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The present investigation is based on a new quadratic equation that links the principal stresses σ1 and σ3 at the moment of failure, which allows to obtain the strength in rocks and brittle materials such as concrete. Through this new empirical two-dimensional failure criterion, the normal stress acting on the fracture plane is determined in the first calculation phase by solving the first order linear differential equation, and subsequently the shear streng‘th envelope. To facilitate the calculations, the expression that relates the major and minor principal stresses σ1 and σ3 at failure are defined for a parabolic equation and presented in normalized form. Thus, the algebraic curve is expressed as a function of (σ3/σc) and the parameter ξ. Where, σcrepresents the uniaxial compressive strength of intact rock (rock matrix) or concrete at 28 days, $$f_{c}^{{^{\prime}}}$$ , and on the other hand, ξ = (σt/σc) is a dimensionless parameter whose quotient is obtained by dividing the tensile strength σt by σc. Through the procedure described on this research (Focus Procedure) the parameters k1 and k2 that relate the principal stresses are determined with a good approximation grade through the relation ξ = (σt/σc), both in the intact rock condition as a function of rock mass quality index. All of this, with the additional advantage, that it is not required to know the interval of the principal failure stress (σ3, σ1) through the experimental tests.

Through the research it describes an analytic methodology, which allows to determine the minimum safety factor depending on the depth of tensile crack and the inclination of surface failure on the most critical condition, considering at the same time, surcharge, seismic effect and water pressure. Therefore, it studies the stability of rock slopes considering that the potential of failure surface it is constituted by two blocks with different inclinations. The superior block is limited by a tensile crack that is represented by a fracture without displacement. On the other hand, on the inferior block, geometry is formed by a potential slide plane of α inclination with the horizontal axis, in which are acting shear stresses. Fracture on superior block is characterized by a normal-tensile stresses field that act over the crack whose presence originates when the rock loses its original cohesion. Finally, comparisons are made through examples with the limit equilibrium method and finite elements method, where it determines the safety factor on dry state, water and seism, being all of them too similar. Besides, the methodology compares the track depth and the distance between the intersection point of tensile crack and the edge of the slope face, results shows that the analytic methodology is very conservative and throws the less values of this distances.

This study investigates the technique of variational calculus applied to estimate the slope stability considering the mechanism of planar failure. The critical plane failure surface should be determined because it theoretically indicates the most unfavorable plane to be considered when stabilizing a slope to rectify the instability generated by several statistically possible planes. This generates integrals that can be solved by numerical methods, such as the Newton Cotes and the finite differences methods. Additionally, a system of nonlinear equations is obtained and solved. The surface of the critical planar failure is determined by applying the condition of transversality in mobile boundaries, for which various examples are provided. The number of slices is varied in one of the examples, while the surface of the critical planar failure is determined in the others. Results are compared using analytical methods through axis rotations. All the results obtained by considering normal stress, safety factors, and critical planar failure are nearly the same; however, in this research, a study is carried out for “n” number of slices using programming methods. Sub-routines are important because they can be applied in slopes with different geometry, surcharge, interstitial pressure, and pseudo-static load.

This research presents the application of Blake´s method, which is based on the logarithmic spiral equation, to determine the strain ammonites in rocks, whose internal structure is very similar to the graphical representation of the equation. The geodynamic processes acting inside the earth generate strains according to the magnitude of the stress, producing lengthening , shortening and change in the angular structures relationship of these fossil species contained in the rocks, which is studied analytically using this method according to the longitudinal strain, the shear strain and the ratio between the ellipse axes forming strain, taking into account the angle and the internal shear strain. The initial lengths of the Ammonites without deforming, compared with the lengths of deformed ammonites are measured in the field, determining that there are lengths lengthened which coincide with the major axis of the ellipsoid strain and minor principal stress between 0,34 to 7,68 cm, while the values in the shortening direction, coincide with the minor axis of the same ellipse corresponding to the direction of the major principal stress, between 0.04 to 3.94 cm. These directions of the principal stresses coincide with the convergence direction of the Venezuelan Andes., Resumen La presente investigación, constituye la aplicación del método de Blake, que es basado en la ecuación espiral logarítmica para determinar la deformación de ammonites en rocas, cuya estructura interna es muy similar a la representación gráfi-ca de dicha ecuación. Los procesos geodinámicos que actúan en el interior de la tierra generan deformaciones de acuerdo a la magnitud del esfuerzo, produciendo alargamientos, acortamientos y cambio en las relaciones angulares de las estruc-turas internas de esas especies fósiles contenidas en las rocas, la cual, es estudiada analíticamente con este método de acuerdo a la deformación longitudinal, la deformación por cizalla y la relación existente entre los ejes que conforman la elipse de deformación, tomando en cuenta el ángulo de cizallamiento y la deformación interna. Las longitudes iniciales de los Ammonites sin deformar, se compararon con las longitudes de los ammonites deformados son medidas en el campo, de-terminándose que existen longitudes que se alargaron y que coinciden con el eje mayor del elipsoide de deformación y el esfuerzo principal menor, entre 0,34 – 7,68 cm, mientras que los valores en la dirección de acortamiento, coinciden con el eje menor de la misma elipse que corresponde a la dirección del esfuerzo principal mayor, entre 0,04 – 3,94 cm. Estas di-recciones de los esfuerzos principales, coinciden con la dirección de vergencia de Los Andes Venezolanos. Palabras Claves: Deformación, blake, espiral logarítmica, ammonites.

This research presents the application of Blake´s method, which is based on the logarithmic spiral equation, to determine the strain ammonites in rocks, whose internal structure is very similar to the graphical representation of the equation. The geodynamic processes acting inside the earth generate strains according to the magnitude of the stress, producing lengthening , shortening and change in the angular structures relationship of these fossil species contained in the rocks, which is studied analytically using this method according to the longitudinal strain, the shear strain and the ratio between the ellipse axes forming strain, taking into account the angle and the internal shear strain. The initial lengths of the Ammonites without deforming, compared with the lengths of deformed ammonites are measured in the field, determining that there are lengths lengthened which coincide with the major axis of the ellipsoid strain and minor principal stress between 0,34 to 7,68 cm, while the values in the shortening direction, coincide with the minor axis of the same ellipse corresponding to the direction of the major principal stress, between 0.04 to 3.94 cm. These directions of the principal stresses coincide with the convergence direction of the Venezuelan Andes., Resumen La presente investigación, constituye la aplicación del método de Blake, que es basado en la ecuación espiral logarítmica para determinar la deformación de ammonites en rocas, cuya estructura interna es muy similar a la representación gráfi-ca de dicha ecuación. Los procesos geodinámicos que actúan en el interior de la tierra generan deformaciones de acuerdo a la magnitud del esfuerzo, produciendo alargamientos, acortamientos y cambio en las relaciones angulares de las estruc-turas internas de esas especies fósiles contenidas en las rocas, la cual, es estudiada analíticamente con este método de acuerdo a la deformación longitudinal, la deformación por cizalla y la relación existente entre los ejes que conforman la elipse de deformación, tomando en cuenta el ángulo de cizallamiento y la deformación interna. Las longitudes iniciales de los Ammonites sin deformar, se compararon con las longitudes de los ammonites deformados son medidas en el campo, de-terminándose que existen longitudes que se alargaron y que coinciden con el eje mayor del elipsoide de deformación y el esfuerzo principal menor, entre 0,34 – 7,68 cm, mientras que los valores en la dirección de acortamiento, coinciden con el eje menor de la misma elipse que corresponde a la dirección del esfuerzo principal mayor, entre 0,04 – 3,94 cm. Estas di-recciones de los esfuerzos principales, coinciden con la dirección de vergencia de Los Andes Venezolanos. Palabras Claves: Deformación, blake, espiral logarítmica, ammonites.

This paper aims to determine the main stresses direction in the Boconó Fault through a numerical method, using vectoranalysis and stereographic projections with the program Stereo32. This requires the strikes and dips of the surfaces of twofaults, the Boconó Fault and a synthetic fault to this, which for this study, is the Quebrada Higuerón Fault. The necessarycondition for using this methodology is that the two faults must have the same connotation indicated by the Anderson diagrams.The analyzed faults data are transcurrent with a high dip angle of its surfaces with a general N-NE course., Este trabajo pretende determinar la dirección de los esfuerzos principales ejercidos en la Falla de Boconó a través de unmétodo numérico, aplicando análisis vectorial y proyecciones estereográficas con el programa Stereo32. Para ello se necesitanlos rumbos y buzamientos de las superficies de dos fallas, la Falla de Boconó y una falla sintética a ésta, que para esteestudio, es la Falla Quebrada Higuerón. La condición necesaria para utilizar esta metodología es que las dos fallas debentener la misma connotación indicada por los diagramas de Anderson. Los datos de las fallas analizadas, sontranscurrentes con un ángulo de buzamiento alto de sus superficies con un rumbo general N- NE

This paper aims to determine the main stresses direction in the Boconó Fault through a numerical method, using vectoranalysis and stereographic projections with the program Stereo32. This requires the strikes and dips of the surfaces of twofaults, the Boconó Fault and a synthetic fault to this, which for this study, is the Quebrada Higuerón Fault. The necessarycondition for using this methodology is that the two faults must have the same connotation indicated by the Anderson diagrams.The analyzed faults data are transcurrent with a high dip angle of its surfaces with a general N-NE course., Este trabajo pretende determinar la dirección de los esfuerzos principales ejercidos en la Falla de Boconó a través de unmétodo numérico, aplicando análisis vectorial y proyecciones estereográficas con el programa Stereo32. Para ello se necesitanlos rumbos y buzamientos de las superficies de dos fallas, la Falla de Boconó y una falla sintética a ésta, que para esteestudio, es la Falla Quebrada Higuerón. La condición necesaria para utilizar esta metodología es que las dos fallas debentener la misma connotación indicada por los diagramas de Anderson. Los datos de las fallas analizadas, sontranscurrentes con un ángulo de buzamiento alto de sus superficies con un rumbo general N- NE

The cutting test allows determining the shear stress of a soil, through the parameters, cohesion (C) and internal friction angle (φ) of the material. In this work, based on the results of this test in a soil, the magnitude and direction of the main stresses were obtained, making the graphical representation by means of three failure points, whose coordinates have been obtained using the direct cut test in the condition "Not drained." Subsequently, these points were represented, obtaining the failure envelope, then the axes rotation was applied, relying on the well-known expression of the failure angle αf = (45º + φ / 2), which is the angle that forms the failure plane with the minor main stress, which allowed to relate the axes x and y to the axes x 'and y' by means of the direct cosines, in this way, the rotation matrix was found. This procedure was carried out with the purpose of comparing it with the analytical solution that relates the main stress when using the Mohr - Coulomb equation, with the axis rotation method, observing that the error is negligible when applying the proposed methodology., El ensayo de corte permite determinar la resistencia al esfuerzo cortante de un suelo, mediante los parámetros de cohesión (C) y ángulo de fricción interna (φ) del material. En este trabajo a partir de los resultados de dicho ensayo en un suelo, se obtuvo la magnitud y dirección de los esfuerzos principales, realizando la representación gráfica mediante tres puntos de rotura, cuyas coordenadas se han obtenido utilizando el ensayo de corte directo en la condición " no drenada ". Posteriormente, se representaron dichos puntos, obteniéndose la envolvente de falla, luego se aplicó la rotación de ejes, apoyándose en la bien conocida expresión del ángulo de falla α f = (45º+φ/2), que es el ángulo que forma el plano de falla con el esfuerzo principal menor, la cual permitió relacionar los ejes x e y con los ejes x' e y' por medio de los cosenos directores, de esta manera, se halló la matriz de rotación. Este procedimiento se realizó con la finalidad de compararlo con la solución analítica que relaciona los esfuerzos principales al utilizar la ecuación de Mohr – Coulomb, con el método de rotación de ejes, observándose que el error es despreciable al aplicar la metodología propuesta.

The cutting test allows determining the shear stress of a soil, through the parameters, cohesion (C) and internal friction angle (φ) of the material. In this work, based on the results of this test in a soil, the magnitude and direction of the main stresses were obtained, making the graphical representation by means of three failure points, whose coordinates have been obtained using the direct cut test in the condition "Not drained." Subsequently, these points were represented, obtaining the failure envelope, then the axes rotation was applied, relying on the well-known expression of the failure angle αf = (45º + φ / 2), which is the angle that forms the failure plane with the minor main stress, which allowed to relate the axes x and y to the axes x 'and y' by means of the direct cosines, in this way, the rotation matrix was found. This procedure was carried out with the purpose of comparing it with the analytical solution that relates the main stress when using the Mohr - Coulomb equation, with the axis rotation method, observing that the error is negligible when applying the proposed methodology., El ensayo de corte permite determinar la resistencia al esfuerzo cortante de un suelo, mediante los parámetros de cohesión (C) y ángulo de fricción interna (φ) del material. En este trabajo a partir de los resultados de dicho ensayo en un suelo, se obtuvo la magnitud y dirección de los esfuerzos principales, realizando la representación gráfica mediante tres puntos de rotura, cuyas coordenadas se han obtenido utilizando el ensayo de corte directo en la condición " no drenada ". Posteriormente, se representaron dichos puntos, obteniéndose la envolvente de falla, luego se aplicó la rotación de ejes, apoyándose en la bien conocida expresión del ángulo de falla α f = (45º+φ/2), que es el ángulo que forma el plano de falla con el esfuerzo principal menor, la cual permitió relacionar los ejes x e y con los ejes x' e y' por medio de los cosenos directores, de esta manera, se halló la matriz de rotación. Este procedimiento se realizó con la finalidad de compararlo con la solución analítica que relaciona los esfuerzos principales al utilizar la ecuación de Mohr – Coulomb, con el método de rotación de ejes, observándose que el error es despreciable al aplicar la metodología propuesta.

In this work, an analytic expression has been developed that enables the shear strength to be determined using the empirical failure criterion developed by Hoek and Brown. Applying this new hypothesis, along with elasticity theory and basic mathematical concepts for obtaining the failure envelope of a family of failure circles, the shear strength has been determined to be such that τα=f(σα,σc,m,s), where σα=the normal stress on the discontinuity, σc the uniaxial compressive strength of the intact rock and the parameters m and s are constants dependent upon the properties of the rock. An expression of this type will help us develop new methods of analysis of slope stability and support of galleries, design of anchor support systems in rock masses, estimation of the skin friction of poor quality rock in driven pile foundations, as well as in other innumerable related areas of geotechnology.