Dynamic interaction effects in underground traffic systems

Dynamic interaction effects in underground traffic systems O. yon Estorff

Industrieanlagen-Betriebsgesellschaft mb H, D-8012 Ottobrunn, Germany A. A. Stamos and D. E. Beskos

Department of Civil Engineering, University of Patras, GR-26110 Patras, Greece H. Antes

lnstitut fiir Angewandte Mechanik, Technische Universitiit Braunschweig, D-3300 Braunschweig, Germany Dynamic interaction effects in tunnel systems subjected to above and below ground traffic loads are studied numerically under conditions of plane strain. Linear elastic or viscoelastic material behaviour for both the structure and the soil is assumed. Two numerical schemes are employed in the computations for comparison purposes. The first works in the time domain and combines the finite element method for the structure with the boundary element method for the soil, while the second works in the frequency domain and uses the boundary element method for both the structure and the soil. The dynamic behaviour of a typical tunnel system is studied in detail for a variety of dynamic and geometric parameters in order to assess their effects on the system. Key Words: boundary elements, dynamic response, finite elements, frequency domain, time domain, tunnels, underground traffic. INTRODUCTION The motion of above ground and below ground vehicles creates waves which are transmitted through the ground and adversely affect nearby structures and people. A successful design of urban traffic systems involving road traffic at the ground surface level and railway traffic in tunnels at a level below ground, is a very difficult task due to the dynamic interaction phenomena between the components of the system. A very good knowledge of the dynamic behaviour of such a complex traffic system, as well as the existence of efficient and accurate methods for predicting its dynamic response, are the most important aspects for the design engineer. This paper provides information concerning these two basic aspects. The heart of the problem at hand is the underground tunnel and its dynamic behaviour. An extended literature study on the subject of tunnel dynamics and underground structures in general can be found in the works of, e.g., Okamoto 1, Manolis 2 and Manolis and Beskos 3"4. In general, the existing methods for dynamic analysis and design of tunnels include the quasi-static method in conjunction with the beam on elastic foundation model (e.g. Kuesel 5, Aoki and Hayashi 6, Constantopoulos et al.~), the dynamic method in conjunction with concentrated masses, springs and dashpots for modelling the soil (e.g., Dawkins a, Yuan and Walker 9, Manolis and Beskos3), experimental dynamic methods (e.g., Costantino and Vey t°, Okamoto and Tamura tt, Hamadat2), analytical dynamic methods (e,g., Garnet and Crouzet-Pascal ta, E1

© 1991 ElsevierSciencePublishers Ltd.

Akily and Datta t4"15, Datta and Shahtr), numerical methods, such as the Finite Difference Method (FDM) and the Finite Element Method (FEM) which require a discretization of the domain in addition to the surface discretization (e.g., Robinson tT, Hwang and Lysmer tS, Bayo and Wilson t9, Lee and Welsey2°, Balendra et al. 2t), the Boundary Element Method (BEM), which requires only a surface discretization for linear problems (e.g., Kobayashi and Nishimura 22'23, Manolis and Beskos24, Vardoulakis et al. 25, Kitahara e t a / . 26) and hybrid numerical methods combining the FEM for the structure and an analytic method or the BEM for the soil (e.g., Underwood and Geers 27, Datta et al. 2s, Wong et al. 29, von Estorff and Kausel 3° and yon Estorff and Antesat). It is apparent that for an accurate and inexpensive dynamic analysis of realistic tunnel problems, resort should be made to numerical methods of solution. Domain type of methods, such as the FDM and the FEM, model the infinite or semi-infinite soil medium as a finite body and, in order to avoid wave reflections at the artificial boundaries, employ either a very large uneconomical mesh or expensive and not always general nonreflecting boundaries. The BEM, which takes automatically into account the radiation condition at infinity, is free of this disadvantage. Hybrid schemes employing the FEM for the structure and the BEM for the soil succeed in combining the best characteristics of both methods, while reducing or eliminating their disadvantages. In conclusion, dynamic analysis of tunnels can be

Engineering Analysis with Boundary Elements, 1991, VoL 8, No. 4

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Dynamic interaction effects in underground traffic systems: O. yon Estorff et al. accurately and efficiently done by either a hybrid FEM-BEM scheme, or by the BEM for both the structure and the soil, especially under plane strain conditions. This work deals with the study of the dynamic interaction effects in urban traffic systems involving above ground road traffic and below ground railway traffic in tunnels and how these systems affect nearby structures. Linear elastic or viscoelastic material behaviour for both the tunnel and the soil is assumed and the problem is solved under plane strain conditions. Both harmonic and transient disturbances are considered. The analysis is done numerically by two schemes, the first one being a hybrid FEM-BEM scheme in the time domain and the second one a BEM scheme in the frequency domain. Thus, a second purpose of this paper is to critically compare these two schemes. The only works similar to the present one are the one by Balendra et al.Z~, which is restricted to harmonic railway traffic only and employs the FEM in conjunction with nonreflecting viscous boundaries and the very recent one by von Estorff and Antes 3t, which is restricted to transient above and below ground traffic and employs a time domain hybrid FEMBEM scheme. TIME DOMAIN FEM-BEM The time domain FEM-BEM hybrid scheme combines appropriately the FEM, used to model the structure and a portion of the surrounding soil, with the BEM, used to model the outer infinite soil domain. According to the displacement FEM as applied to elastodynamics, the domain is discretized into a finite number of elements over which the displacement vector u(x, t) is approximated as u(x, t) = N ( x ) . fi(t)

(1)

where N,~x) are the shape function (linear in this case) and t2~(t) are the nodal displacement amplitudes, x is a point in the element and t represents time. Through a variational or a Galerkin weighted residual formulation, an assemblage_of the individual finite elements and a collocation at time nat, one can obtain the equations of motion of the whole structural domain in the matrix form (e.g. Bathe 32) M" ii~"~+ K. u(") = P("~

(2)

where u("~=fi(nAt) denotes the nodal displacement amplitudes at the current time nat, the overdots indicate time differentiation, M and K are the consistent mass and stiffness matrices of the elastic structure, respectively, which are constant over time, and P~"~ is the vector of the time-dependent nodal force. The time integration of equation (2) is accomplished here by the single-step time marching scheme of Newmark, after this system of finite element equations has been coupled with the subsequent boundary element equations. According to the time domain direct BEM, the starting point is the boundary integral equation

-~ ui(~, t) =

[u*(r, t')tj(x, z ) - t*(r, t')uj(x, z)-IdVdr

(3) under zero initial conditions and body forces, where t'=t-z, r=lx-¢[ with x and ¢ being points at the

boundary F, u; and t, are the displacements and tractions, respectively and u~ and t~ are the time-dependent full-space fundamental displacement and traction tensors, respectively. The numerical treatment of equation (3) is accomplished through a discretization of the boundary F into a number of line elements and a discretization of the time axis into a number of equal time increments At. Then, appropriate shape functions along the boundary elements and the time intervals are chosen. Finally, by an evaluation of the integral terms over each boundary element (numerically) and over each time interval (analytically), and by collocation at each boundary node and at all time steps, the discretized version of equation (3) is obtained in the matrix form (e.g., Antes 33"3., Manolis and Beskos~): (I)

[I)

T" u ~n~ = U" t t"~ - d " - t~

(4)

where u t"J and t ¢"~are the nodal displacement and traction vectors, respectively, at time nat, the vector r ~"-1~ represents the influence of all the previous time steps on (1)

(1)

the current time step, and T and U are the fundamental traction and displacement influence matrices, respectively, at the first time step. Since equation (4) is to be coupled with equation (2), both the displacements and the tractions should be continuous along the interface boundary elements. Thus, linear shape functions are used for the spatial discretization of the isoparametric boundary elements. In addition, the simplest time interpolation of displacements and tractions, appropriate to the integral equation, is stepwise linear and constant, respectively. For these approximations, the time integrations can be done analytically (e.g., Antes33'3*). The coupling of the FEM with the BEM is accomplished by enforcing compatibility of displacements and equilibrium of tractions along the common FEMBEM boundary. For consistency, the boundary element interface tractions have to be transformed to resultant nodal forces. Then, the solution of this coupled system -can be found by employing Newmark's time integration scheme (Bathe32). This leads to the final time-stepping system of equations

K~ss" u~">= ~,SSD~"~

(5)

where Kef f and -essDc"~stand for the so-called effective stiffness matrix and effective load vector, respectively. Displacements, velocities and accelerations are computed at every time step. More details about this coupling procedure can be found in von Estorff and Prabucki 3~.

FREQUENCY DOMAIN BEM Application of Fourier or Laplace transform with respect to time onto the governing equations of motion for an elastic body, reduces them into a static-like form in the transformed domain with obvious gains in formulation and solution procedures. The time t becomes merely a parameter, the real frequency co in the Fourier or the complex frequency s in the Laplace transform domain. One can easily go from the Fourier to Laplace transform domain by simply replacing 'co' by 'is' in this formulation, where i = x/-S-l. The Fourier transformed equations are identical in form with the equations obtained for the case of harmonically varying with time

168 Enoineerin9 Analysis with Boundary Elements, 1991, Vol. 8, No. 4

Dynamic interaction effects in underground traffic systems: O. yon Estorff et al. motion if one interprets the transformed quantities as amplitudes of the harmonic motion. In general, if the motion is transient, it is preferable to work in the complex frequency or Laplace transformed domain, while for the case of harmonic motion one should work in the real frequency domain. In the former case the time domain response is obtained by a numerical Laplace inversion of the transformed solution (e.g. Manolis and Beskos4). The transformed or frequency domain solution of an elastodynamic problem is numerically obtained by the BEM. The starting point is the frequency domain boundary integral equation

......

-~.

.5

.5

.5 .5 (6)

where q~ can be either co or s, an overbar denotes amplitude or transformed quantity and a star denotes a full space fundamental solution component. A spatial discretization of equation (6) in the manner described in the previous section, finally results in the frequency domain matrix equations (e.g., Manolis and Beskos4) T. ~ = O. T

where fi and I are the nodal displacement and traction vectors, respectively, and 1' and U are the fundamental traction and displacement influence matrices, respectively. In this work, quadratic shape functions are used for the spatial discretization of the isoparametric boundary elements and the integrations are done, in general, numerically. In this case the domain of interest, the tunnel system, consists of two subdomains, the tunnel structure and the soil medium. Each domain is modeled by an equation like equation (7) and the two equations are coupled together by enforcing compatibility of displacements and equilibrium of tractions at the soil-structure interface. This leads to the final system of equations (e.g., Manolis and Beskos4)

R,:: ~ Pe:: =

Fig. 1. Geometry of the tunnel in an elastic halfspace

(7)

(8)

in the frequency domain. Equation (8) is solved for Q for a sequence of values of the frequency parameter. Velocities and accelerations have the form si and s2fi, respectively, under zero initial conditions. The time domain response u(x, t) can finally be obtained by a numerical inversion. For the case of Laplace transform, the inversion algorithm of Durbin a6 has been found to provide highly accurate results. More details about the frequency domain BEM formulation and solution procedure, as well as about the coupling of BEM domains can be found in Manolis and Beskos 24"4.

f(t) at the bottom of the tunnel liner are modeled for simplicity by spatially uniform loads of two meters long symmetrically placed about points A and D, as shown in Fig. 1. Their time variation f(t) may be harmonic or transient of any kind. The geometrical data of the system are given in meters in Fig. 1, where d represents the variable tunnel depth. The discretization of the tunnel liner and the surrounding soil medium for the case of the FEM-BEM scheme is shown in detail in Fig. 2(a), while that for the case of the BEM for both the soil and the liner is restricted to the soil surface and to the inner and outer surface of the liner, as shown in Fig. 2(b). The time domain and frequency domain approaches described in the previous two sections are employed here ~m

A¸

!

Y-t

8

1 liliTIT

TUNNEL SYSTEM DATA AND COMPARISON OF METHODS Consider the tunnel system of Fig. 1 under conditions of plane strain consisting of an homogeneous, isotropic and linearly elastic soil medium with ES= 2.66.105 kN/m 2, vs = 0.33 and p" = 2000 kg/m 3 and a concrete elastic liner (in perfect bonding with the soil) with E r = 3.0" 10 7 kN/m 2, v r = 0.25 and p r = 2000 kg/m 3, where E, v and p stand for modulus of elasticity, Poisson's ratio and mass density, respectively. The truck traffic load pA = poa f(t) on the soil surface and the railway traffic load pO = pD °

Fig. 2 ( a ). Discreti-ation of the tunnel and the surrounding soil with boundary and finite elements for the time domain calculation

Engineering Analysis with Boundary Elements, 1991, Vol. 8, No. 4

169

Dynamic &teraction effects in underground traffic systems: O. yon Estorff et al. l.Sm

t

..o.

.

.

.

.

t l s's" t .

.

.

.

.

.

t

,90.

.

--°-

Point A I 0 ~ + I / F ~ I

..... --

P~hl A

lil=M

~l~uAd tIN+~I

--

POint 0 (OEM/FEMI [ s,,,,~x+ Paint D IOEMJ | ,...,.,t

.... o

i

Fig. 2 (b ). Discretization of the tunnel and the surrounding soil with boundary elements for the frequency domain calculation

and critically compared for the case of a transient time variation of the Ricker wavelet type depicted in Fig. 3. The analytic representation of a Ricker wavelet is

c; '0.00

0.04

O.Oe

TIM(

f(t)

=

(1 - 2z 2) e-::

0.20

0.Z4

0.28

CSEC]

(9) Fig. 4. Vertical movement at point A and D due to underground and surface traffic, respectively: check of the reciprocity for both methods

in the time domain, and ] { s ) = A 2 2a + Sto

0. IG

0.1;

- a2 +

- ( ~ ) 2 .v/~- erfc(a + ~ ) }

(10)

in the Laplace transform domain, where erfc(..) means the complementary error function, A = toe-"°, a = - tJto and z = (t - t,)/to

(11)

with t~ = 0.10 secs and to = 0.03 secs in the present case. An accuracy test is conducted first for both methods and the results are presented in Fig. 4, which depicts the displacement history at points D and A due to pa and pO, respectively, for d = 4 m. The two curves from both methods are within plotting accuracy indicating that the dynamic reciprocity theorem is satisfied. However, satisfaction of this theorem for d = 8 m requires a slight increase of surface discretization (2 more elements of 4 m on either side) when the BEM is used. Further comparison studies are portrayed in Figs 5-7 which show the vertical displacement history of points A, B and D,

respectively, due to pa and pn loads. In all cases the agreement between the two methods is excellent• The above computations were performed with poa = pc° = 1 kN, d = 4 m and At = 0.0025 secs. The time domain scheme, even though more complex than the frequency domain one from the conceptual and computer implementation point of view, is more advantageous in that it can easily model inhomogeneous soil regions and can be extended to nonlinear problems. On the other hand, material damping can be easily included in the frequency domain by simply replacing the elastic moduli 2 and p by their complex counterparts 2"=2(1+2i~), # * = # ( 1 + 2 i ~ )

(12)

where/7 is the hysteretic damping coefficient independent of frequency. This is not the case, however, with the time domain approach where damping can be easily included

'T 0

o

o

J

>

lJ

?

=.' m 0 Load

function: 2

f(t} - H-2x

-1:2

......... --.....

) e

B E N / F E N I s~l=c. BEN .o.+c B E N / F E N | Und.,~,+wnd BEN I ,,G,,c

where " ~ , [ t - t s ) / t o

To.co

0.04

O.Oe

0.12 TIME

~.3.

0.16

I I

0.29

0.24

I '0.~

0.04

0.06

0.12 TIME

O. 116

O.ZO

0.24

0.28

(SEC]

0,211

CS~C]

Ricker wavelet (to = 0.03 secs, t, = 0.10 secs)

Fig. 5. Vertical movement at point A due to surface traffic and underground traffic: comparison of the B E M / F E M results with the B E M solution

170 Engineerin9 Analysis with Boundary Elements, 1991, Vol. 8, No. 4

Dynamic interaction effects in underground traffic systems: O. yon Estorff et al.

I

l

I BEMIFEN BEN

•. . . . . . .

I I s.,~,¢, t t,*-,c

I o

~ 0m

?

i

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

,, '..../--,~,;..-:~

i

-

......

...... 0

?

•

o

--_: ,,..I.., o ' 0.00

0.04

0.08

O. 12

O. II;

0.2~1

0.24

Fig. 6. Vertical movement at point B due to surface traffic and underground traffic." comparison of the BEM/FEM results with the BEM solution in the FEM but not in the BEM. Wolf 37 suggests to employ time domain fundamental solutions obtained by a numerical inversion of the frequency domain ones which can easily incorporate damping as (12) indicates. When one works in the real frequency domain with BEM, a practical and easy way to overcome the problem of fictitious eigenfrequeneies 4 is to incorporate very small amounts of damping. When one works in the complex frequency (Laplace transform) domain, there is no problem of fictitious eigenfrequencies. PARAMETRIC STUDIES FOR TRANSIENT EXCITATIONS In order to assess the influence of the geometric and dynamic parameters on the behaviour of the tunnel system of Fig. 1, this section provides results for three

.*%

0

I

.........

BEHIFEM BEll

......

B E M I FEI',4 BEI,4

0.04

0 .(~

0.~

.•. . . . .

0. I0

,..""1.

O. IZ

....

O. t4

I

--

O. IG

0.20

Till((S(C)

i

O.OZ

.....

Fig. 8. Transient vertical response at point A due to an impulsive surface and underground load: influence of the depth d of the tunnel different tunnel depths and for two different transient loads pa and pO with a time variation that of a rectangular impulse of amplitude 1 kN/m and duration 0.02 sees. Figures 8 to 11 depict the vertical displacement history of points A, B, C and D, respectively, for d = 4 m, 8 m and d ~ ~ and for the two loading cases pA and pO. In the case of the tunnel located at infinite depth (d ---, ~), a halfspace without the tunnel was used to compute displacements due to surface traffic, while for underground traffic the tunnel in an infinite space was considered• All the computations for this transient case were done with the aid of the FEM-BEM time domain approach. Considering Figs 8 and 11 one can observe that the maximum deformations at points A and D are caused by the surface and underground traffic, respectively, and that the depth of the tunnel has only little influence on these maxima. When, on the other hand, the load is applied at the surface in the case of point D, or inside the tunnel in the case of point A, the curves clearly show a delay of the response corresponding to the time it takes the waves, induced by the impulsive load at t = 0, to travel from A

I ( su,t,=, t ',*",¢ U nd.ql . = . . = '= ¢

. t*

I

I

I

I

I

I

'

,-

o

\

9

6

7":,- q "%

o

i/

3

q

=

R o '0.00

0.04

O.Oe

0,12 T[H(

O. I I

0.20

0.24

0.211

?o.,~

O.O2

O.O4

O.OS

0.01

0. I0

O. 12

O. 14

O. 16

Tlt~£ C5£C1

CS~C]

Fig• 7. Vertical movement at point D due to surface traffic and underground traffic: comparison of the BEM/FEM results with the BEM solution

Fig. 9 Transient vertical response at point B due to an impulsive surface and underground load: influence of the depth d of the tunnel

Enoineerin9 Analysis with Boundary Elements, 1991, Vol. 8, No. 4

171

Dynamic interaction effects in underground traffic systems.• O. yon Estorff et al. 2

;- -

,¢~

I --

I

J

L

m surf°tunn

--

"~.

J

L

J

6 m sutftlufln

.L

:

i

..... lnf - surf.loaO

";::'......... : ::::!.. ii

. . . . . . . . . /.1

.'.~..."

D

l\\l//

!;-v

,v /

....

'." I 0.04

0,~

0.1~ title

O. tO

I

O, 12

I

O. 14

0

>

t/(

=.

i

TIME [SEC]

Fig. 12. Transient vertical response at point B due to surface and underground load acting simultaneously." influence of the depth d of the tunnel disturbance on humans is usually measured in terms of accelerations, Fig. 13 provides the vertical acceleration history of point B due to the combined loading.

PARAMETRIC STUDIES FOR H A R M O N I C EXCITATIONS This section provides results for the tunnel system of Fig. 1 with the same geometric and dynamic parameters as in the previous section with the exception that the time variation of the loads is here harmonic and not transient as before. This clearly suggests the use of the real frequency domain BEM in the computations, with a very small amount of damping (fl = 0.001 or 0.0001) to avoid any influence of fictitious eigenfrequencies. The frequency range of interest is here 0.01 Hz~
I I

~...-....

".,%

i

,

0.012 0.024 0.0350.0480.0000,072 0.084 o.og5 0.108 0.120 0.132 0.144 0.158

O. I$

?

#

1

CSEC]

to D (or D to A). The maximum deformation decreases with increasing depth of the tunnel and considerable reflections at the free surface of the halfspace are obvious if one considers the displacements due to underground traffic for d = 4 m and d = 8 m with d---, oo (see Fig. 11). Figures 9 and 10 show the response curves of the points B and C, respectively, which are not directly located beneath one of the two loadings. In both cases the surface traffic causes higher displacement amplitudes than the underground loading• This is certainly unexpected for point C which is, in the case of d = 8 m, closer located to the underground than to the surface load. Moreover, the deformations at point B are only very little influenced by the depth of the tunnel, while at point C the influence of the distance d between the surface and the tunnel seems to be rather significant• If one is interested in the general influence of traffic induced vibrations on nearby buildings (see Fig. 1), the observation point B is of particular importance. Therefore, for this point the combined response to surface load at A and A' and underground traffic acting simultaneously is depicted in Fig. 12. Finally, because the adversity of a

o

~

V

,

-I0

Fig. 10. Transient vertical response at point C due to an impulsive surface and underground load: influence of the depth d of the tunnel

-

r

cl = t.m

::: ,.'.' I ......

I

/ 0.~

!

~

-8 t.m

%.00

e2-e

I

I

4 m - surf°tunn

t

J --

I

I

8 m - surf*tunn

I

i

t

i

..... Int - |urt.loeO J

t

0.05

.°

kt '/~Y \Ii I

~"

0

i ,,,,', ^

-"

A

An /"

~ -0.06

%.00

0.~

0.04

I

0.~1~

0,~

I

0,|0

I

0.1~

I

O,II

,,,. O. Ig

0.012 0.024 0.0300.0480.0e00.0720.0840.0~8

TI~E CSEC]

Fig. 11. Transient vertical response at point D due to an impulsive surface and underground load: influence of the depth d of the tunnel

172

0.108 0.120 0.132 0.144 0.155

TIME [SEC]

Fig. 13. Transient vertical acceleration at point B due to surface and underground load acting simultaneously." influence of the depth d of the tunnel

Engineering Analysis with Boundary Elements, 1991, VoL 8, No. 4

Dynamic interaction effects in underground traffic systems: O. yon Estorff et al. I

~

[

t

~

I

J

t

I

t

3 --~ ---

9 LU O --

~ trl -

surf,10l(~

'~

Inf - s u r f , l o I 0

I

I

I

I

• m - tunn, loo~ --

8 m-

4 m - SUtf.lo~l

"'*'" 0 m -

I

I

t

tunn.loId

/ ,,

,.~ I

~ 1.8

.J 0. ~O

~

~5

•~ "~-'%°.

(F"

..... ~...',~ [ -

05

o50.1 ¢B <:

1

0 0.01 0.08 O.Og

0.4

3

08

7

11

FREQUENCY

18

19

23

o 0,01 0.05 0.09

27

3

I

I

2.8

I

I

I

m - Sorf,lold

a

I

i

..

I

I

i

i

i

....

4

> 1.8

CO

t 1

05 0.5

0 0.01 0.08 0.09

0.4

o.e 3 r FREQUENCY

~1 18 [Hz]

~

23

2r

Fig. 15. Vertical response at point B due to an harmonic surface and underground load: influence of the depth d of the tunnel

I 3 t

~

I

I

4m-

tun~loe¢l

I

I

I

~

8 m - luni~.lOId

I "*"

I

I

t

111f- tuNtkloIcI

3

7

11

18

27

Inspection of Figs 14 and 17 reveals that for the case of the tunnel load the displacements at A and D decrease for increasing depth. For the case of the surface load, the same trend is observed for the displacement at D, while the opposite is true at A except for high frequencies. This phenomenon of increasing surface load response at A for increasing depth is due to the presence of the stiff liner in the soft soil. In addition, displacements at A (D) due to the tunnel load are smaller (bigger) than those due to the surface load, with the only exception being the response at D due to the tunnel load when d--* oo. In general, one can observe that for every case of load the influence of the depth is very small with the only exception being again that of the response at D due to the tunnel load when d --, ~ . Finally, it is clear that the influence of the frequency on the response is only significant for very small frequencies and that for intermediate and high frequencies the response is almost independent of frequency, even though some reversals of the afore-mentioned trends for point A are observable. Figure 15 indicates that for very small frequencies there is practically no influence of the tunnel depth on the response, irrespective of the kind of load, while for intermediate and high frequencies a behaviour similar to r

I

~

I

I

I

I

I

i

1

..... In( - tunn,Io~ L

I

t~

2.8

2

I

-.°°

. 1.8

23

Fig. 17. Vertical response at point D due to an harmonic surface and underground load: influence of the depth d of the tunnel

/

!

lg

4.

[Hz]

I--"

It m - l u r f . l o l d

4 m " IUrI.IoI0

I

0.8

FREQUENCY

Inf ° s u r f , l o l ¢ l

[.

0.4

[Hz]

Fig. 14. Vertical response at point A due to an harmonic surface and underground load: influence of the depth d of the tunnel

~

!

0.3

2

t'-I

Inf - tunn, lOIlO

Iurf.lol~

~ 2.5

D~ 0,2

J O. ¢/)

I

/

0.4

g3

:.,

60 ~ <

0.8 0 0.01 0.05 0.09

<~ 2

........~-3~ 0.4

0.8

3

7

FREQUENCY

11

16

lg

23

27

0 0.01

%'°"°'

0.06 0.09

'0.4

""~..~.~,

0.8

3

7

FREQUENCY

[Hz]

Fig. 16. Vertical response at point C due to an harmonic surface and underground load." influence of the depth d of the tunnel

1 11

15

19

23

27

[Hz]

Fig. 18. Harmonic vertical response at point B due to surface and underground load acting sim,dtaneously: influence of the depth d of the tunnel

Engineerin9 Analysis with Boundary Elements, 1991, Vol. 8, No. 4

173

Dynamic interaction effects in underground traffic systems: O. von Estorff et al. ACKNOWLEDGEMENTS

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The authors are grateful to the G r e e k - G e r m a n Bilateral Scientific C o o p e r a t i o n P r o g r a m for supporting this work. The first and fourth a u t h o r are also grateful to the Deutsche Forschungsgemeinschaft for supporting their work on B E M / F E M coupling through scholarship III 02-Es 70/1-I, while the second and third a u t h o r to the Greek General secretariat of Research and Technology for supporting their work on underground structures t h r o u g h the grant 7821/19-5-88.

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REFERENCES 1

Fig. 19. Harmonic vertical acceleration at point B due to surface and underground load acting simultaneously: influence o f the depth d o f the tunnel

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that in Fig. 14 is observed. The set of curves in Fig. 16 has a shape similar to that in Fig. 17. However, the surface load response here is bigger than the tunnel load response, because point C is closer to the surface load than D. In general, both Figs 15 and 16 show a frequency influence, which is again restricted to very small frequencies. Finally, Figs 18 and 19 provide the absolute displacement and acceleration, respectively, at B due to the combination of surface loads at A and A' and the tunnel load at D.

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10 CONCLUSIONS O n the basis of the preceding developments the following conclusions can be drawn: 1. T w o different computational schemes have been presented for the dynamic analysis of u n d e r g r o u n d traffic systems under conditions of plane strain. The first works in the time d o m a i n and combines the F E M for the structure and the B E M for the soil, while the second works in the frequency d o m a i n and employs the BEM for both the structure and the soil. 2. A comparison of the two methods revealed that both can efficiently provide results of high accuracy provided that a discretization of appropriate degree and extent is employed. Depending on the particular problem to be solved, one m e t h o d can be more preferable than the other. 3. Both methods have been successfully used for the study of the dynamic behaviour of a typical tunnel system under transient (time domain F E M - B E M ) or harmonic (frequency domain BEM) surface and tunnel load combinations in order to assess the effect of the dynamic and geometric parameters of the problem on the response of the system. 4. More extensive parametric studies are required for all possible load combinations, more realistic kinds of loads and various combinations of liner and soil material in order to fully understand the dynamic behaviour of the system and formulate useful design guidelines.

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