Serious uneven settlement of the tunnel may directly cause safety problems. At this stage, the deformation of the tunnel is predicted and analyzed mainly by numerical simulation, while the commonly used finite element method (FEM) uses low-order continuous elements. Therefore, the accuracy of tunnel settlement prediction is not enough. In this paper, a method is proposed to study the vertical deformation of the tunnel by using the combination of isogeometric analysis (IGA) and Bézier extraction operator. Compared with the traditional IGA method, this method can be easily integrated into the existing FEM framework, and ensure the same accuracy. A numerical example of an elastic foundation beam subjected to uniformly distributed load and an engineering example of an equivalent elastic foundation beam of the tunnel are given. The results show that the solution of the IGA method is closer to the theoretical solution of the initial-parameter method than the FEM, and the accuracy and reliability of the proposed model are verified. Moreover, it not only provides some theoretical support for the longitudinal design of the tunnel, but also provides a new way for the application and popularization of IGA in tunnel engineering.

Elastic foundation beams are widely utilized in the foundation of industrial, civic, and agricultural building facilities. Analyzing the deformation of elastic foundation beams has always been a research hotspot [

The elastic foundation model mainly includes the Winkler model, semi-infinite elastic foundation model, and two-parameter foundation model proposed by Filonelko-Borodich et al. [

The numerical simulation based on the finite element approach has been widely employed in the settlement analysis of elastic foundation beams, and several academics [

As for other works, Yin [

Because the general thin beam plate and shell structure require C^{1} continuity displacement interpolation function, and the non-uniform rational B-spline [^{1} continuity in the interior of each shell element. After that, Benson et al. [^{k} continuity of arbitrary order. Finally, Li et al. [

Both FEA and IGA employ the isoparametric concept, which means that the same basis is used for geometry and analysis. One obstacle, however, is that NURBS is not as straightforward as Lagrange polynomials. In order to solve this problem, many scholars have used IGA based on the Bézier extraction method which is an isometric analysis element structure similar to FEM and simplifies the implementation of isometric analysis in the finite element environment to solve crack [

Although IGA has the advantages of high geometric accuracy, high continuity, and high precision, it is different from the C^{0} continuous shape function of the conventional FEM, and its programming is complicated. In this paper, we introduce a method combining IGA and Bezier extraction operators to study the vertical deformation of tunnels. The method is to decompose the NURBS function into a linear combination of Bernstein polynomials, thereby realizing the decomposition of NURBS elements into C^{0} continuous Bézier elements, which are similar to Lagrangian elements, making it easy to integrate into existing FEM frameworks and ensure the same accuracy. In addition, applying it to the simplified tunnel model based on the elastic foundation beam can effectively exert its advantages of higher accuracy than the FEM, thereby providing a theoretical basis for the longitudinal design of the tunnel.

The B-spline basis function is composed of a non-subtractive real sequence of node values, called a node vector

The B-spline curve can be evaluated by basis function, and control point coordinates set

The B-spline curve can be extended to the B-spline surface. The node vectors of the two directions are

The quadratic NURBS basis functions and curves are shown in

The NURBS curve is defined as:

The NURBS surface can be obtained from the tensor product of

A sequence of NURBS basis functions is decomposed into linear combinations of Bernstein polynomials using the Bézier extraction procedure. Thus, the NURBS element is decomposed into a C^{0} continuous Bézier element. The Bernstein polynomial is defined as [

The expression of the Bézier curve is as follows:

The node

The shape of the B-spline curve is the same as that of the Bézier curve if the existing nodes are inserted into the original B-spline’s node vector and the degree of repetition is equivalent to the curve’s order. At this time, the continuity of the curve and the continuity between the elements do not change [

Bézier decomposition is a node embedding operation. After getting the expression of

The change of control point after node embedding is as follows:

Let the final control point

According to the B-spline curve

We can deduce the link between the B-spline basis function and the Bernstein polynomial from the preceding equation.

According to

For the denominator of the NURBS basis function, let

A circular beam which is a cantilever beam is used as a numerical example to demonstrate the validity of both the classic IGA and the IGA based on the Bézier extraction method. At the free end, the beam is subjected to the specified displacement

Mesh | Lagrange |
NURBS |
Bézier |
---|---|---|---|

6 × 12 | 0.03042038175071 | 0.02965740783282 | 0.02964986407434 |

12 × 24 | 0.02984351371323 | 0.02964999723578 | 0.02964966945433 |

24 × 48 | 0.02969820784232 | 0.02964968556157 | 0.02964966845279 |

48 × 96 | 0.02966180825828 | 0.02964966942255 | 0.02964966844251 |

96 × 192 | 0.02965270370808 | 0.02964966850106 | 0.029649668442378 |

The deformation of the elastic foundation beam and the soil is consistent under load. As per the elastic foundation’s local deformation law, as shown in

The first four terms on the right side of the equation are the strain energy of the beam, the distributed load potential energy, the concentrated load potential energy, and the concentrated moment load potential energy. Where

After substituting

The form of unit superposition is as follows:

Suppose the element displacement mode of the beam is as follows:

Simultaneous

By substituting the total potential energy

The additional term of foundation stiffness can be calculated by

The expressions of nodal force and nodal displacement of beams on elastic foundation under total stiffness are as follows:

The equilibrium differential equation of the element is as follows:

The right end of the equation is the equivalent internal force. The stiffness matrix

Concentrated force: assuming that there is a concentrated load

Surface force: there is a surface force

Body force:

In this section, the bottom of the foundation is fully constrained, and the beam is coupled with the foundation. Therefore, this model does not consider the separation of the beam and the foundation. The structural parameters of beams on elastic foundations are shown in

0.5 | 0.6 | 4 | 6.84 | 5000 | 0.2 |

This part investigates the deformation behaviour of the shield tunnel longitudinal structure, which serves as a reference basis for longitudinal design, using the equivalent continuous model and the theory of beam on elastic foundation. The actual tunnel structure is a tubular structure formed by bolted segments. The moment of inertia and bending stiffness of the section should be computed according to the actual section and material of the tunnel structure in order to simplify it to an elastic beam. The model assumes that tunnel materials are equally distributed in the transverse direction, and tunnel stiffness and structural features are the same as the simplified model in the longitudinal direction. The expression of equivalent elastic bending stiffness is as follows:

The following equation determines the position of the neutral axis:

The two ends of a subway tunnel are connected with the station, and the soil at the front and rear of the entrance and exit section is reinforced by Φ 800 mm cement jet grouting pile. The reinforcement range is 7 m on both sides, as shown in

Description | Value |
---|---|

Segment ring width | 1000 mm |

Elastic modulus of the segment | |

Bolt length | 400 mm |

Elastic modulus of bolt | |

Bolt yield stress | |

Ultimate stress of bolt |

The length of the interval tunnel is _{1}, the other is the homogeneous soil layer _{2}, and the last is shown in

The foundation reaction coefficient is _{1}.

The foundation reaction coefficient is _{2}.

The foundation reaction coefficient is _{2}_{1}

The 10 × 10, 20 × 20, and 30 × 30 meshing of quadratic, cubic, and quartic Bezier elements is utilized in the computation to depict the displacement history of each control point of the tunnel model. The working condition combinations of three different foundation reaction coefficients are depicted in _{1}_{2}_{2}_{1}_{1}

When the foundation reaction coefficient is _{1}_{2}_{1} _{2}

Mesh | |||
---|---|---|---|

2 | 3 | 4 | |

10 × 10 | 0.002274 | 0.002210 | 0.002082 |

20 × 20 | 0.002260 | 0.002181 | 0.002073 |

30 × 30 | 0.002248 | 0.002178 | 0.002061 |

Mesh | Lagrange | |
---|---|---|

Q4 | Q9 | |

10 × 10 | 0.002412 | 0.002291 |

20 × 20 | 0.002335 | 0.002208 |

30 × 30 | 0.002269 | 0.002195 |

Elements | |||
---|---|---|---|

2 | 3 | 4 | |

10 × 10 | 0.01656 | 0.01651 | 0.01647 |

20 × 20 | 0.01642 | 0.01625 | 0.01618 |

30 × 30 | 0.01639 | 0.01620 | 0.01585 |

Mesh | Lagrange | |
---|---|---|

Q4 | Q9 | |

10 × 10 | 0.01748 | 0.01712 |

20 × 20 | 0.01695 | 0.01674 |

30 × 30 | 0.01673 | 0.01638 |

Elements | |||
---|---|---|---|

2 | 3 | 4 | |

10 × 10 | 0.01533 | 0.01511 | 0.01505 |

20 × 20 | 0.01501 | 0.01487 | 0.01482 |

30 × 30 | 0.01493 | 0.01482 | 0.01478 |

Mesh | Lagrange | |
---|---|---|

Q4 | Q9 | |

10 × 10 | 0.01661 | 0.01574 |

20 × 20 | 0.01593 | 0.01508 |

30 × x30 | 0.01527 | 0.01497 |

The Bézier C^{0} element is used to analyze the vertical displacement of beams on an elastic basis in this work.

The investigation of the convergence and accuracy of vertical deformation of beams on elastic foundations under simple loads is established to validate the efficiency of this method.

After obtaining the stiffness of the homogeneous cylinder from the equivalent continuous calculation model, the shield tunnel is simplified to a uniform continuous beam with equivalent stiffness. The simulation results are in good agreement with the initial-parameter solution, and the accuracy is higher than that of the FEM solution, providing a foundation for longitudinal design.

The foundation reaction coefficient has a major influence on tunnel settlement, as shown by the comparison of the three different workings.

The authors gratefully acknowledge the support from the National Natural Science Foundation of China (52079128).