Non Linear Elasto-Plastic Analysis of Cylindrical Cavity in Rock Mass Using a Hoek-Brown Criterion

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 computacionais 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. σ1 and σ3, as: modulus (E) and Poisson coefficient ν( ). This paper adopts the generalized The vector a is the gradient of the yield Hoek-Brown failure criterion (Hoek et al

a (15) (σ1 - σ3 ) = σci [ mb ( σ3 / σci ) + S ]

In which is the uniaxial compres- sive strength of intact rock, and , and are constants defined as: σci mb a s

(GSI - 100)/(28 - 14D) (16) mb = mi e

(GSI - 100)/(9 - 3D) (17) s = e

(18) a = 0.5 + (e - GSI/15)/(9 - 3D) - e-20/3)/6

Where, is a constant of Hoek-Brown very disturbed rock mass, and GSI is the As no hardening occurs in a perfect- mi criterion intact rock; D is a disturbance Geological Strength Index, which takes ly plastic constitutive model and the yield coefficient which varies from 0.0 for into account the geological condition of concept merges with the failure concept, undisturbed in situ rock mass to 1.0 for the rock mass. the yield function can be written as:

(19) F = F (σ) = 0 or, by using the Hoek-Brown failure criterion (Equation 15):

REM: R. Esc. Minas, Ouro Preto, 68(2), 145-152, apr. jun. | 2015 147 Non linear elasto-plastic analysis of cylindrical cavity in rock mass using a Hoek-Brown criterion

1/a ⎡ ⎤ I 2D (2cos θ) ⎛ 1 ⎞ mbI1 F(I1,I 2D ,θ) = σ ci ⎢ ⎥ + I 2D mb ⎜cosθ+ senθ⎟ − − σ cis = 0 (20) ⎣⎢ σ ci ⎦⎥ ⎝ 3 ⎠ 3 where, I is the first invariant of the equations, the compatibility and bound- integration algorithm. stress tensor; is the second invariant ary conditions, and the constitutive It is possible, by adopting the ex- I2D of the deviator stress tensor, and θ is the equations. However, the constitutive plicit stress integration scheme as sug- Lode angle that depends on the second equation integration is not trivial, even gested by Sloan et al (2001), to increase and the third ( ) invariant of deviator if one knows the incremental strain the precision of the stress calculation. I3D stress tensor (Owen and Hinton 1989). magnitude on each iterative cycle, what According to these authors the incre- In theory, at each increment and is still unknown is the way it varies ment of stress can be divided into two for a selected tolerance, the iterative across the incremental path. It is nec- parcels: elastic, , and elastic plastic, Δσe scheme satisfies the global equilibrium essary then to use an accurate stress , such as: Δσep

k + Δσ = Δσe Δσep = α DeΔε + Δσep (21) where, α is a scalar that varies from 0 to erates an elastic plastic stress increment. purely elastic stress increment. The α sca- unity. For α = 0 the strain increment gen- For α = 1 the strain increment generates a lar value is obtained by solving iteratively:

|F (σn + α DeΔε| ≤ FTOL (22) where FTOL is the tolerance sug- defined, the α scalar, the stress state upon gested by Sloan et al, (2001) as 10-5. Once the yield surface, is updated according to:

σint = σn + Δσe (23)

The increment of elastic plastic stress is obtained by:

j j - 1 j σ = σ + dσep (24)

j j j dσep = Dep (σ )dεep (25)

j j dεep = ΔT Δεep (26)

Δεep = ( 1- α ) Δε (27) where ΔTj is a scalar known as increment exceed a STOL tolerance. The first trial at the end of each sub increment starting of pseudo-time (T) that varies from zero is conducted by adopting ΔT = 1. The from stress state upon the yield function to unity and is evaluated while taking into procedure is controlled by pseudo-time ( 0 ). The procedure described in T σ = σint account the local error committed during (0≤T≤1) at the end of each subincrement, this paper was implemented into ANLOG the stress integration. This error cannot ΣΔT = T = 1. The stress state is updated (Nogueira 2010) by Simão (2014). 3. Cylindrical cavity in elasto-plastic rock mass

This item describes the numerical tropic, and subjected to a isotropic initial Kim (2006) presented an analytical solu-

simulation of a cylindrical cavity in a stress state, σ0. The stress-strain behavior tion for this problem considering the rock semi-infinite rock mass. The main goal of the rock mass is represented by using mass, under isotropic initial stress state, of this simulation is to establish the plastic non-linear elastic-perfectly plastic, with as elastic-brittle-plastic material which and elastic zones around the cavity and its associate plasticity, based on the Hoek- presents a sudden loss of strength after stress distribution, highlighting the influ- Brown criterion constitutive model as reach its maximal value. In this paper, this ence of the pressure acting internally on presented in the previous item. The elasto- solution is adapted to consider an elastic the cavity wall. plastic transition radius, R, defines a zone perfectly plastic material which does not The problem is depicted in Figure with plastic behavior around the present this sudden loss of strength. Then, (R-ri) 1 and consists of a cylindrical cavity of cavity and the external radius, , defines the radial and circumferential stress dis- re internal radius, ri, deeply conducted in a the dominium of the problem. tribution on the plastic zone, , is ri

2 m σ ⎡ ⎛ r ⎞⎤ ⎡ ⎛ r ⎞⎤ b ci ⎜ ⎟ ⎜ ⎟ 2 σr (r) = ⎢ln⎜ ⎟⎥ + ⎢ln⎜ ⎟⎥ mbσci pi + sσ c i + pi (28) 4 ⎣ ⎝ rι ⎠⎦ ⎣ ⎝ ri ⎠⎦

148 REM: R. Esc. Minas, Ouro Preto, 68(2), 145-152, apr. jun. | 2015 Jefferson Tales Simão and Christianne de Lyra Nogueira

m σ ⎛ r ⎞ (r) m p s 2 b ci ln⎜ ⎟ (29) σθ = σr + bσci i + σci + ⎜ ⎟ 2 ⎝ ri ⎠ On the elastic zone, , the radial and circumferential stress distribution is given by: R < r < re

2 ⎛ R ⎞ (30) σr (r) = σ0 − ⎜ ⎟ (σ0 − σR ) ⎝ r ⎠

2 ⎛ R ⎞ (31) σθ(r) = σ0 − ⎜ ⎟ (σ0 − σR ) ⎝ r ⎠

where, σR is a radial stress on the transition radius, R, defined as:

⎛ F −F ⎞ ⎜ 1 2 ⎟ ⎜ ⎟ ⎝ 2mbσci ⎠ (32) R = rIe in which 2 (33) F1 = σ ci (F0 + mbσ ci − 2mb σ ci F0 )

2 (34) F2 = 4 mbσ ci pi + s σ ci

2 (35) F0 = 16s σ ci + mbσ ci +16mbσ 0

(a) (b)

y, θ y

elastic r ri pi σ0 σθ σ r x,r

plastic R re σ0 re=100m

ri=5m

Physical problen Finite element model Figure 1 Deep cylindrical cavity in rock mass

-6 -2 Figure 1 presents the finite element Δλmin = 10 ; Δλmax = 10 ) and a Forward Figure 2b shows an elastic analyti- mesh used in the numerical simulation; Euler stress integration (FTOL = 10 -5 and cal solution provided by Kirsch (Poulos it is composed of 220 quadrilateral qua- STOL = 10 -2). and Davis 1972) of a circular opening dratic isoparametric elements (Q8) and Figure 2a illustrates the analytical considering a horizontal stress coefficient 661 nodal points. The following constitu- and numerical (y = 0) results along the of one. In this case, the circumferential tive parameters were adopted: GPa; radial direction in terms of the radial ( ) stress decreases along the radial direction, E = 5.5 σr = 0.25; = 30MPa; = 1.7; = and circumferential ( ) stresses, consid- while the radial normal stress increases ν σci mb s σθ 0.0039; =0.5 (which correspond to a ering an initial isotropic stress state ( ) in this direction. The elastic analysis a σ0 = 50 and = 10, approximately). of 30MPa and a null internal pressure overestimates the circumferential stress GSI mi The study uses a modified Newton- ( ). Table 1 presents the normalized on the wall cavity. The elasto-plastic so- pi Raphson incremental iterative procedure elasto-plastic transition radius ( ) and lution presents an abrupt decrease in the R/ri with automatic increment of load ( = stresses. As can be observed, numerical circumferential stress on the elasto-plastic Id 10, miter = 20; toler = 0.1%; Δλ0 = 0.01; and analytical solutions agree strongly. transition zone. REM: R. Esc. Minas, Ouro Preto, 68(2), 145-152, apr. jun. | 2015 149 Non linear elasto-plastic analysis of cylindrical cavity in rock mass using a Hoek-Brown criterion

Solution R/ri σr/σ0 σθ /σ0 σz /σ0

Analytical 2.833 0.526 1.474 0.500 Table 1 Numerical 2.893 0.568 1.400 0.492 Elasto-plastic transition radius and stress (pi = 0MPa and ri = 5m)

(a) 2 Elastic solution 1.75

s 1.5 s e r t 1.25 S

d /

e 0

z 1 i l

a r/ 0 Elastic-plastic solutions

m 0.75 r / (Analytical)

o 0

N 0.5 r/ 0 (Analytical) 0.25 / 0 (Numerical) r/ 0 (Numerical) 0 0 1 2 3 4 5 6 7 8 9 10 r/r p / = 0 i σ0 (b) 2 Numerical elastic-plastic solutions / 0 1.75 pi/ 0 = 0.0

s 1.5 p/ = 0.2

s i 0 e r t 1.25 pi/ 0 = 0.5 S

d e

z 1 i l a

m 0.75 r r/ 0 o

N 0.5 pi/ 0 = 0.0

pi/ 0 = 0.2 0.25 pi/ 0 = 0.5 0 0 1 2 3 4 5 6 7 8 9 10 r/r i Figure 2 pi/σ0= varying Stress distribution Figure 3 presents a stress distribution of stress components. The highest vertical of the cavity while the highest horizontal around the cavity in terms of the isocurve stress level is observed near the lateral wall stress level is observed near its roof (Fig. 3)

(a) (b)

Figure 3

Stress distributions – pi = 0 150 REM: R. Esc. Minas, Ouro Preto, 68(2), 145-152, apr. jun. | 2015 Jefferson Tales Simão and Christianne de Lyra Nogueira

Figure 4 shows the influence of served in an unsupported excavation, radial and circumferential stresses the internal pressures acting on the in- represented in this paper by a null in- at this point. As was expected, the ternal wall cavity on the elasto-plastic ternal pressure. In this case the radius elasto-plastic transition radius de- transition radius. No plastic zone is depends on the property’s material and creases as the GSI increases. Related observed from the internal pressure on the cavity radius. to normal stresses, the radial stress the order of magnitude around half of Figures 5a and 5b show the influ- decreases as the GSI increases, while the initial isotropic stress. The highest ence of the GSI on the elasto-plastic the circumferential stress increases as elasto-plastic transition radius is ob- transition radius and on the normal the GSI increases.

4 ri=5m σ0=30MPa 3 i

r Analytical /

R Numerical 2

0 0.1 0 .2 0.3 0.4 0.5

Figure 4 p i/σ 0

Elasto-plastic transition radius - pi varying

(a)

3 i

r r =5m / i

R pi/σ0=0 2 Analytical Numerical 1

10 20 30 40 50 60 70 80 90 10 0 G S I Elasto-plastic transition radius (b) 3 Analytical ri=5m p /σ = 0 Numerical 2 i 0

σθ/σ0 1

Normalized stress σr/σ0

10 2 0 30 40 5 0 60 70 80 90 100

G S I Figure 5 Normalized stress on the elasto-plastic Influence of GSI transition radius

5. Conclusion

The results presented in this paper rock masses. It was shown, for instance, sible to perform parametric studies for a demonstrated the importance of using that the stress distribution on the support wide variety of materials and geometrical the FEM and elastic-plastic constitutive structure changes significantly. By using configuration to improve the design of model to simulate the opening cavity in the computer program ANLOG, it is pos- tunnel support.

REM: R. Esc. Minas, Ouro Preto, 68(2), 145-152, apr. jun. | 2015 151 Non linear elasto-plastic analysis of cylindrical cavity in rock mass using a Hoek-Brown criterion

6. Acknowledgments

The authors are grateful for the CNPq, and FAPEMIG. The first author is They also acknowledge Prof. John P. financial support received from CAPES, grateful for the grant received by CAPES. White for the editorial review of this text.

7. References

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Received: 19 March 2014 - Accepted: 18 August 2014. 152 REM: R. Esc. Minas, Ouro Preto, 68(2), 145-152, apr. jun. | 2015