Stability of subsea circular tunnels using finite element limit analysis and adaptive neuro-fuzzy inference system

Abstract

Construction of subsea tunnels has grown significantly in recent decades around the world. Hence, ensuring the stability of these tunnels is very important. However, investigating the stability of subsea tunnels has not received enough attention. The goal of the present study is to use the finite element limit analysis (FELA) for determining the internal pressure needed to maintain the stability of circular subsea tunnels embedded in a Tresca material by considering various values for the influential parameters. The problem was modeled and investigated in a plane strain condition. It was observed that the required internal pressure, in a dimensionless form, increased linearly by increasing soil cover and water depth. The failure mechanism was also investigated. It was observed that although the water depth did not have a considerable effect on the failure type, the mechanism did not cover the tunnel’s roof for lower values of soil cover and soil’s undrained shear strength. Both the adaptive neuro-fuzzy inference system (ANFIS) and the multiple linear regression (MLR) were implemented to predict the required dimensionless internal pressure. The performance of both methods was assessed by using multiple statistical measures such as the root-mean-square-error and the Bland-Altman plot. It was observed that the predictive capability of the ANFIS was far better than the MLR.

Appendix

The number of independent variables (N_v) was equal to six including D, h_w, h_s, C_u, γ, and σ_t.

The reference dimensions of the variables are: D $\stackrel{.}{=}$ [L], h_w $\stackrel{.}{=}$ [L], h_s $\stackrel{.}{=}$ [L], C_u $\stackrel{.}{=}$ [FL⁻²]; γ $\stackrel{.}{=}$ [FL⁻³]; and σ_t $\stackrel{.}{=}$ [FL⁻²], where F and L are the dimensions of the force and the length, respectively; and the notation $\stackrel{.}{=}$ shows the dimension of the variable. Hence, the number of reference dimensions (N_D) is equal to two including F and L.

The Buckingham’s Pi theorem (Buckingham 1914) indicates that the number of dimensionless products (N_p) is 6–2 = 4. The authors selected the variables D and C_u as repeating variables.

For determining the first dimensionless product (π₁), σ_t was selected as the nonrepeating variable. So:

$${\pi }_{1} ={\sigma }_{t}.{D}^{a}.{C}_{u}^{b}$$

(12)

Since π₁ is dimensionless:

$$\left(F{L}^{-2}\right)\left({L}^{a}\right){\left(F{L}^{-2}\right)}^{b} = {L}^{0}{F}^{0}$$

(13)

Setting the power terms to zero leads to:

$${\pi }_{1} =\frac{{\sigma }_{t}}{{C}_{u}}$$

(14)

In the next step, we considered γ to obtain the second dimensionless variable (π₂) as follows:

$${\pi }_{2} = \gamma .{D}^{c}.{C}_{u}^{d}$$

(15)

As a result:

$$\left({FL}^{-3}\right)\left( {L}^{c}\right){\left({FL}^{-2}\right)}^{d} = {F}^{0}{L}^{0}$$

(16)

As a result: $1+d=0$, and $-3+c-2d=0$. The resulting dimensionless product is:

$${\pi }_{2} = \frac{\gamma D}{{C}_{u}}$$

(17)

Using h_w for obtaining the third dimensionless product (π₃) leads to:

$${\pi }_{3} = {h}_{w}.{D}^{\alpha }.{C}_{u}^{\beta }$$

(18)

Since π₃ is dimensionless:

$$\left(L\right)\left( {L}^{\alpha }\right){\left({FL}^{-2}\right)}^{\beta } = {F}^{0}{L}^{0}$$

(19)

The above equation leads to: β = 0, and α = −1. Consequently:

$${\pi }_{3} = \frac{{h}_{w}}{D}$$

(20)

Since h_s has the same dimension as h_w, the fourth dimensionless product (π₄) is:

$${\pi }_{4} = \frac{{h}_{s}}{D}$$

(21)