MODELING OF NONLINEAR DYNAMIC SOIL-STRUCTURE INTERACTION PROBLEMS

Abstract

1.0 INTRODUCTION It has been well recognized that the soil on which a structure is constructed interacts dynamically with the structure during earthquake excitation, especially when the soil is relatively soft and the structure is stiff. This kind of dynamic soil-structure interaction can significantly alter the deflections and stresses in the whole structural system. Two important characteristics that distinguish a dynamic soil-structure interaction system from other general dynamic structural systems; are the unbounded nature and the nonlinearity of the foundation medium. In the present study, an attempt has been made to investigate some important modeling aspects of a dynamic soil-structure interaction system. Among these, attention has been paid to radiation damping of the unbounded medium and the slidingandseparation at soil-structure interface. In addition, nonlinearity characteristics of the foundation medium and the structure and possible cracking of the structure have been also considered in the present study. Nonlinear earthquake response of the non-overflow monolith of Pine Flat Damto Taft groundmotion has been investigated as a case study. 2.0 BRIEF REVIEW OF EARLIER WORK 2.1 Modeling of Radiation Damping One of the primary obstacles in formulating an accurate yet inexpensive procedure for dynamic interaction analysis is the modeling ofthe unbounded medium beneath the structure. Different techniques have been reported in the literature to model the unbounded soil medium so as to overcome the problem of reflection ofthe waves at the truncated boundary. Acritical review of the existing literature suggests that various available techniques could be classified into a rigorous approach and an approximate approach. The rigorous boundaries (Lysmer and Wass, 1972; Manolis and Beskos, 1988;Wolf and Song, 1996b and Song and Wolf, 1997a; Song and Wolf, 1996b; Basu and Chopra, 2003a,b) are highly accurate and thus may be used with a small bounded domain. These are typically formulated in the frequency domain and thus neglect the nonlinearity of the soil medium. On the other hand, the corresponding time domain formulation (Kausel, 1994; Antes and Von, Estorff iii Synopsis Modeling ofNonlinear Dynamic Soil-Structure Interaction Problems 1994,Yazadchi et al, 1999, Von Estorff and Firuziaan, 2000; Wolf and Song, 1995, Wolf, 2003; Basu and Chopra, 2004) may be computationally very expensive and may not be applicable to all problems of interest. Approximate boundaries (Lysmer and Kuhlemeyer, 1969; Smith, 1974; Clayton and Enqguist, 1977; Liao and Wong, 1984; Higdon, 1992; Song and Wolf, 1994; Wolf and Song, 1995b) are used in the direct method of analysis in time domain and they are local in space and time. Although the approximate approach can model the radiation damping approximately, accurate results could be obtained by applying a suitable transmitting boundary at a sufficient distance away from the structure. Among the various transmitting boundaries, extrapolation algorithm seems to be a very efficient boundary for simulation of the semi-infinite media. While removing the shortcomings associated with this boundary, a transient transmitting boundary based on an extrapolation algorithm in two-dimensions has been proposed recently by Garg (1998). This new formulation in two dimensions was applied to real-life structures and it has been reported to yield very good results (Garg, 1998) 2.2 Modeling of Sliding and Separation Dynamic soil-structure interaction may be defined as a phenomenon of transmitting kinematic energy between soil and the structure through the interface zone. It is obvious that, besides nonlinear behavior of soil and structure, the interface properties also influence the resulting interaction and sometimes may govern the response of the system. In order to account for this interfacial kinematics, modeling of soil-structure-interface is essential. A critical review of literature suggests that commonly used interface elements could be classified into zero or finite thickness interface elements and thin layer interface elements. The zero thickness interface element (Goodman et al, 1968; Desai, 1981; Fishman and Desai, 1988; Ma and Desai, 1990; Morrison, 1995; Karabatakis, 2000) suffers from numerical difficulties like ill-conditioning of the stiffness matrix due to either very large or very small diagonal terms. In addition, this interface element possesses a certain deficiency associated with tangential response, called kinematic inconsistency. In the case ofa thin layer element (Zaman et al, 1984; Desai et al, 1986; Desai and Fishman, 1991; Sharma and Desai, 1992; Desai and Rigby, 1997), besides ill-conditioning problem, kinematic inconsistency also comes into play (Coutinho et al, 2003). IV Synopsis Modeling ofNonlinear Dynamic Soil-Structure Interaction Problems Out of the above two interface element families, zero thickness interface elements have been widely used, due to their simple formulation, ease of numerical implementation, robust normal response, and few parameters required to define it. However, this element needs some modification to model properly the sliding and separation during dynamic loading. Maekawa and his co-workers (Maekawa et al, 2003) used higher order zero thickness interface element to model sliding and separation during seismic event. This model has been applied to real-life structures, such as underground RC-soil system and RC vertical duct-soil system and has been reported to yield good results (Shawky and Maekawa, 1996a, 1996b; An and Maekawa, 1997; An et al, 1997; Maekawa et al, 2003). 3.0 CRITICAL COMMENTS AND PROBLEM IDENTIFICATION A detailed and critical review of the literature suggests that: i) Numerical modeling of dynamic soil-structure interaction is still in its course of development and there are still almost no benchmark numerical models available. Various models are no longer restricted to only the time domain or the frequency domain. Techniques are not restricted to the finite element method or the boundary element method. On the contrary, all these are always incorporated one way or another, and some new analysis techniques have been introduced into the problem solution such as infinite element method and coupled finite element-boundary element method, ii) The future research of dynamic soil-structure interaction modeling tends to be concentrated in time domain because the problem of nonlinearity can be better simulated in time domain rather than in the frequency domain. Moreover, typical structural and geotechnical analysts are not accustomed to working in the frequency domain, their natural approach is to consider the sequence of developments from one time step to the next, i.e., to apply the time domain concept, iii) Direct method of analysis in time domain may be more attractive to research workers and engineers than the substructure method as it can deal with the nonlinearity ofthe unbounded medium through a proper constitutive law as long as sufficiently accurate transmitting boundaries can be provided. Synopsis Modeling ofNonlinear Dynamic Soil-Structure Interaction Problems iv) Two important characteristics that distinguish a dynamic soil-structure interaction system from other general dynamic structural systems are the unbounded nature and the nonlinearity of the soil medium. These modeling aspects, besides sliding and separation that may occur at the interfacial zone are considered to be essential for a realistic behavior of the dynamic interaction system. v) The rigorous procedures are global in space and time and it can model radiation damping exactly. However, in most cases, the rigorous approach is formulated in frequency domain and thus restricted to linear analysis only. Solution in time domain involves convolution process, which makes the method computationally very expansive. On the other hand, approximated procedures are used in direct method of analysis in time domain and are local in space and time. Although, the approximate approach, which is local in space and time, can model radiation damping approximately, accurate results can be obtained by applying a suitable transmitting boundary at a sufficient distance away from the structure. vi) When applying the direct method of analysis to dynamic unbounded mediumstructure- interaction, location of the artificial boundary has to be selected. Available literature on this important issue is scanty. Therefore, further research is required to specify the appropriate location of the interaction horizon beyond which there is no need to discretize the soil domain in the finite element analysis. Also, the percentage error may be calculated if insufficient soil domain is included in the analysis. vii) Since soil and structure display different stiffness characteristics, complete contact at the interface is not always assured. Separation phenomenon may occur at the soilstructure interface, where tensile stress is generated. Moreover, sliding phenomenon may also occur during strong earthquake motion. Therefore, the possibility ofsliding and separation along the interfacial zone should be considered in any dynamic interaction analysis to predicate the realistic behavior. viii)For a dynamic soil-structure system, different materials are involved. These different materials provide distinctly different energy-loss mechanisms in various parts ofthe system. Therefore, the distribution of damping forces will not be similar to the distribution ofthe inertial forces and the elastic forces. In other words, the resulting damping will be nonproportional and therefore it has to be taken into account in any dynamic interaction analysis. vi r- Synopsis Modeling ofNonlinear Dynamic Soil-Structure Interaction Problems 4.0 OBJECTIVES OF THE PRESENT STUDY In view of the inadequacies identified, the objectives of the present work are: i) To implement and apply the transient transmitting boundary for two-dimensional problems so as to simulate a non-reflecting boundary in a dynamic interaction analysis. Further, to modify this boundary so as to take into account the earthquake excitation and other modeling aspects of a dynamic soil-interaction system. ii) To establish proper guidelines for the location of the interaction horizon of the transient transmitting boundary, based on which the extent of soil domain could be decided before a time domain interaction analysis is undertaken. iii) To implement and apply a higher order interface element in order to simulate the sliding and separation at soil-structure interface during strong ground motion. iv) To propose a dynamic model incorporating various modeling aspects including radiation damping at far field and the modeling of sliding and separation at soilstructure interface. v) To develop a software code for nonlinear dynamic soil-structure interaction analysis in time domain under earthquake excitation incorporating various nonlinear constitutive laws of materials participating in the dynamic interaction system and other modeling aspects such as nonproportional damping. vi) To apply the proposed model and the software to study the behavior of some structures of practical engineering importance like a dam-foundation problem, etc. 5.0 PRESENT WORK The present study deals with the numerical techniques required for the nonlinear dynamic soil-structure interaction analysis for earthquake excitation in time domain. The following are the main features of the proposed model: 5.1 Transient Transmitting Boundary A transient transmitting boundary based on an extrapolation algorithm has been presented in detail for two-dimensional applications. This boundary is simple in concept and is easy to understand and implement in the finite element software. Moreover, additional computational efforts required during the execution are negligible in comparison to other operations performed in the finite element program, making it computationally inexpensive. vn Synopsis Modeling ofNonlinear Dynamic Soil-Structure Interaction Problems 5.2 Modeling of Interface Element with Sliding and Separation Asimple and an efficient higher order interface element have been proposed to model the sliding and separation phenomenon between the soil and the structure. Abilinear bond in open/close mode, which has been suggested by Maekawa et al (2003), has been adopted in the present study, since it has been reported to predict the behavior of the real-life soilstructure system reasonably well. 5.3 Software Development An attempt has been made in the present work to develop acomputer program for the nonlinear dynamic soil-structure interaction problems under earthquake excitation. Formulation of the problem has been carried out in time domain using direct method of analysis. The salient features of the software as follows: > Acomplete formulation of transient transmitting boundary has been incorporated in the software to model a non-reflecting boundary and it has been checked for its reliability. Amodified profile solver has been developed which can take into account non-zero prescribed displacements at the transient transmitting boundary. > Acomplete formulation of higher order interface element has also been implemented in the software to model both sliding and separation at the soil-structure interface where tensile stresses are likely to be concentrated especially during astrong ground motion. > Efforts have been made to include various constitutive laws of materials participating in the soil-structure interaction, e.g., soil, concrete, reinforcement and interface zone, so as to enable arealistic analysis of the soil-structure system of practical importance. The concrete tends to develop cracks under strong ground motion and therefore this aspect of the behavior of concrete has been considered in modeling. > In nonlinear dynamic analysis, the effect of initial stresses due to the prevalent static loads along with the occurrence of dynamic loads cannot be neglected. Therefore, the provision has been made in the software for carrying out a nonlinear static analysis before undertaking a dynamic interaction analysis. > To support the main program, pre-processors for mesh/data generation and graphical checking of the input data and post-processors for graphical presentation of the output data have been developed. vin Synopsis Modeling ofNonlinear Dynamic Soil-Structure Interaction Problems 5.4 Validation of Software The software developed for the purpose has been validated by solving some suitably selected test problems reflecting different modeling aspects for which the results are available in the literature. 5.5 Model Proposed for Dynamic Soil-Structure Interaction Analysis Various modeling aspects have been considered in the present study; in one model the transient transmitting boundary was applied and in the other, an elementary boundary was used. In both the models, the mass of the foundation plays a vital role where it is excluded from the loading part due to seismic excitation. Massless foundation model was also examined and evaluated against the model proposed in the present study. 5.6 Location of Interaction Horizon A criterion has been developed to specify the proper location of the interaction horizon beyond which there is no need to discretize the soil domain in finite element model. This would help to fix the transient transmitting boundary before a time domain interaction analysis is undertaken. Further, a percentage error may be calculated if insufficient soil domain included in the analysis. 5.7 Nonlinear Analysis of Pine Flat Dam Using various modeling aspects developed in the present work including the nonlinearity of the system and the location of the interaction horizon, nonlinear earthquake response of the tallest, non-overflow monolith of Pine Flat Dam to Taft ground motion has been investigated as a case study. A parametric study has been conducted to investigate the effect of the hydrodynamic pressure and the vertical component of ground motion on the response of the dam. 5.8 Nonlinear Analysis of Pine Flat Dam including Base Sliding and Separation Nonlinear earthquake response analysis of the non-overflow monolith of Pine Flat Dam to Taft ground motion, scaled up to 0.5g, including base sliding and separation, is presented to investigate the possibility of sliding and separation at the dam base and to study its effect on the dynamic response of dam-foundation system. A parametric study has been performed to investigate the factors that influence the base sliding and opening/rocking ix Synopsis Modeling ofNonlinear Dynamic Soil-Structure Interaction Problems displacements. These factors include the flexibility of the foundation rock, properties of the interface element and ground motion characteristics. 6.0 SIGNIFICANT CONCLUSIONS i) The model proposed in the present study in which the seismic loading is not acting on the foundation mass and the radiation damping has been modeled by using the transient transmitting boundary is capable of predicting the response of the gravity dam-foundation interaction system quite accurately. ii) The response based on elementary boundary simulation is found to be inadequate to predicate the response of the dam-foundation system even if large foundation domain, such as 6B from the heel of the dam (where B is the base width of the dam), is considered. iii) Damping plays a major role in deciding the proper location of the interaction horizon. For higher damping values the transient transmitting boundary could be placed at a closer distance. iv) For high foundation stiffness, the domain of influence required for convergence reduces significantly. It has been found that for hard foundation, domain with extent upto 1.0B is enough while reliable results for weak foundation require a domain extent to2.5B. v) A design chart has been proposed to serve as a guide for deciding the location of the transient transmitting boundary, with full reliability, before a time domain dynamic interaction analysis could be undertaken. vi) Linear elastic analysis may be used to evaluate the behavior of the dam-foundation system for moderate to low ground motion and may serve as a "crutch" while interpreting the complicated nonlinear results. However, for moderate to strong ground motion, a nonlinear elasto-plastic model is essential to predict the behavior of the system realistically and to ensure the safety of the dam. vii) The deformations are much higher in case of nonlinear analysis, and increase further with the development ofcracks. Also, it is a crucial issue to include the hydrodynamic pressure in a nonlinear analysis since these have a major role in formation of cracks in the body of the dam thereby increasing the deformation of the dam-foundation system. ,4 Synopsis Modeling ofNonlinearDynamicSoil-Structure Interaction Problems viii)Modeling of both sliding and separation modes at the interface zone has a significant influence in reducing the deformations as well as the principal stresses in the dam body. Therefore a pre-formed joint, if properly constructed, may be able to reduce significantly both deformations and the principal stresses. This significant reduction in the principal stresses has been suggested using what is called the partially unkeyed foundation surface as a defensive design measure against earthquake. ix) The crest deformation reduces significantly as the foundation stiffness was increased. However, for very stiff foundation rock, the crest horizontal displacement has been found to increase significantly due to significant sliding displacement which governs the dam response in this case. x) The crest deformation reduces significantly as the normal to shear stiffness ratio Kn/ Ks of the interface becomes less than 10. However, the horizontal displacement increases drastically for the interface stiffness ratio of 100 where interface shear stiffness becomes very small thereby allowing the whole dam to slide on its base, xi) Uplift pressure is an important factor that affects the sliding and separation response of the dam in a different way depending upon the distribution of the uplift pressure and the condition at the base of the dam. For pure sliding case and drain effective condition, the uplift pressure increase the crest deformation towards downstream side and the sliding displacement at heel, generally, increases. However, when the drain is ineffective condition, response depends on the time history of the applied ground acceleration and it has been found that the sliding displacement at heel generally decreases. xi