Ground deformation

lower bound solution, Qs, Qγ and QT are calculated by using Eq. 1.6 and Ns and Nγ are calculated as follows:

𝑁_𝑠 = 𝐾_𝐴

Eq. 1.6 𝑁_𝛾 = 𝐾_𝐴(𝐶

𝐷+ 1)

Figure 1.11 A lower bound solution for collapse (Leca & Dormieux, 1990).

As mentioned above, many theoretical models had been proposed to interpret the heading face mechanism of conventional tunnel face. However, there are few studies on the tunnel face stability in frozen ground, much less of pipe jacking in cold regions.

In the following Chapter 2, the parametric study was conducted by using numerical simulation method. At the same time, the face deformation mechanism was analyzed and the optimal face support pressure with various conditions were calculated by using a new proposed calculation model.

For conventional tunneling, R.K.Row and Kack (1983) conducted a parametric study to identify significant factors affecting the prediction of surface settlement due to tunnel construction, including elastic modulus, plastic failure, initial stresses, ground loss, and the injection of clay grouting into the tailpiece void. In particular, a parameter called the gap is defined in the paper to incorporate practical tunneling experience into the calculation of surface settlement. Taking the ground loss near surface into consideration in incompressible materials, Sagaseta (1988) presented a new approach to determine the strain field using a virtual image technique, which reproduces the observed patterns of movement distribution fairly well with the same accuracy of more sophisticated and expensive numerical methods. Because of the over cutting area, the difference between diameter of tunneling and external diameter of the pipes, an elastic solutions are presented by Park (2004, 2005) to predict the tunneling induced undrained ground movements for shallow and deep circular tunnels in soft ground, by imposing the oval-shaped ground deformation pattern as the boundary condition of the displacement around the tunnel opening. Except for the theoretical analysis, the finite element numerical models (De Farias, Júnior, & De Assis, 2004; Yoo & Lee, 2008) and discrete numerical models (Maynar & Rodríguez, 2005) are also presented in the prediction of ground deformation induced by tunneling.

For the pipe jacking tunneling, relatively small amount of data from trenchless pipelaying operations are reported in the literatures, the most notable measurements being presented by Milligan and Marshall (1995) and Marshall (1996, 1998). Rogers and Chapman (1998) compared three alternative analyses suggested to ground movements around tunnel, and recommended the modified Sagaseta analysis to predict the ground movements occurring around pipe jacking and pipe bursting operations.

Based on Roger’s opinion, Chapman (1999) proposed a graphical method for predicting ground movements from pipe jacking. Shui-Long Shen et al. (2012) used the Mindlin solution to vertical and horizontal force in order to calculate the ground deformation caused by the support pressure and the lateral friction resistance, respectively. Besides, they also presented ground response of a case history of the successful application of

observational method to instruction parallel microtunneling with successive pipe jacking (Shen, Cui, Ho, & Xu, 2016). In order to prevent loss and disaster caused by surface deformation, the finite element method was applied to analyze the impact of various parameters during pipe jacking construction on ground deformation (F. Li, Fang,

& Li, 2007).

However, there are not so many documented observations on the ground deformation of pipe jacking in frozen ground. Therefore, a database of surface movements is required to verify the appropriateness of existing, widely accepted in conventional tunneling, design solutions to the ground deformation induced by pipe jacking construction in frozen ground.

1.4.2.1 General predictive solutions of ground settlement

Tunneling induced ground movement can cause distress and possibly failure in adjacent services and structures. Therefore, it is important to be able to predict these movements by using theoretical models prior to construction, and also to enable evaluation of developments in equipment design (Rogers & Chapman, 1998). Many researchers have already proposed the theoretical analysis methods for the prediction of ground movements during trenchless operations, such as error function analysis (O'reilly &

New, 1982), modified Sagaseta analysis (Sagaseta, 1988) and Vafaeian’s analysis (Vafaeian, 1991).

1.4.2.2 Error function analysis

Following many researches and analysis of tunnel settlement, it is widely accepted that the shape of the transverse settlement profile immediately behind a tunnel can be well described by using an error function curve shown in Figure 1.12. The error function analysis for subsurface ground movements, as originally proposed, uses the assumption that the movements are radial and directed towards the center of the tunnel (Rogers &

Chapman, 1998). Surface subsidence, S, is defined as

𝑆 = 𝑆_𝑚𝑎𝑥𝑒𝑥𝑝 (−𝑥²

2𝑖²) Eq. 1.7

where, Smax presents the maximum settlement (at x=0) and i is the standard deviation of the curve. The width of the settlement profile is conveniently defined by i, which is the distance from the tunnel centerline to the point of inflexion. O' Reilly and New (1982) reviewed settlement data from tunnels in the UK to show that trough width parameter, i, was an approximately linear function of depth z0 and for most practical purposes can be simplified to

𝑖 = 𝐾𝑧₀ Eq. 1.8

where K is dependent on the type of soil and for tunnels in clays, and sands or gravels may be taken as 0.5 and 2.5, respectively.

The assumed relationship between horizontal and vertical components of ground displacement at a lateral distance x from the pipe centerline and vertical distance y above the pipe axis is given by

𝐻_(𝑥,𝑦) = (𝑥

𝑦) 𝑆_(𝑥,𝑦) Eq. 1.9

Figure 1.12 Geometry of the tunnel induced settlement trough (Attewell, Yeates, &

Selby, 1986).

The above analysis was applied to tunneling field data, including both surface and

subsurface measurements, and found reasonably good agreement above the tunnel springing level in terms of both the magnitudes of the movements predicted and the patterns of observed movement. However, below the tunnel springing level the predictions were not very precise.

1.4.2.3 Modified Sagaseta’s method

This analysis was proposed by Sagaseta (1988) as a general method for modeling ground movements. The method assumes that the soil behaves as a fluid and flows radially towards a point at which ground volume loss is happening. Formulae were developed for an infinitely deep tunnel and then the effect of a ground surface are considered by using a virtual image technique. The general idea behind this analysis is shown in Figure 1.13. The radial displacement around a cavity is given by the following:

𝑊_𝑟= 𝑘 (𝑎 2) (𝑎

𝑟)

𝛽 Eq. 1.10

This formula can be for both a sink, that is representing a volume loss and where the movements are directed towards it, and a source, that is representing a volume gain and where the movements are directed away from it. Therefore, the general equations for the horizontal and vertical components of the displacements, at point P, shown in Figure 1.13, for a pipe jacking operation are:

𝑊_𝑥= [𝑄 (𝑥 𝛼)] (1

𝑟₁^𝛼− 1

𝑟₂^𝛼) Eq. 1.11

𝑊_𝑦 = (𝑄 𝛼[𝑦

𝑟₁^𝛼+ (2𝑧₀− 𝑦

𝑟₂^𝛼 )]) Eq. 1.12

where 𝑄 = 𝑘[(2𝑅𝛿𝑅)^0.5]^𝛼 , 𝑟₁ = (𝑥²+ 𝑦²)^0.5 and 𝑟₂ = (𝑥² + (2𝑧 − 𝑦)²)^0.5 . These equations determine the vertical, Wx, and horizontal, Wy, ground displacements at a point (x, y) relative to the tunnel axis.

Figure 1.13 Application of the modified Sagaseta’s analysis to pipe jacking operations (Rogers & Chapman, 1998) (a) schematic section through pipe jack with

definitions; and (b) analytical framework.

1.4.2.4 Vafaeian’s method

Vafaeian (1991) presented a simple method of the behavior of non-dilating ground through an excavated circular shallow tunnel, which is based on mathematical relationships that account for observed soil behavior, with the following assumption: 1) no volume change occurs in the soil as the movements extend to the surface; 2) radial movements occur above the tunnel springline, whereas movements blow the springline are assumed to be zero; 3) the extent of the surface settlement trough lies at an angle of approximately 45° to the vertical centerline. Field observations show this angle varies between 30° and 75°, the lower values relating to less cohesive or cohesionless soils and the higher values to cohesive soils (Rogers & Chapman, 1998). The general formulae for vertical and horizontal components of soil deformation are shown as follows:

𝑊_𝑦 = 𝑊_{𝑚𝑎𝑥,𝑦}𝑐𝑜𝑠²𝛽𝑐𝑜𝑠 (𝜋𝛽

2𝜂) Eq. 1.13

𝑊_𝑥 = 𝑊_{𝑚𝑎𝑥,𝑥}𝑐𝑜𝑠𝛽𝑠𝑖𝑛𝛽𝑐𝑜𝑠 (𝜋𝛽

2𝜂) Eq. 1.14

where 𝜂 = 𝑡𝑎𝑛⁻¹(2𝑖 𝑧⁄ ) and 𝛽 = 𝑡𝑎𝑛⁻¹(𝑥 (𝑧 − 𝑦)⁄ ) . The surface trough width parameter, i is defined as in the error function analysis.

In Chapter 3, the ground deformation was analyzed based on the parametric study by using the numerical simulation as well. In addition, the numerical results are compared with the theoretical methods mentioned above to validate the validity of simulated results and to predict the ground deformation mechanism with different ground temperature, pipe diameter and cover depth, cohesion and friction angle of frozen ground.