2 Marco referencial
2.2 Bases teóricas
2.2.3 Proceso educativo
We present here a method for calculating the frictional forces during soil-pipe contact. This method consisting of a bibliographical summary will then be compared to the experimental results.
The approach considered here examines in a first step the stability of the excavation made by the boring machine (see Figure 6.4):
– if this is unstable, the loosened ground comes into contact with the entire pipeline. The frictional forces F are then calculated by multiplying the contact stresses N exerted on the pipes by the friction coefficient
µ
characterizing the state of roughness of the soil-pipe interface;– if the excavation is stable and if the consistency of the soil is less than the thickness of the annular space, the pipeline slides on its base, inside the open annular space. The frictional forces are equal to the product of the actual weight of the pipes and the soil-pipe friction coefficient;
– if the excavation is stable and if the consistency of the soil exceeds the width of the annular space, we return to the first case and calculate the frictional forces from the contact stresses exerted on the pipe.
Figure 6.4. Stability of the excavation
6.2.3.1. Verification of the stability of the excavation
The English Pipe Jacking Association has suggested a method to estimate the confining pressure required (
σ
T) to ensure the stability of the overcut.a) In the case of cohesive soil, the short-term stability is linked to the undrained cohesion and the pressure that will be necessary to maintain a stable excavation is given by the following relation:
.( ) .
T 2
e e u
H D T c
σ =γ + −
with:
– γ: specific weight of the soil above the pipes;
– H: height of the overburden over the pipes;
– De: diameter of the excavation;
– cu: non-dewatered cohesion;
– Tc: stability coefficient of cohesive soil (values given in Figure 6.5 depending on H/De).
The excavation is stable if this internal pressure jT is less than or equal to 0.
Figure 6.5. Values of the stability coefficients Tγ, Tc, Ts
b) In the case of non-cohesive soil, the stability depends on the internal angle of friction of the soil ϕ. In this case there is no simple general solution and the PJA proposes to consider the following two configurations for calculating σT:
– σ = γT .D Te γ in the absence of an overload above the pipeline, Tγ 伊represents the stability coefficient indicated in Figure 6.5 depending on ϕ;
– σ =T q .Ts s in the presence of a significant overload qs and at a low depth. The weight of the soil is then neglected and Ts represents the stability coefficient given in Figure 6.5 depending on ϕ and H/De.
Based on these relations when the soil is purely frictional, σT is always positive;
as a result, the excavation is always unstable in the absence of a confinement pressure.
6.2.3.2. Ground convergence effect
The overcut represents the difference of radius between the excavation and the pipeline. Even in the case of a stable excavation, the ground can cave in on the pipes due to its flexible discharge. The vertical and horizontal decrease in the diameter of the excavation resulting in the flexible discharge of soil are calculated according to the state of initial stresses by adopting a law of flexible behavior of the ground, which leads to the following relations:
2s
V e V h
s
1 v .D .(3.j j ) E
= − −
and:
–
∆
vreduction in the diameter of the excavation in the vertical direction, – ∆hreduction in the diameter of the excavation in the horizontal direction, –V
sPoisson’s coefficient of the soil,–
E
sYoung’s modulus of the soil.If a pressure p is applied inside of the overcut (in a case where a lubricant is injected), this leads to a constant increase ∆p in the excavation diameter:
1 . '.
where p represents the effective internal pressure in the annular space (equal to the confinement pressure p reduced by the pore pressure of the ground).
According to the thickness of the overcut(s) made by the cutting wheel of the boring machine, two types of Figures can be shown:
– if ∆v
畏
and伊
∆h尉伊
– ∆p < s, there is not contact between the soil and the pipe, the annular space remains open and the friction is caused only by the boring machine’s own weight;– if ∆v
(
and ∆h)異伊∆p ≥ s, there is contact between the soil and pipe, the annular space remains closed, and the frictional forces are linked to the stresses exerted by the soil on the pipes.6.2.3.3. Calculation of frictional forces for unstable excavation in granular soil The frictional forces are calculated by multiplying the total normal stress (N) that the soil exerts on the pipes by the frictional coefficient µ.
6.2.3.3.1. Determining the normal stress
The normal stress (N) acting on the outer surface of the boring machine is obtained by integrating the normal stress σn acting on a surface element dS. This is determined from the principal vertical (σv) and horizontal (σh) stresses of the ground. At a given point of the pipe, these stresses are given by (Figure 6.6):
( / 2 )
– y: ordinate of the point P with respect to the centre of the pipe,
The normal stress n acting on the surface of a pipe per linear meter, obtained by integration of the normal stress over the entire surface, is therefore defined as follows:
Digging the microtunnel will disturb the initial state of the stresses around the excavation. This new state of stress, caused by the relaxation of the soil as a result of an overcut, can only be determined using a model. Studies carried until now as part of the National Project have shown that the Terzaghi model provides satisfactory results, close to actual values (Pellet, 1997, Phelipot, 2000).
The Terzaghi model (1951) assumes that the soil located above the pipe “slides”
with respect to two vertical planes. These movements are sufficiently significant to lead to the creation of shear planes (see Figure 6.7).
The resolution of the differential equation of the equilibrium of a section of horizontal soil subject to shear stresses according to slip planes described earlier gives the expression of the vertical stress of the soil on the roof of the pipe:
2. . tan . /
with:
– H: Height of the overburden on the roof of the pipeline, – γ: specific weight of the overlaying strata,
– K: horizontal pressure coefficient of the soil above the excavation, – b: width of the affected ground,
– δ: friction angle of soil in place/decompressed soil above the excavation, – c: cohesion of the soil.
Figure 6.7. Shear corners, Terzaghi model
A certain number of assumptions (roughness of the slip planes, geometry of the shear corners) need to be considered in order to define the experimental parameters K, b and δ. Studies carried out to date (Pellet, 1997, FSTT, RS 25) have shown the adequacy of the model with the results obtained during follow-ups of the microtunneling sites, for parameters defined in the following manner:
1
K =
δ = ϕ伊伊伊伊
[1 2 tan( )]4 2 b=De + π ϕ−
This formula is slightly different from the basic Terzaghi model (see paragraph 6.4.3.1) because it has been derived from experimental results.
The vertical stress on the roof of the pipeline σEV can be represented by defining a coefficient k less than 1, which, by reducing the weight of the soil γH, represents the arching of the ground (see Figure 6.8 and Figure 6.9):
H
EV k.
γ
.σ
=Internal friction angle (ϕ°)
Figure 6.8. Reduction coefficient k depending on ϕ
Figure 6.9. Reduction coefficient k depending on H/De
For granular soil, as the cohesion is zero, the coefficient k becomes:
Figure 6.8 and Figure 6.9 illustrate the variation of the coefficient k for granular soil depending on the ratio H/De and the internal friction angle of the ground ϕ.
It is accepted that when the height of the ground above the pipeline is low (H/b <
1), the decompression movements caused by the excavation act on the entire mass of the ground covering the microtunnel. The arching is thus neglected and the total mass of the soil above the pipeline is considered for the calculation of σEV (Szechy, 1970, AFTES, 1982).
6.2.3.3.3. Determining the frictional force
The frictional force is finally obtained by multiplying the normal stress N applied on the surface of the pipes by the soil-pipe frictional coefficientµ. The choice of the coefficient is discussed in paragraph 6.2.5.1.
6.2.3.4. Calculation of frictional forces for unstable excavation in cohesive soil The shearing stresses caused by the contact between the clayey soil and the pipeline is dependent on the undrained cohesion cu of the soil, of the coefficient β characterizing the soil-pipe adhesion (which depends on the type of pipe surface) and of the total surface of the pipes jacked. In fact, in the case of relatively large soil-pipe displacements caused by jacking, the clay in contact with the pipeline is greatly reworked and it is better to consider the undrained cohesion of reworked clay cur,that is:
ur ext
F = .c . .D .L β π
The undrained cohesion in the reworked state curcan be estimated from the flow index IL = (w – wp)/IP in the abacus presented in Figure 6.10 (Leroueil et al., 1983).
However, this approach assumes that the natural water content of clay in contact with the pipeline has not been modified by percolations of the mucking liquid or injection liquid.
The value of β, which characterizes the soil-pipe interface, was the object of various studies for piles (DTU 13.2). An average value of 0.6 will be considered (in the case of piles drilled in concrete of large diameter).
Figure 6.10. Estimation of the undrained cohesion of reworked clay
6.2.3.5. Calculation of frictional forces for a stable excavation
If the excavation remains stable but the convergence is such that the soil comes in contact with the pipeline, the conditions mentioned in paragraph 6.2.3.3 or 6.2.3.4 are encountered once more.
If the excavation remains stable and the convergence is less than the annular space, the pipe train slides on its base inside the open annular space. The frictional forces then depend on the nature of contact between the soil and the pipe.
6.2.3.5.1. For frictional soil
The frictional forces are equal to the product of the dead weight of the pipes and the soil-pipe frictional coefficient:
. . F = µLW with:
– W: dead weight of the pipeline per linear meter, – µ: soil-pipe frictional coefficient,
– L: Total length of the pipeline jacked.
In the case of a stable excavation located below the groundwater table (γw), or when the annular space is entirely filled with bentonite slurry (γb), the pipe string is subjected to buoyancy, which directly opposes its own weight. It is then advisable to consider the saturated weight of the pipeline. If it is negative, then the pipeline will float and the friction will act on the crown, hence the following general formula:
. 4
6.2.3.5.2. For cohesive soil
The instrumentation used at the actual site in England (Milligan, 1995) has shown that there exists a frictional type relation between the shear stress and the total normal stress in all types of soil except soft clay. Consequently, the formula related to frictional soil is used to calculate the frictional forces. This frictional behavior of most of the clayey soil is explained by Milligan and Norris (1999) with the help of an excavated surface, which is not smooth but relatively rough with sporadic and irregular contacts.
In highly special cases of soft clay, the theory of Haslem can be referred to, where the frictional forces result from the undrained cohesive soil (Haslem, 1986).