UNIVERSIT DEGLI STUDI DI NAPOLI FEDERICO II

SCUOLA POLITECNICA E DELLE SCIENZE DI BASE

CORSO DI LAUREA MAGISTRALE IN INGEGNERIA STRUTTURALE E GEOTECNICA

MASTER IN EMERGING TECHNOLOGIES FOR CONSTRUCTION

Lecture notes of the Course on

Tunnelling and underground structures

(Dr Emilio BILOTTA)

about

GROUND MOVEMENTS INDUCED BY TUNNELLING

written by Eng. Andrea Paolillo

2014-15

CONTENTS

INTRODUCTION ......................................................................................................... 1

1 Tunnelling-induced soil movements ................................................................... 2

2 The prediction of ground movements due to tunnelling .................................... 3

2.1 Empirical methods .................................................................................................................................... 3

2.2 Theoretical solutions .............................................................................................................................. 11

2.3 Numerical analyses ................................................................................................................................. 11

3 Tunnelling induced soil-structure interaction ................................................... 15

3.1 Building deformation parameters ....................................................................................................... 15

3.2 Field data and experimental results .................................................................................................... 16

3.3 Equivalent solids for studying tunnelling induced soil-structure interaction ............................ 17

4 Structural damage evaluation due to tunneling ............................................... 22

4.1 Damage criteria ....................................................................................................................................... 22

4.2 Damage evaluation process .................................................................................................................. 26

REFERENCES ............................................................................................................. 30

1

INTRODUCTION

The always increasing demand in greater and quicker connections routes in urban area, in the last

years, has determined the necessity to use the underground environment for new transportation

lines. Underground tunnels started to be dug in many of the bigger cities, leading to a large number

of deep and shallow excavations inducing not negligible effects on the preexisting buildings.

The ability to predict the tunneling induced settlements and the associated impact on the structures

represents a key point to estimate potential damages and to design protective measures, when

needed. Prediction of displacements induced on a building by tunnel excavation in soft ground is a

typical soil-structure interaction problem. Building stiness and weight are expected to alter the

displacement eld that would be caused by tunneling operations in so-called greeneld conditions.

This work presents a literature review of the methods usually used to study the displacement field

due to shallow tunnel excavation in soft soils. Initially, phenomenology of tunnelling induced

movements in greeneld conditions is described. A quick review of empirical, analytical and

numerical methods commonly used to predict greeneld displacements is given. Then, a description

of the eects of soil-structure interaction is provided and the procedures used to consider the

presence of the structure are provided. Finally, the methodology commonly employed to assess the

expected damage on a building is presented.

2

1 Tunnelling-induced soil movements

The displacements field induced by a real shield tunnelling process in soft ground, is strongly

affected by many phenomena, sketched in Figure 1 such as:

1. Face deformations due to stress release at the excavation front. It can be minimized by

application of a controlled face pressure, using EPB or slurry shield excavations;

2. Movements induced by shield in advancing, due to the over-cutting, useful to reduce friction

between shield and ground. They are strongly depend by the overcutting edge thickness and

by any steering problems in maintaining the alignment of the shield;

3. Tail void loss due to the physical gap between the lining and the shield tail. This can be

reduced by using grout injections;

4. Lining deflections due to the ground loading, generally smaller than the other movement

components if the lining is stiff enough;

5. Long term deformations due to consolidation process in fine grained soils. Can be very

important especially in soft clays. It results from the fact that the construction process

changes the stress regime locally around the tunnel. The dissipation of the pore pressure

changes induced by the undrained excavation is a primary source of time dependent

settlement. Another source of delayed settlements may be the change of pore pressures

due to a draining effect to the tunnel in case of permeable lining.

Fig. 1- Ground movements induced by tunnel excavations (after Attewell et al., 1986)

3

2 The prediction of ground movements due to tunnelling

Many methods were implemented to quantify tunnelling induced ground movements: from the

empirical and analytical methods to predict displacements in green-field conditions, going through

more sophisticated numerical models (e.g. Finite Element Methods), up to the rarely used physical

models, such as centrifuge tests, reproducing in small scale the in situ situation.

For the numerical analyses a sufficiently accurate constitutive model for the soil is required and the

tunnelling process has to be represented with an acceptable degree of accuracy.

2.1 Empirical methods

Advancement of the excavation front in greenfield conditions induces a settlement trough at the

ground surface, diagrammatically sketched in Figure 2 for the simple case of a single tunnel with

straight axis at constant depth z0. The white arrow in the figure indicates the direction of tunnel face

advancement. It is widely accepted that a transverse section of the Greenfield settlement trough

can be described with good approximation by a reversed Gaussian curve. Thus, the analytical

expression of the transverse settlement trough shown in Figure 3 is:

Fig. 2- 3D greenfield settlement trough (from Attewell et al., 1986).

() = exp {2

22} (1)

4

Where:

wmax is the maximum settlement above the tunnel axis;

ix is the distance between the inflection point of the curve, where the trough has its

maximum slope, and the central axis of the tunnel; it separates the sagging from the hogging

zone of the curve;

w(x) is the settlement at distance x from the tunnel axis.

Assuming the tunnel face is at sucient distance ahead of the examined section, no more

settlements develop for further front advancement. This also implies that, referring to Figure 2,

starting from a certain distance y behind the excavation front, settlements are constant for a given

x, implying that the longitudinal section of the settlement trough is horizontal.

Fig. 3- Transverse settlement trough.

The volume per unit length of the surface settlement trough VS is numerically equal to the area

underlying the Gaussian curve in Figure 3. It results:

= ()

= 2 (2)

It is strictly dependent by the ground affected by the excavation. In grounds with low permeability,

such as stiff-clays, with initially undrained response to the excavation process (which means not

allowed changing in volume) the volume of settlements trough has to correspond to the excess

excavated volume of ground to the theoretical volume of the tunnel. It is usual to define the extra

excavated ground as Volume Loss, given by:

(%) =

2

4

(%) (3)

5

Where D is the outer tunnel diameter. VL is usually defined as a proportion of theoretical tunnel

volume per unit length, expressed as a percentage of it.

Combining (2) to (3), the transverse settlements profile can be expressed in terms of Volume Loss

as:

() =

2

2

4

2

22 (4)

It shows that for a given tunnel diameter D, the settlements profile only depends by the Volume

Loss VL and by the trough width ix, two crucial parameters that need to be defined to know the

settlements field induced by tunnelling.

According to Attewell and Woodman (1982) and Attewell at al. (1986) the profile of settlements in

longitudinal direction can be represented by the Cumulative Gaussian Distribution function (or

complementary error function)

=

1

2[1 erf(

2)] =

1

2 (

2) (5)

The shape of longitudinal displacements curve is shown in Figure 4. It indicates the minimum and

maximum values of the longitudinal settlements reached respectively at y=- (ahead the tunnel

face) and y=+ (behind the tunnel face), while above the tunnel face (y=0) it is w=wmax/2.

For completely defining the longitudinal settlement profile, it is important to know about the curve

width, defined by the iy value. Attewell at al. (1986) compared the magnitudes of ix and iy for a range

of case studies; they observed that usually ix is bigger than iy (the transverse settlements troughs

were longer than the longitudinal ones); on the basis of field data coming from the tunnel

construction of the Jubilee Line Extension beneath St. Jamess Park in London, Nyren (1998)

observed the same behavior translated into the ratio ix/iy=1.3. However, despite this discrepancy, it

is common practice to consider ix=iy=i.

Fig. 4- Longitudinal settlement trough.

6

Attewell et al. (1986) assumed that for open face tunnelling, the settlement above the tunnel axis

(x=0) is 50% of the maximum settlement reached behind the tunnel face, while for closed face

tunnelling, where significant face support is provided, the displacements ahead the tunnel face

reduce significantly. Mair and Taylor (1997) concluded that for closed face tunnelling the

settlements above the tunnel axis is 25% - 30% the maximum settlement; this leads to a longitudinal

displacements curve translated as shown in Figure 5.

Fig. 5. Longitudinal settlement profile for open face and closed face tunnelling after Mair & Taylor (1997)

Based on the analyses of tunnel induced displacements in the United Kingdom, Craig & Muir Wood

(1978) stated that, for each point of observation, depending on the ground, for open face tunnelling,

nearly the 80% - 90% of the settlement is reached when the face is one to two times the tunnel

depth. Table 1 indicates the conclusions reached by Craig & Muir Wood on the development of the

settlement profile.

Tab. 1- Development of settlement profile (Craig & Muir Wood, 1978)

In the transverse direction to the tunnel construction, the surface (and subsurface) horizontal

displacements can be estimated using various assumptions. The simplest is to consider that ground

movements are radial, i.e. directed toward the tunnel axis

() =

0() =

0 exp {

2

22} (6)

7

Figure 6 clearly shows as the maximum horizontal displacements occur at the point of

Inflection of settlements trough (x=ix). Simply by derivation of the horizontal displacement the

horizontal strain can be calculated

() =

() (7)

Fig. 6- Horizontal surface displacements and strains in transverse direction together with settlement trough

It is possible to observe the development of a compression zone between the two points of

inflections ix and iy and of a tensile zone outside them; the maximum compression strain develops

at x=0, while the maximum tensile strain develops at x=3 ix

Based on experimental evidences, Taylor (1995) proposed that the vector of displacement does not

point to the tunnel axis but to a point below the tunnel axis as shown in Figure 7 b).

() =

(1+0.175

0.325)0

() (8)

Fig. 7- a) Vector of displacement pointed to the tunnel axis; b) Vector of displacement pointed below the tunnel axis Taylor (1995)

8

All the above defined displacements and strains strictly depend on the trough width parameter i

and the Volume Loss VL, it is so necessary to dwell on them.

The trough width parameter i is related to the size of the trough, it mostly depends on the type of

soil and is largely independent of the tunnel construction technique.

Based on 19 case histories for cohesive grounds and on 16 for coehsionless soils (all in the United

Kingdom), OReilly & New (1982) showed a linear dependence between the trough width parameter

i and the tunnel depth z0

= 0 (9)

The Authors pointed that k can vary between 0.7 to 0.4 going from soft to stiff clay. In case tunnels

is dug in ground which comprises layers of coarse and fine graded soil Selby (1988) and New and

OReilly (1991) suggested that the trough width parameter can be estimated as:

= (10)

Field observations of surface settlements profiles of stratified soils where sand is overlain by a clay

layer indicate wider profiles than would be obtained if the tunnel is only in sand (according to the

equation). Less evidence is available of sand overlaying clay, where the narrowing predicted by the

equations has not been observed (e.g. Grant and Taylor, 1996).

In urban areas there is often the need to estimate the settlements below the ground surface at a

generic depth z from the ground surface. Gaussian profile can also reasonably approximate the

subsurface settlement profiles, provided that the narrowing of the settlements trough with depth

is well modelled.

= (0 ) (11)

According to Mair et al. (1993), the parameter K is not constant with depth, to get a more realistic

wider subsurface trough at depth they proposed for clay

=0,175+0,325(1

0)

(1

0)

(12)

Moh at al. (1996) have proposed a slightly different formulation for i (z)

() = (

2) (

0

)0,8

(0

0)

(13)

9

With m=0.4 for silty sand and m=0.8 for silty clay.

Finally Bilotta and Russo (2012) on the basis of the formulation of Moh et al. (1996) have proposed,

for the ground interested by the low part of Line 1 of the Naples Metro a parameter i:

= (0

)

(14)

With b = 0.8 e m = 0.2.

The volume loss is a measure of total disturbance of the ground caused by tunnelling.

It is usually referred to as a percentage of the excavated volume of tunnel VT

(%) =

100 (15)

For over consolidated clays, Dimmock and Mair proposed:

(%) = 0.234.80( 0.2) (16)

Where LF=N/Nc

If the excavation occur in undrained conditions, such as in clay, the volume of the ground above the

tunnel Vs (volume of the settlement trough) can be considered equal to the VL. Instead in coarse

grained soils the volume of the settlements trough VS is generally lower than the volume loss VL,

due to the dilatancy as shown in Figure 8.

10

Fig. 8- Relation between Volume loss and Volume of settlement trough in Coarse-grained soils, (Attewell, 1977).

The value of V ', essentially depends on the type of soil and the method of excavation. For excavation

with hydroshield in loose soils below the water table and with an adequate conduct of operations,

the value of V 'should be not greater than 0.5%. In the case of clays of reduced consistency with

excavation shield balanced pressure values were up to 3%.

In Table 2 is reported the suggested volume loss values based on engineering judgment and

experience from previous project in similar ground.

Tab. 2- Suggested volume loss values

11

2.2 Theoretical solutions

A number of closed form solutions have been proposed to calculate the displacement eld induced

by tunnel excavation in greeneld conditions. Most of the proposed solutions have been obtained

assuming axial symmetry about the tunnel axis, which is seldom realistic especially for shallow

tunnels. The Sagaseta (1987) method is based on incompressible irrotational uid ow solutions.

The basic case considers the action of a point sink which extracts a finite volume of soil at some

depth h below the top surface. The method has proven to yield settlement troughs much wider than

those predicted by the Gaussian relation but similar maximum settlement. Verruijt & Booker (1996)

extended the solution of Sagaseta by considering the ground compressible (0.5) and taking into

account tunnel ovalisation using a parameter . They nd out that imposing an oval deformed shape

to the tunnel boundary results in settlement troughs in acceptable agreement with those predicted

by the empirical relations and observed in the eld. Loganathan & Poulos (1998) also proposed an

approach based on tunnel boundary radial contraction in an elastic-plastic medium. Predictions with

this method give higher than maximum eld settlements and a wider trough.

2.3 Numerical analyses

Empirical relations presented in Section 2.1 give results in good agreement with ground recorded

data in the following conditions:

Greeneld conditions. The presence of pre-existing structures may affect the displacement

eld induced by tunnelling.

Single tunnel or multiple tunnels without interactions.

Homogeneous ground conditions.

Short term conditions. In ne grained soils displacements evolve with time due to

consolidation.

Moreover a good judgment is required in the selection of an appropriate value of volume loss, for

this reason the method has a particular high practical value in cases where previous tunneling in

similar ground conditions and with similar construction techniques has been performed.

If one of the above conditions is unsatised, prediction of tunnel induced displacements must be

performed with numerical methods. The benefit of the numerical methods over analytical or closed

form solutions are considerable (Potts and Zdravkovic, 2001):

Simulate the construction sequence;

Deal with complex ground conditions;

12

Model realistic soil behavior;

Handle complex hydraulic conditions;

Deal with ground treatment;

Account for adjacent services and structures;

Simulate intermediate and long-term conditions;

Deal with multiple tunnels;

It is worth to recall the techniques most commonly used to simulate tunnel excavation in numerical

analyses.

2.3.1 2D Analyses

Although one of the major peculiarities of the tunnelling process is its three-dimensional nature,

numerical analyses are often performed in two dimensions assuming plane strain conditions. Two-

dimensional analyses are undoubtedly quicker and require less computational power. The

simulation techniques most commonly used to simulate tunnel excavation in 2D are shortly

described here.

Convergence and connement method (Panet & Guenot, 1982). In this method the ratio of stress

unloading prior to lining installation d is prescribed. At a generic excavation increment an internal

forces vector (1)F0 is applied at the nodes on the tunnel boundary, being F0 the nodal force vector

corresponding to the initial stress state 0. At the beginning of the excavation stage it is =0 and soil

elements inside the tunnel boundary are instantaneously removed, then is incrementally

increased up to = d. At this point the lining is activated and increased further until = 1 at the

end of the excavation stage.

Volume loss control method (Addenbrooke et al., 1997). This is very similar to the convergence-

connement method. Excavation is carried out in n increments and the volume loss is calculated at

each analysis increment. Lining elements are activated at increment nL, when a VL slightly lower than

the desired value is obtained. The main dierence between the convergence-connement and the

volume loss control method is that in the latter VL is a prescribed value, whereas in the former it is

an analysis result, depending on the choice of d.

Progressive softening method (Swoboda, 1979). The stiness of the soil inside the tunnel boundary

is multiplied by a reduction factor . Then, excavation nodal forces are incrementally applied to the

tunnel boundary. As with the previous method the lining is activated at a predened excavation

increment.

13

Gap method (Rowe et al., 1983). In the FE mesh, a predened void is introduced between the

excavation boundary and the lining, the area of this void representing the expected volume loss.

The vertical distance between the lining and the excavation boundary is called gap parameter.

Stresses at the excavation boundary are incrementally reduced, as in the previous methods, and at

the same time nodal displacements are monitored. When nodal displacements indicate gap closure

at a point, the soil-lining interaction is activated for that node. The main diculty with this method

is the estimation of the gap value. Indications on how to estimate the gap parameter are given in

Lee et al. (1992).

Many authors argue that realistic results in terms of settlements at the ground surface can only be

obtained in 2D analyses if soil pre-failure nonlinearity is adequately modelled capturing the

following fundamental aspects of soil behavior:

pre-failure non-linearity with high stiness at very small strains;

anisotropy (if present);

Stress path dependent stiness, with the capability to distinguish between load and unload

conditions, at least.

2.3.2 3D analyses

Three-dimensional FE analyses allow to capture the peculiar features of the tunnelling process,

mainly related to the progressive advancement of the excavation front. Furthermore, 3D analyses

may be used to study more complex cases than those of tunnels with straight axis at constant depth,

which 2D simulations are limited to. Finally, when used to study soil-structure interaction problems,

3D analyses allow studying all sorts of building layouts with any orientation respect to the tunnel

axis. Here, three techniques for simulating tunnel excavation in 3D are outlined, in ascending order

of complexity.

Simultaneous excavation method. Tunnel excavation up to desired face position is simulated in one

step only, using either a force or a displacement controlled technique. This method overcomes the

geometry limitations of plane strain analyses but tunnelling is only partly simulated as a 3D process,

as progressive front advancement is not reproduced. Compared to other 3D simulation techniques,

calculation times are greatly reduced.

Step-by-step excavation. At each calculation increment, excavation is simulated by removing soil

elements over an excavation length Lexc ahead of the tunnel face. Lining elements are usually

activated at some distance behind the excavation front. A face support pressure may be applied. In

14

some analyses, rather than leaving the soil between the lining and the excavation head

unsupported, a support pressure or a prescribed displacement eld may be applied to the tunnel

boundary. With this method it is possible to reproduce the development of the settlement trough

as the excavation front advances. This is particularly important when the eects of tunnel

excavation on buildings have to be evaluated. Overlaying buildings, in fact, are undergoing dierent

deformed congurations at each stage of the analysis and usually it is not possible to know a priori

which is the most severe for the examined structure.

Detailed tunnelling simulation. Most details of the tunnelling process are reproduced. As far as

mechanized excavation is concerned, the model can include details of the TBM shield, magnitude

and distribution of the face support pressure, hydraulic jacks thrust, tail grouting volume and

pressure, etc. Clearly, analyses of this kind are the most demanding, usually requiring detailed

geometrical modelling, advanced numerical techniques and high computational power.

15

3 Tunnelling induced soil-structure interaction

In urban area, due to the ground movements produced by tunneling, the existing structure induced

deformations have to be evaluated. It is common practice in a first stage assessment to assume that

the structure follows the displacements in greenfield conditions, neglecting in this way the effect of

building stiffness and weight on altering the deformations field. This section deals with the soil-

structure interaction and the procedure used for a fair evaluation of risk of damage to buildings.

3.1 Building deformation parameters

In order to quantify the foundations movements induced Burland and Wroth (1974) introduced the

following in-plane parameters related to a generic foundation deformation sketched in Figure 9:

Fig. 9- Building deformation parameters (a, b, and c)

Sv the absolute settlement of a point;

Sv (or Sv) the dierential settlement between two points;

Rotation or slope = Sv/L the angle between the line joining the regarded points and the

horizontal distance;

Angular strain the algebraic dierence of slopes of two consecutive segments (e.g. AB and

BC). Conventionally, is taken positive in sagging and negative in hogging;

16

Relative deection the maximum vertical displacement relative to the line joining two

points. Those points usually separate parts of the building deforming entirely in hogging or

in sagging. They could also dene dierent building units, i.e. sections between two columns

or cross walls, parts with dierent stiness or geometry, etc. It is common to dene

positive in sagging (sag) and negative in hogging (hog);

Deection ratio is the ratio DR = /L in sagging (DRsag) or in hogging (DRhog);

Relative rotation or angular distortion is the rotation of the line joining two consecutive

points respect to the rigid body rotation (tilt) of the whole structure ;

Average horizontal strain h = L/L the ratio between the change in length and the initial

length;

It will be shown that the relevant parameters for the damage assessment are the deflection ratio

(DR) and the maximum horizontal strain (h).

3.2 Field data and experimental results

Figures 10 and 11 show monitoring data recorded during excavation of the Jubilee Line Extension

tunnels in London Clay (JLE project). In particular Figure 10 shows the settlements observed at the

foundation level along a longitudinal section of Elizabeth House compared to numerical predictions.

For practical purposes, numerical results in the gure can be thought as being representative of

greeneld conditions. Results are plotted both at the end of construction and at long term. The

building settlement prole can be seen to follow the numerical greeneld curve very closely,

especially in the sagging zone. Elizabeth House is a framed reinforced concrete structure relatively

long and low shaped, thus quite slender in the longitudinal direction. Its clear that as the building

was relatively flexible in longitudinal direction, it follows the Greenfield profile of settlements.

Fig. 10- Elizabeth House in London Comparison of predicted and measured settlements due to tunnel excavation (after Mair, 2003).

17

In Figure 11 are shown the settlements measured for Neptune House following excavation of twin

tunnels compared with results of numerical analyses. In the gure computed results are shown both

for a greeneld analysis and for an interaction analysis in which the building is modelled in a

simplied way, as will be explained in the following sections. Neptune House is an ordinary masonry

building. The observed settlement distribution indicates a sti behavior for the building in the

sagging zone, showing smaller relative deection respect to the predicted greeneld prole. On the

contrary, in the hogging zone a less rigid response is observed as the settlement prole matches the

greeneld predictions quite closely. This behavior, reported in many other case histories, conrms

Burland et al. (1977) observations, indicating that masonry buildings often behave more exibly

when deforming in hogging. The same result is put in evidence by scale model tests of masonry

facades adjacent to deep excavations by Son & Cording (2005).

Fig. 11- Neptune House in London Comparison of predicted and measured settlements due to tunnel excavation (after Mair, 2003).

3.3 Equivalent solids for studying tunnelling induced soil-structure interaction

An equivalent solid can be dened as a simplied building model able to reproduce the behavior of

the real structure in soil-structure interaction analyses. Clearly, the use of an equivalent solid implies

a great degree of simplication in the analysis, as detailed modelling of the building is avoided.

Furthermore, the equivalent model allows reduction of calculation time and computational power.

Thus, it facilitates performing parametric studies of soil-structure interaction problems, aiming to

evaluate the relative inuence of dierent factors on the interaction phenomenon. Potts &

Addenbrooke (1997) carried out a parametric study of the influence of building stiffness on ground

18

movements induced by tunneling using a 2D FE analyses with a no linear elastic-plastic soil model.

The building was represented by an equivalent beam with three input parameters the Youngs

modulus E, the cross-sectional area A and the exural moment of inertia I. Building weight is not

considered in their numerical models and the interface between the beam and the soil is perfectly

rough. Tunnel excavation is simulated through the volume loss control method (see Section 2.3)

using a zone with reduced K0 around the tunnel boundary. Analysis results in terms of settlements

and horizontal strains at the ground surface are presented in function of two measures of relative

building-soil stiness: the relative bending stiness (17) and the relative axial stiness (18):

=

(/2)4 (17)

=

(/2)4 (18)

Where Es is a measure of soil stiness and B is the width of the building. In Figure 12 is reported a

graphic of the result obtained by Potts & Addenbrooke, it is possible to note that increasing the

beam stiffness, the maximum settlement reached decrease and there is a reduction of the

deflection ratio and horizontal strain both in hogging and sagging.

Fig. 12- Parametric study of the influence of building stiffness on ground movements; Potts & Addenbrooke (1997)

On the basis on their study, they proposed modication factors of the corresponding greeneld

deformation parameters:

=

(19)

=

(20)

19

, =,

, (21)

, =,

, (22)

Where h,c and h,t are respectively the maximum tensile and compressive horizontal strains along

the beam and the superscript gf stands for the corresponding greeneld result. Potts &

Addenbrooke (1997) provided design charts for modication factors as functions of the relative

stiness parameters for increasing values of building eccentricity respect to the tunnel centerline

e/B, as shown in Figure 13.

Fig. 13- Charts for modication factors (Potts & Addenbrooke, 1997).

20

For masonry load bearing walls the building can be modelled by an elastic beam located on the

ground surface. The length of the beam L is assumed equal to the full length of the building faade,

B. The height of the beam is H, its thickness t. Determining so an Inertial value (23) and an Area (24)

necessary to define * and *.

=3

12 (23)

= (24)

For framed structures of m storeys (m+1 slab) it is possible to calculate the second moment of area

for the equivalent single beam using the parallel axis theorem (Timoshenko, 1995) considering the

neutral axis to be at mid-height of the building (25), and the axial stiffness (26)

() = ( + 2 )+11 (25)

() = ( + 1) (26)

hm is the distance between the building neutral axis and the slab neutral axis

A different approach was proposed by Pickhaver (2006) for masonry facades with opening.

It differs from Burland and Wroth (1974) (who noted that differing amounts of openings might be

allowed for by manipulation of the ratio E/G directly) considering a more appropriate bending and

shear stiffness of the equivalent beam.

Starting from the typical facade in Figure 14 it is possible to divide it in different strips and through

considerations on shear and bending deformation define respectively a value of the geometric

proprieties Area A* (27) and Inertia I*(28)

=

=1

(27)

n vertical strips of net cross section Ai and length Li

= (

3

12+

2)=1 (28)

n horizontal strips of height hj and thickness t

bj distance to the neutral axis

21

Fig. 14- Schemes for calculating geometrical properties of the equivalent beam (after Pickhaver, 2006).

Franzius et al., (2004) studied the influence of the building self-weight in the results obtained by

Potts and Addenbrooke (1997) concluding that the load of the building alters the deformation

behavior of the soil in two distinct zone: at tunnel depth and in proximity to the foundation of the

building. In particular they presented the complex character of the interaction problem: the load of

the building changes the stress regime which influences the deformation mode of the soil around

the tunnel witch than affects the response of the building to the tunnel induced subsidence.

Absolute and dierential settlements generally increase respect to results for an equivalent plate

with no weight but the eect in terms of modication factors dened in expressions (19) to (22) is

minimal.

22

4 Structural damage evaluation due to tunneling

4.1 Damage criteria

Underground or open excavations unavoidably induce displacements on pre-existing buildings. A

qualitative classication of damage level must be related to objective (i.e. measurable) indicators of

building deformation. Many authors studied the problem of relating observed damage on a

structure to its deformed conguration, either through empirical methods or using theoretical

models in the general framework of continuum mechanics. Skempton & MacDonald (1956), through

examination of a big number of real cases, mainly concerning framed construction buildings

deforming under their self-weight, provided some design indications about maximum admissible

settlements likely to cause either architectonic or structural damage. The Authors recognize that

curvature of the settlement prole of the foundations is related to damage. They chose the

maximum relative rotation max dened in Figure 9 as an indicator of damage on the building.

Limiting values of max causing architectonic or structural damage are shown in Table 3, while Table

4 shows correlations between maximum settlement (either absolute or dierential) and max. In

Table 4 cases for rafts and isolated foundations on either sandy or clayey soil are separated. Hence

the Authors implicitly recognize the key role of relative stiness between the structure and the soil

and of deformation modes related to dierent foundation layouts in determining damage on the

building.

Tab. 3- Maximum admissible relative rotation (after Skempton & MacDonald, 1956).

Tab. 4- Relations between maximum absolute or dierential displacements and maximum relative rotation (after Skempton & MacDonald, 1956).

Another damage classication consists in separating aesthetic, functional and structural damage

(Burland et al., 1977). Those big classes may be further subdivided in categories creating a scale of

damage severity. Burland et al. (1977) proposed the damage classication reported in Table 6 at the

23

end of this work. A critical crack width is also associated to each damage category. Specic values

of limit tensile strain lim can be related to each damage category with reference to a given

construction material. From examination of real cases and model tests on masonry buildings the

values of lim indicated in Table 5 for each damage category were obtained (Boscardin & Cording,

1989; Burland, 1995).

Tab. 5- Relation between category of damage and limiting tensile strain (after Boscardin & Cording, 1989; Burland, 1995).

Using the elastic deep beam theory (Timoshenko, 1955) Burland & Wroth (1974) developed a semi-

empirical method to relate settlements of the foundations to the onset of visible cracking in the

building. The building is idealized as an isotropic, linear elastic deep beam of length L and weight H.

The problem is to calculate the deflection ration value /L at witch is reached a maximum value of

the tensile strain imposed. The distribution of the strains is logically related to the deformation

shape of the beam so it is possible to consider two extreme case of pure bending and pure strain

deformation sketched in Figure 15.

Fig. 15- Cracking of a simple beam in different modes of deformation (after Burland & Wroth, 1974)

24

In pure bending the maximum tensile strain b,max is horizontal, instead in shear it is oriented at 45

and is indicated as d,max (d stands for diagonal).

Starting from the general Timoshenko results for the total mid-span deflection of a centrally load

beam (29) characterized by a Youngs modulus E, a shear modulus G, a second moment of area I

and a point load P, it is possible to re-write the equation in terms of deflection ration /L and

maximum extreme fibre strain b,max (30) and maximum diagonal strain (31)

=3

48(1 +

18

2) (29)

= , (

12+

3

2) (30)

= , (1 +

2

18) (31)

t is the distance of the neutral axis from the edge of the beam in tension.

Similar expression are obtained for the case of uniformly distributed load. Therefore, the maximum

tensile strains are much more sensitive to the value of /L than to the distribution of load.

Assuming E/G=2.6 which corresponds to a =0.3 considering an isotropic behavior and imposing

max= crit, either in bending or in shear, the previous relations can be plotted in terms of (/L)/crit

against L/H as shown in Figure 16. Figure 16a is referred to a neutral axis in the middle of the beam,

16b to a n.a. at the bottom.

a) n.a. at mid-height b) n.a. at the bottom

Fig. 16- Relation between (/L)/crit and L/H for E/G = 2.6, according to the deep beam model.

It should be noted that since foundations offer significant restraint to their deformations, it can be

more realistic to consider the neutral axis at the lower extreme fiber of the beam placed at ground

25

level. In this case for L/H>1.5 the rupture is governed by bending strain instead for L/H

26

Fig. 18- Damage chart for E/G = 2.6, L/H = 1.0 (after Burland, 1995).

4.2 Damage evaluation process

In the design practice for projects involving tunnelling in the urban environment, evaluation of

expected damage on a given building is usually undertaken in three subsequent stages with

increasing level of detail and complexity. If in one stage a negligible risk of damage is predicted for

a specic building, then no further investigations are required for that building. On the contrary, if

in one stage a signicant damage level is indicated, then it is necessary to move on to the next, less

conservative, stage of the process. The three stages are summarized here:

1. Preliminary (or rst level) evaluation: In this stage the presence of the building is not

considered. The settlement prole induced by tunnel excavations in greeneld conditions is

calculated through empirical relations like those introduced in Section 2.1. Rotation and

maximum absolute settlements are calculated on the building footprint. These indicators are

compared to limit values. Rankin (1988) suggests to use = 1/500 and Sv,max = 10mm.

2. Second level evaluation: This stage can be further subdivided in two sub-stages. First, the

hypothesis of a building with no stiness is still assumed. Greeneld displacement proles

are used to calculate kinematic indicators of damage on the building. Using /L and h in

damage charts similar to that drawn in Figure 18, for instance, it is possible to extrapolate

the expected category of damage for the building. If this stage still yields an unacceptable

damage level (usually is considered the damage level=2 as the upper bound) the building

stiness can be accounted for in a simplied way using charts like those proposed by Potts

& Addenbrooke (1997) (Figure 13) to obtain modication factors to reduce the greeneld

values of DR and h.

27

3. Detailed evaluation (or third level): If evaluation of expected damage in the rst two stages

of this process does not give acceptable results for the examined building, it is necessary to

perform detailed analyses of the soil-structure interaction problem. This last stage of

analysis is usually very resources demanding and time consuming as accounting for details

of both the examined building and the tunnel excavation process is required. Typically, it is

required to properly include the following aspects in the analysis:

structural details of the building, including foundations;

geometry of the building and relative position respect to tunnel axis;

tunnel excavation technique.

In some cases it is also necessary to consider the three-dimensional character of the examined

problem. This stage of the damage assessment process is usually carried out with numerical

analyses. These could include either a detailed building model or a simplied model description. If

even with such detailed analyses an unacceptable damage is predicted for the building, design of

protective and remedial measures is required. A schematic diagram of the three stage approach for

damage risk evaluation in reported in Figure 19.

Fig. 19-Schematic diagram of the three stage approach for damage risk evaluation.

28

This procedure, commonly used in the design for a rapid assessment of the damage to the structures

concerned by tunnelling, after a first check on the maximum settlement and slope, associates the

level of damage of the structure exclusively to the inflection of foundation through two parameters

the deflection ratio and the horizontal strain (Figure 18), neglecting the effects related to the rigid

translation and rigid rotation. It results definitely valid for ordinary buildings, but it is important to

know that can lead to not appropriate valuations in particular cases as for old structures of historical

or artistic interest. That kind of structures, may have undergone various changes over the years, and

may be characterized by a significant weight and by the presence of different structural and

ornamental elements to preserve, unusual for the common buildings. In this cases also small

absolute settlements and rigid rotations that usually do not affect the stability of a structure can

lead to significant damage to the artistic heritage. Moreover the simplified approach typically used

to take into account the soil structure interaction through the Potts and Addenbrooke approach

may be not conservative because obtained from 2D analysis in absence of the structural weight. In

these cases realizing a 3D FEM model that allows to study the specific case more efficiently should

therefore be more appropriate.

29

Tab. 6-Classification of visible damage (after Burland at al., 1977)

30

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