Flac Phase2

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Influence of stress path on tunnel excavation response Numericaltool selection and modeling strategy

M. Cai

Geomechanics Research Centre, MIRARCO, Laurentian University, Sudbury, Ontario, Canada P3E 2C6

Received 18 October 2007; received in revised form 26 November 2007; accepted 27 November 2007Available online 21 February 2008

Abstract

The actual stress path in a rock mass during tunnel excavation is complex. To capture the correct tunnel excavation response, it isimportant to correctly resemble the stress path in situ in the numerical tools.

FLAC and Phase2 are two powerful two-dimensional continuum codes for modeling soil, rock, and structural behavior, in the fields ofgeotechnical, geomechanics and in civil and mining engineering. FLAC is based on explicit finite difference formulation while Phase2 isbased on implicit finite element formulation. When the two codes are applied to the analysis of tunnel excavation problems, difference inresults might occur simply due to the different formulation methodologies used in these codes. It is shown that for linear elastic tunnelexcavation problems, both codes provide the same result because stress path is unimportant. For tunnel excavation in elasto-plasticmaterials using long-round drill and blast method, there is significant difference in terms of yielding zone distribution by the two codesif conventional modeling approach is used, especially when the rock strength is low relative to the in situ stress magnitude. The mech-anism of the difference is investigated and recommendation provided for choosing appropriate tools and modeling strategies for tunnelexcavation problems. The importance of honoring the true stress path in tunnel excavation response simulation is illustrated using a fewexamples. 2007 Elsevier Ltd. All rights reserved.

Keywords: Stress path; Tunnel excavation; FLAC; Phase2; FEM; FDM; Numerical model

1. Introduction

FLAC (Itasca, 2005) and Phase2 (Rocscience Inc., 2004)are two popular and powerful numerical analysis tools formodeling soil, rock, and structural behavior, in the fields ofgeotechnical, geomechanics and in civil and mining engi-

neering. These codes provide material models such asMohrCoulomb, HoekBrown failure criteria which aresuitable for geotechnical materials, supply with featuresto simulate underground excavation, and have structuralelements that can represent the soil and rock support sys-tems installed. Some of the applications of the tools in rockmechanics and rock engineering can be found in Hoek(2001), Martin (1993), Cai et al. (2001, 2007a,b).

Young and new professional engineers can not under-stand the hardship in the 1970s and 1980s to perform a sim-ple 2D numerical analysis of underground excavations.Two to four weeks were often required for just preparingthe model mesh (Brady and Johnson, 1989). Nowadays,easy-to-use user interface in the numerical tools has made

it possible for someone with or without strong knowledgebackground of numerical modeling theory to conduct atunnel excavation analysis in just a few minutes. Colorfuloutputs make the interpretation job much easier but atthe same time they also create an illusion that the obtainedresults are remarkable and correct, regardless what inputparameters and modeling approaches are used in theunderlying analysis.

Another thing that is often taken for granted by someusers of the numerical tools is the lack of a complete under-standing of the solution schemes used in the tools. By just

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looking at the user interface and modeling procedures ofFLAC and Phase2, an impression that they are the sameor at least similar can be formed. The fact that Phase2

employs the implicit Finite Element Method (FEM) whileFLAC uses the explicit Finite Difference Method (FDM)may be forgotten or unrecognized by some users. Further-

more, the general attitude towards the relative merits ofFEM and FDM methods is that although the governingequations are derived differently, the resulting equationsare identical for the two methods. The underlying implica-tion is that for a specific boundary value problem in tunnel-ing, FLAC and Phase2 should give the same result. But isthis always true?

In underground engineering such as civil tunnel con-struction and mine excavation, rocks can be removed byemploying different excavation techniques such as full-facedrilling and blasting, TBM, mechanical excavation usingroadheaders, and staged excavation etc. During excavationby conventional drill and blast method, the work face is

perforated by long drill holes. These holes are then filledwith explosives and blasted. A dynamic unloading condi-tion is often created by this type of excavation method.TBM excavation uses several rolling cutters and feeds highpressure of the head against the face of the tunnel, andwhen combined with a rotation of the head, the excavationforce can lead to the crushing of the rock at the face, thusthe excavation. Stress redistribution occurs due to the exca-vation and is often modeled by FLAC or Phase2 for rocksupport design. The question to be asked is: will thesetwo numerical tools, which are widely used in the rockmechanics community, produce the same result for a tunnel

excavation problem?This paper strives to illustrate that the results obtained

by FLAC and Phase2 can be very different, depending onthe material properties and the method of excavation used.It will be explained that the difference can stem simply fromthe solution scheme difference adopted in these two tools,which has not been discussed in depth in the rock engineer-ing community. Knowing such a difference is important ininterpreting obtained numerical results and choosing theright tool and modeling approach for a particular problemunder investigation.

2. Difference in solution schemes between FLAC and Phase2

2.1. FLAC

FLAC uses dynamic equations of motion in its explicit,time-marching scheme, even for static problems. The solu-tion of solid body problems in FLAC invokes the equa-tions of motion (Newtons law of motion), constitutiverelations, and boundary conditions. The solid body isdivided into a finite difference mesh composed of quadrilat-eral elements. In a calculation cycle, the new velocities anddisplacements are obtained from stresses and forces usingthe equations of motion, and then, strain rates are obtained

from velocities and new stresses from strain rates. Since the

calculation cycle requires that the neighboring elementsshould not affect each other, the adopted timestep mustbe smaller than a critical value for numerical stability.

For plasticity analysis, FLAC checks the element state(elastic or plastic) in each cycling step. First, an elastic trialfor the stress increment is computed from the total strain

increment and the stresses are checked against the yield cri-terion. If the corresponding stresses violate the yield crite-ria, plastic deformation takes place. The stresses arecorrected by using the plastic flow rule to ensure that theylie on the yield surface. For an elastic problem, a total of30005000 cycling steps are needed to reduce the unbal-anced force to a negligible value and thus a static solutionis obtained. For an elasto-plastic problem, plastic yieldingis checked once the cycling starts. Hence, the stress pathdiffers significantly from the elastic one if yielding isdetected and stress state corrected. A typical elasto-plasticproblem requires 800010,000 cycling steps to accomplishthe solution.

2.2. Phase2

Phase2 is an implicit FEM program developed initiallyfor underground excavation simulation, and subsequentlynew functions and capabilities have been added allow itto be used for slope stability analysis and groundwater flowsimulation. In the FEM approach, the domain is discret-ized into a finite number of elements with a fixed numberof nodes. Displacements inside an element are approxi-mated using a shape function that links the nodal displace-ment values, which are the system unknowns. The original

partial differential equations are replaced by an assembly ofalgebraic system of equations. A global stiffness matrix isformed and stored. Solving the system of equations deter-mines the node displacement values which in turn can beused to obtain stresses and strains in each element.

For plasticity analysis, Phase2 tackles the problem by aseries of static equilibrium solutions involving iterative pro-cedures. An elastic solution is first obtained and the stressstate in each element is checked against the yield criteria. If

Table 1Comparison of FLAC and Phase2

FLAC Phase2

Solution scheme Explicit ImplicitComputer memory

requirementLow High

Non-linear problemhandling

No iteration necessary Iteration requiredComputationally stable Diverge may occur

Physical process Always follow the physicsif the timestep criterion isguaranteed

Need to bedemonstrated that itfollows the physicalprocess

Excavation method Delete or assign nullelement

Excavation (assignvery low modulus)

Structural elements Yes YesDiscontinuity model Yes (interface element) Yes (joint element)User interface Good ExcellentFirst release 1986 1990

M. Cai / Tunnelling and Underground Space Technology 23 (2008) 618628 619


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the stresses violate the yield criteria, plastic deformationtakes place. The final solution is obtained by iterationand the quality of the solution depends on the convergencecriterion and the algorithm (solver type) used to returnstresses to the yield surface.

2.3. Comparison

A comparison of the explicit and implicit solution meth-ods can be found in the FLAC manual Theory and Back-ground. A comparison, specific to features of FLAC andPhase2, is presented in Table 1. One thing that is of interestfor discussion is the ability of the tools to model the truephysical process in underground excavation. Since thestructure response is stress path depended for nonlinearmaterials (Cai et al., 2002; Kaiser et al., 2001; Ruistuenand Teufel, 1996), if a solution scheme cannot generatestress paths that represent the actual ones, then, a differentstructural response should be expected. This is the case for

numerical tools that employ explicit and implicit solutionschemes. Explicit schemes can always follow the true stresspath during deformation but implicit schemes cannot. Thiscan be demonstrated by stress propagation in a long beamsubjected to the loading and boundary conditions shown inFig. 1. Assume that the load (F) is applied to the left end Ainstantly at time T0. Before the load is applied, the axialstress in the beam is zero. At the time when the load isapplied, load at end A is equal to Fbut load at end B is stillzero. A stress wave is generated in the beam right after the

F

A L B

T

Vp

E, ,

F

t

T0

Fig. 1. Stress wave propagation in a beam.

Tunnel

Host rock

x

y

Excavated by deleting in

the numerical modeling

Fig. 2. Tunnel excavation simulation.

Phase2

FLAC

Minimum principal stress (Pa)

0.00E+00

0.0-3.0-6.0-9.0

-12.0-15.0-18.0-21.0-24.0-27.0-30.0

3 (MPa)

6.00E+06

1.20E+07

1.80E+07

2.40E+07

3.00E+07

Contour interval=3.00E+06

Fig. 3. Comparison of minimum principal stress distribution in Phase2 and FLAC (rx = 60 MPa, ry = 11 MPa; elastic response).

620 M. Cai / Tunnelling and Underground Space Technology 23 (2008) 618628


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load is applied and the compressive stress front will travel

at a speed ofVp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1mE

q1m2m2

qin the beam, where E, m, q are

the Youngs modulus, Poissons ratio, and density of thebeam, respectively. It will take T= L/Vp for the stress waveto travel from A to B, where L is the beam length. Once thestress wave reaches B, a portion of the compressive stress

wave will be reflected and the stress wave travels in thebeam back and forth a few times before final equilibriumis reached. This example shows that applying a load to abeam creates a dynamic loading process; the time involvedis so short that most people would think it as a staticproblem.

The physical process described above can be capturedby FLAC. If Phase2 is used, one obtains only the finalresult of the equilibrium state and the process of stresswave propagation cannot be captured. Since Phase2 is anFEM program, it does not mean that FEM programs can-not simulate the physical process described above. In fact,

FEM codes with explicit solution algorithm, such as ABA-QUS (www.abaqus.com), ELFEN (www.rockfield.co.uk),and ANSYS (www.ansys.com) can simulate this type ofdynamic loading processes as well.

In underground excavation, dynamic loading andunloading conditions often exist. For example, a full-face,long-round drill and blast excavation of a tunnel in rockswill create a dynamic unloading condition to the host rocks.If FLAC and Phase2 are used to model a tunnel excavationproblem (Fig. 2), the tunnel region is simply deleted or

assigned to an excavated material in FLAC and Phase2

,respectively, by following standard or conventionalmodeling approach. For reasons discussed above, it is antic-ipated that results from FLAC and Phase2 could be verydifferent when the conventional modeling approach is used.The difference in modeling results by these two tools areillustrated and discussed in the next section.

3. Yield zone distribution simulation in tunnel excavation by

FLAC and Phase2 illustration examples

3.1. Simulation model

A circular 10 m diameter tunnel excavation problem isconsidered. The outer boundary width and height aretwenty times of the tunnel diameter with fixed boundary

t = 5 MPa

Shear

Tension

state

Elastic

Elastic, Yield in Past

At Yield in Tension

t = 5 MPa

Phase2 FLAC

t = 10 MPa t = 10 MPa

t = 15 MPa t = 15 MPa

a b

c d

e f

Fig. 4. Comparison of tensile yielding zone distribution between Phase2 and FLAC models for rt = 5, 10, 15 MPa (rx = 60 MPa, ry = 11 MPa).

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condition. The Youngs modulus and Poissons ratio of therock mass are 60 GPa and 0.25, respectively. It is assumedthat the tunnel is excavated by long-round drill and blastmethod so that the tunnel portion in the model is deletedor excavation in one stage. In situ stress field and tensileand shear strength are chosen as variable parameters to

study the excavation response simulated by FLAC andPhase2. 4-node and 3-node elements are used in FLACand Phase2 models, respectively. Ideally, the same meshshould be used in both the FLAC and Phase2 models toisolate the effect of mesh size (and shape) but it is deemeddifficult to do so due to the different mesh generation algo-rithms used in these two codes. As an alternative, very finemeshes are used in both models to minimize the mesh influ-ence. As can be seen from Fig. 3, the distributions of theminimum principal stress in both models are very similar(elastic response), except that contours near the tunnelboundary in the FLAC model are missing due to a defi-ciency in contour generation in FLAC.

3.2. Yielding in tension

Figs. 4 and 5 present the distributions of tensile yield ele-ments in the FLAC (right column) and Phase2 (left col-umn) models. The horizontal, vertical, and tunnel axisparallel in situ stress components are assumed to be

rx = 60 MPa, ry = 11 MPa, rz = 45 MPa, respectively. Inthis simulation, the shear strength is set to a high valueto ensure that only tensile yielding occurs in both models.Zero residual tension model is used, which means that ifthe rock fails in tension, its residual tensile strength is zero.The peak tensile strength (rt) varies from 5 to 30 MPa inthe simulation. It is seen that when the rock tensile strengthis low at 5 MPa, the difference of the yielding zone distribu-tion is large between the FLAC and Phase2 models (Fig. 4aand b). As the tensile strength increases, the yield zone pat-terns in the two models gradually converge. In addition,both Phase2 and FLAC predict the tensile yielding in thesidewall center where the tensile stress is the highest. How-

Shear

Tension

state

Elastic


At Yield in Tension

Phase2 FLAC

t = 20 MPat = 20 MPa

t = 25 MPat = 25 MPa

t = 30 MPa t = 30 MPa

a b

c d

e f

Fig. 5. Comparison of tensile yielding zone distribution between Phase2 and FLAC models for rt = 20, 25, 30 MPa (rx = 60 MPa, ry = 11 MPa).



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ever, the shallow tensile failure on the sidewall, distributedover a large region away from the sidewall center, is notcaptured by the Phase2 model. When the peak tensilestrength is 30 MPa, both models predict no tensile yielding(Fig. 5e and f).

To explain the result difference shown in Figs. 4 and 5,

we need to further examine the solution scheme differencein FLAC and Phase2, which had been briefly discussed inSection 2. In reality, when a tunnel is excavated by long-round drill and blast method, some of the strain energyin the system is converted into kinetic energy that needsto be dissipated. A sudden excavation creates large unbal-anced forces right at the excavation boundary and theunbalanced forces need to be dissipated. FLAC, by itsexplicit nature, can model the stress redistribution processdirectly because inertial terms are included. To illustratethe stress propagation and dissipation process after excava-tion, the minimum principal stress distributions in a linearelastic model over several cycle steps (10, 30, 60, 100, 200)

are shown in Fig. 6. Large tensile stresses are observed overa wide area at step 10. If a plastic tensile failure model(Rankine model) is used, immediate yielding would occurif the tensile stress is larger than the tensile strength. After200 steps, the r3P 5 MPa contour extended to a depth of0.5 m. To reach a final solution shown in Fig. 3b, about5000 cycling steps are required. For the results shown inFigs. 4 and 5, the plastic tensile model is active right atthe beginning in the FLAC model cycling, which meansthat yield elements are generated at an early stage in thecycling. In contrast, Phase2 detects the yield element basedon an initial estimate using the linear stiffness matrix,

employing the initial minimum principal stress distribution

shown in Fig. 3a. To proof this point, we first set the tensilestrength in the FLAC model to a high value and cycle themodel 8000 steps to reach equilibrium. An elastic solutionis thus obtained and the minimum principal stress distribu-

Minimum principal stress (Pa)

0.00E+00

2.00E+06

4.00E+06

6.00E+06

8.00E+06

1.00E+07

Contour interval= 1.00E+06

Step 10 Step 30 Step 60

Step 100Step 200

Fig. 6. Distribution of the minimum principal stress in the FLAC model at cycling step 10, 30, 60, 100, and 200 (elastic response, rx = 60 MPa,

ry = 11 MPa).

t = 5 MPa

Shear

Tension

state

Elastic


At Yield in Tension

t = 5 MPa

Phase2

FLAC

Fig. 7. Comparison of tensile yielding in the Phase2 and FLAC models(rx = 60 MPa, ry = 11 MPa). In the FLAC model, an initial high tensilestrength is set and after an equilibrium state is reached, the tensile strength

is set to 5 MPa and the model is cycled to equilibrium.



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tion is shown in Fig. 3b. Next, we change the rock tensilestrength to 5 MPa and run the model for another 5000steps. Immediately after cycling following the change oftensile strength properties, tensile failure is detected andplastic correction conducted. The resulting tensile yield ele-ment distribution is presented in Fig. 7b, which is different

from the pattern shown in Fig. 4b but very similar to thedistribution pattern obtained by Phase2 shown in Fig. 7a.In other words, if an elastic solution is first obtained inFLAC, followed by conducting the plastic solution, then,the FLAC result will be similar to that obtained from thePhase2 model.

Hence, the observed difference in yielding zone distribu-tion by the FLAC and Phase2 models is attributed to thesolution scheme difference adopted in the numerical tools.The solution scheme difference leads to a stress path differ-ence. To further demonstrate the point, the influence ofin situ horizontal to vertical stress ratio (Ko) is conductedand the results are presented in Fig. 8. When Ko = 5.45,

both the FLAC and Phase2 models predict tensile yieldingon the sidewalls. When Ko is less than 3, the FLAC modelstill predicts tensile yielding due to dynamic unloading gen-erated by tunnel excavation but the Phase2 model shows notensile failure in the model (not shown in Fig. 8). The evo-lution of the minimum principal stress in the first 200cycling steps, at the center of the sidewall for Ko = 1, isshown in Fig. 9. It is seen that during the first 20 cycling

steps, the minimum principal stress is tensile. If the rocktensile strength is low, tensile failure will result at an earlystage, before the final equilibrium is reached.

stateElastic


At Yield in Tension

t = 5 MPa

FLAC

(a) Ko= 5.45

(x = 60 MPa y = 11 MPa)

FLAC

(c) Ko= 2

(x = 60 MPa y = 30 MPa)

(d) Ko= 1

(x = 60 MPa y = 60 MPa)

(b) Ko= 3

(x = 60 MPa y = 20 MPa)

Fig. 8. Influence of in situ stress ratio on the tensile yielding in the FLAC model (for Phase2 model, no tensile yielded elements exist for the stress condition

shown in (b)(d)).

2 4 6 8 10 12 14 16 18 20

-1.200

-0.800

-0.400

0.000

0.400

0.800

(107)

Cycling step (x 10)

Minimump

rincipalstress(Pa) Tensile

CompressiveTunnel

Fig. 9. Evaluation of the minimum principal stress at the sidewall centerduring the first 200 steps in the FLAC model (rx = 60 MPa, ry = 60 MPa;elastic solution).



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3.3. Yielding in shear

To understand the impact of dynamic unloading on shearfailure of rocks in tunnel excavation by drill and blastmethod, the same circular tunnel excavation simulationwas carried out using reduced shear strength parameters.

The MohrCoulomb failure criterion is used with a perfectplastic model that has three strength parameters: c cohe-sion;/ friction angle;w dilation angle. The in situ stressesused in the calculations are: rx = 60 MPa, ry = 11 MPa,rz = 45 MPa. The tensile strength is intentionally set to ahigh value so that only shear failure is observed in the mod-eling results.

Fig. 10 presents the simulation results by FLAC (rightcolumn) and Phase2 (left column), showing the shearyielding distribution around the tunnel. It is seen thatwhen the shear strength is relatively high (c = 30 MPa),

the yielding zone distributions in both models are similar(Fig. 10a and b). When the shear strength is reduced to 25and 20 MPa, the shear failure patterns in both modelsstart to differ. Some shallow shear yielding elements areobserved in the FLAC model all over the tunnel bound-ary but not consistently show up in the Phase2 model.

The deepest shear failure depths on the roof and flooras well as in the sidewalls in both models seem to agreewith each other. The reason for this agreement is thatto cause rocks on the roof and floor to fail in shear, themaximum tangential stress has to reach a higher value(e.g., 85.8 MPa for c = 20 MPa and / = 40). Thisrequires approximately 400 cycling steps in FLAC. At thispoint, the overall stress distribution in the FLAC model isclose to the elastic model result by Phase2. Subsequentcycling in FLAC leads to stress redistribution in themodel in a fashion that resembles the stress iteration in

c = 30 MPa

= 40o

= 10o

Phase2 FLAC

Shear

Tension

state

Elastic

At Yield in Shear or Vol.


c = 25 MPa

= 40o

= 10o

c = 20 MPa

= 40o

= 10o

a b

c d

e f

Fig. 10. Comparison of shear yielding zone distribution between Phase2 and FLAC models for different strength levels (rx = 60 MPa, ry = 11 MPa).



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Phase2 so that the deep located shear failure zones in bothmodels are similar.

3.4. Yielding in both tension and shear

Fig. 11 presents the tensile and shear yielding zone distri-butions in both FLAC and Phase2 models for c = 20 MPa,/ = 40, w = 10, rt = 5 MPa, under the in situ stressfield of rx = 60 MPa, ry = 11 MPa, rz = 45 MPa. Again,

dynamic unloading is assumed and the tensile yielding zoneis concentrated on the sidewalls and shear yielding zone onthe roof and floor. The resulting yield zone is very close toa simple composition of the yield zones in tension and shear(Figs. 4b and 10f). The major difference between the twomodeling results is the tensile yielding on the sidewalls.

3.5. Discussion

Modeling of tunnel excavation using Phase2 and FLACby deleting the excavation area is like excavating the tunnelby using long-round drill and blast construction method. Ifa dynamic unloading condition exists and when rocks

behave in an elasto-plastic manner, the results obtainedfrom these two numerical tools can be drastically different,especially when tensile strength of the rock is low. FLAC,due to its explicit modeling scheme, seems to capture therock dynamic unloading process well. The insight from thisinvestigation also explains why more rock damage isexpected in tunnels excavated by drill and blast method.Previous models for describing the mechanisms by whichblasting damages rock have either rated the role of thedynamic stress wave as more important in blasting fragmen-tation (Dally et al., 1975) or the gas action (Nilson et al.,

Phase2

Shear

Tension

state

Elastic

At Yield in Shear or Vol.


At Yield in Tension

FLAC

Fig. 11. Comparison of tensile and shear yielding zone distributionbetween Phase2 and FLAC models (rx = 60 MPa, ry = 11 MPa).

t = 5 MPa

0

10

20

30

40

50

60

70

0 1 2 3 4 5 6 7 8 9 10

Stage

E

(GPa)Modulus softening

Tensile yielding zone

E=60 GPa

E=60 0 GPa

over 10 stages

Phase2

FLAC

a

b

c

Tensile yielding zone

t = 5 MPa

Fig. 12. Distribution of tensile yielding zone by material softening method in: (a) FLAC and (b) Phase2 (rx = 60 MPa, ry = 11 MPa).



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1985; Coursen, 1979). It is seen that in addition to dynamicstress loading and gas action, there is another important ele-ment by dynamic unloading that will contribute to theblasting-induced rock damage. In general, dynamic unload-ing-induced rock damage is shallow (


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