Tunnelling and Underground Space Technology 26 (2011) 524–533

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Tunnelling and Underground Space Technology

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An equivalent beam model for the analysis of tunnel-building interaction

M. Maleki ⇑, H. Sereshteh, M. Mousivand, M. BayatDepartment of Civil Engineering, Bu-Ali Sina University, Hamedan, Iran

a r t i c l e i n f o a b s t r a c t

Article history:Received 7 September 2010Received in revised form 9 January 2011Accepted 15 February 2011Available online 8 March 2011

Keywords:Shallow tunnelAdjacent structureGround movementNumerical modeling

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⇑ Corresponding author. Tel.: +98 811 8257410; faxE-mail address: Maleki@basu.ac.ir (M. Maleki).

The aim of this work is to study the effect of structural characteristics, including stiffness, geometry andweight on tunnel–adjacent structure interaction. Ground materials, tunnel geometry and excavatordevice are related to a part of metro tunnel of Tehran. To describe the ground behavior due to tunneling,a 3D FE code with an elastoplastic soil model was used. The adjacent building was modeled in two ways:one as an equivalent beam or shell and the other as a real geometry (3D frames). The obtained resultsfrom this theoretical work indicate particularly that the stiffness of adjacent structure controls theground movement distribution induced by tunnel excavation which in agree with other researchers.As it was predicatively, increasing in structure weight leads to create the large displacement componentsin the ground. The structure width plays also a significant role in displacement distribution of ground.The comparison of the obtained results using two methods of structure modeling shows a very good con-formity between them.

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1. Introduction

Underground transportation systems have been in demand inmany major cities. These systems require a tunnel which is con-structed in urban areas, particularly in soft ground and in shallowzones. Measurement, designing and performing of undergroundstructure can be known as the most important civil engineer’schallenge (Bernat and Cambou, 1998; Liu et al., 2008).

Influence on adjacent buildings is of major interest for tunnel-ing operations in urban areas, due to the high interaction betweentunneling and existing structures (Pickhavar et al., 2010; Dimmockand Mair, 2008). This problem/issue was previously analyzed usinga combination of in situ observations and numerical modeling.Analysis of previous case histories paved the way for the establish-ment of various empirical relationships between tunneling in-duced ground movement and associated structure damage(Burland and Wroth, 1974; Boscardin and Cording, 1989; Burland,1995; Mair et al., 1996). These methods are widely used in practice.

In reality, a rigorous analysis of the tunneling-structure interac-tion problem is a hard task, due to (I) the high interaction betweentunneling and adjacent structure, (II) 3D nature of this problemand (III) the non-linear geometrical behavior involved that leadsto use an appreciate numerical method (Mroueh and Shahrour,2003). Different approaches have been used to represent the build-ing with varying level of details in the numerical methods. Accord-ing to the simplified operations are executed in two consecutives

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steps. The first step addresses the determination of ground move-ment induced by tunneling using empirical (Peck, 1969), analytical(Sagasta, 1987; Verruijt and Booker, 1996) or numerical methodswhich was widely studied by many researchers (for instance Gon-zález and Sagaseta (2001) and ITA/AITES Report (2007) studies);and in the second step structural analysis of building subjectedto the ground settlement calculated in the first step is done.According to the work of Mroueh and Shahrour (2003) the simpli-fied approach can be considered as very conservative. In the fully3D FE modeling, details of building can be modeled. The advantageof such 3D model is that the building can be taken into account inany geometrical configuration with respect to tunnel axis. Threeexamples of such analyses are the works of Mroueh and Shahrour(2003), Burd et al. (2000) and Keshuan and Lieyun (2008). In theplane strain analysis, the building is described by its width andheight and details can be incorporated in the model. The obtainedresults from 2-D finite element analyzes show that the narrowestsettlement troughs were predicted when the nonlinearity of soilsat small strains was modeled (Chow, 1994).

Furthermore, there is recommended another approach based onsoil-structure relative stiffness which takes into account easily theeffect of structure stiffness in tunnel-structure interaction (Franzi-us, 2003). In this method, the structure is modeled as an equivalentelastic beam or shell having bending and axial stiffness.

In spite of various works existing in the literature, it misses yeta clear parametric study concerning the intensity of influence ofstiffness and the other structure characteristics in tunnel–adjacentstructure interaction problem.

The present work is defined in the framework of tunnel–adjacent structure subject. It focuses particularity on parametric

M. Maleki et al. / Tunnelling and Underground Space Technology 26 (2011) 524–533 525

study of structural characteristics effects of adjacent building onground movement induced by tunnel excavation. The adjacentbuilding is modeled firstly as an equivalent beam that allows useasily to achieve parametric study operation. In the second partof work, the structure is considered with its real geometry. Thestudy of interaction between tunnel and adjacent structure wasdone using an indirect method developed in this work. PLAXISand SAP (powerful in structural problems analysis) codes wereused with the consideration of displacement field compatibilityat soil media and structure interface. Tunnel geometry and groundproperties are corresponding to a part of metro tunnel of Tehrancity which was constructed using a slurry shield machine withan outside diameter of 9.0 m. The obtained results by two methodsstructure modeling were finally compared.

2. Modeling and parameters

PLAXIS 3D code only generates the triangle mesh, but it can usethe meshes in very fine size. Meshing is introduced in five modes:Very coarse, coarse, medium, fine and very fine. The importantability of code is to make finer meshing regarding a region andor surround of a line. However, the precision is increased by useof finer mesh in a region but causes time to add for run problem(PLAXIS code manual, 2005).

Medium mesh mode is used in present work and in more sensi-tive zones, mesh dimension gets finer. Selection of this size of meshis not worrying, because coarse meshes have been used in 3D settle-ment analysis by PLAXIS 3D code in some projects such as Renn-steig tunnel in Thuringia city. Also, for modeling Steinhaldenfeldtunnel in Stutgart city, very coarse mesh with hardening elastoplas-tic constitutive model was used that had a good agreement withreal value from in situ information (Mair et al., 1996). Fig. 1 showsFE mesh and also the lines of displacements measurement.

In this study, the most suitable constitutive model presented inPLAXIS code was selected. This model is elastoplastic with the iso-tropic hardening mechanisms. It can be considered as developmentof non-associated Mohr–Coulomb model. In fact, major limitationsof Mohr–Coulomb model are removed by adding a cap surface todescribe plastification under isotropic stress, and an isotropic hard-ening mechanism to express non-linear plastic behavior before thefailure. Evolution of yield surface in deviatoric mechanism iscontrolled via deviatoric plastic strain. Volumetric plastic straincontrols the cap evolution. The plastic hardening and elasticmodulus are properly considered as function of confining pressure.Basic properties of this model are:

Fig. 1. FE mesh and lines of soi

– Hardening plastic and elastic modulus is dependent on confin-ing stress according to the exponential rule (exponential depen-dence of stiffness on stress).

– Parabolic relationship between deviatoric stress and strain.– Separation of initial loading from unloading–reloading.– Coincidence of failure surface on Mohr–Coulomb criteria.

Nonetheless, this model is useful in monotonic loadings onlyand some of important soil behavior aspects, such as failure surfacedependence on confining pressure and critical state concept arenot taken into account.

This model has eight parameters, fortunately all of which haveclearly physical meanings and are determined easily by the classi-cal laboratory tests. Parameters of model are:

c: Soil cohesion.u: Maximum internal friction angle.w: Dilation angle.E50

ref : Secant modulus in standard triaxial test at the referenceconfining pressure (r3 = pref).E50

ode: Tangent modulus related to the consolidation test.Eref

ur : Modulus related to the unloading and reloading states.m: Controls the dependence of plastic and elastic modulus onconfining stress.mur: Poisson ratio in unloading–reloading state.

In the PLAXIS code, the mobilized shear strength in interface bondis a function of shear strength of soil. This option is controlledusing the parameter Rinter that is equal to or less than 1.0, for realsoil-structure interaction the interface is weaker and more flexiblethan the associated soil layer, which means that the value of Rinter

should be less than 1.0. The Rinter in this study is supposed to be 0.7.Because of the interface behavior before yielding is consideredelastic, the gapping or overlapping (i.e. relative displacements per-pendicular to the interface) could be expected to occur. On theother hand, the gap can be developed between the equivalentbeam and ground surface. In the present work, the gap in certaincase appeared, however, its value was very small without an effectconsiderable on settlement profile in ground surface.

A section of line 1 of Teheran metro near 7tir square station wasmodeled to achieve the aims of this study. Shield method was usedfor tunnel construction. The information concerning the soil prop-erties, tunnel geometry and tunneling device were taken from Teh-ran urban and suburban railway organization. Concerning thegeological aspects, 7tir station is located in the end part of non-homogeneous alluvial formation in Tehran north and its lithologi-

l movement measurement.

Fig. 2. Schematic cross section of geometry and material of line 1 of Teheran metro near 7tir square station.

Table 1Soil physical properties.

Rinter Eresur (kN/m2) Eref

oed (kN/m2) Eref50 (kN/m2) mur w (�) u (�) C (kN/m2)

0.7 1.7E5 5.6E4 5.67E4 0.2 10 40 0.25

Table 2Mechanical parameters of tunnel lining.

EA (kN/m) EI (kN/m) Tunnel lining (cm)

8.05E6 8.218E4 35

Table 3Parameters of modeled structure.

Equivalent structure Structure Row

W (kN/m/m) EAstruct (kN/m) EIstruct (kN m2/m)

20 1.035E7 7.97E7 2-Storey 140 1.725E7 3.989E8 4-Storey 280 3.105E7 2.393E9 8-Storey 3

160 5.865E7 1.627E10 16-Storey 4

Poison ratio for equivalent beam element to load modeling assume to be 0.25.

526 M. Maleki et al. / Tunnelling and Underground Space Technology 26 (2011) 524–533

cal composition consists of sand, gravel, cobblestone and clay. For-mation of this area is of a good permeability and depth of ground-water table is 74 m. Geotechnical data of this station shows thein situ alternative layers of GP, GW, GC, SC and SM. Values of geo-technical parameters are obtained based on jacking and directshear tests (Fig. 2).

According to geotechnical information, the parameters of con-stitutive model were estimated. The values of parameters are listedin Table 1.

Thickness and mechanical parameters of tunnel lining are pre-sented in Table 2.

3. Equivalent beam consideration

The structure of adjacent building is considered by an equiva-lent elastic beam with length of L and width of B (Fig. 1). Bending

stiffness (EI) and axial stiffness (EA) represent the overall stiffnessof the structure. The advantages of this method are; simplicity inconsidering adjacent building stiffness according to structural sys-tem and weight of building and also, in 2D conditions, the smallamount of computational resources is required and therefore theability to perform extensive parametric studies can be achieved.The second moment of area for the equivalent beam was then, cal-culated using the parallel axes theorem (Appendix A). Consideredstructures in analyses were as 2, 4, 8 and 16-storeys. Diverseparameters of structures have been presented in Table 3.

Different steps of FE calculations performed in the first part ofthis work can be concluded as:

– Analysis of ground for considering the gravity.– Introducing the equivalent beam and ground analysis.– Effacing the deformation field engendered by the beam.– Performing the tunnel and capturing the deformation.

4. Discussion on equivalent beam results analysis

In this section, the results of analyses are presented. The focus ison ground horizontal displacement and ground settlement distri-bution. There are presented, in Figs. 3 and 4, the profiles of hori-zontal ground movement at 6 m offset from the tunnel centerline in the cases of with and without consideration of structurestiffness, respectively. The depth of tunnel is 17.7 m and the widthof buildings has been considered to be 100 m. Each of curves cor-responds to one building with specific storey and stiffness. In addi-tion, soil movement corresponding to Greenfield (GF) analysis hasbeen included. The maximum horizontal movement in GF condi-tions along the vertical line is reached at z = 18.14 m. From thereit reduces towards the surface so that at z = 11.50 m horizontal dis-placement is zero. In continuing, the movement toward groundsurface increases so that horizontal displacement of 2.5 mm occursat the ground surface. The presence of building causes a reductionin horizontal movement at ground surface. For example, it changesfrom 2.5 mm corresponding to GF to 0.9 mm for a 2-storeys build-ing. Although horizontal movement is less than GF at ground sur-face, it increases with depth and becomes larger than that obtainedfor GF conditions. These two figures indicate that with the addition

M. Maleki et al. / Tunnelling and Underground Space Technology 26 (2011) 524–533 527

of building storey, soil movement increases. With comparison ofcurves in the cases of with and without consideration of stiffness,it is observed that consideration of building stiffness leads todecrease horizontal soil movement. It can be generally said thatstructure stiffness is a factor that resists against horizontal defor-mation of soil induced by tunnel excavation. The presented resultsin Figs. 3 and 4 indicate that the horizontal movement in the sur-face has the opposite direction in comparison with movement attunnel depth. That is physically real because the region nearground surface is in contraction state due to tunnel excavation.On the other hand, the horizontal movement is toward the tunnelcenterline, whereas in depth the ground movement is outward ofcenterline. Settlement distribution of ground surface in two casesof with and without consideration of building stiffness has been

Fig. 5. Vertical soil movement profile of ground surface without consideration ofthe building stiffness.

Fig. 6. Vertical soil movement profile of ground surface with applying the buildingstiffness.

Fig. 7. Horizontal soil movement profile in 6 m distance from tunnel axis forbuilding with different widths.

Fig. 4. Horizontal movement profile of soil in distance of 6 m from tunnel axis withapplying the building stiffness.

Fig. 3. Horizontal movement of soil profile without applying the building stiffnessin distance of 6 m from tunnel axis.

presented in Figs. 5 and 6. It is clear that settlement distributionis effectively influenced by building stiffness. In fact, building stiff-ness causes the uniform settlement beneath the foundation, so thisuniformity is increased by increase in stiffness. One of the mostimportant interests of equivalent beam method is that for a givenadjacent building, we can attribute the stiffness to building withrespect to its structural characteristics. For example, for a weightymasonry building an insignificant stiffness can be considered.

Fig. 8. Vertical soil movement profile in ground surface for 4-storey building withdifferent widths.

Fig. 9. Horizontal soil movement profiles in ±6 m distance from tunnel axis wherethe eccentricity is 7.5 m.

Fig. 10. Horizontal soil movement profiles in ±6 m distance from tunnel axis wherethe eccentricity is 12.5 m.

Fig. 11. Vertical soil movement profiles in ground surface for 4-storey structurewith different lengths of building.

Fig. 12. Structure columns plane.

Fig. 13. Process of interaction between SAP and PLAXIS.

528 M. Maleki et al. / Tunnelling and Underground Space Technology 26 (2011) 524–533

Fig. 7 shows the effect of building width on ground horizontalmovement due to tunnel excavation. In this figure ground horizon-tal movement has been drawn versus depth for vertical line placed

at 6 m from tunnel center. The building is considered to be 4-sto-reys with 15, 30 and 60 m width. For comparative purposes, the re-sults of green field analysis for B = 100 m are included. Thesecurves indicate that horizontal movement is increased with in-crease in building width. The profiles of soil vertical movementin ground surface for different building widths have been pre-sented in Fig. 8. These results indicate that for the great values ofwidth the maximum settlement is decreased but a large domainof ground is influenced due to tunnel excavation.

Fig. 9 shows the soil horizontal movement profiles of verticalline placed in 6 m distance from centerline of tunnel. These curvesare for a 4-storeys building with 15 m width and 7.5 m eccentricity(building center in comparison with tunnel center). For compara-tive purposes, the results of green field analysis are included. These

Table 4Reaction forces of structure in initial step.

Force (kN m) step 1 G F E D C

Fx Fy Fz Fx Fy Fz Fx Fy Fz Fx Fy Fz Fx Fy Fz

1 �8.4 �1262 �8.9 7.8 �1670 �11.3 �9.5 �1806 �11.4 6.7 �2196 �14.4 1.6 �1451 �16.52 �6.6 �1596 �10.3 4.7 �2076 �6.2 �129.5 �2297 �5.9 130.1 �2711 �2.4 5.9 �1840 1.53 �7.1 �1638 �94.6 5.3 �2160 �6.3 �138.1 �2413 �6.5 134.9 �2828 �10.4 5.7 �1941 �124.24 �7.3 �2180 120.1 3.6 �2721 �2 �5.3 �2914 �2.9 �3.2 �3620 0.8 11 �2524 1495 �1.8 �1446 16.2 �3.1 �1910 12.8 1.7 �2029 13.9 �12.6 �2507 13.5 13.5 �1660 5.8

Table 5Reaction forces of structure in final step.

Force (kN m) step 5 G F E D C

Fx Fy Fz Fx Fy Fz Fx Fy Fz Fx Fy Fz Fx Fy Fz

1 �10.3 �1242 �5 6.6 �1644 �7.7 �10.1 �1775 �7.1 7.1 �2167 �10.9 3 �1433 �13.42 �7.9 �1600 �6.3 3.8 �2075 �2.5 �135.1 �2305 �2 13.4 �2700 1.3 7.2 �1840 15.13 �8.4 �1715 �90.3 4.7 �2187 �4.5 �142.5 �2442 �4.7 143.2 �2852 �8.7 7.1 �2006 �119.74 �8.8 �2143 99.9 3 �2731 �3.4 �5.3 �2928 �4.3 �2.6 �3631 �0.4 12.3 �2501 12.35 �3.1 �14277 13.4 �3.5 �1897 10.5 1.7 �2012 11.4 �11.8 �2494 11.4 15.2 �1649 15.2

Table 6Displacement of beneath a structure columns in initial step.

Displacement (mm) step 1 G F E D C

Ux Uy Uz Ux Uy Uz Ux Uy Uz Ux Uy Uz Ux Uy Uz

1 �0.27 �13.10 �0.34 �0.27 �14.71 �0.38 �0.26 �16.3 �0.41 �0.26 �17.38 �0.43 �0.25 �18.95 �0.482 �0.31 �14.41 �0.34 �0.30 �16.43 �0.37 �0.3 �17.73 �0.40 �0.30 �18.85 �0.42 �0.3 �20.52 �0.463 �0.35 �14.81 �0.33 �0.35 �16.41 �0.36 �0.35 �18.11 �0.40 �0.36 �19.25 �0.42 �0.36 �21.01 �0.464 �0.41 �14.12 �0.35 �0.42 �15.72 �0.38 �0.44 �17.36 �0.41 �0.45 �18.46 �0.43 �0.46 �20.09 �0.465 �0.47 �11.99 �0.35 �0.48 �13.55 �0.4 �0.50 �14.99 �0.43 �0.51 �15.99 �0.45 �0.55 �17.41 �0.49

Table 7Displacement of the beneath a structure columns in final step.

Displacement (mm) step 5 G F E D C

Ux Uy Uz Ux Uy Uz Ux Uy Uz Ux Uy Uz Ux Uy Uz

1 �0.23 �14.08 0.02 �0.23 �14.72 �0.03 �0.24 �15.4 �0.06 �0.24 �15.86 �0.08 �0.24 �16.52 �0.112 �0.27 �15.33 0.004 �0.27 �16 �0.02 �0.27 �16.75 �0.06 �0.28 �17.24 �0.07 �0.29 �18 �0.13 �0.32 �15.71 0.01 �0.32 �16.33 �0.02 �0.33 �17.1 �0.05 �0.33 �17.62 �0.08 �0.33 �18.46 �0.114 �0.39 �14.96 �0.003 �0.4 �15.62 �0.03 �0.41 �16.35 �0.06 �0.41 �16.83 �0.08 �0.42 �17.58 �0.115 �0.46 �12.75 0.006 �0.45 �13.43 �0.05 �0.46 �14.04 �0.08 �0.47 �14.46 �0.09 �0.5 �15.03 �0.13

Fig. 14. 3D FE mesh for PLAXIS code.

M. Maleki et al. / Tunnelling and Underground Space Technology 26 (2011) 524–533 529

results show that eccentricity of building influences the soil hori-zontal movement around tunnel. In fact, an asymmetrical geome-try of structure with respect to tunnel centerline results inasymmetrical displacement field of soil. When eccentricity takes

the great values, the effect of building on ground movement dueto tunneling will be negligible (Fig. 10).

Possibility of accurate studies on tunnel front behavior withrespect to construction methods is one of the most important

Fig. 15. Bending moment distribution in tunnel lining, (a is front section plan of structure b is middle section and c is rear section of structure).

530 M. Maleki et al. / Tunnelling and Underground Space Technology 26 (2011) 524–533

interests of 3D FE analyses. In the present work, the effect of build-ing length on ground movement around tunnel front has beenstudied. To do this, a 4-storeys building with 15 m width was con-sidered. The results of analyses for two different lengths of 10 and50 m of building have been presented in Fig. 11. It can be seen thatground vertical movement in direction of tunnel excavation isinfluenced by building length. In fact, for the case of building withsmaller length, the distribution of longitudinal vertical movementis sharper than the case of building with the bigger length.

Fig. 16. Vertical displacement in the ground surface (a is front section pl

5. Consideration of structure as real geometry

Consideration of structure as real geometry and stiffness in soil-structure interaction problems can give more realistic response incomparison with the equivalent beam. Although such consider-ation necessitates us to have the accurate information from presentadjacent building. This information is related to materials, geome-try and structural system that are generally difficult to obtain.The preparation of data for existing old buildings will be more

an of structure b is middle section and c is rear section of structure).

M. Maleki et al. / Tunnelling and Underground Space Technology 26 (2011) 524–533 531

difficult. Therefore, it is reasonable to use other methods to modeladjacent buildings in tunneling or excavation problems. Thesemethods must be simple and have a good agreement in comparisonwith the results, while the adjacent structure is modeled as its realform. In the previous sections the adjacent building was modeled asan equivalent beam that allowed us to study the effect of variousbuilding properties such as stiffness, geometry, weight, width,length and eccentricity of building with respect to tunnel axis, onthe tunnel-structure interaction problem easily. The question aris-ing is how much equivalent beam properly faces to interactionproblems. To answer this question, one 10-storeys steel structureis considered as real structure, and its results are compared withthe results of equivalent beam. The resistance system of structureis bending frames in two orthogonal directions, and the connectionof columns to foundation is considered as pin. Footing is consideredto be as mat with 20 m � 20 m � 1.2 m dimensions. The columnplane of structure has been shown in Fig. 12.

The SAP code was used for the analysis of structures. The deadand live loads combination is only considered in this study.

The excavation of tunnel creates a displacement field for theground. For shallow tunnels, the created displacement field influ-ences the adjacent building and urban services. On the other part,the presence of adjacent building depending on geometry, weight,and its stiffness controls the ground displacement field due to tun-neling. To study this important interaction between tunnel and

Fig. 17. Ground horizontal movement profiles (a is front section plan

adjacent structure, it is recommended to use a unique finite ele-ment calculation code in which the behavior of ground materialand so the behavior of structure are properly described. At least,in the practical works in geotechnical engineering there is rarelyexistence of such general finite element code. In this paper, theinteraction between tunnel and real adjacent structure is studiedusing an indirect method for which two finite element codes SAP(strong in structural analysis) and PLAXIS (strong in geotechnicalengineering problems) have been performed alternately. Toachieve this aim, it will be necessary to use an iterative process be-tween two codes. In Fig. 13, the analysis process using two codesschematically is presented. As can be seen, the process is basedon transportation of total forces (Fx, Fy, Fz) and total displacements(Ux, Uy, Uz) between two software i.e. SAP and PLAXIS 3D respec-tively; this process is updated in each steps. In the first stage, thestructure is analyzed by SAP code and the forces of support pointsare saved to be sent to the PLAXIS code. Now PLAXIS code is exe-cuted and gives displacements distribution for mat foundation liedon ground surface. The displacements at the columns supportpoints are saved and sent to SAP code. These displacements are in-duced to support points of structure then SAP code is executed andgives a set of new forces at the support points. The necessary con-dition for stopping the iteration process is to satisfy the displace-ment field in the interface of structure and ground. On the otherhand, this defines the compatibility condition applied between

of structure b is middle section and c is rear section of structure).

Fig. 18. (a): Geometrical idealization of structure, (b): consideration of structure as equivalent beam.

532 M. Maleki et al. / Tunnelling and Underground Space Technology 26 (2011) 524–533

structure and ground. This iterative process is continued until thedifferences of two sequence steps were about zero. The reactions ofcolumns and their displacements components for the first and finalstep of analysis have been presented in Tables 4–7.

Fig. 14 shows the 3D FE mesh for PLAXIS code in which the posi-tions of foundation and columns have been specified.

6. Comparison of two models

In this part of work the internal forces particularity bendingmoment of tunnel lining, ground surface settlement and its distri-bution, and also ground horizontal movement profile, obtained bytwo methods of building modeling are compared.

Bending moment distribution of lining versus central angle ofthe tunnel sections, for two methods of building modeling hasbeen presented in Fig. 15.

The central angle is measured anticlockwise from horizontalplane. There can be seen a very good conformity between obtainedresults by two methods.

The ground surface settlement is an important factor that mustbe controlled in interaction problems. The ground surface settle-ment versus mesh width for two methods has been shown inFig. 16.

It is clear that the response of equivalent beam method is veryclose to the obtained results from analysis with real geometry ofstructure. There is a small difference between two methods in loca-tion of structure foundation. This difference is due to uniform dis-tribution of building weight in the first method in comparison withthe concentrated load of columns in second method.

Ground horizontal movement profiles obtained by two methodshave been presented in Fig. 17. These profiles are located at 6 mdistance from centerline of tunnel. From this figure we can con-clude a very good conformity between the equivalent beam andthe real geometry methods.

There is a small difference (less than 0.2 mm) between the ob-tained results by two methods particularly in section G that is dueto special manner of structure modeling.

7. Conclusion

In this paper a set of FE analyses were performed to study theeffects of adjacent building characteristics on interaction betweentunnel and adjacent structure. The adjacent structure was modeledby two different methods. The analysis results obtained by these

methods were then compared with each other. The conclusionsfrom this study are summarized as following:

(1) Structure stiffness plays an important role in tunnel-struc-ture interaction problem. In fact ground movement due totunneling is controlled by the structure stiffness, in particu-lar, neglecting the structural stiffness yields to unrealisticground surface settlement.

(2) Weight of structure is a very fundamental factor in groundmovement caused by tunnel excavation. Ground movementsare generally increased due to increase in the structureweight.

(3) The obtained results indicate that horizontal movement isincreased with increase in building width and also for thegreat values of width the maximum settlement is decreasedbut a large domain of ground is influenced due to tunnelexcavation.

(4) Eccentricity of building from the tunnel centerline is also animportant factor. Asymmetrical deformations are the firsteffect of structural eccentricity.

(5) Adjacent building was modeled by two methods: equivalentbeam and real geometry. The comparison of obtained analy-sis results indicates that the equivalent beam method forpractical purposes can be used as a simple way for introduc-ing the adjacent building characteristics in tunnel–adjacentstructure interaction problems.

Appendix A. Calculations of equivalent beam characteristics

The building is modeled by an equivalent elastic beam (in 2Danalyses) or shell (in 3D analyses) that lies on the ground surface.Young’s modulus E, second moment of area I, and cross section Aare the structural properties of equivalent beam. Each storey ofbuilding is considered as a slab, therefore, considering one slabfor footing, one m storey building can be modeled as m + 1 slabs(Fig. 18). If the vertical mean distance between slabs is H and thethickness of each slab is tslab then, using the parallel axes theorem(Timoshenko, 1955), second moment of area of equivalent beamcan be calculated. Second moment of area I and the area A for slabare defined as:

Islab ¼t3

slabL12

Aslab ¼ tslabL ð1Þ

M. Maleki et al. / Tunnelling and Underground Space Technology 26 (2011) 524–533 533

where L is out-of-plane dimension of the slab. Assuming the neutralaxis to be at the mid-height of the building, the bending stiffness forthe equivalent beam is then calculated as following:

ðEcIÞbeam ¼ EC

Xmþ1

1

Islab þ Aslabh2m

� �ð2Þ

In which hm is the vertical distance between the structure’s neu-tral axis and the mth slab’s neutral axis. Axial stiffness for equiva-lent beam is obtained by the following expression:

ðEcAÞbeam ¼ ðmþ 1ÞðEcAÞslab ð3Þ

Fig. 18 shows the geometry of structure with respect to tunnelposition (Franzius, 2003).

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