Seismic analysis of underground structures

Seismic analysis of underground structures ALBERTO GOMEZ-MASS6 Consulting Engineer, Berliner Str. 290, 6050 Offenbach, West Germany IBRAHIM ATTALLA Kraftwerk Union AG, Berliner Str. 295-299, 6050 Offenbach, West Germany ELIAS KISIRIDIS Medussa GmbH, Berliner Str. 282, 6050 Offenbach, West Germany The present paper describes a soil-structure interaction analysis of underground tunnels. Use has been made of available computer codes in order to study the problem in the three significant sections through the three coordinate planes. The seismic environment consists of a combination of S- and P-waves obtained by conventional methods based on the theory of one-dimensional wave propagation through linearly viscoelastic layered systems. The response of the soil-structure system has been analysed by linearly viscoelastic finite elements. Results of Section A across the tunnel axis indicated only a low level of tunnel-soil interaction, and this only in the high frequency range. Consequently the stresses obtained were small. Results of Section B along a vertical plane through the tunnel axis revealed strong soil-structure interaction as well as important effects of the buildings response on to that of the tunnel. Significant stresses which will govern the structural design were also obtained. The behaviour of the tunnel in a horizontal plane through the axis was studied in Section C. Although some tunnel-soil interaction was observed, the calculated stresses were not as high as in Section B. Comparison of the finite element stresses and the stresses obtained by simplified methods provided reasonable upper and lower limits for the finite element results.

INTRODUCTION

METHODOLOGY

Recently wide attention has been given to the study of soil structure interaction problems for nuclear power plants in active seismic regions. In contrast, the underground constructions for these plants as tunnels, ducts or pipelines, for which the soil structure interaction is a major influence on their dynamic response, has not been given the necessary attention. However, some studies ~ have been carried out and proposals for a simplified analysis have been made. These studies considered tunnels in the free field subjected to different seismic wave types. The stresses are calculated by simplified equations based on the velocity of the considered wave type as given by half space theory, or by multi.supported beams on soft springs. This paper describes a finite element analysis of an underground tunnel with special attention given the soil structure interaction between the tunnel itself, the surrounding soil and buildings. The soil motion was conventionally assumed to be due to simultaneous vertically propagating shear waves and compression waves. Nevertheless, the computer code used can accommodate other wave types. The principle of superposition and the equivalent linear method were used. The following sections describe the adopted methodology and the computer program used. The results obtained were, whenever possible, compared with simplified solutions.

Computational method

0261-7277/83/0200101-09 $2.00 © 1983CMLPublieations

The method of analysis used here consists of a finite element procedure which considers structures embedded in a multi-layered linearly viscoelastic soil system, and a seismic environment consisting of any combination of surface and inclined body waves. This method of analysis is described in detail by G6mez-Mass6 et al) The method consists essentially of the superposition of the free field motions, uf, and the interaction motions, ui, to obtain the total motions, u, as"

u(x,y, t) = uf(x,y, t) + ui(x,y, t)

(1)

According to the complex response method, the harmonic motions at all nodal points can be expressed as: {u(t)} = {O(w)} exp(iwt)

(2)

where {U(w)} contains the displacement amplitudes and w is the frequency of excitation. If the boundaries are sufficiently far from the structure, then the boundary forces Qf are the same in both the free field and the soft-structure systems, and the following complex harmonic equilibrium equations apply:

SoilDynamicsandEarthquakeEngineering, 1983, Vol. 2, No. 2

101

Seismic analysis o f underground structures: A. GOmez-Mass6, 1. Attalla and £L Kisiridis Complete model

([K] -- w 2 [M]) {L T} = {Q/}

(3)

([Kf] -- w 2 [My] ) {Uy} = {Qf}

(4)

Free field model

where [M] and [K] are the mass and stiffness matrices respectively. The latter include viscous material damping as imaginary parts. Subtraction of equation (4) from equation (3) leads to: [Ki] -- w 2 [Mi] ) {Ui} = {Qi}

(5)

where: [Ki] = [K] -- [Kf]

(6)

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(8)

and equation (5) expresses nothing but a source problem. In order to adequately model the boundaries, use is made of the transmitting boundaries for the layered free field and of viscous dashpots for the underlying half space, respectively. These two boundaries are complex matrices assembled into the complete global stiffness matrix [K]. The entire solution is obtained in the two following steps:

Step A: The free fieM analysis. The types of seismic waves considered are vertically propagating S- and P-waves in a layered soil overlying a viscoelastic half space. The free field motions are calculated using program EQSYS 3 for both vertically propagating S. and P- waves. This program is a modified version of program SHAKE, 4 and it calculates the horizontal and vertical motion at each point of the finite element mesh. These motions will be used as indicated by equations (1) and (8) as explained in the next step of analysis. Step B: The finite element analysis. Once the free field motions are obtained, the interaction motions are calculated using equation (5) for each frequency. Finally the total motions are calculated by superposition of the free field motions and interaction motions as follows:

and combining results appropriately. The horizontal decot~volution requires a few iterations until strain compatibility of the soil properties is achieved. The constrained moduii corresponding to the obtained strain-compatible shear moduli are used (with the sole exception of submerged soils for which the constrained modulus is calculated using the P-wave velocity of water) without further ilerations. The implicit assumption is that the variation of the soil properties at any given point in the free field depends only on the shear strains, whereas the effects of the compressive strains are neglected. The combined effects of shear and compressive dynamic strains are the subject of current investigation. Recent work by Griffin and Houston 6 shows some effect of the compressive strains under rather limited test conditions. Neglecting the effect of compressive strains in the free field is, however, a widely accepted practice and, therefore, it is followed in this study. The statistical independence of the horizontal and vertical input motions is determined following the guidelines established by the US Nuclear Regulatory Commission (NRC) Regulatory Guide 1.92. 7 Accordingly, the horizontal motion is taken as the given control motion scaled to a peak acceleration 0.1 (g). The vertical motion is taken as the control motion scaled to a peak acceleration of 0.05 (g), but shifted by a small time increment in order to achieve the desired statistical independence. Shift intervals between 0.6 s and 0.12 s (with an optimum value of about 0.08s) will yield the desired effect. The use of the time shift technique implies that according to equation (2) each individual frequency component will be affected by a different phase shift. This is taken into account by the program EQSYS when the vertical free field motions are calculated. The time shift technique has the important advantage, however, that the finite element analysis is performed with only one set of Fourier coefficients for both motions and, consequently, interpolation on the transfer functions is possible. D E V E L O P M E N T OF A N A L Y T I C A L MODELS Inspection of Fig. 1 indicates that the tunnel behaviour may be significantly affected by the reactor and auxiliary

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A few iterations may be again performed to reach strain compatibility using the equivalent linear method. The computer program CREAM 5 has been developed to accept an arbitrary seismic input, calculate the interaction motions and obtain the total motions according to equation (9).

The seismic environment The specified control motion is an artificial accelerogram with a broad frequency content. The seismic analysis is to be performed for horizontal and vertical earthquakes scaled to peak acceleration values of 0.1 (g) and 0.05 (g) respectively. Furthermore, the earthquakes must be simultaneous and statistically independent. The simultaneousness of the horizontal and the vertical earthquakes is achieved directly in the finite element analysis. In the free field this is achieved simply by performing two independent deconvolutions in the horizontal and vertical directions using the corresponding earthquakes,

102

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Soil Dynamics and Earthquake Engineering, 1983, Vol. 2, No. 2

Seismic analysis o f underground structures: A. G6mez-Mass6, L Attalla and E. Kisiridis

by the neighbouring structures. The analytical model is shown in Fig. 3 and consists of a discretization of the tunnel structure by beam elements. The empty spaces in the tunnel are modelled with elements of zero mass and stiffness.

buildings, even though the tunnel is not linked directly to these two structures. Therefore, the significant sections of the tunnel will be as follows: A. Section A across the tunnel in the Z - Y plane B. Section B in the X - Y plane as indicated by the side view in Fig. 1 C. Section C in the Z - X plane as indicated by the plan view in Fig. 1

Section A - results Comparison between response spectra at tunnel points and at the corresponding free field elevations show some interaction effects especially in the horizontal direction at frequencies higher than about 2.5 Hz. Because the tunnel cross section is a relatively stiff structure of small dimensions, only the high frequencies will produce some deformations and hence some forces. The maximum beam forces obtained from their corresponding time histories are presented in Fig. 4. It is interesting to calculate the maximum beam forces using frame analysis theory and the previously obtained free field strains. Two case assumptions for the soil-structure interaction effects are made: Case I - Interaction effects are restricted and beam forces are calculated as if the frame strains were equal to the free field strains. This will provide a conservative upper limit. Case II - Interaction effects are accounted for by using linear springs equal to the subgrade soil reaction. The frame strains are now lower than the free field values and the forces so obtained will provide for a reasonable approximation. A comparison of the beam forces so obtained against the finite element beam forces is shown in Table 1. The agreement is good.

It must also be pointed out that, unlike other finite element codes of widespread use and acceptance in this kind of analysis, such as FLUSH, LUSH etc., program CREAM does not require the input of the earthquake excitation at the bedrock, but considers the free field motion corresponding to each nodal point of the mesh, and on this basis it calculates the corresponding interaction motions. Consequently, the finite element models need not be very deep, but only include the volume of soil affected by the presence of the structure. Furthermore, the use of a viscous dashpot base as proposed by Lysmer and Khulemeyer s provides for an adequate simulation of the half space underlying the model. FREE FIELD ANALYSIS The given soil profile consists essentially of two main layers. The upper layer has a thickness of some 24.5 m and has higher stiffness properties than the lower layer which, in turn, extends to several hundred meters in depth. Both layers, however, follow the same average degradation curves. Figure (2) summarizes the soil properties and results for the horizontal and vertical earthquakes. The strain compatible soil properties obtained from the free field analyses are input as initial values for the finite element analyses, during which one or two more iterations have been allowed to account for the influence of the structures.

Section B - model The purpose of this analysis is to calculate the maximum tunnel forces and stresses along the tunnel axis including the effects of the UJA and ULB buildings which are each situated at one end of the tunnel. The finite element for the entire Section is given in Fig. 5. The grid is small for the tunnel and the adjacent soil at the center of the model (see Fig. 6 for details), and becomes coarser away from the tunnel zone.

FINITE ELEMENT ANALYSIS Selection A - model

The purpose of this analysis is to calculate the maximum wall forces per meter of tunnel length on a typical cross section of the tunnel which is assumed to remain unaffected

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Soil Dynamics and Earthquake Engineering, 1983, VoL 2, No. 2

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Special consideration must be given to the material properties and stresses of the plane strain model as related to those in the actual system. As indicated in the plan view diagram in Fig. 1, a plane strain modelling of the entire system presents the difficulty of the different actual widths for each of the three existing structures. In the event of any horizontal or vertical earthquakes in the X - Y plane, motions will occur in both buildings, the tunnel and the soil mass in between. If it is assumed that all points along axis Z will move in phase, then the actual forces occurring on a soiltunnel cross section by a plane parallel to the Y-Z plane can be calculated by the following three steps: 1. 'Spread' proportionally element properties over the entire width of analysis which in this case is that of the reactor building. 2. Solve for plane strain stresses. 3. 'Concentrate' plane strain stresses over the actual structure width in a reverse fashion as in Step 1. Accordingly, the concrete structure of the tunnel and the soil mass moving with it are separately modelled by superimposing two identical but separate meshes as shown in Figs. 6 and 7. Solid elements are used to model the soil mass as well as the concrete elements walls. In addition, the upper and lower portions of the tunnel are discretized by means of beam elements.

104

The existing gaps at the ends of the tunnel are modelled by elements with zero mass and stiffness. Section B - results A comparison of response spectra from the model and the free field showed significant interaction effects affecting a wide frequency range. A comparison of response spectra obtained at points in the buildings close to the tunnel, and at tunnel ends and midpoint indicated that the tunnel [esPonse is strongly influenced by the buildings. The absolute time-maxima values for the three stresses in the X - Y plane at each section of the tunnel walls are given in Fig. 8. Higher stress values are observed in the left half of the tunnel near the reactor building, probably due to the combination effect of a larger, heavier, embedded structure and the particular shape of the tunnel. The effect of the gaps is noticeable, especially in the distribution of horizonal compressive stresses which show a tendency to increase away from the tunnel ends due to shear stresses tran~nitted through the soil mass. The time maxima of the actual beam force and stresses are shown in Fig. 9. As observed in the tunnel wall elements, higher maxima are found in the left half of the tunnel. Simplified hand calculations were also carried out to verify the shear forces obtained along the tunnel, on the

Soil Dynamics and Earthquake Engineering, 1983, Vol. 2, No. 2

Seismic analysis o f underground structures: A. G6mez-Mass6, L Attalla and E. KisirMis

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Soil Dynamics and Earthquake Engineering, 1983, Iiol. 2, No. 2

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Seismic analysis o f underground structures. A. GOmez-Massc), L Attalla and 1E Kisiridis

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Soil Dynamics and Earthquake Engineering, 1983, Vol. 2, No. 2

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Seismic analysis o f underground structures: A. GGmez-Mass6, L Attalla and E. Kisiridis

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produced by the neighbouring structures which is transmitted through the soil. The diagram of maximum compressive stresses along the tunnel is given in Fig. 10. These curves show again the absolute values o f time maxima, and again indicate a clear predominance o f the compressive stresses, due to axial forces, over the bending stresses.

Section C - model The purpose o f this analysis is to assess the stresses produced by seismic excitations in the horizontal Z - X plane in Fig. l, which are acting in the entire tunnel structure. The stresses o f interest are now those caused by the bending of the tunnel cross section around the Y axis. The tunnel structure has been idealized as the rectangular beam shown in Fig. 11, and analyzed by means o f pro-

Soil Dynamics and Earthquake Engineering, 1 983, VoL 2, No. 2

107

Seismic analysis of underground structures. A. Gomez-MassO, L Attalla and E. Kisiridis gram STARDYNE 9 for three acceleratlol~ t i m e h i s t o r m ~ given at the tunnel ends and midpoint. The Z-acceleration time histories used at the tunnel ends will be obtained at the corresponding points in the UJA and ULB buildings. The implied assumption is that the effec~ of the buildings in this case is only local. Therefore. the Zmotion used at the tunnel midspan is tire one obtained from the Section A analysis.

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Case

Stresses

. ...................................

[

[[

81.0

28.0

III

STARDYNE

33.0

33.0

-

-

._;iiiilliillllii)iiiilllliiiillllliiiiii_iilliiiilll Case

I

=

Case

II

=

Case

~es6cl

_~

__

_

Ill

Sozl

Structure

=

Beam

with

=

Aver.

Interaction

Restricted. Allowed.

Fixed

Ends. 2

LOWER IlEAl,IS

03

.

m2)

09* '

12i

.

~. . . . . . . . . . .

M

1 9(MNI

.

Forces

and 'or

Qbb

----

0

Max.

Bending

Stress

( MN/m

)

Shear ELEMENT

Figure 10.

108

Maximum total stressesalong tunnel (Section B)

Tabh" 2. Compar~on o]' maximum tunnel Jorces and stre~es in ~,ction C hv simplified methods and hy STARD YNE

Soil Dvnamics and Earthquake Engineering, 1983, Vol. 2, No. 2

Seismic analysis o f underground structures: A. G6mez-Mass6, 1. Attalla and E. Kisiridis The desired motions at the tunnel ends have been obtained from two separate finite element analysis in the Z - Y plane shown in Fig. 1 o f the reactor and auxiliary buildings.

Section C - results Program STARDYNE has been used to carry out multipoint excitation analyses by the large mass method o f the free-free beam shown in Fig. 11, and b y using simultaneous Z and Y acceleration at the three input points. Simplified calculations have also been carried out using the method suggested by Constantopoulos et al. ~ This method essentially consists o f imposing a displacement boundary condition on a continuous multi-supported beam which represents the tunnel. The beam is connected to a rigid base by soil springs to be determined from the theory o f subgrade reaction. Two sets o f results have been obtained from this simplified method. Cases I and II respectively correspond to the soilstructure interaction being restricted and being allowed, and they establish the upper and the lower bounds o f beam forces respectively. Still another upper bound for the beam forces has been established by analyzing, in Case III, the beam model with fixed ends when subjected to the relative displacements obtained from the calculated Z-motions at the tunnel ends. This is essentially a static analysis, and the maximum relative displacement is obtained by double integration o f the relative accelerations. A comparison o f all three cases is shown in Table 2 where the agreement between the STARDYNE results and

the other cases is good. The STARDYNE results fall between the upper and lower limits established by Cases 1 and II, as expected. Furthermore, results from Case III establish an even lower upper limit which is still above the STARDYNE results. Therefore, the STARDYNE results seem quite reasonable. REFERENCES 1 Constantopoulos, I. V., MotherweU, J. T. and Hall, J. R. Dynamic analysis of tunnels, Third lntl. Conf. on Numerical Methods in Geomechanics, Aachen, April 1979 2 G6mez-Mass6, A., Lysmer, 1., Chen, J. C. and Seed, H. B. Soilstructure interaction in different seismic environments, Report No. UCB/EER C-79/18, Univ. of California, Berkeley, 1979 3 G6mez-Mass6, A. EQSYS- a computer program for free field analysis with simultaneous time-shifted horizontal and vertical earthquakes, unpublished, 1982 4 Schnabei, P. B., Lysmer, J. and Seed, H. B. SHAKE - a computer program for earthquake response analysis of horizontally layered sites, Report No. EERC 72-12, Univ. of California. Berkeley, 1972 5 G6mez-Mass6, A., Lysmer, J. and Seed, H. B. CREAM - a computer program for soil-structure interaction in an arbitrary seismic environment, to be published by the Earthquake Engineering Research Center, Univ. of California, Berkeley. 1979 6 Griffin, P. M. and Houston, W. N. Interaction effects of simultaneous torsion and compression cyclic loading of sand, Report No. UCB/EERC. 79/34, Univ. of California, Berkeley, 1979 7 US Nuclear Regulatory Commission. Combining modal responses and spatial components in seismic response analysis. Regulator), GuMe 1.92, February 1976 8 Lysmer, J. and Khulemeyer, R. L. Finite dynamic model for infinite media, J. Eng. Mech. Div., ASCE 1969, 95 (EM4) 9 STARDYNE-3. System Development Corporation, 360 New Sepulveda Blvd., El Segundo, California (90245), 1977

Soil Dynamics and Earthquake Engineering, 1983, Vol. 2, No. 2

109