Rem: Revista Escola de Minas ISSN: 0370-4467 [email protected] Universidade Federal de Ouro Preto Brasil Tales Simão, Jefferson; de Lyra Nogueira, Christianne Non linear elasto-plastic analysis of cylindrical cavity in rock mass using a Hoek-Brown criterion Rem: Revista Escola de Minas, vol. 68, núm. 2, abril-junio, 2015, pp. 145-152 Universidade Federal de Ouro Preto Ouro Preto, Brasil Available in: http://www.redalyc.org/articulo.oa?id=56439476002 How to cite Complete issue Scientific Information System More information about this article Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Journal's homepage in redalyc.org Non-profit academic project, developed under the open access initiative Jefferson Tales Simão and Christianne de Lyra Nogueira Civil EngineeringEngenharia Civil Non linear elasto-plastic analysis of cylindrical cavity in rock mass using a Hoek-Brown criterion Análise não linear elastoplástica de cavidades cilíndricas em maciços http://dx.doi.org/10.1590/0370-44672015680155 rochosos usando o critério de Hoek-Brown Jefferson Tales Simão Abstract Mestrando Universidade Federal de Ouro Preto – Escola de This paper aims to present an elastic, perfectly plastic, constitutive model Minas – Departamento de Engenharia de Minas based on the Hoek-Brown failure criterion and with non-associative plasticity. – Programa de Pós-Graduação em Engenharia de The objective is to apply the model to the non-linear analysis of geotechnical Minas – PPGEM problems like excavations in rock mass. The computational implementation was [email protected] carried out with a computational program called ANLOG (Non-Linear Analy- sis of Geotechnical Problem) system based on a displacement formulation of the Christianne de Lyra Nogueira finite element method. Due to the non-linear nature of the constitutive model, Professora Associada IV da Universidade Federal the study adopts an incremental iterative Newton-Raphson procedure with auto- de Ouro Preto – Escola de Minas – Departamento matic load increments to guarantee the global level equilibrium. In addition, to de Engenharia de Minas – Programa de Pós guarantee the consistency condition at the local level, the study adopts, for the Graduação em Engenharia de Minas – PPGEM stress integration, an explicit algorithm with automatic sub-increments of strain. [email protected] To validate the computational implementation and applicability of the numerical model, the study uses theoretical results to compare with ones obtained with the numerical simulation of cylindrical cavity in rock mass. Keywords: Hoek-Brown failure criterion, finite element method, elastoplasticity, stress integration algorithm, cylindrical cavity, tunnel, rock mass. Resumo Esse artigo apresenta um modelo constitutivo elástico perfeitamente plás- tico com plasticidade associada e com base no critério de resistência de Hoek- -Brown. O objetivo é aplicar esse modelo para análise não linear de problemas geotécnicos como escavações em maciços rochosos. As implementações compu- tacionais foram realizadas no sistema ANLOG (Análise não linear de obras ge- otécnicas) com base na formulação em deslocamento do método dos elementos finitos. Devido à natureza não linear do modelo constitutivo, o estudo adota um procedimento incremental iterativo do tipo Newton-Raphson com incrementos automáticos de modo a garantir o equilíbrio em nível global. Além disto, para garantir a condição de consistência em nível local, o estudo adota um esquema de explicito de integração de tensão com subincrementos automáticos de defor- mação. Para validar as implementações computacionais e a aplicabilidade do modelo numérico gerado, o estudo usa os resultados da simulação numérica de uma cavidade cilíndrica em maciços rochosos. Palavras-chave: Critério de ruptura de Hoek-Brown, método dos elementos finitos, elastoplasticidade, algoritmo de integração de tensão, cavidade cilíndrica, túnel, maciço rochoso. REM: R. Esc. Minas, Ouro Preto, 68(2), 145-152, apr. jun. | 2015 145 Non linear elasto-plastic analysis of cylindrical cavity in rock mass using a Hoek-Brown criterion 1. Introduction The application of the finite ele- a rock mass. Such a model is capable of 1988; Hoek et al 1992, 1995, 2002). ment method (FEM), which considers providing more realistic results when us- The use of a Hoek-Brown failure a continuous media, to analyze the me- ing a displacement formulation of FEM. criterion, as the yield function in an chanical behavior of a rock mass has been In the early 1980’s, the Hoek- elastic-plastic analysis, leads to the appli- restricted to hard rock or non-fractured Brown failure criterion was developed cation of an incremental iterative proce- rock mass. Due to the increasing number for hard rock (Hoek and Brown 1980). dure at the global level of a FEM analysis of geotechnical works carried out on frac- Since then, several versions have been and the application of a stress integration tured rock masses, it has become neces- published in order to include the influence scheme at the local level (Sharan 2003; sary to use a constitutive model that takes of geological conditions on the failure pa- Sharan 2005; Clausen and Dumkilde into account the geological condition of rameter of rock masses (Hoek and Brown 2008; Wang and Yin 2011). 2. Hoek-Brown elasto-plastic model formulation The equilibrium equations of a (Teixeira et al 2012). During a given rium path. mechanical problem, in static condi- equilibrium path, the variation on the Based on the displacement finite tion, describe a non-linear equation displacement, strain, and stress fields element formulation, the equilibrium system when adopting an elasto-plastic depends on the stress and strain levels equations can be written as: stress-strain-strength relationship and their history through the equilib- Fint = Fext (1) Where, is the external nodal force vec- of the element external nodal force vector Fext tor that represents the global arrangement ee defined as: Fext e e e e Fext= FS + Fb + Fδ (2) e in which ffffFS represents the parcel of represents the parcel of external nodal arrangement of the element internal external nodal force due to surface load; force due to non-null prescribed dis- nodal force vector ffffe equivalent to the Fint ffffe represents the parcel of external placements, . is the internal nodal stress state in a given element that is Fb δ Fint σ nodal force due to body force, ffffe and force vector that represents the global defined as: Fδ e T (3) Fint = ∫ B σdVe Ve B is the cinematic matrix which depends on the strain-displacement relationship. Due to the non-linear nature of the configuration n, where the stress and ΔÛ) until reaching a new equilibrium equation system represented by Equation strain states are known, a predicted configuration n+1 (Crisfield 1991). In (1), an incremental-iterative procedure incremental solution in terms of the this strategy, the problem solution is should be used in order to obtain the global displacement o is obtained. obtained by updating the nodal displace- (ΔÛ n) displacement, strain, and stress fields. This predicted approximation should ment vector (Û) in each new equilibrium Then, starting from a given equilibrium be corrected by successive iteration (δ configuration, by doing: k Ûn+1 = Ûn + Û (4) iter k ο k ΔÛ = ΔÛ n + ∑δΔÛ k =1 (5) Where, o -1 ΔÛ n = [Kep] Δ λ Fext (6) k -1 k δ ΔÛ = [Kep] Ψ (7) k k k Ψ = Fext - Fint (8) iter is the necessary iterative cycle cally defined starting from the initial represents the global arrangement number to reach convergence at the trial provided by the user (Nogueira of the element stiffness matrix e K ep current step, while Δλ is the increment 1998; Oliveira 2006; Simão 2014). defined by: of load factor, which can be automati- is the global stiffness matrix that Kep 146 REM: R. Esc. Minas, Ouro Preto, 68(2), 145-152, apr. jun. | 2015 Jefferson Tales Simão and Christianne de Lyra Nogueira e T (9) Kep= ∫ B Dep BdVe where, is the elasto-plastic constitu- strain-strength relationship. According to scheme the global stiffness matrix is kept Dep tive matrix which depends on the stress- the Modified Newton-Raphson iterative constant during the iterative cycles. The vector k represents the exter- and kept constant throughout the iterative iterative scheme. This vector is updated Fext nal nodal force applied at each load step cycles, according to the Newton-Raphson at the beginning of a given step load by: k (10) Fext = Fext n + Δ λ Fex where is the external nodal force . The internal nodal force vector k is on the stress state evaluated at this itera- Fext n n Fint vector at a given equilibrium configuration evaluated at each iterative cycle depending tive cycle, σ k. At the end of each iterative cycle, a the external nodal force vector. Thus, for stitutive relationships. convergence state of the solution is veri- a given tolerance and at each increment, This iterative scheme involves the fied by using a criterion that relates the the iterative scheme ensures the overall stress state evaluation at each iterative Euclidian norm of the unbalance nodal balance by satisfying the compatibility cycle. Then, in each element, the stress force vector with the Euclidian norm of conditions, boundary conditions and con- vector σ k is obtained by: k k (11) σ = σn + Δσ Where, k k (12) σ = Dep Δε (13) Δεk = -B ΔÛk and, ΔÛk is the incremental displacement vector at element level and updated at the current iterative cycle By adopting linear elastic, per- the associated plasticity (in which the elasto-plastic constitutive matrix can fectly plastic (which is free of harden- potential plastic and yield functions be written as: ing during the plastic flow) and with are the same) constitutive model, the T (14) T (a a) Dep = De - De T De (a Dea) where, is the elastic constitu- function ( ) which depends on the failure 2002) written in terms of principal stress, De F tive matrix which depends on the young criterion used.