Simple method for the design of jet grouted umbrellas in tunneling

Simple Method for the Design of Jet GroutedUmbrellas in Tunneling

G. P. Lignola1; A. Flora2; and G. Manfredi3

Abstract: Tunnel excavation in cohesionless soils implies the use of a temporary supporting structure prior to lining installation. Thistemporary structure has to couple safety and economy, and can be conveniently realized using ground improvement techniques �forinstance, by creating an arch of partially overlapped subhorizontal jet grouted columns�. The adoption of ground improvement techniquesresults in structures far from having a perfect shape because they are intrinsically affected by defects �in both geometrical and mechanicalcharacteristics�, and therefore their design may hide unforeseen risks. As a consequence, this is the typical case in which sophisticatednumerical analyses may just give the illusion of being refined, if possible defects are not correctly taken into account. In this paper asimple yet rational analytical method for the design of a nonclosed tunnel supporting structure that may be of some help to this aim ispresented. It is done with reference to a simple two-dimensional scheme. In the first part of the paper, a design chart of optimal shape andminimum structural thickness of the cross section of the supporting structure is shown. In the second part, an iterative procedure to verifythe stability or to design the minimum structural thickness of an existing supporting structure with a predefined shape is described. Thismethod, coupled with the analysis of structural demand, allows one in principle to plot design charts. This approach can easily take intoaccount structural defects with a semiprobabilistic approach and therefore with a chosen risk level, which is of great help to the designerat least in a preliminary design stage. The proposed semiprobabilistic procedure is applied to the case of a temporary supporting structurerealized by partially overlapped subhorizontal jet grouted columns, intrinsically affected by defects in diameter and position. Thevariability of these geometrical parameters was considered based on the large quantity of experimental evidence collected in field trials bythe writers and published elsewhere.

DOI: 10.1061/�ASCE�1090-0241�2008�134:12�1778�

CE Database subject headings: Arches; Jet grouting; Limit analysis; Probability; Structural design; Tunnel supports; Thrust;Tunneling.

Introduction

Even though a large number of papers in literature are devoted totunneling in general and to lining design in particular �Panet andGuenot 1982�, there seems to be a lack of information about thebest design approach for some of the most widespread soil im-provement structures adopted as temporary supporting means. Areason can be found in the difficulty of defining the geometricaland mechanical properties of the improved soil body. The ex-ample of most grouting techniques is striking �permeation grout-ing, jet grouting�, for which the final result of soil treatment isaffected by large uncertainties in terms of geometrical extensionand mechanical properties. Often, this difficulty is overlooked at

1Assistant Professor, Dept. of Structural Engineering, Univ. of NaplesFederico II, Via Claudio 21, Naples I-80125, Italy �correspondingauthor�. E-mail: [email protected]

2Associate Professor, Dept. of Geotechnical Engineering, Univ. ofNaples Federico II, Via Claudio 21, Naples I-80125, Italy. E-mail: [email protected]

3Full Professor, Dept. of Structural Engineering, Univ. of NaplesFederico II, Via Claudio 21, Naples I-80125, Italy. E-mail: [email protected]

Note. Discussion open until May 1, 2009. Separate discussions mustbe submitted for individual papers. The manuscript for this paper wassubmitted for review and possible publication on May 1, 2006; approvedon November 12, 2007. This paper is part of the Journal of Geotechnicaland Geoenvironmental Engineering, Vol. 134, No. 12, December 1,
2008. ©ASCE, ISSN 1090-0241/2008/12-1778–1790/$25.00.
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the design stage, and either a very conservative design approachis adopted �i.e., an excess of treatment with a massive approach�,or an undesired and hidden risk is introduced �i.e., a structureunable to carry soil loads with the desired factor of safety�. Mosttimes, cost effectiveness of the design of such temporary support-ing structures is not rationally evaluated. In order to overcomethis limitation, in the writers opinion, two steps have to be under-taken at the research stage: �1� there must be an effort to getinformation from trial fields and from any site application to im-prove the capability of predicting the effect of soil treatment andthe relevance and extension of defects; and �2� there is a need tothink of simple yet rational methods to design and verify thetemporary supporting structures which result from soil treatment,taking into account their unavoidable defects.

In this paper, the latter issue is approached, considering rela-tively thin temporary supporting structures. In particular, explicitreference is made to structures �from now on called umbrellas�made of overlapped jet grouted columns �Croce et al. 2004�, re-alized from within the tunnel with the typical iterative construc-tion sequence shown in Fig. 1. In order to allow the constructionof subsequent umbrellas, these structures are realized withslightly diverging columns, thus giving the shape of a three-dimensional frustum of cone to umbrellas, whose diameter in thecross section increases along the tunnel axis. The simple methodproposed hereafter refers to the cross section of the umbrella�whose dimensions vary along tunnel axis both in terms of diam-eter and thickness�. Even though tunnel excavation is a fully
three-dimensional �3D� problem, it is typical in tunnel lining de-
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sign to adopt a simpler 2D approach. The key issue in doing thisis obviously the appropriate choice of a stress reduction factor. Nosoil structure interaction is explicitly considered.

The method is rather general, yet specifically conceived for jetgrouted umbrellas, whose geometrical characteristics and me-chanical properties are intrinsically affected by some uncertainties�Croce and Flora 2000; Croce et al. 2001; and Flora et al. 2007�.

Stress State around Tunnels Front for ProvisionalLining Design

In the framework of bidimensional plane strain models, a widelyused, simplified method to account for stress release in the tran-sition zone around the tunnel front is the so called convergence-confinement method on which for instance the so called newAustrian tunneling method �NATM� is based �Mussger et al.1987; Panet 1986�. With this method, 3D effects on stress distri-bution are accounted for by a fictitious pressure � f = �1−�cc� ·�soil, where �cc increases from the zero value which refersto undisturbed soil to the maximum value allowed by a givenconstruction series ��cc=1 for unlined tunnels�, thus simulatingtunneling effects. A realistic simulation largely relies on thechoice of an appropriate reduction factor �cc, which in a givencross section depends on ground parameters, excavation length,overburden and tunnel cross section, supporting structures instal-lation sequence, etc. Indications are given in the literature �Panet1995� for some typical situations.

Ground movements at the tunnel depth usually start about oneradium in front of the tunnel face. If the temporary supportingstructure is installed behind the tunnel front �for instance withshotcrete and steel ribs�, some displacements have already takenplace in the soil in front of the facing, and stresses have alreadybeen partially redistributed, then, �cc�0 at the front and the lin-ing sustains only a portion of the initial soil stresses, no matterwhat its installation distance from the front. If a jet grouted um-brella is installed prior to excavation, it certainly modifies soilmovements beyond the excavation front, and differences with thecase of sprayed linings must be expected. Very few field data areavailable to give some hint on this aspect �Russo and Modoni2006�.

The convergence of the tunnel can be computed as a functionof the applied earth pressure, and structural lining design canfinally be carried out. The reduction of the earth pressure and theevolution of the lining load �both as equivalent radial stresses�with tunnel convergence are defined by convergence and confine-ment curves, respectively. With the convergence-confinementmethod, the intersection of these curves gives the final state ofequilibrium.

Starting from this simple scheme, a large number of methodshave been developed, considering elastic, plastic, and elastoplas-tic behavior of both soil and lining �Clough and Schmidt 1981;

Fig. 1. Construction sequence of jet grouted umbrellas

Muir Wood 1975; Atkinson and Potts 1977�.

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Viable Mechanical Approach: Thrust Line AnalysisMethod

In this paper a method to analyze supporting nonclosed structuresof tunnels is presented. With such a method, it is possible tochoose at the design stage an optimal shape of the cross section ofsupporting structures and to define its minimum needed structuralthickness, once a given reference stress state is considered. On theother hand, the method can be adopted with an iterative procedureto check the stability of an existing supporting structure with agiven shape, or to calculate its minimum structural thickness,again on the basis of a reference stress state. Even though the firstuse of the method is, in principle, possible the method has beenconceived for the second goal, specifically thinking to jet groutedumbrellas. In fact, for such structures, it is necessary, at least at apreliminary stage, to decide the number of jet grouting columnsand their mutual overlapping, and a simple criteria is better than acompletely empirical rule of thumb.

In this work, a method to evaluate the limit load of a givennonclosed tunnel supporting structure �or, conversely, the mini-mum required structural thickness for a given load� is proposedusing a classical limit analysis approach. A rigid-perfectly plasticbehavior is assumed for the structure, with a given value of thelimit compressive strength and a null tensile strength �i.e., thestructure is conservatively considered unable to withstand tensilestresses�. Collapse is assumed to take place when, in at least onepoint of the supporting structure, the internal stress state �definedby a combination of axial thrust and bending moment� lies on theyielding limit surface.

Geometrical nonlinearity and deflection dependent earth pres-sure distributions are not expressly considered. The influence ofsoil stress anisotropy around the tunnel is considered only via aparametric analysis, assuming different values of the ratio be-tween vertical and horizontal stresses. The writers are well awareof the importance of soil structure interaction in general, and ofthe increasing consideration it is getting in tunnel lining design inparticular. It is argued, however, that for geometrically imperfectstructures as jet grouted umbrellas, there are other unavoidablesimplifications far more relevant �one for all: the true shape�.Considerations on the possible influence of this important hypoth-esis will be done in the following sections.

The analysis is carried out in the tunnel cross section. Withreference to the open frustum of cone shape of jet grouted um-brellas defined in Fig. 2, the current diameter of the structurevaries within the span, being the largest at the end of it. Frictionalrestraints are considered at the base of the supporting structure.

The proposed approach is based on the use of the thrust line�TL� introduced by Hooke in 1695 �Lancaster 2005�, that by defi-nition is the locus of points in an arch through which the resultantforces pass. Because of its simple geometrical meaning, the use ofTLs in arch design dates back to the old times of civil engineer-ing. Typically, in masonry arch design it was possible to overtake

Fig. 2. Scheme of generic longitudinal and cross-sectional resistantelements, and definition of angle of opening �

the hyperstatic indeterminacy, modeling TLs first, and then plac-

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ing structural material around them, keeping the thrust in the cen-ter of the structures. With such an approach, engineers applied thelower bound theorem �LBT� of limit analysis, even before it wasexplicitly stated. In fact LBT states that, if it is possible to find aload distribution on a structure corresponding to internal stresseswithin the yield surface and satisfying the equilibrium as well asthe static boundary conditions for the acting load, then this loadwill not be able to cause collapse of the structure. The importanceof this theorem is that it is not necessary to determine the actualstate of the structure; any satisfactory equilibrium solution dem-onstrates stability.

A simple TL is not sufficient for dealing with structures ascomplex as jet grouted umbrellas. Failure of those arches is re-lated to the strength of materials and the shape of the structure. Itis also important to point out that a proportional increasing load�i.e., increasing depth h, or soil unit weight � or the distance fromthe front� cannot cause any change in the shape of the TL, but itdoes change the stress field.

Jet Grouted Umbrella Geometric Idealization

From a structural point of view, the three-dimensional jet groutedumbrella can be conveniently thought of as being made of longi-tudinal elements and cross-sectional arches �see Fig. 2�. Arches’supporting reactions on the longitudinal elements are due to in-plane stiffness of arches themselves, and can be modeled assprings �having the same stiffness of the arches�. Then, longitu-dinal elements lay on a continuous field of springs with decreas-ing stiffness �as the arch diameter increases along each span�, andare essentially loaded in bending. Within this modeling scheme,often adopted in the analyses of cylindrical tubes, the bearingcapacity of the structure depends on the load capacity and in-plane stiffness of arches as well as on the strength of longitudinalbeams.

Even in the ideal case of jet grouted umbrellas made of per-fectly cylindrical columns, the thickness of the umbrella, given bycolumns overlapping, decreases along the axis. There is large ex-perimental evidence that the perfectly cylindrical shape of jetgrouting columns is rarely the case �see, for instance, Flora et al.2007�, and columns have a number of defects linked to a randomvariation of diameter along the axis, a deviation of column axisfrom the ideal position, and a mechanical nonhomogeneity ofsoilcrete. Because of these defects in each single column, thecontinuity and regularity of the structure is not always granted,and a structural thickness smaller than ideal must be expected. Iffull overlapping among columns is ensured throughout the wholespan, and the stiffness of the supporting arches is large enough,the beamlike behavior can be reasonably overlooked, and the sim-pler bidimensional plane strain scheme of a series of compressedadjacent arches can be considered. Then, a properly constructednonclosed supporting structure behaves as a constrained arch tak-ing advantage of the lateral confining earth pressure. If the con-dition of full overlapping among columns fails, and at least one ofthe cross arches is not continuous, then the stability of the um-brella should be analyzed by looking to the flexural behavior ofcolumns in the longitudinal direction. In such a scheme, the bedof springs supporting the columns would be incomplete, withmissing springs in the sections where the arch is not complete.This condition would result in the necessity of using some kind oflongitudinal reinforcement to compensate for the no-tensile be-havior of soilcrete. Since in the following reference will be done
to nonreinforced jet grouted columns, stability of the umbrella


will be considered possible only if full overlapping is ensured inany cross section. Then, it is reasonable to analyze the umbrella’sbehavior via a simple arch scheme.

For each arch the local origin is set at the left support �pinjoint� �Fig. 3�, where y� is the coordinate along the tunnel axis�span direction�, and x� and z�, respectively, are the horizontal andupwards vertical coordinates in the plane of the arch. Each archcorresponds to a given value of y, its diameter increasing with y.

Frictional restraints have been adopted at the arch base. Then,the pin joints behave as hinges if at the soilcrete-soil interface��max, and as simple supports if ��max �in this case the struc-ture is supposed to be unstable and therefore to fail�, where�max=� · tan �. The soilcrete-soil interface friction angle is as-sumed to be equal to the soil friction angle as suggested by someexperimental evidence �Croce and Flora 1998�.

The resulting two-pin arch is a statically indeterminate struc-ture. The horizontal unknown force H can be determined in theelastic range, for example, using Castigliano’s theorem �i.e., ne-glecting axial and shear deformation�, but due to the complexityof the problem �nonlinear no-tensile behavior of soilcrete� thecrack pattern likely to occur may modify the evaluated elasticstress field.

Mechanical Analysis

The vertical stress in the ground around the tunnel linearly in-creases with depth. The horizontal stress is proportional to it via aparameter k, kept constant throughout the analysis. The coeffi-cient k was assumed constant and different analyses were carriedout for values of k included in the active-passive range �kakkp�. Structural displacements may strongly influence the valueof k. The proposed approach, intended to be adopted in straight-forward calculations �for instance at a preliminary design stage�,is conservative in this respect, as will be shown in the nextsection.

The vertical and horizontal load components qv and qh actingon the arch are therefore

qv = � · �h − z�

qh = � · k · �h − z� �1�

As previously mentioned, reference is made to an ideal depth hjust for the ease of calculation, which by no means must be in-tended as a true tunnel depth. In fact, because of 3D archingeffects, the real stress values around the advancing tunnel frontare much lower than the lythostatic ones; in the analysis, total andeffective stresses are assumed to coincide �i.e., pore pressure is

Fig. 3. Soil loads on generic cross section of umbrella

zero�.



By writing the TL equation and solving it �see the Appendix�,the normal and shear stresses N and T in any section of the archcan be calculated as

N�x,z� = �Rv2 + Rh

2 cos� − ��

T�x,z� = �Rv2 + Rh

2 sin� − �� 2�

For the sake of simplicity, the definition of all the variables isreported in the Notation.

Is The Approach Conservative and Appropriate?

In the previous sections, explicit reference has been made to theLBT of limit analysis, valid only for proportional loading paths,which is for stress paths in which vertical and horizontal stressesare changed, while keeping a constant ratio �Chen 1975�. Becauseof soil structure interaction and consequential stress changesaround the jet grouted umbrella, the proportionality does notstrictly hold in reality, and therefore the use of LBT may not seemthe best suited for this case. However, it can be shown that, eventhough the true loading path is not proportional �depending on thestiffness of the supporting structure relative to that of soil�, theassumptions adopted in this paper are conservative if an appro-priate choice is made in selecting the stress parameter k in Eq. �1�.In this section, considerations on this aspect will be drawn.

To this aim, let us consider the horizontal component of stressacting on the umbrella �Eq. �1��. Soil structure interaction affectsthe k coefficient, which is of course kakkp �where the sub-scripts a and p stand for active and passive limit conditions�.Depending on the deformed shape of the supporting structure,different values of k will be consequently attained along the arch.Usually, the displacements of the arch �supposed to have a regularshape� are in the inward direction above the haunches up to thecrown, and in the outward direction at the springings �Oreste2007�. Since the inward displacements of the structure cause areduction in the horizontal pressures �k reduces�, while the out-ward displacements give an increase in the horizontal pressure,the qualitative changes in the coefficient k shown in Fig. 4 mustbe expected. The depth at which k maintains the original valuecorresponds to the point in the arch having no horizontal displace-ment. Then, because of this expected behavior, soil structure in-teraction results in larger confinement at the springings and lowerloads above the haunches. Both variations tend to reduce the flex-ural moments within the structure, if compared with the conditionof no soil structure interaction �corresponding to a constant khorizontal load component�. Such a benefic effect is linked to thereduction in eccentricity of the resulting load along the wholearch. Coupled to this effect, there is a variation of the axial loadand therefore of the compressive stresses within the arch. The

Fig. 4. Effect of soil-structure interaction on earth pressure

effect of this variation will be discussed in the following section.



Because of these considerations, the adoption of the approachproposed in this paper �which assumes proportional loading pathsconsistent with the LBT of limit analysis� is conservative if boththe limit conditions of active and passive horizontal load compo-nents are independently analyzed putting, respectively, k=ka andk=kp in two different analyses in Eq. �1�. By using ka, the lowestconfinement is given to the structure along with the lowest pos-sible normal stresses. On the contrary, by using kp, the largestaxial loads are imposed on the structure. As far as eccentricitiesare concerned, both limit conditions may be critical, as shown inthe following section. The two limit analyses then allow us toanalyze the most critical conditions in terms of both eccentricitiesand axial loads. The minimum necessary structural thickness Swill be the largest of the values inferred from the two independentanalyses.

Even though the above considerations ensure the conservative-ness of the approach, they do not allow us to evaluate how con-servative it is. This will depend on the values of k, ka, and kp.Indeed, in the writers’ opinion further refinements on this aspectwould be useless because the uncertainties connected with thetrue shape �far from being the ideal one� and properties of jetgrouted umbrellas �discussed in the last part of the paper� wouldmake it just an academic exercise. The proposed range approachis just meant to be of help in the preliminary design of theumbrella �number of columns and rows�, and in this sense it isappropriate.

Design Procedure

Due to the complexity of the problem, the design procedure isiterative: upon selecting a geometrical configuration �shape,thickness�, the designer has to verify the structural performanceof the supporting structure �i.e., strength, stability�.

In this section, a procedure to design such a supporting struc-ture �considered as a 2D arch with a TL shape� is introduced anda design chart finally proposed, while a checking method isshown in detail in the next section.

The arch diameter is Dtun�y� and the rise at the crown isDtun�y� /2, both varying along the tunnel axis y if jet groutedumbrellas are going to be adopted. In this case, the goal is to letthe thrust pass in the center of each cross section of the arch,therefore having �=Dtun�y� /2.

Introducing the normalized variables

=x

Dtun�y�, � =

z

Dtun�y�and � =

h

Dtun�y�

with the constraint ��1 /2 needed to ensure that the tunnelis underground, the TL implicit equation �in the domain �Dtun�y� /2� become

4 · 3 − 12� · 2 + �12� − 3�

+ 4k · �3 − 12k� · �2 + �6k� − k − 6� − 2�� = 0 �3�

and the tangent equation is

tan = −12 · 2 − 24� · + �12� − 3�

12k · �2 − 24k� · � + �6k� − k − 6� − 2��4�

The stress ratio k has a much larger influence of relative depth �on TL shape �Figs. 5 and 6�. The analyses �see Fig. 5 where k=0.5 and 1��30� clearly shows that the value �=2 defines atransition in behavior, because for �2 depth does play a majorrole in TL shape, while this is not the case for ��2. This result is
in accordance with the typical empirical rule of considering the


value �=2 as the separation between shallow and deep tunnels.For k� �6�−2� / �6�−1� the TL shaped arch enters the quad-

rant of negative �Fig. 6�.Once the shape of the supporting structure is known, the axial

load N is also needed to define the thickness of the arch �shearand flexural moments being zero by definition of TL�, accordingto Eq. �2�. The normalized vertical and horizontal resultants canbe evaluated as

Rv� �� · Dtun�y�2 = −

1

2· 2 + � · −

4� − 1

8

Rh�� · Dtun�y�2 = −

k

2· �2 + k� · � −

6k� − k − 6� + 2

24�5�

The normalized axial load �=N /� ·Dtun�y�2, where N is given byEqs. �2�, is plotted in Fig. 7 for some values of k and �. It can beseen that for typical values of these parameters the curves areroughly monotonic, being therefore reasonable to consider that inthis kind of arche, the maximum axial load �to be used in theevaluation of arch thickness S� is usually attained either at thehinge or at the crown. In the spirit of the proposed approach, then,a preliminary design can be based on the � curves relative to thehinge and to the crown. Fig. 8 shows a design chart where themaximum values of the normalized axial force � are plotted for

Fig. 5. Effect of � on thrust line shape �k=0.5�

Fig. 6. Effect of k on thrust line shape ��=2�



some values of k and �. The critical values of � are attained at thehinge for low values of k �say k1�.

In compressed members, the axial strength per unit thicknessNo=�oS, where �o=compressive strength of the soilcrete, is lin-early related to the minimum thickness S of the section. Then

� =No

� · Dtun�y�2 =�o · S

� · Dtun�y�2 =�o

� · Dtun�y�· � S

Dtun�y��is also representative of the minimum normalized thickness of thesupporting structure.

Then, given soil and soilcrete properties, geometrical dimen-sions, and load conditions, it is possible to design the minimumthickness of the supporting structure �being � in the design chartshown in Fig. 8 directly related to S� having the shape of TL �i.e.,Figs. 5 and 6�. For the given conditions, this will be the mostefficient arch without the adoption of tensile reinforcement.

Checking Procedure

The analysis of the performance of a given jet grouted arch, what-ever the shape, can be done by dividing the arch in smaller parts�for instance each part being the volume included between theaxes of two adjacent columns�. Then the forces generated by andon each element can be computed, and the development and flowof forces throughout the structure can be traced. Forces are di-vided into components and a tabular calculation is usefully madeby adding the components �and computing moments they havewith reference to a chosen origin�. As a general equilibriumthumb rule, the sum of all the forces acting on a part of thestructure are represented by a resultant which is a component ofthe forces acting on the remaining part.

The checking method is iterative �Fig. 9�, needing to be re-peated for each arch of the jet grouted umbrella, corresponding toa given value of y. All numerical tests have shown that conver-gence is very fast.

Referring to the centroid �evaluation point� of the two sides ofeach element, the internal forces can be computed. The flexuralmoment M�x ,z� is computed on any user-defined arch shape witha given Dtun�y� span, according to the left side of Eq. �13�. Theunknown H is given by Eq. �14� as a function of the geometricparameter � �ordinate of the TL at the arch crown, see the Appen-dix�. The horizontal Rh and vertical Rv internal force resultants

Fig. 7. Normalized axial load � versus for some values of param-eters k and �

are derived by the flexural moment as





Rv�x� =�M�x,z�

�xand Rh�z� =

�M�x,z��z

Then the axial N and shear T forces are evaluated along the archaccording to Eqs. �2�, in which tan �x ,z�=Rv /Rh=tangent to theTL and tan ��x ,z�=dz /dx=tangent to the arch.

Once the internal forces are known for each value of the pa-rameter �, the thickness S of the arch can be checked according toa no tensile material M-N interaction diagram �where it is at leastnecessary that the internal force resultant is acting inside the crosssection� at every evaluation point.

Introducing the normalized variables

m =6 · M

S · No=

6 · M

S2 · �o

and

k for some values of parameter �

Fig. 10. Thrust line and stress filed visualization

Fig. 8. Normalized axial load � versus

Fig. 9. Iterative procedure



n =N

No=

N

S · �o

the yield surface of the rectangular cross section in the �m ,n�plane is formed by two parabolic arches

�m� � 3 · n�1 − n� �6�

In the elastic field, two situations can be found �see Fig. 10�: if theinternal force resultant is inside the kernel of the cross section,there should be no cracks because the section is fully compressed,while if the resultant is outside the kernel �however inside thecross section�, some cracks are expected �that is, the section is notfully compressed� but the overall stability is still granted. Theresultant position can be easily checked by visual inspection: theTL is the envelope of internal axial load resultants and the eccen-tricity M /N of the axial load is equal to the distance between thearch axis and the TL. As previously stated, if the influence ofwater is overlooked �i.e., zero pore pressure or total stress analy-sis� the TL, and consequently the M /N ratio, do not depend onsoil unit weight � or on a proportional coefficient like �cc, whilethe values of N and M are strictly dependent.

The M-N interaction diagram of a fully compressed elasticrectangular cross section is defined by two intersecting lines, onecorresponding to the stress limitations ��0 �allowable stress�and ��0 �no tension�, respectively

�m� � �1 − n�

�m� � n �7�

An extension of this interaction domain within which stability isstill granted but cracks are allowed to develop �with rectangularcross sections not fully compressed� is formed by two parabolicarches

m � n�3 − 4 · n� �8a�

These three normalized interaction diagrams are shown togetherin Fig. 11. If n�0.5 or m /n�1 �that is e=M /N�S /6� thencracking cannot occur. In any case, the stress state must respectthe constrain m /n�3, which means that the eccentricity of theresultant force cannot exceed the value e=M /N�S /2.

A final check must be carried out at the base frictional re-straints: according to the classical shear theory �Timoshenko andGoodier 1987�, the shear stress can be evaluated in the base rect-angular section as �=1.5· �H� /S: if frictional failure is attained at

Fig. 11. Normalized interaction diagram �m ,n� for rectangular crosssection

the base restraint ��max� then the arch is supposed to be un-



stable and therefore to fail. Reminding us of the true 3D shape ofthe jet grouted umbrella, this condition is very conservative, be-cause a global structural failure caused by a local shear failure ina single section is unrealistic.

The checking procedure can then be carried out as follows:after deciding which critical situation should be analyzed �eitherthe full elastic compression condition, Eq. �7�; or the crackingcondition, Eq. �8a� and �8b�; or the yielding condition, Eq. �6��the parameter � is changed until the assumed condition is satisfiedat each evaluation point within the arch. The above proceduremust be carried out in the range �Dtun�y�−S� /2� �Dtun�y�−S� /2, in any case representing the constrain for the TL of beinginside the structure at the crown.

If a uniform thickness S is required, the minimum necessarythickness S of the arch is the maximum thickness S�� foundamong the evaluation points at each value of parameter �. This isa nonlinear minimization problem and the procedure is repeated,changing the geometrical parameter � until the minimum valueSmin of S�� is found.

The three conditions �Eqs. �6�, �7�, �8a�, and �8b�� can beexplicitly expressed in terms of S, i.e., Eq. �8a� becomes

S � 2 ·�M�N

+4 · N

3 · �0�8b�

This equation �Eq. �8b�� holds as far as S�6�M� /N; larger valueswould result in a fully compressed section �Eq. �7��. To speed upthe convergence, a new estimate of �new can be done subtractingthe average eccentricity �computed as the mean value of the cal-culated maximum and minimum eccentricities �M /N�max

�M /N�min� found at the previous step in the �prev estimation. Thisprocedure can be roughly seen as an eccentricity minimizationprocess. The new value �new is then

�new = �prev −�M/N�min + �M/N�max

2�9�

Probabilistic Analysis of Jet Grouted Umbrellas

The jet grouted umbrella works as an arch as far as the single jetgrouting columns are overlapped by a thickness sufficient to en-sure mutual interaction. In the ideal case of perfect columns, sucha thickness depends on the column diameter, initial overlapping,and opening angle of the frustum of the cone �for each jet grout-ing span, the opening angle must be such to allow the treatment ofthe following span, see Figs. 1 and 2�. In practice, columns over-lapping �and therefore arch thickness� strongly depend on columndefects, and the overall structural performance must be consid-ered by taking into account the possible variations of both diam-eter and axis inclination. Furthermore, mechanical properties ofsoilcrete have to be carefully calibrated.

In a previous work �Flora et al. 2007� the authors carried out aseries of Monte Carlo simulations to determine the influence ofpossible defects within the columns in the umbrella performance,starting from the analysis of site data that some of the authorscontributed to publish in the past �Croce and Flora 2000�.

These defects are the reason why jet grouted columns are farfrom being perfectly cylindrical, homogeneous bodies. Therefore,the real shape of jet grouted umbrellas is not a regular frustum ofcone, and overlapping decreases along the span more dramaticallythen it would in the ideal case of columns having no defects.
There is a critical length of the span after which structural conti-


nuity is difficult to obtain, depending on the characteristics of thejet columns. Defects in axis entation play a major role in thiscritical length, but diameter variations are also relevant. The criti-cal length is also obviously influenced by the initial columns setup, becoming larger as jet grouted columns diameter �Dcol� andinitial overlapping thickness become larger.

Defects of jet columns are usually overlooked by practitioners.However, structural performance is strongly influenced by thesedefects, and the only possible way to take them into account is theadoption of a probabilistic or semiprobabilistic approach. As faras the diameter is considered, the experimental evidence shows�for instance Croce et al. 2004� that normal probabilistic distribu-tions are found in practice along a single column. Then, the sta-tistical interpretation of site data can be done by using twovariables: the average diameter and the coefficient of variationCV�Dcol�=CVd. The column axis may deviate from its theoreticalposition due to a number of reasons linked to soil properties andtechnological limitations. For the sake of simplicity, it is conve-nient to quantify this deviation by using two additional angles �and �, which for each column represent the components of devia-tion in the horizontal and vertical plane, respectively. Such devia-tions must be taken into account, as the overlapping of adjacentcolumns and the continuity of jet grouted structures strongly de-pend on them. Subhorizontal columns are more likely to deviatefrom their ideal position than vertical columns. It is assumed thatthe column axis deviations in the vertical and horizontal direc-tions are normally distributed and have identical standard devia-tions, thus neglecting any preferential trend in the deviation fromthe ideal column’s position. The two deviations are also un-coupled, in the sense that they have no mutual influence. Sincethe average value of deviation is zero �ideal position of the col-umn�, in the following the standard deviation �� ,�� will be usedinstead of the coefficient of variation.

Furthermore, physical and mechanical properties of soilcretemay vary within the column both along the axis and along theradius. Even though the differences along the radius are muchmore difficult to spot, a general variability must be expected.Soilcrete can be assumed to behave similarly to a low qualityconcrete, with mechanical properties strongly depending on thetreated soil. The soilcrete uniaxial compressive strength �o is as-sumed to have a lognormal distribution, depending on the meanvalue and the variance of the logarithm of �o �Croce et al. 2004�.From a statistical point of view such a distribution indicates thatsoilcrete is not mechanically homogenous, and as a consequencethe scatter in values of compressive strength is due not only to arandomness unavoidable when this technique is used. Then, it isreasonable to assume a unique, deterministic value for soilcretewith a given level of confidence.

As well known from laboratory testing and site evidence, jetgrouting is more effective in sand than in clay both in terms ofdimensions �i.e., for a given set of treatment parameters largercolumns are obtained� and mechanical properties �i.e., a largervalue of �o is obtained�. In the simulations, therefore, differentaverage values will be assumed.

Sets of Simulations

The proposed method is applied to verify the simple case of a jetgrouted umbrella in a semicircular shaped tunnel having Dtun�y=0�=8 m. A length of each frustum of cone of ymax=12 m wasassumed, with an overlapping among adjacent frustums of coneof 3 m, as typical in Italian practice.

The geometrical, mechanical, and statistical parameters as-



sumed for both jet grouting columns and soils are reported inTables 1 and 2. The parameters pertaining to jet grouted columnshave been selected as typical of monofluid jet grouting, based onthe available experimental evidence �Croce et al. 2004� and theauthors’ experience. As an example, 12 combinations of � and khave been considered, the latter ranging between the ka and kp.Calculations in the following have been carried out using thevalue �o,5%, with a 95% level of confidence typical for construc-tion materials �e.g., concrete�. The initial overlapping betweenadjacent columns at the beginning of the span is 80% of thecolumn diameter.

Due to variations in position and diameter of the columns, thegeometrical shapes generated by Monte Carlo simulations are im-perfect, in the sense previously described, and therefore have nei-ther semicircular shape nor constant overlapping thickness S ineach cross section. Each simulation gives rise to a different shape,even starting from the same set of input data in terms of meanvalue and coefficient of variation of column’s diameter CVd orstandard deviation of angular deviation �� ,��. Due to the largeamount of different combinations to be analyzed via the MonteCarlo procedure, a simplified analysis criterion had to be adopted�Flora et al. 2007�: in this case, the rather irregular shape wasassumed to be equivalent to a semicircular arch having a cross-sectional constant thickness S5%. Such an overlapping thickness�S5%�, which changes along tunnel axis, is the value having a 95%probability of being exceeded in the arch considered. The as-sumption of a certain probability to be exceeded is arbitrary andhas no intent of being a general rule. Simply, it is a referencecase, and the procedure can be applied with any other choice. Fig.12 shows the normalized value S5% /Dtun�y� versus the normalizedy /ymax in the case of sandy soil. As expected, the lack of overlap-ping �geometrical problem� is basically governed by defects in theposition of the columns’ axis, and the variation of column diam-eter plays a minor role. If �� ,��=0 �ideal position�, overlappingis always granted, while if �� ,��=1 then columns start to looseoverlapping in the range 0.4�y /ymax�0.6 �around midspan� de-pending also on Dcol and CVd.

Table 1. Assumed Geometric, Mechanical, and Statistical Parameters forJet Columns

Soil

Geometrical Statistical Mechanical

�Dcol

�m��°� CVd �� ,�� CV�0

�0,5%

�MPa�

Sand 0.60 5.71 6.70

1 0.00 0.0

2 0.05 0.1

4 0.10 0.5 0.1

6 0.20 1.0

Clay 0.40 3.81 2.74

Table 2. Soil Properties Adapted in Analyses

Soil�

�kN /m3��°�

c�MPa� ko ka kp k

Sand 35 0.43 0.27 3.70 ka

18 0 ko

Clay 28 0.53 0.36 2.78 kp

Note: Horizontal stresses have been computed as k ·�v.



Safety Assessment

The safety assessment requires many uneasily definable param-eters. The structural stability under soil loads can be evaluated byfully locating at least one TL within the arch profile and satisfyingstress requirements. In such a way a conservative estimate of theminimum thickness Smin needed for the equilibrium of the struc-ture is obtained. If the actual thickness S5% �capacity� is greaterthan Smin �demand� throughout the whole span �that is, for anyvalue of y�ymax�, then the structure is safe. Actually, since sub-sequent umbrellas have a mutual overlapping, continuity is notstrictly needed at the end of the span. The ratio S5% /Smin can beassumed as a measure of safety. The greater this ratio, the largerthe loads that can be carried by the structure.

A minimum uniform average thickness of the jet grouted um-brella necessary for equilibrium can then the found with the pro-

Fig. 12. Normalized S5% /Dtun�y� v

Fig. 13. Normalized S /Dtun�y� versus y /



posed method. The results of these structural analyses aresummarized for sand in the form of normalized charts�S�y� /Dtun�y� versus y /ymax� in Figs. 13 and 14 for some values ofk and �. The same results can also be plotted as continuous nor-malized response surfaces of minimum structural thicknesses�S�y� /Dtun�y� versus k, � at y=0� in Figs. 15�a and b� for sand, inthe range of values of the statistical parameters considered. Theresponse surfaces clearly indicate that if cracking is permitted,lower values of Smin thickness will usually result. Furthermore,for large values of � and k�1, a cracked condition is usuallyreplaced by a fully compressed condition as the critical condition.

This is because the larger the �, the more uniform �but withlarger values� the load distribution within the arch. Furthermore,the closer to one is k, the smaller the gap between horizontal andvertical load components �i.e., the TL tends to become circular�.

normalized y /ymax in case of sand

case of sand �frictional restraint check�

ersus

ymax in



Therefore, because of this evidence, larger values of � and ex-treme �small or high� values of k result in larger values of Smin,because axial load �highly related to � and slightly to k� andeccentricity of the internal force resultant �highly related to k�both increase. Thus full compression condition is more conserva-tive than the cracked condition �Fig. 15�. In the present numericalinvestigation the cracked condition is fully satisfied only in thecase of �=1 and k=0.43.

It is pointed out that in the case of TL shaped arches, the effectof � is much more relevant than that of k, because, the archthickness is related to the thrust and not to the eccentricity of theload �which is virtually zero�.

The respect of shear strength at the arch base frictional re-straint is generally satisfied, becoming less and less crucial as �reduces and k tends to one �i.e., the horizontal reaction reduces�.

It is useful to distinguish among the three following possiblecases �Fig. 16�:1. For any value of y�ymax, the columns are always overlapped

and stability condition is satisfied; both geometric continuityand structural performance are guaranteed for the wholespan;

2. The columns are overlapped for any value of y�ymax, butthe stability condition is satisfied up to a value y1ymax;

Fig. 14. Normalized S /Dtun�y� versus

Fig. 15. Normalized response surface at y=0 for sa



geometric continuity is guaranteed for the whole span, butthe structure fails to carry soil loads for y�y1; and

3. The columns are overlapped to a value y2ymax, and thestability condition is satisfied up to a value y1ymax, whichis always y1y2. Then, neither geometric continuity norstructural performance are guaranteed to the end of the span.

Case �1� is the ideal situation, in which the umbrella’s struc-tural performance is guaranteed for the whole span. Case �2� is acandidate case in which the designer can also use steel reinforce-ment to increase the structural capacity of the umbrella that is notable to carry the loads on the whole span. In Case �3� the designercan decide to lower the span length ymax to y2 but will also needto increase the opening angle of the cone to allow for the over-lapping of adjacent vaults. In both cases �1� and �2� a furtheralternative solution may be the use of a multilayered umbrella�Flora et al. 2007�.

It is important to remember that the clear span of a singleumbrella cone to be considered is smaller than ymax, because theusual overlapping of adjacent vaults is about 2–3 m, and thatarea is not excavated until a new successive umbrella is realized�Fig. 1�.

ax in case of sand �cracked condition�

� cracked condition; �b� fully compressed condition

y /ym

nd: �a



Conclusions

In this work a simple method to analyze supporting nonclosedstructures of both deep and shallow tunnels has been presented.The procedure was conceived to be adopted in structures whosegeometrical and mechanical properties cannot be perfectly as-sessed, like jet grouted umbrellas. In such cases, FEM analysesmay not be the best design tool, as they may just give the illusionof being more refined. Simpler approaches like the one describedin this paper may be of some help.

In the first part a direct method to design an optimal shape ofthe cross section of supporting structures and define the minimumstructural thickness was shown. In the second part, an iterativeprocedure to verify the stability of an existing supporting structureor to design the minimum structural thickness of the cross sectionfor a given structure with a predefined shape was described.

The lower bound plastic theorem was used, and considerationswere carried out on the limits of the approach. It was shown thatby carrying out two analyses with the extreme values ka and kp ofthe horizontal pressure coefficient k, conservative solutions arefound considering both bending moments and axial loads withinthe structure. The value k=kp is usually the most critical one.However, it must be pointed out that this case has little physicalmeaning, and may then be too conservative of a choice.

Geometrical nonlinearity and deflection dependent earthpressure distributions do play a role in the performance of a tunnelsupporting arch. However, in the writers’ opinion the real shape ofa jet grouted umbrella �far from being a perfect arch� has to befirst considered in the calculation if soil-structure interaction hasto be taken into account. This activity is under course.

In the hypotheses adopted in this work, the semicircular shapeis not suited to support all the considered soil loads. The bestpossible shape has the equation of the thrust line itself, whichwould ideally bring it to zero eccentricity, thus corresponding to aperfectly centered axial load in the cross section. Because the loadvariation along the arch is not very large, the cross section in thiscase is always fully compressed.

Static analyses were carried out assuming a simplified 2D archscheme. It is argued that the validity of this approach is notaffected by the hypothesis on the stress field around the tunnel,and that the 2D arch scheme is realistic when the supportingstructure has no flexural capacity in the longitudinal tunneldirection, as is often the case in practice.

By combining geometric and static considerations, a simple yetconservative method to evaluate tunnel linings structural

Fig. 16. Definition of different curves of design chart

performance �Smin thickness� has been proposed. This method,



coupled with an analysis of structural demand �S5%� carried outvia a probabilistic approach with reference to jet groutedumbrellas with defects, has allowed us to plot design charts thatmay help in the design of temporary supporting structures. It isshown that column defects mainly influence the supportingstructure stability, and this method allows us to plan the number ofjet grouted columns and their initial overlapping, also inmultilayered layouts, to be adopted with a given level ofconfidence once the tunnel average diameter is given.

Appendix

The TL equation �where the TL is the representative line of anideal structure that loaded with the prescribed components has noflexural moment or, that is the same, is the line representing theinternal forces resultant� can be derived setting the bending mo-ment equation in a generic point P�x ,z�, the point of the TL,equal to zero

M�x,z� =�x��

x

qvdx�dx +�z��

z

qhdz�dz = 0 �10�

Four boundary conditions are needed to carry out this integration.In the left restraint the vertical reaction �vertical resultant Rv inthe origin� =V and the horizontal reaction �horizontal resultant Rh

in the origin�, =H, while the flexural moment in both hinges iszero

� �M�x,z��x

�x=0

z=0

= �Rv�x=0 = V � �M�x,z��z

�x=0

z=0

= �Rh�z=0 = H

M�x,z�x=0

z=0= 0 M�x,z�x=Dtun�y�

z=0= 0 �11�

Based on equilibrium and symmetry considerations, the verticalreaction V in the hinge is

V = −�0

Dtun�y�/2

qv�x�dx = � ·�Dtun�y� − 4 · h�

8· Dtun�y� �12�

where qv is supposed to vary linearly along the support ceiling, so

z�x� =Dtun�y�

2− Dtun�y�

2− x

�see Fig. 3� and Dtun�y�=diameter of the tunnel at the longitudinalcoordinate y.

From this point on, for the sake of simplicity only the lefthalf of the arch �x�Dtun�y� /2� is considered, due to symmetryconditions.

The TL implicit equation �MP=0�

−�

6· x3 +

h · �

2· x2 +

Dtun�y� · �Dtun�y� − 4 · h� · �

8· x −

k · �

6· z3

+k · h · �

2· z2 + H · z = 0 �13�

is still undetermined, because the H reaction is unknown. It ismore convenient to have H depending on the geometric parameter�, the ordinate z of the TL at midspan: x=Dtun�y� /2. Therefore,
substituting x=Dtun�y� /2 and z=�, Eq. �13� becomes


H = −�Dtun�y�3 − 3 · Dtun�y�2 · h + 4 · k · �2 · �3 · h − �� · �

24 · �

�14�

The TL equation, putting Eq. �14� into Eq. �13�, is independent of� �or a proportional coefficient like the fictitious pressure coeffi-cient �cc�, as previously noted.

The last step is to evaluate the axial N and shear T forces alongthe arch. Given tan �x ,z�=Rv /Rh as the tangent to the TL andtan ��x ,z�=dz /dx as the tangent to the arch, they are given byEqs. �2�.

If the arch has a TL shape then =� and shear forces are zero.

Notation

The following symbols are used in this paper:CVd � coefficient of variation of diameter of jet

grouted columns;CV�o � coefficient of variation of soilcrete uniaxial

compressive strength;c � cohesion;

Dcol � diameter of jet grouted columns;Dtun � diameter of frustum of cone �umbrella� and

arch;e � eccentricity of axial resultant force;

H � horizontal reaction at base �unknown�;h � depth of tunnel base from ground level;k � horizontal over vertical stress in soil ratio;

ka � active earth pressure coefficient;ko � coefficient at rest;kp � passive earth pressure coefficient;M � flexural moment;m � normalized flexural moment;N � axial force;

No � axial strength of soilcrete per unit depth;n � normalized axial force;

P�x ,z� � coordinate of generic point of arch;qh � horizontal load component on arch;qv � vertical load component on arch;Rh � horizontal force acting in arch;Rv � vertical force acting in arch;S � minimum effective thickness due to

overlapping;Smin � minimum structural thickness according to

strength check;S5% � overlapping between any couple of adjacent

columns having 5% level of confidence;S5%,tot � overlapping between any couple of adjacent

multilayered columns having 5% level ofconfidence;

T � shear force;tan �x ,z� � Rv /Rh-tangent to thrust line;tan ��x ,z� � dz /dx-tangent to arch;

V � vertical reaction at base;x � abscissa of point �element� of arch;y � tunnel axis abscissa;

ymax � frustum of cone �umbrella� length;y1 � abscissa y so that strength conditions are

satisfied for yy1;



y2 � abscissa y so that overlapping conditions aresatisfied for yy2;

z � ordinate of point �element� of arch;z�x� � relationship to be inserted in load components

equation;� � opening angle;� � soil unit weight;� � ordinate of thrust line at crown;v � normalized axial force �also representative of

normalized thickness�;X � normalized abscissa of point �element� of

arch;�new � new estimation of ordinate of thrust line at

midspan;�prev � previous estimation of ordinate of thrust line

at midspan;� � normalized ordinate of point �element� of

arch;� � normalized depth of tunnel base from ground

level;�cc � convergence-confinement method coefficient;

� � deviation angle in yz plane;� � normal stress to evaluate limit shear stress

given by Mohr–Coulomb criterion;�f � fictitious stress in soil;�h � horizontal stress in soil;�o � soilcrete uniaxial compressive strength;

�o,5% � uniaxial compressive strength with 95%confidence level;

�soil � stress in soil;�v � vertical stress in soil;

�� ,�� standard deviation of � and � angle deviation;� � shear stress at base;

�max � limit shear stress value given byMohr–Coulomb criterion;

� � friction angle of soil; and� � deviation angle of axis in xy plane.

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Simple method for the design of jet grouted umbrellas in tunneling