Application and Effect Evaluation of Elastic Foundation Beam Method in Deformation Analysis of Underground Passage Underneath Excavated Tunnel

Abstract

Tunneling under existing underground buildings has become a common issue in densely populated urban areas. The current elastic analysis for ground displacement from new tunnel construction focuses on circular tunnels. However, theoretical analysis for non-circular chamber excavations is lacking. This paper aims to develop an elastic analysis for ground deformation from non-circular chamber excavations using the ‘equivalent radius’ theory, Verruijt’s formula, and Park’s model. It adjusts the ellipticization parameter δ and uniform radial displacement ε in Verruijt’s formula. Additionally, equivalent stiffness values for diverse existing hollow structures are considered using various approaches. A comparison of the theoretical results from field measurements and simulations shows the modified equations predict ground deformation well. Finally, parametric analysis explores the impacts of non-circular cavern excavations on existing structures under different factors.

1. Introduction

Rapid urban development has resulted in many deep excavation projects for construction and underground systems. Constraints of underground space often force excavation works to be carried out near or directly under existing underground structures. Excavation-induced soil movements interacting with the tunnel can cause adjacent existing structures to experience insufficient substrate-bearing capacity, leading to settlement and internal force changes. These, in turn, compromise the safety of the existing structures [1,2]. Therefore, evaluating the adverse impacts of tunnel excavation on existing underground structures is crucial.

In terms of model tests [3,4,5,6], extensive research has been conducted, including investigations into the stress and deformation characteristics of existing foundations under cyclic load in new tunnels [7], the mechanical behavior of existing tunnels due to pebble stratum shield undercutting [8], and the effects of shield underpenetrating on tunnels with varying lining types [9]. For numerical simulations [10,11,12,13,14,15,16], studies include the effect of new tunnel underpasses on the ballastless track of existing high-speed railway tunnels [17], the torsional deformation of existing structures due to shield underpasses in rectangular lanes [18], and the influence of double-shield tunnel excavation on the displacement and internal forces of existing buildings [19]. Considerable and valuable efforts have been made to delve into the effects of tunnel excavation on existing underground structures. However, while model tests offer a superior reflection of actual stresses in new and existing structures, their high equipment and operational costs hinder widespread adoption among researchers. Furthermore, numerical simulations, despite comprehensively illustrating transverse and longitudinal deformations and internal force distributions in existing structures influenced by new tunnels, face challenges such as complex modeling processes, time-consuming calculations, and intricate parameter determinations for tunnel structures and soil constitutive models. Consequently, practical engineering evaluations find limited applications for numerical simulations, and swiftly and efficiently calculating the response of existing structures remains challenging.

Compared with model tests and numerical simulations, the theoretical method boasts clear conceptual mechanisms, straightforward calculations, and easy generalization. It is suitable for analyzing the response of existing structures during the preliminary engineering evaluation stage, offering efficient and stable calculation results. Theoretical analysis studies encompass elastic foundation models [20,21,22,23,24,25,26,27], including the Euler–Bernoulli beam calculation for vertical deformation of existing tunnels under underpasses, derived from the Winkler model [28], and settlement methods for homogeneous structures during tunnel excavation [29]. For example, based on the analytical solution of shield tunnels influenced by additional ground loads derived from the Pasternak model [30], dynamic modeling of partially embedded structures’ PNF in soil was performed [31]. Furthermore, Timoshenko beams have been employed by researchers to model the response of existing tunnels to shield crossing [32,33,34,35] alongside the application of the Fuss–Winkler model [36]. However, these studies primarily focus on analyzing the influence of circular tunnels passing beneath structures, with a lack of accurate analysis of irregular cavern sections. In fact, as urbanization progresses, irregular cavity cross-sections have become widely used [37,38,39,40]. The excavation of these irregular sections’ impact on existing structures warrants further exploration.

In this paper, an underground subway tunnel is used as an example to analyze existing structures from non-circular chamber excavation utilizing the two-stage method incorporating the “equivalent radius” theory. The interaction between the new tunnel, soil mass, and existing structure is decoupled into two separate analysis stages. First, the existing structures are ignored, and only the new tunnel is considered. The free-soil displacement at the location of the future existing tunnel, due to the excavation, is analyzed and computed. In the second stage, the additional displacement load arising from this free-soil displacement is applied to the existing structure. Subsequently, the interaction between the soil and the existing structure is analyzed to determine the mechanical response of the latter. Field monitoring and numerical simulation validate the method. Parametric studies investigate the effects of factors like the clear distance between the new single-cavity tunnel and the existing underground structure’s floor, the excavation area of the new tunnel, and the spacing between twin-cavity tunnels. The presented analysis and theoretical models offer valuable insights for engineers tackling non-circular cavern excavations through existing hollow structures.

2. Displacement Field of Soil Caused by the Excavation of Non-Circular Caverns

2.1. Equivalent Radius Theory

Based on the principles of elastic–plastic mechanics and rock mechanics, the stress and displacement fields of surrounding rock show significant stress concentration due to angularity in irregular caverns. Given the current limitations in the development of these mechanics, precise analysis is only feasible for tunnel sections like circular and elliptical caverns. Irregular sections, such as horseshoe and square shapes, lack accurate analysis. The shape effect on stress distribution around irregular sections is concentrated in the corners, minimally impacting the overall stratum’s stress and displacement field evolution. Thus, circular quantization of non-circular tunnels is an effective method to consider the size and shape of non-circular tunnel sections.

For calculating the equivalent radius of the non-circular caverns, Liu and Zhai [41] introduced two different circular quantification standards of the non-circular caverns:

(1)

Equivalent radius folding algorithm

To calculate the stress distribution around deep non-circular caverns, the equivalent radius r₀ is replaced by the radius of a virtualized circle with the same area. The equation for this is given below:

$$r_0=k\cdot {\left({\displaystyle \frac{S}{\pi }}\right)}^{{\displaystyle \frac{1}{2}}},$$

(1)

where r₀ is the virtual circle radius, S is the actual tunnel section size, and k is the correction coefficient of the section shape. The correction factor k is enlisted in

Table 1. Values of correction coefficient k for different shapes.

Shape	Oval	Arch	Rectangle	Square	Positive Trapezoid
k	1.05	1.10	1.20	1.15	1.20

.

(2)

Circumscribed circle method

The irregular section’s circumscribed circle is constructed, and its radius is used to represent the virtual radius of the actual section. This virtual radius can be obtained via geometric graphing or analytical calculation, as shown in Figure 1.

2.2. Correction of Soil Displacement Owing to Soil Loss

Based on Sagaseta’s [42] source and sink theory, Verruijt [21] assumes the soil to be a linear elastic material. The deformation mechanism during tunnel excavation is equivalent to radial shrinkage on the tunnel surface and long-term elliptical deformation under initial in-situ stress. For any Poisson’s ratio, the displacement and stress fields in a semi-infinite space result from the superposition of these two effects. The settlement at the existing tunnel due to excavation at a free site can be expressed as follows:

$$\begin{array}{cc}\hfill U_z\left(x\right)={\displaystyle \frac{2\epsilon R^2}{m}}& \left({\displaystyle \frac{\left(m+1\right)z_2}{r_2^2}}-{\displaystyle \frac{mz\left(x^2-z_2^2\right)}{r_2^4}}\right)\hfill \\ \hfill & -2\delta R^{2}h\left({\displaystyle \frac{x^2-z_2^2}{r_2^4}}+{\displaystyle \frac{m}{m+1}}{\displaystyle \frac{2z{z}_2\left(3x^2-z_2^2\right)}{r_2^6}}\right),\hfill \end{array}$$

(2)

$$r_1=\sqrt{x^2+{\left(z-h\right)}^2};r_2=\sqrt{x^2+{\left(z+h\right)}^2},$$

(3)

where ε is uniform radial contraction, ε = u₁/R; u₁ is uniform radial displacement, δ is the tunnel elliptical, R is the tunnel radius, h is the buried depth of the tunnel, x is the horizontal coordinate from the tunnel axis, ν is the Poisson ratio, and z is the vertical coordinate, z₂ = z + h, and m = 1/(1 − 2ν).

The key parameters for Verruijt’s [21] formula are the values of the radial shrinkage ε and the ovalization parameter δ. Verruijt [21] proposes the calculation formula of radial shrinkage ε as follows:

$$\epsilon ={\displaystyle \frac{A}{4\left(1-\nu \right)\pi R^2}}={\displaystyle \frac{\eta }{4\left(1-\nu \right)}},$$

(4)

where A is the area of surface settling trough per unit length, and η represents the rate of formation loss.

Sagaseta [42] presents the relationship between the two parameters as shown below:

$$\delta =4(1-v)(1-2v)\epsilon ,$$

(5)

The radial shrinkage ε and ovalization parameter δ proposed by Verruijt [21] and Sagasta [42] are often tested for calculating ground settlement. However, these differ from the measured ones after being verified through actual project results. Thus, these two parameters require adjustment.

Park [43] proposed four models for soil movement around tunnels to calculate stratum deformation in shallow buried tunnels, as illustrated in Figure 2. Among these, both B.C-3 and B.C-4 consider tunnel ovalization deformation.

In conjunction with measured data and Park’s tunnel convergence model, it was determined that the tunnel’s elliptic convergence aligns with the B.C-4 model. Consequently, the choice of convergence model and the validation of measured data play crucial roles in optimizing elliptic parameters, with their adjustments significantly affecting the accuracy of elliptic parameters in non-circular caverns. By considering the B.C-4 model as a benchmark for the long-term elliptical deformation of the tunnel, the tunnel ellipticity parameter δ can be calculated as follows:

$$\delta ={\displaystyle \frac{D_{\mathrm{h}}-D_{\mathrm{v}}}{D}}={\displaystyle \frac{1.5u_0}{2R}},$$

(6)

where D_h is the horizontal diameter after deformation, D_v is the vertical diameter after deformation, and D is the tunnel design diameter.

Regarding the radial contraction ε suggested by Verruijt [21], it has been found through verification and calculation that directly applying Verruijt’s [21] results to non-circular cavities typically results in underestimated soil displacement. Therefore, adjustments are necessary. Based on practical engineering projects, the calculation is refined as follows:

$$\epsilon ={\displaystyle \frac{\eta }{2\left(1-\nu \right)}},$$

(7)

2.3. Validation of the Modified Soil Displacement Equation

To verify the applicability of the iso-substituted circle method in step excavation and the reliability of the corrected values for ε and δ in Verruijt’s [21] study, statistics from step excavation works were compiled in

Table 2. Mining method construction case.

Cases	New Tunnel	Geological Condition	Construction Method	H (m)	R (m)	Smax (mm)	V (%)
1	Urumqi Metro Line 1 [44]	Silty clay, muddy siltstone	Bench method	14.83	3.34	−21.4	1.13
2	Xishipo Tunnel [45]	Gravel soil, sandstone, strong weathered mudstone	Bench method	28.23	3.23	−11.33	1.22
3	Trial tunnel [46]	London clay, River Thames gravel	CRD	20.8	4.15	−14.84	0.713
4	Beijing Subway Line 4 [47]	Round gravel pebbles, pebbles, silty clay	Bench method	12.80	3.30	−19.03	0.887

Where H is the buried depth of the tunnel axis, R is the virtual radius of the new tunnel, S_max is the maximum settlement, and V is the formation loss rate.

. The modified formula presented in this paper is compared with measured data, and the results are contrasted with the relationship calculation results introduced by Sagaseta [42].

Figure 3 highlights the surface settlement curves for several cases. From the figure, it can be seen that there is no significant difference in curve shape and deformation pattern between the modified parameters in this paper and Sagaseta’s [42] relational formula for computing soil displacement, with only minor numerical discrepancies. The comparison reveals that incorporating ε and δ from Sagaseta [42] into the Verruijt [21] formula typically yields a soil displacement smaller than the measured value. At the maximum difference, the calculated value is only 71.46% of the measured maximum settlement. The adjusted parameter values align more closely with actual measurements, indicating that the corrected soil displacement for non-circular chamber excavation in semi-infinite space is applicable in the step method. Based on these results, the improved analytical solution and Peck formula [48] show good predictability for tunnel excavation-induced surface subsidence. However, the Peck formula [48] oversimplifies the influence of tunnel elliptical deformation on strata in later stages. The calculation method used in this study is slightly larger than the actual settlement, resulting in more conservative conclusions that ensure project safety.

3. Analysis of Deformation Characteristics of Underground Structure

3.1. Stress Analysis of Existing Structures on the Pasternak Foundation Model

The Pasternak model [22], built upon the Winkler material model, assumes the existence of shear action between foundation units. This shear action is achieved through a shear layer that can only produce lateral shear deformation and is incompressible, as shown in Figure 4. The governing equation for the impact of additional soil deformation on the existing structure can be derived as follows:

$$EI{\displaystyle \frac{d^4\omega }{d{x}^4}}-G_{\mathrm{p}}b{\displaystyle \frac{d^2\omega }{d{x}^2}}+kb\omega =bq(x),$$

(8)

where EI represents the flexural stiffness of the extant structure, E is the elastic modulus of the existing structure, I represents the moment of inertia of the section of the existing structure, q(x) denotes the additional load applied to the beam, G_p indicates the foundation shear stiffness, k is the foundation spring stiffness, ω is the deflection of the beam, and b is the beam cross-section width.

Regarding stiffness equivalence of existing structures, the structure is often assumed to be a rectangular beam for calculation, as shown in Figure 5. Equating them to a rectangular beam using Equation (14) gives the following:

$$I={\displaystyle \frac{1}{12}}\left(b{h}^3-b_1{h}_1^3\right),$$

(9)

where b is the width of the existing structure, h is the existing structural height, and b – b₁ is the thickness of the existing structure.

Vesic [49] and Tanahash [50] introduced formulas for the foundation shear stiffness G_p and the foundation spring stiffness k, respectively.

$$k={\displaystyle \frac{0.65}{b}}{\left({\displaystyle \frac{E_{\mathrm{s}}{b}^4}{EI}}\right)}^{{\displaystyle \frac{1}{12}}}{\displaystyle \frac{E_{\mathrm{s}}}{1-{\nu }_{\mathrm{s}}^2}},$$

(10)

$$G_{\mathrm{p}}={\displaystyle \frac{E_{s}t}{6\left(1+{\nu }_{\mathrm{s}}\right)}},$$

(11)

where E_s and ν_s are the elastic modulus and Poisson ratio of soils, respectively, and t is the thickness of the shear layer.

Regarding Equation (8), Zhang [25] utilized the finite difference method to derive the Pasternak foundation model for tunnel–soil–beam interaction.

Based on the assumption that the finite long beam is divided into n equal segments, the ordinary differential equation can be transformed into a difference equation by applying the finite difference method:

$$\lambda {\omega }_{i-2}+\mu {\omega }_{i-1}+\xi {\omega }_i+\mu {\omega }_{i+1}+\lambda {\omega }_{i+2}=q_i\left(x\right),$$

(12)

where i = 0, 1, 2, 3,…, n; and coefficients λ, μ, and ξ follow the following relationship:

$$\left[\begin{array}{c}\lambda \\ \mu \\ \xi \end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ -4& -1& 0\\ 6& 2& 1\end{array}\right]\cdot {\left\{\begin{array}{ccc}{\displaystyle \frac{EI}{b{h}^4}}& {\displaystyle \frac{G}{h^2}}& k\end{array}\right\}}^T,$$

(13)

Taking Equation (11) into account, the relationship between the displacement of the foundation beam and the additional load can be derived as follows:

$${\left\{\omega \right\}}_{\left(n+1\right)\times 1}={\left\{K\right\}}_{\left(n+1\right)\times \left(n+1\right)}^{-1}\cdot {\left\{P\right\}}_{\left(n+1\right)\times 1},$$

(14)

where {ω} is the displacement matrix, {K} is the transverse stiffness matrix of the foundation beam, and {P} is the additional load matrix. The calculation method is mentioned in Li [51].

3.2. Equivalent Stiffness Analysis of Elastic Foundation Beam

In the case of box structures, equivalent calculation often results in overly large stiffness for the elastic foundation beam. Therefore, it is crucial to discuss the equivalent stiffness of box or hollow tubular structures with varying sections and materials.

The statistics for shallow buried cut-and-cover tunnels beneath existing structures of various cross-sections and materials in China are provided in

Table 3. Construction cases of underpassing existing structures.

Project	New Tunnel	Construction Method	Excavation Area (m2)	z	Existing Structure	h0	L × H	V
1 [51]	Culvert (double-line)	Bench method	20.43	3.667	Wukesong Station	4.833	18.3 × 7.8	V1 = 0.284
								V2 = 0.284
2 [52]	Heat distribution tunnel (single-line)	Bench method	18.56	1.6	Chongwenmen-Beijing Station interval	2	10 × 6	0.0552
3 [53]	Beijing Subway Line 6 (double-line)	Bench method	31.67	2.6	Beijing Subway Line 4	11.27	6.0 × 6.0	V1 = 0.264
								V2 = 0.243
4 [54]	Beijing Subway Line 8 (double-line)	Bench method	32.17	2.456	Beijing Subway Line 10	15.144	6.1 × 6.1	V1 = 0.133
								V2 = 0.123
5 [55]	Subway entrance (single-line)	Bench method	38.48	1.888	Sewage pipe	5.97	2.8 × 2.8	0.215
6 [56]	Qingdao Subway Line 3 (single-line)	Three-step method	37.58	9.5	Water supply pipe	1.8	1.2 × 1.2	1.645
7 [57]	Dalian Subway Line 4 (single-line)	Bench method	29.27	2.7	Cast iron pipe	3.26	1.8 × 1.8	0.309

Where z indicates the net distance between the new tunnel and floor, h₀ denotes the thickness of coated soil of the existing structure, L is the width of the existing structure, H represents the height of the existing structure, V is the stratigraphic loss rate, z₀ indicates the height of the cover layer on the new tunnel, S_h is the horizontal clear distance between the tunnel and the pile, D represents the pile diameter, and L₁ is the pile length.

. Projects 1 and 2 feature ground box structures, while Projects 3 and 4 have large-diameter subway tunnels. Project 5 involves a small-diameter cast iron pipe, and Projects 6 and 7 utilize small-diameter municipal concrete pipes. To explore the foundation beam stiffness equivalence for the hollow structures in Projects 1 to 7, the box-type structure foundation beams are equated to unit width using 1D, 2D (D representing the base plate thickness), and the overall structure. For tubular structures, finite length beams are equated to unit width using 1D, 2D (D denoting the wall thickness), and the overall structure. Then, the two-stage theory is applied to calculate the equivalent foundation beam through various methods. By comparing the computed existing structural deformation with the additional bending moment, a reasonable equivalent method is sought. The foundation beam stiffness values are presented in

Table 4. Equivalent beam stiffness of the existing structure.

Project	Existing Structure Forms	Material	1D (N·m−2)	2D (N·m−2)	Overall Equivalent Stiffness of Structure (N·m−2)
1	Box-like structures	C40	2.85 × 109	2.3 × 1010	2.24 × 1012
2	Box-like structures	C40	1.76 × 109	1.18 × 1010	3.65 × 1012
3	Tunnels of metro	C50	1.27 × 108	1.01 × 109	8.58 × 1011
4	Tunnels of metro	C50	1.27 × 108	1.01 × 109	9.04 × 1011
5	Sewage pipe	C25	2.875 × 106	1.87 × 107	1.47 × 109
6	Sewage pipe	C25	2.3 × 107	1.84 × 108	3.88 × 1010
7	Cast iron pipe	Rolled steel	1.71 × 104	1.37 × 105	4.6 × 109

.

3.3. Existing Structure Deformation Analysis

Figure 6 compares the results of numerical predictions, field monitoring, and equivalent calculations for the deformation of an existing structure due to a new tunnel undercrossing. The figure shows that the structural settlement curve of the hollow structure, calculated by equating the foundation beam to the thickness (1D), aligns well with measured and simulated results.

The settlement curves for Projects 1 to 7, calculated based on overall structural equivalence, differ significantly from measured data due to excessive stiffness. In Project 1, the existing structural floor deformation abruptly changes at expansion joints and distributes linearly. Notably, a distinct difference arises between Projects 1 and 2 when the structure is equivalent to twice the thickness of the floor or twice the thickness of the wall (2D). Conversely, Projects 3 to 7 show good matching. The settlement curves for Projects 5 to 7, computed by equivalent foundation beams according to 1D and 2D, respectively, coincide. This is because minor stiffness increases in an elastic foundation beam with low stiffness do not significantly alter deformation until a critical stiffness value is reached.

By comparing Project 2 and Project 4, it can be observed that, based on the 2D equivalent foundation beam versus the 1D, the deformation of the existing structure is reduced by 39.4%, 4.22%, and 3.26%, respectively. These results suggest limited deformation reduction for Project 3 and Project 4. Considering the existing structural stiffness data in Table 4, structural deformation calculated from small changes in foundation beam stiffness only occurs when the stiffness reaches 1.0 × 10⁹ N·m⁻². Thus, determining the equivalent stiffness of foundation beams solely from existing structural deformation is insufficient. Integrating these results into force analysis is necessary.

3.4. Internal Force Analysis of Existing Structure

As shown in Table 3, seven cases are simulated using the finite element method to analyze the force issue of the newly built structure crossing beneath the existing underground structure, which varies in materials and sections. The calculated additional bending moments of the existing structure are then compared with the results from the two stages.

Figure 7 depicts the comparison of additional bending moments in various existing structures. In this figure, a positive bending moment signifies tension on the lower side, while a negative bending moment indicates tension on the upper side. Due to the excessively large overall equivalent stiffness of the structure, the calculated additional bending moment of the existing structure is excessively large and is therefore not analyzed here. Observing the figure, it is evident that, in terms of curve trend or additional bending moment value, the results of the 1D equivalent foundation beam align closely with numerical simulation outcomes. When using a 2D equivalent foundation beam, the additional bending moment value is several times that of the 1D. This implies that, when analyzing new tunnels crossing box or hollow tubular structures using elastic foundation beams, the calculation can be approximated by considering the thickness of the box structure’s bottom plate or, one time, the wall thickness of the structure, equivalent to a foundation beam of unit width.

4. Actual Project Study

To verify the accuracy of the analytical solution for stratum displacement caused by non-circular tunnel excavation and the equivalent stiffness value method of elastic foundation beams discussed, this paper uses the underground excavation tunnel in China Jinan CBD, crossing under existing structures, as the engineering background. As shown in the comparison, theoretical calculation results are contrasted with field measurements and numerical simulation results. This section emphasizes, through theoretical analysis, the impact of new tunnel parameters and their spatial relationships with existing structures on the latter.

4.1. Project Overview

The Silk Belt Park—Ligeng Road interval lies within the CBD of Jinan. The cut-and-cover tunnel in this interval sits beneath a single-layer concrete frame structure. Retaining piles, reserved for open excavation, flank the existing structure on both sides, with a diameter of 0.8 m and a length of 14.25 m. The tunnel vault stands roughly 2.22 m to 8.18 m away from the existing structure floor, and a minimum horizontal clearance of about 3.9 m separates the tunnel vault from the retaining pile.

The bench method is utilized for constructing the section, with the grouting leading small pipe placed within a 120° angle of the tunnel top for pre-reinforcement. As excavation proceeds, the height of the upper step is 4.5 m, advancing 2.5 m to 3.0 m per ring, while the lower step progresses 5 m per cycle. The distance between the palm faces of the upper and lower steps ranges from 15 m to 20 m. At the grating’s base, two single-length 3 m locking foot anchors are installed. The initial support comprises shotcrete C25 reinforced with a grating steel frame network, designed to be 0.25 m thick. Additionally, two Φ22 mm, 3 m long anchors with a 14° horizontal inclination are placed at the grating’s base as locking foot anchors. Figure 8 illustrates the spatial relationship between the tunnel and the existing structure.

4.2. Establishment of Finite Element Model

The finite element method is utilized to assess the dynamic response of the existing structure due to the construction of the new tunnel crossing, with the results being compared to theoretical calculations. As depicted in Figure 9, the MIDAS GTS was used to create a three-dimensional finite element model for the cut-and-cover tunnel project of the Jinan Silk Belt Park—Ligeng Road interval underneath the existing structure. In accordance with the survey results, the formations primarily impacted by tunnel excavation encompass silty clay, consolidated conglomerate, fully weathered marl, and moderately weathered marl. In this paper, with reference to these soil properties and the applicability of various constitutive models, the modified Mohr–Coulomb model, combining the nonlinear elastic and elastoplastic models, is adopted.

Figure 9b depicts the model of the existing structure. The roof and floor of the underpass, along with the tunnel’s initial support, are simulated using slab elements. The envelope piles, columns, and grid steel frames are simulated using beam elements. The lock-foot bolt and lead conduit are simulated using implantable trusses. The interface unit emulates the relative movement or separation between the envelope pile, steel grid frame, and solid unit. For beam, plate, column, pile, and lock anchor, the linear elastic model is adopted.

To mitigate the influence of boundary conditions, the model’s length and height are extended outward by 3R (where R is the tunnel span) from the tunnel edge and arch bottom, respectively. The modeling dimensions are 62.4 m × 40 m × 36 m, as shown in Figure 9a. In this model, the Z-axis represents the vertical direction, the Y-axis aligns with the tunnel axis with the positive direction being the tunnel excavation direction, and the X-axis is perpendicular to the tunnel axis. The top surface is a free surface, while normal constraints are applied to the other four sides and the bottom.

4.3. Physical and Mechanical Indexes

Based on the geological exploration results, the physical and mechanical indexes of the formations and existing structures within the study scope are shown in

Table 5. Physical and mechanical parameters of the formations.

Formations	Compression Modulus (Mpa)	Internal Friction Angle (°)	Poisson Ratio	Unit Weight (kN/m3)	Cohesion (kPa)
Loess	4.10	16	0.37	18.0	34.9
Silty clay	5.17	15	0.42	16.4	39.5
Conglomerate	35.00	30	0.30	18.8	50.0
Fully weathered marl	55	20	0.27	24.1	48.0
Highly weathered diorite	104.3	45	0.27	26.4	300

and

Table 6. Physical and mechanical parameters of the structural members.

Component	Elasticity Modulus (GPa)	Unit Weight (kN/m3)	Poisson Ratio	Sectional Area (m2)
Substructure	45	32	0.28	1.38 × 10−3
Fender piles	210	78.4	0.3	3.14 × 10−4
Primary support	23	22	0.2	-
Condulet	30	25	0.24	0.502
Feet-lock bolt	34.5	25	0.23	-

.

4.4. Simulate Working Circumstances and Excavation Procedures

Tunnel construction is a continuous and dynamic process. However, due to limitations in analysis theories and methods, the entire continuous construction of mine tunnels can be divided into sequential stages. The dismantling process can be summarized as follows: for each construction step corresponding to each advancement of the tunneling face, the solid unit is deactivated, while the initial support, grid steel frame, retractable anchor rod, and condulet are activated.

In the actual construction process, the excavation footage for each ring of the upper step varies from 2.2 m to 3.0 m, while that of the lower step is approximately 5.0 m. Consequently, during the simulation, the upper step is excavated in increments of 2.25 m each time. When the spacing between the excavation faces of the upper and lower steps reaches 15 m, the lower step is excavated in increments of 6.75 m. Considering that the underground structure and surrounding soil have consolidated due to self-weight in practical engineering, tunnel construction is deemed to be the primary cause of soil disturbance. The construction sequence of the hidden tunnel is depicted in Figure 10.

4.5. Comparison of Calculation Results

Figure 11 displays the settlement deformation curve of the bottom slab, while Figure 12 illustrates the additional bending moment in the slab due to tunnel excavation. Notably, both analytical and finite element results, as well as measured data, exhibit similar deformation and bending moment patterns. In terms of quantities, the analytical solution predicts a maximum settlement of 3.58 mm, with a measured value of 3.1 mm, differing by 0.48 mm. Numerical simulation and analytical solutions yield additional bending moments of 84.13 kN∙m and 98.34 kN∙m, respectively, with a 14.27% error. These comparisons validate the feasibility of using the proposed elastic foundation beam method.

4.6. Comparative Analysis of Simulation Results with Different Parameters

4.6.1. Influence Analysis of Excavation Area of New Single Tunnel

The excavation area A of the new tunnel is calculated by 0.75S, 1.0S, 1.25S, 1.5S, 1.75S, and 2.0S (S is the excavation area of the tunnel in this project, m²). Figure 13 shows the mechanical response of the existing structure. From this figure, it can be observed that the maximum settlement (S_max) and the maximum additional bending moment (M_max) of the existing structure vary linearly with the excavation area (A), indicating an increase in both as the excavation area increases. Based on Peck’s [48] proposed relationship between S_max, A, and the stratum loss rate, reducing the excavation area can effectively control the deformation of the existing structure.

4.6.2. Analysis of the Impact of the Clear Distance Between the New Single-Hole Tunnel and the Bottom Plate

The net distance S_v0 between the tunnel crown and the bottom plate is taken as 1.5 m, 2.25 m, 3.0 m, 4.5 m, 6.67 m, and 8.74 m, respectively, for calculation.

Figure 14 shows the relationship of clear distance between the newly built tunnel and the floor and the forces and deformations. From this figure, it can be seen that as the clear distance between the new tunnel and the existing structure floor increases, the forces and deformations on the existing structure decrease. When the net distance S_v0 of the bottom plate rises from 1.5 m to 6.67 m, the decay of deformation and force in the existing structure is relatively rapid; from 6.67 m to 8.74 m, this decay slows down. This suggests that the impact of the new tunnel on the existing structure can be effectively mitigated when S_v0 exceeds twice the tunnel span. As illustrated in Figure 13c, the maximum bending moment M_max reaches 214.9 kN∙m at S_v0 = 6.67 m and jumps to 480.96 kN∙m at S_v0 = 3.0 m, marking a 2.24 times increase and a rapid growth. Hence, in this project, it is advisable to maintain a spacing between the tunnel and the existing structure of at least 3.0 m; otherwise, reinforcement of the bottom slab is necessary to guarantee structural safety.

4.6.3. Influence of Spacing Between the Twin Tunnels

The new tunnel is a two-lane separated type, and the mechanical response of the twin tunnels’ horizontal spacing L change on the existing structure is highlighted in Figure 14. Here, the spacing L is taken as 3.0 m~12 m. As shown in Figure 15a, as the distance L between the two tunnel lines increases, the settlement curve of the existing structure’s floor shifts outward, transitioning from a “V” to a “W” shape. The settlement trough becomes shallower and wider. For tunnel spacings L < 6.0 m (roughly equal to the tunnel span), the bottom plate deformation curve is significantly influenced by L. When L exceeds 9.0 m, the deformation curve starts to change from “V” shaped to “W” shaped. This is attributed to the close proximity and strong mutual disturbance between the successively excavated tunnels, intensifying soil deformation in the commonly disturbed area and exhibiting a “V”-shaped settlement trough.

From Figure 15c, the maximum bending moment M_max of the existing structure negatively correlates with the spacing L between the tunnels. However, as shown in Figure 14b, when L equals 9 m, the maximum positive bending moment + M_max reaches 51.91 kN∙m. In contrast, at L = 12 m, +M_max is 40.51 kN∙m, and the maximum negative bending moment − M_max is −39.71 kN∙m, resulting in a more complex structural response. Therefore, the tunnel is excavated underneath the existing structure, not only to increase the tunnel spacing to reduce the impact on the existing structure but also to identify a reasonable spatial layout relationship.

5. Discussion

During urban construction, constructing new underground tunnels is necessary. However, limited underground space inevitably results in a close distance between the new tunnel and the existing structure, leading to interaction. In such situations, while short-term mutual influence is unavoidable, conducting analyses and adopting control measures can minimize the impact on the existing structure. In the long term, new tunnels constructed near existing structures significantly aid urban development by maximizing underground space utilization and forming a comprehensive underground transportation system, which boosts transportation and economic development. However, the influence of the new tunnel on the existing structure may compromise its safety, necessitating analysis to ensure the existing structure’s internal forces and deformations remain manageable.

Using the equivalent radius method, this study improved the application of Verruijt’s formula for excavating non-circular caverns. It analyzed the impact of irregularly sectioned new tunnel excavations on existing structures. The ellipticity parameter and radial shrinkage were optimized. This method applies to situations with rigid formations or low water levels and short consolidation periods. Long-term effects of pore water pressure dissipation on the underground structure are not considered. As shown through field monitoring and numerical simulations, the proposed method is effective, providing useful guidance for engineers excavating non-circular caves within existing hollow structures.

6. Conclusions

In this study, considering the subway tunnel project supported by the Jinan CBD Municipality, the impact of undercrossing non-circular tunnels on existing underground structures was systematically analyzed using theoretical calculation and numerical simulation. The following conclusions were derived:

(1)

By integrating the equivalent radius of non-circular caverns, the Park soil movement model, and the Verruijt formula, the analytical solution for stratum deformation from non-circular cavern excavation is derived. Compared to the measured results, the revised formula exhibits good predictive accuracy for stratum deformation.

(2)

By employing various equivalent methods, the stiffness of tubular or box hollow structures with different cross-sections and materials is determined and verified through numerical simulation. The results indicate that the deformation and additional bending moment of the foundation beam, equivalent to the wall thickness of the hollow structure, align with finite element findings.

(3)

During the construction of the new tunnel undercrossing the existing structure, the excavation face area of the new tunnel and the net distance between its vault and the existing structure must be tightly controlled. Additionally, considering the spatial relationship between the tunnel and the existing structure, an appropriate axis distance for the twin tunnels should be chosen to minimize any disturbance to the latter.

(4)

When analyzing the deformation of the existing structure in the crossing project using an elastic foundation beam, the sensitivity of the beam’s deformation to the equivalent stiffness is low. However, when approaching a critical stiffness value (approximately 1 × 10⁹ N·m⁻²), small variations in the foundation beam’s stiffness led to significant differences in the calculated deformation of the existing structure. Notably, the structural stress is highly sensitive to these stiffness changes, with even minor adjustments resulting in considerable differences.