1 Introduction

Simplifying complex practical engineering problems with mathematical models is the essence of theoretical research [1,2,3,4,5,6,7,8,9,10]. In tunnel engineering, settlement deformation of the existing tunnel is one of the key controlling factors in building a new tunnel. The existing tunnels are generally treated as an Euler–Bernoulli beam to study their bending deflection due to the perturbance generated by the newly constructed adjacent shield tunneling [11,12,13]. Wang et al. [14] comprehensively studied the bending deflection of the Euler–Bernoulli beam with a series of boundary constraints, including the simply supported, clamped, free and elastic constraint, and presented the explicit solutions. In the derivation procedure, a point can be reached is that an appropriate filed function is firstly required to be assumed in these analytical methods, which also yields complex and cumbersome integral operations. Thus, these analytical methods are limited to some simple working conditions.

Therefore, many new methods have been used in theoretical research. Liang et al. [15] employed a semi-analytical method to evaluate the deformation behavior of the Euler–Bernoulli beam resting on the foundation with local subsidence. Liang [16] used the finite difference analysis method to study the corresponding effect of shield tunneling on existing tunnels. Based on this method, a theoretical analysis model of the beam subjected to the jacking force was developed [17]. In addition, the Rayleigh–Ritz method [18] was widely used to assume displacement functions in the early stages of theoretical research. However, there are several problematic issues encountered during the research process. For example, the continuing behavior for selecting the appropriate displacement functions matching the various boundary constraints can be summarized from the assumptions of boundary conditions in some research work [16, 19,20,21]. Nevertheless, obtaining the expression under some boundary conditions will not be smooth when adopting the analytical method. The admissible functions satisfying all system constraints are difficult to find when adopting the Rayleigh–Ritz method, which greatly limits its applicability [22]. To overcome the difficulties mentioned above, Deng et al. [22] employed the null space method (NSM) to facilitate the constraints in Rayleigh–Ritz method framework. And the results showed that the NSM exhibits outstanding advantages in terms of computational accuracy and cost.

In this work, the NSM based on the mechanical equilibrium is introduced to the Euler–Bernoulli beam model. This approach can satisfy arbitrary cases by filling the elements of the 'key matrix' depending on required constraint conditions. Then, a set of fundamental solution was acquired by adopting the NSM. The final response consisted of the specific solutions, so that it could well meet the specified boundary condition requirements. The applicability and accuracy of this method are corroborated by comparing the mechanical behaviours with the exact analytical solutions under the same conditions, including the constraints and load forms. The parameter values in the NSM and its effect are also discussed to give recommended values to attain greater accuracy and lower computing costs. Then, the NSM is extended to a practical engineering project of a new tunnel crossing the existing tunnel, which confirms its applicability in bending deformation assessment. Moreover, the difference in the displacement of the beam under two commonly assumed constraint conditions is also studied.

2 Methodology

The displacement function is assumed as given in Eq. (1)

$$ w = {\varvec{\alpha}}_{1 \times N} {\varvec{f}}_{N \times 1} $$
(1)

The assumed displacement function is similar to the Rayleigh–Ritz method [18], as given in Eqs. (2) and (3).

$$ {\varvec{\alpha}} = \left[ {\alpha_{1} ,\alpha_{2} , \cdots ,\alpha_{N - 1} ,\alpha_{N} } \right]_{1 \times N} $$
(2)
$$ f = \left[ {\sin \frac{{\pi x_{i} }}{L},\;\sin \frac{{2\pi x_{i} }}{L}, \cdots ,\sin \frac{{n_{s} \pi x_{i} }}{L},\;1,\;\cos \frac{{\pi x_{i} }}{L},\;\cos \frac{{2\pi x_{i} }}{L}, \cdots ,\cos \frac{{\left( {N - 1} \right)\pi x_{i} }}{L}} \right]_{{}}^{T} $$
(3)

where, α is the vector of undetermined coefficient with N order, f is the trigonometric series vector with N × 1 order, xi is the abscissa value (i = 1, 2,…, NN + 1), NN is the number of segment points assumed, ns is the maximum number of sin function (ns < N, the suggested values will be discussed later), L is the determined calculation length.

Boundary conditions concerning displacement, bending angle, bending moment, and shear force can be written in a unified form as Eq. (4).

$$ \left\{ {\begin{array}{*{20}c} {w_{\left( 0 \right)}^{{\left( {n_{1} } \right)}} = 0,w_{\left( L \right)}^{{\left( {n_{1} } \right)}} = 0} \\ {w_{\left( 0 \right)}^{{\left( {n_{2} } \right)}} = 0,w_{\left( L \right)}^{{\left( {n_{1} } \right)}} = 0} \\ \end{array} } \right. $$
(4)

where, n1 and n2 are the numbers of derivative order.

Equation (4) can be rewritten as Eq. (5).

$$ G^{*} \alpha^{T} = 0 $$
(5)

where G* is the 'key matrix', given in Eq. (6).

$$ G^{*} = \left[ {f\left( 0 \right)^{{\left( {n_{1} } \right)}} ,f\left( L \right)^{{\left( {n_{1} } \right)}} ,f\left( 0 \right)^{{\left( {n_{2} } \right)}} ,f\left( L \right)^{{\left( {n_{2} } \right)}} } \right]^{T} $$
(6)

where f(n1) and f(n2) are the corresponding vectors after derivative operation according to Eq. (4).

Then the maximum independent vector group can be obtained by the so-called null space technique, as given in Eq. (7).

$$ Z_{N \times H} = {\text{nullspace}}\left( {G^{*} } \right) $$
(7)

where Z is the required maximum independent solution (H < N, and the value of H is dependent of the numbers for assuming boundary conditions).

According to the Euler–Bernoulli beam theory [23], the classical governing equation about displacement function w is easily obtained by Eq. (8).

$$ EI\frac{{d^{4} w}}{{dx^{4} }} = P $$
(8)

where EI is the bending stiffness of the beam, and P is the uniform load acting on the beam.

Substituting Eq. (1) into Eq. (8) yields,

$$ EI\alpha f^{\left( 4 \right)} = P $$
(9)

One intermediate parameter, i.e., the initial stiffness matrix kf, must be introduced, as given in Eq. (10).

$$ k_{f} = EIf^{\left( 4 \right)} $$
(10)

Then, the other two intermediate parameters, i.e., advanced stiffness matrix ks and load vector Ps, can be written as presented in Eq. (11).

$$ \left\{ {\begin{array}{*{20}c} {k_{sN \times N} = fk_{{_{f} }}^{T} } \\ {P_{sN \times 1} = fP^{T} } \\ \end{array} } \right. $$
(11)

It should be noted that ks is the singular matrix, in which there are linearly dependent relationships in the rows or columns. With the introduction of the solution Z mentioned in Eq. (7), the singularity phenomenon in matrix ks would be eliminated. The manufacturing process is given in Eq. (12).

$$ \left\{ {\begin{array}{*{20}c} {K_{H \times H} = Z^{T} k_{s} Z} \\ {Q_{H \times 1} = Z^{T} P_{s} } \\ \end{array} } \right. $$
(12)

where, K and Q are the final stiffness matrix and load vector without any independent elements, respectively. Therefore, the displacement vector β without impurities can be solved as,

$$ K\beta = Q \Rightarrow \beta = K^{ - 1} Q $$
(13)

In the same way, the undetermined coefficient assumed at the beginning of the work is obtained as Eq. (14).

$$ \alpha^{T} = Z\beta $$
(14)

Substituting Eq. (14) into Eq. (1) yields the final solution for the displacement w, i.e., Eq. (15).

$$ w = \left( {\alpha^{T} } \right)^{T} f = \alpha f $$
(15)

The detailed flow chart of the proposed analytical method is shown in Fig. 1.

Fig. 1
figure 1

Flow chart of the proposed method

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3 Validation of the NSM

For comparing the mechanical response results (MRR) acquired by the NSM and analytical method, a classical Euler–Bernoulli beam is imposed by various constraints, including simply supported and clamped constraints. In addition, three forms of external load, uniform, locally uniform and linear distributed load, are applied to act on the beam.

3.1 Simply-Supported Constraint

The Euler–Bernoulli beam is imposed by a simply supported constraint under three kinds of external load (see Fig. 2). The related parameters values using in examples are referred to Morfidis’s report [20], EI = 248,675 kN m2, L = 7.5 m, p = 36.5 kN/m, supposing a = 0.6 L and b = 0.4 L. Then the mechanical deformation obtained by the NSM and analytical solution method, including the bending angle, shear force, and error between the two methods, are presented in Table 1. The calculation formulae of the analytical solution are provided in Appendix A.

Fig.2
figure 2

The simply supported Euler–Bernoulli beam under three kinds of load

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Table 1 Results and comparisons of the two methods
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It is seen that even though one specific value has a relatively large error, the overall calculation results by the NSM are in good agreement with the other by the analytic method, even though some error values are close to 0. An error of about 5% is an acceptable result in practical engineering. Thus, the proposed method can obtain perfect calculation accuracy.

3.2 Clamped Constraint

All conditions are consistent with Sect.  3.1 apart from the changing in ends constraint and supposing a = 0.4 L and b = 0.6 L (see Fig. 3). The results and comparisons of the NSM and analytic method are listed in Table 2, including the bending moment, shear force and error values between the two methods. It is found that the results obtained by the NSM are consistent with the analytical method, confirming the accuracy of the proposed method.

Fig.3
figure 3

The Euler–Bernoulli beam imposed by clamped constraint under three kinds of linear load

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Table 2 Results and comparison of the two methods
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4 Discussion on the Parameter Selection in the NSM

Here, the two vital parameters in the calculation process, i.e., the total number of selected items N and partition point ns, are discussed to determine the influence on the final results. The discussion is divided into two sections: simply supported and clamped constraints. Due to space limitations, the detailed calculation results are provided in Appendix B. It is shown that the precision degree of the calculation results does not continuously increase with the increasing number of items N and ns, which is against our common sense. The relationship between N and ns has not been found yet, so achieving the best accuracy is still a problem. However, the case with clamped constraint needs more items of N and ns to approach the exact answers than the case with simply support. Therefore, while employing the NSM to assess the mechanical behavior of the Euler–Bernoulli beam, the total number of selected items N and partition point ns are suggested to be in a certain range near 15 and 5 under the simply constraint and near 20 and 15 under the clamped constraint. The issue will come under the spotlight in future work.

5 Application in Settlement of Existing Tunnel

Compared with traditional theoretical methods, the present method provides more room for improvement to solve engineering problems. In this paper, the case that settlement evaluation of existing tunnel induced by new shield tunneling is taken as a case study.

5.1 The NMS Development

Based on the elastic foundation theory, the existing tunnel is regarded as an Euler–Bernoulli beam resting on the Pasternak foundation [24], so the specific characteristic of continuous deformation in foundation can be considered. The analytical model of excavation under crossing tunnels using the Pasternak foundation is shown in Fig. 4. The governing equation thus is written as [25].

$$ \left( {EI} \right)_{eq} \frac{{d^{4} w}}{{dx^{4} }} - GD\frac{{d^{2} w}}{{dx^{2} }} + kDw = PD $$
(16)

where, (EI)eq and D are the equivalent bending stiffness and width of the beam, k and G are the stiffness of the springs and shear stiffness of the shear layer, and P is the loading acting on the tunnel due to excavation of the new tunnel.

Fig. 4
figure 4

The settlement model

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The displacement function w is introduced in Eq. (16), which leads to Eq. (17).

$$ \left( {EI} \right)_{eq} \alpha f^{\left( 4 \right)} - GD\alpha f^{\prime\prime} + kD\alpha f = DP $$
(17)

The initial stiffness matrix is given in Eq. (18).

$$ k_{f} = EIf^{\left( 4 \right)} - GDf^{\prime\prime} + kDf $$
(18)

Referring to Liang's work [16], the two ends of the beam satisfy the free boundary condition, thus the following condition exists,

$$ \left\{ {\begin{array}{*{20}c} {M_{0} = M_{N} = 0} \\ {Q_{0} = Q_{N} = 0} \\ \end{array} } \right. $$
(19)

Equation (19) can be further written as Eq. (20).

$$ \left\{ {\begin{array}{*{20}c} {w_{\left( 0 \right)}^{^{\prime\prime}} = 0,w_{\left( L \right)}^{^{\prime\prime}} = 0} \\ {w_{\left( 0 \right)}^{\left( 3 \right)} = 0,w_{\left( L \right)}^{\left( 3 \right)} = 0} \\ \end{array} } \right. $$
(20)

Hence, the G* is obtained as Eq. (21).

$$ G^{*} = \left[ {f\left( 0 \right)^{\prime\prime},f\left( L \right)^{\prime\prime},f\left( 0 \right)^{\left( 3 \right)} ,f\left( L \right)^{\left( 3 \right)} } \right]^{T} $$
(21)

It is noteworthy that the calculation of external load P is different from the general way. It has its specific approach [26], as given in Eq. (22).

$$ P = ku_{z} - G\frac{{d^{2} u_{z} }}{{dx^{2} }} $$
(22)

where, uz is the free-field soil movement due to tunnel excavation.

According to Ref. [27], soil movement can be expressed as Eq. (23).

$$ \begin{gathered} u_{z} = R^{2} \left\{ { - \frac{z - H}{{x^{2} + (z - H)^{2} }} - \frac{{2z\left[ {x^{2} - (z + H)^{2} } \right]}}{{\left[ {x^{2} + (z + H)^{2} } \right]}}} \right. + (3 - 4v)\left. { \cdot \frac{z + H}{{x^{2} + (z + H)^{2} }}} \right\} \hfill \\ \begin{array}{*{20}c} {} & {} \\ \end{array} \cdot \varepsilon_{0} \exp \left\{ { - \left[ {\frac{{1.38x^{2} }}{{(H + R)^{2} }} + \frac{{0.69z^{2} }}{{H^{2} }}} \right]} \right\} \hfill \\ \end{gathered} $$
(23)

where, R is the radius of the new tunnel, ε0 is the ground loss caused by excavation, H is the vertical distance between the axis of the new tunnel and the surface, and z is the depth measured from the ground surface.

However, the plane relationship between the new and old tunnels is generally not vertical. If Eq. (23) is still adopted to estimate the soil displacement, it leads to erroneous results, not according with fact. Then, the intersection angle factor is taken into consideration by Liang et al. [28], as given in Eq. (24).

$$ \begin{gathered} u_{z} = R^{2} \left\{ { - \frac{z - H}{{\left( {x\sin \theta } \right)^{2} + (z - H)^{2} }}} \right. + (3 - 4v) \cdot \frac{z + H}{{\left( {x\sin \cdot \theta } \right)^{2} + (z + H)^{2} }} \hfill \\ \begin{array}{*{20}c} {} & {} \\ \end{array} \left. { - \frac{{2z\left[ {\left( {x\sin \theta } \right)^{2} - (z + H)^{2} } \right]}}{{\left[ {\left( {x\sin \theta } \right)^{2} + (z + H)^{2} } \right]}}} \right\} \cdot \varepsilon_{0} \exp \left\{ { - \left[ {\frac{{1.38\left( {x\sin \theta } \right)^{2} }}{{(H + R)^{2} }} + \frac{{0.69z^{2} }}{{H^{2} }}} \right]} \right\} \hfill \\ \end{gathered} $$
(24)

where, θ is the intersection angle between the new and existing tunnel.

The foundation parameters, k and G, can be obtained by Eqs. (25) and (26) [17, 29,30,31].

$$ k = \frac{{1.3E_{s} }}{{\eta D\left( {1 - v^{2} } \right)}}\sqrt[{12}]{{\frac{{E_{s} D^{4} }}{{\left( {EI} \right)_{eq} }}}} $$
(25)
$$ G = \frac{{E_{s} t}}{{6\left( {1 + v} \right)}} $$
(26)

where, Es is the elastic modulus of soil, v is the Poisson's ratio, η is a coefficient considering the influence of the embedment depth of the beam, and t is the depth of the elastic layer (t = 2.5D, as suggested in a previous study [32]). The next operations are the same as those mentioned in Sect. 2, thus, these are not repeated here.

5.2 FEM Simulation

A three-dimensional model is established to investigate the settlement of the existing tunnel induced by the new tunnel construction using the finite element method (FEM), PLAXIS 3D software. The new tunnel drives perpendicularly beneath the existing tunnel. In order to eliminate the boundary effect [16], the model dimensions are selected to be 100 m (length) × 75 m (width) × 60 m (depth). The new shield tunnel with an outer diameter of 6.2 m and segment thickness of 0.35 m is buried at a depth of 29 m, and the existing structure of the same size is buried at 17 m depth. The elastic moduli for the tunnel lining and earth pressure balance-type machine (EPB) are set as 35.5 GPa and 200 GPa, respectively. The elastic modulus of the existing tunnel in the longitudinal direction is set as 2.85 GPa to consider the reduction in tunnel stiffness. The Poisson's ratio is taken as 0.2 [16]. The tunnel excavation leads to a ground loss of 0.75%.

It is well-known that accurately predicting the structure deformation in geotechnical engineering is a difficult task due to the lack of reasonable application in the deformation characteristics of soil, especially in the small strain. However, the Hardening Soil model with small-strain stiffness (HSS model) can consider the above-mentioned issue, promoting the results toward a high degree of accuracy. Thus, the HSS model has been widely adopted in predicting deformation in soft soil areas. The HSS model parameters of Shanghai soft clay used in PLAXIS 3D software are summarized in Table 3, according to Gu et al. [33].

Table 3 HSS model parameters of soft clay used in the FEM
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Comparisons between the FEM simulation and the prediction results under two different boundary conditions for the settlement of the existing tunnel are shown in Fig. 5, including the free boundary (F–B) and simply supported (S–S). The results under both assumptions are basically consistent with the FEM results on the whole, even though this study slightly overestimates the settlement range of the existing tunnel. The calculation results under two conditions are very close, and the error at the maximum settlement is only 0.646%.

Fig. 5
figure 5

Comparison of the finite element results and predictions

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5.3 Comparison with Field Measurements

5.3.1 Metro Line-11 Crossing Metro Line-4 Project

Zhang et al. [34] reported a project of the new tunnel of Metro Line-11 crossing the existing tunnel of Metro Line-4 in Shanghai, China. Both up-line and down-line shields start from Xujiahui station and end at Shanghai gymnasium station. The EPB with an outside diameter of 6.43 m is employed in tunnel construction. The Metro Line-11 passes through the Metro Line-4 at an angle of 75°, while the EPB advances to Lingling Road. The inner and outside diameters of the tunnel segment are 5.5 m and 6.2 m, respectively. The tunnel ring width and thickness are 1.2 m is 0.35 m, respectively [34]. The parameter values used in the calculations are given in Table 4.

Table 4 Parameter values used in the calculations
Full size table

The settlement values are obtained using the NSM under two different boundary conditions, F–B and S–S (see Fig. 6). A fairly good agreement is found between the results obtained by the NSM and the measured values. Although careful observation can reveal that there still are some gaps between the two curves at some intervals, the difference is much smaller (about 0.678% at the maximum displacement), which is an ideal level in the engineering field.

Fig. 6
figure 6

Comparison of the observations and predictions

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5.3.2 Shanghai Metro Tunnel Project

A new tunnel is built to cross under the existing tunnel in Shanghai, China [35]. During the construction period, the new tunnel advanced nearly perpendicularly through the old structure. The existing tunnel with an outer diameter of 6.2 m and lining thickness of 0.35 m is buried at 9.1 m, and the new one with the same size is buried at 20.1 m [35]. The parameter values used in calculations are listed in Table 5.

Table 5 Parameter values used in calculations
Full size table

As shown in Fig. 7, while the settlement results by applying two boundary conditions are consistent with the measured data, the two curves are close to each other, with a difference of 0.584% at the maximum displacement. Therefore, considering the tunnel as an Euler–Bernoulli beam restrained by simply supported constraints is reasonable, as has been done in previous studies. Both boundary conditions, simply supported and free boundary, can be well employed to solve for the beam displacement.

Fig. 7
figure 7

Comparison of the observations and predictions

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6 Conclusions

In this study, a new method, the null space technique, is introduced to estimate the mechanical response of the Euler–Bernoulli beam model. The applicability and accuracy of this method are validated by comparing the analytic method results and the error analysis between the two methods. Based on error analysis adopting various values of N and ns under the conditions that the beam is imposed by different boundary constraints, it is found that the values significantly impact the accuracy of mechanical responses. And for different cases, the recommended values of two parameters are given. Then, the combination of the Euler–Bernoulli beam theory and the NSM is applied to practical engineering for predicting the settlement of the existing tunnel due to new shield tunneling. The rationality of this interdisciplinary approach is verified by the FEM results adopting PLAXIS 3D software and the field-measured data from published literatures. In addition, it is found that the settlement curves under two kinds of boundary conditions, free boundary and simply support, are close to each other, and the difference between the two cases is relatively small at the maximum displacement. This phenomenon thus proves that it is feasible and reasonable to regard tunnels as an Euler–Bernoulli beam with simply support or free boundary condition in the previous studies.