Ronald Y. S. Pak

Seismic soil-structure interaction analysis by direct boundary element methods

Abstract Upon establishing the mathematical framework necessary for a proper understanding of the analytical theory, a regularized form of the conventional direct boundary integral equation formulation for three-dimensional elastodynamics is presented for a general anisotropic medium. Founded on the basis of a full decomposition of the Greens functions into regular and singular parts, the alternative boundary integral equation format is compact and without demand ingmathematical and numerical complexities such as Cauchy principal values. Extended to deal with general seismic soil-structure interaction problems in semi-infinite media, the formulation is implemented computationally together with a rigorous treatment of singular dynamic multi-layered viscoelastic half-space Greens functions and interfacial boundary tractions arising in typical soil-structure-foundation configurations. A set of new benchmark numerical results are included.

Static fundamental solutions for a bi-material full-space

In this paper, the complete static response of two joined dissimilar half-spaces due to an arbitrary interior point load is derived. By means of a method of displacement potentials and integral transforms, a dual format of the solution in the form of a Hankel integral representation and algebraic closed-form expressions is presented. As illustrations, its analytical degeneration to benchmark solutions for a homogeneous medium as well as its variation under general geometric and material conditions are shown. The importance of the dual format of the bi-material solution in connection with the method of asymptotic decomposition to the development of a rigorous treatment of its dynamic counterpart is also demonstrated.

Experimental developments for studying static and seismic behavior of retaining walls with liquefiable backfills

The effects of earthquakes on cantilever retaining walls with liquefiable backfills were studied. The experimental techniques utilized in this study are discussed here. A series of centrifuge tests was conducted on aluminum, fixed-base, cantilever wall models retaining saturated, cohesionless backfills. Accelerations on the walls and in the backfill, static and excess pore pressures in the soil, and deflections and bending strains in the wall were measured. In addition, direct measurements of static and dynamic lateral earth pressures were made. In some tests, sand backfills were saturated with the substitute pore fluid metolose. Modeling of model type experiments were conducted. The experimental measurements were found internally consistent and repeatable. Both static and dynamic earth pressure measurements were determined to be reliable. It was also observed that for the test configuration adopted, a special boundary treatment such as the use of duxseal is optional. Static and seismic modeling of models were also successful, which indicated that the assumed scaling relations were essentially correct.

Scattering of vertically-incident P-waves by an embedded pile

An exact theoretical formulation is presented for the analysis of a thin-walled pile embedded in an elastic half-space under vertically-incident P-wave excitation. In the framework of three-dimensional elastodynamics and a shell theory, the axisymmetrical wave-scattering problem is shown to be reducible to a set of Fredholm boundary integral equations. With the incorporation of the singular characteristics of the wave-induced contact load distributions into the solution scheme, a computational boundary element method is developed for a rigorous treatment of the seismic soil-structure interaction problem. Typical results for the dynamic contact load distributions, displacements, complex-valued foundation input motion functions, and resonant pile foundation response are included for direct engineering applications.

Three-dimensional wave propagation analysis of a smoothly heterogeneous solid

Abstract A method of analysis is presented for three-dimensional wave propagation problems of a vertically-heterogeneous half-space with a linear shear wave velocity profile. With the aid of a displacement-potential representation, Hankel transforms and Fourier decompositions, the dynamic response of the semi-infinite solid to an arbitrarily distributed buried source is shown to admit integral representation in terms of modified Bessel functions. Specific aspects of the problem such as the multiple poles along the inversion path on the complex plane and the characteristics of the wave propagation in the vertical direction are elucidated. Apart from its intrinsic interest, the solution can be degenerated to ring-load and point-load Greens functions which are fundamental to boundary integral equation formulations.

Elastodynamic Green's functions for a smoothly heterogeneous half-space

In this paper, the response of a vertically heterogeneous elastic half-space with a smooth modulus variation under a set of time-harmonic ring- and point-sources is derived analytically. A method of evaluation via asymptotic decomposition for the singular Greens functions is presented. In the technique, the Greens functions are decomposed into an analytical part and a residual component. Capturing the corresponding singular behavior, the analytical parts of the ring- and point-load Greens functions are expressible in terms of the elliptic integrals and algebraic functions, respectively. The residual integrals which are regular can be evaluated by numerical contour integration. To obtain correct results, one must note and take into account the existence of multiple poles along the formal path of the inversion integrals, the details of which are discussed in the paper. To highlight the various aspects of the physical problem, a set of illustrative numerical results is included.

Rocking rotation of a rigid disc in a half-space

Abstract An analysis is presented for the determination of the rotational response of a rigid circular disc embedded in a semi-infinite elastic medium under the action of a rocking moment. With the aid of Hankel transforms, a relaxed treatment of the mixed boundary value problem is formulated as dual integral equations. On reduction of the dual integral equations to a Fredholm integral equation which features a closed-form kernel, solutions to the inclusion problem are computed. In addition to providing a unified view of existing solutions for zero and infinite embedments, the present analysis reveals a severe boundary-layer phenomenon which is apt to be of significance to this class of problem in general. As illustrations, numerical results on the load-displacement relation and the response of the embedding medium, as well as the contact load distribution, are included.

Torsion of a rigid disc in a half-space

Abstract A mathematical treatment is presented for the torsional response of a rigid disc embedded in a semi-infinite elastic medium. By means of the theory of Hankel transforms, an exact formulation for the mixed boundary value problem is obtained in the form of dual integral equations. On reduction of the dual integral equations to a Fredholm integral equation of the second kind, solutions to the problem are computed. In addition to providing a unified view of past solutions for surface and infinite embedments, the present analysis reveals a severe boundary-layer phenomenon which is apt to be of relevance to inclusion problems in general. As illustrations, selected numerical results on the torque-rotation relationship, the stress and displacement fields, as well as the contact distribution are provided.

ONR MURI project on soil blast modeling and simulation

Current computational modeling methods for simulating blast and ejecta in soils resulting from the detonation of buried explosives rely heavily on continuum approaches such as Arbitrary Lagrangian-Eulerian (ALE) and pure Eulerian shock-physics techniques. These methods approximate the soil as a Lagrangian solid continuum when deforming (but not flowing) or an Eulerian non-Newtonian fluid continuum when deforming and flowing at high strain rates. These two extremes do not properly account for the transition from solid to fluid-like behavior and vice versa in soil, nor properly address advection of internal state variables and fabric tensors in the Eulerian approaches. To address these deficiencies on the modeling side, we are developing a multiscale multiphase hybrid Lagrangian particle-continuum computational approach, in conjunction with coordinated laboratory experiments for parameter calibration and model validation. This paper provides an overview of the research approach and current progress for this Office of Naval Research (ONR) Multidisciplinary University Research Initiative (MURI) project.

Axisymmetric stress-transfers from an embedded elastic cylindrical shell to a half-space

Within the framework of three-dimensional classical elastostatics and thin shell theories, a rigorous mathematical formulation is presented for the torsionless axisymmetric stress-transfer problem of a cylindrical shell of finite length embedded in a semi-infinite solid. By virtue of a set of ring-load Green’s functions for the shell and a group of fundamental solutions for the half-space, the mechanical interaction problem is shown to be reducible to a pair of Fredholm integral equations. Through the analysis of an auxiliary set of Cauchy integral equations, the singularities of the resultant contact stress distributions are rendered explicit, the results of which are incorporated in a numerical procedure. Typical solutions for the axial and radial load-transfers, contact stress distributions, as well as other related responses are included as illustrations. In addition to furnishing results of direct relevance to a number of engineering applications, the present treatment is apt to be useful as a basis of assessment for various approximate methods for this class of contact problems.