In this paper general equation system for linear dynamic soil-structure interaction (SSI) in frequency domain is presented. The main objective of the paper is to provide and investigate a possibility to use spectral elements in SSI domain. Spectral elements reduce considerably number of unknowns and in some cases, e.g. in frame structures coupled with analytically obtained impedance functions of sub-grade, produce no modelling errors. Results for two different shallow-founded beam structures with identical foundations excited by harmonic free-field motions are presented.

With the computer power available an analysis of a total soil-structure system has become possible. In this paper the advantages of an analysis performed in an frequency domain is shown: the condensation to the primary degrees of freedom, thereby conserving all the characteristic properties of the system, is straightforward; material damping based on rheological models can be considered easily; and even nonlinear problems may be investigated. The formulation is developed in a total displacement formulation, where the motion is driven by forces acting at the soil-structure interface, or more generally, at so-called interaction nodes. The total displacement formulation is compared with the classical approach, where the unknowns are the relative deformations with respect to the rigid base of the structure. These unknows are produced by inertial forces acting on the masses of the structure. In the conclusion the total displacement formulation in the frequency domain is suggested for structures with embedded elastic foundation built on a sub-grade which requires extended modeling for the mechanical representation of the structure and the sub-grade.

Special dynamic soilstructure analysis procedures demonstrated for two tower-like structuresMany problems in Earthquake Engineering require the modeling of the structure as a dynamic system including the sub-grade. A structural engineer is usually familiar with the Finite Element Method but has a problem modeling the sub-grade when its infinite extension and wave propagation are the essential features. If the dynamic equation of a soil-structure system is written in a frequency domain and the variables of the system are total displacements, then the governing equations are given as in statics. The dynamic stiffness matrix of the system is obtained as the sum of the stiffnesses of the structure and sub-grade sub-structures. To illustrate the influence of the sub-grade on the dynamic behavior of the structure, the frequency response of two tower-like structures excited by a seismic harmonic wave field is shown. The sub-grade is modeled as an elastic homogeneous half-space. The structure is modeled as a finite beam element with lumped masses

The dual reciprocity boundary element method (DRBEM) is studied thoroughly in the present paper. To the best knowledge of the authors, the DRBEM has never been applied to 3D half-space dynamics previously. In the present paper, the mathematical derivation of the method is presented, with stress placed on peculiarities of the method when applied to transient half-space dynamics. It has been found that semi-infinite domains (half-space) are more difficult to simulate, since the truncation of the discretization of the half-space surface by boundary elements results in excessive spurious wave reflection on the border of the discretized region. Mathematical derivation of the method is followed by its numerical implementation. Wave propagation due to various kinds of loading in the time-domain is studied afterward, aimed at the validation of the method. The main advantage of the DRBEM over its counterparts, used to model infinite and semi-infinite domains (such as classical BEM formulation, integral transformations, thin layer method), is that it produces time- and frequency-independent matrices (mass and stiffness matrix), by preserving a boundary-only discretization (no internal nodes necessary). This makes the method very attractive, since this feature is very close to the common engineering understanding and the final equation of motion has a similar form like the one known from the finite element method. Moreover, the formulation allows for seamless incorporation of non-homogeneous initial conditions, i.e. non-zero initial displacements and velocities and surface tractions can be prescribed.

An analysis procedure is developed in order to describe the system-dynamics of railroad track, substructure and soil. It allows to model the dynamic behavior of rails, elastic pads, sleepers, railroad earthworks and subsoil. By means of the developed program the dynamic features of various track-models under harmonic load as well as under moving load can be analyzed. A model is developed for a passing train which allows the simulation of the influence of the moving load by means of a frequency- and time-domain transformation. Rails, sleepers and elastic pads are modelled by finite elements while the system-components extending to infinity, such as rigid track, ballast and subsoil, for which the wave radiation is essential, are described with boundary elements. The developed model is validated by a comparison with other numerical models in the frame of a benchmark test.

The rapid development of high-speed trains like the TGV or the ICE in recent years results in high dynamic loads causing vibrations which propagate from the train-track structure into the ground and further into nearby buildings. In this context it is important to develop rigid tracks with improved dynamic behaviour and to investigate possible means of vibration reduction. The boundary element method in frequency and time domain is used to simulate train-track structures subjected to dynamic loading and the reduction of vibrations which for instance can be achieved via a trench running parallel to the rigid track. In this context the non-causality error, which arises when the time-domain BEM algorithm is applied to mathematically concave domains, is discussed and the substructure method is proposed as a solution to this problem. A two-layered cylindrical elastic structure on a half-space with a trench is added as an example of a possible application.

1 Department of Civil Engineering, Zagazig University, Cairo, Egypt. Email: mariadam@frcu.eun.eg, Fax: +(20) 13-601-510 2 Graduiertenkolleg Ruhr University Bochum, Germany. Email: pflanz@sim.bi.ruhr-uni-bochum.de 3 Department of Civil Engineering, Ruhr University Bochum, Germany. Email: schmid@sim.bi.ruhr-uni-bochum.de TWO-AND THREE-DIMENSIONAL TRANSIENT RESPONSES OF HALF-SPACE UNDER DYNAMIC STRIP LOADS AND TRAIN TRACK LOADS

In this paper a new method is presented, which permits consideration of effects of wave propagation in the soil and local nonlinearities. By using a step-by-step method in the Laplace domain the calculation will be expanded for problems of nonlinear foundation uplift. The nonlinear calculation is performed by a number of linear steps, where each linear step satisfies the superposition condition. The nonlinearity is dealt with in the time domain, while the remaining calculations are performed in the Laplace domain. FEM and BEM techniques are used to model the structure and the soil, respectively.