# Mindlin-Reissner Analytical Model with Curvature for Tunnel Ventilation Shafts Analysis

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## Abstract

**:**

## 1. Introduction

- The model involves a bending of a flat plate $\left(\frac{1}{R}=0\right)$ plus a membrane behavior without curvature (Figure 2);
- Nonexistence of the constitutive coupling between the membrane forces and the bending moments for any slenderness ratio $\left(\frac{h}{R}\right)$;
- Shear correction factor (${\alpha}_{n}$) is employed in the constant shear force, independently of the shell slenderness ratio.

## 2. Coupled Constitutive Model for General Bending of a Coaxial Cylinder

## 3. New Mathematical Operational Model

_{1}and B

_{1}from the Equation (19) characterize the influence of the shear and the membrane force facing the bending respectively.

## 4. Particular Cases

## 5. Analytical Resolution of the New Mathematical Model

_{1}). The general solution of the model is the sum of two solutions due to its linearity, Equation (24), i.e., the homogeneous solution, $\theta c,$ and the particular solution, $\theta p$ [21,22].

## 6. Numerical Results and Discussions

## 7. Conclusions

- When the isotropic shell is thin, $\left(\frac{h}{R}\le 0.05\right)$ its predominant resistant mechanism is the circumferential membrane force with an inversion of bending moments in the both main directions. From an increase in the slenderness ratio, the flexural contribution in the two main directions dominated the internal equilibrium of the shell and the inserting of the constitutive curvature acquires the biggest importance in the structural response of the shell. For analysis and tunnel ventilation shafts design with slenderness ratio of $\frac{h}{R}\ge 0.12$ it is recommendable to insert the constitutive curvature;
- The equations and the general methodology displayed in this paper might be usefully employed in the analysis and design of the cylindrical shell (isotropic and orthotropic) under general distribution of axial-symmetric pressures. The mathematical model formulated in terms of the internal forces per unit of arc longitude allows to solve differential equation systems of multiple degrees of freedom and to model the complex boundary conditions by the Saint-Venant simplification. For this study case, the equations can be applied by means of the basic spreadsheets as tools to assist in the design.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

${q}_{{n}_{1}},{q}_{{n}_{2}}$ | Lateral earth pressure plus hydrostatic pressure plus overload on the shaft |

$D,\text{}d$ | External and internal diameter of the shaft, respectively |

${q}_{{z}_{pp}}$ | Body load |

$h$ | Thickness of the wall-shaft (shell thickness) |

$R$ | Principal radius of curvature |

${M}_{z},{M}_{\theta}$ | Bending moments |

${Q}_{z}$ | Shear force |

${N}_{\theta},\text{}{N}_{z}$ | Membrane-axial forces |

${\kappa}_{z},{\kappa}_{\theta},{\gamma}_{zN},{\epsilon}_{\theta},\text{}{\epsilon}_{z}$ | Deformations |

${U}_{n},\text{}{U}_{z},\text{}{\psi}_{z}$ | Displacement |

${\gamma}_{m}$ | Shaft material weight |

${\gamma}_{a}$ | Water weight |

${\gamma}_{s}$ | Soil weight |

${K}_{0}$ | Coefficient of lateral earth pressure at rest |

$H$ | Shaft height |

${D}_{f}$ | Cylindrical flexion stiffness of a plate |

$K$ | Membrane stiffness |

$E$ | Elasticity modulus |

$\mu $ | Poisson ratio |

$C$ | Characteristic longitude of the shell |

$D$ | Discriminant |

${\alpha}_{n}$ | Shear correction factor |

$e$ | exp |

${F}_{CG}\left(z\right)$ | Function with constitutive curvature |

${F}_{SG}\left(z\right)$ | Homogeneous function without constitutive curvature |

$\left({\alpha}_{1},{\alpha}_{2},\text{}{\alpha}_{3}\right)\in {\mathbb{R}}^{3}$ | Orthogonal curvilinear coordinates |

$\theta c\left(z\right)$ | Solution of the homogeneous equation |

$\theta p\left(z\right)$ | Solution of the particular equation. Indeterminate coefficient method |

Acronyms used | |

$\left(F.S.S\right)$ | Fundamental system solution |

$\left(HD\right)$ | Hyperstatic Degree |

$\left(DF\right)$ | Degree of freedom |

$\left(ICM\right)$ | Indeterminate coefficient method |

$\left(FEM\right)$ | Finite element method |

$\left(NTL\right)$ | Natural terrain level |

$\left(PL\right)$ | Phreatic Level |

(BC) | Boundary condition |

$\left(PWP\right)$ | Porous water pressure |

Note: The term “Constitutive curvature” refers to the inclusion of the shell curvature in the constitutive equations relating the resultant internal forces per unit of arc longitude and the middle shell deformations, Equations (7) and (9). |

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**Figure 1.**Mechanical model of tunnel ventilation shafts building by “floating shafts” construction system, (

**a**) Analysis scheme for shafts, (

**b**) Free body diagram of shaft differential slice

**Figure 2.**Linear distribution of the normal stress $\left({\sigma}_{z}{}^{*}\right),$ disregarding the constitutive curvature of the shell.

**Figure 3.**Geometrical details for modeling the shell curvature in the free body diagram of the tunnel ventilation shaft differential slide.

**Figure 8.**Circumferential stress in extrados for thin, medium and thick shells, considering different models.

**Figure 9.**Circumferential stress in intrados for thin, medium and thick shells, considering different models.

**Figure 10.**Longitudinal stress in intrados for thin, medium and thick shells, considering different models.

**Figure 11.**Longitudinal stress in extrados for thin, medium and thick shells, considering different models.

**Figure 12.**Longitudinal flexural moment for thin, medium and thick shells, considering different models.

**Figure 13.**Circumferential flexural moment for thin, medium and thick shells, considering different models.

**Table 1.**Comparison between the new shear correction factor for tunnel ventilation shafts, and the shear correction factor for plates for different slenderness ratios $\left(\frac{h}{R}\right)$ .

$\left(\frac{\mathit{h}}{\mathit{R}}\right)$ | ${\mathit{\alpha}}_{\mathit{n}}=\frac{5}{6}\text{}\left(\mathbf{Plate}\right)$ | ${\mathit{\alpha}}_{\mathit{n}}\text{}\left(\mathbf{Shafts}\right)$ | Relative Error (%) |
---|---|---|---|

$0.05$ | $0.83\overline{3}$ | $0.869$ | $4.142$ |

$0.10$ | $0.83\overline{3}$ | $0.907$ | $8.159$ |

$0.12$ | $0.83\overline{3}$ | $0.923$ | $9.751$ |

$0.15$ | $0.83\overline{3}$ | $0.947$ | $12.038$ |

$0.20$ | $0.83\overline{3}$ | $0.990$ | $15.859$ |

$0.22$ | $0.83\overline{3}$ | $1.008$ | $17.361$ |

$0.25$ | $0.83\overline{3}$ | $1.035$ | $19.517$ |

**Table 2.**Comparison between the new shear correction factor for tunnel ventilation shafts, and the shear correction factor for plates for different slenderness ratio $\left(\frac{h}{R}\right)$ .

ID | Simple Term | $\mathbf{Radial}\text{}\mathbf{Displacement}\text{}\left({\mathit{U}}_{\mathit{n}}\right)$ $\mathbf{Twist}\text{}\mathbf{Angle}\text{}\left({\mathit{\psi}}_{\mathit{z}}\right)$ $\mathbf{Shear}\text{}\left({\mathit{Q}}_{\mathit{z}}\right)$ $\mathbf{Bending}\text{}\mathbf{Moment}\text{}\left({\mathit{M}}_{\mathit{z}}\right)$ | $\mathbf{Axial}\text{}\mathbf{Displacement}\text{}\left({\mathit{U}}_{\mathit{z}}\right)$ |
---|---|---|---|

BC1 r | Clamped | ${U}_{n}=0$ and ${\psi}_{z}=0$ | ${U}_{z}=0$ |

BC1 f | ${U}_{n}=0$ and ${M}_{z}=0$ | ${U}_{z}=0$ | |

BC2 r | Pinned | ${U}_{n}=0$ and ${\psi}_{z}=0$ | ${U}_{z}\ne 0$ |

BC2 f | ${U}_{n}=0$ and ${M}_{z}=0$ | ${U}_{z}\ne 0$ | |

BC3 | Free edge | ${M}_{z}=0$ and ${Q}_{z}=0$ | ${U}_{z}\ne 0$ |

**Table 3.**Physical, mechanical and geometrical properties of the tunnel ventilation shaft N°1 of the project-“Río La Compañia” from Espejel.

Physical and Mechanical Properties |

$E=2.378\times {10}^{7}\frac{kN}{{m}^{2}}$${\gamma}_{a}=10\frac{kN}{{m}^{3}}$${\gamma}_{m}=20.46\frac{kN}{{m}^{3}}$${\gamma}_{s}=10.23\frac{kN}{{m}^{3}}$$\begin{array}{c}\mu =0.2\\ {K}_{0}=0.8\end{array}$${q}_{{n}_{1}}=206.735\text{}\frac{kN}{{m}^{2}}$${q}_{{n}_{2}}=0$ |

Geometrical Properties |

$h=0.70\text{}m$$H=20.30\text{}m$$R=6\text{}m$$\text{}\mathrm{Ratio}\text{}\left(\frac{h}{R}\right)=0.12$ |

Finite Element Type Description | $\frac{\mathit{h}}{\mathit{R}}=0.05$ | $\frac{\mathit{h}}{\mathit{R}}=0.12$ | $\frac{\mathit{h}}{\mathit{R}}=0.25$ | ||||
---|---|---|---|---|---|---|---|

Nodes | Elements | Nodes | Elements | Nodes | Elements | ||

Solid elements 3D | C3D20R: A 20-node quadratic brick, reduced integration | 59,644 | 8432 | 289,708 | 57,528 | 326,636 | 71,928 |

C3D8R: An 8-node linear brick, reduced integration, hourglass control | 17,112 | 8432 | 77,456 | 57,528 | 84,952 | 71,928 | |

Shell elements | S8R: An 8-node doubly curved thick shell, reduced integration | 8694 | 8568 | 8694 | 8568 | 8694 | 8568 |

S4R: A 4-node doubly curved thin or thick shell, reduced integration, hourglass control, finite membrane strains | 25,956 | 8568 | 25,956 | 8568 | 25,956 | 8568 |

$\left(\frac{\mathit{h}}{\mathit{R}}\right)$ | Length (m) | $\mathbf{Maximum}\text{}\mathbf{Circumferential}\text{}\mathbf{Stress}\text{}\mathbf{in}\text{}\mathbf{Extrados}\text{}\left({\mathit{\sigma}}_{\mathit{\theta}}\right)\text{}\left(\raisebox{1ex}{$\mathit{k}\mathit{N}$}\!\left/ \!\raisebox{-1ex}{${\mathit{m}}^{\mathbf{2}}$}\right.\right)$ | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Analytical Solution | FEM Shells | FEM 3D | ||||||||||

With CC | Without CC | % EA | S4R | % ES4R | S8R | % ES8R | C3D8R | % EC3D8R | C3D20R | % EA-C3D20R | ||

0.05 | 2.57 | 3833.002 | 3923.135 | 2.35 | 3596.92 | 6.16 | 3773.11 | 1.59 | 3730.3 | 0.514 | 3749.56 | 2.225 |

0.12 | 3.73 | 1523.094 | 1601.1821 | 5.13 | 1447.13 | 4.99 | 1447.63 | 5.21 | 1640.83 | 0.787 | 1628.02 | 6.445 |

0.25 | 5.1 | 644.19 | 706.634 | 9.69 | 586.868 | 8.90 | 587.301 | 9.69 | 752.557 | 0.557 | 748.391 | 13.923 |

Where: | CC: Constitutive curvature; EA: relative error between the analytical results (pattern: analytical solution with CC); ES4R: relative error between the analytical result with CC and shell element S4R (pattern: analytical solution with CC); ES8R: relative error between the analytical result with CC and shell element S8R (pattern: analytical solution with CC); EC3D8R: relative error between the elements C3D8R and C3D20R (pattern: C3D20R); EA-C3D20R: relative error between the analytical result with CC and the element C3D20R (pattern: C3D20R). |

$\left(\frac{\mathit{h}}{\mathit{R}}\right)$ | Length (m) | $\mathbf{Maximum}\text{}\mathbf{Circumferential}\text{}\mathbf{Stress}\text{}\mathbf{in}\text{}\mathbf{Intrados}\text{}\left({\mathit{\sigma}}_{\mathit{\theta}}\right)\text{}\left(\raisebox{1ex}{$\mathit{k}\mathit{N}$}\!\left/ \!\raisebox{-1ex}{${\mathit{m}}^{\mathbf{2}}$}\right.\right)$ | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Analytical Solution | FEM Shells | FEM 3D | ||||||||||

With CC | Without CC | % EA | S4R | % ES4R | S8R | % ES8R | C3D8R | % EC3D8R | C3D20R | % EA-C3D20R | ||

0.05 | 3.06 | 3770.237 | 3673.337 | 2.57 | 3834.26 | 1.70 | 3599.85 | 4.52 | 3619.5 | 4.517 | 3790.72 | 0.543 |

0.12 | 4.38 | 1560.739 | 1471.875 | 5.69 | 1335.59 | 14.43 | 1336.09 | 14.39 | 1665.84 | 0.471 | 1673.72 | 7.239 |

0.25 | 5.85 | 712.314 | 634.328 | 10.95 | 649.409 | 8.83 | 649.813 | 8.77 | 828.22 | 1.459 | 840.48 | 17.993 |

See Table 5 for legend |

$\left(\frac{\mathit{h}}{\mathit{R}}\right)$ | $\mathbf{Maximum}\text{}\mathbf{Longitudinal}\text{}\mathbf{Stress}\text{}\mathbf{in}\text{}\mathbf{Extrados}\text{}\left({\mathit{\sigma}}_{\mathit{z}}\right)\text{}\left(\raisebox{1ex}{$\mathit{k}\mathit{N}$}\!\left/ \!\raisebox{-1ex}{${\mathit{m}}^{2}$}\right.\right)\text{}\mathbf{z}=0$ | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Analytical Solution | FEM Shells | FEM 3D | |||||||||

With CC | Without CC | % EA | S4R | % ES4R | S8R | % ES8R | C3D8R | % EC3D8R | C3D20R | % EA-C3D20R | |

0.05 | 6500.02 | 6556.31 | 0.87 | 4429.12 | 31.860 | 6006.91 | 7.586 | 412.305 | 93.334 | 6184.77 | 5.097 |

0.12 | 2452.9 | 2506.165 | 2.17 | 1623.46 | 33.815 | 1917.51 | 21.827 | 1265.24 | 53.086 | 2696.94 | 9.049 |

0.25 | 861.4145 | 910.11 | 5.65 | 432.582 | 49.782 | 537.565 | 37.595 | 535.274 | 57.637 | 1263.55 | 31.826 |

See Table 5 for legend |

$\left(\frac{\mathit{h}}{\mathit{R}}\right)$ | $\mathbf{Maximum}\text{}\mathbf{Longitudinal}\text{}\mathbf{Stress}\text{}\mathbf{in}\text{}\mathbf{Intrados}\text{}\left({\mathit{\sigma}}_{\mathit{z}}\right)\text{}\left(\raisebox{1ex}{$\mathit{k}\mathit{N}$}\!\left/ \!\raisebox{-1ex}{${\mathit{m}}^{\mathbf{2}}$}\right.\right)\text{}\mathbf{z}=\mathbf{0}$ | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Analytical Solution | FEM Shells | FEM 3D | |||||||||

With CC | Without CC | % EA | S4R | % ES4R | S8R | % ES8R | C3D8R | % EC3D8R | C3D20R | % EA-C3D20R | |

0.05 | 7444.063 | 7386.986 | 0.77 | 5253.69 | 29.424 | 6837.6 | 8.147 | 412.305 | 94.214 | 7126.37 | 4.458 |

0.12 | 3390.943 | 3336.8407 | 1.60 | 2450.06 | 27.747 | 2748.18 | 18.955 | 2179.69 | 40.199 | 3644.93 | 6.968 |

0.25 | 1790.302 | 1740.786 | 2.77 | 1258.13 | 29.725 | 1368.24 | 23.575 | 1444.26 | 33.889 | 2184.61 | 18.049 |

See Table 5 for legend |

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**MDPI and ACS Style**

Álvarez-Pérez, J.; Peña, F.
Mindlin-Reissner Analytical Model with Curvature for Tunnel Ventilation Shafts Analysis. *Mathematics* **2021**, *9*, 1096.
https://doi.org/10.3390/math9101096

**AMA Style**

Álvarez-Pérez J, Peña F.
Mindlin-Reissner Analytical Model with Curvature for Tunnel Ventilation Shafts Analysis. *Mathematics*. 2021; 9(10):1096.
https://doi.org/10.3390/math9101096

**Chicago/Turabian Style**

Álvarez-Pérez, José, and Fernando Peña.
2021. "Mindlin-Reissner Analytical Model with Curvature for Tunnel Ventilation Shafts Analysis" *Mathematics* 9, no. 10: 1096.
https://doi.org/10.3390/math9101096