# Experimental and Theoretical Investigations of the Mechanical Behavior of Column-Free Quasi-Rectangular Segmental Tunnel Linings

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## Abstract

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## Featured Application

**Engineering applications of non-circular segmental tunnels.**

## Abstract

## 1. Introduction

## 2. Prototype Experiment

#### 2.1. Project Background and Test Specimen

#### 2.2. Test Setup

- All test loads must fulfill a self-balancing condition [33] to ensure the safety and stability of the prototype test platform during the test loading process. If the design of the test load is unbalanced, it will cause unexpected relative motion between the centers of the specimen and the counterforce frame, thereby affecting the stability of the platform;
- It is necessary to ensure that the deviation of target structural responses (deformation or internal force) between the design loading state and the test loading state remains within an acceptable range. Considering that the design of segments and joints is mainly governed by bending moment, greater emphasis should be placed on bending moment rather than axial force and shear force when determining the test loads.

- Structural deformation;
- Joint deformation and bolt axial force.

#### 2.3. Experimental Results

#### 2.3.1. Structural Deformation

#### 2.3.2. Joint Response

_{in}and u

_{ex}are the axial opening/closing deformations of the joint intrados and joint extrados, respectively, and b is the thickness of the segment.

#### 2.4. Analysis of Structural Behavior

- (1)
- The structural convergence deformation along the short axis at the top and bottom is relatively larger than that along the long axis at the waists. This phenomenon is attributed to the higher vertical load levels compared to the horizontal load levels. Moreover, the span of the structure primarily extends along the long axis, resulting in greater deformations in the vertical direction, corresponding to the structural convergence deformation along the short axis.
- (2)
- Asymmetry in structural deformation. Considering that longitudinal joints represent the most significant characteristic of segmental tunnel linings, the asymmetry in the structure mainly originated from the distribution of joints. Specifically, Joints 1 and 5 are closer to the short axis, while Joints 3 and 8 are closer to the long axis, causing local deformations at four positions (top, bottom, left waist, and right waist) shifting towards these joint positions. According to the given distribution of joints, the deformations at the top and bottom exhibit axial symmetry with respect to the long axis of the structure, while the deformations at the waists exhibit central symmetry with respect to the center of the structure. These deformation characteristics are also consistent with the experimental results of joint responses.

## 3. Simulation and Analysis of Structural Behavior

#### 3.1. Input Parameters for Models

#### 3.1.1. ESHR Model

#### 3.1.2. BS Model

#### 3.1.3. MBS Model

_{0}). Considering the deformation within a certain distance l from the joint, the stiffness reduction factor ρ can be obtained. Therefore, for segments near joints, with a length of the influence area set to 450 mm (the thickness of the segment [41,42]), the reduction factor is determined to be 0.82.

#### 3.2. Simulation Process and Results

#### 3.3. Deformation Mechanism

_{i}) represent the bending moments of the segment and the joints of the actual test structure, respectively. M’(φ) and M’(φ

_{i}) represent those of the virtual structure subjected to unit loads. k

_{i}is the bending stiffness of Joint No. i (ranging from 1 to 10). Hence, the actual joint bending moment M(φ

_{i}) divided by the joint bending stiffness k

_{i}yields the actual joint rotation angle θ

_{i}obtained from the test. Therefore, Equation (4) can be transformed into Equation (5).

## 4. Structural Design

#### 4.1. Structural Design during the Construction Process

- Synchronous grouting condition. After the tunnel structure comes out from the shield tail, the external soil–water pressure and grouting pressure are resisted by the column-free QRST structure and the temporary support, the load-bearing state of which is similar to that of the typical QRST structure, as shown in Figure 15.
- Stable load condition. As the grouting pressure gradually dissipates and the external load stabilizes, the structure undergoes an unloading process while the structural form remains unchanged.
- Column removal condition. Subsequently, the temporary column is removed slowly to ignore the dynamic structural response. During this process, soil–structure interaction is simulated by a set of foundation springs surrounding the tunnel structure. The final state of the structure is the same as that depicted in Figure 6b.

#### 4.2. Sensitivity Analysis of Key Load Parameters

#### 4.3. Engineering Recommendations

- Joints: All joint contact plates can be fully extended with a width of 1200 mm. For positive moment joints, change the positions of M48 bolts and M30 bolts to fully exploit the load-bearing capacity of M48 bolts. For negative moment joints with waterproofing permitted, the position of bolts is suggested to be fully moved toward the external surface direction.
- Temporary support: Considering the additional loads during the construction stage beyond the loads considered in the service stage, it is suggested that temporary support be employed to ensure structural safety.
- Segmentation and joint positions: After canceling interior columns, the joints are now situated at relatively unfavorable positions compared to typical QRSTs with interior columns. It is recommended that larger segments be used at the top, bottom, and waists to mitigate the joints’ exposure to high bending moments, thereby further reducing overall structural deformation.
- Ring-to-ring interaction: The experiments and simulations conducted are focused on single-ring structures and do not account for the interaction effects between rings in real tunnels. In practice, QRSTs are assembled in a staggered fashion. With the help of circumferential joints, structural deformation can be reduced. It is suggested that tongue-and-groove or shear keys be used to ensure the overall integrity of the tunnel system.

## 5. Conclusions

- At the service stage with a designed buried depth of 13 m, the inward vertical convergence deformation along the short axis of the structure is 108.64 mm, while the outward horizontal convergence deformation along the long axis is 49.03 mm. The column-free QRST structure predominantly experiences inward deformation;
- The development of joint deformations and bolt axial forces indicates that as the load level increases, among the positive moment joints, deformations and internal forces of Joints 1 and 5 develop faster than those of Joints 6 and 10. For negative moment joints, Joints 3 and 8 exhibit faster development compared to Joints 2 and 7, with Joints 4 and 9 exhibiting only rapid initial development. Based on the experimental results, it is recommended that the bolt dimension and bolt position for negative moment joints be optimized, and the rows of M48 and M30 bolts be repositioned for positive moment joints, with the help of sophisticated finite element models;
- The experimental results indicate that the deformation and internal force distribution of the structure are mainly determined by the shape of the tunnel cross-section and joint distribution. The span of the structure in the horizontal direction results in significant convergence deformation along the short axis at the top and bottom. Based on the given distribution of the joints, the structural deformations at the top and bottom exhibit an axis symmetricity with respect to the long axis of the structure, while those at the waists exhibit a central symmetry with respect to the center of the structure;
- The ESHR model, BS model, and MBS model are capable of reflecting the mechanical performance of column-free QRST structures. The BS and MBS models exhibit better simulation effects, particularly for local deformations near joints, while the ESHR model can serve as a design model for structures with appropriate calculation parameters;
- The deformations contributed by joints in column-free QRST structures account for approximately 65–70%, indicating optimization potentials for joints;
- When using the ESHR model to design column-free QRST structures subjected to the construction process, the results calculated by the three-stage calculation method are more reliable. For ease of engineering applications, structural responses calculated by the direct calculation method, with the response magnified by 10%, is sufficient to ensure structural safety;
- The influence of structural buried depth on structural response is generally proportional. In cases of changes in offset load and lateral pressure coefficient, the negative bending moment regions at the waists of the structure are more sensitive. Considering unexpected conditions such as ground surface surcharge or adjacent construction activities during the service period, special attention should be paid to the structural safety reserves of the negative bending moment regions at the waists.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**Diagram of joint surfaces (mm): (

**a**) Schematic diagram of positive moment joint; (

**b**) Schematic diagram of negative moment joint; (

**c**) Photo of positive moment joint; (

**d**) Photo of negative moment joint.

**Figure 7.**Development of structural deformations: (

**a**) Overall structural deformation; (

**b**) Convergence deformations of short and long axes.

**Figure 8.**Development of joint rotation angles: (

**a**) Positive moment joints; (

**b**) Negative moment joints.

**Figure 11.**Illustration of the joint influence zone: (

**a**) Configurations of segments and distribution of stress zones; (

**b**) Photo of segment in the joint vicinity.

**Figure 12.**Simulation results of the MBS model under the 13 m service stage condition: (

**a**) Bending moment; (

**b**) Axial force; (

**c**) Shear force.

**Figure 13.**Comparison of overall structural deformations between experimental results and simulation results based on the three models.

**Figure 14.**Illustration of the structure under virtual unit loads used for the calculation of convergence deformations.

**Figure 15.**Bearing state of the column-free QRST structure with the temporary column subjected to simultaneous grouting condition: (

**a**) Loading state; (

**b**) 3D illustration.

**Figure 17.**Influence of different load parameters on bending moments at critical positions on the structure: (

**a**) Lateral pressure coefficient η; (

**b**) Offset load; (

**c**) Buried depth.

Soil Layer | Layer Thickness (m) | Water Content (%) | Unit Weight (kN/m ^{3}) | Internal Friction Angle (°) | Cohesion (kPa) |
---|---|---|---|---|---|

① 2-Clay | 0.6~3.4 | 27.9 | 19.3 | 18.9 | 28.7 |

① 3c-Silty clay | 0.5~4.4 | 38.0 | 18.0 | 19.5 | 7.2 |

② 2c-Silty clay | 1.0~6.6 | 39.7 | 17.7 | 21.8 | 5.5 |

② 2b-Silty clay | 15.1~20.0 | 47.9 | 16.9 | 13.4 | 12.5 |

② 2T-Silt sand | 0.6~4.5 | 28 | 19.0 | 18.9 | 28.7 |

③ 1b-Silt sand | 4.0~8.7 | 22.4 | 19.6 | 19.5 | 7.2 |

Loading Step | P1 (kN) | P2 (kN) | P3 (kN) |
---|---|---|---|

1 | 25 | 23 | 22 |

2 | 49 | 46 | 44 |

3 | 74 | 70 | 67 |

4 | 86 | 81 | 78 |

5 | 98 | 93 | 89 |

6 | 105 | 99 | 94 |

7 | 111 | 104 | 100 |

8 | 117 | 110 | 105 |

9 (9 m service stage) | 123 | 116 | 111 |

10 | 130 | 122 | 137 |

11 (10 m service stage) | 137 | 129 | 164 |

12 | 141 | 114 | 144 |

13 | 146 | 100 | 125 |

14 | 150 | 85 | 106 |

15 | 157 | 110 | 119 |

16 | 165 | 134 | 132 |

17 (13 m service stage) | 172 | 159 | 145 |

Parameter | Unit | Value |
---|---|---|

Segment | ||

Thickness | mm | 450 |

Width | mm | 1200 |

Thickness of internal and external steel plates | mm | 30 |

Total thickness of web plates | mm | 90 |

Area of the segment A_{0} | m^{2} | 1.071 × 10^{−1} |

Moment of inertia of the segment I_{0} | m^{4} | 3.625 × 10^{−3} |

Material properties of steel | ||

Young’s modulus E | GPa | 206 |

Poisson ratio ν | / | 0.31 |

Stiffness of segment | ||

Axial compressive stiffness EA_{0} | N | 2.206 × 10^{10} |

Bending stiffness EI_{0} | N·m^{2} | 7.468 × 10^{8} |

**Table 4.**Comparison of convergence deformations of different axes between experiment results and simulation results based on three models (mm).

Location | Experimental Results | ESHR Model | BS Model | MBS Model |
---|---|---|---|---|

Short axis | −108.64 | −108.69 (0.04%) | −105.55 (−2.84%) | −108.72 (0.07%) |

Long axis | 49.03 | 52.17 (6.4%) | 51.56 (5.15%) | 53.13 (8.36%) |

Left short axis | −104.67 | −106.99 (2.22%) | −102.26 (−2.31%) | −105.34 (0.64%) |

Right short axis | −110.82 | −106.99 (−3.45%) | −108 (−2.54%) | −111.25 (0.38%) |

Condition | Deformation of Short Axis (mm) | |
---|---|---|

Analysis Total (Joint Contribution) | Test | |

9 m service stage | 79.73 (52.74) | 72.00 |

10 m service stage | 95.59 (66.35) | 94.88 |

13 m service stage | 111.31 (74.49) | 108.64 |

**Table 6.**Comparison of internal forces between the three-stage calculation method and the direct calculation method.

Position | Direct Calculation Method | Three-Stage Calculation Method | ||
---|---|---|---|---|

Bending Moment (kN·m) | Axial Force (kN) | Bending Moment (Kn·M) | Axial Force (kN) | |

Top | 1764 | −982 | 1772 (+0.5%) | −992 |

Right waist | −1861 | −2066 | −1890 (+1.6%) | −2089 |

Bottom | 1629 | −1480 | 1759 (+8.0%) | −1492 |

Left waist | −1660 | −1950 | −1688 (+1.7%) | −1972 |

Load Parameter | Unit | Design Value | Values for Sensitivity Analysis |
---|---|---|---|

Lateral pressure coefficient η | / | 0.54 | 0.4, 0.5, 0.54, 0.6, 0.7 |

Offset load | kPa | 30 | 0, 15, 30, 45, 60 |

Buried depth | m | 13 | 5, 10, 13, 15, 20 |

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## Share and Cite

**MDPI and ACS Style**

Liu, Z.; Chen, Y.; Wu, Y.; Liu, X.
Experimental and Theoretical Investigations of the Mechanical Behavior of Column-Free Quasi-Rectangular Segmental Tunnel Linings. *Appl. Sci.* **2024**, *14*, 2896.
https://doi.org/10.3390/app14072896

**AMA Style**

Liu Z, Chen Y, Wu Y, Liu X.
Experimental and Theoretical Investigations of the Mechanical Behavior of Column-Free Quasi-Rectangular Segmental Tunnel Linings. *Applied Sciences*. 2024; 14(7):2896.
https://doi.org/10.3390/app14072896

**Chicago/Turabian Style**

Liu, Zhen, Yizheng Chen, Yuebin Wu, and Xian Liu.
2024. "Experimental and Theoretical Investigations of the Mechanical Behavior of Column-Free Quasi-Rectangular Segmental Tunnel Linings" *Applied Sciences* 14, no. 7: 2896.
https://doi.org/10.3390/app14072896