Keywords

1 Introduction

Hydraulic tunnels are crucial components in water infrastructure projects, serving important functions such as diversion and water conveyance. However, the stability of the rock masses directly affects the safety of tunnel structures. Due to the complex geological nature of rock masses, which consist of various types of rocks and structural planes, their stress states are also influenced by groundwater and in-situ stress conditions. Therefore, the stability analysis of rock masses in hydraulic tunnels is an uncertainty problem that involves fuzziness and randomness [1].Currently, in the analysis of rock mass stability in hydraulic tunnels, methods such as the safety factor method and finite element method are commonly used. These methods share the common characteristic of representing many design parameters with single values, often failing to reflect the actual stress conditions of the structures.

Fuzzy random reliability theory considers the randomness of rock mechanics parameters, dimensions of structures, loads, and lining material properties. Engineering structural reliability regards these engineering variables as random variables [2, 3]. Zou Shanshan proposed a probability-based Miner’s rule using fuzzy theory to predict the fatigue life of critical components under random load stress and obtained a fatigue life prediction model [4].

Currently, in water infrastructure projects, there are still many factors that impact hydraulic structures, and the design parameters vary greatly, while the number of statistically available parameters is limited. Therefore, reliability theory is less commonly used in hydraulic structures [5, 6]. This paper primarily focuses on the fuzzy random reliability theory and establishes a fuzzy random reliability model for the stability of rock masses in hydraulic tunnels. The calculation method of fuzzy random reliability for rock masses is considered when support structures are present.

2 Establishment of the Fuzzy Random Reliability Calculation Model for the Stability of Rock Masses in Hydraulic Tunnels

2.1 The Construction of Function Function

Establishment of safety reserve function:

In order to facilitate analysis, computation, and practical application, a functional relationship is constructed to represent the safety margin of rock masses in hydraulic tunnels. The safety margin is defined as:

$$ Z = R - S $$
(1)

S—Load Effects on Rock Masses; R—Structural Resistance.

2.2 Basic Assumptions and Fundamental Mechanical Models.

Basic Assumptions and Fundamental Mechanical Models. In order to avoid excessive complexity in the established model and solution process, assumptions are adopted for the analytical process inclueded: the rock mass is considered as an isotropic continuum;the initial stress field in the rock mass is due to self-weight and is in a state of hydrostatic pressure (λ = 1);the tunnel is relatively long, and the analysis is conducted based on the assumption of plane strain, within the small deformation range;the tunnel cross-section is assumed to be circular; after yielding, the rock mass still satisfies the Mohr-Coulomb criterion; only the overall deformation and failure of the rock mass are considered.

Determination of Equivalent Strain Values around the Tunnel Periphery.

For axisymmetric problems, when body forces are neglected, the equilibrium differential equation is given by:

$$ \frac{{\partial \sigma_{r} }}{\partial r} + \frac{{\sigma_{r} - \sigma_{\theta } }}{r} = 0 $$
(2)

The stress within the plastic zone satisfies the plasticity criterion:

$$ \sigma_{\theta p}= \frac{1+\sin\,\varphi}{1-\sin\,\varphi}\sigma_{rp}+\frac{2c \times \cos\,\varphi}{1-\sin\,\varphi}$$
(3)

When the tunnel is freshly excavated without lining, the radial load at the inner edge of the plastic zone, which is the tunnel periphery, is zero. This serves as the boundary condition at this stage. By using Eqs. (2) and (3) along with the boundary condition, the stress field and displacement field of the surrounding rock can be determined:

$$ \varepsilon_{\theta p} = { - }u\left| {_{{R_{p} }} } \right.\frac{{R_{p} }}{{r^{2} }} = \frac{1 + \mu }{E} \cdot \left( {p \cdot \sin \varphi + c \cdot \cos \varphi } \right)\frac{{R_{p}^{2} }}{{r^{2} }};\overline{\varepsilon } = \sqrt {\frac{2}{3} \cdot e_{ij} \cdot e_{ij} } ;\varepsilon_{1p} = \varepsilon_{rp} ,\varepsilon_{2p} = 0,\varepsilon_{3p} = \varepsilon_{\theta p} $$
(4)

In the equation, \(R_{p}\) represents the radius of the plastic zone, \(c\) is the cohesive strength, \(\varphi \) is the internal friction angle, \(r\) is the radius of the tunnel.

Thus, the strain field of the surrounding rock in the plastic zone is obtained as:

$$ \varepsilon_{rp} = u\left| {_{{R_{p} }} } \right.\frac{{R_{p} }}{{r^{2} }} = \frac{1 + \mu }{E} \cdot \left( {p \cdot \sin \varphi + c \cdot \cos \varphi } \right)\frac{{R_{p}^{2} }}{{r^{2} }};\varepsilon_{\theta p} = { - }u\left| {_{{R_{p} }} } \right.\frac{{R_{p} }}{{r^{2} }} = \frac{1 + \mu }{E} \cdot \left( {p \cdot \sin \varphi + c \cdot \cos \varphi } \right)\frac{{R_{p}^{2} }}{{r^{2} }} $$
(5)

According to the plasticity theory, the equivalent shear strain is obtained as:

$$ \overline{\varepsilon } = \sqrt {\frac{2}{3} \cdot e_{ij} \cdot e_{ij} } $$
(6)

As the tunnel is a plane strain problem, \(\varepsilon_{1p} = \varepsilon_{rp} ,\varepsilon_{2p} = 0,\varepsilon_{3p} = \varepsilon_{\theta p}\).

So the equivalent variation field is obtained as:

$$ \overline{\varepsilon } = \frac{2\sqrt 3 }{3} \cdot \frac{1 + \mu }{E} \cdot \frac{{R_{0}^{2} }}{{r^{2} }} \cdot \left[ {\frac{(p + c \cdot ctg\varphi )(1 - \sin \varphi )}{{d \cdot ctg\varphi }}} \right]^{{\frac{1 - \sin \varphi }{{\sin \varphi }}}} \cdot (p \cdot \sin \varphi + c \cdot \cos \varphi ) $$
(7)

Since \(\left( {\overline{\varepsilon } } \right)_{{p_{i} }} = 0\) is maximum at the perimeter of the tunnel, decreasing with increasing radius r, In Eq. (7) let \(r = R_{0}\) then we have:

$$ \overline{\varepsilon } = \frac{2\sqrt 3 }{3} \cdot \frac{1 + \mu }{E} \cdot \frac{{R_{0}^{2} }}{{r^{2} }} \cdot \left[ {\frac{(p + c \cdot ctg\varphi )(1 - \sin \varphi )}{{d \cdot ctg\varphi }}} \right]^{{\frac{1 - \sin \varphi }{{\sin \varphi }}}} \cdot (p \cdot \sin \varphi + c \cdot \cos \varphi ) $$
(8)

The Fuzzy Random Reliability Calculation Process of Surrounding Rock Stability.

Using the limit strain criterion, the limit state equation of surrounding rock stability can be obtained:

$$ Z = \varepsilon_{0} { - }\overline{\varepsilon } = \frac{2\sqrt 3 }{3} \cdot \frac{1 + \mu }{E} \cdot \frac{{R_{0}^{2} }}{{r^{2} }} \cdot \left[ {\frac{(p + c \cdot ctg\varphi )(1 - \sin \varphi )}{{d \cdot ctg\varphi }}} \right]^{{\frac{1 - \sin \varphi }{{\sin \varphi }}}} \cdot (p \cdot \sin \varphi + c \cdot \cos \varphi ) $$
(9)

The formula contains fuzzy random variables \(E,c,\varphi ,\varepsilon_{0} ,\mu ,p\), Considering the actual situation of the surrounding rock’s Poisson ratio \(\mu\), the variation of bulk density \(\gamma\)  is generally one order of magnitude smaller than \(c,\varphi\), so only \(E,c,\varphi ,\varepsilon_{0}\) are taken as basic fuzzy random variables in the above formula. In geotechnical engineering, most of the variables related to geotechnical parameters are normally distributed, so \(E,c,\varphi ,\varepsilon_{0} ,\overline{\varepsilon }\) are fuzzy random variables with normal distribution, The fuzzy random reliability is calculated by using the composite function derivation rule and Harlin method.

Then the limit state equation is:

$$ Z\left( {\overline{x} } \right) = x_{1} - \left( {\frac{{x{}_{2}}}{{x{}_{3}}}} \right)^{{x_{4} }} \cdot x_{5} $$
(10)

The limit state equation contains five substitution variables \(x_{i} (i = 1,2, \cdots 5)\), and each substitution variable is a function of four variables \(E,c,\varphi ,\varepsilon_{0}\).

Let \(E,c,\varphi ,\varepsilon_{0}\) correspond to variables \(y_{i} = (i = 1,2,3,4)\) respectively, namely:

\(E = y_{1} ,c = y_{2} ,\varphi = y_{3} ,\varepsilon_{0} = y_{4}\).

then get:

$$ \frac{\partial Z}{{\partial y_{i} }}\left| {_{{y^{*} }} } \right. = \sum\limits_{j = 1}^{5} {\frac{\partial Z}{{\partial x_{j} }}} \cdot \frac{{\partial x_{j} }}{{\partial y_{i} }}\left| {{}_{{y^{*} }}} \right. $$
(11)

As long as the reliability index \(\beta\) is obtained, the fuzzy random reliability can be obtained. Harlin method believes that the geometric meaning of the reliability index is the shortest distance from the origin to the limit state equation in the standard normal coordinate system,

If the basic fuzzy random variables are mutually independent, they all follow a normal distribution, and have mean values \(u_{i}\) and mean square deviations \(\sigma_{i}\) respectively, and the functional function is a linear combination of basic fuzzy random variables, namely:

$$ Z(\overline{x} ) = a_{0} + \sum\limits_{i = 1}^{n} {a_{i} } ,\quad \beta = \frac{{a_{0} + \sum\limits_{i}^{n} {a_{i} \cdot u_{i} } }}{{\left( {\sum\limits_{{}}^{{}} {a_{i}^{2} \cdot \sigma_{i}^{2} } } \right)^{\frac{1}{2}} }} $$
(12)

If the functional function is a nonlinear combination of basic variables, then the functional function can be expanded into a Taylor series at the verification point \(p^{*}\) as follows:

$$ Z\left( {\overline{x} } \right) = Z\left( {x_{1}^{*} ,x_{2}^{*} , \cdots x_{n}^{*} } \right) + \sum\limits_{i = 1}^{n} {\left( {\frac{\partial Z}{{\partial x_{i} }}} \right)}_{{p^{*} }} \cdot \left( {x_{i} - x_{i}^{*} } \right) $$
(13)

Thus, it can be obtained:

$$ \beta = \frac{{\sum\limits_{i = 1}^{n} {\left( {\frac{\partial Z}{{\partial x_{i} }}} \right)_{{p^{*} }} \cdot \left( {u_{i} - x_{i}^{*} } \right)} }}{{\left( {\sum\limits_{i = 1}^{n} {\left( {\frac{\partial Z}{{\partial x_{i} }}} \right)^{2}_{{p^{*} }} \cdot \sigma_{i}^{2} } } \right)^{\frac{1}{2}} }};x_{i}^{*} = u_{i} + \sigma_{i} \lambda_{i} \beta ; $$
(14)
$$ \lambda_{i} = \frac{{ - \left( {\frac{\partial Z}{{\partial x_{i} }}} \right)_{{p^{*} }} \cdot \sigma_{i} }}{{\left( {\sum\limits_{i = 1}^{n} {\left( {\frac{\partial Z}{{\partial x_{i} }}} \right)^{2}_{{p^{*} }} \cdot \sigma_{i}^{2} } } \right)^{\frac{1}{2}} }};Z\left( {x{}_{1}^{*} ,x{}_{2}^{*} \cdots ,x{}_{n}^{*} } \right) = 0; $$
(15)

Since the verification point is unknown in advance, when expanding into a Taylor series, a point must be assumed in advance, such as the mean value point of each basic variable. In the calculation process, use the iteration method to gradually approach the real verification point and correct the obtained \(\beta\) value until the result converges satisfactorily. Assume the initial value of \(x_{i}^{*}\), generally take the average value \(u_{i}\).

Hydraulic tunnels are more complex than above-ground structures, making the working state of the surrounding rock very fuzzy. Therefore, using the exact judgment criterion of “either this or that” sometimes does not match the actual situation. If considering the fuzziness of the judgment criterion, it is necessary to fuzzify the limit strain judgment criterion and make it a fuzzy subset \(\Omega\), whose membership function is shown in the following formula:

$$ u_{\Omega } \left( {\varepsilon_{0} ,\overline{\varepsilon } } \right) = \left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 & {\varepsilon_{0} - \overline{\varepsilon } \prec { - }L} \\ \end{array} } \\ {\begin{array}{*{20}c} {\frac{{\varepsilon_{0} - \overline{\varepsilon } + L}}{2L}} & { - L \le \varepsilon_{0} - \overline{\varepsilon } \le L} \\ \end{array} } \\ {\begin{array}{*{20}c} 1 & {\varepsilon_{0} - \overline{\varepsilon } \succ L} \\ \end{array} } \\ \end{array} } \right. $$
(16)

The value of L in the formula can be taken according to the actual situation, and here L can be set to 0.05%.

3 Example

A circular hydraulic tunnel with a diameter of 6 m, a burial depth of 300 m, located in limestone strata, sprayed C30 concrete immediately after tunnel excavation, with a concrete thickness of 8 cm. Through the indoor test of rock samples, the mechanical parameters of surrounding rock samples and the mechanical parameters of concrete used in working state, after fuzzy random processing, the statistical characteristic values of normal distribution are shown in Table 1. Since the variation of bulk density \(\gamma\), Poisson’s ratio \(\mu\) of rock and elastic modulus \(E_{1}\) and Poisson’s ratio \(\mu_{1}\) of concrete are much smaller than that of other parameters, they are considered as constants.

Table 1. Fuzzy random statistical values of mechanical parameters
Full size table

The vertical component of the initial ground stress is calculated from the above data as follows:

$$ P_{v} = \gamma \cdot H = 2.2 \times 300 = 6.6\;{\rm MPa}\;\lambda = \frac{\mu }{1 - \mu } = \frac{0.34}{{1 - 0.34}} = 0.52 $$
(17)

The average ground stress is: \(P = \frac{1 + 0.52}{2} \times 6.6 = 5.0\,{\rm MPa}\).

According to Eq. (5), we get \(D = 53.323,F = 3.602\), and rearrange to get.

$$ 1194.05\left( {\frac{{R_{p} }}{{R_{0} }}} \right)^{5.602} + 6345.43\left( {\frac{{R_{p} }}{{R_{0} }}} \right)^{3.602} - 11783.48 = 0 $$
(18)

Using Newton’s iteration method, we get after 3 iterations: \(\frac{{R_{p} }}{{R_{0} }} = 1.12\)

Therefore, the plastic radius is: \(R_{p} = 3.36\,{\rm m}\)，then: \(P_{r} = \frac{{706 \times \left( {10 - c \cdot ctg\varphi } \right)}}{E + 2123}\).

According to Eq. (4), the tangential stress on the inner side of the lining is:

$$ \sigma_{\theta } = \frac{{26828 \times \left( {10 - c \cdot ctg\varphi } \right)}}{E + 2123} $$
(19)

According to the limit strength criterion, the functional function is:

$$ Z = R_{c} - \sigma_{\theta } = R_{c} - \frac{{26828 \times \left( {10 - c \cdot ctg\varphi } \right)}}{E + 2123} $$
(20)

This functional function contains four basic variables, namely \(R_{c}\), \(E\), \(c\), \(\varphi\). And this function is a nonlinear function of these four variables. Therefore, this time we use the Halin method to solve the reliability index \(\beta\).

After iteration, the reliability index of the lining is finally determined to be \(\beta = 3.949\), and the support structure is stable and reliable.

4 Conclusion

This paper mainly expounds the fuzzy stochastic reliability analysis method of surrounding rock stability of hydraulic tunnel. Based on the fuzzy stochastic reliability theory, a fuzzy stochastic reliability calculation model for surrounding rock stability of hydraulic tunnel is established, and the accuracy of the reliability calculation model is verified by practical engineering cases:

  1. (1)

    On the basis of considering the fuzziness and randomness of surrounding rock parameters, a fuzzy stochastic model for surrounding rock reliability analysis of hydraulic tunnel is proposed.

  2. (2)

    On the basis of the established fuzzy stochastic reliability model for surrounding rock stability, the influence of support structure on surrounding rock stability is considered, and the fuzzy stochastic reliability of support structure is calculated.

  3. (3)

    The fuzzy stochastic reliability model of surrounding rock stability of hydraulic tunnel is applied to a circular hydraulic tunnel, and the surrounding rock reliability considering the support structure is calculated, and its reliability index \(\beta = 3.949\) is obtained. The support structure is reliable and consistent with the actual situation. This model considers both the fuzziness and randomness of surrounding rock, which is more scientific and reasonable than the traditional reliability method.