Applications of infinite elements to dynamic soil-structure interaction problems

153

APPLICATIONS OF INFINITE ELEMENTS TO DYNAMIC SOIL-STRUCTURE INTERACTION PROBLEMS Chongbin Zhao (CSIRO Division of Exploration and Mining, Perth, WA 6009, Australia)

Abstract

This paper presents a brief summary on the research, which was mainly related to the applications of infinite elements to dynamic soil-structure interaction problems and carried out by the author and his coworkers during the last decade. Owing to the complex and complicated nature of practical engineering problems, infinite elements often need to be coupled with finite elements to deal with such problems. This has enabled the combination of infinite elements and finite elements to be used to successfully solve many practical problems in various engineering fields, such as earthquake engineering, structural engineering, geotechnical engineering, dam engineering and so forth. Several examples are given in this paper to show the applications of infinite elements to dynamic soil-structure interaction problems in engineering practice. 1 Introduction It is well known that compared to the structural size in engineering practice, the Earth's crust is vast on the geometrical side, and therefore can be treated as an infinite medium on the mathematical analysis side. This poses a challenge problem for the conventional finite element method because the modelling domain must be finite in the conventional finite element analysis. The early treatment of this problem is to simply cut a finite region of the Earth's crust as the foundation medium of a structure, and then to model the structure and its foundation medium of finite size using the finite element method. This treatment may be acceptable for static problems, provided the foundation medium of a structure is taken large enough. However, this treatment does not work at all for dynamic problems because it cannot avoid the wave reflection and refraction behaviour on the artificially truncated boundary, no matter how large the foundation of a structure is taken. To solve the above-mentioned problem more effectively and efficiently, great efforts have been made during the last a few decades. Among them the infinite element is one of the most powerful techniques to tackle the above problem [ 1-21 ]. The theory of infinite elements can be found in many open literatures [ 14, 6-7, 13] and will not be repeated in this paper. The infinite element was initially proposed for dealing with static problems in infinite elastic media [ 1, 4] and later extended to the solution of steady-state wave propagation problems in infinite elastic media [2-3, 6-7, 13]. Only very recently, the transient infinite element was developed for solving truly transient heat transfer, mass transport and pore-fluid flow problems in infinite media [11, 14, 18, 19]. Considering the relevance to the topic of this paper, we only present some numerical results obtained from using the combination of finite elements and infinite elements for solving dynamic soil-structure interaction problems. The basic idea behind using the combination of finite and infinite elements is as follows. The finite element is used to effectively model the geometric irregularities and material varieties in a structure and the near field of its foundation, while the infinite element is used to effectively and efficiently model the wave propagation behaviour in the far field of the foundation. To demonstrate the applicability of infinite elements to various practical engineering problems, several examples are given to show how to use the combination of finite and infinite elements for solving the dynamic soil-structure

154 interaction problems in earthquake engineering, structural engineering, geotechnical engineering and dam engineering in the following sections.

2 Applications of Infinite Elements to Wave Propagation Problems in Earthquake Engineering The free field distribution of a site due to an earthquake is a wave scattering problem in the infinite medium. This free field distribution is often affected by the site geometrical and geological conditions. Applications of the coupled method of finite elements and infinite elements to this kind of problem have been reported in several publications [7, 10]. An example are only given below to demonstrate the applicability of using infinite elements to solve some practical problems in earthquake engineering. For the above-mentioned purpose, a V-shaped canyon with different ratios of top width to height is considered. H and L are used to stand for the height and top width of the canyon respectively. As shown in Fig. 1, the near field of the canyon is simulated by eight-node isoparametric finite elements, while the far field of the canyon is modelled by dynamic infinite elements. Since this study aims to investigate the displacement distribution pattern along the canyon surface, it is appropriate to use a unit plane harmonic wave in the analysis. To meet different needs for the study, the unit wave may be considered to have different wave types (i.e., P-wave or SV-wave), different circular frequencies and different incident angles. The horizontal line (Y=0) is chosen as the wave input boundary where the incident harmonic wave is transformed into dynamic loads using the elastic wave theory. The angle between the normal of the front of the plane harmonic wave and the vertical line is defined as the wave incident angle. According to this definition, 0 = 0 means that the harmonic wave is vertically propagating onto the wave input boundary, while 0 ~ 0 implies that the harmonic wave is obliquely propagating onto the wave input boundary. In order to reflect the effects of wave incident direction on free field motion, different incident angles have been considered in the analysis. The following parameters are used in the computation" the height of the canyon is 100 m; the elastic modulus of the canyon rock mass is 24 x 106 kPa ; the unit weight and Poisson' s ratio of the canyon rock mass are 2 4 k N / m 3 and 113 respectively. For the purpose of investigating the effect of the circular frequency of the incident wave, the dimensionless frequency is defined as

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(1)

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155

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Fig. 3 Distributions of displacement amplitudes along V-shaped canyons due to SV wave oblique incidence Fig. 2 and Fig. 3 show the displacement amplitude distribution along the V-shaped canyon due to harmonic SV wave incidences with different incident angles. In these figures, A v and Av are the displacement amplitudes in the horizontal and vertical directions respectively. It is clear that in the case of vertical incident waves (Fig. 2), the symmetric nature of the displacement pattern along the canyon surface is maintained for the SV wave incidence. Both the maximum value and the pattern of the displacement amplitudes are different for different ratios o f canyon width to height (/JH), especially for higher frequency wave incidences (a o = 1.0). This indicates that the canyon topographic condition has significant effects on free field motions along the canyon surface. The maximum value of the displacement amplitude can reach over 3 for the SV wave vertical incidence. This maximum value appears at the top of the narrower canyon (L/H=I). Even though the input waves are vertical, both the

156

horizontal and vertical displacement components are not zero for the SV wave incidences due to wave mode conversion. In the case of oblique incident waves (Fig. 3), the related results show that the pattern of the displacement amplitude appears to be asymmetric and its distribution along the canyon surface is also different due to different ratios of canyon width to height (/A-/). This phenomenon can also be attributed to wave mode conversion along the canyon surface in the case of SV wave incidences.

3 Applications of Infinite Elements to Dynamic Soil-Structure Interaction Problems in Structural Engineering To investigate the effect of foundation flexibility on the dynamic response of a structure with the soilstructure interaction included, a three dimensional multistorey frame structure with a plate foundation resting on a rock medium is considered. The frame structure is modelled using three dimensional frame elements, while the plate and the near field of the rock mass are modelled using the thick plate elements and solid elements respectively. To reflect the wave propagation between the near and far fields, the far field of the rock mass is modelled using three dimensional dynamic infinite elements [13]. Fig. 4 shows the discretized model of the frame structure-plate foundation-rock mass system. It needs to be pointed out that owing to the symmetric nature of the problem, only a quarter of the structure, the plate and the rock mass medium is considered and shown in Fig. 4. It is also assumed that only horizontal periodic loading is applied at the center of the plate (point O in Fig. 4). The plate foundation is considered both flexible and rigid so that the related results can be compared with each other.

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Fig. 5 shows the dimensionless horizontal displacement distribution of the frame structure. In this figure, the solid triangle, the solid dot and the solid square represent the numerical results for column 1, column 2 and column 3 respectively. H is the height of the frame. It is observed that when the frame is subjected to a horizontal movement induced by the dynamic load on the plate, the maximum value of the displacement difference within each storey of the frame structure occurs in the ground storey of the frame for both rigid and flexible plate foundations. This implies that the safety of the columns in the ground storey is the controlling factor in the seismic design of frame structures. This is due to the fact that the

157 repeated reflection within the columns between the ground floor and the first floor takes place and consequently, the wave energy is trapped in these regions. In addition, the flexibility of the plate foundation has a significant effect on the displacement response of the ground storey of the frame structure, as can be clearly seen in Fig. 5.

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4 Applications of Infinite Elements to Dynamic Soil-Structure Interaction Problems in Geotechnical Engineering

Retaining walls are frequently used in geotechnical engineering. Once a retaining wall undergoes an earthquake event, the dynamic soil-structure interaction should be included in the seismic design of the retaining wall. Generally, a retaining wall is a finite structure, while its foundation soil is a infinite medium. Thus, the coupled method of finite and infinite elements is very suitable for solving this kind of problem. Fig. 6 shows a typical soil-retaining wall system in geotechnical engineering. The near field of this problem consists of the retaining wall, the backfill soil and part of the natural soil, while the far field is comprised of the rest of the natural soil and the base rock mass. As was mentioned before, f'mite elements and dynamic infinite elements are used to model the near and far fields of the natural soil and rock mass respectively. The parameters used for the retaining wall, backfill soil, natural soil and rock mass can be found in a previous publication [12]. Fig. 7 shows the effect of different backfill soils on the acceleration amplitude distribution of the retaining wall due to plane SV wave vertical incidence. In this figure,//a and i;a are the amplitudes of the horizontal and vertical acceleration components of the retaining wall; co is the circular frequency of the incident harmonic wave; Stations 1, 2 and 3 represent the top, the middle and the bottom of the wall; Cases 1, 2 and 3 are corresponding to the softer, the medium and the stiffer backfill soil situations. It is clear that although the backfill soil has negligible influence on the dynamic response of the retaining wall in the case of low frequency wave incidences, it has a considerable effect on the response of the wall during high frequency wave incidences. This indicates that the change in mechanical properties of the backfill soil should be accounted for in the seismic design of a retaining wall. Since unit harmonic waves are used in the calculation, the acceleration amplitudes obtained here can serve as amplification factors of the wall to incident harmonic waves. It is concluded from the related results that the configuration of a retaining wall may affect significantly the amplification factor of the retaining wall to an input earthquake because an earthquake wave can be decomposed into several harmonic waves.

158

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5 Applications of Infinite Elements to Dynamic Soil-Structure Interaction Problems in Dam Engineering

In this section, an example is used to show how the infinite elements are applied to solve dynamic soilstructure interaction problems in dam engineering. A typical embankment dam with either a central clay core or an upstream inclined concrete apron is considered and shown in Fig. 8, where the dam and the near

159 field of its foundation medium are modelled using finite elements, but the far field is modelled using dynamic infinite elements. The details about the configuration and the parameters of the dam-soil foundation system can be found in an open literature [17]. Only some results are briefly given and discussed below. legend ~

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Station 1 (crest)

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160 In terms of the amplification factors of the system due to different impervious members, it has recognized that the types of impervious members have a significant influence on the dynamic response of the system in the low frequency range of excitation. The reason for this is that the material damping of the system plays a considerable role in the dynamic response of the system for low frequency excitation. In the case of the embankment dam with an upstream inclined concrete apron, the total material damping of the system decreases as the hysteretic coefficients of the related materials decrease, compared with the central clay core case. Thus, the amplification factors as well as the dynamic response of the dam with an upstream inclined concrete apron increase in the low frequency range. This leads to the conclusion that central clay cores are more suitable as impervious members for embankment dams, from the seismic resistant point of view. Since the dynamic response of a dam is dominated by the radiation damping of the system for high frequency excitation, the amplification factors for the dam with either an upstream inclined concrete apron or a central clay core are nearly the same due to the identical radiation damping of the foundation. 6 Conclusions

The dynamic infinite element has the following two main advantages. Firstly, the concept of the infinite element is very clear in physics. Secondly, the formulation of the infinite element is very easy to be include into the existing finite element codes. Thus, infinite elements have been widely used to solve dynamic soil-structure interaction problems in earthquake engineering, structural engineering, geotechnical engineering, dam engineering and so forth. However, further research is needed to develop the infinite element for solving dynamic soil-structure interaction problems in time domain, since the current dynamic infinite element can only be used to solve dynamic soil-structure interaction problems in frequency domain. References

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