Determination of the Shear Failure Envelope by Adjusting with the Statistical Method of Error in Variables through the Relationship between the Principal Stresses

• Orlando Zambrano Mendoza Programa de Ingeniería de Petróleo, Escuela de Petróleo. Facultad de Ingeniería, Universidad del Zulia, Sector Grano de Oro, Apartado postal 4011-A-526. Maracaibo, Zulia, Venezuela https://orcid.org/0009-0006-6895-8102
• Peter P. Valko Harold Vance Department of Petroleum Engineering Texas A&M University 3116 TAMU College Station, TX 77843-3116 https://orcid.org/0000-0001-9942-2403
• James E. Russell Harold Vance Department of Petroleum Engineering Texas A&M University 3116 TAMU College Station, TX 77843-3116

DOI: https://doi.org/10.22209/rt.v46a13

Keywords: EIV, failure envelope, objective function, principal stress plane, transformation

Abstract

This work is based on developing the parametric representation of the failure envelope to Mohr's circles in intact rock as a function of the principal stresses. In the proposed method, the stresses are adjusted using the statistical method EIV (error-in-variables), which does not make artificial distinctions between the independent and dependent variables. To accomplish the transformation from the principal stress plane to the Mohr plane, Balmer's method was used by applying computational algebraic analysis. To illustrate and verify the application of this proposed methodology, the well-documented dataset collected from previous work by Pincus and Sheorey is used. To test the improvement provided by this method, the calculated objective function (likelihood of erroneous decision) have been compared with the parametric equation representation obtained using various least squares methods. It was found that our proposed methodology, and the transformation method of Balmer, has two advantages: i) It simplifies the process of creating a failure envelope for practical applications, and ii) It minimizes the likelihood of erroneous judgment during applications (i.e. indicating failure in a stable state or vice versa.

 

References

Anand, A., Kumar, S. (2015). Application of multi-objective optimization techniques to geotechnical engineering problems. M. Tech. Thesis. Rourkela: National Institute of Technology.

Balmer, G. (1952). A general analytical solution for Mohr’s envelope. Proceedings of American Society of Test Materials, 52, 1260-1271.

Britt, H., Luecke, R. (1973). The estimation of parameters in nonlinear, implicit models. Technometrics, 15, 233.

Coulomb, C. (1776). Essai sur une application des regles des maximis et minimis a quelquels problemesde statique relatifs, a la architecture. Memoires de L´Academie Royale des Sciences - Par Divers Savans, 7, 343-387.

Deming, W. (1943). Statistical adjustment of data. New York: Willey.

Edgar, T., Liebman, M., Kim, I. (1990). Robust error-in-variables estimation using nonlinear programming techniques. AIChE Journal, 36(7), 985-993.

Esposito, W., Floudas, C. (1998). Parameter estimation in nonlinear algebraic models via global optimization. Computers and Chemical Engineering, 22, 213-220.

Hobbs, D. (1964). The strength and the stress-strain characteristics of coal in triaxial compression. Journal of Geology, 72, 214.

Hoek, E., Brown, T. (1980a). Empirical strength criterion for rock masses. Journal of the Geotechnical Engineering Division - ASCE, 106(GT9), 1013-1035.

Hoek, E., Brown, E. (1980b). Underground excavation in rock. London: CRC Press.

Jiefei, G., Puhui, C. (2018). A failure criterion for isotropic materials based on Mohr’s failure plane theory. Mechanics Research Communications, 87, 1-6.

Jiefei, G., Ke, L., Lei, S. (2020). Modified nonlinear Mohr–Coulomb fracture criteria for isotropic materials and transversely isotropic UD composites. Mechanics of Materials, 151,103649.

Jiefei, G., Puhui, C., Ke, L., Lei. S. (2019). A macroscopic strength criterion for isotropic metals based on the concept of fracture plane. Metal, 9, 634-647.

Kumar, S., Kumar, P. (2011). Parameter optimization of rock failure criterion using error-in-variables approach. International Journal of Geomechanics, 11(1), 36-43.

Liebman, M., Edgar, T. (1988). Data reconciliation for nonlinear processes. Proceedings of the AIChE Annual Meeting. Washington, DC: American Institute of Chemical Engineers (AIChE), 137.

Mostyn, G., Douglas, K. (2002). Strength of intact rock and rock masses [on line] available from: http://geotle.t.u-tokyo.ac.jp/towhata/lecture/rock/mostyn.pdf [accessed: 1 August 2002].

Nelder, J., Mead, R. (1965). A simple method for function minimization. Computer Journal, 7, 308-313.

O’Neil, M., Sinclair, I., Smith, F. (1969). Polynomial curve fitting when abscissas and ordinates are both subject to error. Computer Journal, 12, 52-56.

Peneloux, A., Deyrieux, E., Neau, E. (1976). The maximum likelihood test and the estimation of experimental inaccuracies: Application to data reduction for vapor-liquid equilibrium. Journal of Computers, 73, 706-716.

Pincus, H. (1993). Interlaboratory testing program for rock properties, round one-longitudinal and transverse pulse velocities, unconfined compressive strength, uniaxial elastic modulus, and splitting tensile strength. Geotechnical Testing Journal, 16(1), 138-163.

Pincus, H. (1994). Addendum to interlaboratory testing program for rock properties, round one. Geotechnical Testing Journal, 17(2), 256-258.

Pincus, H. (1996). Interlaboratory testing program for rock properties, round two-confined compression: Young’s modulus, Poisson’s ratio, and ultimate strength. Geotechnical Testing Journal, 19(3), 321-336.

Reilly, P., Patino-Leal, H. (1981). Bayesian study of the error-in-variables model. Technometrics, 23(3), 221.

Schwetlick, H., Tiller, V. (1985). Numerical methods for estimating parameters in nonlinear models with error in the variables. Technometrics, 27(1), 17-24.

Sheorey, P. (1997). Empirical rock failure criteria. Rotterdam: A. A. Balkema.

Southwell, W. (1969). Fitting experimental data. Journal of Computational Physics, 4, 465-474.

Ucar, R. (2019). La resistencia al corte en macizos rocosos y en el hormigón. Una metodología reciente de cálculo. Madrid: Bellisco Ediciones.

Ucar, R. (2021). Determination of a new failure criterion for rock mass and concrete. Geotechnical and Geological Engineering, 39, 3795-3813.

Valkó, P., Vajda, S. (1987). An extended Marquardt-type procedure for fitting error-in-variables models. Computational Chemical Engineering, 11(1), 37-43.

van Huffel. S., Lemmerling. P. (2013). Total least squares and errors-in-variables modeling: analysis, algorithms and applications. Berlin: Springer Science & Business Media.

Willianson, J. H. (1968). Least squares fitting of a straight line. Canadian Journal of Physics, 46, 1845-1847.

York, P. (1966). Least squares fitting of a straight line. Canadian Journal of Physics, 44, 1079.

Zambrano Mendoza, O., Valkó, P., Russell, J. (2003). Error-in-variables for rock failure envelope. International Journal of Rock Mechanics and Mining Sciences, 40(1), 137-143.