This Compilation of TAC Papers was prepared courtesy of

Rational Assessment of Tunnel Liner Capacity

P.K. KAISER

Department of Civil Engineering, University of Alberta, Edmonton, Alberta

I.P. BARLOW

Department of Civil Engineering, University of Alberta, Edmonton, Alberta

Abstract:

For the structural design of retaining structures, such as tunnel linings, it is necessary to solve an

often complex ground -structure interaction problem because ground pressures and their

distribution depend on the displacement history of the structure. The magnitude and location of

bending moments and shear or axial forces in these retaining structures are strongly influenced by

the flexibility of the structure and the variability of the ground pressure around the circumference

of the tunnel. Simplified pressure distribution functions or envelopes, covering all reasonably

possible variations of ground and water pressure, are commonly prepared for the structural design

of the liner. It is demonstrated that major inconsistencies in the most common design approaches

may lead to unacceptable results for the design of tunnel liners, primarily because unconservative

bending moments are predicted in this manner. Tunnel liner design techniques resulting in

symmetric, steadily changing ground pressure distribution envelopes are conceptually wrong. A

more rational design approach considering the structural interaction diagram (moment versus

thrust) and the effects of ground pressure variability is evaluated by comparison with

measurements from two case histories. The relevance of this approach and the practical

implications are discussed. This paper is a revised version of the presentation given at the 5th

Annual Canadian Tunnelling Conference in Montreal, 1985.

Keywords: current tunnel liner design, pressure envelopes leading to unconservative envelopes

of bending moments; bending moments must be considered in subgrade reaction; assess the effect

of pressure differential on bending moments; compare resulting range of linear thrusts and

bending moments with a reduced linear capacity; comparisons of bending moments on the

Edmonton Sewer Tunnel, and the Elbe Tunnel in Hamburg, West Germany.

Kaiser, P.K., Barlow, I.P. Rational Assessment of Tunnel Liner Capacity. 1986 Tunnelling Association of Canada Annual Publication.

Kaiser, Barlow 31

Rational Assessment of Tunnel Liner Capacity

P.K. KAISER Department of Civil Engineering, University of Alberta, Edmonton, Alberta

I.P. BARLOW Department of Civil Engineering, University of Alberta, Edmonton, Alberta

Abstract

For the structural design of retaining structures, such as tunnel linings, it is necessary to solve an often complex ground -structure interaction problem because ground pressures and their distribution depend on the displacement history of the structure. The magnitude and location of bending moments and shear or axial forces in these retaining structures are strongly influenced by the flexibility of the structure and the variability of the ground pressure around the circumference of the tunnel. Simplified pressure distribution functions or envelopes, covering all reasonably possible variations of ground and water pressure, are commonly prepared for the structural design of the liner. It is demonstrated that major inconsistencies in the most common design approaches may lead to unacceptable results for the design of tunnel liners, primarily because unconservative bending moments are predicted in this manner. Tunnel liner design techniques resulting in symmetric, steadily changing ground pressure distribution envelopes are conceptually wrong. A more rational design approach considering the structural interaction diagram (moment versus thrust) and the effects of ground pressure variability is evaluated by comparison with measurements from two case histories. The relevance of this approach and the practical implications are discussed. This paper is a revised version of the presentation given at the 5th Annual Canadian Tunnelling Conference in Montreal, 1985.

1. Introduction

The development of new tunnel support techniques such as extruded linings, the continuing economic pressure for more cost effective tunnelling methods and the need to excavate larger openings for subway stations, requires more accurate design techniques to ensure safe and economic tunnel construction. Fortunately, few tunnels have failed, deformed excessively or performed in an unsatisfactory manner. Hence, one might be tempted to conclude that tunnel liner design methods are adequate. The following discussion will attempt to demonstrate that this is not necessarily the case and that there is much need for an improved and more rational tunnel lining design approach. The primary reason for this can

be attributed to the fact that current techniques do not properly predict the bending moments in the tunnel support. Even though bending moments are often reduced by construction measures (e.g., stress raisers), this should not prevent the designer from conducting a proper analysis of the expected loading path in terms of thrust and bending moments. Furthermore, the fact that yielding of a lining must occur at several points before a collapse mechanism becomes possible, should not be taken as a justification for following design procedures that produce erroneous results. For example, if the support yields, the thrust capacity may be reduced at the flexural yield location, shear failure may more readily occur at points of high bending stresses, or water inflow and leakage may be enhanced due to

32 CANADIAN TUNNELLING CANADIEN 1986

excessive cracking of the liner. Because high shear stresses exist where the bending moment gradient is highest, a proper calculation of the expected bending moments will help to prevent distress due

to shear.

Tunnels are complex structures because their lin~ar nature often requires crossing of variable ground conditions, which makes it rather difficult to arrive at a lining design approach that is generally applicable. The following discussion will concentrate primarily on the design of temporary supports for soft ground tunnels. It will deal with the specific problem of designing a liner with adequate load bearing capacity. Other design considerations such as control of settlements are not covered. The objective of design is to predict the expected stresses in the liner and to select the dimensions and material properties such that overstressing due to thrust, shear forces and bending moments is prevented. While water

THRUST (MN/m)

pressures, stresses due to construction and from other sources of load (for example, foundation pressures or surface surcharge) must be considered, this paper deals only with the assessment of liner thrust and bending moments resulting from ground pressures.

2. Design Criteria from structural viewpoint

2.1 Liner Capacity

The capacity of a tunnel support (steel sets, cast-in-place concrete rings, concrete segments or reinforced shotcrete rings) can best be represented by a moment - thrust interaction diagram as shown in Fig. l.a for a 250 mm thick concrete liner with No.6 reinforcement (after Hansmire, 1984) and in Fig. l.b for steel sets (4 x 4 M13 H -profile) used as part of a rib and lagging support for a sewer tunnel in Edmonton.

THRUST (MN/m)

Figure 1. Moment-thrust interaction diagram for (a) a 250 mm concrete liner (after Hansmire, 1984) and (b) steel sets installed in a sewer tunnel in Edmonton

Kaiser, Barlow 33

Current structural design procedures based on limit state design require that the capacity be reduced by a strength reduction factor 0 and that the actual expected load be increased by a certain load factor (LF) (i.e., ACI building code requirements for reinforced concrete, ACI 318-77). Hence, the reduced capacity must be equal to or in excess of the thrusts and bending moments caused by the factored loads (as shown schematically in Fig. l.b) if any overstressing is to be prevented: However, the liner is restrained by the surroundmg ground and can only fail by axial thrust, by shear, or, less likely, if many hinges create a kinematically possible collapse mechanism.

Sgouros (1982) conducted laboratory tests and numerical calculations, and found that a load path as shown schematically in Fig. l.a is followed for a concrete liner. After the yield point (Y) is reached, undesirable bending moments are reduced but the path does not remain on the 'full capacity' curve. The zero moment line is approached quickly without a significant increase in thrust. The ultimate thrust capacity of a flexurally yielded concrete liner may be much less than that of a liner crushed in pure thrust. Consequently, for a proper assessment of the thrust capacity, it is essential to know where the capacity line is reached so that the proper ultimate thrust can be selected.

2.2 Thrust and Bending Moment Determination

One of the five approaches summarized in Fig. 2 is commonly adopted for the determination of thrusts and bending moments in a tunnel lining:

a. Distributed ground pressure assumption (e.g., Hewett and Johannesson, 1922);

b. Subgrade reaction model (e.g., Schulze and Duddeck, 1964);

c. Continuum (ground - support interaction) approach (e.g., Einstein and Schwartz, 1979);

d. Numerical, continuum or discontinuum modelling (e.g., by finite elements, boundary elements, ... );

e. Convergence - Confinement Method (e.g., Brown et al., 1983).

These approaches, with the exception of (a), consider the interaction of the ground with the support and the effect of the initial in situ Stresses (KG). However, because of the assumed homogeneous ground, these techniques lead to symmetric and steadily changing pressure distributions (with or without tangential pressure components) which do not correspond with the actual ground pressures. Even numerical models seldom simulate heterogeneities in sufficient detail and do not produce much more realistic pressure distributions. It will be shown later that the consequences of this simplification may be detrimental to the stability of a liner and it is the purpose of this contribution to discuss the reasons for it. A more rational design procedure that respects these deficiencies is then proposed.

Three-dimensional numerical models are capable of properly simulating the situation near the tunnel face but only at great expense for realistic ground conditions and properties. Consequently, the two-dimensional design models shown in Fig. 2 are commonly employed for design and the. near face effects are neglected or adjustments are made by application of empirical correction factors (for a,b and c) or approximated (for d and e) by various techniques (e.g., Laabmayr and Swoboda, 1979). The convergence - confinement method (e) demonstrates clearly that a significant pressure reduction occurs unless the tunnel is constructed in a soil mass with time - dependent properties or with swelling potential. Hence, stress reductions due to face effects must be considered separately for most conventional design models.

3. DeSign Criteria from geotechnical viewpoint

The prime concern of a geotechnical engineer is to design and install a liner such that a collapse of the ground above the tunnel is prevented and that ground displacements near the tunnel are adequately controlled.

ilL

34 CANADIAN TUNNELLING CANADIEN 1986

"('= 0 or not

@] 111111111111111111111111111111 p v

or not

w cr: ::J (fJ (fJ

w cr: Il..

I

_ L _--I:!:d::r.:~ I '"\= 0 or not

CONVERGENCE CURVE

DISPLACEMENT

Figure 2. Conventional tunnel lining design models

Kaiser, Barlow 35

The pressure assumed to act on a liner is selected to satisfy one of the following three design criteria illustrated by the schematic convergence curves in

Fig. 3:

(I) Allowable Stress Design: - no yielding of ground is permitted;

(II) Limit State Design: - a) to prevent onset of rupture • or - b) to provide at least the ultimate

pressure (reached after yielding);

(III) Serviceability:

w a: :::l (fl

(fl

w a: c.

- to control displacements to less than an allowable limit.

p(ll.b)

ALLOWABLE DISPLACEMENT u{l!!)

YIELDING --t>-

DISPLACEMENT u

Figure 3. Schematic convergence curve

These deSign criteria are often hidden in empirical design rules or guidelines that are largely based on practical experience (e.g .• Deere et aI., 1969, or Schmidt, 1984; ultimate distortion ranges of 6D/D < 0.25 to 0.75% depending on soil type).

Large and often uneconomic pressures, p(I), are required if overstressing of the ground is to be eXcluded; therefore, some yielding is desirable. However, rupture due to propagation of a localized yield zone to the ground surface and associated pressure increases must be prevented;

p>p(II.a) (Wong and Kaiser, 1986). Design based on an ultimate pressure p(ILb) for shallow tunnels is not acceptable because excessive deformations could cause damage to nearby structures. For relatively deep tunnels, however, a propagation of failure to the surface is not possible and displacement control requirements often dominate the design such that displacement limits u(III) dictate a minimum support pressure p(III) .

Design pressures determined by anyone of the techniques listed in Fig. 2 do not correspond with expected or measured ground pressures. They should represent an upper limit or an envelope to the actual pressures which should always fall inside a pressure design envelope (see Fig. 4).

4. Rational design approach

4.1 Inconsistency in Current Design Approaches

A gra ve inconsistency in design logic exists for all methods listed in Fig. 2 because geotechnical design requirements are represented by a pressure envelope while the structural design is based on thrust and bending moment envelopes. For the design of foundations and retaining walls this difference does not lead to unsafe results but it may for tunnel linings.

Fig. 4 compares the commonly adopted design sequence for a strutted retaining wall and a tunnel in soft ground. Field observations of earth pressures behind retaining walls led Terzaghi to recommend rectangular or trapezoidal pressure envelopes (Fig. 4.a). If these pressures are applied to a multi -supported beam. they also provide an envelope to the bending moments (Fig. 4.b) created by the actual, variable earth pressures. Due to wall movements, the assumed earth pressure is less than the pressure at rest. The axial forces in the wall depend on the friction at the wall/soil interface and these are in general small compared to the thrust capacity as demonstrated by the interaction diagram in Fig. 4.c. The reSUlting range of moments and

I

36 CANADIAN TUNNELLING CANADIEN 1986

thrusts is at a level where the interaction diagram is widest and a relatively large safety margin between actual stresses and capacity exists.

For tunnels, the ground pressures are also not uniform inside an assumed or calculated design envelope (Fig. 4.d). Contrary to the thrust in a retaining wall, the thrust forces in a tunnel liner are overestimated and, more importantly, the bending moments are drastically underestimated if determined from the pressure envelope (Fig. 4.e). The thrust is overestimated, because it is related to the sum of the pressure acting on the liner which is always less than the integral pressure given by the pressure envelope. The bending moments are underestimated because they are related to the pressure variation along the circumference of the liner which is not properly reflected by a pressure envelope. The moment distribution from the pressure envelope falls inside the range bounded by a minimum moment M(min) and a maximum moment M(max) (causing tension inside and outside, respectively). A rational determination of the required liner capacity is not possible unless the variation in pressure is known (in magnitude, location and extent) and the extreme moment envelopes M(min) and M(max) are calculated (Fig. 4.e).

Ground movements ahead of the tunnel face and before interaction of the liner with the ground cause a pressure reduction that must be considered for an economic design. Consequently, for a tunnel, the range of expected thrusts is lower than calculated from a pressure envelope but the moments are much larger leading to a more critical situation (Fig. 4.f). The safety margin between the liner capacity and the actual range of thrusts and bending moments is much less than predicted by conventional techniques.

4.2 Comparison of Predicted and Measured Thrusts and Bending Moments

The significance of this inconsistency in design logic can best be assessed by comparison with observations from tunnels where extensive

measurements of thrusts and moments were made. Results from two tunnels are presented in the following. A detailed description of these projects is given by Corbett (1984) and Wong and Kaiser (1986) for the Edmonton Sewer Tunnel and by Duddeck and Janssen (1980) for the Elbe Tunnel in Hamburg, West Germany.

At the Edmonton Sewer Tunnel, the ground pressure on wood lagging was measured and the thrusts and bending moments were then calculated in the roof segment (± 60°) of the steel set ring consisting of three segments. The ranges of these thrusts and moments are plotted for 14 cross-sections along the tunnel in Fig. 5.a together with the thrusts and moments calculated by the methods proposed by Einstein and Schwartz (1979) and Schulze and Duddeck (1964). Some of the sections were in good ground (stiff Edmonton Till) with little thrust and moment development and others in intermediate quality ground. At three sections, in extremely poor flowing saturated sand, pressures approaching full overburden load and surface settlements of up to 300 mm were recorded .. In these latter sections, the steel sets yielded (dashed lines with question marks) and local shoring was required to prevent tunnel collapse.

It is of interest to note that the magnitude of moments in areas of good ground with low support pressures «0.05 MN/m) are in the order of those predicted under full load. From this and other studies, we concluded that in expanded steel supports, construction related bending moments alone (M, at zero thrust or zero ground pressure) are approximately equal to those predicted by conventional continuum approaches.

The range of measurements at two cross -sections of the Elbe Tunnel together with the predicted thrusts and moments are shown in Fig. S.b. At this tunnel, thrusts and moments were measured by strain gauges in three consecutive support rings consisting of eight steel segments. Duddeck and Janssen (1980) processed the data to separate load components that were not related to ground

Kaiser, Barlow

MEASURED GROUND

PRESSURE ENVELOPE

BENDING MOMENTS

~-- DUE TO PRESSURE ENVELOPE

MOMENT

M(max)

DUE TO M(max) AND M(min)

CAPACITY '""-OF LINING

MOMENT Figure 4. Comparison of design sequence for retaining wall and tunnel lining

37

CANADIAN TUNNELLING CANADIEN 1986

38 of data shOwn are after

The ranges 'b h pressures, pure ground 10adlUg, In ot 'gorduetO 'ld

processln I I ted capacity for Yle thecacua ' diagramS, It can be seen, that despite

, ' t' on is shOwn, ed inltla I dicted thrusts, measur

the lower than pre rl'tical than those predicted , ' are more c ' condltlOns d by Einstein and Schwartz

by the methOd proPosned Duddeck (1964), This is

) d Schulze a ' (1979 an use apparently conservatlve

unexpected beca de for both techniques. tions are ma 1" ' assump h tz aSSume that the Inlng IS

, and Sc war ' Einstein nnel excavation, uo=O In installed befor; tU Schulze and Duddeck full fig. 2.e, and ~I taken as the vertical pressure overburden loa hiS f the bedded ring (Fig. 2.b).

, 'the arc 0 ' acting In he actual moments were lU excess In some cases, t

THRUST (IIIN/m)

.0,03

0.8

0.02 ..,,01 -0\ .. \, ~ 0,01

.0.02 -4"\1''0--III 0 III E N T (IIINm/m)

SCHuLZEJDUDDECK, fUll slip ..... EINSTEINISCHWARTZ, fUll slip

:: pJNGECfM~ENTS ".... YIELDUMffOFSTEELSEl'

0.03

of six times the moments calculated by the two-dimensional techniques mentioned above. The reason for this is, as explained earlier, that the influence of ground pressure variability is neglected. The same divergence of predicted and measured values is found when these observations are compared to results of numerical models of

tunnels in homogeneous ground.

4.3 Alternatives for Improved and More Rational

Design Approach

There are several options to improve current

design techniques:

1. Correct numerical modelling of the interaction of tunnel liners with

heterogeneous ground: this is seldom practical

THRUST

THRUST DUE TO FULL OVERBURDEN

w c

~ z o

I MOM E N T (MNm/m)

__ SCHULZEIDUDDECK, lull slip

_ EINSTEIN/SCHWARTZ, lull slip ...... YIELD LIMIT OF STEEL SEGMENTS RANGE OF MEASUREMENTS AT STATION:

~MP01 ~ MP07

III ~ z Q (J) z w ...

0.6

M ment.thrust diagrams for temporary supports of two tunnels: (a) sewer tunnel in Edmonton, and (b)

fignre~. 0 t nnel in !Iamburg, West Germany (after Duddeck and Janssen, 1980)

Elbe !Ilghway u

Kaiser, Barlow 39

because conditions change drastically along a tunnel and around its circumference, and the actual ground properties as well as their distribution are often not known with sufficient accuracy.

2. Introduction of additional or increased load factors (LF) to cover for the observed,

large bending moments: this is incorrect because it increases the predicted thrust forces beyond reasonable values. Extreme conditions for thrusts are already assumed by giving a pressure design envelope. Furthermore, this approach would only increase the bending moments proportional to the increase in thrust. Our analyses showed that the moments predicted in this fashion are still too small.

3. Adjustment of the predicted moments by multiplication of the calculated moments by

an empirical moment factor (MF): unfortunately, only few case histories with sufficient and accurate measurements of moments and thrusts are available to establish a generally applicable moment adjustment factor. Such a moment factor would have to reflect the local ground conditions, the support type, the construction sequence and many other factors which can not be formulated in a general manner. Nevertheless, for the two projects described earlier, it was found that a much larger than expected moment factor, between MF = 3 to 5, should be used to increase the predicted moments for temporary supports of

ribs or steel segments, if yielding must be JlfllVenr"tl. Further research is necessary before l.ene:ral!ly applicable guidelines can be specified for

SUpport types.

Supplementation of current design techniques by separate determination of

moment increments 11M resulting from pressure variability: this can be best

by specifying a ground pressure ~re,"til" that may exist at any location around .cjf(:UlIlfel·enlce of the tunnel. This pressure :rential increases or decreases the assumed

pressure (determined by techniques

summarized in Fig. 2) by liP = m . po as shown in Fig. 6.a (Po = yH is the overburden pressure at the tunnel centre line). The shape, amplitude liP and wave length, given by an arc length liS or the angle 11/3, depends primarily on the local ground condition, the type of ground/support interaction and the adopted construction procedure. Unfortunately, there is scarce data available to properly assess these parameters. Hence, they must be determined based on engineering judgement and in close cooperation with a qualified geotechnical engineer. In the next section, preliminary results are reported to provide some guidance for selecting an appropriate differential design pressure lip.

It must be pointed out that this last approach is consistent with current structural design practice for structures such as slabs and shells where the effect of a distributed load of limited extent is to be assessed. Furthermore, this design logic can be extended, without much difficulty, to the predominantly three-dimensional conditions near the tunnel face where the effect of a differential load on the bending moments near the end of the support tube are to be evaluated.

While it is beyond the scope of this contribution to give a step-by-step design procedure that follows the rationale presented above, the evidence given demonstrates that the influence of ground pressure variations on the bending moments in the tunnel support should be assessed properly and in a more accurate manner. Only then can the designer evaluate whether and to what extent yielding (or cracking) will occur, whether the ultimate thrust capacity (e.g., for the design of a shotcrete liner) must be reduced, or whether special construction procedures must be used to prevent undesirable collapse mechanisms (Le., shear failure).

The most logical tunnel lining design approach, illustrated by the schematic diagram in Fig. 6.b, includes the following three components: first, a prediction of the most likely and expected range of thrusts and bending moments which should correspond to the conditions actually encountered; second,' a prediction of thrusts and bending

I

CANADIAN TUNNELLING CANADIEN 1986

40

D.P = m • Po n-y·H 1"0 -

THRUST

~--FULL CAPACITY

~r--RI,DUCED CAPACITY

MOMENT

PREDICTED RANGES ~ ACTUAL LOADS ~ FACTORED LOADS

~ FACTORED MOMENTS

. emalic diagram of (a) press~re differential FIgure 6. Sch d termination of moment mcrement, and assumption ~~ u:t interaction diagram for actual, (b) moment'd ~oment adjusted conditions factored, an

moments under factored loads; and third, a determination of the moment increment I>M due to variable ground pressures or a pressure differential (Fig. 6.a). If yielding is to be prevented completely, the resulting area of moments and thrusts must fall inside the reduced capacity line. However, as indicated earlier, for most tunnel linings, bending moments will be reduced when the capacity line of the lining is approached. Consequently, moments calculated assuming elastic material behaviour may fall outside. Collapse will not occur (see example in Fig. 5.a) and such conditions may be acceptable if the lining is able to maintain an adequate thrust capacity (e.g., if buckling of steel sets is prevented). A significantly reduced thrust capacity may, however, have to be assumed for deSign of concrete liners where crushing reduces the thrust capacity, or special construction measures such as shear reinforcements (e.g., mesh, steel ribs or lattice girders in shotecrete) may have to be implemented to prevent loss of strength.

4.4 Prediction of Moment Increment I>M

A constant pressure differential I>P = m . Po was applied to a fully bedded support ring as shown in Fig. 7.a. Equal and opposite differential pressures over equal arc lengths were assumed to maintain balanced ground pressures. The resulting moment distribution for a support ring bedded in ground with uniform and constant subgrade reaction modulus is also shown in Fig. 7.a. Extreme positive (maximum) and negative (minimum) moments are almost equal for this particular case. Since this pressure variation could occur anywhere around the tunnel periphery. a constant moment increment I>M equal to the largest maximum or minimum moment should be considered for design and added to the moments calculated under average pressure conditions.

Parametric studies were conducted for the two case histories presented earlier. to provide guidance for selecting the parameters 8/3 and m. The arc angle 1>{3 was varied from 60' to 180' under a differential pressure of 10% of full overburden pressure (m = 0.1) and the results are plotted in Fig. 7.b and c.

Kaiser, Barlow 41

From these figures the following preliminary conclusions can be drawn: . _ positive and negative moment increments

have equal extremes even though at different 11/3 angles;

160'

so· 100' 120' 140' 160' &(3

M"velOiellt distribution around bedded and (c) Maximum moment variation for m = 0.1 ( P = m.p )

except for very localized loads (11/3<60') the extreme moment increment 11M (maxium of I M(min) I or M(max)) is relatively constant at 3 kNm/m for the Edmonton Tunnel and at 67 kNm/m for the Elbe Tunnel. Hence, the extreme moment increment can be determined assuming arc angles of 11/3 > 60 to 80'; and the moment increment 11M is proportional to the magnitude of I1p.

As a result, the problem of assessing the moment increment has been reduced to the prediction of the magnitude of the pressure variation I1P (or m).

For the purpose of estimating a meaningful magnitude of m, ranges of calculated moments (following the proposed design rationale) have been superimposed on Figs 5.a and b to produce Figs 8.a and b. It was assumed that a basic or construction moment Mo, equal to those predicted by Einstein and Schwartz (1979) or Schulze and Duddeck (1964), exist at any thrust level (see Section 4.2). The extreme moment increment due to pressure differentials (Fig. 7.b and c) was then added to this basic moment and limits of expected extreme moments were constructed. For the Edmonton Tunnel, limits for m = 0.3, 0.6, and 0.9 are shown in Fig. 7.b but only on one side (tension inside) because extreme moments near the springline area were not measured. A pressure variation I1P of between 60 and 90% of Po (m = 0.6 to 0.9) is required to predict the observed extreme moments. These values are in good agreement with the pressure variations reported by Corbett (1984) and Wong and Kaiser (1986).

Similarly, in Fig. 8.b, for the Elbe Tunnel, ranges of m = 0.2 to 0.6 with a construction moment Mo are shown. Even though only two test sections are presented here, it can be seen that m-values in the order of 0.4 to 0.6 are required to cover the extreme moments.

The vertically shaded areas in Figs 8.a and b represent the zones of load paths that might be encountered for these two tunnels. Accordingly, at the Elbe Tunnel, yield initiation is to be expected

I

42

CANADIAN TUNNELLING CANADIEN 1986

.0,03

THRUST (MN/m)

THRUST DUE TO FULL OVERBURDEN (NO WATER)

0,3

0,2

.0,01 YM

LOI .0.02 ---1 '1

MOM E N T (MNm/m)

~ SCHULZE!DUDOECK, full slip

rpn:I. EINSTEIN/SCHWARTZ, full slip _ RANGE OF MEASUREMENTS

_ YIELD LIMIT oF STEEL SET

m=O.9

Ul 0 Vi II % 0 Vi % Ul t-

'" 9 l!! ::> 0

% 0 iii % Ul t-

·0.6

THRUST (MN/m)

~M~·2 MOM E N T (MNm/m)

_ SCHULZEIDUDDECK. full sHp

m=0,2/0.4/0.6

"'"'" EINSTEIN/SCHWARTZ, full slip ....... YIELD liMIT OF STEEL SEGMENTS RANGE OF MEASUREMENTS AT STATION: '

~ MPOI V?"':'{j MP07

F' 8 Moment.thrust diagrams for temporary supports of two tunnels (as in Fig.5) with limits of expected Ignre 'bendl'ng moments' (a)sewer tunnel in Edmonton, and (b) Elbe Highway tunnel

extreme .

h average pressure reaches about 70 to 80%

when t e ' f full overburden pressure. while at the o t Tunnel this point was to be reached at Edmon on •

50 to 70% of full overburden load,

about for expanded steel supports. m-values of

Hence. 'd d f I

tO 4 to 0,6 should be conSl ere or ~ ed . d" df

d' ting extreme moment con Itlons an or

pretC 'ld"'t' Th , g the point of Yle mltla Ion. e assessln , nee at the Edmonton Tunnel. where expene ' ed . '

1 ntal Strutting was reqUlr m one sectlon

suppeme '

d to losS of passive ground restramt.

ue d ' h' demonstrates that the proposed eSlgn approac IS

rational.

Conclusions

A major inconsistency in current tunnel liner design techniques has been identified. It arises from the fact that pressure envelopes resulting from most commonly employed design techniques lead to unconservative envelopes of bending moments. An evaluation of results from two case histories showed that bending moments caused by ground pressure variations (or changes in subgrade reaction) must be considered for safe and economic tunnel lining design. Of several options that might be considered to improve current design techniques. it was concluded that it is, and

Kaiser, Barlow 43

consistent with current limit state design practice, to assess the effect of a pressure differential on the bending moments separately and to compare the resulting range of liner thrusts and bending moments with a reduced liner capacity. The extreme moments predicted by a given pressure differential were found to be relatively insensitive. to the arc length over which the pressure differential acts. For the two case histories of expanded steel supports, pressure differentials bP of at least 0.4 to 0.6 Po would have been required to predict the magnitude of the measured extreme moments.

The most important practial implication of these findings is that more information, than commonly provided, is required for a proper design of a tunnel lining. It is not sufficient to request from a geotechnical engineer average or extreme ground properties (e.g., E, c, ~) and in situ stresses (e.g., Ko). An assessment of the expected magnitude and extent of variations in ground pressure or subgrade reaction modulus must be made even if this, at the present time, may only be possible based on engineering judgement.

Acknowledgements

investigation was partially funded by NSERC, Natural Sciences and Engineering Research

of Canada. The practicality and eUU lcance of the concepts presented were

in close cooperation with D.D. Dunbar Anderson Associates Ltd. The contributions of

Dunbar and his colleagues through fruitful CUS;SiOllS is respectfully acknowledged.

References

Brown, E.T., Bray,J.W., Ladanyi,B. and Hoek,E., 1983. Ground response curves for rock tunnels, Journal of the Geotechnical Division, ASCE, 109, pp. 15·39.

Corbett, I., 1984. Load and Displacement Variation along a Soft Ground Tunnel M.Sc. Thesis, Department of Civil ' Engineering, University of Alberta, Edmonton, 246 p.

Duddeck, H. and Janssen,P., 1980. Autobahn -Elbetunnel Hamburg; Messungen der Beanspruchung der Schildstrecke' Ergebnisse und Auswertungen, Beri:ht Nr. 80-33, Institute foer Statik Technische Universitaet Braunschweig, 147 p.

Einstein, H.H. and Schwartz,C.W., 1979. Simplified analysis for tunnel supports, Journal of the Geotechnical Division ASCE lOS, pp. 499-518. "

Hansmire, W.H., 1984. Example analysis of circular tunnel lining, Tunnelling in Soil and Rock, GEOTECH'84, ASCE, pp. 30-45.

Hewett, B.H.M. and Johannesson,S., 1922. Shield and compressed air tunnelling. McGraw-Hill, New York.

Laabmayr, F. and Swoboda,G., 1979. Zusammenhang zwischen Elektronischer Berechnung und Messung; Stand der Entwicklung fuer seichtliegende Tunnel. Rock Mechanics, Suppl. 8, pp. 29-42.

Deere, D.U., Peck, R.B., Monsees,J.E. and Schmidt,B., 1969. Design of Tunnel Liners and Support Systems, Report for U.S. Department of Transportation, Contract No. 3-0152, 287 p. plus appendicies.

Schmidt, B., 1984. Tunnel lining design - Do the theories work? 4th Australia - New Zealand Conference on Geomechanics, Perth, 12p.

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CANADIAN TUNNELLING CANADIEN 1986 44

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