Evaluation of impending collapse in circular tunnels by analytical and numerical approaches

Tunnelling and Underground Space Technology 26 (2011) 507–516

Contents lists available at ScienceDirect

Tunnelling and Underground Space Technology

journal homepage: www.elsevier .com/ locate/ tust

Evaluation of impending collapse in circular tunnels by analyticaland numerical approaches

M. Fraldi ⇑, F. GuarracinoDipartimento di Ingegneria Strutturale, Università di Napoli ‘‘Federico II’’, Italy

a r t i c l e i n f o

Article history:Received 2 August 2010Received in revised form 4 March 2011Accepted 13 March 2011Available online 3 April 2011

Keywords:Circular tunnelsImpending collapsePlasticityLimit analysisNumerical analysis

0886-7798/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.tust.2011.03.003

⇑ Corresponding author.E-mail address: [email protected] (M. Fraldi).

a b s t r a c t

On the basis of a straightforward analytical approach which has been recently proposed by the presentauthors, a comparison with numerical procedures to predict plastic collapse in circular rock tunnels isreported. In fact, numerical modeling of the evolution of progressive failure leading to collapse in tunnelsis a quite complicated matter and requires great care in modeling the problem and interpretating theresults. In order to provide a guide to engineers facing rock limit state problems, a few examples bymeans of three commercial packages are presented, discussed in details and confronted with the analyt-ical results.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

The possible collapse of a tunnel is a rather complex problembecause it is strongly affected by the random variability of themechanical properties of the rock in situ and from the presenceof cracks and fractures in the rock banks. Therefore, since Terzaghi(1946) several systems have been developed to estimate the de-gree of safety and, due to their simplicity, empirical methods arestill widely used. However, the results can vary very significantlyand their applicability strongly relies on the judgement and exper-tise of the designers.

Boundary Elements Methods, Finite Element-based analysesand other numerical strategies are routinely employed to assessthe degree of safety of such problems, but pose trouble in model-ling and validation of results. In fact, the main difficulties arisingin conventional numerical methods, such as FEM, are constitutedby introducing heterogeneity of rock parameters into the modeland simulating non-linear behavior in rock, processing discontinu-um mechanics problems by continuum mechanics methods andrecording the event-rate of failed elements. Several analysis tech-niques have been proposed in the past years for evaluating stabilityof tunnels and investigating the so-called arching effect both forsoft soils (see, for example Lee and et al. (2006), Osman et al.(2006)) and hard rocks and some of these techniques rely uponspecific pre-partitioning of the soil domain, by invoking the theo-rems of limit analysis (Drescher and Detournay, 1993). Very re-

ll rights reserved.

cently, simulations of fracture and fragmentation of geologicalmaterials have been obtained by coupling FEM and DEM strategies(Morris and et al., 2006) and procedures based on a translationalthree-dimensional multi-block failure mechanism have also beenproposed, with the aim of determining the face collapse pressureof a circular tunnel driven by a pressurized shield (Mollon et al.,2010). Mollon et al. (2010) provided as well some design chartsin the case of a frictional and cohesive soil. However, many ap-proaches in the current practice focus on the onset of plasticity(strength ratio or other plasticity indicators) rather than on thekinematics of collapse and this fact can be misleading since in elas-tic analyses, as well as in Finite Elements analyses, it often happensthat the first yielding does not correspond to the actual collapsemechanism.

As a matter of fact, most of these approaches are very specialisticand turn quite difficult for practical use in several problems in geo-technical engineering. Indeed, numerical simulation of progressivefailure leading to collapse in underground openings is a complicatedtask which requires great care and experience in judging results.

Following two previous works in which a straightforward ana-lytical approach has been proposed by the present authors (Fraldiand Guarracino, 2009, 2010) and on account of the recurrenceand importance in civil engineering of circular tunnels, in the pres-ent paper a few examples are analyzed by means of three commer-cial packages, discussed in details and compared to the analyticalresults. In fact, in their first work on the subject (Fraldi and Guarra-cino, 2009), the present authors introduced an exact solution in therealm of the plasticity for the evaluation of collapse mechanisms inrectangular cavities and some numerical examples were also pro-

http://dx.doi.org/10.1016/j.tust.2011.03.003

mailto:[email protected]

http://dx.doi.org/10.1016/j.tust.2011.03.003

http://www.sciencedirect.com/science/journal/08867798

http://www.elsevier.com/locate/tust

508 M. Fraldi, F. Guarracino / Tunnelling and Underground Space Technology 26 (2011) 507–516

vided, while in their second work (Fraldi and Guarracino, 2010) thesolution was extended to tunnels with arbitrary cross section butno comparisons with widely used software packages wereperformed.

For this reason, in order to provide a guide to engineers facingrock limit state problems, some examples concerning widely em-ployed circular tunnels are presented here and it is confirmed thatthe analytical solution can represent an important guidance aboutthe parameters characterising the overall collapse mechanism andcontribute to shed light on the basic mechanics underlying this

Fig. 1. A MATHEMATICA� notebook for the

area of design, thus constituting a useful complementary tool tonumerical analyses.

2. Symbolic implementation of the analytical results from limitanalysis

The present authors have recently proposed an analytical proce-dure to estimate the stability of the ceiling in tunnels rectangularor with an arbitrary, yet symmetric with respect to a vertical axis,

evaluation of the analytical formulae.

y

xo

f (x)

H

h

Hoek-Brownrock mass

sym

L

Collapsingblock

Tunnel

Free boundary

Δ hc(x)

R

Fig. 2. The collapse mechanism.

Fig. 3. Determination and plot of the effective collapsing block profile for differentvalues of the parameter B (3/4, 4/5, 5/6, 1) by means of MATHEMATICA�.

M. Fraldi, F. Guarracino / Tunnelling and Underground Space Technology 26 (2011) 507–516 509

shape (Fraldi and Guarracino, 2009, 2010). The problem was con-sidered plane and the solution was achieved by means of theGreenberg minimum principle (Maugin, 1992) and of the calculusof variations with reference to a Coulomb-type mechanism, mak-ing reference to the Hoek–Brown criterion (Hoek and Brown,1980, 1997).

It is now shown that with the aid of a symbolic mathematicalpackage such as Wolfram’s MATHEMATICA� (Wolfram, 2003) theproposed formuale can be straightforwarldy employed to studythe collapse problem for circular tunnels and to derive some re-sults for a comparison with numerical solutions.

Fig. 1 shows the first part of a simple MATHEMATICA notebook,which encompasses the geometrical and mechanical parametersand the formulae relative to a circular tunnel for a sample casestudy. A and B are dimensionless parameters characterising therock mass and rc and rt, to be intended as absolute values, arethe compressive and tensile stresses at failure. �rt and �rc are twodimensionless strength parameters, and �p is a specific generalizedpressure, defined as follows

�rt � q�1rt ; �rc � q�1rt; �p � q�1 1L

Z L

0qcðxÞdx ð1Þ

where q is the rock mass density and L the intersection of thedetaching curve, f(x), with the tunnel profile (see Fig. 2). H is theheight of the collapsing block with respect to a line passing throughthe intersection between the tunnel profile c(x) and the x-axis.

Fig. 3 shows the second part of the MATHEMATICA notebook,where the calculation of the falling block profile is carried outand the results are plotted for different values of the parameter B.

The easiness of the procedure is evident and by this means it ispossible to perform several types of parametric studies, useful bothat the design and assessment stage. For example, Fig. 4 shows theweight and the width of the collapsing block for varying tunnel ra-dii and Fig. 5 shows a logarithmic plot of the weight of the collaps-ing block for varying tunnel radii and Hoek–Brown parameter B.These results can be very helpful in practical applications, as theygive a direct estimate of the overall burden on the lining.

On the same basis the examples for the comparisons withnumerical methods discussed in the next Sections have beenderived.

3. Numerical modelling

The numerical analyses have been performed by means of threedifferent packages: Examine2D by Rocscience (2009), ANSYS11(2009) and PLAXIS8 (2002). Notwithstanding the existence of tensof specialized geotechnical programs, for the sake of clarity theattention has deliberately been focused on interactive and widelyused codes, commonly employed for performing parametric analy-sis, preliminary design and verification procedures. Also, they areoften used as educational tools for both practicing engineers and

Fig. 4. Collapsing block for circular tunnels with increasing radii (top), dimensionless width of the collapsing block, L/R (bottom-left), and weight of the collapsing block, P(bottom-right) [A = 2/3, B = 3/4, rc = 2.5 MPa, rt = rc/100, q = 25 K Nm�3].

Fig. 5. Logarithmic plot of the weight of the collapsing blocks, P, for varying tunnel radii and Hoek–Brown parameter, B [A = 2/3, rc = 2.5 MPa, rt = rc/100, q = 25 K Nm�3].


Fig. 6. Analytical results for impending collapse of a circular tunnel (R = 3.75 m). The rock mass parameters are: A = 2/3, B = {3/4, 4/5, 5/6, 1}, rc = 2.5 MPa, rt = rc/100, q = 25 K Nm�3.

Fig. 7. Hoek–Brown (continuous line) and equivalent Mohr–Coulomb (dotted line, calculated according to the linear regression in Fraldi and Guarracino (2009)) yield loci inthe r–s plane.


Table 1Hoek–Brown and equivalent Mohr–Coulomb parameters (calculated according to thelinear regression in Fraldi and Guarracino (2009)).

Hoek–Brown Mohr–Coulomb

A = 2/3, B = 3/4, mb = 3.1, s = 0.035,a = 0.62, rc = 2.5 MPa, rt = rc/100

u ¼ 33:8 � c ¼ 0:17 MPa


u ¼ 34:3 � c ¼ 0:13 MPa


u ¼ 34:4 � c ¼ 0:1 MPa

A = 2/3, B = 1, mb = 2.65, s = 0.026,a = 0.95, rc = 2.5 MPa, rt = rc/100

u ¼ 34:6 � c ¼ 0:03 MPa


students. In fact, these softwares can be employed to make it easyfor the engineer to understand the basic principles of stress analy-sis and its application to the modelling of undergroundexcavations.

Examine2D by Rockscience is a two-dimensional indirectboundary element program for calculating stresses and displace-ments around underground and surface excavations in rock. Unlikethe Finite Element and Finite Difference methods, the boundaryelement method only requires meshing around the excavationand the rock mass boundaries, eliminating the need for complexvolume mesh generation.

Therefore Examine2D provides an integrated graphical environ-ment for data entry and visualization and also a real time stressanalysis capability, which allows to manipulate model parameters,and in real-time, visualize the effect on a stress analysis. It canadopt both Mohr–Coulomb and generalized Hoek–Brown strengthcriteria and contoured results include principal stresses, displace-ments, and strength factors.

ANSYS package is a general purpose FEM code, which can beemployed for elasto-plastic analyses of both small and large defor-mation problems. The main features which render ANSYS packagesuitable to the numerical modelling of collapse mechanisms intunnels are its capabilities of performing analyses of small andlarge elasto-plastic deformations using Drucker–Prager or Mohr–Coulomb yield criteria and of simulating the construction process,

Fig. 8. Mean stress plot from E

including excavation, by deactivating selected elements. In addi-tion, ANSYS can also simulate the effects of progressive collapseby providing dynamic restraint and killing elements options.

Additionally, it is worth to notice that ANSYS is a package withvery advanced numerical capabilities. It can adopt different typesof solvers and iterative procedures for nonlinear problems, suchas Adaptive Descent, a technique which switches to a ‘‘stiffer’’ ma-trix if convergence difficulties are encountered, Line Search, a tech-nique which attempts to improve a Newton–Raphson solution byscaling the solution vector by a scalar value termed the line searchparameter, and Arc-Length Method, a method which involves thetracing of a complex path in the load–displacement response.

On the contrary, PLAXIS is intended to provide a tool for practicalanalysis to be used by geotechnical engineers who are not necessar-ily numerical specialists. In fact, quite often practising engineersconsider non-linear finite element computations cumbersomeand time-consuming and PLAXIS, according to the producer, hasbeen designed to provide robust and theoretically sound computa-tional procedures encapsulated in a logical and easy-to-use shell.Many geotechnical engineers world-wide have adopted this prod-uct and use it for design purposes. The implicit integration of differ-ential plasticity models is a dedicated one and relies upon the use ofthe scheme by Vermeer (1979), which overcomes the requirementto update the stress to the yield surface in the case of a transitionfrom elastic to elasto-plastic behavior. Moreover, a specialisedautomatic step size procedure, as introduced by Van Langen andVermeer (1989), is implemented in the package.

On these bases, the examples for a comparison between analyt-ical and numerical solutions will be analysed in the next Section.

4. Examples and discussion

Fig. 6 shows, from left to right, the collapsing blocks corre-sponding to increasing values of the parameter B (3/4, 4/5, 5/6, 1)for a typical rock mass (A = 2/3, rc = 2.5 MPa, rt = rc/100, q =25 KN/m3) and a tunnel radius of 3.75 m, evaluated according tothe analytical formulation of Section 2.

xamine2D elastic analysis.

Fig. 9. Stress factors and failure trajectories from Examine2D elastic analysis. The rock mass parameters are (from top left to bottom right): A = 2/3, B = {3/4, 4/5, 5/6, 1}, rc = 2.5 MPa, rt = rc/100, q = 25 K Nm�3.


From the results it appears evident that, along with the increas-ing values of B, the width L of the collapsing block tends to de-crease and its height, on the contrary, tends to increase. Thevalue B = 1 coincides with the Mohr–Coulomb yield criterion.

Fig. 10. Element mesh for ANS

Fig. 7 shows the Hoek–Brown (continuous line) and the equiv-alent Mohr–Coulomb yield loci (dotted line, calculated accordingto the linear regression in Hoek and Brown (1997)) in the r–splane. The Hoek–Brown and the Mohr–Coulomb parameters are

YS elasto-plastic analysis.

Fig. 11. Equivalent plastic strains from ANSYS elasto-plastic analysis. The rock mass parameters are (from top left to bottom right): A = 2/3, B = {3/4, 4/5, 5/6, 1}, rc = 2.5 MPa, rt = rc/100, q = 25 K Nm�3.

Fig. 12. Element mesh for PLAXIS elasto-plastic analysis.


listed in Table 1. The equivalence is needed because the models inANSYS and PLAXIS make reference to the Drucker-Prager andMohr–Coulomb material models, respectively.

Fig. 8 shows the results from Examine2D elastic analysis interms of mean stress. It is immediate to notice that the stress con-tour plot at the top of the cavity indicates a stress intensificationbut it cannot be related to the collapsing block shapes found bymeans of the analytical solution, which are dependent on theHoek–Brown rock mass parameters. In fact, Examine2D analysesgive indications about the onset of plasticity only through the cal-culated strength ratios, as shown in Fig. 9, where the contour plotsof the stress factors, which represent the ratios of material strengthto induced stress at a given point, are represented. In the same pic-tures, failure trajectories, which indicate the locations at which theinduced elastic stresses exceed the strength envelope of the mate-rial, are also shown. This corresponds to regions where thestrength factor contours are less than 1. Shear failure is indicatedby two intersecting lines, where the orientations of the lines showthe theoretical orientation of the shear failure planes, relative tothe principal stress directions. Tensile failure is indicated by a sin-gle line, displayed perpendicular to the direction of tensile failure.

It is immediate to notice that the locations at which the inducedelastic stresses exceed the yield locus of the material tend to in-crease with the value of the parameter B, that is with the decre-ment of tensile strength in the r–s plane. This fact, can also besomehow inferred from the analytical analysis (see Fig. 6), wherethe weight of the collapsing blocks tends to increase with the valueof the parameter B. However, in the cases of Fig. 9 no collapsing re-gions can be identified at the top of the excavation. This is not sur-prising, given the characteristic of the underlying analysis, which isincapable of simulating the effects of plastic strain evolution.

It is worth noticing that in the numerical analyses conductedwith Examine2D, as well as the subsequent ones by means ofANSYS and PLAXIS, the stress field defined by the in situ stressstate in the rock mass prior to excavation has been taken into ac-count on the basis of an assumed depth of the excavations belowground surface of about 12 m. In this respect it is worth pointingout that, with reference to the employed analytical formulation,

Table 2Computed volumes of the plastic zones at impending collapse from ANSYS andPLAXIS analyses versus those of the collapsing blocks from analytical results.

B Volume of plastic region (m3) Volume of detaching block (m3)

Ansys Plaxis Analytical

3/4 0.3079 0.2568 0.27814/5 0.8005 0.6662 0.67765/6 0.9051 0.7574 0.79791 0.9492 0.8242 0.9147


these data only influence the failure domain, i.e. the parametersgoverning the Hoek–Brown criterion, but from the standpoint oflimit analysis they are irrelevant to the collapse mechanism.

Fig. 10 shows the mesh employed for the ANSYS nonlinear FEanalysis. The failure criterion has been replicated by means of anExtended Drucker-Prager (EDP) model, available in the ANSYSmaterial library. In fact, both yield surface and the flow potential,can be taken as linear, hyperbolic and power law independently,and thus results in either an associated or non-associated flow rule.However, given the fundamentals underlying the analytical treat-ment of the problem and in order to obtain a straightforward com-parison, the analysis has been conducted according to anassociative flow rule without significant loss of generality (Fraldiand Guarracino, 2009). Basically, ANSYS makes reference to theBesseling model, also called the sub-layer or overlay model and

Fig. 13. Plastic points from PLAXIS elasto-plastic analysis. The rock mass parameters are (q = 25 K Nm�3.

in the plastic range the material response can be represented bymultiple layers of perfectly plastic material, where individualweights are derived from the uniaxial stress–strain curve. Further,a Riks arc-length procedure has been employed in conjunction

from top left to bottom right): A = 2/3, B = {3/4, 4/5, 5/6, 1}, rc = 2.5 MPa, rt = rc/100,


with PLANE42 elements to ensure an accurate convergence of theanalysis in the plastic range. In fact, the problem with impendingcollapse modes is essentially that of analysing a force-loaded struc-ture that locally passes through a singular (rigid motion) configu-ration. This is a very complicated task and both nonlinearstabilization and the arc-length method have been employed tosimulate the collapsing of blocks. Moreover, the system has beenartificially restrained during intermediate load steps in order toprevent unrealistically large displacements from being calculated.

In particular, Fig. 11 shows the equivalent plastic strains atimpending collapse. It can be noticed that in this case, if the col-lapsing blocks are identified with the plastic zones from the FEanalyses, the results tend to resemble, although with differentshapes, those by the analytical procedure. In fact, the collapsingareas tend to increase with the value of the parameter B.

The same happens with the results from the PLAXIS analyses. Asstated before, PLAXIS employs an implicit integration scheme byVermeer (1979) and a specialised automatic step size procedureby Van Langen and Vermeer (1989), thus reducing the optionsavailable to the user and, therefore, the knowledge required. Themesh is shown in Fig. 12. The mesh generator is a special trianglemesh generator based on a robust triangulation procedure, whichresults in ‘unstructured’ meshes. According to the PLAXIS producer,these meshes may look disorderly, but the numerical performanceof such meshes is usually better than for structured meshes. Fig. 13shows the plastic stress location for the examples at hand. It isimmediate to notice that again the plastic areas tend to increasewith the value of the parameter B, but their extensions are to someextent larger than those resulting from ANSYS analyses.

In order to make a quantitative comparison between the resultsfrom analytical and FE procedures, Table 2 shows the computedvolumes of the fully plastic zones at impending collapse fromANSYS and PLAXIS analyses versus those of the collapsing blocksfrom analytical results. It is evident that the results from ANSYSare on average 10% larger than those from analytical analysis andthese latter are, in turn, about 10% larger than those from PLAXISanalyses.

It can be concluded that all the performed analyses (limit,numerically elastic and numerically elasto-plastic) tend to agreeon the influence of the Hoek–Brown parameter B with respect tothe impending collapse of the tunnel ceiling. However, when itcomes to the evaluation of the dimensions of the collapsing blocksand, therefore, to the design of the liner, the results of the two dif-ferent numerical elasto-plastic analyses differ on average 20%, withthe results of the limit analysis falling in the middle.

It is thus confirmed that, given the intrinsic difficulty in per-forming a collapse analysis by means of commercial FE codesand in correctly judging the obtained results, the proposed analyt-ical method can constitute a simple, direct and important guidanceabout the parameters characterising the overall collapse mecha-nism and a useful auxiliary tool to these analyses.

5. Conclusion

The study has presented a comparison between numerical andanalytical approaches to modeling plastic collapse in circular rocktunnels. With the aid of commercial packages, it has been shownthat the numerical modeling of the evolution of progressive failureleading to collapse in such tunnels remains a complicated matter,which requires great care in preparing the model and analyzing theresults. In this respect, the usefulness of the straightforward ana-lytical approach proposed by the present authors as a complemen-tary tool to these analyses has been confirmed. In fact, theproposed analytical solution is firmly rooted in the classical theoryof limit analysis, which is believed to be still able to offer somevaluable information for many engineering problems, notwith-standing the availability of many flexible and powerful numericaltools (Fraldi et al., 2010).

Acknowledgments

This study was partially supported by MIUR through grant PRIN20078MHYS4.

References

ANSYS 11.0, 2009. User’s Manual. ANSYS, Inc., Canonsburg, PA 15317, USADrescher, A., Detournay, E., 1993. Limit load in translational failure mechanisms for

associative and non-associative materials. Géotechnique 3, 443–456.Examine2D, 2009. User’s Manual. Rocscience Inc., Toronto, Ontario, Canada.Fraldi, M., Guarracino, F., 2009. Limit analysis of collapse mechanisms in cavities

and tunnels according to the Hoek–Brown failure criterion. Int. J. Rock Mech.Mining Sci. 46, 665–673.

Fraldi, M., Guarracino, F., 2010. Analytical solutions for collapse mechanisms intunnels with arbitrary cross sections. Int. J. Solids Struct. 47 (2), 216–223.

Fraldi, M., Nunziante, L., Gesualdo, A., Guarracino, F., 2010. On the bounding of limitmultipliers for combined loading. Proc. R. Soc. A 466, 493–514.

Hoek, E., Brown, E.T., 1980. Empirical strength criterion for rock masses. J. Geotech.Eng. Div. ASCE 106 (GT9), 1013–1035.

Hoek, E., Brown, E.T., 1997. Practical estimates of rock mass strength. Int. J. Mech.Min. Sci. 34 (8), 1165–1186.

Lee, C.J. et al., 2006. Tunnel stability and arching effects during tunnelling in softclayey soil. Tunn. Undergr. Sp. Tech. 21 (2), 119–132.

Maugin, G., 1992. The Thermomechanics of Plasticity and Fracture. CambridgeUniversity Press, Cambridge.

Mollon, G., Dias, D., Soubra, A.H., 2010. Face stability analysis of circular tunnelsdriven by a pressurized shield. J. Geotech. Geoenviron. Eng. 136 (1), 215–229.

Morris, J.P. et al., 2006. Simulations of fracture and fragmentation of geologicalmaterials using combined FEM/DEM analysis. Int. J. Impact Eng. 33, 463–473.

Osman, A.S., Mair, R.J., Bolton, M.D., 2006. On the kinematics of 2D tunnel collapsein undreined clay. Geotechnique 56 (9), 585–595.

PLAXIS2D v 8, 2002. User’s Manual. Balkema, Rotterdam.Terzaghi, K., 1946. Rock defects and loads on tunnel supports. In: Proctor, R.V.,

White, T.L. (Eds.), Rock Tunnelling with Steel Supports, vol. 1. CommercialShearing and Stamping Company, Youngstown, OH, pp. 17–99.

Van Langen, H., Vermeer, P.A., 1989. Automatic step size correction fornonassociated plasticity problems. Int. J. Num. Meth. Eng. 29, 579–598.

Vermeer, P.A., 1979. A modified initial strain method for plasticity problems. In:Proc. 3rd Int. Conf. Num. Meth. Geomech., Rotterdam, Balkema, pp. 377–387

Wolfram, S., 2003. The Mathematica Book, Fifth ed. Wolfram Media, Champaign,USA.

Documents

Evaluation of impending collapse in circular tunnels by analytical and numerical approaches