Theoretical model for predicting deformation of pipe curtain support structure in underground engineering

Abstract

Pipe curtains are widely employed in underground engineering as support structures to control stratum deformation during excavation. However, a comprehensive theoretical model for accurately predicting pipe curtain deformation remains lacking. This study establishes a deformation prediction model for pipe curtains based on the small-deflection elastic plate theory, incorporating the actual stress characteristics of pipe curtains and the effects of overlying loads. The calculation method for the bending stiffness of pipe curtains under various arrangements is derived. Using in situ monitoring data from the Pinganli Station and the Shifu Station, the model’s effectiveness is validated. Furthermore, the influence of key parameters on pipe curtain deformation under different arrangements is systematically analyzed. The results show that the proposed calculation methods achieve satisfactory accuracy for both transverse and longitudinal arrangements, with average errors of 0.9% and 19.8%, respectively, making them suitable for practical engineering applications. In addition, the transverse arrangement effectively reduces the deformation of pipe curtains induced by excavation. Among all factors, the excavation span exerts the most significant influence on pipe curtain deformation. Specifically, the maximum deformation decreases exponentially with increasing steel pipe diameter, decreases linearly with the stiffness of the grouting body between pipes, and increases exponentially with the excavation span.

Graphical abstract

Keywords

underground engineering / pipe curtain deformation / excavation / calculation model / sensitivity analysis

1 Introduction

In recent years, underground space construction has undergone rapid development worldwide. During the construction process, controlling ground deformation induced by excavation has become a critical concern [1–4]. Owing to its high stiffness and bearing capacity, the pipe curtain method can effectively mitigate excavation-induced disturbances [5–8]. Consequently, it has been widely adopted in various projects, including subway stations, underground passages, and underpasses beneath existing structures [9,10]. When pipe curtains are used as support structures, surface settlement during construction is primarily caused by their deformation [11]. Therefore, accurately predicting the deformation behavior of pipe curtains is essential for controlling stratum deformation.

Many scholars have extensively investigated the mechanisms of pipe curtain deformation through laboratory tests, in situ monitoring, and numerical simulations. Yang et al. [12] and Wang et al. [13] examined the deformation behavior of pipe curtains during the pipe jacking process, analyzed the effects of flange plate position and steel pipe diameter on deformation, and established functional relationships among soil pressure, structural deformation, and these parameters. Li et al. [14] and Jia et al. [15] utilized monitoring data to investigate the deformation patterns of pipe curtains throughout the entire subway station construction process. They identified key construction stages that could cause significant deformation and subsequently optimized the construction methods to enhance deformation resistance. Zhang et al. [16] simplified the pipe curtain structure based on the principle of area equivalence, simulated the interaction between the pipe curtain and surrounding soils using spring models, and analyzed deformation induced by large-section excavation through numerical simulations and laboratory experiments. Although these studies primarily focus on specific projects or cases, their conclusions lack broad generality. Nevertheless, these investigations provide valuable data and theoretical insights that support further research and parametric analyses of pipe curtain deformation.

Regarding the theoretical analysis of pipe curtain deformation, scholars have often simplified the pipe curtain structure into an elastic foundation beam model for computational convenience. Prediction models for pipe curtain deformation under various working conditions have been developed based on Winkler or Pasternak foundation beam theories, enabling analysis of the effects of sensitive parameters on deformation [17–19]. Although these beam models offer clear mechanical interpretations and relatively simple calculations, they fail to fully represent the actual stress characteristics of pipe curtain structures. This limitation arises because adjacent pipes are transversely connected through steel joints or grouting bodies, resulting in coordinated deformation among structural components in both transverse and longitudinal directions. To address this issue, Wu et al. [20] and Zhang et al. [21] modeled the pipe curtain as a rectangular thin plate and derived corresponding deformation formulas, demonstrating that the plate theory provides a more accurate representation of pipe curtain behavior. In addition, some pipe curtain systems, such as NTR and STS types, enhance bending resistance through locking or interlocking connections between adjacent steel pipes [22–24]. When the ratio of structural thickness to short-span length ranges between 1/80 and 1/5, the pipe curtain can also be approximated as an elastic thin plate (ETP) [25]. Most of the above studies simplified the overlying load as a uniformly distributed load during calculation. However, existing research indicates that an arching effect may develop above the pipe curtain due to excavation-induced unloading, thereby altering the load distribution [26,27]. When the soil beneath the pipe curtain is excavated, the structure deforms under stress, leading to uneven settlement of the overlying soil. This results in the development of a soil arching effect, which in turn modifies the soil pressure acting on the structure. Nevertheless, previous studies did not account for the iterative process of load redistribution and pipe curtain deformation, leading to noticeable discrepancies between analytical predictions and actual observations. Moreover, earlier studies typically used equivalent bending or compressive stiffness to estimate the overall stiffness of the pipe curtain. In reality, the bearing capacity and stiffness differ significantly between transverse and longitudinal arrangements, and the calculation should reflect these structural distinctions.

Based on the practical stress characteristics of pipe curtains, a deformation calculation model that accounts for the soil arching effect is established using ETP theory. Methods for calculating the bending stiffness of pipe curtains under different arrangement modes are then proposed. The proposed calculation model is validated through comparison with in situ monitoring data. Finally, the influence of key parameters on pipe curtain deformation is analyzed, including the pipe diameter, the stiffness of grouting body between pipes, and the excavation span.

2 Mechanism of pipe curtain structure

2.1 Model establishment

Figure 1 illustrates several underground projects in China constructed using pipe curtains. In practice, a pipe curtain consists of steel pipes, internal concrete filling, and grouting bodies between adjacent pipes. Owing to the high overall stiffness of the pipe curtain structure, its deformation induced by excavation generally remains within the elastic range. According to elastic theory, when the ratio of the long span to the short span does not exceed 2, the stress state of the structure approximates that of a two-way slab [25]. In practical engineering, the excavation span supported by pipe curtains typically ranges from 20 to 25 m, while the structural length is about 40 to 50 m, thus falling within the two-way slab category. Even when the span ratio exceeds 2, the structure is supported on all four sides, allowing transverse and longitudinal forces to act cooperatively; in such cases, it can still be analyzed as a two-way slab [20,21,28]. When the pipe curtain method is employed to support underground structures, the curtain bears both the pressure from overlying loads and the elastic resistance of the unexcavated soil near the tunnel face. Therefore, based on the principle of stiffness equivalence, the steel pipes, the grouting body between pipes, and the concrete within the pipes can be approximated as a rectangular thin plate subjected to specific boundary constraints.

Taking Pinganli Station on Beijing Metro Line 19 and Shifu Road Station on Shenyang Metro Line 4 as examples, during excavation under pipe curtain support, the left and right ends of the pipe curtain are connected to the initial support structure. Under the influence of overburden loads, the settlement at the junction between the pipe curtain and the initial support can be considered negligible. However, the pipe curtain is capable of slight rotational movement at these connections. Therefore, its boundary condition with the initial support can be reasonably simplified as a simply supported edge. When excavation reaches the tunnel face, the settlement of the pipe curtain at that location is constrained by the unexcavated soil and temporary support. Therefore, the boundary at the tunnel face can be simplified as a simply supported edge. At the excavation end, the pipe curtain typically lacks temporary support and can therefore be regarded as a free edge. Consequently, as the soil beneath the pipe curtain is excavated, the structural boundary condition can be described as “simply supported on three ends and free on one end”. Based on this condition, a mechanical model of the pipe curtain is established, as illustrated in Fig. 2. In this model, the x-direction represents the excavation width, while the y-direction corresponds to the excavation direction, with y = 0 denoting the tunnel face. The parameters a and b denote the plate width and plate length, respectively.

The main assumptions adopted in the calculation of pipe curtain deformation are as follows.

1) After deformation, all cross-sections perpendicular to the mid-plane of the thin plate maintain a mutually parallel relationship. Therefore, the shear strain at any point within the thin plate is zero.

2) During the bending deformation process, the mid-plane of the pipe curtain remains a neutral surface without in-plane displacement. In other words, each point on the mid-plane undergoes only out-of-plane displacement in the z-direction, with no displacement in the x-direction or y-direction.

3) All points along a line perpendicular to the mid-plane within the thickness of the pipe curtain experience identical out-of-plane displacement in the z-direction, which is a function only of the x and y coordinates.

2.2 Solution methodology for pipe curtain deformation

The pipe curtain structure possesses a relatively high bearing capacity, and its maximum deflection under overlying loads generally does not exceed 100 mm. According to elastic mechanics theory, the deformation of the pipe curtain can therefore be calculated using the small deflection theory [29]. The loads acting on the thin plate can be decomposed into longitudinal loads parallel to the mid-plane and transverse loads perpendicular to the mid-plane. During construction, the soil load above the pipe curtain is regarded as a transverse load, and the corresponding stress and deformation can be analyzed as a thin plate bending problem. The fundamental governing equation for the bending of a rectangular thin plate is expressed as follows:

(1)\frac{{\mathrm{\partial }}^4\omega \left(x,y\right)}{\mathrm{\partial }{x}^4}+2\frac{{\mathrm{\partial }}^4\omega \left(x,y\right)}{\mathrm{\partial }{x}^2\mathrm{\partial }{y}^2}+\frac{{\mathrm{\partial }}^4\omega \left(x,y\right)}{\mathrm{\partial }{y}^4}=\frac{q\left(x,y\right)}{D},

where ω(x, y) represents the deformation of the pipe curtain, D denotes the bending stiffness of the plate, which is related to the arrangement of the pipe curtain, and q(x, y) is the overlying load. To obtain the general solution of the fourth-order partial differential Eq. (1), it is usually necessary to perform a series expansion of q(x, y). Before calculating the overlying load, the geometry of the sliding surface above the pipe curtain must be defined. According to relevant research [30], the sliding surface above the pipe curtain be approximated as a trapezoid when the burial depth-to-span ratio (H/a) is less than 1.5, and as a rectangle when H/a exceeds 1.5. Wang et al. [31] derived formulas for calculating the earth pressure corresponding to different sliding surface geometries based on trapdoor test results. Accordingly, when the sliding surface is trapezoidal, the overlying load can be determined using Eqs. (2) and (3):

(2)

(3)\begin{array}{rl}{K}_1=& {\displaystyle \frac{3\left[K_0{\cos }^2\left(\theta +{\displaystyle \frac{\pi }{2}}-\alpha \right)+{\sin }^2\left(\theta +{\displaystyle \frac{\pi }{2}}-\alpha \right)\right]}{\left(K_0-1\right){\cos }^2\left({\displaystyle \frac{\pi }{2}}-\theta \right)+3}}\\ -& {\displaystyle \frac{3\left[K_0{\cos }^2\left(\theta +{\displaystyle \frac{\pi }{2}}-\alpha \right)+{\sin }^2\left(\theta +{\displaystyle \frac{\pi }{2}}-\alpha \right)\right]\tan \phi }{\cot \alpha \left[\left(K_0-1\right){\cos }^2\left({\displaystyle \frac{\pi }{2}}-\theta \right)+3\right]}}-1,\end{array}

where σ₁ is the earth pressure, \gamma is the soil unit weight, H is the burial depth of the pipe curtain, θ is the angle between the maximum principal stress and the horizontal surface, α is the angle between the sliding surface and the horizontal surface, K₀ is the passive earth pressure coefficient, and φ is the angle of internal friction.

When the sliding surface is rectangular, the overlying load can be calculated by combining Eqs. (4) and (5):

(4){\sigma }_2=\frac{\gamma a}{2K_2}\left(1-{\mathrm{e}}^{-\frac{2K_{2}H}{a}}\right),H/a>1.5,

(5)K_2=\frac{3\left[K_0{\cos }^2\theta +{\sin }^2\theta \right]\tan \phi }{\left(K_0-1\right){\cos }^2\left({\displaystyle \frac{\pi }{2}}-\theta \right)+3}.

Based on Ref. [32], K₀, θ, and α can be solved using the following equations.

(6)\{\begin{array}{rl}& K_0={\tan }^2\left(\frac{\pi }{4}+\frac{\phi }{2}\right),\\ & \theta =\frac{4\arctan \left({\displaystyle \frac{4a\omega }{a^2+4{\omega }^2}}\right)}{\pi }+{\displaystyle \frac{\pi }{4}}-\frac{\phi }{2},\\ & \alpha =\frac{4\arctan \left({\displaystyle \frac{4a\omega }{{\mathrm{a}}^2+4{\omega }^2}}\right)}{\pi }+{\displaystyle \frac{\pi }{2}}-\phi .\end{array}

According to elasticity theory, the deflection of a rectangular plate with simply supported edges at x = 0 and x = a (where a is the width of the plate) can be calculated using Eq. (7). In this equation, K_n is a function of y that satisfies the boundary conditions along both sides of x = 0 and x = a.

(7)\omega =\sum _{n=1}^{\mathrm{\infty }}{K}_{\mathrm{n}}\sin \frac{n\pi x}{a},n=1,2,3,\dots .

Substituting Eq. (7) into Eq. (1), it yields:

(8)D\sum _{n=1}^{\mathrm{\infty }}\left[\frac{{\mathrm{d}}^4{K}_n}{\mathrm{d}{y}^4}-2{\left(\frac{n\pi }{a}\right)}^2\frac{{\mathrm{d}}^2{K}_n}{\mathrm{d}{y}^2}+{\left(\frac{n\pi }{a}\right)}^4{K}_n\right]\sin \frac{n\pi x}{a}=q\left(x,y\right).

The term of q (x,y) in Eq. (8) is expanded into a triangular series:

(9)q\left(x,y\right)=\sum _{n=1}^{\mathrm{\infty }}{L}_n\sin \frac{n\pi x}{a},

where L_n can be calculated according to the following equation:

(10)L_n=\frac{2}{a}\underset{0}{\overset{a}{\int }}q\left(x,y\right)\sin \frac{n\pi x}{a}\mathrm{d}x.

Substituting Eqs. (9) and (10) into Eq. (8), it yields:

(11)D\left[\frac{{\mathrm{d}}^4{K}_n}{\mathrm{d}{y}^4}-2{\left(\frac{n\pi }{a}\right)}^2\frac{{\mathrm{d}}^2{K}_n}{\mathrm{d}{y}^2}+{\left(\frac{n\pi }{a}\right)}^4{K}_n\right]=\frac{2}{a}\underset{0}{\overset{a}{\int }}q\left(x,y\right)\sin \frac{n\pi x}{a}\mathrm{d}x.

The right-hand side of Eq. (11) is integrated with respect to y. The general solution of Eq. (11) can thus be expressed as follows, where A_n, B_n, C_n, and D_n are undetermined coefficients that will be determined using the remaining two boundary conditions of the pipe curtain.

(12)\begin{array}{rl}{K}_n=& A_n\mathrm{c}\mathrm{h}\frac{n\pi y}{a}+B_n\frac{n\pi y}{a}\mathrm{s}\mathrm{h}\frac{n\pi y}{a}+C_n\mathrm{s}\mathrm{h}\frac{n\pi y}{a}\\ & +D_n\frac{n\pi y}{a}\mathrm{c}\mathrm{h}\frac{n\pi y}{a}+F_n\left(y\right),\end{array}

where F_n(y) is a particular solution of Eq. (11), determined according to the types of loads, and is expressed as follows:

(13)F_n\left(y\right)=\frac{4q{a}^4}{D{n}^5{\pi }^5},n=1,3,5,\dots .

During the iteration process, it was found that when n takes adjacent even values, F_n(y) terms cancel each other out. Therefore, only odd values of n, starting from 1, are considered.

At this point, the general solution for the pipe curtain deformation can be written as:

(14)\begin{array}{rl}\omega =& \sum _{n=1}^{\mathrm{\infty }}(A_n\mathrm{c}\mathrm{h}\frac{n\pi y}{a}+B_n\frac{n\pi y}{a}\mathrm{s}\mathrm{h}\frac{n\pi y}{a}+C_n\mathrm{s}\mathrm{h}\frac{n\pi y}{a}\\ & +D_n\frac{n\pi y}{a}\mathrm{c}\mathrm{h}\frac{n\pi y}{a})\sin \frac{n\pi x}{a}+\frac{4q{a}^4}{D{\pi }^5}\sum _{n=1,3,5}^{\mathrm{\infty }}\frac{1}{{\mathrm{n}}^5}\sin \frac{n\pi x}{a}.\end{array}

To solve Eq. (14), only the boundary conditions at the excavation end of the pipe curtain and the tunnel face are required. For the case of “simply supported at three ends and free at one end”, the boundary conditions of the pipe curtain at y = 0 and y = b are as follows:

(15){\omega }_{y=0}=0,{\left(\frac{{\mathrm{\partial }}^2\omega }{\mathrm{\partial }{y}^2}\right)}_{y=0}=0,

(16)\{\begin{array}{rl}& {\left(M_y\right)}_{y=b}={\left[-D\left(\frac{{\mathrm{\partial }}^2\omega }{\mathrm{\partial }{y}^2}+\mu \frac{{\mathrm{\partial }}^2\omega }{\mathrm{\partial }{x}^2}\right)\right]}_{y=b}=0,\\ & {\left(V_y\right)}_{y=b}={\left[-D\left(\frac{{\mathrm{\partial }}^3\omega }{\mathrm{\partial }{y}^3}+\left(2-\mu \right)\frac{{\mathrm{\partial }}^3\omega }{\mathrm{\partial }y{x}^2}\right)\right]}_{y=b}=0,\end{array}

where M_y and V_y are the internal forces of the ETP, respectively, and μ is the Poisson’s ratio of the plate.

Substituting the boundary condition at y = 0 into Eq. (14), it yields:

(17)A_n=-\frac{4q{a}^4}{D{\pi }^5{n}^5},A_n+2B_n=0.

Substituting the boundary condition at y = b into Eq. (14), and noting that α_n = nπb/a, it yields:

(18)\begin{array}{rl}& [A_n\mathrm{c}\mathrm{h}{\alpha }_n+B_n\left(2\mathrm{c}\mathrm{h}{\alpha }_n+{\alpha }_n\mathrm{s}\mathrm{h}{\alpha }_n\right)+C_n\mathrm{s}\mathrm{h}{\alpha }_n\\ & +D_n\left(2\mathrm{s}\mathrm{h}{\alpha }_n+{\alpha }_n\mathrm{c}\mathrm{h}{\alpha }_n\right)]-\mu [A_n\mathrm{c}\mathrm{h}{\alpha }_n+B_n{\alpha }_n\mathrm{s}\mathrm{h}{\alpha }_n\\ & +C_n\mathrm{s}\mathrm{h}{\alpha }_n+D_n{\alpha }_n\mathrm{c}\mathrm{h}{\alpha }_n]-\frac{\mu 4q{a}^4}{D{\pi }^5{n}^5}=0\end{array},

(19)\begin{array}{rl}& [A_n\mathrm{s}\mathrm{h}{\alpha }_n+B_n\left(3\mathrm{s}\mathrm{h}{\alpha }_n+{\alpha }_n\mathrm{c}\mathrm{h}{\alpha }_n\right)+C_n\mathrm{c}\mathrm{h}{\alpha }_n\\ & +D_n\left(3\mathrm{c}\mathrm{h}{\alpha }_n+{\alpha }_n\mathrm{s}\mathrm{h}{\alpha }_n\right)]-\left(2-\mu \right)[A_n\mathrm{s}\mathrm{h}{\alpha }_n\\ & +B_n\left(\mathrm{s}\mathrm{h}{\alpha }_n+{\alpha }_n\mathrm{c}\mathrm{h}{\alpha }_n\right)+C_n\mathrm{c}\mathrm{h}{\alpha }_n+D_n\left(\mathrm{c}\mathrm{h}{\alpha }_n+{\alpha }_n\mathrm{s}\mathrm{h}{\alpha }_n\right)]=0.\end{array}

By combining Eqs. (17) to (19), it yields:

(20)\{\begin{array}{rl}& A_n=-\frac{4q{a}^4}{D{\pi }^5{n}^5},\\ & B_n=\frac{2q{a}^4}{D{\pi }^5{n}^5},\\ & C_n=\frac{4q{a}^4}{D{\pi }^5{n}^5}\frac{\left\{3+{\mu }^2-2\mu \left(\mu +1\right)\mathrm{c}\mathrm{h}{\alpha }_n+\left(\mu -1\right)\left(\mu +3\right)\mathrm{c}\mathrm{h}2{\alpha }_n+\left(\mu -1\right){\alpha }_n\left[\left(\mu -1\right){\alpha }_n-2\mu \mathrm{s}\mathrm{h}{\alpha }_n\right]\right\}}{\left(\mu -1\right)\left[\left(\mu +3\right)\mathrm{s}\mathrm{h}2{\alpha }_n-2\left(\mu -1\right){\alpha }_n\right]},\\ & D_n=-\frac{4q{a}^4}{D{\pi }^5{n}^5}{\displaystyle \frac{2\left[\left(\mu +3\right)\mathrm{c}\mathrm{h}{\alpha }_n-\left(\mu -3\right)\right]\mathrm{s}\mathrm{h}{\left({\displaystyle \frac{{\alpha }_n}{2}}\right)}^2}{\left(\mu +3\right)\mathrm{s}\mathrm{h}2{\alpha }_n-2\left(\mu -1\right){\alpha }_n}}.\end{array}

Substituting Eq. (20) into Eq. (14), the calculation equation for the pipe curtain deformation with the boundary condition of “simply supported at three ends and free at one end” can be obtained.

From the pipe curtain deformation formula, it can be observed that the stiffness of the pipe curtain and the earth pressure are the primary factors influencing its deformation. Greater pipe curtain stiffness results in smaller deformation, a weaker soil arching effect, and consequently higher earth pressure acting on the pipe curtain. During excavation, the deformation of the pipe curtain and the earth pressure interact with each other; therefore, an iterative calculation method is required to achieve accurate results. Initially, the pipe curtain deformation is assumed to be ω₀, which can be estimated based on engineering experience. For example, during the construction of station pilot tunnels, the deformation of the pipe curtain caused by excavation typically ranges from 10 to 20 mm. An intermediate value of 15 mm can be adopted to calculate the geometric parameters θ₀ and α₀ using Eq. (6). Subsequently, the earth pressure above the pipe curtain is calculated using Eq. (2) or Eq. (4), and this value is substituted into Eq. (14) to determine the corresponding deformation ω. If ω₀ = ω, the calculation process is terminated. Otherwise, the newly obtained deformation ω is substituted back into Eq. (2) or Eq. (4) to recalculate the earth pressure, which is then reintroduced into Eq. (14) to update the deformation value ω′. This iterative process is repeated until convergence is achieved, and the final value of ω is obtained. In practice, 5 iterations are generally sufficient to achieve an accurate result.

2.3 Limitations of the calculation model

1) This model is more applicable to strata capable of forming soil arches, such as sandy soils with low cohesion and fractured rock masses with poor structural integrity.

2) The proposed calculation model is established based on ETP theory, assuming that there is no shear deformation occurs under the action of overlying loads.

3) This method is derived based on elastic theory, while plastic deformation effects are excluded from consideration.

4) The calculation model does not account for the influence of groundwater. In water-rich strata, it is necessary to adopt a soil pressure calculation method that considers the effects of water content. The resulting deformation behavior becomes more complex and requires further investigation.

3 Calculation of bending stiffness under different arrangements

3.1 Pipe curtain arrangements

In calculating pipe curtain deformation, the bending stiffness D is a key parameter that significantly influences the accuracy of the results. When applying the pipe curtain method in construction, factors such as geological conditions, surrounding environment, and construction complexity must be considered. The pipe curtain structure is generally arranged in two forms: transverse arrangement and longitudinal arrangement, as shown in Fig. 3. The transverse arrangement refers to the configuration in which the axis of the pipe curtain structure is perpendicular to the excavation direction, whereas the longitudinal arrangement corresponds to the case where the axis is parallel to the excavation direction. Under these different arrangements, the bearing capacity and corresponding bending stiffness D vary accordingly. Therefore, this section presents a method for calculating the D value for both transverse and longitudinal arrangements.

3.2 Transverse arrangement

Transverse pipe curtains are commonly employed in subway station projects, such as the Olympic Sports Center Station on Jinan Metro Line 4 and the previously mentioned Pinganli Station. The length of the transverse pipe curtain is typically shorter, which facilitates greater control over the precision of steel pipe jacking. Based on experimental data and numerical simulation results [33,34], under the transverse arrangement, the longitudinal bending resistance (from the concrete-filled steel pipe structure) of the pipe curtain plays a primary load-bearing role, while the transverse bending resistance (from the grouting body) contributes as a secondary load-bearing factor. According to the stiffness equivalence method in elastic theory, the stiffness of ribs on stiffening plates can be treated as that of thin plates [20]. When applying the pipe curtain method in construction, the grouting body can be regarded as an isotropic plate, and the concrete-filled steel pipes function as stiffening ribs. The combined system of concrete-filled steel pipes and the grouting body can therefore be idealized as an ETP with stiffening ribs. The corresponding equivalent mechanical model is shown in Fig. 4. The stiffness of the concrete-filled steel pipe can be uniformly distributed over the thin plate, and the bending stiffness of the transverse pipe curtain can be calculated using Eq. (21):

(21)D_{\mathrm{T}}=\frac{E_1{t}^3}{12\left(1-{\mu }_{\mathrm{T}}{}^2\right)}+\frac{E_{2}I}{S},

where D_T is the bending stiffness of the pipe curtain under the transverse arrangement (kN·m), E₁ and E₂ are the elastic moduli of grouting body and concrete-filled steel pipe structure (MPa), respectively, S is the center-to-center spacing between adjacent steel pipes (m), t is the thickness of the grouting zone (m), μ_T is the Poisson’s ratio, and I is the moment of inertia of the section (kg·m²). It can be observed that D_T is uniformly distributed over the spacing S, and its equivalent bending stiffness within this range is represented by D_T.

3.3 Longitudinal arrangement

Longitudinal pipe curtains are commonly employed in projects such as the previously mentioned Shifu Station, the Gongbei Tunnel of the Hong Kong-Zhuhai-Macao Bridge, and various urban underpass projects. Longitudinal pipe curtains are generally longer, with maximum lengths reaching approximately 300 m, which makes it more difficult to control the precision of steel pipe jacking. Under the longitudinal arrangement, the transverse bending resistance provided by the grouting body serves as the primary load-bearing component, whereas the longitudinal bending resistance contributed by the concrete-filled steel pipe structure acts as a secondary factor. When the design parameters of the pipe curtain are identical, the overall deformation resistance of the longitudinal arrangement is significantly lower than that of the transverse arrangement. Consequently, a greater number of temporary supports are typically required beneath the longitudinal pipe curtain during excavation. Therefore, when calculating the D of the longitudinal pipe curtain, both the grouting body between the pipes and the steel–concrete structure have played a role in resisting bending. The overall structure of the pipe curtain can be approximated as a thin plate composed of grouting body and steel–concrete structure, and its bending stiffness can be calculated using the superposition method [35,36]. However, due to the weak contribution of steel–concrete structure to flexural performance, it needs to be multiplied by a reduction factor. Based on elasticity theory, the bending stiffness of pipe curtains under longitudinal arrangement can be calculated using Eq. (22):

(22)D_{\mathrm{L}}=\frac{E_{\mathrm{L}1}{t}_1{}^3}{12\left(1-{\mu }_1{}^2\right)}+\frac{\eta E_{\mathrm{L}}{}_2{t}_2{}^3}{12\left(1-{\mu }_2{}^2\right)},

where D_L is the bending stiffness of the pipe curtain under the longitudinal arrangement, E_L1 and E_L2 are the elastic moduli of the grouting body and the concrete-filled steel pipe structure, respectively, t₁ and t₂ are the thicknesses of the grouting body and the concrete-filled steel pipe structure, and μ₁ and μ₂ are their corresponding Poisson’s ratios. ƞ is a reduction factor that depends on the boundary conditions of the pipe curtain structure and the actual bending resistance of the concrete-filled steel pipe component. According to previous research findings [37,38], when the boundary condition of the pipe curtain is simply supported on all four sides, both the grouting body and the concrete-filled steel pipe contribute to the overall stiffness, and the contribution coefficient ƞ can be taken as 1. When the pipe curtain is simply supported or free at both ends, the load-bearing capacity is primarily provided by the grouting body, and ƞ can be taken as 0. Therefore, to obtain more conservative calculation results, this study assumes that the concrete-filled steel pipe structure contributes 50% of its bearing capacity (ƞ = 0.5) under the boundary condition of three simply supported sides and one free side.

In summary, for practical applications, the bending stiffness calculation formula should be selected according to the arrangement form of the pipe curtain. The resulting stiffness value can then be substituted into Eq. (20) to determine the corresponding pipe curtain deformation.

4 Evaluation of the theoretical model

4.1 Evaluation of transverse pipe curtain deformation

To assess the prediction accuracy of the proposed calculation model, two engineering projects, the Pinganli Station [21] and the Shifu Station [39], are analyzed as case studies.

The diagram of Pinganli Station is shown in Fig. 5(a). Excavation was first carried out for the pilot tunnels, after which closely spaced small-diameter pipe curtains were installed into the soil on both sides of the tunnels. The remaining portion of the station was then excavated under the protection of the pipe curtain support structure. The pipe curtain axis is perpendicular to the excavation direction, corresponding to a transverse arrangement. The settlement monitoring points located on the crown of the steel arch inside Pilot Tunnel 2 are shown in Fig. 5(b). Since the steel arch is in close contact with the pipe curtain, the crown settlement can be approximately regarded as the deformation of the pipe curtain. The main parameters of the project are as follows: the pipe curtain has a diameter of 402 mm, a center-to-center spacing of 450 mm, and a wall thickness of 16 mm. The steel pipes are filled with liquid cement. The elastic modulus of the steel pipe is 210 GPa, while that of the liquid cement is 2 GPa. The elastic modulus of the grouting material is also taken as 2 GPa, with a Poisson’s ratio of 0.3. The overlying soil above the pipe curtain has a thickness of 7 m, an internal friction angle of 33°, a Poisson’s ratio of 0.28, and a weight of 20 kN/m³. The span between the two pilot tunnels is 4.4 m, and the total length of the pipe curtain is 40 m.

The pipe curtain at Pinganli Station is connected to the initial support of the pilot tunnels on both sides, with no support provided at the excavation end. Therefore, the boundary conditions during pilot tunnel excavation can be regarded as “simply supported on three sides and free on one side”. The pipe curtain deformation is calculated using Eq. (20), while its bending stiffness is determined using Eq. (21). To evaluate the performance of the proposed model, the calculated results are compared with on-site monitoring data and with predictions obtained from the simply supported beam model [40], the elastic foundation beam model [41], and the ETP model [21], as illustrated in Fig. 6. During the construction of the pilot tunnel, when the distance between the tunnel face and the monitoring point is less than 6 m, the pipe curtain deformation increases rapidly as excavation progresses. When this distance exceeds 6 m, the deformation gradually stabilizes and fluctuates within a limited range. The maximum measured settlement of the pipe curtain is 4.4 mm. The deformation curves obtained from the simply supported beam model and the elastic foundation beam model are both linear, indicating that the calculated deformation remains constant throughout the excavation process. In contrast, the deformation trend predicted by the ETP model closely agrees with the monitored values, demonstrating good consistency with the observed deformation behavior.

The final deformation values at the monitoring point calculated using the four models are –3.8 mm (simply supported beam model), –3.6 (elastic foundation beam model), –4.09 (ETP model), and –4.36 mm (the proposed model). A comparison between the theoretical and monitoring results indicates that the ETP model accurately reflects the monitored deformation, both in terms of the maximum deformation and its variation trend during excavation. In contrast, the simply supported beam and elastic foundation beam models yield deformation values close to the final measured value but fail to capture the evolving deformation behavior of the pipe curtain throughout the excavation process. Compared with the ETP model, the theoretical calculation model proposed in this study demonstrates higher accuracy, with a maximum deviation of only 0.9% from the monitoring data.

4.2 Evaluation of longitudinal pipe curtain deformation

The diagram of the Shifu Station is shown in Fig. 7(a). During construction, multiple steel pipes were first jacked into place to form the pipe curtain structure, after which pilot tunnels were excavated beneath the support system. The axis of the pipe curtain is parallel to the excavation direction, corresponding to a longitudinal arrangement. The deformation data recorded at monitoring points A and B are analyzed, as shown in Fig. 7(b). The pipe curtain structure consists of Q235 seamless steel pipes with an outer diameter of 402 mm, a wall thickness of 16 mm, and a net spacing of 50 mm between adjacent pipes. The steel pipes are filled with C30 self-compacting concrete. The grouting body between pipes is made of liquid cement, with an elastic modulus of 2 GPa and a Poisson’s ratio of 0.3. The soil parameters are listed in Table 1. The overlying soil above the pipe curtain is primarily composed of sand, with a thickness of 4.2 m and a weight of 20 kN/m³.

Based on the parameters of the Shifu Station and Eq. (22), the bending stiffness D of the pipe curtains was calculated, and the result was substituted into Eq. (20) to determine the pipe curtain deformation. As shown in Fig. 8, the theoretical results were compared with the in situ monitoring data. The variation trends of the calculated and measured deformations exhibit good agreement. When the distance between the tunnel face and the monitoring points is within 30 m, the pipe curtain deformation increases rapidly as the tunnel advances. However, when the distance exceeds 30 m, the deformation gradually stabilizes. A comparison of the deformation at monitoring points A and B shows that the deformation at point A is smaller because it is located closer to the pipe curtain boundary and therefore receives greater structural support. For monitoring point A, the final deformation values are 2.21 mm (theoretical) and 2.88 mm (measured), corresponding to a maximum error of 23.3%. For monitoring point B, the respective values are 11.31 mm (theoretical) and 13.52 mm (measured), with a maximum error of 16.3%. The average error is 19.8%. These results indicate that the predictions are in close agreement with the monitoring data, thereby validating the accuracy of the proposed deformation calculation method for pipe curtains under the longitudinal arrangement.

5 Sensitivity analysis

5.1 Analysis scheme

The pipe curtain consists of steel pipes, the grouting body between the pipes, and concrete filling within the pipes. According to previous studies conducted by the authors’ research team, variations in the concrete properties inside the pipes have a negligible effect on the overall bending stiffness of the pipe curtain [22,26,36]. Therefore, this section focuses on the effects of the steel pipes and the grouting body between them. Since steel pipes are primarily composed of steel, their elastic modulus remains relatively constant; hence, the influence of the steel pipe on bending stiffness can be investigated by varying its diameter. The elastic modulus of the grouting body between pipes also affects the flexural stiffness of the structure. In addition, pipe curtain deformation tends to increase with the enlargement of the excavation span, although the specific influence requires further investigation. Therefore, this study selects three key parameters: the steel pipe diameter, the elastic modulus of the grouting body, and the excavation span, for detailed analysis. The stiffness of the pipe curtain cis calculated according to Eqs. (21) and (22), and the value range of the relevant parameters are listed in Table 2. For ease of comparison of the deformation behavior of the pipe curtain under different parameter conditions, the span and length are set to 12 and 20 m, respectively. The pipe curtain structure consists of Q235 seamless steel pipes with a wall thickness of 16 mm and a net spacing of 50 mm. The steel pipes are filled with C30 self-compacting concrete, which has an elastic modulus of 30 GPa and a Poisson’s ratio of 0.2. The grouting body between adjacent pipes is made of liquid cement, with an elastic modulus of 2 GPa and a Poisson’s ratio of 0.3. The overlying soil above the pipe curtain has a thickness of approximately 4.2 m and a unit weight of 20 kN/m³.

5.2 Steel pipe diameter

The deformation results of the pipe curtain caused by excavation under different pipe diameters are shown in Fig. 9. For the transverse arrangement, when d decreases from 600 to 500 mm, the pipe curtain deformation increases slightly, but the overall deformation trend remains nearly unchanged. As d further decreases to 400 mm, the deformation increases noticeably, and the deformation profile of the pipe curtain exhibits a more pronounced curvature. When d is reduced to 300 mm, the deformation rises sharply, forming a distinct “V” shaped surface. Under the same pipe diameter, the deformation patterns of pipe curtains in both transverse and longitudinal arrangements are similar and exhibit symmetry along the structural centerline. The maximum deformation occurs at the excavation end and gradually decreases toward the opposite end.

The maximum deformation values of the pipe curtain for different pipe diameters, as shown in Fig. 10. As d increases, the maximum deformation of the pipe curtain exhibits an exponential decreasing trend. Taking the transverse arrangement as an example, when d ranges from 300 to 600 mm, the corresponding maximum deformations after excavation are –207.75, –85.23, –40.26, and –22.66 mm. Compared with the case where d = 300 mm, increasing d to 400, 500, and 600 mm reduces the maximum deformation by 58.97%, 80.62%, and 89.09%, respectively, indicating a reduction of nearly 90% at the largest diameter. These results demonstrate that increasing d significantly enhances the bending stiffness of the pipe curtain, thereby reducing deformation. However, when d increases from 500 to 600 mm, the fitted curve gradually flattens, suggesting that further enlargement of the pipe diameter has a limited effect on controlling deformation. Therefore, the theoretical model proposed in this study can be used to calculate pipe curtain deformation under various working conditions and to determine the optimal pipe diameter that satisfies design control requirements.

5.3 Stiffness of the grouting body between pipes

The deformation results of the pipe curtain for different elastic moduli of the grouting body are shown in Fig. 11. After excavation, the overall deformation trends of the pipe curtain under both transverse and longitudinal arrangements are similar. For the same elastic modulus, the deformation of the pipe curtain in the transverse arrangement is smaller. As the elastic modulus of grouting bodies increases, the deformation of the pipe curtain gradually decreases, and the rate of reduction remains relatively uniform.

The maximum deformation values of the pipe curtain for different elastic moduli of the grouting body are shown in Fig. 12. As the elastic modulus of grouting body increases, the deformation decreases approximately linearly. When the elastic modulus ranges from 5 to 20 GPa, the maximum deformation values under the longitudinal arrangement are –105.07, –93.53, –84.55, and –76.84 mm, while those under the transverse arrangement are –76.38, –70.20, –64.97, and –60.44 mm. For every 5 GPa increase in elastic modulus, the maximum deformation of the pipe curtain is reduced by approximately 9.35 mm in the longitudinal arrangement and 5.30 mm in the transverse arrangement. When the elastic modulus increases from 5 to 20 GPa, the maximum deformation reductions reach 26.87% and 20.87% for the longitudinal and transverse arrangements, respectively. These results indicate that variations in the elastic modulus of the grouting body have a relatively minor influence on pipe curtain deformation. Therefore, the strength of the grouting material can be appropriately adjusted to optimize construction costs without significantly affecting structural performance.

5.4 Excavation span

The deformation results of the pipe curtain for different excavation spans are shown in Fig. 13. For the same excavation span, the deformation curve under the longitudinal arrangement exhibits more pronounced curvature. As the excavation span increases, the deformation of the pipe curtain increases significantly, and both the amplitude and the extent of the deformation gradually expand.

The maximum deformation values of the pipe curtain for different excavation spans are shown Fig. 14. As the excavation span increases, the maximum deformation exhibits an exponential growth trend. When the span increases from 6 to 12 m, the maximum deformation of the pipe curtain under the longitudinal arrangement is –9.75, –22.84, –55.57, and –113.28 mm, while under the transverse arrangement, the corresponding values are –7.93, –16.13, –39.49, and –80.63 mm. Compared with the case of a 6 m excavation span, increasing the span to 12 m results in deformation increases of approximately 11.6 times and 10.2 times under the longitudinal and transverse arrangements, respectively. These results clearly demonstrate that the excavation span has a substantial influence on pipe curtain deformation. When the excavation span exceeds 8 m, the deformation of the pipe curtain increases sharply in both arrangements. Therefore, for large-span excavation conditions, the multi-pilot tunnel method is recommended to reduce the span of each pilot tunnel and effectively control pipe curtain deformation. It is suggested that the pilot tunnel span be limited to within 8 m to ensure structural stability and safety.

6 Conclusions

1) A deformation calculation model for pipe curtains was established based on the small-deflection theory of ETPs. The proposed model solves the deformation equation under the boundary condition of “simply supported at three ends and free at one end”. The methods for calculating the bending stiffness of pipe curtains under both transverse and longitudinal arrangements were also provided.

2) Comparison with existing theoretical models and in situ monitoring data shows that the results of the proposed model exhibit better agreement with the observed deformation, both in trend and magnitude. The average errors under the transverse and longitudinal arrangements are 0.9% and 19.8%, respectively. Although the results are slightly conservative, they are suitable for direct application in engineering practice. During construction, the deformation of pipe curtains under the transverse arrangement is smaller than that under the longitudinal arrangement. Therefore, where conditions allow, the transverse arrangement is recommended for engineering projects.

3) Increasing the pipe diameter effectively reduces excavation-induced deformation of the pipe curtain, however, this control effect diminishes as the diameter continues to increase. When the elastic modulus of grouting body increases from 5 to 20 GPa, the maximum reduction in pipe curtain deformation is within 30%, indicating that the elastic modulus has a relatively minor influence. On the other hand, the excavation span has a pronounced effect on deformation. For large-span excavations, the multi-pilot tunnel method is recommended to reduce the span of individual tunnels and improve deformation control. It is recommended to control the maximum span of a single pilot tunnel within 8 m.

4) The maximum pipe curtain deformation decreases exponentially with increasing steel pipe diameter, decreases linearly with the elastic modulus of the grouting body between pipes, and increases exponentially with the excavation span. The relative influence of these three parameters on pipe curtain deformation is ranked as follows: excavation span > steel pipe diameter > elastic modulus of the grouting body between pipes.

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