Interacion of Circular Holes in an Infinite Elastic Medium - C.B. KOII, A. VERRUIJT

8/12/2019 Interacion of Circular Holes in an Infinite Elastic Medium - C.B. KOII, A. VERRUIJT

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.Tunnelling and Underground Space Technology 16 2001 5962

Interaction of circular holes in an infinite elastic medium

C.B. Kooi, A. Verruijt

Delft Uniersity of Technology, Department of Ciil Engineering, 2628 CN Delft, The Netherlands

Received 11 December 2000; accepted 26 February 2001

Abstract

Problems of a single cavity or a series of cavities in an infinite or semi-infinite elastic medium are of considerable interest forthe analysis of stress concentration factors near the boundaries of the cavities. In the mining industry, interest in problems of thistype mainly derives from displacements that occur at the soil surface, which may cause considerable damage. Similar damage mayoccur when constructing tunnels in densely populated areas. Analytical expressions for the subsidence due to multi-tunnellingactivities usually do not match very well with the measured field results. One of the possible explanations for the discrepancybetween the two is the interaction of the tunnels. Simple superposition of the effects of each tunnel may give reasonable results ifthe distance between the tunnels is large, but not when the tunnels are constructed close together. To gain more insight into thephenomenon of interaction, an analytical solution is presented in this paper for an infinite elastic medium. Comparison of thesolution with two computer packages, FLAC 2D and PLAXIS, supports the finding that the interaction of circular holes indeedcauses an increase in the displacements. 2001 Elsevier Science Ltd. All rights reserved.

Keywords: Elastic medium; Tunnels; Circular holes

1. Introduction

The problem of a single circular hole in an infiniteelastic plate, with a uniform state of stress at infinity,has a simple solution, described by Timoshenko and

.Goodier 1970 . This solution can be used as a firstapproximation for the vertical displacement of the sur-face in case of a single cavity in an elastic half plane.This approximation is applicable for a tunnel at a

sufficiently large depth. For tunnels at shallow depth,the influence of the boundary condition at the freesurface must be taken into account. A full analyticalsolution of this problem using complex variable analysis

.has been given by Verruijt 1998 , and an approxima-tion using an imaging technique was presented by

.Sagaseta 1987 . The present paper addresses the prob-lem of two parallel tunnels, constructed close together,

Corresponding author. Tel.: 31-15-278-5280; fax: 31-15-278-

3328. .E-mail address: [email protected] A. Verruijt .

at sufficient depth to justify neglecting the effect of thefree surface, see Fig. 1. For this case, the calculation ofthe stresses can be performed using bipolar co-ordinates

.as introduced by Jeffery 1921 . The stress concentra-tion factors in the vicinity of the two cavities have been

. .discussed by Howland and Knight 1939 , Green 1940 .and Ling 1948 . The method of bipolar co-ordinates

fails to give explicit expressions for the displacements,however.

In order to derive expressions for the displacementsas well, the problem may be solved using Schwarz

.alternating method, see Sokolnikoff 1956 , in whichsuccessive approximations are made by alternately sat-isfying the boundary conditions on each of the twocavities. A second order approximation using this

.method has been given by Zimmerman 1988 . Thispaper presents a generalisation of this method using asmany terms as required for sufficient accuracy. In orderto validate the results, data will be compared with theresults of numerical analysis using a finite difference ora finite element solution.

0886-779801$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. .PII: S 0 8 8 6 - 7 7 9 8 0 1 0 0 0 2 7 - X


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( )C.B. Kooi, A. Verruijt Tunnelling and Underground Space Technology 16 2001 596260

Fig. 1. Definitions for two tunnels.

2. The solution method

The general solution for a circular cavity in an infi-nite medium, with an arbitrary load at the cavityboundary has been given by Timoshenko and Goodier .1970 . If the stresses at infinity are supposed to vanish,this solution can be written as

j1a .u A B j24r j j / r

j0

j1a acosj, / 5r

j1a .u A B j44 j j / rj0

j1a asinj, / 5r

where a is the radius of the cavity, and where B B0 10. In case of plane strain deformations the stressescorresponding to this displacement field are

j2 ar r . A j1 j /2 rj0

ja . .B j2 j1 cosj,j /5r

j2 a . A j1 j /2 rj0

ja . .B j2 j1 cosj,j /5r j2 ar . A j1 j /2 r

j0

ja .B j j1 sinj.j /5r

The procedure now is that firstly, only a single termis taken in the boundary conditions for the first cavity,representing a tensile radial stress at this cavity, balanc-

.ing the uniform compressive stress p in the undis-0turbed medium. The influence of this load on the radialnormal stress and the shear stress on the boundary ofthe second cavity can then be determined from thegeneral solution given above, and it will appear thatthese stresses are unequal to zero. These two stressdistributions can then be expanded into a Fourierseries in terms of polar co-ordinates around the centreof the second cavity, and these stresses can then bebalanced by a loading of opposite sign at the boundaryof the second cavity, using the same series expansion asgiven above, but with r now representing the distance

from the second cavity. The resulting stresses on theboundary of the first cavity can then be calculated,which will again appear to be unequal to zero. Againthese stresses can be expanded into a Fourier series interms of polar co-ordinates around the centre of thefirst cavity, and, in its turn, these stresses can bebalanced by another load at the boundary of the firstcavity. This process can be continued until the correc-tion to be applied is practically zero. The method hasbeen used to construct a second order approximation

.by Zimmerman 1988 , with a single correction of thestresses at the boundary of the second cavity. By gener-alising the procedure, using the general expressions for

the field of stresses and displacements around onecavity, and using Fourier series expansions of the re-sulting stresses on the boundary of the other cavity, allperformed in a relatively simple computer program, theerrors in the approximation can be made arbitrarilysmall. The number of successive alternative iterationsneeded to achieve a certain accuracy depends upon thedistance of the two cavities, of course. In the computerprogram, the iterations stop when the relative error ofthe constants in the expansions is less than 1010 . Thismay require up to 50 iterations when the cavities are

close together with a gap of only 5% of the radius of

.the cavities .

3. Stresses

The method described above has been tested forvalues of the distance d between the centres of the twocavities in the range da2.110.0. For very largevalues of the distance between the two tunnels, theirinteraction appears to be so small that a simple super-position of the stresses produced by each of the tunnelsin an infinite medium, is accurate enough. In order tocheck the performance of the method for tunnels at


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( )C.B. Kooi, A. Verruijt Tunnelling and Underground Space Technology 16 2001 5962 61

Fig. 2. Results compared with FLAC2D and PLAXIS.

smaller distances, the tangential stress along the linebetween the two tunnels has been calculated, and com-pared with the numerical results obtained using the

computer programs FLAC2D Itasca Consulting Group. Inc., 1995 and PLAXIS Brinkgreve and Vermeer,

.1998 . The results from the various programs for thecase da4 are shown in Fig. 2. The results of thetwo computer programs are very close to those of theanalytical solution using the full interaction procedure,although the numerical values are slightly larger thanthe analytical ones, especially near the boundaries ofthe cavities.

The stress concentration factors in the points on thecavity boundary on the horizontal axis are shown inTable 1, as calculated by three methods: simple super-position of the solutions for two single cavities; theSchwarz alternating method described in this paper;and the method of bipolar co-ordinates used by Ling .1948 .

4. Displacements

Perhaps the main advantage of the method of suc-

cessive approximations presented in this paper is that italso gives closed form expressions for the displace-ments. As an example, the vertical displacements at the

Table 1Maximum stress concentration factors on the cavity boundary

da Superposition Interaction Ling

3 1.2500 1.8873 1.8874 1.1111 1.4108 1.4116 1.0400 1.1546 1.155

10 1.0123 1.0488 1.049 1.0000 1.0000 1.000

Fig. 3. Scaled displacements for yd.

level yd are shown in Fig. 3, for a value of Poissonsratio of 0.25 and for da4, which means thatthe gap between the two cavities is equal to the sum oftheir radii. The data obtained by the computer pro-gram, taking into account full interaction of the twocavities are shown as a fully drawn curve, whereas thedashed curve indicates the results obtained by simplysuperimposing the effects of the two cavities. It appearsthat in this case, the interaction of the cavities leads to15% larger displacements in the centre. The effectstrongly depends upon the relative distance of thecavities as illustrated in Table 2, which shows themultiplication factor of the maximum vertical displace-ment due to the interaction effect of the two cavities,for 0.25, various values of da, and for two valuesof the depth of the cavities below the surface. Itappears that the interaction effect is significant only ifthe gap between the two cavities is small. For cavitiesvery close to each other, the effect may be that thedisplacements at the surface are almost doubled. Itmay be noted that such small distances occur fre-quently in engineering practice when two parallel tun-nels are being constructed, especially near under-ground stations.

Table 2The effect of interaction on the displacements

da yd y5d

2.1 1.837 1.8573.0 1.268 1.3114.0 1.143 1.1755.0 1.090 1.1146.0 1.062 1.0807.0 1.045 1.0598.0 1.035 1.045


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5. Conclusions

It has been shown that the displacements of the soilsurface due to the creation of a tunnel a very long

.circular cavity may be considerably larger in case ofthe presence of an existing parallel tunnel. The interac-

tion effect can be calculated using a relatively simpleiteration procedure.

References

Brinkgreve, R.B.J., Vermeer, P.A., 1998. PLAXIS Manual Version 7.Balkema, Rotterdam.

Green, A.E., 1940. General biharmonic analysis for a plate contain-ing circular holes. Proc. Roy. Soc. A 176, 121132.

Howland, R.C.J., Knight, R.C., 1939. Stress functions for a platecontaining groups of circular holes. Phil. Trans. A 238, 357 392.

Itasca Consulting Group Inc, 1995. FLAC Users Manual Version 3.3.Itasca, Minneapolis.

Jeffery, G.B., 1921. Plane stress and plane strain in bipolar co-ordinates. Phil. Trans. A 221, 265293.

Ling, C.B., 1948. On the stresses in a plate containing two holes. J.Appl. Phys. 19, 7782.

Sagaseta, S., 1987. Analysis of undrained soil deformation due to

ground loss. Geotechnique 37, 301320.Sokolnikoff, I.S., 1956. Mathematical Theory of Elasticity, 2nd Ed.

McGraw-Hill, New York.

Timoshenko, S.P., Goodier, J.N., 1970. Theory of Elasticity, 3rd Ed.McGraw-Hill, New York.

Verruijt, A., 1998. Deformations of an elastic half plane with acircular cavity. Int. J. Solids Structures 35, 27952804.

Zimmerman, R.W., 1988. Second-order approximation for the com-

pression of an elastic plate containing a pair of circular holes.ZAMM 68, 575577.

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Interacion of Circular Holes in an Infinite Elastic Medium - C.B. KOII, A. VERRUIJT