki, 2022; Zamanian, Kheyroddin and Mortezaei, 2022; Akbari, Rozbahani and Isari, 2021). Therefore, further investigation is warranted in this regard. This study aims to evaluate how near and far field earthquake events affect floor spectral accelerations. This study uses the dynamic interaction between an elastic PS and SS to examine how the SS performs seismically. To evaluate how secondary structures respond to seismic activity, the floor response spectra (FRS) incorporating dynamic interaction between the primary structure (PS) and secondary structure (SS) are analyzed. The study analyses the effects of various factors on the FRS, including the mass ratio, the time period of the PS, and the secondary structure’s damping ratio. Finally, comparisons are drawn between the floor response spectra obtained from this study and those derived from existing code-based formulations.

The structure of the paper is as follows: Section 2 provides a brief summary of the modeling of coupled and uncoupled systems. Section 3 addresses the selection of ground motions, along with other relevant details for this study. Section 4 discusses the study results, with a focus on floor response spectra. Finally, Section 5 offers brief concluding remarks.

Modelling and Analysis

This study uses a single-degree-of-freedom (SDOF) system for elastic PS and SS. Coupled analysis accounts for the dynamic interaction between the PS and SS, whereas uncoupled analysis neglects this relationship. Figure 1 illustrates the PS connected to an acceleration-sensitive SS. It is assumed in this study that the primary structure’s damping ratio  is 5%.

Figure 1 Secondary structure attached to a Primary structure

2.1. Uncoupled Analysis

This analysis method ignores the PS and SS dynamic interaction (see Figure 1). Equation 1 may be used to determine how the PS will respond dynamically to a given earthquake loading.

where m_p, c_p, and k_p are the mass, damping, and stiffness of the primary structure: is the given primary structure’s frequency; are the relative displacement, velocity, and acceleration of the primary structure with reference to the ground; is the acceleration of the ground motion;  is the primary structure’s absolute acceleration response. Equation 2 may be used to compute the SS response, which is then used to assess the SS.

where k_s, c_s and m_s, are the stiffness, damping, and mass of the secondary structure:  are the damping ratio and frequency of the SS; are the relative displacement, velocity, and acceleration of the SS, respectively. Figure 2 shows how to generate the floor response spectrum.

2.2. Coupled Analysis

This method studies the dynamic relationship between the structures. The PS and SS’s response to a certain dynamic loading can be calculated using Equations 3 and 4, respectively.

The matrix version of Equations 3 & 4 is as follows:

Figure 2 The process of creating floor response spectrum using uncoupled analysis

2.3. Fourth-Order Runge-Kutta Method for Solving Differential Equations

In order to solve ordinary differential equations (ODEs) and especially address the dynamic behavior described by the second-order differential equations given in Equations 1– 4, we utilize the Fourth-Order Runge-Kutta (RK4) technique as a numerical tool in this work. The RK4 approach makes it easier to derive dynamic solutions that are necessary to understand how the system reacts to external influences (Challagulla et al., 2023b; Challagulla, Parimi and Thiruvikraman, 2020; Reyes et al., 2020; 2016; Smith-Pardo et al., 2015). This numerical method approximates solutions to ODEs by utilizing known initial conditions. By breaking the problem into smaller segments, the method calculates slopes at various points within each step. The approximation is progressively refined until the desired endpoint is reached by updating the solution at the end of each step. Smaller step sizes lead to greater accuracy, and the precision can be adjusted by modifying the step size (Yaakub and Evans, 1999). The RK4 approach is utilized extensively in several scientific and engineering fields to solve ODEs in situations when precise analytical solutions are not available. This paper presents the steps involved in solving the second-order differential equation given in Equation 1. This is how a first-order system that is equal to the ordinary differential equation in Equation 1 might be redefined Equation 6:

The system of autonomous first-order ordinary differential equations (Equations 7 and 7) that follows is produced by combining Equations 8 and 9 with the initial condition .

The numerical solution of Equation 10 was achieved by the development of a MATLAB code that employs the explicit RK4 technique. Other ODEs, such as Equations 2 – 4, can be solved using the same method described above.

3.    Selection and Scaling of Ground Motions

In the context of seismic response assessment, realistic responses are generated by utilizing actual ground-motion records, readily accessible from the Pacific Earthquake Engineering Research Centre (PEER) NGA-West2 Database. Therefore, for the current research, we have incorporated 20 horizontal ground motion excitations, as specified by ASCE 7-16 (ASCE, 2016) tailored for hard soil conditions with a shear wave velocity (V_s30) greater than 350 m/sec. Additionally, for this study, we have chosen to employ a set of ground-motion records recommended in FEMA P695 (FEMA, 2009). These records will be used to carry out both linear and nonlinear dynamic analyses on the building structures under consideration, as detailed in Table 1. According to the classification in FEMA P695, the far-field record set comprises ground motions originating from sites situated at a distance equal to or greater than 10 km from the fault rupture. In contrast, the near-field record set includes ground motions recorded at sites located within a distance of less than 10 km from the fault rupture, as determined by the Joyner-Boore distance (R_jb). The ground-motion records under consideration were obtained from sites with rock soil conditions, specifically falling within NEHRP site classes B and C. These records are associated with moment magnitudes (M_w) ranging from 6.69 to 7.62, with an average magnitude of 7.05. Among the selected records, the closest distances to the fault rupture, calculated as the average Joyner-Boore distance, span from 0 to 26 km, with an average distance of 8.11 km. The epicentral distances (R_epi) for this chosen set of ground motions vary between 4.5 and 86 km, with an average distance of 33.4 km. The peak ground acceleration (PGA) values of these selected records range from 0.22 to 1.49 g, and their average PGA is 0.494 g. For more comprehensive information regarding these ground motions, further details can be found in FEMA P695. To achieve compatibility with the target response spectrum, which is the Zone V elastic design spectrum of IS 1893 (Part 1): 2016 (Indian Standard, 2016), the chosen ground motion records were subjected to scaling. The process employed for this purpose involved the utilization of a time-domain spectral matching approach to generate earthquake excitations that align with the desired spectrum. The target spectrum, according to IS 1893:2016 (Indian Standard, 2016), is shown in Figure 3, which is linked to 5% damping, along with the mean spectra of ground excitations. In accordance with ASCE 7-16 standards, the mean spectra must not fall below 90% of the target spectrum across the entire period range. It is evident from the figure that the mean spectra comfortably exceed this 90% threshold.

Table 1 Details of near-field and far-field records

Figure 3 Scaled ground motions mean spectra and the target spectrum (a) near-field data, (b) far-field data

The subsequent sections delve into an examination of secondary structures' behavior. The acceleration time-history response of the secondary structure is studied in a few cases. The key response parameter used to characterize the performance of the secondary structure includes the Floor Response Spectrum (FRS).

4.1. Time History Response

The secondary structure's acceleration response is displayed in this section, as seen in Figure 4. The system depicted in Figure 1 is exposed to ground motions in order to study the impact of the dynamic interaction on the dynamic behavior of the SS. For this particular analysis, two ground motions (one near-field: RSN 802 and one far-field: RSN 1111) were selected from Table 1. These ground motions were chosen to have identical PGA and duration. The impact of the mass ratio  and the SS damping ratio  on the SS's acceleration response is examined. The ratio of the mass of the secondary structure to the mass of the primary structure is known as the mass ratio For the coupled analysis, the values of  are 0.001, 0.01, 0.1, and 0.5 are considered. For the purpose of this analysis, the PS (T_p) and SS (T_s) vibration periods are assumed to be 0.5 seconds. The SS damping ratios are assumed to be 1% and 5%. As expected, the acceleration response's amplitude increases as the SS's damping ratio decreases. The seismic response of the secondary structure (SS) is not significantly affected by the dynamic interaction between the PS and SS when the mass ratio is as low as 0.001 (0.1%). This is evident as the acceleration response at ? = 0.001 closely resembles that of the uncoupled system. Therefore, at this mass ratio, the seismic demands on the secondary structure can be computed using the uncoupled analysis. The dynamic interaction between the PS and SS has a significant effect on the SS's acceleration response as the  A similar conclusion was observed in the study by (Kaiyuan et al., 2023).

The peak acceleration response of the SS is shown in Table 2 for = 5%, as peak values of any seismic response quantity provide important information about the bbehaviorof the structure. In comparison to the uncoupled evaluation conducted under NF and FF ground motions, Table 2 clearly demonstrates that for 0.01, 0.1, and 0.5, the peak acceleration response of the SS has decreased significantly. Under near-field (NF) excitation, the peak acceleration of the secondary structure (SS) experiences reductions of 15.7%, 68.3%, and 79.1% compared to the uncoupled case. Additionally, when compared to the uncoupled case, the peak acceleration of the secondary structure (SS) under far-field (FF) excitation decreases by 15.2%, 68.9%, and 78.8%. Overall, in both near- and far-field excitations, the secondary structure coupled with the primary structure shows notable reductions in peak accelerations compared to the uncoupled scenario. Furthermore, it is worth mentioning that near-field (NF) excitation tends to impose higher seismic demands on the secondary structure (SS), as evidenced by the data presented in Figure 4 and Table 2. In the specific scenario outlined in Table 2, it is observed that the response of the secondary structure (SS) exhibits an average increase of 10% under NF excitations compared to FF excitations. This finding underscores the notion that NF excitation imposes greater seismic loads on the SS (Bravo-Haro, Virreira and Elghazouli, 2021; Zhai et al., 2016) in contrast to FF excitation. This trend highlights the importance of considering the specific characteristics of ground motion in assessing seismic performance. A similar pattern was seen for the  = 1% also. 

Table 2 Peak acceleration of the SS () for  = 5%

Figure 4 Time-history response (a) near-field data, (b) far-field data

4.2. Floor Response Spectrum

The maximum design forces for the design of the SS can be obtained from the floor response spectrum (FRS) approach (Haymes, Sullivan and Chandramohan, 2020). The FRS method disregards the PS and SS's dynamic interaction (Surana, Singh and Lang, 2018b; Adam, Furtmüller and Moschen, 2013). As a result, the current study tried to examine the FRS by considering the coupling effect. The floor response spectrum represents the SS’s peak responses to input ground motion. The effects of the vibration period (Tp) of the PS, the mass ratio , and the damping ratio on the floor response spectrum are studied. For this purpose, scaled near- and far-field ground motions are input data for time history analyses. Absolute acceleration responses are individually obtained and subsequently utilized to calculate the corresponding FRS. These FRS are derived using a 5% damping ratio of the PS, and the mean results are then plotted and analyzed. The graphs depicted in Figures 5 and 6 illustrate the correlation between the average spectral acceleration  of an SS and the tuning ratio is the vibration period of the SS. Analysis of Figures 5 and 6 reveals that dynamic interaction significantly influences the FRS within the range of  the impact of dynamic interaction appears negligible across all considered values of the damping ratio () and mass ratio (). Thus, it may be said that the coupling effect is noteworthy only when the  closely aligns with that of the PS, and a similar conclusion is observed in the study by (Zheng, Shi, and Sui, 2023). Regardless of the  and type of ground motions, the  reduces with an increase in the primary structure's vibration period for a given damping ratio of the SS.

When the SS has lower damping ratios, there is a noticeable coupling impact on the FRS. These findings indicate that as the damping in the SS decreases, the dynamic interaction has a more significant impact on the FRS. Put differently, as the  decreases in the SS, the significance of the interaction in shaping the FRS becomes more pronounced. This observation underscores the importance of considering damping ratios, particularly in the SS, when analyzing the dynamic response of structures subjected to seismic forces. Peaks (in the vicinity of ) in the Floor Response Spectrum (FRS) show how the SS responds to earthquakes. When we look at these peaks at different mass ratios, we see that earthquakes nearby (near-field) make bigger peaks than those far away (far-field) at some combination of damping ratios and mass ratios.

As an illustration, consider the FRS value associated with the  to see the effect of near-field (NF) and far-field (FF) records on the seismic response of the SS. The analysis was conducted for different periods of vibration (), including 0.1 sec, 0.5 sec, 1 sec, and 2 sec. For each , the impact of NF and FF records on FRS was investigated across various  and . At  = 0.1 sec, it was observed that NF records consistently resulted in a decrease in  compared to FF records across all damping and mass ratios. The magnitude of these decreases varied, with higher damping ratios generally leading to larger percentage decreases.

At  = 0.5 sec, the influence of NF records on  exhibited variability depending on the damping and mass ratios. For  = 0.01 and  = 0.5, NF records consistently yielded higher  values, indicating a significant effect. However, for  = 0.1, NF records showed a slight decrease in Sa compared to FF records, suggesting a less pronounced impact. At  = 1 sec, similar variability in the influence of NF records on  was observed. For  = 0.01 and  = 0.1, NF records consistently led to lower  values, indicating a significant effect. Conversely, for  = 0.5, NF records showed a slight increase in  compared to FF records. Finally, at  = 2 sec, the pattern of influence of NF records on  varied based on the damping and mass ratios. NF records consistently resulted in higher  values for  = 0.01 and  = 0.5, indicating a significant effect. However, for  = 0.1, NF records showed a slight decrease in  compared to FF records, suggesting a less pronounced impact. These findings underscore the complex interplay between ground motion characteristics, , and the  in influencing secondary structural response to seismic events.

Figure 5 FRS Vs. tuning ratio under near-field records for (a)  = 0.1 sec, (b)  = 0.5 sec, (c)  = 1 sec, (d)  = 2 sec

The vibration period of the primary structure is a critical factor in determining its response to ground motion. Structures with shorter periods are more susceptible to the high-frequency motions characteristic of near-field (NF) records, whereas longer-period structures are predominantly influenced by the lower-frequency content typical of far-field (FF) records. It is essential to emphasize the significance of considering both NF and FF records in seismic hazard assessment and structural design. From Sections 4.1 and 4.2, the important points or analysis results can be summarized as follows:

• The dynamic interaction between the primary structure (PS) and secondary structure (SS) significantly affects the acceleration response of the SS, particularly when the mass ratio (?) is high. For low mass ratios ( = 0.001), the acceleration response of the SS is similar to that of the uncoupled system, indicating minimal impact of dynamic interaction.
• Near-field (NF) ground motions tend to impose higher seismic demands on the SS compared to far-field (FF) ground motions. 
• Lower damping ratios in the SS result in a more pronounced coupling impact on the FRS. This emphasizes the importance of considering the damping ratio of the SS when analyzing the dynamic response of structures subjected to seismic forces.
• At certain vibration periods and mass ratios, NF records unexpectedly result in lower spectral accelerations () compared to FF records, which is contrary to the general trend observed. This suggests that NF ground motions do not always impose higher seismic demands, highlighting the complexity of seismic response behavior.

Figure 6 FRS Vs. tuning ratio under far-field records for (a)  = 0.1 sec, (b)  = 0.5 sec, (c)  = 1 sec, (d)  = 2 sec

The findings underscore the need for site-specific analyses that account for the complex interplay between ground motion characteristics and structural response. This conclusion underscores the importance of considering the source-to-site distance and the ground motion characteristics when assessing the seismic response of PS and SS to withstand earthquakes.

4.3. Comparing the FRS with Eurocode 8 (EC8) Formula

A comparative examination between the FRS generated in this study using near-field (NF) and far-field (FF) records and the approach outlined in Eurocode 8 (NSAI, 2005) is conducted in the present section. Eurocode 8 provides a formulation for computing the spectral acceleration () applied to a SS, as depicted in Equation 11:

where  represents the ratio of ground acceleration to gravity , while  denotes a soil amplification factor. The term  indicates the relative height of the structure where the component is located,  stands for the period of the secondary structure and  represents the primary structure’s vibration period.

Two fundamental factors affect the design of FRS: the tuning ratio () and the relative height of the SS. The formula provided by Eurocode provides a series of curves that show the highest spectral acceleration values for each floor when  equals . Plots of the elastic generated and the Eurocode 8 (EC8) spectra are shown for the PS under consideration in Figure 7 and 8.

When the mass of SS is negligible ( = 0.01), the EC8 approach tends to underestimate the maximum demand for SSs at  under both near-field (NF) and far-field (FF) seismic records, assuming a primary structure with  = 0.5 sec and varying . For  = 1 sec, under NF and FF records, the EC8 formulation underestimates the maximum demand for SSs with damping ratios of  = 0.1% and 0.5%. Conversely, for damping ratios of  = 2% and 10%, the EC8 tends to overestimate the maximum demand for SSs. In contrast, for SSs with mass ratios,  = 0.1 & 0.5, in both near-field (NF) and far-field (FF) records, the EC8 formulation tends to predict higher floor acceleration demands across various  of the SS within the same  range. If  falls below 0.5, the EC8 consistently predicts higher demands on SSs without regard to mass ratios, damping ratios, primary structure type, or record type. Conversely, when  surpasses 2.5, the EC8 consistently underestimates demands on SSs, regardless of mass ratios and damping ratios for a flexible primary structure ( = 1 sec). The dynamic interaction between the primary and secondary structures is thus not considered by the EC8 formulation. Figure 7 and 8 show a notable discrepancy in which the floor spectral acceleration is either greatly overestimated or underestimated by the criteria given in EC8. This inconsistency emphasizes the requirement of updating the code-based approach. To enhance the accuracy of the formulation, it is crucial to incorporate the effects of dynamic interaction between the PS and SS. This adjustment is vital for improving the seismic evaluation of secondary structures, as it aligns code-based predictions more closely with observed spectrum accelerations.

Figure 7 Comparing the predicted FRS of a PS (Tp = 0.5 sec) with code-based FRS for (a) near-field records and (b) far-field records

4.4. Engineering Application

        The above analysis results can be effectively utilized in practical engineering applications, such as the seismic design of hospital equipment. Hospitals house numerous critical secondary structures (SS), such as medical equipment, electrical panels, and storage units, which are essential for patient care and hospital operations. Ensuring the functionality of these structures during and after an earthquake is crucial. The dynamic interaction between the hospital building (PS) and these secondary structures can significantly impact their seismic response. The present research findings can be applied in the practical field as follows:

·      Hospital equipment can be modeled as single-degree-of-freedom (SDOF) systems, characterized by parameters such as mass ratio (), damping ratio (), and vibration period (). This modeling allows for a detailed understanding of the equipment’s behavior during seismic events.

·      Ground motion records, including both near-field (NF) and far-field (FF) events, can be used to simulate the seismic response of the hospital building and its equipment. Numerical tools like MATLAB can facilitate these simulations, providing realistic scenarios of seismic activity.

·      The research shows how the peak acceleration response of secondary structures varies with different mass and damping ratios. These insights help in predicting the seismic demands on the equipment.

·      Understanding that dynamic interaction is minimal at very low mass ratios (? = 0.001) and significant at higher mass ratios allows engineers to optimize the design of equipment supports and anchors. Robust anchoring systems can be designed for equipment with higher mass ratios to ensure they can withstand seismic forces effectively.

·      The study’s findings on the Floor Response Spectrum (FRS) can be applied to derive maximum design forces for hospital equipment. By considering the coupling effects, the FRS method provides a more accurate estimate of seismic demands, preventing over- or under-design.

Figure 8 Comparing the predicted FRS of a PS (Tp= 1 sec) with code-based FRS for (a) near-field records and (b) far-field records

The purpose of this article is to examine how dynamic interaction affects secondary structures' seismic needs. This paper delves into a parametric investigation of the dynamic interaction between primary and secondary structures. The dynamic interaction exhibits a notable impact on the acceleration demands of the SS with increasing mass ratio. Near-Field excitations impose greater seismic loads on the SS in contrast to Far-Field excitations. At a very low mass ratio , the dynamic interaction is negligible. For values of 0.01, 0.1, and 0.5, the peak acceleration response of the SS under near-field (NF) excitation decreased by 15.7%, 68.3%, and 79.1%, respectively, and by 15.2%, 68.9%, and 78.8% under far-field (FF) excitation, compared to the uncoupled case. Only in the cases when the secondary structure closely matches the main structure's vibration period—that is, in the region of  coupled analysis required. Conversely, for  the influence of interaction on the FRS appears insignificant across all examined values of 

Lower damping ratios  in the SS increase the significance of dynamic interaction on the FRS. Peaks in the FRS within the range of  show higher responses to NF excitations compared to FF excitations. The finding that lower damping ratios in SS enhance the impact of dynamic interaction on FRS was unexpected. This underscores the need for detailed consideration of damping properties in seismic analysis and design, as it significantly affects the seismic response of SS. The Floor Response Spectrum (FRS) provides essential insights into the seismic response of secondary structures (SSs), particularly within the range of  where peaks are most prominent. Near-field (NF) earthquakes tend to generate larger peaks in FRS compared to far-field (FF) events, especially at specific combinations of damping and mass ratios. Our analysis across various vibration periods (Tp) highlights the nuanced impact of NF and FF records on spectral accelerations (Sa). Notably, NF records consistently lead to higher Sa values for shorter periods (Tp= 0.1 sec), while the influence varies at longer periods.

The design of FRS is influenced by the relationship between the period of secondary structure (SS) and the primary structural system. However, the current Eurocode 8 (EC8) formulation doesn't fully account for this dynamic interaction. Our analysis found that for lightweight SS  EC8 underestimates maximum floor acceleration demands in certain scenarios, while for heavier SS , it tends to overestimate them. Additionally, EC8 consistently overestimates acceleration demands when and underestimates them when . This discrepancy highlights the need to revise EC8 to consider the effects of dynamic interaction. Incorporating these effects will improve the accuracy of predictions and ensure better seismic performance of secondary structures.

    The authors would like to acknowledge the support of Prince Sultan University in paying the Article Processing Charges (APC) for this publication.

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