The French recommendations for the structural design of large sewer linings

1. INTRODUCTION

To further the knowledge in the domain of sewer system rehabilitation, the French ministry of public works and the profession created a National project named RERAU (Rehabilitation of Urban Network Sewers). The purpose of a National Project is to unite around a given research topic the various organisms concerned and to allow each participant to benefit from all the results, while contributing only a part of the total financing. The French ministry contributes to 20% of the total program cost. The main objectives of RERAU program are:

• Improve the quality of structure diagnoses;
• Define the field and the conditions of use of renovation techniques;
• Establish a rational design methodology for sewer rehabilitations. 

RERAU is divided in several operations each devoted to one topic:
The first operation concerns the diagnosis and the testing of man entry sewers. The second operation is devoted to the sprayed concrete (with or without reinforcement); a rehabilitation technique used in large man entry sewers.

The fourth operation is devoted to prefabricated channel linings (GRP and GRC), including mono-piece, multi-piece and invert only liners. And to this last operation, the design methodology was added.

The RERAU design methodology is a limit state design including partial safety factors on loads and on material properties, published in 2004 [1]. RERAU 4 has focused on non-circular lining because man-entry sewers are generally non-circular.

From the mechanical standpoint, a lining can be effected by hydrostatic pressure when the sewer is under the ground water and by ground-traffic loading. Deflections of buried structures are generally small and, in the linings, produce stress that are almost negligible. Conversely, external pressure due to ground water can easily cause lining failure by geometric instability or material breakdown. The lining system is designed to act as a flexible pipe with the old sewer, annulus grout (when appropriate) and soil providing the necessary support to maintain stability. The prime design requirements on the lining are therefore:

• Ability to sustain the grouting pressure during installation (when appropriate).
• Ability to sustain the external head of groundwater pressure that must be considered to rise, once hydraulic integrity is restored.
• And eventually ability to sustain soil loading transfer, if the sewer loses its hoop compressive stiffness after lining. 

2. THE DESIGN FOR GROUND-WATER PRESSURE

Because lining restores hydraulic integrity and because the bond between the lining and the host cannot be relied-on in the long term, it is necessary to consider the effect of external water pressure acting on the lining. The grout is considered likely to crack due to small ground movements. Groundwater may percolate through the cracks and act at the interface between the lining and the host (Figure 2).

2.1. Shapes and imperfections

The lining shape has a great importance for the resistance to the external hydrostatic pressure. For prefabricated linings (GRP), the design shape is simply the manufactured shape. In this case, imperfections of the host structure do not have any consequences, on the design. For cure in place linings, the design shape is the most unfavourable shape of the host structure. In this case, imperfections of the host structure may have important consequences on the design.

The design shape is transformed into an idealized shape which is a simpler geometrical shape of convex form (see Figure 3). The imperfections are then the deviations compared to the idealized shape.

They are classified in two types : global or local. Global imperfections are uniformly distributed around the perimeter like annular gap. Local imperfections are distributed on a limited angular sector like local intrusion (see Figure 4). In all the cases, the distribution of the imperfections in the longitudinal direction is supposed to be constant, that is the most unfavourable configuration. These imperfections are not always measurable, and default values must be carefully chosen.

2.2. The behavior of a close-fit lining subject to external pressure

Normally one or two lobes can develops according to the geometrical and the mechanical symmetries. For the egg-shaped lining of the figure 5a) two lobes develop at the middle of the two straight sections (the straight section is the arc between the springing and the invert). For the oval linings of the figure 5c) two lobes develop at the crown and the invert. This two configurations are both symmetric. For the horseshoe-shaped lining of the figure 5b) one lobe develops at the middle of the invert. This configuration is also symmetric. The figures 5d) and 5e) are the anti symmetric configurations corresponding to figure 5a) and 5c). It is a switch from two lobes to single lobe.
The buckling pressure of the one lobe mode is lower than the buckling pressure of the two lobes. So the designer shows generally the one lobe mode.

2.3. Critical or sub-critical shape

When le pressure increases, one can observe two opposite behaviors for the lobes development according to the shape:
On one hand, if the curvature and the extension angle of the arc where the lobe develops are sufficient, the deflection at the center of the lobe increases but the lobe angle remains constant or decreases, the lobe does not extends. There is a limit pressure where the lobe buckles, see Figure 6(a).
On the other hand, for instance in the case of an oval shape or an egg shape with straight sides, the lobe may extend continuously over the entire lining and there is no buckling pressure. The first behavior is named “ critical ” and the second “ sub-critical ”.

Figure 7 illustrates the difference between critical and sub-criticalshape. The mechanical characteristics of these 2 shapes are identical (same thickness and same modulus). There height and their width are also identical, it is only the curvature of the straight section which changes. The curvature is infinite for the shape a) and equal to 1.5 time the height for the shape b).

The graph on the right of the figure shows the deflection at the middle of the straight section calculated with the finite element method for the two shapes. One can see that the deflection curve of the shape b) shows a bend where the pressure is maximum (this is the buckling pressure) and after the bend, the stiffness becomes negative which causes the instability. For the shape a) there is no bend, the deflection increases continuously with the pressure and the stiffness decreases. It is also obvious that the stiffness of the shape a) is much lower than the stiffness of b).

2.4. The design of critical shape (egg-shape, oval shape…)

The calculation method is based on an analytical solution obtained by O. Thépot [2]. It extends Glock ’s analysis [3]. The industry has known for some time that the data from buckling tests indicates that the Glock’s model predicts more accurately the buckling behavior of encased liners. Bernhard Falter [4], [5], David Hall [6], Ian Moore [7] and many other researchers have confirmed this.

The shape must be convex, comprising a succession or arcs tangent at their points of contact. For elliptic shape we simply consider the largest circle tangent to the ellipsoid. We can see in the Figure 3 examples of shapes which can be calculated by the method : a) elliptic shape, b) egg-shape, c) horseshoe-shape. There is no bond between the lining and the host pipe, that means that tangential and tensile forces at the interface are assumed as zero. In practice bond is often present and is beneficial. This bond may be purely frictional in some circumstances.

The buckling pressure is given by the following formula:

P_cr = 2.02 ⋅ K^0.4 ⋅ [(EI^0.6EA^0.4) / (p^0.4R^1.8)]

Where k is the number of lobe (1 or 2), P is the mean perimeter of the lining, R is the radius of the arc where the lobe develops, EI is the flexural stiffness and EA is the compressive stiffness of the wall.
Note that the formula calls into play both the flexural stiffness (at power 0.6) and the compressive stiffness (at power 0.4), whereas Timoshenko’s formula uses only the flexural stiffness. This is because the lining must shorten in order to separate from the host, which calls its compressive stiffness into play. In the case of a plain wall and homogeneous material of t, the formula can be simplified as follow:

p_cr = 0.455 ⋅ k^0.4 ⋅ E_L ⋅ [(t^2.2) / (P^0.4R^1.8)]

Where t is the thickness of the lining and EL is the long-term flexural stiffness. Notice that the buckling pressure of the two lobes mode is higher than those of the one lobe mode by a factor of 1.3 (precisely 2 ex 0.4). It is also easy to calculate the buckling pressure or a particular shape, for example the 3x2 egg-shape:

P_cr = 0.308 ⋅ E_L ⋅ [t / H]^2.2

Where H is the height of the shape (R=H, P=2.6433xH, k=1) Or simply the circular shape of diameter D (R=D/2, P=3.1416xD, k=1):

P_cr = 1.0 ⋅ E_L ⋅ [t / D]^2.2

The bending moment is maximal at the middle of the lobe (Figure 9). The axial force is constant all over the perimeter (if the friction between the lining and the host can be neglected). The deflection is also maximal et the center of the lobe. These three quantities are given by the following formulas (see also Figure 10):

M(p_W) = [0.5 ⋅ (p_W/p_cr) M_cr] / [1 - 0.5 ⋅ (p_W/p_cr)²]

N(p_W) = (p_W/p_cr) N_cr

d(p_W) = d_cr [1 - [1 - (p_W/p_cr)]^0.5]

where p_w is the design pressure, Mcr is the critical bending moment and Ncr is the critical axial force:

N_cr = 1.26p_crR

M_cr = 1.2 (EI / R) 

Imperfections are taken into account using reduction or amplification factors which apply to the calculated quantities (critical pressure, deflections, moments). The combination of the imperfections is carried out by the multiplication of the factors.

2.5. The design of sub-critical shapes (oval-shape, horseshoe shape, rectangular…)

The calculation method for sub critical linings (in fact oval shaped linings) is based on an analytical solution obtained by O. Thépot and published in Thin-Walled Structures Journal [8]. This method gives an explicit solution for the pressure function of the maximum deflection at the center of the lobe which is a serviceability requirement (normally 2 or 3% of the “straight” section but a different value may be used).

The steps of the design procedure are summarized in Fig. 9. Firstly one chooses the limit deflection d (this is a serviceability requirement generally 3% of the straight section). Secondly one calculates the gap angles α1 and α2 at the crown and the invert. Thirdly one calculates the limit pressure p_λ, the maximum bending moment M_λ and the axial force N_λ functions of the gap angles. And finally one does the checks: the design pressure must be less that the limit pressure and the maximum stress must be inferior to the yield stress divided by a safety factor.

The Geometrical parameters are:

r1: The radius of the crown
r2: The radius of the invert
L: The length of the straight section
P: The perimeter of lining
EA: The compressive stiffness of the lining
EI: The Flexural stiffness
λ: The relative deflection of the straight section (3%)

With these 7 parameters, 6 non-dimensional quantities are defined:

θ₁ = [r₁ / L]

θ₂ = [r₂ / L]

λ = [d_L / L]

β = [EI_d / EA_d] ˙ [P / L³]

m = [EI_d / L³]

g₁₂ = 1 + √(r₁/r₂)

The gap angle is then the positive solution of a polynomial equation of degree 5 whose coefficients are function of the 6 non-dimensional quantities:

120β + 9λ² + g₁₂θ₂(120β + 18λ²)α₂ + (12λθ₂ + 9λ²g²₁₂θ²₂)α²₂ - 12λg²₁₂θ³₂α⁴₂  - 16g₁₂θ₂³α₂⁵ = 0

This equation can be solved by a Newton-Raphson iterative method (It should be pointed out that α₂ has a finite value different from zero though d equals zero).

Then two quantities are defined:

η = 1 + α₂θ₂g₁₂ and γ = g₁₂θ₂2α₂³

Finally, the expressions of the pressure, the axial force, the bending moment and the bending stress may be written in terms of λ, β,m,η,γ,:

Expression of the pressure:

p_λ = [(4π⁴m) / η⁴) ⋅ λ [(1+ (γ² / 9βη)) - (γ / 4β)γ + (η / 8β)λ²]

Expression of the axial force:

N_λ = - [π²EA / 12] ⋅ (L/P) (λ/η²) [4γ - 3ηλ]

Expression of the maximum bending moment:

M_λ = [(2π²EI) / L] [λ / η²]

The length of the straight section must be greater than 1.5 times the radius of the crown.

The gap angle α1 must be lower than half the arc angle of the invert.

3. THE DESIGN FOR GROUTING

As a minimum requirement any lining system should be capable of supporting the pressure of grout filling the annulus during installation. In the case of a large-sized lining, internal struts combined with grouting in several stages makes it possible to avoid the excessive deformation of the panels.
For man entry linings, staged or partial grouting are generally adopted rather than full grouting. The first stage involves grouting the annulus up to a third or a quarter of the vertical height, and this is followed by a second stage, carried out after the grout of stage one as set; finally a third possible stage supplements the filling of annular space.

The performance of non-circular linings is particularly sensitive to the following factors:

• The degree of restraint provided during grouting. Hardwood wedges are normally used as packing. Restrains of the vertical height of the lining by an internal strut may be very important.
• The height of grouting at the first stage.
• The shape of the lining and particularly the curvature of the straight section.
• The bending stiffness of the lining material (not the hoop stiffness).
• The unit weight of the grout.

Circular linings with simple boundary conditions such as GRP pipes can be designed using analytical formulas based on Timoshenko approach (Figure 14).

At present for non-circular linings and/or special boundary conditions, and especially for man entry linings, non-linear finite element analysis with large deflection and appropriate formulation of the boundary conditions may be used.

4. THE DESIGN FOR GROUND AND TRAFFIC LOADING

In most cases the host structure continues to sustain the ground and traffic loading and there is normally no requirement to consider directly the transfer of any loading from the soil to the lining. But in certain circumstances, significant soil loads can be transferred to the linings:

• If the host sewer loses its hoop compressive stiffness after lining.
• If a subsequent excavation near the renovated pipe is undertaken.
• If the host sewer is obviously in an unstable ground, where the source of instability is not eliminated by lining. 

For the design of ground loading, RERAU recommends the Finite element method.

The lining is modeled like a tunnel lining, not like a buried pipe (see Figure 15):

• Firstly initial stresses are generated in the soil by specifying the head of soil, the specific weight of soil and the coefficient of earth pressure at rest.
• Secondly the equilibrium forces are calculated at the interface between the soil and the lining. These equilibrium force are actually sustained by the host structure.
• And thirdly the reversed forces, are applied to the lining.

These tree stages model the transfer of ground pressure from the host structure to the lining, considering that the host structure loses its hoop compressive stiffness after lining.

5. REFERENCES

[1] RERAU, 2004, Restructuration des collecteurs visitables, Lavoisier, Paris.
[2] Thépot O., 2000, A new design method for non-circular sewer linings. Trenchless Technology Research, Vol. 15, No. 1, 25-41.
[3] Glock D., 1977, Behavior of liners for rigid pipeline under external water pressure and thermal expansion, Der Stahlbau, Vol. 46, No. 7, 212-217.
[4] Falter B.,1996, Structural analysis of sewer linings, Trenchless Technology Research, Vol. 11, No. 2, 27-41.
[5] Falter B., 2001, Structural design of linings, Proceedings of the International Conference on Underground Infrastructure Research, Kitchener, Ontario, Balkema publishers, 49-58.
[6] Hall D.E., Zhu M., 2001, Creep induced contact and stress evolution in thin-walled pipe liners, Thin-Walled Structures, Vol. 39, 939-959.
[7] Moore ID., 1998, Tests for pipe liner stability: what we can and cannot leran, Proceedings of North American No-Dig 98, NASTT, Albuquerque, 443-457.
[8] Thépot O. 2001, Structural design of oval-shaped sewer linings, Thin-Walled Structures, Vol. 39, 499-518.