This study calculated 5% damped elastic seismic response spectra acceleration(Sa) for each point on the surface of the work condition and free-field condition, respectively. As shown in Fig. 7, upon observing the enveloped area of C5 condition at site S1(referred to as S1C5 site), it could be noted that significant disturbances in ground motion of the surrounding area occur after building participation. These disturbances are primarily concentrated in the short-period segment of 0-0.5s of Sa, manifested by the fact that the enveloped line of the work condition response Sa is entirely below the free-field condition Sa curve, indicating suppression of ground motion response. At the 0.5-1s period range, Sa approaches the level of the free-field condition. In the medium to long periods greater than 1s, significant amplification of Sa occurs in the site condition, with the maximum amplification approaching twice the level of the free-field condition. This finding is generally consistent with the amplification factor trends obtained by Evangelia et al. (2020), but it shows some discrepancies with actual records in the long-period range. This is explained by the fact that plane waves emanating from the foundation propagate outward and affect the free-field records. Figure 8 illustrates significant differences in Sa on different sites: on sites S4 and S5 representing hard soil conditions, both long-period amplification and short-period attenuation are significantly less than those on site S1 representing soft soil conditions. Overall, compared to the free-field condition, the Sa of the site condition exhibits amplification in the long-period segments and attenuation in the short-period segments.
As the shear wave velocity increases, the envelope area of Sa curves near the fundamental period of the site gradually disappears which means the spatial variability of amplification and attenuation effects in the Sa near the fundamental period of the site is significantly reduced. Simultaneously, the fundamental period progressively shifts towards the short-period direction, and the influence on the nearby Sa values becomes increasingly pronounced: In the case of the S5C5 site with a shear wave velocity of 1000m/s, the Sa near the fundamental period of the site shows no significant difference compared to the free-field condition. As shown in Fig. 9, on sites with higher shear wave velocities, the standard deviation of Sa under the same conditions is smaller compared to other sites. This indicates that as the shear wave velocity of the site increases, the spatial variability of Sa gradually decreases. Specifically, the standard deviation at the S5C5 site with a shear wave velocity of 1000 m/s is reduced by approximately 54% compared to the S2C5 site with a shear wave velocity of 400 m/s. Moreover, there is a strong correspondence between site wave velocity and the overall spatial variability of Sa. However, on the S1 site with a shear wave velocity of 300 m/s, the standard deviation between the S1C5 and S2C5 sites, which have similar shear wave velocities, differs by 32% due to soil nonlinearity. This indicates that soil nonlinearity significantly increases the spatial variability of building-induced Sa disturbances.
Equation 9 defines ratio Sa, which means the Sa amplification of work condition relative to free-field condition, referred to as \(\:{R}_{Sa}\), where, \(\:{S}_{a}^{SP}\) is Sa of specific ground-motion point, \(\:{S}_{a}^{FF}\) is Sa of free-field condition in the same point. As shown in Fig. 10, \(\:{R}_{Sa}\) curves become increasingly close to 1, which represents the level of free-field condition, with the shear wave velocity of the site increased. This indicates that the Sa disruption caused by buildings in the surroundings gradually diminishes across all periods. On the S1 site with the lowest shear wave velocity, a maximum decrease of 48% was observed. Furthermore, the period at which the Sa of different shear-wave velocity sites reaches the free-field level varies among scenarios. Generally speaking, the corresponding period at which the response spectrum value reaches the free-field level is longer for sites with a greater shear wave velocity.
\(\:{R}_{Sa}\left(T\right)={S}_{a}^{SP}\left(T\right)/{S}_{a}^{FF}\left(T\right)\) | (9) |
Average \(\:{R}_{Sa}\) at each site exhibit a strong linear correlation with the mass of the building. The building exerts greater disturbance on the surrounding seismic motion as the mass increases. As shown in Figs. 11 and 12, within the short-period range of 0–1 seconds, the greater the mass of the building, the greater the amplification compared to the free-field condition. Conversely, within the long-period range of 1–6 seconds, the greater the mass of the building, the greater the suppression compared to the free-field. In all simulation results, a crossing period can be identified, denoted by the black dashed line in Fig. 12, different buildings on the same site exhibit a consistent influence on the Sa in the surrounding area at this period, thereby entering an inverse spectral amplification mode. The crossing period occurs consistently across different operational conditions on the same site and remains unaffected by variations in the building mass, demonstrating the dominant role of site conditions in the BSI effect. As shown in Fig. 12b, the crossing period slightly increases with increasing shear wave velocity of the site, generally falling within the medium to long period range and does not exceed 2s.
As shown in Fig. 13, the \(\:{R}_{Sa}\)(T0) continually increases with increasing distance, where, \(\:{R}_{Sa}\)(T0) means the Sa value at the fundamental period of the site. Considering that there is a more significant suppression at the fundamental periods of various sites compared to the free-field, this phenomenon indicates that as the distance increases, the attenuation effect of the buildings on the Sa near the fundamental period gradually diminishes. It is noted that resonance phenomena at the site fundamental periods expected in the Sa under operational conditions were not observed. Instead, a suppression effect was uniformly present across all sites compared to the free-field conditions. Among them, the S1 site with the lowest shear wave velocity of 300 m/s exhibited a reduction of 38%. Even at the S5 site with the highest shear wave velocity, also at 300 m/s, a 15% reduction in the Sa was observed. This phenomenon arises from considering the BSI effect where the building of varying mass contributes to seismic responses and leads to changes in the fundamental period of the site compared to its previous state, Furthermore, the resonance point is offset from the fundamental period of the site. As shown in Fig. 14a and Fig. 14b, the Sa value at various distances from the operational site exhibits amplification relative to the site center in medium to short periods, while demonstrating attenuation in periods exceeding 3 seconds compared to the site center. This contrast with Fig. 14c and Fig. 14d, depicting the ratio of Sa between the site center and free-field conditions, illustrates amplification in the short periods and a trend of suppression in the high-frequency range. This indicates that with increasing distance, the influence exerted by buildings gradually diminishes.
Figure 15 presents contour plots illustrating the spatial distribution of Sa and \(\:{R}_{Sa}\) relative to the corresponding free-field conditions across different distances and periods. On the contour map representing the S5 site with a shear wave velocity of 1000 m/s which is characteristic of stiff soil conditions, the \(\:{R}_{Sa}\) exhibit a consistent distribution throughout the site. Compared to the free-field Sa, which shows suppression in short periods but no significant amplification or reduction in other periods, the Sa across the entire site remains largely consistent. In contrast, on the contour map representing the S1 site with a shear wave velocity of 300 m/s, typical of soft soil conditions, noticeable fluctuations in the free-field Sa with distance can be observed which can be explained by the nonlinear behavior of the soft S1 site under seismic loading. At the S1C5 site, the presence of buildings mitigates the seismic response in the surrounding areas, resulting in reduced nonlinear behavior of the soil compared to the free-field conditions. Consequently, the short-period fluctuations in the Sa at site S1C5 are less pronounced. Furthermore, in the comparison of the \(\:{R}_{Sa}\) between site S1C5 site and the free-field condition it is observed that periodic fluctuations in the Sa appear more frequently in the longer periods, indicates that considering the BSI effect exhibit additional amplification effects on the Sa at longer periods, potentially leading to an underestimation of the seismic performance requirements for long-period structures on soft soil sites under nonlinear conditions.
Figure 16 illustrates on the contour map of \(\:{R}_{Sa}\) a noticeable reduction across all periods at the central location of the site. This indicates that under seismic loading, the interaction with buildings at this location dissipates seismic energy significantly more than in other areas of the site, resulting in a substantial decrease in seismic response and thereby affecting the Sa. However, this phenomenon is relatively weak on the site represented by S5, characteristic of stiff soil conditions. The reason lies in the BSI effect, where the building primarily feeds back vibrational energy into the soil in the form of stress wave radiation. On the one hand, due to the lower shear wave velocity at site S1, this stress wave radiation cannot propagate rapidly into the external infinite domain. On the other hand, this portion of energy cannot be dissipated effectively as in the linear phase due to the damping characteristics of the soil mass changed by the nonlinear behavior of the soil. Instead, it interacts significantly with seismic motions within the soil, leading to alterations in the seismic response of the site. However, these factors are weak on stiff soil sites like Sthe 5 site, hence significant building effects cannot be observed prominently in the \(\:{R}_{Sa}\) contour maps.
Based on the previous analysis, the most significant impact of buildings on the ground surface Sa is located at the center of the site. In this simulation, as a frequency domain analysis method for ground motion, the response spectrum exhibits considerable variability in spatial distance and frequency domain intensity. To determine the impact range of building-induced disturbances, the 90% level of the maximum diversity factor at the site center is chosen as the standard. Here, the maximum diversity factor at the site center, denoted as \(\:{DIV}_{Rsa}\), is determined by Eq. (11), where \(\:{R}_{Sa}\) represents the ratio of the response spectra between the scenario site and the free field; and \(\:{T}_{I}\) is the period corresponding to the maximum value of the acceleration response spectrum ratio at the specified seismic point at the site center.
As shown in Fig. 17a, it can be observed that the maximum influence range of buildings decreases continuously as the shear wave velocity of the site increases. In the influence range of buildings, the \(\:{R}_{Sa}\) tends to develop slowly in a linear trend as the distance increases. However, outside the influence range, the perturbation from buildings rapidly decays towards the free field level. Additionally, as depicted in Fig. 17b, the differences in perturbation range caused by different buildings on the same site are minimal, and the perturbation range from buildings under different conditions remains almost unchanged. This indicates that the perturbation range of the site's Sa is primarily influenced by the characteristics of the site itself when considering the BSI effect, which is consistent with the conclusion drawn by Chen et al. (2015): site variations have a significant impact on the seismic response of the entire system. As the site shear wave velocity decreases, the influence of the building on the surrounding seismic field becomes more pronounced and the influence range becomes larger.