Forced Vertical Vibration of Rigid Circular Disc on a Transversely Isotropic Half-Space

Vertical vibration of a rigid circular disc attached to the surface of a transversely isotropic half-space is considered in such a way that the axis of material symmetry is normal to the surface of the half-space and parallel to the vibration direction. By using Hankel integral transforms, the mixed boundary-value problem is transformed to a pair of integral equations termed dual integral equations in the literature, which generally can be reduced to a Fredholm integral equation of the second kind. With the aid of complex variable or contour integration the governing integral equation is numerically solved in the general dynamic case. The reduced static case of the dual integral equations is solved analytically and the vertical displacement, the contact pressure, and the static impedance/compliance function are explicitly solved. The dynamic contact pressure under the disc and the impedance function are numerically evaluated, and it is shown that the singularity that exists at the edge of the disc is the same as the one obtained for the static case. In addition, the impedance functions evaluated here are identical to the solution given by Luco and Mita for the isotropic domain. To show the effect of different material anisotropy, the numerical evaluations are given for some different transversely isotropic materials and compared.     

Three-Dimensional Nonlinearly Varying Rectangular Loads on a Transversely Isotropic Half-Space

In engineering situations, loads applied to the four corners of a rectangle might have different values and might not be uniformly or linearly distributed. A configuration of linearly or nonlinearly varying loads with different contact pressures at each corner can be represented as a superposition of various loading types. The loading types include uniform, linearly varying in the x direction, linearly varying in the y direction, nonlinearly varying in the x direction, and nonlinearly varying in the y direction. This work newly presents the first and second loading solutions, and derives the others therefrom. These solutions are directly obtained by integrating the point load solutions in a transversely isotropic half-space. The presented solutions are concise and easy to use; they specify that the type and degree of material anisotropy, the dimensions of the loaded region, and the loading types decisively affect the displacements and stresses in a transversely isotropic half-space. The proposed solutions can simulate realistically the actual loading problem in many engineering situations.     

Displacements and Stresses Due to a Uniform Vertical Circular Load in an Inhomogeneous Cross-Anisotropic Half-Space

In this work, we present the solutions for displacements and stresses along the centerline of a uniform vertical circular load in an inhomogeneous cross-anisotropic half-space with its Young’s and shear moduli varying exponentially with depth. The planes of cross anisotropy are assumed to be parallel to the horizontal surface. The presented solutions can be directly integrated from the point load solution in a cylindrical coordinate system, which were derived by the writers. However, the resulting integrals of the circular solution for displacements and stresses cannot be given in closed form; hence, numerical integrations are required. For a homogeneous cross-anisotropic half-space, the numerical results agree very well with the exact solutions of Hanson and Puja, published in 1996. Two examples are given to elucidate the effect of inhomogeneity, and the type and degree of soil anisotropy on the vertical displacement and vertical normal stress in the inhomogeneous isotropic/cross-anisotropic soils subjected to a uniform vertical circular load acting on the surface. The proposed solutions can more realistically simulate the actual stratum of loading problem in many areas of engineering practice.     

Asymmetric Dynamic Green’s Functions in a Two-Layered Transversely Isotropic Half-Space

By virtue of a complete representation using two displacement potentials, an analytical derivation of the elastodynamic Green’s functions for a transversely isotropic layer underlain by a transversely isotropic half-space is presented. Three-dimensional point-load and patch-load Green’s functions for stresses and displacements are given in the complex-plane line-integral representations. The formulation includes a complete set of transformed stress-potential and displacement-potential relations in the framework of Fourier expansions and Hankel integral transforms, that is useful in a variety of elastodynamic as well as elastostatic problems. For the numerical computation of the integrals, a robust and effective methodology is laid out. Comparisons with the existing numerical solutions for a two-layered transversely isotropic half-space under static surface load, and a homogeneous transversely isotropic half-space subjected to buried time-harmonic load are made to confirm the accuracy of the present solutions. Selected numerical results for displacement and stress Green’s functions are presented to portray the dependence of the response of the two-layered half-space on the frequency of excitation and the role of the upper layer.     

Stress Solution of Multiple Elliptic Hole Problem in Plane Elasticity

Using Schwarz’s alternating method and Muskhelishvili’s complex variable function techniques, this paper presents an iterative algorithm method for the effective and accurate calculation of the stresses in an elastic solid of infinite extent containing multiple elliptic holes and subject to external loading at the infinity. The elliptic holes can have different dimensions and locate at any points while their axis orientations must be orthogonal. The proposed iterative algorithm method is based on the approximation of the resultant force vector on each elliptic hole boundary as a series of complex variable. As a result, exact closed-form analytical stress solutions can then be obtained for the solid with a single elliptic hole whose boundary is subject to the reverse resultant force vector in the forms of complex series. The numerical results presented in the paper show that the iterative solution converges quickly and stably. The proposed convergent criterion ensures the satisfaction of the required accuracy of the stress results. The stress concentration at elliptic holes can then be evaluated with high accuracy.     

Vertical Vibrations of a Rigid Foundation Embedded in a Poroelastic Half Space

This paper considers the steady-state vertical vibrations of a rigid, cylindrical massive foundation embedded in a poroelastic soil. The foundation is subjected to time-harmonic vertical loading and is perfectly bonded to the surrounding soil. The contact surface between the foundation and the soil is assumed to be smooth and fully permeable. Biot’s poroelastodynamic theory is used in the analysis. The soil underlying the foundation base is assumed to be a homogeneous poroelastic half space while the soil along the side of the foundation is assumed to consist of a series of infinitesimally thin layers. The dynamic interaction problem is solved by a simplified analytical method. The accuracy of the present solution is verified by comparisons with existing solutions for both elastodynamic and poroelastodynamic interaction problems. Selected numerical results for the vertical dynamic impedance and response factor of the rigid foundation are presented to demonstrate the influence of nondimensional frequency of excitation, depth ratio, mass ratio, shear modulus of the backfill, and poroelastic material properties on dynamic interaction between an embedded foundation and a poroelastic half space.     

Elastodynamic Potential Method for Transversely Isotropic Solid

A theoretical formulation is presented for the determination of the displacements, strains, and stresses in a three-dimensional transversely isotropic linearly elastic medium. By means of a complete representation using two displacement potentials, it is shown that the governing equations of motion for this class of problems can be uncoupled into a fourth-order and a second-order partial differential equation in terms of the spatial and time coordinate under general conditions. Compatible with Fourier expansions and Hankel transforms in a cylindrical coordinate system, the formulation includes a complete set of transformed displacement-potential, strain-potential, and stress-potential relations that can be useful in a variety of elastodynamic as well as elastostatic problems. As an illustration of the application of the method, the solution for a half-space under the action of arbitrarily distributed, time-harmonic surface traction is derived, including its specialization to uniform patch loads and point forces. To confirm the accuracy of the numerical evaluation of the integrals involved, numerical results are also included for cases of different degree of the material anisotropy, frequency of excitation, and compared with existing solutions.     

Propagation of Love Waves in an Inhomogeneous Fluid Saturated Porous Layered Half-Space with Properties Varying Exponentially

Based on Biot’s theory for transversely isotropic fluid saturated porous media, the complex dispersion equation for Love waves in a transversely isotropic fluid-saturated porous layered half-space is derived with the consideration of the inhomogeneity of the layer. The equation is solved by an iterative method. Detailed numerical calculation is presented for an inhomogeneous fluid-saturated porous layer overlying a purely elastic half-space. The dispersion and attenuation of Love waves are discussed. In addition, the upper and lower bounds of Love wave speed are also explored.     

Automatically Computing the Displacements and Stresses Induced by Three-Dimensional Arbitrarily Shaped Loads in a Transversely Isotropic Medium

Since the planes of foundations are not usually regularly shaped, and the loads are often applied on the anisotropic materials, such as transversely isotropic soils or rocks, calculating the induced displacements and stresses by an arbitrarily shaped load for a transversely isotropic medium is rather tedious and time consuming. Hence, how to estimate those values correctly and quickly by computer was the major objective in constructing the fast anisotropic displacements and stresses (FADAS). FADAS is based on the solutions of displacements and stresses in a transversely isotropic half space subjected to three-dimensional buried right-angled triangular loads, which were derived by the first writer. Utilizing these solutions, the displacements and stresses for a general triangular region at any point can be obtained by superposition. An illustrative example is given to demonstrate a few features of FADAS, and to elucidate how to compute the vertical displacement induced by a uniform vertical circular load in an equivalent medium. Results from FADAS reveal that the usage of it is correct, easy, and very fast to offer a good tool for practitioners.     

Impulse Response of Elastic Half-Space in the Wave Number–Time Domain

This paper presents formulas for the response functions in the mixed wave number–time domain for a homogeneous, elastic half-space subjected to impulsive, spatially harmonic sources on its surface. These functions are useful when obtaining the wave field in a half space elicited by dynamic surface sources of arbitrary spatial distribution on the surface, in either two or three dimensions. The formulas in this paper are obtained by contour integration of the Green’s functions in the frequency–wave number domain. The correctness and accuracy of these solutions is then assessed by comparison with the results of the well-known transient response functions for suddenly applied loads in both two and three dimensions.     

Analytical Elastic Solution Based on Fourier Series for a Laterally Confined Granular Column

This paper presents an analytical solution methodology for the complete stress and displacement fields of a laterally confined granular column loaded from the top end. The granular column is idealized as a homogeneous isotropic elastic medium with Coulomb’s friction at the lateral boundary. The solution methodology consists of an analytical procedure that incorporates a potential approach with trigonometric series and Bessel functions, finite Fourier transforms and the superposition method, and an iterative algorithm to satisfy the Coulomb’s friction condition at the lateral boundary. Stress and displacement fields are computed for a specific example and found completely consistent with corresponding finite element results. Key characteristics, computational errors, the convergence behavior, and restrictions of the present approach are discussed. The methodology developed herein can be beneficially applied in the validation process of numerical simulation techniques in granular mechanics such as finite or discrete element methods.     

Three-Dimensional Green’s Functions for a Multilayered Half-Space in Displacement Potentials

To advance the mathematical and computational treatments of mixed boundary value problems involving multilayered media, a new derivation of the fundamental Green’s functions for the elastodynamic problem is presented. By virtue of a method of displacement potentials, it is shown that there is an elegant mathematical structure underlying this class of three-dimensional elastodynamic problems which warrant further attention. Constituted by proper algebraic factorizations, a set of generalized transmission-reflection matrices and internal source fields that are free of any numerically unstable exponential terms common in past solution formats are proposed for effective computations of the potential solution. To encompass both elastic and viscoelastic cases, point-load Green’s functions for stresses and displacements are generalized into complex-plane line-integral representations. An accompanying rigorous treatment of the singularity of the fundamental solution for arbitrary source-receiver locations via an asymptotic decomposition of the transmission-reflection matrices is also highlighted.     

Early-Time Solution for a Radial Hydraulic Fracture

The small-time asymptotic solution for a penny-shaped fluid-driven fracture is obtained semianalytically. Scaling considerations indicate that the portion of the fracture that is filled with fluid increases with time according to a power law. The problem is shown to be self-similar at the length scale of the small fluid-filled region and to depend on only the mean fluid pressure at the length scale of the fracture. This similarity solution is unusual as the two length scales of the problem—the radius of the fracture and the radius of the fluid front—evolve according to two different power laws of time.     

Finite-Dynamic Model for Infinite Media: Corrected Solution of Viscous Boundary Efficiency

This paper presents the corrected development of viscous boundary efficiency initially proposed by Lysmer and Kuhlemeyer. The expressions of the energy ratio are given, in accordance with the original numerical results, and confirm the authors’ recommendations to minimize the reflected energy of body waves that impinge on the artificial boundary.     

Anisotropic Nonlinear Elastic Model for Particulate Materials

This paper presents the development of an elastic model for particulate materials based on micromechanics considerations. A particulate material is considered as an assembly of particles. The stress–strain relationship for an assembly can be determined by integrating the behavior of the interparticle contacts in all orientations and using a static hypothesis which relates the average stress of the granular assembly to a mean field of particle contact forces. Hypothesizing a Hertz–Mindlin law for the particle contacts leads to an elastic nonlinear behavior of the particulate material, we were able to determine the elastic constants of the granular assembly based on the properties of the particle contacts. The numerical predictions, compared to the results obtained during experimental studies on different granular materials, show that the model is capable of taking into account both the influence of the inherent anisotropy and the influence of the stress-induced anisotropy for different stress conditions.     

Elastic Solution for Tunneling-Induced Ground Movements in Clays

This paper presents the elastic solutions for the prediction of the tunneling-induced ground deformations for shallow and deep tunnels in the soft ground. The oval-shaped ground deformation pattern is incorporated as the boundary condition of the displacement around the tunnel section. The difference between uniform radial and oval-shaped ground deformation patterns on surface and subsurface settlements and lateral deformation is investigated. Five case studies are used to check the applicability of the proposed analytical solutions. Generally the good agreement of the predicted ground deformations can be seen with field observations for tunnels in uniform clay.     

Comparison between Coupled and Uncoupled Consolidation Analysis of a Rigid Sphere in a Porous Elastic Infinite Space

Linear consolidation analyses are usually treated either by means of Terzaghi-Rendulic uncoupled theory or Biot’s consolidation theory. In this note, the problem of consolidation displacements around an axially loaded sphere was considered. It is demonstrated that both the uncoupled analysis and the coupled analysis give the same governing equation for pore fluid pressure dissipation with time. A simplified procedure for deriving transient strain components is illustrated. A general solution for time-dependent displacements is obtained using uncoupled consolidation analysis. Close agreement is evident between the new approximate uncoupled analysis solution and the existing coupled analysis solution with a maximum error of less than 0.5%.     

Stress Reduction Coefficient and Amplification Factor for Seismic Response of Ground

Incorporating the increase of shear modulus with depth (z) by using a depth exponent p, the response of linear visco-elastic ground under steady state harmonic vibration was examined. The soil mass lying above the rigid base was discretized into a large number of horizontal layers. For a given vertical thickness (Hr) of the upper ground material in the first mode of resonance, it was noted that the resonant frequency (fr) as well as the values of amplification factor (Mf) and the stress reduction coefficient (Cd) at a given z/Hr can be computed as a function of p and D, where D is the damping ratio of the material. An increase in D leads to (1) an increase in Cd; (2) a very little increase in fr; and (3) a decrease in Mf. On the other hand, an increase in p causes: (1) an increase in fr; (2) a marginal increase in Mf; and (3) a decrease in Cd. It was evident that the values of Mf and Cd at a given depth will be affected significantly by changes in the chosen thickness of the soil deposit in resonance.     

In-Plane Nonlinear Buckling Analysis of Deep Circular Arches Incorporating Transverse Stresses

Large discrepancies exist among current classical theories for the in-plane buckling of arches that are subjected to a constant-directed radial load uniformly distributed around the arch axis. Discrepancies also exist between the classical solutions and nonlinear finite-element results. A new theory is developed in this paper for the nonlinear analysis of circular arches in which the nonlinear strain-displacement relationship is based on finite displacement theory. In the resulting variational equilibrium equation, the energy terms due to both nonlinear shear and transverse stresses are included. This paper also derives a set of linearized equations for the elastic in-plane buckling of arches, and presents a detailed analysis of the buckling of deep circular arches under constant-directed uniform radial loading including the effects of shear and transverse stresses, and of the prebuckling deformations. The solutions of the new theory agree very well with nonlinear finite-element results. Various assumptions often used by other researchers, in particular the assumption of inextensibility of the arch axis, are examined. The discrepancies among the current theories are clarified in the paper.     

Upper Bound for Cylinder Movement Using “Elastic” Fields and Its Possible Application to Pile Deformation Analysis

A new upper bound failure mechanism for the problem of rigid cylinder motion is presented. The velocity field associated with the mechanism is derived from a known elastic solution by similitude of the deformation field. The obtained upper bound value is 21% higher than the exact solution. However, the failure mechanism is continuous, involving no discontinuity, not even on the cylinder perimeter. The solution has a certain advantage if one, for example, wishes to combine its mechanism with a strain path approach to investigate the T-bar penetration problem. The absence of discontinuities in the mechanism also allows evolution of deformation under serviceability conditions, by associating a mobilized strength as a function of an average strain. Based on this approach, a load transfer function for lateral loading of piles in an undrained clay is suggested. This load transfer function involves nonlinear scaling of a stress-strain curve obtained from a triaxial compression test. An analytical, closed form, solution is given for the case of a hyperbolic stress-strain curve.