Jyh-Jong Liao

Elastic solutions for a transversely isotropic half‐space subjected to a point load

SUMMARY We rederive and present the complete closed-form solutions of the displacements and stresses subjected to a point load in a transversely isotropic elastic half-space. The half-space is bounded by a horizontal surface, and the plane of transverse isotropy of the medium is parallel to the horizontal surface. The solutions are obtained by superposing the solutions of two infinite spaces, one acting a point load in its interior and the other being free loading. The Fourier and Hankel transforms in a cylindrical co-ordinate system are employed for deriving the analytical solutions. These solutions are identical with the Mindlin and Boussinesq solutions if the half-space is homogeneous, linear elastic, and isotropic. Also, the Lekhnitskii solution for a transversely isotropic half-space subjected to a vertical point load on its horizontal surface is one of these solutions. Furthermore, an illustrative example is given to show the e⁄ect of degree of rock anisotropy on the vertical surface displacement and vertical stress that are induced by a single vertical concentrated force acting on the surface. The results indicate that the displacement and stress accounted for rock anisotropy are quite di⁄erent for the displacement and stress calculated from isotropic solutions. ( 1998 John Wiley & Sons, Ltd.

Direct tensile behavior of a transversely isotropic rock

Abstract The tensile behavior of a transversely isotropic rock is investigated by a series of direct tensile tests on cylindrical argillite specimens. To study the deformability of argillite under tension, two components of an electrically resistant type of strain gage with a parallel arrangement, or a semiconductor strain gage, are adopted for measuring the small transverse strain observed on specimens during testing. The curves of axial stress vs axial strain and vs average volumetric strain are presented for argillite specimens with differently inclined angles of foliation. Experimental results indicate that the stress-strain behavior depends on the foliation inclination of specimens with respect to the loading direction. The five elastic constants of argillite are calculated by measuring two cylindrical specimens in the manner recommended by Wei and Hudson. Based on theoretical analysis results, the range of the foliation inclination of the specimens tested is investigated for feasibility obtaining the five elastic moduli. A dipping angle of the foliations (θ) of 30–60° with respect to the plane normal to the loading direction is recommended. The final failure modes of the specimens are investigated in detail. A sawtoothed failure plane occurs for the specimens with a high inclination of foliation with respect to the plane perpendicular to the loading direction. On the other hand, a smooth plane occurs along the foliation for specimens with low inclination of foliation with respect to the plane normal to the loading direction. A conceptual failure criterion of tensile strength is proposed for specimens with a high inclination of foliation.

Displacements and stresses due to a vertical point load in an inhomogeneous transversely isotropic half-space

We present the solutions for displacements and stresses subjected to a vertical point load in a continuously inhomogeneous transversely isotropic half-space with Young’s and shear moduli varying exponentially with depth. Planes of transverse isotropy are assumed to be parallel to the horizontal surface. The solutions for the half-space are obtained by superposing the solutions of two full spaces, one with a point load in its interior and the other with opposite traction of the first full space along the z ¼ 0 plane. The Hankel transform in a cylindrical co-ordinate system is employed for deriving the solutions. However, the resulting integrals for displacements and stresses involve polynomial, exponential function, and Bessel function that cannot be given in closed form; hence, numerical techniques are adopted in this work. In order to check the accuracy of numerical procedures, the comparisons are carried out with the homogeneous solutions of Liao and Wang, and the calculated results agree with those to nine decimal places. Furthermore, two illustrative examples are presented to elucidate the effect of inhomogeneity, and the type and degree of rock anisotropy on the vertical surface displacement and vertical normal stress in the inhomogeneous isotropic/ transversely isotropic rocks subjected to a vertical concentrated force acting on the surface. The calculated results show that the induced displacement and stress are decisively influenced by the inhomogeneity, and the degree and type of material anisotropy. The proposed solutions can more realistically simulate the actual stratum of loading problem in many areas of engineering practice. r 2003 Elsevier Ltd. All rights reserved.

Elastic solutions for a transversely isotropic half-space subjected to buried asymmetric-loads

Elastic closed-form solutions for the displacements and stresses in a transversely isotropic half-space subjected to various buried loading types are presented. The loading types include finite line loads and asymmetric loads (such as uniform and linearly varying rectangular loads, or trapezoidal loads). The planes of transverse isotropy are assumed to be parallel to its horizontal surface. These solutions are directly obtained from integrating the point load solutions in a transversely isotropic half-space, which were derived using the principle of superposition, Fourier and Hankel transformation techniques. The solutions for the displacements and stresses in transversely isotropic half-spaces subjected to linearly variable loads on a rectangular region are never mentioned in literature. These exact solutions indicate that the displacements and stresses are influenced by several factors, such as the buried depth, the loading types, and the degree and type of rock anisotropy. Two illustrative examples, a vertical uniform and a vertical linearly varying rectangular load acting on the surface of transversely isotropic rock masses, are presented to show the effect of various parameters on the vertical surface displacement and vertical stress. The results indicate that the displacement and stress distributions accounted for rock anisotropy are quite different for those calculated from isotropic solutions. Copyright

The influence of surface ruptures on building damage in the 1999 Chi-Chi earthquake: a case study in Fengyuan City

Abstract In addition to the main surface rupture along the Chelungpu fault associated with the 1999 Chi-Chi, Taiwan earthquake, numerous secondary or branch ruptures on hangingwall were also observed. These secondary surface ruptures are parallel or sub-parallel to the main rupture within a distance of a few meters to 1–2 km. The rupture length of these secondary ruptures varies from a few tens of meters up to 5 km. The surface deformation resulted in serious damages of buildings. The present work studied the features of surface deformation on the hangingwall around the Chung-Cheng Park, Fengyuan, Taichung. Three distinct surface ruptures, minor ruptures and tension cracks were observed in this area. The observed distribution and types of building damage on the hangingwall are demonstrated. Due to the difference in geological condition and complex pattern of surface deformation, the resulted building damages on the hangingwall vary. A series of site investigation including field survey, drilling, seismic prospecting, P–S logging tests and laboratory tests were carried out in the interested area. A geological structure model was proposed on the basis of the results of site investigation. Numerical simulation was carried out to model the surface deformation as well as subsurface potential damage zone of an active fault. It reasonably explains the observed pattern of surface deformation and indicates that the surface deformation zone during a catastrophic earthquake is predictable.

Stress influence Charts for transversely isotropic rocks

A graphical procedure to calculate stresses in a transversely isotropic half-space subjected to a three-dimensional surface load has been developed. The surface load can be distributed on an irregularly-shaped area. The planes of transverse isotropy are assumed to be parallel to the horizontal surface of the half-space. The closed-form solutions for stresses at a point under the vertex of a loading sector, with a unit load intensity are presented first. Based on these solutions, five influence charts are constructed for calculating the six components of a stress tensor at any given point in the half-space. The charts are composed of unit blocks. Each unit block is bounded by two adjacent radii and arcs, and contributes the same level of influence to the stress within the half-space. An example is presented to demonstrate the use of the new graphical method. For the case analyzed, results from the new graphical method agree with those of analytical solutions within 3%. The new influence charts can be a practical alternative to the existing analytical or numerical solutions, and provides results with reasonable accuracy.

Deriving landslide dam geometry from remote sensing images for the rapid assessment of critical parameters related to dam-breach hazards

Dam-breaches that cause outburst floods may induce downstream hazards. Because landslide dams can breach soon after they are formed, it is critical to assess the stability quickly to enable prompt action. However, dam geometry, an essential component of hazard evaluation, is not available in most cases. Our research proposes a procedure that utilizes post-landslide orthorectified remote sensing images and the pre-landslide Digital Terrain Model in the Geographic Information System to estimate the geometry of a particular dam. The procedure includes the following three modules: (1) the selection of the reference points on the dam and lake boundaries, (2) the interpolation of the dam-crest elevation, and (3) the estimation of dam-geometry parameters (i.e., the height, length, and width), the catchment area, the volumes of barrier lake and landslides dam. This procedure is demonstrated through a case study of the Namasha Landslide Dam in Taiwan. It was shown the dam-surface elevation estimated from the proposed procedure can approximate the elevation derived from profile leveling after the formation of the landslide dam. Thus, it is feasible to assess the critical parameters required for the landslide dam hazard assessment rapidly once the ortho-photo data are available. The proposed procedure is useful for quick and efficient decision making regarding hazard mitigation.

Elastic solutions of displacements for a transversely isotropic half-space subjected to three-dimensional buried parabolic rectangular loads

This paper presents the closed-form solutions for estimating the displacements in a transversely isotropic half-space subjected to three-dimensional buried linearly varying, uniform, and parabolic rectangular loads. The loading types include an upward linearly varying load, a downward linearly varying load, a uniform load, a concave parabolic load, and a convex parabolic load on a rectangle. The planes of transverse isotropy are assumed to be parallel to the horizontal surface of the half-space. The presented solutions are obtained from integrating of the point load solutions in a Cartesian co-ordinate system for the transversely isotropic half-space. The solutions for a transversely isotropic medium subjected to parabolic rectangular loads are never mentioned in the literature. The parabolic loads might be more realistic than linear variable or uniform loads acting on foundations. The buried depth, the dimensions of loaded area, the type and degree of material anisotropy, and the loading type for transversely isotropic half-spaces influence these solutions. An illustrative example is presented to investigate the effect of the type and degree of rock anisotropy, and the loading types on the displacement. The proposed solutions can provide reasonably results to estimate the induced displacements in the isotropic/transversely isotropic geomaterials subjected to three-dimensional buried parabolic rectangular loads for practical purposes.

A systematic approach for the assessment of flooding hazard and risk associated with a landslide dam

Inundation caused by landslide dams may occur in the upstream and downstream of the dams. A proper flooding hazard assessment is required for reaction planning and decision-making to mitigate possible flooding hazards caused by landslide dams. Both quick and detailed procedures can be used to evaluate inundation hazards, depending on the available time and information. This paper presents a systematic approach for the assessment of inundation hazards and risks caused by landslide dam formation and breaches. The approach includes the evaluation of dam-breach probability, assessment of upstream inundation hazard, assessment of downstream inundation hazard, and the classification of flooding risk. The proposed assessment of upstream inundation estimates the potential region of inundation and predicts the overtopping time. The risk level of downstream flooding is evaluated using a joint consideration of the breach probability of a landslide dam and the level of flooding hazard, which is classified using a flooding hazard index that indicates the risk of potential inundation. This paper proposes both quick and detailed procedures for the assessments of inundation in both the upstream and downstream of a landslide dam. An example of a landslide dam case study in southern Taiwan was used to demonstrate the applicability of the systematic approach.

GRAVITATIONAL STRESSES IN ANISOTROPIC RIDGES AND VALLEYS WITH SMALL SLOPES

We present an approximate analytical solution for the effects of topography on near-surface gravity-induced stresses in symmetric anisotropic ridges and valleys. The solution can be used for laterally constrained ridges and valleys consisting of isotropic, transversely isotropic, or orthotropic rock with horizontal or vertical layers. A parametric study on the effect of anisotropy type and degree of rock anisotropy on gravity-induced stresses is presented. The magnitude of the predicted stresses is of the order of the characteristic stress pgH, where H is the height of the ridge or depth of the valley. The approximate solution is limited to ridges and valleys with small slopes not exceeding 10%.