Richard E. Goodman

THE THEORY OF BASE FRICTION MODELS

Abstract The base friction principle is used widely to reproduce the effects of gravity in two dimensional physical models of excavations in rock. The body force of gravity is simulated by the drag of a belt moving along the underside of the model. This paper develops the mathematical principles upon which analogy between gravity and base friction can be examined. It is shown that the equations of motion in the realm of the model are obtainable from those of the real world by replacing any linear or angular acceleration term by corresponding linear or angular velocity term. For limiting equilibrium analysis, in which motion is incipient, the analogy is flawless.

Numerical and Physical Modeling of Flexural Slip Phenomena and Potential for Fault Movement

Numerical and Physical Modeling of Flexural Slip Phenomena and Potential for Fault Movement. In the course of a site evaluation study for a proposed liquified natural gas (LNG) terminal on the West Coast of the United States, geomechanical studies were undertaken to complement conventional geological investigations. The geologic structure under investigation was a flexural fold formed by a compressional tectonic environment. It consists of an approximately 4000 foot thick shale deposit, with bedding planes dipping roughly 45 degrees and outcropping at the rock surface. Continuous folding had produced several reverse bedding plane faults, with slip separations at the bedrock surface of up to 8 feet in the last 100 000 to 200 000 years.

A ground reaction curve based upon block theory

SummaryDiscontinuities in a rock mass can intersect an excavation surface to form discrete blocks (keyblocks) which can be unstable. Once a potentially unstable block is identified, initial normal and shear stresses on each block face are calculated using elastic theory, and are then modified by discontinuity deformations as the keyblock displaces. The modified stresses are summed into resultant forces to evaluate block stability. Since the resultant forces change with displacement, successive increments of block movement are examined to see whether the block ultimately becomes stable or fails. Calculated keyblock stability increases with larger in situ stress magnitudes, larger lateral stress ratios, and larger shear strengths. Discontinuity stiffness controls block displacement more strongly than it does stability itself. Large keyblocks are less stable than small ones, and stability increases as blocks become more slender.

Precursory and coseismic water-pressure variations in stick-slip experiments

We have measured changes in water pressure as a function of shear displacement and time during undrained direct shear of flat joint surfaces in saturated quartz monzonite. During the postpeak shear behavior, the water pressure generally decreases, probably owing to opening of new cracks and enlarging of some of the existing cracks and pores. During the stick part of the stick-slip event, there is an accelerated drop in water pressure ( preseismic ) followed by a momentary rise during the slip ( coseismic ). If it is accepted that the slip part of the stick-slip event is a “laboratory earthquake,” then the pore-pressure variations during the stick-slip may have a direct bearing on earthquake prediction through water-level observations in open wells. It is hereby suggested that for more meaningful measurements of water-pressure variations in wells, pressure transducers with a quick response factor be used with packers, rather than measuring the water-level variations in open wells with appreciable time lag.

Rotational kinematics and equilibrium of blocks in a rock mass

Abstract Under certain conditions, rock blocks “removable” at a free surface but safe against translational (falling or sliding) failure may fail in rotation. This paper examines: (1) the kinematic constraints on rotation of tetrahedral blocks; and (2) rotational stability under gravity. The results are presented graphically, using the full-sphere stereographic projection, in keeping with ideas and notation developed by Goodman and Shi. A rotational stability diagram for varying orientations of gravity is presented. If friction angles are known the sphere can be subdivided (with some grey areas) into failure mode zones corresponding to lifting, sliding on one or two planes, pure rotation or sliding-rotation.

Fault and system stiffnesses and stick-slip phenomena

SummaryThis paper discusses the influence of system stiffness on the dynamic instability of fault surfaces under laboratory conditions for a number of test modes. In conjunction with shear load stiffness, the normal load stiffness, often neglected, is shown to have a considerable effect on the stick-slip process —its presence or absence and its characteristics. Also appropriate stiffnesses are suggested for an earthquake sequence modeled as a growing dislocation.

Electrical Resistivity Changes in Rocks During Frictional Sliding and Fracture

Significant changes of electrical resistivity of saturated rocks and water pressure along sliding surface occurred during stick-slips in our direct shear experiment. Two types of changes of electrical resistivity occurred. In the first, resistivity decreased with increasing shear stress, reached minimum together with a sudden release of shear stress and returned to a higher value immediately afterwards. In the second, resistivity again decreased with increasing stress but, in contrast to the first type of changes, it decreased further upon the sudden drop of shear stress. The magnitude and the direction of the changes of water pressure on the sliding surface during stick-slip were not uniform, indicating local variations of surface deformation.

The influence of rock anisotropy on stress measurements by overcoring techniques

SummaryThe Influence of Rock Anisotropy on Stress Measurements by Overcoring TechniquesA medium is anisotropic if its properties vary with direction. This is the general characteristic of many rocks, for example, schists, slates, gneisses, phyllites and other metamorphic rocks. Bedded and regularly jointed rocks also display anisotropic behavior.This paper is concerned with the influence of rock anisotropy on in-situ stress measurements. It is limited, to stress measurements by overcoring techniques for which strains and displacements are recorded either on the walls of a pilot hole at the end of one or several boreholes or within instrumented solid or hollow inclusions perfectly bonded to the surface of the pilot hole. The rock is described as homogeneous, continuous, anisotropic and linearly elastic.The following questions are answered with special emphasis on rocks that can be classed as transversely isotropic or orthotropic: the number of independent measurements obtainable in a single borehole; the number of boreholes required to determine the in-situ stress field; the influence of rock anisotropy on these numbers; the influence of the anisotropy type and the error involved by neglecting rock anisotropy.

The resolution of stresses in rock using stereographic projection

Abstract This paper shows how to find the shear and normal stress on an arbitrarily oriented plane of weakness in a known three dimensional stress field by manipulations on a stereographic projection. First are found graphically the direction angles of the normal to the plane relevant to co-ordinate axes in principal directions. Knowing the magnitude of the principal stresses and the direction cosines of the plane, the magnitude of the resultant stress across the plane is calculated. The direction of the resultant stress is found by calculating its direction angles and plotting these angles as small circles centered about the principal directions, the point of intersection of the small circles being the stereographic projection of the resultant stress. The angle between the resultant stress and the plane of weakness is read directly from the stereographic projection, facilitating calculation of the magnitudes of the shear and normal stresses of the plane. Finally, the direction of the maximum shear stress in the plane is found graphically, making use of the fact that the normal stress, shear stress, and resultant stress must be coplanar. The principles of stereographic projection are reviewed and an example of stress resolution is worked out.

Determination of the ‘Design Block’ for Tunnel Supports in Highly Jointed Rock

Publisher Summary This chapter presents the determination of design block for tunnel supports in highly jointed rock mass. For an underground excavation through a competent and discontinuous rock mass, a preferred block failure can be expected with particular boundary joint orientations and half-space combinations. This block can be found analytically by using the rock mass discontinuity distributions, the rock mass spacing distributions, and the procedures and methods of block theory. The three major factors that contribute to the block failure are the frequency of its boundary planes in the rock mass, the shape of its joint pyramid, and the instability of its joint pyramid. Block theory is established as a relevant and powerful tool for assessing the influence of discontinuities and tunnel geometry on design factors. The design block for an opening can be found prior to actual construction on the basis of the geological mapping or exploration data at hand. A great variability in joint orientations can be handled by introducing more joint sets to the system.