Parametric Analysis on the Effect of Dynamic Interaction between Nonlinear Soil and Reinforced Concrete Frame

Abstract

The effect of dynamic soil–structure interaction on the seismic demand of a reinforced concrete frame is of great significance to seismic design, retrofit, and damage evaluation. To investigate the degree of influence of the consideration of the soil–structure interaction on the structural seismic response, an efficient numerical model considering the nonlinearities of both a reinforced concrete frame and soil was developed and validated against a shaking table test. Subsequently, detailed parametric analyses on the dynamic soil–structure interaction effect were conducted, where the influences of the length and diameter of the pile, span number and frequency of the structure, soil property, and natural uncertainty of the seismic record were investigated. The research results indicate that the base shear of the pile-supported reinforced concrete frame generally increases with a larger pile length and pile diameter. The influence of the span number and pile diameter on the soil–structure interaction effect is up to 40% in some cases while that of the pile length is within 10% in general. Consideration of the soil–structure interaction can also considerably increase the structural base shear in certain cases and the growth can be greater than 30%. The dynamic soil–structure interaction effect predominantly depends on the structure frequency, spectral characteristic and peak acceleration of the seismic record, and soil shear wave velocity while the influence of the pile diameter and number of spans cannot be neglected in some cases.

1. Introduction

Dynamic interaction between the soil and structure (SSI) [1] is a complex issue due to the fact that its influence factors such as the structure frequency, soil property, and seismic record, etc. are numerous and mutually coupled. Compared with a fixed-base structure, SSI can significantly alter the seismic demands of a structure built on soft soil, which has been widely investigated by scholars. Since SSI can lengthen the natural period and increase the damping of the structure, previous studies [2] generally agreed that the seismic demands of structures considering SSI are smaller than those of fixed-base structures. Therefore, many codes [3,4,5,6] adopt the assumption of a fixed base, where MHURD-PRC, NBCC, and IBC stipulate that the design base shears of structures can be reduced when considering SSI. Moreover, in the earthquake damage evaluation of cities or regions, the SSI effect on the responses of structures has usually been neglected [7,8] to improve the computational efficiency. However, as research continues, increasingly more scholars [9,10] have noticed that the horizontal movement and rotation of the foundation may increase the lateral deformation and story drift of the superstructure, and SSI can provide pejorative conditions for structural seismic vulnerability [11,12,13,14]. Therefore, further investigations on the SSI effect can provide technical support for improving the conventional seismic design method and urban earthquake damage evaluation method, which has significant application value.

Numerical methods considering the SSI effect can be divided into two groups: the direct method [15] and substructure method [16,17]. In the direct method, the soil, structure, and foundation are modeled together to fully consider the coupling effects among the soil, foundation, and superstructure. Since the direct method can consider the complicated model shape and load condition and the nonlinearity of the soil, foundation, superstructure, and soil–foundation contact, it has the advantage of wide application while it has the disadvantage of a long computation time. In the substructure method, the soil, foundation, and superstructures are solved separately, and the coupling among the soil, foundation, and superstructures is considered using the compatibility conditions of the force and displacement on the interface. The substructure method has a high computation efficiency while the mathematical definition of the simplified soil model is complex and the nonlinear coupling effect of SSI systems is difficult to consider [18].

Arboleda-Monsalve et al. [19] evaluated the SSI effect on the seismic performance of hypothetical plane tall buildings and earthquake-induced losses using a direct method, finding that SSI changed the seismic demands such as the maximum story drift and peak story horizontal accelerations, and the addition of the SSI effect to the tall structures triggered larger losses compared with fixed-base structures in normal situations. Galal and Naimi [20] investigated the effects of soil conditions and near-fault earthquakes (NFEs) on the responses of 6-story and 20-story plane RC frame structures using a substructure method. They found that the SSI effect on the responses of structures subjected to NFE ground motions was more pronounced in soft and medium soil properties and the roof displacements and inter-story drifts of structures considering SSI exceeded those of fixed-base structures under certain near-fault records and soil conditions. Zhang and Far [21] studied the seismic responses of 20-, 30-, and 40-story plane high-rise frame-core tube structures constructed on soil with class E_e (based on Australian Standards) under different seismic records. They pointed out that SSI can increase the lateral deflections and story drift and decrease the story shear forces of structures, and the seismic responses of SSI systems under near- and far-field earthquakes were considerably different.

Based on the existing research outcomes, it can be found that comprehensive parametric analyses of the SSI effect on structural seismic demands considering the nonlinearities of structure and soil are rarely reported. To provide a reference for fast regional seismic damage evaluation considering SSI, based on a nonlinear numerical model to account for the SSI effect, detailed parametric analysis considering the influence of the pile length, pile diameter, number of spans, frequency of the structure, seismic records, and soil shear wave velocity on SSI effects were conducted.

2. Numerical Method and Its Validation

2.1. Numerical Model

A two-dimensional finite element model of a soil–structure system was established based on the platform OpenSees, as shown in Figure 1. The soil is simulated by four node quad element, and the beam, column, pile, and pile cap are simulated by dispBeamColumn elements. The dispBeamColumn element is based on the displacement formulation and considers the spread of plasticity along the element, and the default integration along the element is based on the Gauss–Legendre quadrature rule [22]. The soil constitutive model adopts the PressureIndependMultiYield model [23], which is an elastic-plastic material and is suitable for simulating the nonlinear behavior of clay under fast loading conditions. The constitutive models of concrete and reinforcing bar for reinforced concrete (RC) frames are uniaxial materials Concrete02 [24] and Steel02 [25], respectively. The stress–strain relationship of Concrete02 is shown in Figure 2a [26], where E₀ and E_t denote the initial elastic modulus and tension softening modulus; and σ_c0, σ_cu, and σ_t, respectively, denote the compressive strength, crushing strength, and tensile strength; ε_c0 and ε_cu denote the strain at the maximum strength and the strain at the crushing strength; and d denotes the ratio of the unloading slope at ε_cu to the initial elastic modulus E₀. Figure 2b illustrates the stress–strain relationship of Steel02 material, a uniaxial Giuffre-Menegotto-Pinto steel material considering isotropic strain hardening, where fy, E₀, and b denote the yield strength, initial elastic tangent, and ratio between the post-yield tangent and initial elastic tangent, respectively. Regarding the connection between the nodes of the pile cap and soil, the horizontal translational degrees of freedom (DOFs) of the pile cap node and soil node are coupled to simulate the horizontal restraint of soil on the pile cap; the vertical DOFs of the pile cap node and soil node are connected by employing Zerolength elements, and the uniaxial material elastic-no tension (ENT) is assigned to the Zerolength elements to simulate the opening and closing between the soil and pile cap. Two lateral boundaries of plane soil are set as layered shear boundaries [27], where the translational DOFs of the nodes at the same height on the two lateral soil boundaries are coupled. The nodes of the pile and soil at the same position are fully coupled. The soil bottom is considered bedrock and the seismic excitation is conducted by inputting the time history acceleration of the bedrock to the bottom of the soil.

2.2. Validation of Numerical Model

To verify the reliability of the above numerical method, simulations on the free field test and SSI test in a scaled model experiment [28] were conducted using the introduced numerical model. The scaled model experiment was conducted in the Structural Engineering Laboratory of Southeast University, as shown in Figure 3. In the scaled model experiment, the scale factors of the density, elastic modulus, and length are 1:1.5, 1:6, and 1:20, respectively; the size, density, and shear wave velocity of soil in the hypothetical prototype soil are 58 m × 40 m × 25 m, 1.65 ton/m³, and 120 m/s, respectively, and those of scaled soil are 2.9 m × 2.0 m × 1.25 m, 1.1 ton/m³, and 60 m/s, respectively. The height and period of the hypothetical prototype SDOF structure are 12.1 m and 0.5 s, respectively, and those of the scaled SDOF structure are 6.05 m and 0.05 s, respectively. The specimen of the scaled SDOF structure with pile foundations is shown in Figure 4. The scaled soil with a density, height, and shear wave velocity of 1.1 ton/m³, 1.25 m, and 120 m/s was placed in a laminated shear soil box [29,30,31] with a clear size of 2.9 m × 2.0 m × 1.54 m. The time–history curves of the seismic records used in the free field test and SSI test are shown in Figure 5. The acceleration in the center of the surface soil was measured in the free field test, and the acceleration at the top of structure was measured in the SSI test.

Based on the elevation dimension of the scaled model, the plane finite element models of the free field and SSI system were established and are shown in Figure 6, where the element size was set as 0.05 m because this size can ensure the influence of the element size is negligible [32]. Dynamic time-history analyses were conducted on the free field model and SSI system model using the seismic records in Figure 5. The comparisons of the experiment and simulation results of the acceleration in the center of the surface soil and the acceleration at the top of the superstructure are shown in Figure 7a,b, respectively. It is found that the experiment results are in good agreement with the simulation results in the time domain, indicating that the above numerical method is reliable and can be applied to the subsequent numerical study.

3. Influence of the Structure Frequency, Soil Shear Wave Velocity, and Spectral Characteristic of the Seismic Record

3.1. Definition of Evaluation Metrics

The maximum story drift and maximum base shear are two important seismic demands of structures. To evaluate the influence of SSI on the maximum story drift and maximum base shear of structures, the SSI influence coefficients of the story drift and base shear are defined as e_sd and e_bs, respectively. e_sd and e_bs can be calculated by Equations (1) and (2), where the maximum story drift and base shear of the structure considering SSI are labeled as S_sd and S_bs, and those of the fixed-base structure are labeled as D_sd and D_bs, respectively. In this study, the responses of the fixed-base structures were obtained by inputting the ground motions recorded at the surface of the soil to the bases of the structures. The responses of the structures considering SSI were obtained by inputting the bedrock motions to the bottom of the soil, and the bedrock motions were obtained by conducting wave propagation analyses on the ground motions recorded at the soil surface using the program SHAKE [33]:

$$e_{sd}=\frac{S_{sd}-D_{sd}^{}}{D_{sd}^{}}$$

(1)

$$e_{bs}=\frac{S_{bs}-D_{bs}^{}}{D_{bs}^{}}$$

(2)

3.2. Parameter Setup

Considering the soil shear wave velocities (V_s) of most building sites are within the range of 150 to 500 m/s, the vs were respectively set as 150, 250, 350, and 500 m/s to roundly investigate the influence of the soil shear wave velocity on the SSI effect. To investigate the influence of the spectral characteristics of seismic records on the SSI effect, three ground motion records with different spectral characteristics were selected from the PEER Ground Motion Database, and the peak accelerations of the three ground motion records were adjusted as 100 gal (cm/s²). The detailed information of the three records is given in

Table 1. Ground motion records.

Records	Earthquake Name	Year	Station Name	Magnitude	Rjb (km)	Component
RSN-445	New Zealand-01	1984	Turangi Telephone Exchange	5.5	3.76	TUR329.AT2
RSN-587	New Zealand-02	1987	Matahina Dam	6.6	16.09	MAT083.AT2
RSN-1050	Northridge-01	1994	Pacoima Dam	6.69	4.92	PAC175.AT2

, where the R_jb (Joyner–Boore distance) [34] is the shortest distance to the surface projection of the fault plane. The comparison of the Fourier spectra of the three records is shown in Figure 8.

The numbers of floors for RC-frames are generally no more than 15 in reality. Therefore, 1-, 3-, 6-, 9-, 12-, and 15-story RC-frames (Frame-1, Frame-3, Frame-6, Frame-9, Frame-12, and Frame-15) were designed to roundly investigate the influence of the structure height, in which the span lengths and story heights of RC-frames were 6.0 and 3.3 m, respectively. The elevations of the fixed-base RC-frames and RC-frames considering SSI are shown in Figure 9 and Figure 10, respectively. Regarding the RC-frames, the reinforcing bar is the HRB400 with a nominal yield strength of 400 MPa, and the concrete is C35 grade concrete with a nominal cubic compressive strength of 35 MPa [35]. The detailed parameters of the RC-frames and their pile foundations are shown in

Table 2. Detailed parameters of RC-frames and pile foundations.

Frames	Story	Beam		Column		Pile Foundation
		Section (m)	Rebar (mm2) (Top × Bottom)	Section (m)	Rebar (mm2) (Each Side)	Diameter (m)	Length (m)	Total Rebar (mm2)
Frame-1	1	0.3 × 0.6	1080 × 940	0.5 × 0.5	833.2	0.20	1.0	251.2
Frame-3	1 to 3	0.3 × 0.6	1080 × 940	0.5 × 0.5	833.2	0.30	3.0	565.2
Frame-6	1 to 6	0.3 × 0.6	1080 × 940	0.6 × 0.6	1200	0.45	6.0	1272
Frame-9	1 to 5	0.3 × 0.6	1080 × 940	0.65 × 0.65	1408	0.55	9.0	1900
	6 to 9	0.3 × 0.6	1080 × 940	0.55 × 0.55	1008
Frame-12	1 to 4	0.3 × 0.6	1080 × 940	0.7 × 0.7	1633.2	0.65	12.0	2653.6
	5 to 8	0.3 × 0.6	1080 × 940	0.6 × 0.6	1200
	9 to 12	0.3 × 0.6	1080 × 940	0.5 × 0.5	833.2
Frame-15	1 to 5	0.3 × 0.6	1080 × 940	0.75 × 0.75	1875	0.75	15.0	3532.8
	6 to 10	0.3 × 0.6	1080 × 940	0.65 × 0.65	1408
	11 to 15	0.3 × 0.6	1080 × 940	0.55 × 0.55	1008

. Conducting modal analyses on fixed-base RC-frames, it can be obtained that the periods of Frame-1, Frame-3, Frame-6, Frame-9, Frame-12, and Frame-15 are 0.127, 0.384, 0.713, 1.091, 1.493, and 1.876 s, respectively. Regarding the soils in the soil–structure systems, the length of soil parallel to the seismic excitation direction should be greater than five times the width of the structural foundation to ensure the influence of the artificial boundary of the soil is slight [36]; the size of the soil element should not exceed (1/5~1/8)V_s/f_max to prevent the high frequency components of seismic records from being filtered out, where vs. denotes the shear wave velocity of soil and f_max denotes the maximum frequency of the seismic record. Accordingly, the length of soil parallel to the seismic excitation direction was determined as 60 m, and the length of soil element was set as 1.0 m. In addition, the thicknesses of soils in the soil–structure systems were set as 20 m based on the recommendation of the Chinese code [3].

3.3. Result Analysis

Based on the responses of fixed-base structures and structures considering SSI, the e_bs and e_sd of RC-frames with different frequencies under seismic records with different spectral characteristics are shown in Figure 11 and Figure 12, respectively. It can be seen from Figure 11 that in general, the absolute values of e_bs for most structures decrease with the increase in the soil shear wave velocity under three seismic records, and the absolute values of e_bs for most structures are more than 40% when the vs. is 150 m/s and reduced to below 20% when the vs. is 500 m/s. The above description indicates that in general, the effect of SSI on the maximum base shear of the structure decreases with a larger shear wave velocity. Moreover, the values of e_bs for most structures are negative, and the e_bs of the 3-story RC-frame is about −55% when the vs. is 150 m/s and the seismic record is RSN-587, indicating that considering SSI can reduce the maximum base shear of the structure in most cases. Conversely, Figure 11a shows that the values of e_bs in some cases are positive and the e_bs of the 3-story RC-frame is more than 30% when the vs. is 500 m/s. A similar phenomenon can also be seen in Figure 11c, indicating that considering SSI can also increase the maximum base shear of the structure in specific cases, especially for low-rise structures built on tough soil. The reason for the above phenomenon is that in certain cases, the resonance effect occurs when the dominant frequency of the ground motion recorded at the surface of tough soil approaches the frequency of the low-rise structure.

Figure 12 shows that the curves of e_sd have a significant difference compared with the curves of e_bs shown in Figure 11 in some cases, indicating that the influence of SSI on the maximum base shear does not always have a positive correlation with that on the maximum story drift. The maximum value of e_sd of the 1-story RC-frame is up to 95%, as shown in Figure 12c, and a similar phenomenon can also be seen in Figure 12a,b, indicating that considering SSI can significantly increase the maximum story drift of low-rise structures built on tough soil. Figure 11 and Figure 12 show that the influence of SSI on the structural maximum story drift and base shear is closely related to the structure frequency, spectral characteristic of the seismic record, and soil shear wave velocity.

4. Influence of Pile Length and Pile Diameter

4.1. Pile Length

Considering the ratios of the pile length to the structure height for most pile-supported RC frames are between 0.1 and 0.5 in reality, the ratios of the pile length to the structure height of the 3-story, 6-story, and 9-story RC-frames (F3, F6, and F9) were set as 0.1, 0.2, 0.3, 0.4, and 0.5 to investigate the influence of the pile length on the SSI effect. To sufficiently capture the influence of the pile length, the seismic records RSN-587 and RSN-445 with a peak acceleration of 100 gal were employed as seismic inputs, and the values of vs. were, respectively, set as 150 and 250 m/s. It can be seen from Figure 13 that with the increase in the pile length, the values of e_bs generally increase with the increase in the pile length while the values of e_sd may increase or decrease. Moreover, the maximum influence of the pile length on e_bs and e_sd is only about 10%. The above finding indicates that in general, the maximum base shear of the structure considering SSI increases with a larger pile length while the influence of the pile length on the structural base shear and story drift is within 10%.

4.2. Pile Diameter

To investigate the influence of the pile diameter on the SSI effect, the pile diameters of the 3-story, 9-story, and 15-story RC-Frames considering SSI were, respectively, set as 0.25, 0.45, 0.65, 0.85, and 1.05 m for comparison. In this section, the seismic records RSN-587 and RSN-445 with a peak acceleration of 100 gal were used as seismic inputs, and the values of vs. were, respectively, set as 150 and 250 m/s to fully investigate the influence of the pile diameter. Figure 14a,b show that the maximum and minimum values of e_bs for F9 built on soil with vs. of 250 m/s are −25.0% and −46.0%, and those of e_sd for F9 built on soil with vs. of 150 m/s are 37.5% and −4.8%, respectively. The maximum influences of the pile diameter on e_bs and e_sd are, respectively, up to 21% and 42%, indicating that the influence of the pile diameter on the SSI effect is significant in some cases. Moreover, the influences of the pile diameter on the values of e_bs and e_sd with different structure frequencies, soil shear wave velocities, and seismic records are different, which is related to the soil shear wave velocity, structure frequency, and spectral characteristic of the seismic record. In addition, Figure 14 shows that with the increase in the pile diameter, the values of e_bs and e_sd, respectively, increase and decrease in most cases, indicating that in general, the maximum base shear and story drift of RC frames, respectively, increase and decrease with a larger pile diameter. The reason for the above phenomenon is that in general, a larger pile diameter can transmit a bigger seismic force to the superstructure and produce a more significant inhibitory effect on the rocking of the superstructure.

5. Influence of the Seismic Intensity and Structural Span Number

5.1. Span Number of the RC-Frame

The influence of the span number on the SSI effect has commonly been neglected while investigation of the influence of the span number on the SSI effect is of great significance to accurately evaluating the seismic demands of structures with different spans. To investigate the influence of the span number on the seismic demands of different structures, 3-story, 9-story, and 15-story RC-frames with different spans were designed and are shown in Figure 15, in which the span numbers were, respectively, set as 1, 2, 3, 4, and 5; and the periods of the fixed-base structure with different spans and the same height were the same. The seismic record RSN-587 with a peak acceleration of 100 gal was used as seismic input, and the values of vs. were set as 150 and 250 m/s, respectively.

It can be seen from Figure 16 that the maximum influences of the span number on the values of e_bs and e_sd are, respectively, up to 25% and 40% when vs. is 150 m/s and reduced to 15% and 30% when vs. is 250 m/s. With the increase in the span number, the values of e_bs and e_sd decrease in most cases and increase in some specific cases. The results point out that the influence of the span number on the SSI effect can be significant under the condition of soft soil, which decreases with a lager soil shear velocity in general. The influence of the span number on the maximum base shear and story drift has no obvious laws, which is closely related to the structure frequency, soil shear wave velocity, and spectral characteristic of the seismic record.

5.2. Seismic Intensity

The soil will undergo large nonlinear deformation under strong earthquakes, which will inevitably have an impact on the SSI effect. To investigate the influence of the seismic intensity on SSI, the seismic records RSN-587 with a peak acceleration of 50, 100, 150, 200, and 300 gal were employed to conduct seismic excitation on 3-story, 9-story, and 15-story RC-frames considering SSI and without considering SSI. The soil shear wave velocity was set as 300 m/s. It can be found from Figure 17 that the maximum influences of the peak acceleration on the values of e_bs and e_sd are up to 35% and 60% when vs. is 300 m/s. The values of e_bs and e_sd may significantly increase or decrease with the increase in the peak acceleration. The above description indicates that the influence of peak acceleration on the seismic demands of structures considering SSI is significant and irregular, which cannot be neglected when accurately calculating the influence of SSI on the structural seismic demand.

6. Conclusions

To provide a reliable reference for rapid and accurate regional seismic damage evaluation considering SSI, based on a validated finite element model considering the nonlinearity of both the structure and soil, detailed parametric analyses considering the influence of multiple factors such as the pile length, pile diameter, number of spans, frequency of structure, seismic records, and soil shear wave velocity on SSI effects were conducted. Based on the presented works, some conclusions are provided as follows:

(1)

For pile-supported RC-frames considering SSI, the maximum base shear generally increases with a larger pile length and the growth is within 10%; the maximum base shear and story drift, respectively, increases and decreases with the increase in the pile diameter, and the maximum change is up to 20% and 40%, respectively.

(2)

Both the span number and seismic intensity can considerably influence the seismic demands of pile-supported RC-frames in some cases and their influences on the structural seismic demands have no obvious laws. In general, the influence of SSI on the structural seismic demands decreases with a larger soil shear wave velocity.

(3)

SSI can reduce the maximum base shears of pile-supported RC-frames in most cases while it can also significantly increase the structural maximum base shear and story drift in some specific cases, especially for low-rise structures built on tough soil. The SSI effect on the maximum base shear does not always have a positive correlation with that on the maximum story drift.

(4)

The SSI effect on seismic demands predominantly depends on the structural frequency, spectral characteristic, and peak acceleration of the seismic record and the soil shear wave velocity while the influence of the pile diameter and span number of structure still cannot be neglected in some cases.

(5)

The influence of the seismic soil–structure interaction should be fully considered to reduce casualties and economic losses when designing low-rise structures built on tough soil and high-rise structures built on soft soil. Machine learning is expected to solve the seismic soil–structure interaction issue by establishing and training a data set that sufficiently considers all the influencing factors of soil–structure interaction.