George D. Manolis

Green's function for the vector wave equation in a mildly heterogeneous continuum

Abstract In this work, a fundamental solution is derived for the case of time-harmonic elastic waves originating from a point source and propagating in a three-dimensional, unbounded heterogeneous medium with a Poissons ratio of 0.25. The first step in the solution procedure is to transform the displacement vector in the Navier equations of dynamic equilibrium through scaling by the square root of the position-dependent shear modulus. Following imposition of certain constraints that are subsequently used to derive the depth profile of the elastic moduli and of the density, it becomes possible to employ Helmholtzs vector decomposition so as to generate two scalar wave equations for the dilational and rotational components of the wave motion, a process which again generates additional constraints. The corresponding Greens function is then synthesized in the conventional way followed for homogeneous media. Consideration of all intermediate constraints shows that the elastic moduli and the density all have quadratic variation with respect to the depth coordinate z , while the pressure and shear wavespeed profiles are constant and correspond to reference values at z = 0. The present methodology is based on earlier algebraic transformation techniques applied for the case of scalar wave propagation. The methodology is finally illustrated through a number of examples involving a commonly encountered geological medium.

Stochastic soil dynamics

This article is a review of the work that has been done in a sub-field of soil mechanics, namely stochastic soil dynamics, whereby the ground is viewed as a material whose properties are random functions of space and time. This is a relatively new concept, which first emerged in the early 1960s within the context of acoustic waves in the ocean and electromagnetic waves in the atmosphere. Since then, new fields of application, such as biomechanics, non-destructive testing, aeroelasticity, etc have been found. The present article discusses the aforementioned concept within the framework of the various sub-disciplines of soil mechanics, always emphasizing the predominantly dynamic (i.e. transient) nature of these phenomena. Finally, besides presenting an extensive review of the pertinent literature, we have also included some background material on stochastic soil dynamics plus a few thoughts on future trends in this rather novel field, which has been under development in the last 20 years.

Scattering of seismic waves by cracks in multi-layered geological regions. I. Mechanical model

In this work, a hybrid boundary integral equation method (BIEM) is developed, based on both displacement and hypersingular traction formulations, for the analysis of time-harmonic seismic waves propagating through cracked, multi-layered geological regions with surface topography and under plane strain conditions. Specifically, the displacement-based BIEM is used for a multi-layered deposit with interface cracks, while the regularized, traction-based BIEM is used when internal cracks are present within the layers. The standard uni-dimensional boundary element with parabolic shape functions is employed for discretizing the free surface and the layer interfaces, while special discontinuous boundary elements are placed near the crack tips to model the asymptotic behaviour of both displacements and tractions. This formulation yields displacement amplitudes and phase angles on the free surface of a geological deposit, as well as stress intensity factors near the tips of the cracks. Finally, in the companion paper, numerical results are presented which show that both scattered wave and stress concentration fields are sensitive to the incidence seismic wave parameters and to specific site conditions such as surface topography, layering, the presence of cracks and crack interaction.

Seismic assessment and design of R/C bridges with irregular congiguration, including SSI effects

Abstract This work investigates the effect of a modelling approach, also including the interaction phenomenon between supporting ground and the pier plus deck system, on the seismic response of reinforced concrete (R/C) bridges with irregular configuration, as well as its ramifications on the design of the piers. The focus is on a four span highway bridge with piers of unequal height crossing a mountain valley. The bridge and its foundation system, including the surrounding soil, are modelled by finite elements plus the spring/dashpot/added mass discrete parameter system. A hierarchy of finite element meshes is developed, starting with shell elements, and ending with linear elements whose performance as far as dynamic loads are concerned is gauged to be completely satisfactory. Moreover, two basic types of foundations are examined, namely spread footings versus pile groups. Following a preliminary design of the bridge, a series of time history analyses of the combined deck–pier–foundation system are performed, the results of which are used in assessing the influence of foundation compliance on the superstructure. Furthermore, the influence of key construction details such as pier–to–deck connection on the dynamic displacement and force fields that develop, is also examined. Finally, a series of recommendations are given on when and how to account for the influence of the ground in the design of the piers.

A Green's function method to SH-wave motion in a random continuum

Abstract In this work, we develop Greens functions for SH waves in an elastic continuum exhibiting large randomness. These functions can be subsequently used within the context of BEM formulations for wave scattering problems of engineering interest. More specifically, the methodology developed here employs a series expansion for the proposed Greens functions, where the basis functions are orthogonal polynomials of a random argument. The corresponding BEM formulation is then presented in the Fourier transform domain. This way, we depart from earlier BEM derivations based on perturbations, which imply the presence of ‘small’ amounts of randomness in the elastic continuum, and move towards the development of methods that are computationally efficient alternatives to Monte Carlo simulations.

Scattering of seismic waves by cracks in multi-layered geological regions: II. Numerical results

The mechanical model for plane strain, time-harmonic seismic wave propagation problems in cracked, multi-layered geological regions with surface topography and non-parallel interfaces was described in the first part of this work. Here, this model is used to investigate the response of such a region to the presence of traveling elastic waves generated by a seismic source. The computational methodology that was developed in the first part is based on a combination of both the regular (displacement-based) and the hypersingular (traction-based) Boundary Integral Equation Method (BIEM). First, the accuracy and convergence characteristics of this hybrid BIEM are studied. Then, a series of problems involving four different configurations of a reference geological deposit with both interface and internal cracks are solved, for a loading that is due to a seismically-induced pressure wave propagating upwards from the underlying rigid half-plane. The purpose of the numerical study is to investigate the influence of various key parameters of the problem, such as frequency and incidence angle of the incoming wave, size of the surface relief, location and size of the buried cracks, interaction effects between cracks and finally the presence of layers, on both the scattered displacement field and the stress concentration field.

Seismic response of lined tunnels in the half-plane with surface topography

In this work, we examine the seismic response of multiple tunnels reinforced with liners and buried within the elastic homogeneous half-plane in the presence of surface relief. The seismic waves are upward propagating, time-harmonic, horizontally polarized shear (SH) waves. More specifically, we examine: (a) the scattered wave fields along the free surface and inside the half-plane with the embedded tunnels; (b) the dynamic stress concentration factors that develop at the soil-liner interfaces; (c) the stresses and displacements that develop inside the tunnel liners. We use a sub-structuring technique that is based on the direct boundary element method to model each constituent part of the problem separately. Then, assembly of the full problem is accomplished through the imposition of compatibility and equilibrium conditions at all interfaces. Next, a detailed verification study is carried out based on comparisons against available analytical and/or numerical results for a series of test examples. Subsequently, detailed numerical simulations are conducted and the results of these parametric studies reveal the influence of the following key parameters on the soil-tunnel system response: (a) the shape of the free-surface relief; (b) the depth of placement of the tunnels and their separation distance; (c) the SH-wavelength to tunnel diameter ratio; (d) the elastic properties of the tunnel lining rings and (e) the dynamic interaction effects between the free-surface relief and the tunnels.

Dynamic response of unlined tunnels in soil with random properties

The analysis of underground structures under various environmental and mechanical loads is fraught with uncertainty. In this work, a simple perturbation-based probabilistic model is employed for modeling the complex composition of the geological material surrounding the underground opening. In addition, boundary elements are used for modeling the geometry of the problem, which is basically assumed to be that of an unlined cavity in the halfspace under plane strain conditions. The transient nature of the loading is taken into account through use of the Laplace transform with respect to the time variable. In a series of examples depicting some representative cases, the effect of material stochasticity is reflected in the presence of a non-zero, time dependent covariance matrix defined at the nodes of the discretized structure. In essence, the use of a uni-dimensional random field for representing the pressure and shear wave velocities of the geological medium is an alternative way for modeling the combined effects of heterogeneity, the presence of discontinuities, etc., which alter the simple, yet basic assumption that the ground can be modeled as a homogeneous viscoelastic semi-infinite continuum.

Seismic analysis of buried pipeline in a 3D soil continuum

An efficient numerical approach based on both boundary and finite element methods is developed in this work. This development is capable of realistic three dimensional analyses of soil-structure interaction problems in the real time domain and is specifically tailored to buried lifelines. In particular, boundary elements are used in a surface-only representation of the buried cavity problem for determining influence functions at the cavity/pipeline interface for the case of prescribed motions at the control point. Subsequently, these influence functions are converted into loads and are used as input to a finite element model of the pipeline. Following careful validation studies, the present methodology is applied to a real site with known seismological characteristics and the results are gauged against empirical design formulae. It is shown that the seismically induced stress state in a buried pipeline is more pronounced in the case of transverse vibrations than in the case of longitudinal vibrations.

A generalized Helmholtz equation fundamental solution using a conformal mapping and dependent variable transformation

Fundamental solutions to a generalized Helmholtz equation are determined through dependent variable transforms using the material properties and independent variable transforms based on conformal mapping. This allows variable wave speed media to be examined under some fairly broad material property constraints.