Seismic soil-structure interaction by the superposition method

The superposition technique is a simple yet powerful method for soil-structure seismic interaction problems. The method essentially consists of obtaining total motions by superposing free field motions and interaction motions, both previously calculated in separate analyses. Although this method is strictly applicable only to linear systems, the equivalent linear method is easily incorporated in order to approximate soil non-linearities. The superposition technique is valid for 3-D geometries, and allows for a seismic environment consisting of any kind of body or surface waves, or combinations thereof. The method is explained in detail, and three case studies are also summarized as an illustration of the flexibility of the method.     

Special topics on soil-structure interaction

In the design of nuclear power plants, the topic of soil-structure interaction has traditionally been one of great theoretical and practical controversy for several reasons. In the beginning of the era of nuclear power, the science of soil-structure interaction was in its infancy and very little theory or observational data were available to the engineering profession. At the same time, the state of the art of structural dynamics, however, was already developed and engineers had a considerable amount of data on actual dynamic performance of structures, so that a great amount of confidence in the ability to perform a dynamic structural analysis was already in existence. It was only natural to extend the basic fundamentals of dynamic structural modeling to soil-structure interaction. Consequently, many ‘approaches’ and ‘theories’ of how to account for soil-structure interaction were erroneously developed without providing real consideration of basic fundamentals of interaction of a foundation with an elastic continuum, commonly referred to as elastic half-space or continuum solutions. At the same time, while great confusion was developing on how to apply elastic half-space theory to earthquake analysis problems, the dynamic finite element approach was born and was thought by many to be the final panacea to the problem of soil-structure interaction. Of course, as is usually the case, there is no perfect or single best approach to any problem and the end result is that each of the many approaches to account for soil-structure interaction, has its strong and weak points depending on site or structural conditions. This paper is intended to discuss some of the important details of soil-structure interaction theory to provide a common means of comparison and to introduce some new approaches to simplify the solutions of deeply embedded foundations.A review of recent literature on soil-structure interaction reveals several important facts. First, conflicting comparisons between the lumped parameter and finite element solutions apparently exist. In some cases, both approaches give similar results while for others the results vary widely essentially because different models are used for comparison of both methods. Secondly, fundamental errors are committed on finite element mesh size parameters and boundary conditions. Thirdly, misunderstandings on soil mass and foundation or structural modal damping lead to gross errors in the lumped parameter approach. Finally, the limitations of the various approaches are not always understood. It is noted that both the finite element and lumped parameter approaches should yield similar results if they are appropriately used to solve the same problems. A summary of the advantages and limitations of both approaches are presented and discussed with a short presentation regarding the state of the art in the determination of soil stiffness and material damping characteristics.Furthermore, the paper will illustrate one type of analysis technique which uses a hybrid approach of both finite element results and lumped parameter solutions. Such an approach is developed to account more accurately for the influence of embedment. Results using both pure finite element solutions and lumped parameter models show that the influence of embedment can be accurately considered even for deeply embedded structures (depth to width ratio equal to 1.0). The key to the approach lies in establishing the coupling between horizontal and rocking modes of foundation vibration. Once the coupling parameter is accounted for, it is then possible to develop a lumped parameter models that account for the variation of soil motions below the ground sufrace. Previous lumped parameter models have not accounted for these variations from the surface to the depth of the foundation.Details of the lumped parameter approach for embedded foundations and an illustration with numerical examples are provided. Recommendations are then presented on a procedure for soil-structure interaction of deeply embedded foundations. The primary advantages are that: (1) the model is more easily generated than a finite element model, (2) the model is less susceptible to modeling errors, such as mesh size, model size, and boundary influences, (3) parametric studies may be easily conducted since parameters such as damping may be more directly controlled, and (4) the computer costs for analysis are significantly reduced.     

Multi-scattering and boundary effects in soil-structure interaction

This paper presents two applications of a coupled finite element and boundary element method (FEBEM) to two-dimensional, transient problems of scattering of elastic SH waves. One application concerns multi-scattering: examples are shown for scattering from two semi-cylindrical inclusions embedded in a half-plane and separated by a small distance. Responses of one inclusion in the time and frequency domains are compared with those associated with a single inclusion. The other application concerns the effect of the size of the finite element mesh and boundary on accuracy. Response of the flat foundation with rectangular shape and seated on a half-plane is analyzed. It is shown that while simple silent boundaries are quite effective for large models, Rayleigh damping cannot model radiation damping effectively.     

The use of frequency-independent soil-structure interaction parameters

The validity of approximating frequency-independent foundation impedance functions by constant parameters was evaluated for nuclear power plant structures. The soil-structure interaction system with the frequency-dependent impedances was analyzed using the Foss method to uncouple the equations of motion; this closely follows the method developed by Jennings and Bielak. The interaction system with the constant impedances was approximately analyzed by the normal mode method using equivalent modal damping values computed according to a procedure developed by Tsai. The above two methods were applied to simplified containment structural models founded on an idealized elastic half-space, the shear wave velocities being taken to be 600, 1150, 2000 and 10 000 ft/sec. The results such as frequencies, damping, and in-structure response spectra were then compared. It was concluded that frequency-independent foundation impedances can be adequately used for plant sites having relatively deep and uniform overburdens.     

Lotung large-scale seismic experiment and soil-structure interaction method validation

This paper describes the Electric Power Research Institute's (EPRI) current on-site large-scale soil-structure interaction (SSI) research. The objectives of the research are: (1) to obtain an earthquake database which can be used to substantiate SSI models and analysis methods; (2) to develop realistic SSI analysis guidelines and procedures based on experimental-analytical correlation; and (3) to quantify nuclear power plant reactor containment and internal component's seismic margin based on earthquake experience data.To meet the objectives, two model structures were sited in a high seismic region, Lotung, Taiwan, under the joint sponsorship of EPRI and the Taiwan Power Company (Taipower). The model structures which simulate scaled-down nuclear containments ( ) were constructed and instrumented within an existing strong motion array (SMART-1) deployed by U.C. Berkeley under a U.S. National Science Foundation grant. The instrumentation layout associated with the model structures included accelerometers on the models, on the internal components, on the ground surface, and in the ground. These instrumentations are to provide information required for validation and qualification of SSI models. Between September 1985 and November 1986, 18 earthquakes ranging from Richter magnitude 4.5 to 7.0 were recorded.The analysis phase of the research was conducted with the cooperation of the U.S. Nuclear Regulatory Commission (NRC) and Taipower. A round-robin approach was utilized with emphasis on blind predictions and independent assessment of existing methodologies. A total of 13 research teams from the United States, the Republic of China, Japan, and Switzerland participated in the effort. A workshop was held in December 1987 where research results and findings were presented. Further effort is ongoing to synthesize the results and findings for providing technical bases of developing improved SSI analyses guidelines and procedures.     

Effects of horizontally travelling waves in soil-structure interaction

Phenomena related to horizontally travelling waves are normally not considered in soil-structure interaction. Only vertically incident S- and P-waves are commonly assumed. To determine the influence of this very basic assumption, the responses of a massless basemat, of a massless structure, of a basemat with mass and of a mass-spring system connected to a basemat with mass are parametrically analysed for harmonic and transient excitations for all wave forms (SH-, P-, SV- and Rayleigh waves). Comparisons of the results of the same structures, calculated for the standard vertically incident body waves of the same amplitudes are made. Various possibilities of combining P- and SV-waves to create specified horizontal and vertical free-field motions are examined. By way of illustration, a reactor building and the through-soil coupling of a reactor and a reactor auxiliary building are examined in detail, using the J-145 record of the 1971 San Fernando earthquake as a travelling wave. The effect of a flexible basemat on the structural response is discussed.     

Two-dimensional approximations to the three-dimensional soil-structure interaction problem

The feasibility of representing a three-dimensional soil-structure interaction problem by a plane strain model, and the errors involved in such representation, were studied. By comparing the rocking and translational force-displacement relationships for a rigid circular foundation placed on an elastic half-space and the corresponding relationships for a strip footing placed on an elastic half-plane it was found that it is not possible to obtain a two-dimensional representation that will approximate both the dynamic stiffness and radiation damping over a reasonable range of frequencies. Several two-dimensional models were considered and a measure of the errors involved is presented. In general, the two-dimensional models overestimate the radiation damping associated with the three-dimensional problem. To study the effects that the use of a two-dimensional plane strain model may introduce in the solution of the soil-structure interaction problem for typical nuclear power plants, a comparison was made between the system frequencies and modal dampings obtained for three and two-dimensional models. The corresponding response at the top of the containment shell, top of the internal structure, and base slab for a particular earthquake were also compared. It was found that by properly selecting the two-dimensional model it was possible to obtain close approximations to the system frequencies. However, since the dampings associated with the low frequency modes are overestimated, the earthquake response of the structure, as obtained by the two-dimensional model, is underestimated to a significant degree.     

A mixed implicit/explicit procedure for soil-structure interaction

This paper describes an efficient method for the solution of dynamic soil-structure interaction problems. The method which combines implicit and explicit time integration procedures is ideally suited to problems in which the structure is considered linear and the soil non-linear. The equations relating to the linear structures are integrated using an unconditionally stable implicit scheme while the non-linear soil is treated explicitly. The explicit method is ideally suited to non-linear calculations as there is no need for iterative techniques. The structural equations can also be integrated explicitly, but this generally requires a time step that is much smaller than that for the soil. By using an unconditionally stable implicit algorithm for the structure, the complete analysis can be performed using the time step for the soil. The proposed procedure leads to economical solutions with the soil non-linearities handled accurately and efficiently.     

Simplified soil-structure interaction analysis with strain dependent soil properties

Recent work regarding the response of above-ground soil structures, such as dams, has indicated the need to use strain dependent soil properties. Unlike other building materials soil stiffness and damping properties are highly strain dependent. The application of these concepts to problems in soil-structure interaction has also been suggested. Without commenting on the appropriateness of this extension to soil-structure interaction problems, it is suggested that answers similar to those given by the strain dependent solution of finite element models can be obtained more simply by the use of lumped-parameter impedance functions. To establish this equivalence, it is imperative that all other variables in the problem be made equal for both models; that is, the strain dependency problem must be isolated if the comparison of the two approaches is to be meaningful. The proposed method uses a damping value equal to the average strain dependent soil profile damping. The strain dependent soil profile damping values are obtained by the use of a much simpler model using one-dimensional wave propagation theory. From this same one-dimensional model, the strain dependent soil stiffness corresponding to the average top layer of soil with and without an overburden to approximate the superstructure is used in the equivalent simplified model. Several case comparisons indicate the validity of the proposed method.     

Vertical soil-structure interaction of nuclear power plants subjected to seismic excitation

The vertical soil-structure interaction problem is investigated by coupling an N-mass lumped mass structure to a two-dimensional elastic half space. This problem is formulated as an integral equation of the Volterra type. Numerical results are obtained by iteration for an idealized threemass two-mode model of a nuclear power plant containment structure. The effects of interaction are evaluated by comparing free-field acceleration spectrum response curves with similar curves determined from foundation motion.     

Seismic response analysis of soil-structure interactive system using a coupled three-dimensional FE-IE method

This paper proposes a slightly new three-dimensional radial-shaped dynamic infinite elements fully coupled to finite elements for an analysis of soil-structure interaction system in a horizontally layered medium. We then deal with a seismic analysis technique for a three-dimensional soil-structure interactive system, based on the coupled finite-infinite method in frequency domain. The dynamic infinite elements are simulated for the unbounded domain with wave functions propagating multi-generated wave components. The accuracy of the dynamic infinite element and effectiveness of the seismic analysis technique may be demonstrated through a typical compliance analysis of square surface footing, an L-shaped mat concrete footing on layered soil medium and two kinds of practical seismic analysis tests. The practical analyses are (1) a site response analysis of the well-known Hualien site excited by all travelling wave components (primary, shear, Rayleigh waves) and (2) a generation of a floor response spectrum of a nuclear power plant. The obtained dynamic results show good agreement compared with the measured response data and numerical values of other soil-structure interaction analysis package.     

The use of boundary elements to represent the far field in soil-structure interaction

To solve soil-structure interaction problems, the modelling is mostly restricted to a two-dimensional representation of the reality. It is only recently that three-dimensional soil-structure interaction problems are attacked. The solution of these problems by using a straight forward finite element representation is very expensive.This paper describes the combination of finite elements and boundary elements for static computations of foundations. The advantages and disadvantages are given. Afterwards, the developments for dynamic analysis using boundary elements are also described.     

Recent tendency of the practice of soil-structure interaction design analysis in Japan and its theoretical background

Many methods of soil-structure interaction analysis for design calculations of nuclear power plants are available. The validity of methods has often been examined by their application to simulation analysis of shaker tests or seismic observations of nuclear power plant buildings, and forced vibration tests of large-scale foundation blocks. In this paper, such simulation analyses performed in Japan are reviewed and discussed for their practical applications.     

A review of soil-structure interaction effects in the seimic analysis of nuclear power plants

A survey of investigations of soil-structure interaction in the seismic analysis of nuclear power plants is presented. After a discussion of various methods that have been applied to calculate interaction effects, results of various investigations are discussed. Differences in system response associated with various analytical models are pointed out. Conclusions and recommendations for additional studies are made.     

堆芯燃料组件抗震分析简化模型研究

针对堆芯燃料组件在地震作用下可能发生的结构变形及破坏现象,采用简化方法对燃料组件进行时程分析,计算地震工况下格架所受的碰撞载荷以及应力情况,并将计算值与格架的压塌载荷以及导向管的应力限值进行了比较,从而对堆芯燃料组件的结构完整性进行了评估,为日后堆芯燃料组件结构的抗震性能分析计算提供参考。     

Dynamic analysis of VVER type nuclear power plants using different procedures for consideration of soil-structure interaction effects

The dynamic response of structures due to seismic loadings is conventionally analyzed in the time domain using substructure methods (decoupled system models). This procedure uses frequency-independent impedances to represent capabilities of the soil underneath the structure. The soil parameters are tuned to the fundamental frequencies of the soil-structure system. This is a common procedure widely used in the preliminary design of power plant structures which provides conservative results. However, parallel to the rapid progress being made in upgrading the capability of data processing systems, methods and software tools have become available which work also in the frequency domain using complex models (for the soil and the structure) or models in which the soil is represented by frequency-dependent impedances. This procedure (coupled system models) also allows realistic treatment of kinematic interaction effects and especially consideration of the embedment parameters of the building structure. The main goal of the study presented here was to demonstrate the effects of different procedures for consideration of soil-structure interaction on the dynamic response of the structures mentioned above. The analyses were based on appropriate mathematical models of the coupled vibrating structures (reactor building, turbine hall, intermediate building structures of a VVER 440/213 as well as the main building of the VVER 1000) and the layered soil. On the basis of this study, it can be concluded that substructure methods using frequency-independent impedances (equivalent dashpots) and cut-off of modal damping usually provide conservative results. Coupled system models which allow the soil-structure interaction effects to be realistically represented (by coupled models of the soil and the structure or by frequency-dependent impedances) provide more accurate results. The advantage of the analysis using coupled system models will be demonstrated and discussed, based on results obtained for the VVER 440/213 PAKS and VVER 1000 Kozloduy.     

Comparison of results of soil-structure interaction analyses based on time and frequency domain approaches with emphasis on treatment of damping

Comparison of results of soil-structure interaction analyses of the reactor building of a nuclear power plant using different analytical approaches and solution procedures is presented. The emphasis of the comparison was on the treatment of damping in these different approaches and procedures. An axisymmetric model of the reactor building was employed. The analyses were performed for the aircraft impact loadings. Two different locations were used for these loadings.The following four different sets of analyses were performed.

1. (1) Time-domain analysis using frequency-independent soil springs in conjunction with modal damping cut-off.

2. (2) Frequency-domain analysis using frequency-independent soil springs in conjunction with a complex modulus approach.

3. (3) Frequency-domain analysis using frequency-dependent soil-impedance coefficients in conjunction with a complex modulus approach.

4. (4) Frequency-domain analysis using frequency-dependent soil-impedance coefficients in conjunction with Rayleigh damping.

The frequency-independent soil springs were computed using the standard approach based on rigid base supported on an elastic layered half-space. The frequency-dependent soil impedance coefficients were computed in the form of a soil substructure matrix which included the uncoupled as well as the coupled terms. The computations were based on the use of a “flexible” base mat supported on a layered half-space. Unit dynamic loads, for each frequency, were applied to the layered half-space corresponding to each degree of freedom and the displacements were xomputed corresponding to all degrees of freedom. The compliance matrix so computed was inverted to obtain the impedance matrix for each frequency. The computations were repeated for all frequencies of interest for the aircraft impact loading.Floor response spectra were developed and compared at various floor elevations of the reactor building using the above four different sets of analyses. Conclusions were developed as a result of these comparisons.     

Seismic fragilities for nuclear power plant risk studies

Seismic fragilities of critical structures and equipment are developed as families of conditional failure frequency curves plotted against peak ground acceleration. The procedure is based on available data combined with judicious extrapolation of design information on plant structures and equipment. Representative values of fragility parameters for typical modern nuclear power plants are provided. Based on the fragility evaluation for about a dozen nuclear power plants, it is proposed that unnecessary conservatism existing in current seismic design practice could be removed by properly accounting for inelastic energy absorption capabilities of structures. The paper discusses the key contributors to seismic risk and the significance of possible correlation between component failures and potential design and construction errors.     

Seismic probabilistic risk assessment and its impact on margin studies

In recent years a number of seismic probabilistic risk assessments of nuclear power plants have been conducted. These studies have highlighted the significance of seismic events to the overall plant risk and have identified several dominant contributors to the seismic risk. It has been learnt from the seismic PRAs that the uncertainty in the seismic hazard results contribute to the large uncertainty in the core damage and severe release frequencies. A procedure is needed to assess the seismic safety of a plant which is somewhat removed from the influence of the uncertainties in seismic hazard estimates. In the last two years, seismic margin review methodologies have been developed based on the results and insights from the seismic probabilistic risk assessments. They focus on the question of how much larger an earthquake should be beyond the plant design basis before it compromises the safety of the plant. An indicator of the plant seismic capacity called the High Confidence Low Probability of Failure (HCLPF) capacity, is defined as the level of earthquake for which one could state with high confidence that the plant will have a low probability of severe core damage. The seismic margin review methodologies draw from the seismic PRAs, experience in seismic analyses, testing and actual earthquakes in order to minimize the review effort. The salient steps in the review consists of preliminary screening of components and systems, performance of detailed seismic walkdowns and evaluation of seismic margins for components, systems and plant.     

悬挂式储液罐抗震分析

针对悬挂式储液罐建立了2种有限元抗震分析模型。模型1为基于Housner模型和ASCE 4-98规则的质量-弹簧模型。该模型将液动压力对容器底部的弯矩等效为侧壁的压力,并通过提高侧壁压力的高度实现等效。模型2是对Housner模型的改进,是通过公式计算液动压力对容器底部的弯矩,并将弯矩直接施加在容器底部。对所建立的2种模型进行了模态求解和谱分析,并对结果进行对比。结果表明,模型1并不适用于悬挂式储液罐的抗震分析,它忽视了容器底部的弯矩在容器底部到支耳之间引起的应力,其应力结果不符合实际且不保守,而模型2的结果更可靠。