Abstract: The propagation mechanism of radial cracks emanating from the blast holes in any rock blasting operation can be modeled by the displacement discontinuity method which is a version of the broad boundary element method. The stress intensity factors (SIFs) at the crack tips of the radial cracks are related to the displacement discontinuities near the crack ends. The stress intensity factors can be numerically evaluated based on the displacement discontinuity variations near the crack tips. Due to the singularity of the stress and displacement fields at the crack tips a special treatment of the displacement discontinuities near the crack tips can be used to obtain more accurate results of SIFs. In this paper a cubic variation of the displacement discontinuities along the boundaries of the circular blast holes and along the radial cracks have been assumed. For the crack tip treatment, the same cubic variation of the displacement discontinuities is also used but to decrease the singularity effect of the crack tips a special crack tip element variation in the form of the square roof of the crack tip element length multiplied by the general displacement discontinuity variation is formulated and some typical example problems are solved numerically. The numerical results are compared with the corresponding results cited in literature which verify the validity and accuracy of the proposed method. 1. INTRODUCTION The displacement discontinuity method is a version of the indirect boundary element method originally developed by Crouch (1976) [1]. The method has been modified for crack analysis of fracture problems in geomechanics by several researches (e.g. [2–4]). Recently, some mixed mode fracture mechanics problems using special crack tip elements and kink elements have been solved for rock mechanics problems [4–7]. Linear elastic fracture mechanics (LEFM) principles have been widely used in rock fracture mechanics (RFM) [8–12]. Based on LEFM principles, a superposition of the three fracture modes describes the general case of loading called mixed mode loading. For a given cracked body under a certain type of loading, the SIFs are known and the displacements and stresses near the crack tip are accordingly determined. Due to brittle behavior of most rocks, the linear elastic fracture mechanics principles have been used to find the fracture mechanics parameters i.e. the mixed mode (mode I or opening mode and mode II or shearing mode of fractures) SIFs of radial cracks occur in the common blasting operations. A general numerical modeling for quasi static crack analysis in infinite plane is given and as a practical problem, the radial cracks around the blast holes are numerically analyzed. Any number of blast holes with any gas pressurization ratios along the emanating cracks can be studied by this model. Suitable normal gas pressurization ratios along the radial cracks are used, to solve the problem. 2. THE HIGHER ORDER DISPLACEMENT DISCONTINUITY METHOD In this paper, the higher order displacement discontinuity elements is used (i. e linear, quadratic and cubic elements are considered) for analysis of crack problems in finite and infinite bodies.

Abstract The subsidence phenomenon of near surface excavations is numerically treated in this study. The most important mechanical parameters are the induced stresses, displacements and strains around the excavation and also on the ground surface above the excavated area. In this paper the stresses and displacements around the excavations are obtained numerically using an indirect boundary element method. The vertical and horizontal strains and tangential stresses on the ground surface and along the boundary of the excavation can be obtained by implementing a finite difference scheme on the displacement results already obtained by the boundary element analysis. Underground and near surface excavations having rectangular shapes are studied in this context. Several example problems are solved and the results are graphically shown in the related figures. 1. Introduction Underground mining of raw materials is often the cause of ground movements at the surface which can produce considerable damage to the structures located within the area of influence. It is too often considered that damage depends on ground strain which in turn is affected by its geometrical and geomechanical properties. Therefore, structural stresses are evaluated for estimating different mechanical properties of the ground and the structure, as well as for different amplitudes of ground movements. There are several basic methods for predicting subsidence phenomenon which can be grouped as: 1 empirically derived relationships, 2 profile functions, 3 influence functions, 4 analytical and numerical models and 5 physical models [1]. In most of these analyses the ground movements are mainly divided into two parts: ground subsidence and horizontal strains. In this paper, the analytical and numerical modeling procedure is used and the subsidence phenomenon is considered as an elastic problem. The elastic treatment of subsidence for various geometric conditions has recently gained a considerable attraction. In this study, the semianalytical indirect boundary element method known as displacement discontinuity method is used for the elastic analysis of the subsidence [2]. This numerical method has been widely used for the stress and displacement analysis of many engineering problems and its details are explained in the literature [3–9]. 2. Higher order indirect boundaryelement A brief explanation of the higher order indirect boundary element method specially modified for the half plane problems with traction free surfaces is given bellow. The quadratic element displacement discontinuity is based on the analytical integration of quadratic collocation shape functions over collinear, straight-line displacement discontinuity elements [5]. 2.1. . Higher Order Displacement Discontinuity in a Half-plane The subsidence problems can be treated as halfplane problems which can also be solved by infinite boundary element methods. However, a more accurate and economic way for solving semi-infinite problems with a traction free surface, using the method of images as explained by Crouch and Starfield (1983) for the constant element displacement discontinuity method [4]. They used the analytical solution to a constant element displacement discontinuity, over the line segment | x |≤ a, y= 0 in the semi-infinite region y ≤ 0 as shown in Figure 3.

ABSTRACT Beheshtabad Water Transmission Tunnel is going to be driven in the Iranian central plate with an approximate length of 65 kilometers. There are several zones of weak rocks in its rout. Some of these zones are situated at great depths so that the squeezing phenomenon may occur during the tunnel exaction period. In this paper the mechanical behavior of weak rocks are studied and their effects on the stability of tunnel are investigated. Several empirical, theoretical and numerical methods are considered to model the time dependence behavior of the weak rocks. Some of these methods are used to study the time dependent behavior of the rocks surrounding this tunnel. The computed results are compared for different zones of weak rocks with potential of squeezing. These results are tabulated in tables and graphically shown in some related figures of the text. Based on these results, it is concluded that there may be a potential of squeezing causing instability in some part of the tunnel rout during its excavation. 1. Introduction Population growth, decreasing the level of underground water resources and concentration of vital industries in the central part of Iran demands a high amount of water to be transmitted to this region. Therefore, projects of water transmission to Zayande-rood (the Zayand River) are some practical and effective ways to convey the water from the West and North West of Iran to its central regions. The Beheshtabad tunnel project is planning to drive a tunnel of about 64.93 kilometer in length in order to bring water from Karoon's water resources to the Iran's central plate. This tunnel (with a SW-NE trend) will start from Darkesh-Varkesh Valley near the Ardal city and finishes at the Se-Cham Asman. This paper briefly discusses about the squeezing phenomenon associated with some part of the tunnel. Squeezing describes the reduction of the cross section of the tunnel during its excavation stages. It's due to the time dependent deformation of the surrounding rock mass resulted from the redistribution of the stresses in an elasto-visco-plastic medium. Three methods are usually used to evaluate the squeezing behavior of the surrounding rocks in a tunnel excavation sequence: empirical method, Semi-analytical method and Theoretical- Analytical method. In this study, the following methods are used to estimate the squeezing of Beheshtabad tunnel and the corresponding results are compared with each other: -Empirical methods (e.g. Geol (1994)'s approach based on N value). -Semi-analytical methods (e.g. Hoek and Marinos (2000)'s approach). -Analytical methods (e.g. Carranza-Torres & Fairhurst approach (2000) using Hoek-Brown criteria). -Analytical methods (e.g. Duncan-Fama approach (1993) using Mohr-Coulomb criteria). In order to estimate the instability of each section of the tunnel, the first step is to determine that either the rock mass should be considered as continuum or dis-continuum material.

Abstract Underground openings of different shapes are excavated in rocks for several engineering purposes such as mining of mineral deposits, underground roadways (tunnels), underground power houses, etc. In this paper the analysis of stresses and strains around these excavations are of main concern. There are several methods for their analysis but because of the complexity of loadings and geometries of these openings and also different depths and rock types, the general closed form solutions are technically impossible. Therefore, one of the versatile numerical methods such as finite element, boundary element and finite difference methods or a combination of them should be used for these analyses. In this paper, an indirect boundary element method is used to evaluate the stresses and displacements on the boundary and around the excavations. The strains and tangential stresses are also determined by implementing a finite difference scheme based on the results obtained by the boundary element method. Excavations of different cross-sections (e. g. rectangular and circular shapes) are solved numerically by the proposed method. These results are compared with some existing analytical results which prove the validity of the method. The analytical and numerical results are tabulated in several tables and some of them are shown graphically in the related figures of the text. 1. Introduction Underground excavations are designed to solve many complicated problems in mining and civil engineering projects. The stress and strain (or displacement) analysis of these problems is the primary steps of their solution. Several analytical and numerical methods are developed for stress analysis around underground openings. In this research the indirect boundary element method (IBEM) which is a semi-analytical method and can be well matched with other numerical techniques such as finite element, finite difference and boundary collocation methods; is used for the analysis of stress and strain around underground excavations [1, 2]. These indirect boundary element techniques which are divided into two individual methods known as fictitious and displacement discontinuity methods as explain by Crouch and Starfield [3]. As a particular example the fictitious stress method (FSM) is briefly discussed here and for the details of these methods the reader is referred to the references [3–6] at the end of this study. The higher order displacement discontinuity method proposed by Crawford and Curran, and Shou and Crouch can also be used for the stress and displacement analysis of cavity problems [5, 6]. 2. Fictitious stress method The fictitious stress method concerns with constant resultant tractions x x t = P and P y t = are applied to a finite strip in an infinite body (Figure 1). The problem of a constant traction x x t = P and P y t = applied to the line segment x ≤ a , y = 0 in an infinite elastic solid can be solved by integrating the solution to the Kelvin's problem.