Abstract
Because of the limitations of testing facilities and techniques, the seismic performance of soil-structure interaction (SSI) system can only be tested in a quite small scale model in laboratory. Especially for long-span bridge, a smaller tested model is required when SSI phenomenon is considered in the physical test. The scale effect resulting from the small scale model is always coupled with the dynamic performance, so that the seismic performance of bridge considering SSI effect cannot be uncovered accurately by the traditional testing method. This paper presented the implementation of real-time dynamic substructuring (RTDS), involving the combined use of shake table array and computational engines for the seismic simulation of SSI. In RTDS system, the bridge with soil-foundation system is divided into physical and numerical substructures, in which the bridge is seen as physical substructures and the remaining part is seen as numerical substructures. The interface response between the physical and numerical substructures is imposed by shake table and resulting reaction force is fed back to the computational engine. The unique aspect of the method is to simulate the SSI systems subjected to multisupport excitation in terms of a larger physical model. The substructuring strategy and the control performance associated with the real-time substructuring testing for SSI were performed. And the influence of SSI on a long-span bridge was tested by this novel testing method.
1. Introduction
In the seismic analysis of a structure founded on ground, the ground motion passes to the base of structure and then loads on structure. The response of the foundation system affects the response of the structure and vice versa, which is called dynamical soil-structure interaction (SSI). Theoretical results [1, 2] indicate that SSI is sometimes beneficial and sometimes detrimental to structural performance. Therefore, the effect of soil cannot be neglected, because SSI phenomenon is closely related to its dynamic characteristics [3], especially the damping [4] of the whole system. To explore the effect of SSI on the seismic performance of engineering structures, the finite element methods [5] and theoretical analysis [6] are often used. However, the uncertainties [7, 8] and boundary condition [9] existing in SSI system have not been simulated properly by these methods. At present, experimental evidence relating to SSI system is scarce. The SSI system is difficult to be investigated by testing full-size specimen (including both structure and soil-foundation systems) under earthquake loading because of the size and power of the testing facilities, especially, for the structures with large spatial extent. In current testing of SSI, the soil-foundation systems are replaced by laminar shear box with soil [10–12], and the structures are usually scaled down to small size (often scaled to 1/30 or even smaller) or a simple cantilevered mass [10, 11], which results in the inevitable possibility of size effects. In addition, time must be scaled, accompanying with frequency scaled up, in test to provide rapid shaking, often prevented by the frequency bandwidth of shaking table. SSI plays a more important role in the seismic performance of long-span bridge [13]. However, the smaller specimen is required in the laboratory experiment because of its dimension. As a result, simplified models are tested at reduced scales making the interpretation and application of results challenging.
The method of real-time substructuring derived from hardware-in-the-loop test [14] has been paid close attention since the first real-time substructuring test was reported by Nakashima et al. [15], which is a method of dynamically testing a structure without experimentally testing a physical model of the entire system; instead the structure can be split into two coupled parts, the region of particular interest, which is tested experimentally, and the remainder which is tested numerically. It also received interest in SSI systems. To simulate the soil medium numerically in a shaking table substructuring test where the superstructure is tested experimentally, Konagai et al. [16, 17] used analogue electric circuits to generate the transient response of both linear and nonlinear soils. Heath et al. [18] and Wang et al. [19] investigated an inverted substructuring system by modelling the superstructure numerically. However, only very simple soil model and superstructures were adopted in these works. To simulate the SSI phenomenon with bridge experimentally using RTDS, further validation is required involving more complex structures and more complex interactions.
In the research presented in this paper, a framework of substructuring for SSI system with bridge was established firstly; after that the control strategies for the real-time dynamic substructuring testing were analysed. Finally, the influence of SSI on a long-span bridge was tested through this novel testing method.
2. Modelling of RTDS Systems for SSI Simulation
2.1. RTDS Scheme Based on Shaking Table Array
The development of advanced computation and control techniques resulted in the great progress of real-time substructure [20, 21]. The basis of such tests is to divide the overall system into experimental test specimen and the numerical model. Typically the experimental test specimen contains the area of particular interest which behaves nonlinearly and uncertainly; the numerical model contains the remainder of the system which behaves linearly or can be modelled adequately. These two parts are tested in parallel and in real-time with information relating to the interface between the two parts being exchanged between them, allowing the two parts to emulate the whole system. Figure 1 shows the generally schematic representation of substructuring used to emulate dynamics of structures in earthquake engineering. Interface responses are passed from the numerical model to shaking table, which reproduces accurately the responses at the interface of the test specimen. The interface reaction force measured from test specimen then feed back to the numerical model.
Importantly, the most significant advantage of real-time substructuring is that the process is implemented as a time-stepping routine in real-time, which allows that not only can the nonlinear and uncertain behaviour of physical substructure be accurately tested, but also the time-dependent nonlinearity of numerical substructure can be adopted in the testing. The test method mentioned above supplies a novel solution for the SSI testing. As known, compared with soil-foundation system, the behaviour of structure is relatively easily modelled, especially, the nonlinearity including gapping between the foundations and soil and nonlinear stress-strain response of the soil, and so on. However, the real interest in SSI is the response of structure under earthquake loading; meanwhile, the complexity of structural form and material, geometrical, and contacting nonlinearity of structure are also not well described theoretically. Therefore, both the two kinds of substructuring (soil-foundation system or structure modelled numerically) for SSI need to be developed. In this work, the soil-foundation system is treated as numerical substructure and simulated in computer, while the bridge seems as the physical substructure and tested in the laboratory. A diagrammatic representation of RTDS for a bridge is shown in Figure 2. The emulated SSI system shown in Figure 2(a) can be tested by the RTDS strategy shown in Figure 2(b). Unlike other building structures, a single shaking table is enough to carry out the test; the length of bridge is quite large, even the scale model is often more than ten meters long. To overcome this drawback, the shaking table array composed of multishaking tables was constructed and used to test the seismic performance of bridge [22]. Therefore, the shaking table array is adopted to reproduce the interface response of each bridge pier and foundation.
(a) Whole soil-structure system under multisupport excitation
(b) Substructured system:soil-foundation system represented as numerical substructure
2.2. Numerical Model of Soil-Foundation System
The ground motion imposing on superstructure caused by earthquake includes three portions (as illustrated in Figure 3(a)): the movement of free-field soil resulting from earthquake (), the deviation of the foundation movement resulting from the free-field soil motion (), and the deformation of foundation produced by the reaction force from superstructure (inertia interaction: and ). This work focuses on the interaction between soil and structure; consequently, the foundation movement is identical to the free-field soil movement (). In the seismic analysis for SSI, the substructure method is cost-effective, comparing with the direct integrity approach, in which the behaviour of the soil-foundation system (see in Figure 3(a)) was described as an impedance function (also shown in Figure 3(b)):where is horizontal stiffness, is horizontal damping, is circular frequency, and is imaginary unit. However, the impedance function is frequency-dependent, which cannot be used to represent the nonlinear effect of dynamic interaction between soil and structure.
(a) Deformation of soil-pile-foundation
(b) Time-domain difference model
In order to overcome this disadvantage, two main methods [23–25] were developed to transform the frequency-dependent impedance function to time-domain: lumped parameter model approximately fitting to impedance functions using different mass-spring-damping systems and fast integral algorithms (also referred to as recursive model) based on the properties of Fourier inverse transform. Nevertheless, the two methods have their own defects: lumped parameter model [23, 24] can only fit the actual impedance functions in narrow frequency band accurately but cannot reflect the regular component, which corresponds to time-delay effect; meanwhile, recursive model [25] has intrinsically limitation at Nyquist frequency; namely, the imaginary part of filter must be zero at Nyquist frequency.
So as to take the full advantage of the two models and make up for their deficiency as well, a time-domain difference model (TDDM) for soil-foundation system was proposed by Du and Zhao [26], combining lumped parameter model and time-domain recursive model, which can take into account both the singular and regular component of foundation impedance comprehensively. The mechanical model is shown in Figure 3(b), wherewith being numbers of piles, static stiffness of single pile, normalized stiffness, normalized damping, pseudo-force, displacement of foundation, integration time step, and the coefficients of pseudo-force and displacement. From (2) we can see that the determination of parameters is the key point for the time-domain difference model, the best values of which are derived using hybrid optimization based on the theory or numerical solution of impedance function.
Herein, we assume that the bridge pier is constructed on a 3 × 3 pile group foundation, and the parameters are as follows: damping ratio of soil is 0.05, Poisson’s ratio of soil is 0.4, ratio of pile spacing and diameter is 5, Young Modulus ratio of pile to soil is 1000, mass density ratio of pile to soil is 1.42, and ratio of pile length and diameter is 15. Normalized by single pile static stiffness, the impedance function of the pile group foundation can be expressed aswhere is dimensionless frequency, is single pile horizontal static stiffness, is shear wave velocity of soil, is pile diameter. Using the dynamic impedance function developed by Makris and Gazetas [27, 28] for the pile group foundation, the parameters of TDDM were estimated by multiple regression analysis. The evaluated values of all parameters for this model are shown in Table 1. The comparative results of the TDDM and Markris’s analytical model presented in Figure 4 show that TDDM is suitable to represent the soil-foundation system in RTDS.
3. Control Strategies for RTDS
3.1. RTDS Controller Design
In commonly used RTDS system, displacement or force was used to be target signal to drive transfer system; however in some SSI systems, such as sandy soil, the effect of SSI on the acceleration of structure is noticeable, but the effect on displacement is inconspicuous. In that case, the acceleration of interface cannot be reproduced well by using displacement control. In this work, the acceleration driving was attempted to be used in the RTDS for SSI of this work. In this RTDS system, shaking table array was used as the transfer system to produce the interface response of foundation, resulting from the dynamics of which, the response between physical and numerical substructure is always asynchronous, even causing the instability, especially for the low damping system [29]. In standard earthquake testing, the shaking table is driven by a conventional controller, such as PID or three-variable controller. These controllers cannot give a perfect control. Phase lag and magnitude error always exist. In order to impose the interface response calculated from numerical substructure on the physical substructure through shaking table synchronously, an additional controller is necessary to cancel the shaking table dynamics.
Based on TDDM presented in Section 2.2, the emulated system shown in Figure 5(a) can be substructured as the RTDS system shown in Figure 5(b). TDDM model represents the soil-foundation system. The bridge is installed on shaking table array, which is used as the transfer system to simulate the interface acceleration (). As can be seen in Figure 5(b), the interface acceleration resulted from external excitation () and reaction force of superstructure ():From Figure 5(b), the transfer function models for the different parts can be determined.where is integration time step. Normally, the achieved value of interface acceleration is not but after going through shaking table because of its dynamics. That leads to the needs of RTDS controller. As reference [30] mentioned, the shaking table can be approximated to a first-order system such thatwhere is the product of the time constant and the proportional gain of shaking table. And then the reaction force is formulated as represents the transfer function of the total shear force () of superstructure from excitation (), which is determined by the dynamic parameters. From Figure 5(b), can be expressed as where Mp, Cp, and Kp are the mass, damping, and stiffness of bridge, respectively.
(a) Emulated system
(b) Substructured system
To formulate the synthesis procedure of RTDS dynamics and control, a framework of linear substructuring controller (LSC) [31] together with the minimal control synthesis with error feedback (MCSEF) algorithm [32] was proposed. To simplify the LSC for multidegree of freedom system, a modified linear substructuring controller (MLSC) together with MCSEF was developed by Guo et al. [33]. As illustrated in Figure 6, an emulated system is conceptually decomposed into at least two substructures, and . For practical cases represents the numerical model and represents the physical model. Control and excitation signals are denoted by and , respectively. Here, the RTDS dynamics are represented by three generalized blocks, , constraint; , excitation; and , transfer system dynamics. is the relationship between the displacements of the top of the foundation () and the ground motion is the relationship between the control signals () and the acceleration of the top of the foundation () from the reaction force, including the TS dynamics , the reaction force dynamics , and the numerical substructure dynamics is the TS dynamics; thusWith reference to [31], the error dynamics () of an LSC-controlled DSS can be written asTo synchronize the two outputs , the substructuring error is expected to equal zero, the numerator of (10) is set equal to zero, and thenThe control signal and adaptive gains of MCSEF in Figure 6 are generated from (12), where are adaptive weights; the ratio , which has been shown to work well empirically [29], is used here. The term is the output error, generated directly from , according to , where is selected to ensure a strictly positive real dynamic in the hyperstability proof for the MCSEF controller [29]; for first-order control, is normally defined as [29]: , where is the step response time of the implicit reference model.
3.2. Validation of RTDS Controller Performance
For verifying the validity of the RTDS for SSI simulation, further tests of a one-story steel frame including SSI considerations were performed. The parameters for the soil model are shown in Table 1, and = 62600 N/m. The parameters of the physical substructure are as follows: = 115 kg, = 17.67 N/ms, and = 7.56 × 104 N/m. The shaking table used here is a shake table array including nine shaking tables (shown in Figure 7) from Beijing University of Technology (BJUT), and the specifications are shown in Table 2. The estimated parameter of the shaking table used in MLSC design was = 16 Hz. The acceleration record of El Centro earthquake (NS component of the 1940 El Centro, Station: 117 El Centro Array #9, Component 270°) was chosen as the ground motion. Experiment and simulation results are shown in Figure 8, which demonstrate that the RTDS method emulates dynamic soil-structure interaction effectively. That means the MLSC-MCSEF controller compensated the dynamics of shaking table successfully.
4. The Effect of SSI on Seismic Performance of a Long-Span Bridge
4.1. Test Design
The dynamic response of a large-span bridge with the considerations of SSI subjected to earthquake excitation was tested using RTDS method developed in this work. In this RTDS test, the specimen (see in Figure 9) was taken as physical substructure. The parameters of this specimen are mass 4500 kg, length 13.84 m, and height 1.2 m. The time-domain difference model described in Section 2 was chosen as numerical substructure. The parameters of soil model are listed in Table 1. And the Young modulus of pile is 113.4 Gpa, the density of soil is 2000 kg/m3, and diameter of pile is 0.4 m. The photo of the specimen is shown in Figure 10. Herein, four shaking tables are used to produce the interface response of each foundation under piper. The acceleration record of El Centro (See in Figure 11(a)) and Wenchuan earthquake (See in Figure 11(b)) was chosen as the ground motion.
(a) El Centro
(b) Wenchuan
Before conducting RTDS test, a conventional shaking table test with white noise excitation was carried on to identify the dynamic parameters of the physical substructure. The Fourier spectrums of the acceleration measured at pier2 were shown in Figure 12. As can be seen, the frequencies of the first and second mode were 4.76 Hz and 9.58 Hz, and the first mode is the dominated mode. For the design of MLSC controller in this test, the physical substructure still can be treated as a single degree-of-freedom system. The damping ratio of the first mode obtained from Figure 12 is 2.6%.
4.2. Effect of SSI on the Response of the Long-Span Bridge
To evaluate the effect of SSI on the dynamic response of long-span bridge, the soft soil-foundation system with the shear wave velocity = 100 m/s, 200 m/s, and 400 m/s was tested using the MLSC-MCSEF controlled RTDS. For comparison, the hard soil-foundation system was also tested using the conventional shaking table test without the consideration of SSI.
In the experiment, the MLSC-MCSEF strategy was adopted. Figure 13 displayed the desired and achieved acceleration of shaking table in the form of synchronization subplots. It was seen that MLSC-MCSEF supplied an acceptable accuracy for RTDS-SSI test.
The tested results of acceleration and strain response excited by Wenchuan earthquake were presented in Figures 14(a) and 14(b), respectively. As can be seen from Figure 14, the SSI effect enlarged the dynamic response of bridge when the shear wave velocity of soil is 400 m/s and 200 m/s. The amplification ratios of acceleration for = 400 m/s and 200 m/s were around 45% and 30%, respectively, while the corresponding value of strain was around 106% and 86%. However, when the shear wave velocity of soil is too low (e.g., = 100 m/s), the SSI effect reduced the dynamic response of bridge in this case. The acceleration and strain were both reduced to 50% of hard soil case ( = infinity). The clear evidence can be obtained from the acceleration measured on the shaking table shown in Figure 15. Compared with the hard soil case ( = infinity), the excitation imposed on the bridge was changed significantly by soil. Generally, the high frequency components of the seismic wave were filtered by soil, while the low frequency components were amplified. The can be seen clearly from the time history (see Figure 15(a)) and Fourier spectrum (see Figure 15(b)). In the hard soil case, the seismic wave used in this test has abundant components from 4 Hz to 16 Hz. After considering SSI, the components of the frequency more than 10 Hz were lessened, and the components of the frequency less than 8 Hz were enhanced in the cases of = 400 m/s and 200 m/s. However, the components of the low and high frequency were both filtered in the case of = 100 m/s. In this case, the soil is too soft and equivalent to an isolator. When = 400 m/s and 200 m/s, the amplified components were close to the frequency of the first mode of the bridge; hence, the acceleration and the strain of the bridge were enlarged.
(a) Acceleration at the top of pier2
(b) Strain at the bottom of pier2
(a) Time history
(b) Fourier spectrum
Table 3 summarized the maximum strain at the bottom of Pier2 and Pier3 measured from El Centro and Wenchuan earthquake. It can be seen that Pier2 and Pier3 gave the similar conclusion. Whereas almost the same reduction ratio was measured from El Centro excitation when = 100 m/s, the amplification ratio is smaller than Wenchuan excitation when = 200 m/s and 400 m/s. This resulted from the frequency characteristics of the two excitations. The Flourier spectrum plotted in Figure 16 demonstrated that El Centro excitation (see Figure 16(a)) has different frequency components with Wenchuan excitation (see Figure 16(b)). It showed that the characteristics of excitation determined the response of bridge together with the soil properties in SSI system.
(a) El Centro
(b) Wenchuan
5. Conclusions
The purpose of this work is to establish a potential testing method for the simulation of soil-structure interaction based on real-time dynamic substructuring (RTDS), testing bridge experimentally and modelling the remainder part numerically, which make the long-span bridge testing with consideration of SSI possible. The “size effect” resulting from the scaled model in whole system tests because of the size and power of the testing facilities is weakened; meanwhile the nonlinear dynamics of the SSI system can be investigated using this method. The feasibility of this method has been verified experimentally.
The influence of SSI on a long-span bridge was tested through RTDS method. The SSI effect enlarged the dynamic response of bridge when the shear wave velocity of soil is 400 m/s and 200 m/s, while the SSI effect reduced the dynamic response of bridge when the shear wave velocity of soil is too low (e.g., = 100 m/s). In other words, the medium soft soil-foundation may be detrimental to the seismic performance of bridge, and the soft soil may be beneficial to the seismic performance based on the test results of this work.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors gratefully acknowledge the support of the National Science Foundation of China, Grant no. 51608016, and the support of the Beijing Natural Science Foundation (8164050) in the pursuance of this work.