Effect of Soil–Structure Interaction on the Damage Probability of Multistory RC Frame Buildings with Shallow Foundations

Abstract

The purpose of this study is mainly to investigate, through fragility curves, the effect of soil–structure interaction (when it is neglected during design) on damage probability. It also examines how realistic it is to conduct a performance estimation with rapid assessment methods without considering soil–structure interaction. Three RC frame buildings, with varying numbers of stories, were designed according to the Turkish Seismic Design Code, 2007. Incremental dynamic analyses of the considered structures, both with and without soil–structure interaction (SSI), were performed using 21 ground motion records to determine the damage limits. The cone model with springs was used to take soil–structure interaction into account. The discrete damage probabilities of each considered performance level were calculated, using statistical methods, in terms of elastic spectral acceleration, and continuous fragility curves were obtained. The results show that the effect of SSI on fragility was remarkable and that damage probability generally increases when soil–structure interaction is taken into consideration. The effect of site class becomes significant for life safety and collapse prevention performance levels. The increase in the probability of exceeding the collapse prevention performance level can reach up to 72% due to the existence of SSI. Thus, the results of damage estimation made without considering SSI can sometimes be significantly misleading.

1. Introduction

The damage estimation of existing buildings is an important part of disaster response plans, especially for cities with high populations; these estimations are sometimes carried out using rapid assessment methods and, at other times, are carried out with more detailed methods such as performance-based design, including nonlinear structural analysis. When the number of buildings to be assessed increases, rapid assessment methods are frequently used, and these methods do not always take soil–structure interaction into consideration. In addition to this uncertainty, there are a significant number of mid-rise buildings in the existing building stock that were designed and constructed in the past without taking soil–structure interaction (SSI) into account and where soil–structure interaction may have an effect on structural behavior. Thus, this study mainly aims to investigate the effect of SSI on the damage probability of RC frame structures (through fragility curves by taking SSI into consideration) which were designed as fixed-base structures. Furthermore, this study also examined how realistic it is to conduct a performance estimation with rapid assessment methods without considering soil–structure interaction. Fragility curves are commonly used tools for the estimation of damage probability caused by earthquakes. SSI was assumed to be beneficial in the past [1] due to the reduction in internal forces and drifts because of the increased flexibility of the soil. Thus, to be on the safe side, several buildings were designed and constructed in the past without taking soil–structure interaction (SSI) into account. Such buildings may carry an additional risk, since SSI was neglected during the design process. However, studies carried out to take the effect of SSI into consideration showed that the effect of SSI on seismic performance is not always positive [2]. Thus, this study aims to investigate the effect of SSI on the damage probability of RC frame structures through fragility curves by taking SSI into consideration. Fragility curves were used within the scope of several studies that investigate the damage potential of different types of structures through fragility curves by taking SSI into consideration. Rajeev and Tesfamariam [3] investigated the damage probability of non-ductile two-dimensional RC frame structures for only one site class and concluded that SSI could significantly change damage probability. Hoseyni et al. [4] analyzed the effect of soil–structure interaction on the seismic fragility of a power plant containment. They concluded that the effect of SSI should be considered in probabilistic seismic risk analyses and that ignoring it causes uncertainties in the structural response. Karapetrou et al. [5] investigated the fragility curves of a two-dimensional, nine-story frame structure. The results show the significant role of SSI and site effects, under linear or nonlinear soil behavior, in altering the expected structural performance and fragility of high-rise fixed-base structures. They analyzed the structure using both multilayer and single-layer soil models and concluded that SSI increases damage probability. For both RC and steel structures with different numbers of stories, the increase in damage probability is more evident in the case of multilayer acceleration in their study. In their study, Mitropoulo et al. [6] obtained fragility curves in terms of spectral acceleration for both RC and steel structures with different numbers of stories. They considered three different foundation types: fixed-base, SSI interaction, and pile foundation. They concluded that structural performance is not affected by foundation type for structures whose number of stories is four or six. However, the performance of eight-story structures is affected by their foundation type. Banizadeh and Behnamfar [7] investigated the structural behavior of structures with different numbers of stories and different levels of SSI. They concluded that SSI increases structural damage, since plastic deformations are increased in lower stories while the base shear decreases. Mekki et al. [8] studied single-degree-of-freedom systems considering SSI and concluded that damage probability is affected by foundation type and the soil characteristics beneath the structure. Ghandil and Behnamfar [9] investigated the nonlinear response of moment-resisting frames on nonlinear soft soils and showed that SSI increases the drift and ductility demands of the lower stories. Forcellini [10] applied a probabilistic approach to investigate the effects of SSI on the fragility of shallow-founded reinforced concrete buildings with and without masonry infill walls. The results indicate that SSI increases the failure probabilities of both buildings, highlighting the effects of structural stiffness on seismic vulnerability. Akhoondi and Behnamfar [11] investigated the effect of SSI on the damage probability of special moment steel frames. They observed that SSI can increase the collapse probability. Cruz et al. [12] investigated the effects of SSI on the seismic vulnerability of RC buildings. They concluded that increasing foundation flexibility leads to higher seismic damage. Cruz et al. also [13] investigated the effect of SSI on the seismic performance of a real five-story RC building, located on soft soil, which had been designed and constructed prior to the seismic code. The results show that the effect of SSI on fragility is remarkable; for instance, a severe damage ratio can be worsened by up to 38% when SSI is considered. Mata et al. [14] examined the seismic behavior of RC moment frame buildings while accounting for soil–structure interaction (SSI). Their findings indicated that SSI has a notable impact on seismic response by increasing interstory drift ratios and reducing shear forces. Consequently, they concluded that buildings with a flexible base are more likely to collapse compared to those with a fixed base. Wang et al. [15] developed a one-dimensional convolutional neural network model to predict the influence of SSI on RC frame buildings, particularly regarding interstory drifts and base shear. Their results demonstrated that the model accurately estimated the absolute prediction errors for SSI influence coefficients, with maximum base shear force and interstory drift errors ranging between 9.3% and 11.7% for 80% of test cases. Awchat and Monde [16] analyzed the seismic performance of a 10-story building located in various seismic zones in India, incorporating SSI effects. Their study revealed that SSI contributes to an increase in lateral story drift. Additionally, Jamkhaneh [17] focused on the seismic response of two adjacent 20-story reinforced concrete buildings, emphasizing the role of soil–structure interaction. The study found that softer soil conditions resulted in greater deformations and settlements. Three different hypothetical residential RC frame building structures were considered within the scope of this study. They were designed as moment-resisting frames with three different numbers of stories: 5, 10, and 15. Incremental dynamic analyses were carried out, under the effect of 21 ground motions, to determine the limit of the considered performance levels in terms of the interstory drift ratio, which is used as the damage indicator in this study. The cone model with springs was used to take the soil–structure interaction into account. A statistical procedure was followed to obtain the fragility curve of each considered performance level. A lognormal distribution was assumed for fragility curve development, as was done traditionally in previous studies, and the fragility curves were developed in terms of elastic spectral acceleration (S_ae). Finally, the probability of exceeding the considered performance levels was compared for the considered buildings; thus, the effect of SSI on damage probability could be investigated.

2. Materials and Methods

2.1. Sample Buildings

Sample 5-, 10-, and 15-story hypothetical residential buildings were designed according to the 2007 version of the Turkish Seismic Design Code (TSDC) [18] considering different site classes. They have ductile frames in both principal directions. Therefore, they meet the criteria of ductile moment-resisting frames, including provisions such as a strong column–weak beam design principle, capacity design for shear reinforcement, restrictions on the axial force level in columns, minimum requirements for compressive reinforcement in beams, and limitations on the spacing of shear reinforcement. Figure 1 shows the typical ground floor plan and section view of the sample buildings. Although the main portion of the existing building stock does not have such a regular structural system, a regular plan building is used within the scope of this study to simplify the structural behavior; thus, the effect of SSI on structural performance can be clearly compared. In consideration of Turkey’s construction practices, the typical floor dimensions are 16.5 m × 16.5 m, and the story height is typically 3 m. The foundations of the examined structures were designed as mat foundations with thicknesses of 60 cm, 70 cm, and 80 cm, respectively. The characteristic compressive strength of concrete is assumed to be 30 MPa for the design of buildings. The reinforcement type is Grade 420, which has a characteristic yield strength that is equal to 420 MPa.

Table 1. Structural periods of considered buildings.

Number of Stories	Fixed Base	SSI
		B	C	D
5	0.69	0.70	0.72	0.76
10	1.19	1.23	1.28	1.41
15	1.46	1.56	1.69	1.99

shows the vibration period of the first mode of the considered buildings.

As the study focuses on hypothetical RC frame buildings, the assumed characteristic strengths for concrete and reinforcement steel are 30 MPa and 420 MPa, respectively. Since the building under consideration is not a real structure, there is no need for material quality testing or statistical evaluation of test results. However, the analytical approach applied in this study aligns with the methodology used for assessing the performance of existing buildings. Therefore, the mean values of material strengths must be considered. These mean values, required for nonlinear time-history analysis, were determined based on the assumed characteristic strengths in accordance with the provisions of the 2018 version of TSDC [19]. We used a mean compressive strength of concrete that was equal to 1.30 times the characteristic strength; similarly, we used a mean yield strength of reinforcement steel that was 1.2 times the characteristic strength according to TSDC [19] provisions. However, in the case of making a performance evaluation for a real structure, a statistical evaluation is required, as explained in ref. [20].

2.2. Ground Motions

A total of 21 ground motion records of 14 events, provided by Fema (2009) [21], were used in this study, with their two components, to take the random nature of ground motions into consideration. The minimum fault distance was selected as 10 km to eliminate the near-fault effect as much as possible, following the recommendations of Fema P695 (2009) [21]. All the considered earthquakes are shown in

Table 2. Ground motions used in this study.

Event	Mw	Station	Fault Dist. (km)	1st Comp.	PGA (g)	2nd Comp.	PGA (g)	Shear Wave Velocity (m/s)	Site Class
Hect. Mine, 1999	7.1	Hector	10.35	HEC000	0.263	HEC090	0.360	685	B
Kocaeli, 1999	7.7	Arçelik	10.56	ARE000	0.218	ARE090	0.14	523	B
Friuli, 1976	6.5	Tolmezzo	14.97	TMZ000	0.344	TMZ270	0.302	505	B
Loma Pri., 1989	6.9	Anderson	19.90	LOMAP 250	0.250	LOMAP 340	0.262	489	B
Northridge, 1994	6.7	Big Tuj	19.10	TUJ262	0.165	TUJ352	0.255	550	B
Kobe, 1995	6.9	Chihaya	49.90	CHY000	0.094	CHY090	0.106	609	B
Imper. Val., 1979	6.5	Cerro Priet	15.19	CPE147	0.167	CPE237	0.147	472	B
Northern, 1967	5.6	Ferndale	27.36	FRN224	0.268	FRN314	0.110	219	C
Kern Coun., 1952	7.4	Taft	38.42	TAFT021	0.156	TAFT111	0.183	385	C
Imper. Val., 1979	6.5	Delta	22.03	DLT262	0.239	DLT352	0.333	242	C
Kobe, 1995	6.9	Shin	19.14	SHI000	0.238	SHI090	0.212	256	C
Kocaeli, 1999	7.5	Duzce	13.60	DZC180	0.325	DZC270	0.36	282	C
Landers, 1992	7.3	Yermo	23.62	YER270	0.226	YER360	0.148	354	C
Northern, 1954	6.5	Ferndale	26.72	FRN044	0.172	FRN314	0.200	219	C
Imper. Val., 1979	6.5	El Centro	12.56	E11140	0.367	E11230	0.401	196	D
Supers Hills, 1987	6.5	El Centro	18.20	ICC000	0.376	ICC090	0.293	192	D
Chi Chi, 1999	7.6	CHY047	24.13	CHY047N	0.194	CHY047-W	0.163	170	D
Northridge, 1994	6.7	Carson	45.55	WAT180	0.087	WAT270	0.088	161	D
Loma Pri., 1989	6.9	Foster City	43.77	A01000	0.268	A01090	0.291	116	D
Coalinga, 1983	6.4	Parkfield	43.83	H-C02000	0.112	H-C-2090	0.117	173	D
Supers Hills, 1987	6.2	Imper. Val.	17.59	IVW090	0.130	IVW360	0.127	179	D

. The selected ground motions were recorded on Site Class B, C, and D according to the classification of USGS. Site class A was not considered, since it is believed that soil–structure interaction does not exert an effect on the structural behavior of this type of soil. According to USGS classification, Site Class A corresponds to a shear wave velocity higher than 750 m/s, while Site Class B corresponds to a shear wave velocity between 360 m/s and 750 m/s. The shear wave velocity is classified between 180 m/s and 360 m/s for Site Class C, and finally, the shear wave velocity corresponding to Site Class D is lower than 180 m/s.

The average shear wave velocity of the selected ground motions is shown in

Table 3. Mean shear velocity for each site class considered.

Site Class	Mean Shear Wave Velocity (m/s)
B	547
C	280
D	170

for each site class considered.

Figure 2 shows the PGA–magnitude and magnitude–shear wave velocity distributions of the considered ground motions.

The main purpose of this study was to investigate the effect of the actual foundation flexibility on the damage probability in structures designed in the past without considering SSI. Thus, soil layers and their amplification effect were not taken into account in this study, since the records used are not for the bedrock but for the stations located on different soil profiles. Therefore, the randomness originating from the soil profiles was also taken into account in an indirect manner. The time-history and acceleration spectra of all the selected ground motions are shown in Appendix A.

2.3. Damage Levels

Performance levels and their qualitative definitions provided by the 2018 version of the TSDC [19] are used to determine the threshold of each performance level in terms of interstory drift ratio. There are three different levels of section damage defined by Code [19] for RC sections: minimum damage, controlled damage, and collapse prevention. Each one has a strain limit for concrete in compression and for reinforcement steel in tension.

Table 4. Section damage levels and corresponding strain limits.

Section Damage	Strain Limit for Concrete	Strain Limit for Reinforcement
Limited damage	0.0025 (unconfined)	0.0075
Controlled damage	$0.75\left(0.0035+0.04\sqrt{{\omega }_{we}}\right)$ (confined)	0.75 (0.4 Ɛsu)
Collapse prevention	$0.0035+0.04\sqrt{{\omega }_{we}}\le 0.018$ (confined)	0.4 Ɛsu

shows the corresponding limits of each sectional damage level defined by the aforementioned code. ω_we is the transverse reinforcement ratio in Table 4. There are also three section damage zones limited by the aforementioned damage limits. These are the limited damage zone, significant damage zone, and advanced damage zone. The threshold value of each performance level, in terms of the interstory drift ratio, was determined via incremental dynamic analysis (IDA) considering the performance level’s qualitative definitions provided by the 2018 version of the TSDC [19]. The building performance levels defined by the TSDC [19] are as follows: limited damage, controlled damage, and collapse prevention. These performance levels can be considered as the global damage levels of a building. The lowest spectral acceleration value that leads to any of the specified section damages identified through incremental dynamic analysis was associated with the building’s performance levels. This value is considered to be the lower threshold for the corresponding performance level in terms of spectral acceleration. Likewise, the interstory drift ratio determined from the same analysis is regarded as the limit for the building’s performance level in terms of interstory drift ratio. The limited damage performance level, which represents the limited nonlinear behavior in structural elements, is assumed to be the interstory drift ratio corresponding to the first minimum section damage of a structural system element. The interstory drift ratio of the controlled damage performance level is assumed to be reached when a structural system element’s strain reaches the strain limit of the section damage level of controlled damage. Finally, the collapse prevention performance level, which represents the onset of collapse, is assumed to be the interstory drift ratio corresponding to the first collapse prevention section damage of a structural system element.

2.4. Incremental Dynamic Analysis

Incremental dynamic analysis (IDA) has previously been employed to identify damage thresholds corresponding to the performance levels defined in the seismic code [19]. This method is particularly effective in generating curves that illustrate the relationship between a selected ground motion intensity measure and a selected damage indicator. A comprehensive discussion of IDA can be found in the work of Vamvatsikos and Cornell [22]. In this study, 5%-damped elastic spectral acceleration is used as the intensity measure, while the maximum interstory drift ratio serves as the damage indicator. Each ground motion is progressively scaled based on its individual spectral acceleration. Consequently, the maximum interstory drift ratio obtained at each step of the IDA process results from a nonlinear time-history analysis conducted at a specific spectral acceleration level. At first, the relationship between the spectral acceleration and the maximum interstory drift ratio remains linear up to the yield point. However, once yielding occurs, the curve deviates from linearity, leading to a significant reduction in its slope. A structure is considered to have collapsed if the slope of the IDA curve declines to 20% of its initial value [22]. In cases where dynamic instability occurs due to nonconverging analyses before this threshold is reached, the spectral acceleration level of the nonconverging run is accepted as the collapse capacity. Seismostruct [23] was used as the structural analysis software for IDA. Seismostruct uses the Newmark scheme with automatic time step adjustment for optimum accuracy for the direct integration of the equations of motion during time-history analysis of a three-dimensional model. The Mander model [19,24] was used to describe the stress–strain relationship of both confined and unconfined concrete following the provisions of the 2018 version of the TSDC [19]. For the stress–strain relationship of reinforcement steel, the model proposed by this same code, including strain-hardening, is used. Figure 3 shows the stress–strain relationship of the materials explained above.

Beams and columns are modeled as distributed inelasticity frame finite elements and fiber approach is used to represent the cross-sectional behavior, where each fiber has a uniaxial stress-strain relationship. Figure 4 shows a schematic representation of a sample section that describes the fiber section model.

Each fiber has a uniaxial stress-strain relationship of corresponding material. The considered stress-strain relationship and hysteretic behavior of reinforcement and concrete are those provided by the Menegotto and Pinto [25] and Mander model [24], respectively. Figure 5 shows the aforementioned material models.

The analytic model of the 10-story buildings is provided in Figure 6 for both fixed-base and flexible-base structures.

For fixed-base structures, it is not required to define the foundation beneath the structure. However, for structures where interaction is occurring, the cone model requires the shear wave velocity V_s, dilatational wave velocity V_p, mass density ρ, and Poisson’s ratio υ to characterize the soil under the structure. The soil is replaced with a spring-dashpot-mass model, as schematically shown in Figure 7 and within circles in Figure 6. Equations (1)–(4) formulate the stiffness of each spring (K_h and K_θ), and the damping coefficients (C_h and C_θ) used for sway and rocking motions were based on the truncated cone model of Meek and Wolf [26], where A₀ and I₀ are the area and moment of the inertia of foundation, respectively. The main reasons for using the cone model in this study are its simplicity and ability to take damping into account.

$$K_h=K_v=\rho V_s^2{\displaystyle \frac{A_0}{z_0}}$$

(1)

$$K_{\theta }={\displaystyle \frac{3\rho V_s^2{I}_0}{z_0}}$$

(2)

$$C_h=C_v=\rho V_s{A}_0$$

(3)

$$C_{\theta }=\rho V_s{I}_0$$

(4)

Table 5. Mean values of parameters used for cone model.

Site Class	Mean Shear Velocity (m/s)	Soil Density (kN/m3)	Poisson Ratio
B	400	21	0.3
C	250	20	0.4
D	150	18	0.5

shows the parameters used for incremental dynamic analyses [27].

As mentioned before, this study aims to investigate to what extent soil–structure interaction changes the damage probability of structures where soil–structure interaction is not taken into account during the design phase. To evaluate the impact of neglected soil–structure interaction on damage probability, the inelastic behavior of the soil is not considered in this study. The Turkish Seismic Design Code 2007 [18] requires the application of the capacity design approach for determining the required shear reinforcement of structural elements, as previously mentioned. According to this principle, the design shear force is derived from the maximum probable moment capacities at the ends of structural elements, ensuring that these elements attain their ultimate strength through bending failure. As a result, shear hinges are not included in the analysis. Additionally, the code limits the dimensionless axial load level to 0.4 f_ck during the design phase, making axial failure negligible. Consequently, local failures, such as shear or axial failure of structural elements, are excluded from the scope of this study. All the median IDA curves are shown in Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15 and Figure 16.

2.5. Fragility Curves

Fragility curves represent the likelihood of structural damage caused by earthquakes as a function of ground motion parameters. In this study, fragility curves were developed based on elastic spectral acceleration and are described using two-parameter lognormal distribution functions. The probability of reaching or exceeding a specific limit state (LS) at a given earthquake intensity is defined as follows:

$$P\left(LS\right)=P\left(d_{LS}\le d_{max}\right)=1-\mathrm{\Phi }\left(r\right)$$

(5)

d_LS and d_max denote the limit state capacity and the maximum demand, respectively. Assuming a lognormal distribution, the standard normal variable is provided by

$$r={\displaystyle \frac{\ln d_{LS}-{\lambda }_D}{\sqrt{{\xi }_{LS}^2+{\xi }_D^2}}}$$

(6)

Here, λ_D represents the mean value assuming a lognormal distribution and is defined in terms of the mean maximum response (d_max) and its dispersion (ζ_D).

$${\lambda }_D=ln{\overline{d}}_{max}-{\displaystyle \frac{{\xi }_D^2}{2}}$$

(7)

The parameter ξ_LS in Equation (6) corresponds to the lognormal standard deviation of a limit state, derived from the results of incremental dynamic analysis (IDA). The dispersion of maximum demand ξ_D incorporates uncertainties related to demand estimation and is calculated as

$${\zeta }_D=\sqrt{ln\left(1+{\left({\displaystyle \frac{{\sigma }_r}{{\overline{d}}_{max}}}\right)}^2\right)+ln\left(1+{\left({\displaystyle \frac{{\sigma }_c}{{\overline{d}}_{max}}}\right)}^2\right)+ln\left(1+{\left({\displaystyle \frac{{\sigma }_D}{{\overline{d}}_{max}}}\right)}^2\right)}$$

(8)

where σ_r and σ_c represent the standard deviations associated with variability in earthquake records and material properties, respectively. The effect of record randomness is accounted for using 21 ground motion records. The term σ_D denotes the standard deviation of the structural response. Figure 17, Figure 18, Figure 19, Figure 20, Figure 21, Figure 22, Figure 23, Figure 24 and Figure 25 illustrate the fragility curves generated for all the considered buildings.

3. Results

Figure 26, Figure 27 and Figure 28 show the interstory drift ratio and corresponding pseudo-spectral acceleration levels obtained through incremental analysis for each soil type and building at the considered performance levels. The left-hand side of the figures shows interstory drift ratios, while the right-hand side shows corresponding pseudo-spectral acceleration levels in terms of g at considered performance levels.

Table 6. Interstory drift ratio limits of each considered performance level at a 5% probability of exceedance with a 90% confidence level.

Number of Stories	Performance Level	Soil Type
		B	C	D	B	C	D
		Fixed Base			SSI
5	IO	0.0032	0.0052	0.0048	0.0039	0.0037	0.0035
	LS	0.012	0.015	0.014	0.008	0.01	0.0097
	CP	0.018	0.02	0.018	0.01	0.015	0.012
10	IO	0.0051	0.005	0.0049	0.0037	0.0038	0.0039
	LS	0.014	0.013	0.014	0.01	0.018	0.012
	CP	0.016	0.015	0.018	0.015	0.02	0.017
15	IO	0.002	0.0035	0.004	0.001	0.0032	0.0034
	LS	0.011	0.012	0.015	0.009	0.01	0.014
	CP	0.013	0.015	0.017	0.011	0.016	0.018

shows the interstory drift ratio limits of each considered performance level at a 5% probability of exceedance with a confidence level of 90%. If the interstory drift ratio can be maintained below those values, only 5% of sample structures will reach the corresponding performance limit. The minimum interstory drift ratio values are 0.2%, 1.1%, and 1.3% for the minimum damage, safety limit, and collapse performance levels, respectively. The same limits are 0.1%, 0.8%, and 1% for flexible-base structures.

Since the investigated buildings are assumed to be residential buildings, they are expected to satisfy the performance level requirements for immediate occupancy under the effect of an earthquake with a 50% probability of being exceeded (minimum earthquake), with life safety for the earthquake with a 10% probability of being exceeded (design earthquake), and with collapse prevention under the effect of an earthquake with a 2% probability of being exceeded (maximum earthquake).

Table 7. Damage probability of each considered building for different performance levels.

Number of Stories	Performance Level	Soil Type
		B	C	D	B	C	D
		Fixed Base			SSI
5	IO	0.86	0.95	0.98	0.99	1.00	1.00
	LS	0.47	0.72	0.81	0.72	0.90	0.91
	CP	0.36	0.63	0.72	0.62	0.88	0.87
10	IO	0.99	0.98	0.99	1.00	1.00	1.00
	LS	0.32	0.67	0.72	0.54	0.77	0.91
	CP	0.25	0.58	0.63	0.41	0.71	0.82
15	IO	0.97	1.00	1.00	1.00	1.00	1.00
	LS	0.73	0.73	0.83	1.00	0.80	0.93
	CP	0.68	0.62	0.72	0.98	0.70	0.77

shows the probability of exceeding the performance levels of the considered buildings within the scope of this study for the design earthquake, with a 10% probability of being exceeded.

4. Discussion

As seen in Figure 26, Figure 27 and Figure 28, there is a difference between both the interstory drift ratio limits and the corresponding spectral acceleration levels of the examined fixed-base and flexible-base structures. However, the difference observed in spectral acceleration levels is significant. The decrease in the stiffness of flexible-based structures results in a higher interstory drift ratio at a constant spectral acceleration level, as expected. Thus, the spectral acceleration level of a fixed-base structure at different damage levels is almost always higher than that of a flexible-based structure, except for in the case of a 15-story building at an immediate occupancy performance level. The difference between the spectral acceleration levels of the examined buildings at different performance levels decreases with an increasing number of stories. As shown in Table 7, the SSI generally causes an increase in the damage probability.

5. Conclusions

The difference between the probabilities of exceeding a certain performance level limit in the two considered cases generally increases with an increasing damage level. The site class does not exert a significant effect on the damage probability at the immediate occupancy performance level; however, the effect of site class becomes significant for the life safety and collapse prevention performance levels. The spectral acceleration level of a fixed-base structure at different damage levels is almost always higher than that of a flexible-base structure, except in the case of a 15-story building at an immediate occupancy performance level. The difference between the spectral acceleration level of the examined buildings at different performance levels decreases with an increasing number of stories. As shown in Table 7, the SSI generally increases the damage probability. The increases in the probability of exceeding a certain performance level are provided below for each considered building.

For the five-story building, these increases are as follows:

• At the immediate occupancy performance level for Site Class B, the probability of damage for the five-story building increases by a factor of 1.15 (from 86% to 99%) when SSI is taken into account. This ratio is 1.05 (from 95% to 100%) for Site Class C. There is a slight increase in damage probability, equal to 1.02 (from 98% to 100%), for Soil Class D. This slight increase may be misleading, since it is very low and may change under the effect of a different ground motion set and/or building type.
• At the life safety performance level for Site Class B, the probability of damage increases by a factor of 1.53 (from 47% to 72%) when SSI is taken into account. This ratio is 1.25 (from 72% to 90%) for Site Class C and 1.12 (from 81% to 91%) for Soil Class D.
• At the collapse prevention performance level for Site Class B, the probability of damage increases by a factor of 1.72 (from 36% to 62%) when SSI is taken into account. This ratio is 1.40 (from 63% to 88%) for Site Class C and 1.21 (from 72% to 87%) for Site Class D.

For the 10-story building, these increases are as follows:

• At the immediate occupancy performance level, there is a slight difference between the two cases regardless of site class. However, this difference may be misleading, since it is very low and may change under the effect of a different ground motion set and/or building type.
• At the life safety performance level, the probability of damage increases by a factor of 1.69 (from 32% to 54%) for Site Class B, 1.25 (from 67% to 77%) for Site Class C, and 1.12 (from 72% to 91%) for Site Class D when SSI is taken into account.
• When SSI is taken into account, the probability of damage at the collapse prevention performance level for Site Class B increases by a factor of 1.64 (from 25% to 41%). This ratio is 1.22 (from 58% to 71%) for Site Class C and 1.30 (from 63% to 82%) for Site Class D.

For the 15-story building, these increases are as follows:

• At the immediate occupancy performance level for Site Class B, the probability of damage slightly increases, by a factor of 1.03 (from 97% to 100%), when SSI is taken into account. There no is no difference between the probability of the two cases for Soil Classes C and D.
• At the life safety performance level, this ratio is 1.37 (from 73% to 100%) for Site Class B and 1.10 (from 73% to 80%) for Site Class C. However, there is no difference between the probabilities of the two cases for Soil Class D.
• When SSI is taken into account, the probability of damage at the collapse prevention performance level for Site Class B increases by a factor of 1.44 (from 68% to 98%). This ratio is 1.13 (from 62% to 70%) for Site Class C and 1.07 (from 72% to 77%) for Site Class D.

These results show that buildings designed in the past without considering soil–structure interaction have an additional damage risk that cannot be estimated during the risk evaluation without taking into account soil–structure interaction. To carry out damage estimation for these types of buildings, the effect of soil–structure interaction on the probability of damage can be considered, at least in an indirect manner, by reducing the interstory drift ratio limits corresponding to the performance levels. Furthermore, soil has an elastic limit; however, the inelastic behavior of soil was not considered within the scope of this study. The effect of the inelastic behavior of soil on the results can also be evaluated within the scope of future studies.