Tunnelling-induced response of buried pipelines and their effects on ground settlements

Tunnelling and Underground Space Technology 96 (2020) 103193

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Tunnelling-induced response of buried pipelines and their effects on ground settlements

T

Cungang Lina,b,c, Maosong Huanga,b, , Farrokh Nadimc, Zhongqiang Liuc ⁎

a

Department of Geotechnical Engineering, Tongji University, Shanghai 200092, China Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education, Tongji University, Shanghai 200092, China c Norwegian Geotechnical Institute, Oslo 0806, Norway b

ARTICLE INFO

ABSTRACT

Keywords: Soil-pipeline interaction Tunnelling Ground movement tensionless Pasternak model Analytical solution

In urban areas, it is common to construct new tunnels beneath existing buried pipelines. The accurate estimation of the response of pipelines and the overlying ground is of great importance as it is essential for the protection of the pipelines and other facilities. However, this problem is quite difficult as it involves complicated soil-pipeline interaction. In this paper, analytical solutions incorporating the tensionless Pasternak model, which takes full account of the gap formation and the pipeline orientation, have been formulated to estimate the response of the pipeline and the overlying ground. The analytical solutions have been validated by observations from field and model tests, followed by parametric studies to investigate the effects of the gap formation and pipeline orientations. It is found that the formation of a gap beneath the pipeline lowers its deflection and bending moment generated by tunnelling, and a pipeline with a larger intersection angle with respect to the tunnel alignment is anticipated to bear larger bending moment.

1. Introduction In urban areas, buried pipelines are commonly utilized to transport water, sewage, oil, natural gas and other materials. It has been well recognized that ground movement resulting from tunnelling can cause longitudinal bending stress in overlying pipelines. Hence, a rational evaluation of the pipeline’s responses to tunnelling is of great importance for its integrity, especially in regions where large ground subsidence is anticipated. However, this problem is quite complicated since it involves soil-structure interaction, which not only determines the pipeline’s response but also alters the ground displacements. The response of buried pipelines to tunnelling has received greater attention in recent years as such projects have become more common throughout the world. Many efforts have been made to formulate analytical solutions, which are less time-consuming and easier for practical application compared with elaborate numerical simulations (Wang et al., 2011a). Attewell et al. (1986) made a comprehensive investigation of soil-pipeline interaction using the Winkler model with the subgrade modulus proposed by Vesic (1961). Klar et al. (2005a) compared an elastic continuum solution and a closed-form Winkler solution with the Vesic subgrade modulus, and gave an alternative expression of the subgrade modulus. In addition, they provided designoriented dimensionless charts that can be used for any subsurface ⁎

Gaussian settlement trough as long as maximum settlement and inflection point offset are estimated. Klar and Marshall (2008) compared the treatment of the pipeline as a beam with a more rigorous shell representation. Klar and Marshall (2015) examined the tunnel-pipeline interaction problem based on an elastic-continuum approach and proved the principle of volume loss equality between the input greenfield settlement and the generated pipeline deflection. The pipeline was generally regarded as an Euler-Bernoulli beam in these solutions. To take the influence of the pipeline joints into account, Klar et al. (2008) and Zhang et al. (2012) formulated analytical solutions for jointed pipelines by modelling the joints as rational springs. The soil behaviour was idealized as linear elastic in the above solutions. The effects of soil stiffness degradation and local yielding have been further considered (Klar et al., 2007; Klar et al., 2016; Marshall et al., 2010; Vorster et al., 2005). A method taking account of soil nonlinearity using an equivalent linear approach was presented by Vorster et al. (2005) to estimate the bending moment of continuous pipelines. Klar et al. (2007) gave a solution utilizing a boundary integral formulation to describe the elastic continuum, in conjunction with a limiting force to reflect relative pullout failure. Marshall et al. (2010) developed a new method that introduces an out-of-plane shear argument from soil-pipeline interaction into the elastic-continuum solution. Furthermore, Klar et al. (2016) presented a design-oriented

Corresponding author at: Department of Geotechnical Engineering, Tongji University, Shanghai 200092, China. E-mail address: [email protected] (M. Huang).

https://doi.org/10.1016/j.tust.2019.103193 Received 17 June 2018; Received in revised form 2 October 2019; Accepted 13 November 2019 0886-7798/ © 2019 Elsevier Ltd. All rights reserved.

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approach for evaluating the effect of tunnelling on pipelines, in which soil nonlinearity was considered by an iterative calculation of the equivalent stiffness. In addition to analytical solutions, a series of model tests and numerical simulations have been conducted to provide more insight into the dominant mechanisms governing soil-pipeline interaction induced by tunnelling (Shi et al., 2016; Wang et al., 2011a; Wang et al., 2013; Wham et al., 2016; Zhu et al., 2016). The Winkler model is most prevalent for representing the subgrade in the analytical solutions available in the literature owing to its simplicity (Attewell et al., 1986; Klar et al., 2005a; Rajani et al., 1996; Rajani and Tesfamariam, 2004; Wang et al., 2010; Wang et al., 2011b). Nevertheless, it fails to depict the continuous behaviour of real soil masses because soil displacement is assumed to be confined to the loaded region. The elastic-continuum model, as used by Klar et al. (2008) and Zhang et al. (2012) in their analytical solutions, takes account of the continuous behaviour of soil masses. However, its practical application is largely restricted due to the complex mathematical problem it presents (Selvadurai and Gladwell, 1979). In the previous studies, the pipeline and the tunnel were generally set to be perpendicular to each other for this was believed to be the most critical case (Klar et al., 2005a; Klar et al., 2005b; Klar and Marshall, 2008;Klar et al., 2008; Klar and Marshall, 2015; Marshall et al., 2010; Vorster et al., 2005; Wang et al., 2010; Wang et al., 2011b; Wang et al., 2013; Zhu et al., 2016). In reality, however, the pipeline may intersect the tunnel at an arbitrary angle. Moreover, it was observed in the centrifuge tests conducted by Shi et al. (2016) that tunnelling-induced pipeline deflection and strain in the case of an oblique intersection might be greater than that of a perpendicular one. This was also confirmed by finite element analyses (Wang et al., 2011a). Hence, the performance of a pipeline may be overestimated if only the perpendicular condition is considered. However, the effect of the intersection angles has seldom been taken into account, especially in the current analytical solutions. The above analytical solutions rely predominantly on the use of greenfield settlements as an input in order to determine the pipeline’s generated response. With regard to the greenfield settlement at the ground surface, a variety of predictive methods have been established (Avgerinos et al., 2017; Lin et al., 2015). By contrast, when it comes to the subsurface at the pipeline level, no versatile methods applicable to all types of soil are available at present. Mair et al. (1993) proposed a relationship that relates the settlement at depth to that at the ground surface, which was utilized by Klar and Marshall (2008) for analysis of soil-pipeline interaction. However, as this relationship was originally formulated from field and centrifuge model tests in London clay, its applicability to soils of other types still needs further investigations. Regarding tunnelling-induced greenfield displacements in sands, many centrifuge tests have been executed in the UK to investigate the effects of tunnel size, depth, volume loss, soil relative density, and stratification (Franza, 2016; Franza et al., 2019; Marshall, 2009; Marshall et al., 2012; Zhou, 2015). It is worth mentioning that, based on the results from these centrifuge tests, empirical and semi-analytical methods have been derived by Franza and Marshall (2019) to estimate tunnellinginduced ground movements in sands. As observed in several model tests, a gap may occur beneath the pipeline during the generation of tunnel volume losses (Marshall et al., 2010; Vorster et al., 2006; Wang et al., 2014; Wang et al., 2015). The gap consumes part of the volume loss and hence affects the pipeline’s responses. The effect of gap formation beneath foundations of surface buildings has been investigated by both centrifuge modeling (Farrell et al., 2014) and analytical solutions of tunnel-building interaction (Basmaji et al., 2019; Deck and Singh, 2012; Franza and DeJong, 2019). However, such an effect on tunnel-pipeline interaction has not yet been incorporated in the existing analytical solutions. It has been proven by field observations and model tests that tunnelling-induced ground settlement above a pipeline, compared with the

greenfield settlement, is significantly modified due to the pipeline’s existence (Jia et al., 2009; Marshall et al., 2010; Ma et al., 2017a; Ma et al., 2017b; Shi et al., 2016; Wang et al., 2015; Zhao et al., 2015). In areas where multiple underground facilities are located at different depths, for instance when a pipeline is buried beneath some other facilities, it is essential to predict the ground settlement above the pipeline for estimation of the effect on the overlying facilities during the tunnelling process. Nevertheless, there are no analytical solutions available at present to predict the tunnelling-induced ground settlement taking the pipeline’s modification effect into account. This paper aims to provide analytical solutions to predict the response of buried pipelines to tunnelling using the Pasternak model (Pasternak, 1954), which is capable of reflecting the soil’s continuous behaviour and possesses the advantage of mathematical simplicity. Subsequently, a procedure that takes account of the gap formation and the pipeline’s shielding effect is put forward to estimate the ground settlement above the pipeline. In particular, the effect of the pipeline’s orientation with respect to the tunnel alignment is incorporated into the analytical solutions. Finally, the analytical solutions are verified using published observations and parametric studies are conducted for deeper investigation. 2. Theoretical formulations Fig. 1 depicts the intersection of a tunnel and a pipeline with an arbitrary angle θ, which is the problem to be formulated in this study. The formulations are based on the following assumptions: (1) both the tunnel and the pipeline are buried horizontally in a homogeneous isotropic elastic soil; (2) the pipeline and the surrounding soils are represented as an Euler-Bernoulli beam and a tensionless Pasternak model, respectively, for soil-pipeline interaction analysis; (3) contact is always maintained between the pipeline and the surrounding soils except at the region where a gap may occur; (4) a gap most probably occurs beneath the pipeline at the location just above the tunnel centerline; (5) the soil-pipeline interaction forces acting on the pipeline are expected to be downward at the region where a gap exists; (6) the soil above the pipeline is represented as an elastic-continuum model for estimation of its displacement caused by the loading from soil-pipeline interaction; and (7) the responses of the soil and the pipeline to loading from soil-pipeline interaction are both linear elastic. 2.1. Tunnelling-induced greenfield settlements It has been well recognized that tunnelling-induced transverse surface settlement troughs at a greenfield site can generally be described by a Gaussian function as (Peck, 1969)

s (x ) = smax exp

smax =

Rt2 Vl 2 is

x2 2is2

(1a) (1b)

where x′ is the horizontal distance to the tunnel centerline; smax is the maximum ground surface settlement located directly above the tunnel centerline; s(x′) is the ground surface settlement at coordinate x′; is is the trough width factor, denoting the horizontal distance from the tunnel centerline to the inflection point of the settlement trough; Rt is the tunnel radius; and Vl is the volume loss. Numerous field observations have proven that the subsurface settlement trough can also be well depicted by Eqs. (1), with is expressed as (O'reilly and New, 1982)

i s = K (z 0

z)

(2)

where K is the trough width parameter; z0 and z are the depth of the tunnel axis and the subsurface under consideration, respectively. Surface measurements will almost certainly remain more 2

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Fig. 1. Illustration of the intersection of a tunnel and a pipeline.

economical and feasible compared with subsurface measurements. It therefore makes practical sense to predict settlements at depth based on that at the surface. Many such attempts have been made by exploring the variation of K with depth (Dyer et al., 1996; Heath and West, 1996; Hu et al., 2016; Han et al., 2017; Jones, 2010; Lee, 2009; Mair et al., 1993; Moh et al., 1996; Marshall et al., 2012; Pan, 2016; Wan et al., 2017; Zhao, 2004). Based on empiricism gained from field observations or model tests worldwide, a versatile relationship, as described in Eq. (3), is proposed herein for estimation of K with depth in both cohesive and granular soils (Lin and Huang, 2019; Marshall et al., 2012).

K = Ks +

1

z/z 0 z /z 0

additional assumption of equal volume loss at different depths. For tunnels in clay, if no sound experience is available, the formula of Mair et al. (1993) for London clay can be applied for a rough estimation. While for tunnels in sand, a rough estimation could be achieved by assuming Ks = 0.375 and ξ = 0.125, which are obtained from limited observations (Dyer et al., 1996; Hu et al., 2016; Lee, 2009; Moh et al., 1996). However, more data are needed for a reliable prediction. In circumstances where an elaborate estimation is required, the empirical and semi-analytical methods (Franza et al., 2019; Franza and Marshall, 2019) that consider more details (including tunnel volume loss, soil relative density, and geometrical parameters) can be referenced. It still needs to be stated that only tunnelling-induced immediate settlements are concerned, without consideration of their progressive propagation as the tunnel advances or any ongoing development due to consolidation.

(3)

where Ks is the trough width parameter at the ground surface; and ξ is an empirical parameter that determines the changing rate of K with depth. It deserves mentioning that the formula established by Mair et al. (1993) in London clay is a particular case of Eq. (3) for Ks = 0.5 and ξ = 0.175. From the perspective of field applications, K at depth could be estimated with confidence when reliable empirical values of Ks and ξ are available from a large number of field observations or model tests, especially that limited to a specified field site or soil type. For practical purposes, the transverse settlements both at and below the ground surface can be estimated by a combination of Eqs. (1)–(3), with the

2.2. Formulation of the pipeline deflection and bending moment Fig. 2 illustrates the problem to be solved. Herein the pipeline is taken as an Euler-Bernoulli beam. Consider an infinitesimal element of length dx of the beam at an arbitrary coordinate x as shown in Fig. 2(a). The internal forces that act on the element are the vertical force Q, the bending moment M and the subgrade reaction force q(x) resulting from soil-pipeline interaction. Note that q(x) is the resultant of reaction forces from soils both above and below the pipeline. The force imposed by the pipeline on the surrounding soil, p(x), is

Fig. 2. Illustration of tunnelling-induced pipeline deflection in the x-z and y-z planes. 3

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the counterforce of q(x), therefore

p (x ) =

(4)

q (x )

The bending rigidity of the pipeline along the y-direction is assumed to be infinite, thus a rigid body translation u(x) in the z-direction is anticipated for each cross-section of the pipeline perpendicular to the xdirection, as illustrated in Fig. 2(b). In addition, the subgrade is idealized as a Pasternak model, hence ground settlement within the projection plane of the pipeline at the horizontal level of its axis can be expressed as

p (x ) = ku (x )

Gp

d2u (x ) dx 2

(5)

where u(x) is the ground vertical displacement at the depth of the pipeline axis generated by p(x); k is the coefficient of subgrade reaction; and Gp is the subgrade shear stiffness. The equilibrium of the vertical forces acting on the infinitesimal element gives

dQ = dx

q (x ) D p

Fig. 3. Illustration of likely gap formation beneath the pipeline.

As is assumed above, the pipeline keeps in contact with the surrounding soil along its longitudinal direction except within the zone where a gap occurs beneath the pipeline. Thus, for soils above the pipeline, the ground displacement at the pipeline level, g(x), is always equal to the pipeline deflection, w(x), that is

(6)

where Dp is the outer diameter of the pipeline. The moment equilibrium of the infinitesimal element yields

dM + Q dx

q (x ) Dp

(dx ) 2 =0 2

However, for soils beneath the pipeline, Eq. (15) is generally satisfied except within the gap zone. Then, substituting Eqs. (13) and (15) into Eq. (12) results in

Then, neglecting of the terms of the second order of smallness with respect to dx leads to

Q=

dM dx

(8)

Ep Ip

Subsequently, the differentiation of Q with respect to x results in

dQ d2M = dx dx 2

Ep Ip

(9)

d2w (x ) dx 2

(10)

where Ep and Ip are the Young's modulus of the pipeline and the second moment of area of its cross-section, respectively; and w(x) is the pipeline deflection. Furthermore, differentiating Eq. (10) twice with respect to x yields

d2M = dx 2

Ep Ip

d4w (x ) dx 4

(11)

kug (x )

Thus, the differential equation governing the pipeline deflection caused by soil-pipeline interaction due to tunnelling can be obtained by a combination of Eqs. (4)–(6), (9) and (11), which is

d4w (x ) Ep Ip + kDp u (x ) dx 4

d2u (x ) Gp Dp =0 dx 2

(12)

s (x ) =

s (x ) =

2 K (z 0

zp )

exp

Gp Dp

d2s (x ) dx 2

(16)

Gp

d2ug (x ) dx 2

=

rs zp

(17) (18a)

Rt2 Vlt exp 2 K (z 0 z p )

(x sin ) 2 2K 2 (z 0 zp) 2

(18b)

where s′(x) is tunnelling-induced ground settlement at coordinate x at depth zp in the gap zone; and Vlt is the volume loss resulting from tunnelling at the tunnel level. Then, substituting Eq. (18a) into Eq. (17) yields

where s(x) is tunnelling-induced ground settlement at coordinate x at the depth of the pipeline axis. When the intersection angle between the tunnel and the pipeline is θ, as shown in Fig. 1, x' = xsin(θ). Substituting this into Eqs. (1) and (2) yields

(x sin ) 2 2K 2 (z 0 zp) 2

d2w (x ) = kDp s (x ) dx 2

g (x ) = s ( x ) + u g ( x )

(13)

Rt2 Vlp

Gp Dp

Similarly,

Ground displacement at the pipeline level, g(x), is actually generated by the combined effects of tunnelling and soil-pipeline interaction, thus

g (x ) = s (x ) + u ( x )

d4w (x ) + kDp w (x ) dx 4

When a gap is formed beneath the pipeline, as illustrated in Fig. 3, the surface deflection of soils beneath the pipeline in the range of the gap, g′(x), can also be derived by application of the Pasternak model. The initial soil stress at the gap surface is approximated to be rszp, where rs is the depth-averaged unit weight of soils above the pipeline axis. Once the gap emerges, the stress from the soil that separates from the pipeline changes to zero. Thus, the external force imposed on the soil at the gap surface is -rszp with respect to its initial state. Its generated surface deflection, ug(x), is also assumed to be invariable along the y-direction within the pipeline’s vertical projection area. Thus, ug(x) can be formulated as Eq. (17) by application of the Pasternak model for plane strain problems.

The basic equation governing the flexure of an Euler-Bernoulli beam is (Timoshenko, 1940)

M=

(15)

w (x ) = g (x )

(7)

kg (x )

Gp

d2g (x ) = ks (x ) dx 2

Gp

d2s (x ) dx 2

rs zp

(19)

Finally, both the pipeline deflection and the ground displacement at the pipeline level can be determined by solving Eqs. (16) and (19). Subsequently, the generated bending moment of the pipeline can be determined using Eq. (10). However, whether a gap will appear or not should be judged with great caution. The judgment herein is based on the relative magnitude

(14)

where zp is the burial depth of the pipeline axis; and Vlp is the volume loss that takes part in soil-pipeline interaction. 4

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Fig. 4. Illustration of calculation of ground displacement induced by p(x)Dp.

of g′(x) and w(x). If g′(x) ≤ w(x) in the likely gap zone, no gap will occur.

C, du*(x, y, z), can be derived as

2.3. Formulation of the ground settlement above the pipeline

+

It is deemed in this study that the pipeline’s modification effect on tunnelling-induced greenfield settlement stems from two aspects: (1) additional ground displacement generated by soil-pipeline interaction forces; and (2) a decrease of volume loss above the pipeline caused by consumption of volume loss by the gap beneath the pipeline.

Dp

zp ) 2 R13

+

(3

4vs)(z + zp)2

2zzp

R23

+

8(1

vs )2 (3 R2

6zzp (z + zp)2 R25

]

4vs )

(21a)

R1 =

(x

h ) 2 + y 2 + (z

z p) 2

(21b)

R2 =

(x

h ) 2 + y 2 + (z + z p ) 2

(21c)

u (x , y , z ) =

Ep Ip d4w (x ) dx 4

(z

+

where Es and vs are the Young's modulus and Poisson's ratio of soil, respectively. Furthermore, u*(x, y, z) can be obtained by integration of Eqs. (21), that is

2.3.1. The effect of soil-pipeline interaction After w(x) has been estimated, a combination of Eqs. (4), (6), (9) and (11) yields

p (x ) =

(1 + vs ) p (x ) Dp dh 3 4vs [ R 8 Es (1 vs) 1

du (x , y , z ) =

(20)

b a

du (x , y , z )

(22)

where a and b are the coordinates of the pipeline’s left end and right end, respectively.

It is predefined that p(x) is distributed within the projection plane of the pipeline at the horizontal level of its axis. For simplicity in mathematical treatment, p(x) can be equivalently transformed to a line load of p(x)Dp acting upon the pipeline axis. As depicted in Fig. 4, it is attempted to estimate the ground displacement at an arbitrary point C with coordinate (x, y, z), u*(x, y, z), induced by p(x)Dp. Now consider an infinitesimal length dh of p(x)Dp at coordinate (h, 0, zp), which can be regarded as a concentrated load of p(x)Dpdh. Using the Mindlin solution (Mindlin, 1936), the ground displacement caused by p(x)Dpdh at point

2.3.2. The effect of gap formation Herein it is assumed that equal volume loss, Vlt, is generated along the tunnel’s longitudinal direction. However, at the pipeline level, part of the volume loss, Vlg, which is manifested as a gap beneath the pipeline, as demonstrated in Fig. 3, is not expected to propagate to the soils above the pipeline. For analysis of induced settlement of the ground far from the tunnel, the volume loss can be assumed to be

Fig. 5. Illustration of distribution and propagation of volume loss along tunnel alignment. 5

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concentrated at the tunnel axis. Fig. 5 illustrates the distribution and propagation of volume loss that could propagate to the soils above the pipeline along the tunnel’s longitudinal direction. Along the tunnel axis, within the vertical projection of the pipeline’s cross-section, the volume loss is (Vlt-Vlg); while within other sections, it is Vlt. Given the assumption that soil volume loss does not vary with depth between the tunnel and the pipeline, and because of the conservation of volume loss at the pipeline level, it can be expressed that

Vlp = Vlt

(23)

Vlg

The expected ground surface settlement along the tunnel axis, s(y′), can be estimated by Eq. (24), using the virtual image technique proposed by Lin et al. (2014).

s (y ) Dp 2

=

Rt2 Vlt z 02 r12 z0

Dp 2 Dp

d +

Rt2 Vlp z 02 r12 z0

2

d +

Dp 2

Rt2 Vlt z 02 z0 r12

d (24a)

r1 =

=

)2 + z 02

(y

(

(x ) =

1 2

(24b)

)

(24c)

( ) 0

tx

1 exp(

t )dt

(24d)

Fig. 6. Flow chart for calculation of pipeline deflection and ground movement.

where α is the modification coefficient of settlement trough width; η is the conversion coefficient of volume loss; and Γ(x) is the Gamma function. According to Eq. (1b), under the assumption of constant trough width factor is, the volume loss at the ground surface of any transverse section perpendicular to the tunnel axis, Vl(y′), is proportional to s(y′). Thus, Vl(y′) can be simply determined by

Vl (y ) =

s (y ) Vlt s

(1) Determination of Vlt and Ks based on observed greenfield surface settlement. (2) Estimation of K at the depth of the pipeline axis using Eq. (3). (3) A trial method is applied to determine the value of Vlg. First, an initial value of Vlg, which is denoted as Vlg-0, is assumed for the firstround calculation. Then both w(x) and g′(x) can be estimated using Eqs. (14), (16), (18), (19) and (23). As indicated in Fig. 3, the area of the gap, Ag, which is enveloped by the curves of w(x) and g′(x), can be expressed as

(25)

where s∞ is the calculated s(y′) as y′ approaches infinity. Afterwards, for an arbitrary depth z (z ≤ (zp-0.5Dp)), the anticipated volume loss at coordinate y′, Vlz(y′), can be approximated by Eq. (26a) and (26b) using the linear interpolation method.

Vlz (y ) = Vl (y )

Vlzp (y ) =

(Vl (y )

Vlt , y < Vlt

Vlg,

Dp 2 Dp

Vlzp (y )

or y >

2

y

)

z zp

Ag = Rt2 Vlg - c

where Vlg-c is the calculated volume loss occupied by the gap. Herein the criterion of judging the appropriateness of Vlg-0 is described below. If |Vlg-c-Vlg-0| ≤ 0.1Vlt, then Vlg-0 can be approximately taken as the actual value of Vlg. Otherwise, other values (for instance, Vlg-1 = Vlg-0 ± 0.05Vlt) are assigned to Vlg-0 for a new round of calculations until the above criterion is satisfied. Subsequently, w(x) can be determined.

(26a)

Dp 2 Dp 2

(26b)

where Vlzp (y ) is the anticipated volume loss that could propagate to the soils above the pipeline at coordinate y′ at depth of the pipeline axis. After the volume loss at any depth above the pipeline has been determined, tunnelling-induced ground settlement, s*(x), which takes the pipeline’s shielding effect into account, can subsequently be calculated using Eqs. (1)–(3). Thus the ground displacement above the pipeline at an arbitrary point C with coordinate (x, y, z), g*(x, y, z), can be determined by

g (x , y , z ) = u (x , y , z ) + s (x )

(28)

(4) Estimation of the ground displacement above the pipeline. 3.2. Finite difference formulation The finite difference method has been employed to solve the governing differential equations (Eqs. (16) and (19)). Subsequently, g*(x, y, z) can be solved numerically by a discrete method.

(27)

3.2.1. Solution of pipeline deflection As illustrated in Fig. 7, the pipeline of L in length is evenly divided into n elements with (n + 1) difference nodes (i = 0, 1, 2, ···, n). Thus, the length of each element, l, is L/n. In order to establish the difference equations for the nodes at the pipeline ends (i = 0, 1, n-1, n), four virtual difference nodes (i = −2, −1, n + 1, n + 2) are added. The application of the central-difference method (Qian and Yin, 1991) to Eq. (16) yields

3. Method of solution 3.1. Calculation procedure Fig. 6 outlines the procedure for calculation of both the pipeline deflection and the ground settlement, which is described in detail as follows. 6

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L m nodes ...

m1 nodes

Ep Ip

(

wi + 2

4wi + 1 + 6wi l4

) + kD w G D (

4wi 1 + wi 2

= kDp si

p

p

p

si + 1

i

Gp Dp

2si + si 1 l2

)

(

wi + 1

2wi + wi 1 l2

)

t

where wi-2, wi-1, wi, wi+1 and wi+2 are the pipeline deflection at the (i − 2)-th, (i − 1)-th, i-th, (i + 1)-th and (i + 2)-th difference node, respectively; and si-1, si and si+1 are the tunnelling-induced ground settlement at location of the (i − 1)-th, i-th and (i + 1)-th difference node, respectively. Thus, a total of (n + 1) equations can be built at difference nodes of i = 0 ∼ n. The boundary conditions are that both the bending moment (M0 and Mn) and the shear force (Q0 and Qn) at the two ends of the pipeline are zero (subscript relates to difference nodes). Hereinafter, Mi and Qi are referred to the bending moment and the shear force of the pipeline at the i-th difference node, respectively. A combination of Eqs. (8) and (10) results in

Q=

Ep Ip

Ep Ip Ep Ip

w1

wn + 1

w2

1

=0

2wn + wn l2

2w1 + 2w 2l3

wn + 2

1

Mi =

1

w

2wn + 1 + 2wn 2l3

2

1

wn

=0

[K s ]{w} = [Kr ]{s }

[K s ]{s }

k=

=

Es 1

vs

8

Es Ep Ip Dp 4

2.18, zp /Dp 1+

1 1.7zp / Dp

(31d) (32)

qi =

Es Ht 6(1 + vs)

t

(35)

Ep Ip

wi + 1

2wi + wi l2

1

(36)

(37)

(38)

Ep I Dp l 4

(wi + 2

4wi + 1 + 6wi

4wi

1

+ wi 2)

(39)

where qi is the subgrade reaction force acting on the difference nodes of i = 0 ∼ n. Likewise, the subgrade reaction force imposed at the four virtual difference nodes is taken to be zero:

q - 2 = q - 1 = qn + 1 = qn + 2 = 0

(40)

Similarly, Eqs. (39) and (40) can be generalized to a matrix form as

(33a)

{q} = [Cr ]{w}

(41)

where {q} is the vector presenting the subgrade reaction force imposed at every difference node; and [Cr] is the coefficient matrix of subgrade reaction. As can be seen from Eqs. (35), (55), and (59) (Eqs. (55) and (59) are listed in the Appendix), the direction of q(x) along the pipeline keeps constant with variations of Vlp. Therefore, after qi has been obtained from Eq. (41) for any given Vlp, the gap is anticipated in the region where the direction of qi is downward. Thus, the length of the gap zone

(33b)

The subgrade shear stiffness Gp was given by Selvadurai and Gladwell (1979) as

Gp =

[K s ]){s}

3.2.2. Solution of ground displacement at the gap zone As the subgrade is idealized as a tensionless Pasternak model, the pipeline solely bears compression force from the surrounding soils. Thus, according to the assumptions (4) and (5) mentioned above, q(x) is anticipated to be downward at the section of the pipeline where a gap may occur. Substituting Eqs. (4) into Eq. (20), combining with the central-difference method, q(x) can be numerically solved as

0.5

, zp / Dp > 0.5

[K s]) 1 ([Kr ]

where {M} is the moment vector presenting the pipeline bending moment at every difference node; and [Cm] is the coefficient matrix of bending moment. Thus, the pipeline’s deflection and bending moment can both be numerically solved.

where [Kp] is the stiffness matrix of the pipeline; {w} and {s} are the displacement vectors representing the pipeline deflection and the tunnelling-induced ground settlement at every difference node, respectively; [Kr] is the stiffness matrix of subgrade reaction; and [Ks] is the matrix of subgrade shear stiffness. The coefficient of subgrade reaction k formulated by Yu et al. (2013) as expressed in Eqs. (33) is utilized herein. It considers the influence of pipeline embedment depth and has been verified by centrifuge model tests.

3.08

(34b)

{M } = [Cm ]{w}

Eqs. (29) and (31) can be generalized to a matrix form as

[Kp ]{w} + [Kr ]{w}

p

Similarly, M0, Mn, and Eqs. (36) and (37) can be generalized to a matrix form as

(31c) 2

p

M - 2 = M - 1 = Mn + 1 = Mn + 2 = 0

(31b)

=0

Ht

sinh2 (Ht p)

Herein Eq. (36) is applicable at the difference nodes of i = 1 to (n − 1). However, the bending moment at the four virtual difference nodes is deemed to be zero, that is

(31a)

=0

sinh(Ht p) cosh(Ht p)

Furthermore, the application of the central-difference method (Qian and Yin, 1991) to Eq. (10) leads to

(30)

2w0 + w l2

3 2Ht

{w} = ([Kp] + [Kr ]

Then, the application of the central-difference method (Qian and Yin, 1991) to Eqs. (10) and (30) leads to

Ep Ip

=

where Ht is the thickness of the shear layer; and γp is the empirical constant. From Eq. (32), {w} can be deduced as

(29)

d3w (x ) Ep Ip dx 3

Fig. 7. Illustration of division of a pipeline for finite difference scheme.

Possible gap zone l

(34a) 7

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along the pipeline, Lg, can be approximated. Moreover, the number of difference nodes located in the gap zone, m, can be obtained by examining the direction of qi, which are labeled as j = 1, 2, ···, m. Similarly, the number of difference nodes located on the left of the gap zone, m1, can also be determined. Fig. 7 depicts the labels of the difference nodes inside and outside the gap zone. Similarly, the application of the central-difference method Qian and Yin, 1991) to into Eq. (19) results in

kg j

Gp

gj+1

2g j + g j l2

1

= ks j

Gp

sj+1

2s j + s j l2

1

rs zp

After Vlz(y′) has been determined by Eqs. (24)–(26), {s*} can be estimated by

(42)

gm + 1 = wm1+ m

[K s ]{g } = [K r2 ]{s }

where [K′r1] and [K′r2] are the stiffness matrix of subgrade reaction; [K′s] is the matrix of subgrade shear stiffness; {g′} and {s′} are the displacement vectors representing total and tunnelling-induced ground settlement of every difference node at the gap zone, respectively; and {C′} is the vector of constants. Finally, {g′} can be derived from Eq. (44) that

{g } = ([K r1]

[K s ]) 1 [([K r2]

ij

+

(45)

[K s ]){s } + {C }]

(1 + vs ) 3 4v [ R s 8 Es (1 vs) 1

=

(z

zp ) 2 R13

+

(3

+

4vs)(z + zp)2

8(1

2zzp

R23

vs)2 (3 R2

+

4vs)

6zzp (z + zp)2 R25

]

(50a)

R1 =

[(i

j) l]2 + y02 + (z

z p) 2

(50b)

R2 =

[(i

j ) l]2 + y02 + (z + zp)2

(50c)

Observations from field and model tests in the literature are used to verify the analytical solutions formulated above.

where {p} is the vector presenting the force imposed by the pipeline on the soil at every difference node. Hence, Eqs. (21)–(22) and can be alternated by the discrete method. Moreover, s*(x) can also be discretely presented. Thus, g*(x, y, z) can be depicted by the vertical displacements of a series of difference nodes, {g*}, as shown in Fig. 4. Similarly, Eq. (27) can be expressed in a matrix form as

4.1. A centrifuge model test in sand A three-dimensional centrifuge model test was conducted by Ma et al. (2017a) to investigate the effects of tunnelling on a pipeline buried in sands. Herein the measurements in model scale are used for analysis. The general dimensions and characteristics of the model are given in Table 1. Two lines of displacement transducers transverse to the tunnel alignment and parallel to the pipeline were installed to measure the ground surface settlements. One line is right above the pipeline axis, and the second line is at a horizontal distance of 3 Dp from the pipeline axis. In addition, the pipeline’s deflection and longitudinal strain were

(47)

{g } = {u } + {s }

(49)

4. Observation verifications

(46)

[Cr ]{w}

(48)

n+ 5

Subsequently, {u*} and {g*} can be calculated by a combination of Eqs. (47)–(49).

3.2.3. Solution of ground displacement above the pipeline As q(x) has been solved by the finite difference method, p(x) can also be presented in a discrete form. By combining Eqs. (4) and (41), {p} can be deduced as

{p} =

(xn + 2 sin )2 2K2 (z 0 z )2

where [λ] is the matrix of force-displacement coefficient, and λij denotes its (i, j)-th element, which defines the soil displacement at difference node i (labelled as 'Target node' in Fig. 4) at depth z and y = y0 due to a unit point load imposed at the j-th difference node of the pipeline (labelled as 'Loaded node' in Fig. 4); and {P} = Dpl{p}, which is the vector presenting the equivalent concentrated load imposed on the soil at every difference node along the pipeline. With reference to Eq. (21) for x = (i-j)l, y = y0, h = 0, and p(x) Dpdh = 1, λij can be derived as

(44)

[K s ]{s } + {C }

(xi sin )2 2K2 (z 0 z )2

{u } = [ ]{P }

Then, Eqs. (42) and (43) can be expressed as a matrix form as

[K r1 ]{g }

(x - 1 sin )2 2K2 (z 0 z )2

where xi is the x coordinate of the i-th difference node at depth z and y = y0 . Using the discrete method, {u*} can be formulated as

(43b)

2

exp

exp

(43a)

3

(x - 2 sin )2 2K2 (z 0 z )2

Rt2 Vlz (y ) 2 K (z 0 z ) exp

{s } =

where g′j-1, g′j, and g′j+1 are the total ground displacement of the (j − 1)-th, j-th, and (j + 1)-th difference node at the gap zone, respectively; s′j-1, s′j, and s′j+1 are tunnelling-induced ground displacement of the (j − 1)-th, j-th, and (j + 1)-th difference node at the gap zone, respectively. Herein Eq. (42) is applicable to the difference nodes of j = 1 ∼ m. The displacements of another two difference nodes outside the gap zone (j = 0 and (m + 1)), as shown in Fig. 7, can be determined by the boundary conditions. Since these two difference nodes are outside the gap zone, their displacements are incorporated in {w}, thus

g0 = wm1

exp

where {s*}, {u*} and {g*} are the displacement vectors representing the ground settlement of the difference nodes at depth of z and y = y0 due to tunnelling, soil-pipeline interaction forces and their combined effects, respectively. Table 1 Dimensions and characteristics of the model (Ma et al. 2017a). Pipeline

Tunnel

Dp (10−3 m)

tp (10−3 m)

L (10−3 m)

Ep (GPa)

Ip (10−8 m4)

EpIp (kN·m2)

zp (10−3 m)

31.75

2.08

1150

69

2.1439

1.4793

95.875

Rt (m) 0.05

Z0 (m) 0.25

Soil

Es (MPa) 10

vs 0.3

Notes: tp is the thickness of the pipeline wall. 8

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Fig. 8. Observed and calculated results for case 1.

measured. Based on the measured strain, the pipeline bending moment, M, can be calculated by (Timoshenko, 1940)

M=

comparison. It should be mentioned that the effect of gap formation is not considered in the original solution of Zhang et al. (2012). Herein, gap formation is incorporated into their original solution in the same way as in this study. The values of the parameters used in the calculation are listed in Table 2. The value of α is generally found to be around 5 by numerous observations in Hangzhou soft ground (Lin et al., 2014). Following this experience, α is taken to be 5 for calculation of volume loss at any specified depth above the pipeline. As shown in Fig. 8(a) and (b), there is generally good agreement between the calculated and the experimental results. The ground surface settlements observed by the two measurement lines are presented in Fig. 8(c), from which it can be seen that the observed greenfield surface settlement is satisfactorily fitted with Eq. (52). The fitted values of Ks and Vlt are 0.558 and 3.76%, respectively. However, in the analysis of soil-pipeline interaction, the input volume loss at the pipeline level (which is Vlp in Eq. (14)) is lowered to 2.90% for the emergence of a gap at the soil-pipeline interface. The elastic-continuum solution presented in Fig. 8(c) was obtained by a combination of the method of this study and that of Zhang et al. (2012). First of all, {p} was obtained using the method of Zhang et al. (2012), and then {g*} was calculated using Eqs. (47)–(50). In the elastic-continuum solution, the subgrade is always idealized as the elastic-continuum model. As shown in Fig. 8(c), the results by the two

Ep Ip (51)

Rt

where ε is the longitudinal strain measured at the pipeline outer surface. By combining Eqs. (1) and (2) for x' = xsin(θ), tunnelling-induced greenfield settlement at depth z, sz(x), can be derived as

sz (x ) =

Rt2 Vlt exp 2 K (z 0 z )

(x sin ) 2 2K 2 (z 0 z ) 2

(52)

Measurements at the second line can be considered as the greenfield surface settlement solely due to tunnelling, as previous studies have suggested that ground settlement at this distance is slightly affected by the presence of the pipeline (Yeates, 1984). Thus, the values of Ks and Vlt can be determined by fitting the measurement data at the second line using the least square method with Eq. (52). In Fig. 8, experimental results are presented and compared with the results calculated using the above analytical formulas. To further confirm the validity of the solutions in this study, the results calculated using the elastic continuum solution derived by Zhang et al. (2012), which has already been verified experimentally, are also displayed for

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Table 2 Values of the parameters used for calculation (Case 1, 3, 4 and 5). Parameters

Case 1 Case 3

θ (˚)

90 90 60 90 90

Test P Test O

Case 4 Case 5

Ks

0.558 0.367 0.309 0.45 0.292

Volume loss Vlt (%)

Vlp (%)

Vl(0) (%)

3.76 0.94 0.94 0.7 1.52

2.90 0.65 0.65 0.7 1.00

3.45 0.85 0.85 0.7 –

ξ

n

Ht

0.125 0.125 0.125 0.10 0.125

1001 1001 1001 1001 1001

10 10 10 10 10

DP DP DP DP DP

γp (m−1)

α

0.70 0.70 0.70 0.7 0.70

5 5 5 – –

Notes: Vl(0) is the volume loss at the ground surface at y'=0.

Table 3 Dimensions and characteristics of the model (Wang et al. 2014). Pipeline

Tunnel

Table 4 Values of the parameters used for calculation (Case 2).

Dp (m)

tp (m)

L (m)

Ep (GPa)

Ip (10−5 m4)

EpIp (kN·m2)

zp (m)

0.2

0.02

2

2.9

4.637

134.47

0.75

Rt (m 0.5

Z0 (m) 1.5

Soil

Es (MPa) 0.4

vs 0.3

Parameters

θ (˚)

K

90

0.399

Volume loss Vlt (%) Vlp (%) 0.84 0.80

n

Ht

γp (m−1)

1001

10 DP

0.70

analytical methods are close to each other, and both can give a general estimation of the ground surface settlement above the pipeline. In general, both measured and calculated results demonstrate that the existence of the pipeline results in a wider but shallower settlement trough than that of the greenfield. This is consistent with the finding of Wang et al. (2014) in a model test. 4.2. A 1-g model test in sand A three-dimensional model test was conducted by Wang et al. (2014) to investigate the pipeline responses to tunnelling in sand. The general dimensions and characteristics of the model are listed in Table 3. The applied instrumentation is schematically shown in Fig. 9. Measurement line A on the crown of the pipeline is used to monitor the

Fig. 10. Observed and calculated ground settlement and pipeline deflection.

pipeline deflections. Measurement line B is located at a horizontal distance of 4 Dp from the pipeline axis; this is used to reflect the greenfield subsurface settlements at the depth of the pipeline axis. Measurement line C is placed at the depth of the pipeline invert and right below the pipeline sidewall; this is used to observe ground settlements beneath the pipeline. The measurements in model scale are used for analysis. Similarly, the values of K and Vlt are obtained by curve fitting with Eq. (52). The pipeline deflections are calculated using the analytical solutions of this study and Zhang et al. (2012), with the values of the parameters listed in Table 4. The observed and calculated results are shown in Fig. 10, from which it can be seen that Eq. (52) can describe the greenfield settlement with great accuracy, and a high degree of consistency is achieved between the measured and calculated pipeline deflections. As shown in Fig. 10, a gap is formed beneath the pipeline, which can be depicted well by the analytical solutions, in terms of its width and magnitude. 4.3. A centrifuge model test considering the pipeline orientations Centrifuge tests were conducted by Shi et al. (2016) to investigate the three-dimensional responses of the ground and pipeline to tunnelling in sand, considering different pipeline orientations with respect to the tunnelling direction. The dimensions and characteristics of the model are summarized in Table 5.

Fig. 9. Layout of instrumentation (after Wang et al. 2014). 10

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Table 5 Dimensions and characteristics of the model (Shi et al. 2016). Pipeline

Dp (10−3 m) 15.88

tp (10−3 m) 1.65

L (10−3 m) 0.92

Tunnel

Rt (m) 0.152

z0 (m) 0.301

Soil

Ep (GPa) 70

Ip (10−9 m4) 1.8922

EpIp (N m2) 132.4515

Es (MPa) 10

vs 0.3

zp (10−3 m) 37.94

Fig. 11. Observed and calculated results for case 3.

Two tests, referred to as Test P and Test O, were carried out, in which the intersection angles (θ) between the axis of the model pipelines and the tunnel were 90˚ and 60˚, respectively. The ground surface settlement was measured by one linear variable differential transformer (LVDT) installed directly above the pipeline axis and a row of LVDTs at a horizontal distance of 4.8 Dp. As mentioned above, measurement by LVDTs of this row may be considered as the greenfield surface settlement. Thus, the values of Ks and Vlt can be determined by curve fitting with Eq. (52).

At the crown and invert of the pipeline, strain gauges were mounted on the outer surface to measure its longitudinal strain. Instituting the measured strain into Eq. (51), the pipeline bending moment can be obtained. The test results in model scale are used for analysis. The analytical methods of this study and Zhang et al. (2012) are applied to calculate the pipeline bending moment, with values of the parameters listed in Table 2. The measured and calculated pipeline bending moment in Test P and Test O are presented in Fig. 11(a) and (b), respectively, from which it can be seen that the analytical solutions

Table 6 Dimensions and characteristics of the case history (Jia et al. 2009). Pipeline

Dp (m) 3

tp (m) 0.2

L (m) 100

Tunnel

Rt (m) 3

z0 (m) 14.4

Soil

Ep (GPa) 25

11

Ip (m4) 1.7329

EpIp (1010N·m2) 4.3323

Es (MPa) 8.2

vs 0.3

zp (m) 8.7

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Fig. 12(b) shows the observed ground surface settlement above the cable pipeline and the calculated results using the elastic-continuum solution and the method of this study. As shown in Fig. 12(b), the two analytical solutions produce results close to each other, and they can both give a rough prediction of the ground surface settlement above the cable pipeline. It is noted that the values of Vlt and Vlp are equal in this case, which is different from the three previous cases. This implies that no gap is formed beneath the cable pipeline. Compared with the three model tests mentioned above, the initial soil pressure exerted on the cable pipeline is much larger due to its deep burial depth, which may hinder the gap formation. As stated above, the pipeline’s modification to tunnelling-induced greenfield settlements relies on the effects of soil-pipeline interaction and the gap formation. Since no gap is formed in this case, this offers an opportunity to solely quantify the effect of soil-pipeline interaction. Fig. 13 examines the variation of maximum settlements along the tunnel’s longitudinal direction. In Fig. 13, g*max is the estimated ground surface settlements above the tunnel centerline, and sgmax is the maximum greenfield surface settlement, which is obtained from the best-fit curve in Fig. 12(b). As can be seen from Fig. 13, along the tunnel alignment, the modification to greenfield surface settlements from soil-pipeline interaction is mainly concentrated within 5 Dp from the pipeline axis. Such a modification effect would decrease exponentially with the distance from the pipeline. 5. Discussions and parametric studies 5.1. Comparisons with elastic continuum solutions and suggested parameter values The elastic continuum solution derived by Zhang et al. (2012), incorporating the effect of gap formation, gives a reliable prediction of the pipe deflections. In addition, the elastic continuum solution for ground settlements above the pipeline, which combines the methods of Zhang et al. (2012) and this study, also yields satisfactory estimates. In both solutions, the soil media are idealized by an elastic-continuum model, in which the subgrade reaction due to soil-pipeline interaction is estimated using the Mindlin solution (Mindlin, 1936). The elastic-continuum model is capable of depicting the continuous nature of the soil mass, but it involves complex mathematical calculations. The above case studies have provided some validation of the analytical solutions formulated in this study. The calculated results are quite close to that from the elastic continuum solutions. In the Pasternak model, Gp is the critical parameter that describes the soil's shear characteristics, which relies on the values of Ht and γp, as expressed in Eqs. (34). Xu (2005) proposed that Ht is about 2.5 times the tunnel diameter for soil-tunnel interaction analysis. It is found in this study that Ht is nearly 10 times the pipeline diameter and the value of γp is around 0.7 m−1. This is consistent with the proposal of Xu (2005) with regard to the absolute value of Ht.

Fig. 12. Observed and calculated results for case 4.

are applicable in conditions of both oblique and perpendicular intersections. Fig. 11(c) and (d) show the measured and calculated ground surface settlement in Test P and Test O, respectively. As demonstrated in Fig. 11(c) and (d), Eq. (52) gives a good fitting of the greenfield settlement, and the observed ground surface settlement just above the pipeline is close to the calculated results. 4.4. Field observations in sandy clay The deflection of an existing cable pipeline due to shield tunnelling in sandy clay of Shenzhen was observed (Jia et al., 2009). In addition, two rows of monitoring points perpendicular to the tunnel axis were installed to monitor the ground surface settlement. One row is directly above the axis of the cable pipeline, and the second is at a horizontal distance of 50 m. As mentioned above, measurements in the second row can be taken as the greenfield surface settlement. The dimensions and characteristics of this case history are summarized in Table 6. The analytical solutions formulated above and proposed by Zhang et al. (2012) are applied to calculate the deflection of the cable pipeline and the ground displacement above it. The values of the parameters used in the calculation are listed in Table 2. Fig. 12(a) presents the observed and calculated pipeline deflections. As can be seen from Fig. 12(a), the analytical methods of this study and that of Zhang et al. (2012) both give a good estimation.

5.2. Effect of the gap formation What most distinguishes the solutions herein from the previous ones is the full consideration of gap formation. The pipeline, beneath which a gap is formed, imposes a shielding effect on the settlements of the overlying ground. As expressed in Eqs. (24), the shielding effect relies predominately on the values of Vlg and α. Herein α is found to be approximately 5, which is consistent with the empirical value gained through field observations in Hangzhou soft ground (Lin et al., 2014). As mentioned above, the gap consumes part of the volume loss, thus

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Fig. 13. Variation of maximum ground surface settlements along the tunnel alignment.

the remaining portion taking part in the soil-pipeline interaction is lowered. However, the effect of the gap formation has not yet been considered in most available analytical solutions for soil-pipeline interaction. Taking case 1 for instance, Fig. 14 shows a comparison of calculated results with and without account of the gap formation, in terms of the pipeline deflection and bending moment, respectively. As can be seen from Fig. 14, the generated pipeline deflection and bending moment will be overestimated if the gap formation is not taken into account. Marshall et al. (2010) reported gap formations beneath pipelines in centrifuge tests with high-quality measurements. No apparent correlation was observed between the gap formation and the development of the pipeline bending behaviour. As stated above, according to the analytical solutions herein, the portion of volume loss represented as a gap would not contribute to any pipeline bending response resulting from soil-pipeline interaction. After the onset of a gap beneath the pipeline, further increased volume loss was found to be mainly occupied by the gap as it expanded. This may explain why there was no apparent change in the pipeline bending behaviour with ongoing increasing volume loss after a gap formation. The findings from these centrifuge tests demonstrate the effect of gap formation on the pipeline bending behaviour, and they may also validate the analytical solutions in this study. As implied by Eqs. (23), (35), (39), and (55), the pipeline deflections and bending moment increase linearly with volume loss before a gap is formed beneath the pipeline. However, after the onset of the gap, as part of the continuing increased volume loss will be occupied by the gap, the pipeline deflections and bending moment would increase nonlinearly with volume loss. A continuum-based two-stage solution implemented by Franza and DeJong (2019) concluded that tunnelling-induced flexural deformations of surface structures could be overestimated if gap formation is not allowed, as well as that gap formation could induce nonlinear trends of structure deformations with tunnel volume loss. This exactly agrees with the findings in this study, especially the point that the gap

Fig. 14. Comparisons of calculated results with and without account of the gap.

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Table 7 Dimensions and characteristics of the field test (Takagi et al. 1984). Pipeline

Dp (m) 0.165

tp (m) 0.005

L (m) 13.5

Tunnel

Rt (m) 2.42

zt (m) 8.35

Soil

Ep (GPa) 210

Ip (10−6 m4) 8.0503

EpIp (106N·m2) 1.6906

Es (MPa) 10

vs 0.3

zp (10−3 m) 1.5825

Fig. 16. Ratios of maximum pipeline bending moment with different θ.

numerical simulations, with values of the parameters listed in Table 2. As can be seen from Fig. 15(b), there is a high level of consistency between the observed, simulated, and calculated results. Furthermore, a parametric study is carried out with reference to the finite element simulations conducted by Wang et al. (2011a). The ratios of the maximum bending moment with a variation of θ to that at θ = 90˚ is presented in Fig. 16, in which Mmax,θ and Mmax,90˚ are the maximum pipeline bending moment at an arbitrary θ and θ = 90˚, respectively. As can be seen from Fig. 16, the calculated and simulated results agree well with each other and both demonstrate that the pipeline’s maximum bending moment drops substantially with a decrease of the intersection angle. 6. Conclusions Fig. 15. Observed, simulated and calculated pipeline response.

Analytical solutions have been formulated in this study to estimate the response of buried pipelines and the ground resulting from soilpipeline interaction. The following main conclusions have been drawn:

formation may decrease the tunnelling-induced influence in terms of structure displacements and bending moment.

(1) The analytical solutions formulated in this study can estimate the pipeline responses (in terms of deflection and bending moment) and the ground displacements above the pipeline with great accuracy. (2) The tensionless Pasternak model is applicable for analysis of soilpipeline interaction. The suggested values of Ht and γp are 10 Dp and 0.7 m−1, respectively. (3) A gap may occur beneath the pipeline resulting from soil-pipeline interaction. The procedure proposed in this study can give a rough estimation of the width and magnitude of the gap zone. Moreover, the gap formation has the effect of lowering the tunnelling-induced pipeline deflection and bending moment. (4) The account of the pipeline orientations with respect to the tunnel alignment in the analytical solutions has been validated by field observations and numerical simulations. In terms of the generated bending moment, the most significant potential for damage of the pipeline is to be encountered at a higher intersection angle.

5.3. Effect of the intersection angles The pipeline’s orientation with respect to the tunnel alignment has been taken into account in the analytical solutions in this study. The effect of the intersection angles on the pipeline responses, in terms of the pipeline bending moment, is to be further examined through one case study. Takagi et al. (1984) conducted a full-scale field test to investigate the response of a steel pipeline to tunnelling. Table 7 lists the dimensions and characteristics of the field test. Later, Wang et al. (2011a) carried out FE analysis to simulate this field test. Fig. 15(a) shows the measured greenfield surface settlement due to tunnelling. It can be reasonably approximated by Eq. (52) with Ks = 0.292 and Vlt = 1.52%. Fig. 15(b) shows the observed pipeline bending moment, as well as the results from analytical calculations and

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The analytical solutions derived herein may be considered valid as long as interaction forces do not exceed the soil limit load, which is generally represented as the limit uplift force of the buried pipelines. In cases when interaction forces exceed the limit uplift forces, the soil’s nonlinearity could have a remarkable role in decreasing tunnelling-induced bending moment (Klar et al., 2007). Thus, conservative estimates would be achieved by the application of the above analytical solutions. However, for an elaborate examination of the soil’s nonlinear behaviours after yielding, as well as for a more economical evaluation, the effects of the limit uplift forces need further investigation. The approach proposed by Klar et al. (2007) could be employed to incorporate the limit uplift forces into the current analytical solutions. Regarding the effect of intersection angles on pipeline response, the conclusion here is valid under the assumptions in this study. However, when the progressive propagation of the ground settlements and the nonlinear behaviours of the soil and the pipeline are concerned, further investigation is still needed.

Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements This work was financially supported by the National Key R&D Program of China (Grant No. 2016YFC0800200), the National Natural Science Foundation of China (Grant Nos. 51738010 and 41702313), the R&D Program RENSSKADE II/REMEDY at Norwegian Geotechnical Institute, and the China Postdoctoral Science Foundation (Grant Nos. 2018 M630468 and 2018 T110407). The authors would like to thank the reviewers for their many constructive comments, which greatly helped to improve the manuscript. The authors gratefully acknowledge the support provided by Norwegian Geotechnical Institute, the Research Council of Norway, and China Scholarship Council.

Appendix Details of some of the matrices and vectors mentioned in the main text are presented below.

[Kp] =

1

4 1

6 4

4 6

0

1 2

2

1

Ep Ip l4

0 0 {w} = [ w

{s} =

w

2

1

1

0

w0

4

1

z p)

0 0 1

exp

(x - 2 sin )2 2K2 (z 0 zp)2

exp

(x - 1 sin )2 2K2 (z 0 zp)2

exp

(xi sin )2 2K2 (z 0 zp)2

2

1

zp ) 2

0

n+ 5

(55)

(n + 5) × (n + 5)

(56)

0

2 1 1

l2

(54)

1

0 1

(53)

(n + 5) × (n + 5)

(xn + 2 sin )2

2K2 (z 0

1 0 0 0 0 0 0 0 1

2 0

4 1 0 0 1 0 2 1

0

1

[Kr ] = kDp

Gp Dp

1 2

6

wn wn + 1 wn + 2 ]nT+ 5

wi

Rt2 Vlp 2 K (z 0

1 2

0

1

1

exp

[K s] =

1 4

2 1 0 0 0 0 0

(57)

(n + 5) × (n + 5)

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Ep Ip

[Cm] =

0 0 0 0 0 1

2 1

l2

1

0 0 0

0 0 1 [Cr ] =

4 1

Ep Ip Dp

0 0 0 0 0 2 1 0 0 0 0 0

l4

6 4 1

4 6 4

0 0 0 1

1 4 6

1 4 1

1

4 6

(58)

(n + 5) × (n + 5)

0 0 0 0 0 4 1 0 0

(59)

(n + 5) × (n + 5)

0 k k

[K r1] = k k

0 0

1

(m + 2) × (m + 2)

(60)

(m + 2) × (m + 2)

(61)

0 k k

[K r2] = k k

0

{g } = [ g0 g1 g2 0 1 [K s ] =

T

gj

2 1

Gp

1 2 1

0

gm gm + 1]

0 1 2 1

l2

1

2 1

0 {C } = [ wm1

3

rs zp

(62)

m+2

rs zp

1 2 1

rs zp

1 2 1 0 rs zp

(63)

(m + 2) × (m + 2)

wm1+ m 2 ]T m+2

(64)

Appendix A. Supplementary material Supplementary data to this article can be found online at https://doi.org/10.1016/j.tust.2019.103193.

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