Bernard Amadei

Disaster Resilience: A National Imperative

pled with the increasing frequency of billion-dollar disaster events, such as the recent Hurricane Sandy, highlight some of the challenges to hazards and disaster policy in the United States. American society is also facing challenges to its economic, sociocultural, and environmental systems: The national jobless rate is near historic high values, more than one in six Americans now live in poverty, population migration to the coastal communities continues, and environmental degradation due to development, farming practices, or industrial processes and accidents continues to degrade natural defenses against floods, storm surge, and wildfires. Many of these changes are transformative and long lasting and, coupled with the nation’s inability to act decisively to counteract climate change, portend a future where we are more vulnerable to hazards at multiple scales. Extreme natural events (either unprecedented magnitudes or intensities of natural hazards or the unprecedented consequences of more routine hazards) may become normal occurrences under changing climatic conditions or changes in economic circumstances and social conditions.1,2 Low-probability/high-consequence events and highly improbable ones like earthquakes, pandemics, and other kinds of hazards take on more policy interest as these events become more probable.3-5 From a policy perspective, these unlikely events pose significant risk management challenges, starting with how to encourage investments to lessen the impacts of these disasters (Figure 1). by Susan L. Cutter, Joseph A. Ahearn, Bernard Amadei, Patrick Crawford, Elizabeth A. Eide, Gerald E. Galloway, Jr., Michael F. Goodchild, Howard C. Kunreuther, Meredith Li-Vollmer, Monica Schoch-Spana, Susan C. Scrimshaw, Ellis M. Stanley, Sr., Gene Whitney, and Mary Lou Zoback

Importance of anisotropy when estimating and measuring in situ stresses in rock

Foliated metamorphic rocks and laminated, stratified or bedded sedimentary rocks have properties (physical, dynamic, thermal, mechanical, hydraulic) that vary with direction and are said to be anisotropic. Rock mass anisotropy can be found in volcanic formations and sedimentary formations consisting of alternating layers or beds of different rock types. Rock masses cut by one or several regularly spaced joint sets are anisotropic in addition to being discontinuous. This paper deals with the subjects of rock anisotropy and rock stress. Both topics are important in rock engineering and are interrelated. Rock fabric controls the build-up in in situ stresses in the Earths crust, their magnitude and orientation. On the other hand, stresses and in particular compressive stresses tend to close microcracks or discontinuities thus making rock behavior non-linear and rock anisotropy pressure dependent. This paper is divided into three parts. In the first part, existing models to describe the deformability properties of anisotropic rocks as well as the laboratory and field methods to determine those properties are reviewd. Then, it is shown how to account for both rock anisotropy (intact and joint induced) and the curvature of the Earth when estimating in situ stresses in rock masses. Finally, the effect of anisotropy on the analysis of overcoring measurements with cells similar to the CSIR Triaxial Strain Cell is discussed.

Rock Anisotropy and the Theory of Stress Measurements

Keywords: Mecanique des roches Reference Record created on 2004-09-07, modified on 2016-08-08

Extensions of discontinuous deformation analysis for jointed rock masses

Beginning with the original work of Shi [Discontinuous deformation analysis: a new numerical method for the statics and dynamics of block systems. Ph.D. thesis, University of California, Berkeley (1988)], called the Discontinuous Deformation Analysis (DDA) method, a number of extensions to the method have been explored. The extensions consist of improving the contact algorithm, adding block fracturing and sub-blocking capabilities. Contacts between blocks have been modeled using an Augmented Lagrangian Method instead of the penalty method originally proposed by Shi. This allows block-to-block contacts to be enforced more precisely and block contact forces to be determined more accurately. A sub-blocking capability has been developed, whereby blocks are discretized into sub-blocks. The continuity of the sub-block contacts is preserved and the variation of stresses in each large block can be determined. The sub-blocking capability is done using a consistent formulation in which the same methodology is used for the sub-blocks as the original large blocks. This is different from other discrete block methods that imbed finite difference zones or finite elements inside larger blocks. Finally, two block fracturing algorithms have been implemented in the DDA method. Using a three-parameter (cohesion, friction, tensile strength) Mohr-Coulomb criterion, one algorithm allows intact rocks to be broken into smaller blocks. Fracturing can be in shear or tension. The second algorithm allows fractures to propagate in the sub-blocks either in Mode I (tensile fracturing) or Mode II (shear fracturing). All three extensions have been implemented into the original DDA program of Shi. With the three extensions, the DDA method is more applicable to a greater range of rock mechanics problems and other engineering problems involving blocky systems. Examples of application of the method, for plane stress condition, are presented with regard to rock fall, slope stability and underground excavation problems.

Determination of deformability and tensile strength of anisotropic rock using Brazilian tests

This paper is the first of a series of two papers dealing with the determination of the deformability, tensile strength and fracturing of anisotropic rocks by diametral compression (Brazilian test) of discs of rock. It presents a combination of analytical and experimental methods for determining in the laboratory the elastic constants and the indirect (Brazilian) tensile strength of transversely isotropic rocks, i.e. rocks with one dominant direction of planar anisotropy. A computer program based on the complex variable function method and the generalized reduced gradient method was developed to determine the elastic constants of idealized linearly elastic, homogeneous, transversely isotropic media from the strains measured at the center of discs subjected to diametral loading. The complex variable function method was also used to construct charts for determining the indirect tensile strength of anisotropic media from the failure loads measured during diametral loading. Brazilian tests were conducted on four types of bedded sandstones assumed to be transversely isotropic. Based on strain measurements obtained with 45° strain gage rosettes glued at the center of the discs, the five independent elastic constants of the tested rocks could be determined. The elastic constants determined with the Brazilian tests were compared with those obtained from conventional uniaxial compression tests. The indirect (Brazilian) tensile strength of the tested sandstones was found to depend on the angle between the apparent planes of rock anisotropy and the direction of diametral loading.

Modelling rock joints under shear and normal loading

Abstract A model is presented to determine the effect of boundary conditions on the shear behaviour of a dilatant rock joint. The model is given in both graphical and mathematical forms. It relates the normal load-deformation response of a joint to its shear load-deformation and dilatant behaviour. The proposed model predicts the increase in normal deformability of an initially mated joint as it traverses a range of unmated conditions. It provides a tangent formulation for the deformability of a rock joint that fully accounts for the coupling between joint normal and shear response due to dilatancy. Finally, the applicability of the proposed model to predict the behaviour of a rock joint under applied constant normal stiffness boundary conditions is verified using existing experimental results.

Gravitational stresses in anisotropic rock masses

Abstract This paper presents closed-form solutions for the stress field induced by gravity in anisotropic rock masses. These rocks are assumed to be laterally restrained and are modelled as a homogeneous, orthotropic or transversely isotropic, linearly elastic material. The analysis, constrained by the thermodynamic requirement that strain energy be positive definite, gives the following important result: inclusion of anisotropy broadens the range of permissible values of gravity-induced horizontal stresses. In fact, for some ranges of anisotropic rock properties, it is thermodynamically admissible for gravity-induced horizontal stresses to exceed the vertical stress component; this is not possible for the classical isotropic solution. Specific examples are presented to explore the nature of the gravity-induced stress field in anisotropic rocks and its dependence on the type, degree and orientation of anisotropy with respect to the horizontal ground surface.

Modeling the effect of water, excavation sequence and rock reinforcement with discontinuous deformation analysis

A powerful numerical method that can be used for modeling rock-structure interaction is the discontinuous deformation analysis (DDA) method developed by Shi in 1988. In this method, rock masses are treated as systems of finite and deformable blocks. Large rock mass deformations and block movements are allowed. Although various extensions of the DDA method have been proposed in the literature, the method is not capable of modeling water-block interaction, sequential loading or unloading and rock reinforcement; three features that are needed when modeling surface or underground excavation in fractured rock. This paper presents three new extensions to the DDA method. The extensions consist of hydro-mechanical coupling between rock blocks and steady water flow in fractures, sequential loading or unloading, and rock reinforcement by rockbolts, shotcrete or concrete lining. Examples of application of the DDA method with the new extensions are presented. Simulations of the underground excavation of the ‘Unju Tunnel’ in Korea were carried out to evaluate the influence of fracture flow, excavation sequence and reinforcement on the tunnel stability. The results of the present study indicate that fracture flow and improper selection of excavation sequence could have a destabilizing eAect on the tunnel stability. On the other hand, reinforcement by rockbolts and shotcrete can stabilize the tunnel. It is found that, in general, the DDA program with the three new extensions can now be used as a practical tool in the design of underground structures. In particular, phases of construction (excavation, reinforcement) can now be simulated more realistically. However, the method is limited to solving two-dimensional problems. # 1999 Elsevier Science Ltd. All rights reserved.

Green’s functions and boundary element method formulation for 3D anisotropic media

Abstract The implementation of Wang’s theoretical solution is presented for elastostatic displacement Green’s function for three-dimensional solids of general anisotropy. Excerpts from the authors’ fortran code are included. A numerical algorithm for the calculation of the derivatives of the Green’s displacements and stresses is also introduced. These implementations have been incorporated into a boundary element method (BEM) code developed by the authors. The numerical results of Green’s displacements, stresses and stress derivatives are in perfect agreement with closed-form solutions for transversely isotropic solids. The BEM code results are also in very close agreement with both exact solutions and other BEM formulations, even if coarse mesh discretizations are used.

Fracture Mechanics Analysis of Cracked Discs of Anisotropic Rock Using the Boundary Element Method

This paper is the second of a series of two papers dealing with the determination of the deformability, tensile strength and fracturing of anisotropic rocks by diametral compression (Brazilian test) of discs of rock. It is shown how a new formulation of the Boundary Element Method (BEM), proposed recently by the authors, can be used to determine the stress intensity factors (SIFs) and the fracture toughness of anisotropic rocks from the results of diametral compression tests on initially cracked discs. Crack initiation angles and propagation paths can also be predicted using a numerical procedure based on the BEM and maximum tensile stress criterion. Numerical examples of calculation of mixed mode SIFs are presented for both isotropic and anisotropic media. The calculated SIFs for the special isotropic case are found to be in good agreement with those reported by previous authors. Diametral loading tests were conducted on Cracked Straight Through Brazilian Disc (CSTBD) specimens of a shale in order to evaluate its fracture toughness, the angle of crack initiation and the path of crack propagation. It was found that the numerical simulations of crack initiation and propagation in the CSTBD specimens of the shale were in good agreement with the experimental observations.