# Stability Analysis of Tunnel Face Reinforced with Longitudinal Fiberglass Dowels Together with Steel Pipe Umbrella

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## Abstract

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## 1. Introduction

## 2. Stability Analysis of the Tunnel Face Reinforced with Longitudinal Fiberglass Dowels Together with Steel Pipe Umbrella

#### 2.1. The New Analytical Prediction Model

_{s}. The analytical prediction model consists of a wedge ahead of the tunnel face, distributed force acting on the wedge exerted by overlying ground, and the support forces stem from the longitudinal fiberglass dowels. On the one hand, the advanced pre-reinforcement structure of the steel pipe umbrella is deemed the beam on the Winkler elastic foundation, which shows that the existence of the steel pipe umbrella effectively reduces the vertical pressure exerted by overlaying ground. On the other hand, the strengthening effect of the longitudinal fiberglass dowels depends primarily on the tensile bearing capacity of the bolt or the bond strength of the ground–bolt interface. The ground extrusion in front of the tunnel face is effectively reduced for the installation of longitudinal fiberglass dowels. The mechanical force analysis of the new analytical prediction model is shown in Figure 2b.

#### 2.2. Power of the External Loads P_{e}

#### 2.2.1. Power of the Soil Unit Weight

#### 2.2.2. Power Induced by the Friction on Both Sides

#### 2.2.3. Power Induced by the Vertical Stress on the Sliding Wedge Body

- (1)
- Both the beam theory and the theory of beam on Winkler elastic foundation are adopted to investigate the mechanical behavior and characteristics of advanced small pipes in tunnel construction. Beam theory is used to analyze the advanced small pipes that are embedded in soil in the front of the tunnel face, whereas the theory of beam on Winkler elastic foundation is used to analyze the advanced small pipes behind the tunnel face, as rendered in Figure 3.
- (2)
- It is assumed that the fixed end A has certain vertical displacement y
_{0}, which is a known value and is considered as the measured vault subsidence value. - (3)
- The length of the advanced small pipes consists of two parts, which includes unsupported span (length in 1.5a, includes excavation footage 1.0a and the length 0.5a due to support delay effect) and the length of the wedge (length in l), as depicted in Figure 3. The symbol a denotes the excavation footage—that is, the length of the tunnel for each excavation.
- (4)
- In order to simplify the analysis, the horizontal projection length of advanced small pipes is considered.

- (1)
- The remaining length l
_{e}of the pipe in soil is longer than the length l of the wedge (Type I). The stability analysis model I is shown in Figure 3a. The subgrade reaction force that acts on the wedge is p (triangular distribution). - (2)
- The remaining length l
_{e}of the pipe in soil is shorter than the length l of the wedge (Type II). The stability analysis model II is shown in Figure 3b. The load that acts on the wedge can be divided into two parts, one is the subgrade reaction force p (trapezium distribution) along the pipe and the other is the uniform load q that acts on the wedge.

- (1)
- For Type I, the control differential equations of the reinforced foundation beam for different segments are obtained as follows:$$\{\begin{array}{l}AO:\frac{{d}^{4}y}{d{x}^{4}}=\frac{bq\left(x\right)}{EI}\\ OB:\frac{{d}^{4}y}{d{x}^{4}}+4{\lambda}^{4}y=\frac{bq\left(x\right)}{EI}\\ BC:\frac{{d}^{4}y}{d{x}^{4}}+4{\lambda}^{4}y=0\end{array},$$

_{1}, y

_{2}and y

_{3}denote the deflections of the beams AO, OB and BC, respectively; C

_{1}, C

_{2}, C

_{3}, C

_{4}, C

_{5}, C

_{6}, C

_{7}and C

_{8}present the undetermined coefficients of differential equations; y

_{t}is a particular solution to a differential equation for beams OB.

- (2)
- For Type II, the control differential equations of the reinforced foundation beam for different segments are obtained as follows:$$\{\begin{array}{l}AO:\frac{{d}^{4}y}{d{x}^{4}}=\frac{bq\left(x\right)}{EI}\\ OB:\frac{{d}^{4}y}{d{x}^{4}}+4{\lambda}^{4}y=\frac{bq\left(x\right)}{EI}\end{array},$$

#### 2.2.4. Power Induced by the Longitudinal Fiberglass Dowels

_{m}is the bond strength of the soil–grout interface of the longitudinal fiberglass dowels.

#### 2.3. Dissipation Power on Discontinuity Surface Pv

#### 2.4. Critical Reinforcement Density of Longitudinal Fiberglass Dowels

_{1}, n

_{2}and n

_{3}corresponding to three distributions of support pressure) of longitudinal fiberglass dowels is calculated:

## 3. Sensitivity Analysis

#### 3.1. Longitudinal Fiberglass Dowels in the Excavation Face Alone

#### 3.1.1. The Influence Rules of the Cover Depth on Limit Reinforcement Density

_{cr}with the variation of the cohesion c. The results indicate that when the cover depth is greater than the width of the tunnel face, the limit reinforcement density does not increase significantly.

#### 3.1.2. The Influence Rules of the Tunnel Shape on Limit Reinforcement Density

_{cr}with the variation of the cohesion c. The results are markedly different between shapes A, B and C. Specifically, shape C needs a lower reinforcement density than shape A, though they have the same height, which agrees with common sense that the C-D tunneling method is more stable than the full-face tunneling method. Moreover, a comparison between the reinforcement required in the cases of tunnel shapes B and C indicates that the C-D tunneling method is more stable than the benching tunneling method.

#### 3.1.3. The Influence Rules of the Reinforcement Installation Interval on Limit Reinforcement Density

_{cr}with the variation of the cohesion c. The results show that large installation intervals of bolts require greater reinforcement density n

_{cr}.

#### 3.2. Longitudinal Fiberglass Dowels in the Excavation Face Together with Pre-Supports

_{cr}.

## 4. Conclusions

- (1)
- The advanced pre-reinforcement structure of the steel pipe umbrella is considered as the beam on the Winkler elastic foundation, which shows that the existence of the steel pipe umbrella effectively reduces the vertical pressure exerted by overlaying ground. Under general conditions, with the increase in the length of the pre-reinforcement, its promoting effect on tunnel face stability is obvious. However, when the surplus length of the pre-reinforcement structure reaches the critical fracture length, the length of the pre-reinforcement structure on the stability of the tunnel face is no longer a key factor.
- (2)
- The strengthening effect of the longitudinal fiberglass dowels depends primarily on the tensile bearing capacity of the bolt or the bond strength of the ground–bolt interface. The ground extrusion in the front of the tunnel face is effectively reduced for the installation of longitudinal fiberglass dowels. Moreover, the limit reinforcement density of longitudinal fiberglass dowels is assessed under specific lengths with or without the consideration of the steel pipe umbrella.
- (3)
- The results indicate that the required reinforcement density does not increase significantly when the cover depth is greater than the width of the face. The C-D tunneling method is more stable than the full-face tunneling method and benching tunneling method. Moreover, the results show that large installation intervals of bolts require greater reinforcement density.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Face failure reinforced with longitudinal fiberglass dowels together with steel pipe umbrella.

**Figure 4.**Distribution of support pressure for different wedge angles β (Anagnostou and Perazzelli 2015). (

**a**) Condition I; (

**b**) Condition II; (

**c**) Condition III.

**Figure 5.**The influence rules of the cover depth h on limit reinforcement density n

_{cr}with the variation of the cohesion c.

**Figure 6.**The influence rules of the different tunnel shapes on limit reinforcement density n

_{cr}with the variation of the cohesion c.

**Figure 7.**The influence rules of the two installation intervals l on limit reinforcement density n

_{cr}with the variation of the cohesion c.

**Figure 8.**The influence rules of the reduction factor (RF) on limit reinforcement density n

_{cr}with the variation of the cohesion c.

Pre-Reinforcement Technology | Construction Safety | |||||
---|---|---|---|---|---|---|

Stable Arch | Stable Working Face | Stability of Arch Foot | Groundwater Control | |||

Advance payments | Advance bolt | √ | √ | |||

Pipe shed | √ | √ | ||||

Horizontal rotary jet pile | √ | √ | ||||

Reinforcement of excavated surfaces | Core soil reserved for annular excavation | √ | ||||

Shotcrete on excavated surface | √ | |||||

Anchor of excavated surface | √ | |||||

Grouting of excavated surfaces | √ | |||||

Reinforcement of arch foot | Anchor bolt reinforcement | √ | ||||

Reinforcement of locked pile | √ | |||||

Grouting reinforcement of arch foot | √ | |||||

Temporary inverted arch | √ | |||||

Groundwater control | Drainage measures | Surface drainage | √ | √ | √ | |

Drainage | √ | √ | √ | |||

Waterproofing measures | Grouting | √ | √ | √ | ||

Freeze | √ | √ | √ | |||

Formation reinforcement | Contact grouting | √ | √ | √ | ||

Full section grouting | √ | √ | √ | |||

Joint grouting | ||||||

Surface pre-grouting | √ | √ | √ | √ |

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**MDPI and ACS Style**

Han, K.; Wang, X.; Hou, B.; Cao, C.-y.; Lin, X.-T.
Stability Analysis of Tunnel Face Reinforced with Longitudinal Fiberglass Dowels Together with Steel Pipe Umbrella. *Symmetry* **2020**, *12*, 2069.
https://doi.org/10.3390/sym12122069

**AMA Style**

Han K, Wang X, Hou B, Cao C-y, Lin X-T.
Stability Analysis of Tunnel Face Reinforced with Longitudinal Fiberglass Dowels Together with Steel Pipe Umbrella. *Symmetry*. 2020; 12(12):2069.
https://doi.org/10.3390/sym12122069

**Chicago/Turabian Style**

Han, Kaihang, Xuetao Wang, Beibei Hou, Cheng-yong Cao, and Xing-Tao Lin.
2020. "Stability Analysis of Tunnel Face Reinforced with Longitudinal Fiberglass Dowels Together with Steel Pipe Umbrella" *Symmetry* 12, no. 12: 2069.
https://doi.org/10.3390/sym12122069