Stability analysis and protection design of mountain gas pipeline gabion retaining wall based on multi-objective optimization algorithm

The slope ratio of the original design slope is 1:1 for the first grade and 1:1.25 for the second grade, after removing the landslide body, a 5m high gabion retaining wall is set up at the foot of the slope for support; the slope at the top of the wall is sloped according to 1:1.5/1:1.5, and the height of slope of each grade is 8m. 1)

Wall dimensions: wall height 6m; wall top width 2.3m; face slope inclined slope 1:1; back slope inclined slope 1:0; wall bottom inclined slope rate 0:1.

The main content of the calculation of gravity retaining wall stability [27] parameters using Rizheng software includes: sliding stability check of retaining wall, overturning stability check, foundation stress and eccentricity check. 1)

Sliding stability check: sliding stability coefficient of horizontal footing Kc, anti-sliding stability calculation as shown in Figure 1.

Figure 1.

A schematic of anti-sliding stability

Calculation formula: 1Kc=(W+Ey)fEx$${K_c} = \frac{{\left( {W + {E_y}} \right)f}}{{{E_x}}}$$

Kc - Sliding stability coefficient along the base of the foundation;

f - Friction coefficient at the base of the retaining wall;

W - weight of the retaining wall (kN)$$\left( {kN} \right)$$;

E_x - Horizontal component of earth pressure acting on the retaining wall (kN)$$\left( {kN} \right)$$;

E_y - Vertical component of earth pressure acting on the retaining wall (kN)$$\left( {kN} \right)$$. 2)

Calculation of overturning stability

Calculation of overturning stability coefficient [28] K₀, overturning stability calculation sketch shown in Figure 2.

Figure 2.

Capsizing stability calculation schematic

Calculation formula: 2K0=WZw+EyZxExZy$${K_0} = \frac{{W{Z_w} + {E_y}{Z_x}}}{{{E_x}{Z_y}}}$$

K₀ - the stability coefficient of the retaining wall against overturning around the toe of the wall or the toe point of the foundation;

Z_x - the horizontal distance from the vertical component of the earth pressure on the retaining wall to the point of overturning calculation (m)$$\left( m \right)$$;

Z_y - the vertical distance from the horizontal direction of the component force of the earth pressure borne by the retaining wall to the point of overturning calculation (m)$$\left( m \right)$$;

Z_w - the horizontal distance from the center of gravity of the self-weight of the retaining wall to the point of overturning calculation (m)$$\left( m \right)$$. 3)

Foundation Stress and Eccentricity Calculation

When the foundation form of gravity retaining wall is natural foundation, replacement foundation or foundation of other reinforced concrete structure, according to the theory of Rizheng software, it is necessary to carry out the checking calculation of foundation stress and eccentricity distance of retaining wall.

Eccentricity e, eccentricity calculation sketch shown in Figure 3.

Figure 3.

Eccentric distance calculation

The formula is as follows: 3e=B2−Zn=B2−Ma11Wa11$$e = \frac{B}{2} - {Z_n} = \frac{B}{2} - \frac{{{M_{a11}}}}{{{W_{a11}}}}$$ 4Ma11=WZw+EyZ−ExZ$${M_{a11}} = W{Z_w} + EyZ - ExZ$$ 5Wa11=W+Ey$${W_{a11}} = W + {E_y}$$

e-Eccentricity of the bottom section of the retaining wall (m)$$\left( m \right)$$;

B-Width of the bottom section of the retaining wall or foundation (m)$$\left( m \right)$$;

Mall-Bending moment of the toe of the wall or foundation by all the loads acting on the retaining wall (kN⋅m)$$\left( {kN \cdot m} \right)$$, clockwise is positive;

Wall-Sum of all vertical loads acting on the retaining wall (kN)$$\left( {kN} \right)$$, positive downward;

Z_n-Distance from the point of action of the combined force of the foundation reaction to the toe of the retaining wall (m)$$\left( m \right)$$;

W-Self-weight gravity of the retaining wall (kN)$$\left( {kN} \right)$$;

Z_x-Horizontal distance from the component force of earth pressure in the vertical direction on the retaining wall to the point of the toe of the wall (m)$$\left( m \right)$$;

Z_y-Vertical distance from the horizontal direction of the component force of the earth pressure borne by the retaining wall to the point of the toe of the wall (m)$$\left( m \right)$$;

Z_w- Horizontal distance from the center of gravity of the self-weight gravity of the retaining wall to the toe point of the wall (m)$$\left( m \right)$$;

Foundation stress σ: 1)

When e ≤ B/6: 6σ1,2=Wa11B(1±6eB)$${\sigma_{1,2}} = \frac{{{W_{a11}}}}{B}\left( {1 \pm \frac{{6e}}{B}} \right)$$ 7σ1,2≤λ1,2[σ]$${\sigma_{1,2}} \leq {\lambda_{1,2}}[\sigma ]$$ 8σ≤λ3[σ]$$\sigma \leq {\lambda_3}[\sigma ]$$ 9σ=0.5(σ1+σ2)$$\sigma = 0.5\left( {{\sigma_1} + {\sigma_2}} \right)$$

σ_1,2 - Foundation stresses at the toe of the retaining wall, and at the heel of the wall, respectively (KPa)$$\left( {KPa} \right)$$;

σ - the average compressive stress at the base of the retaining wall foundation (KPa)$$\left( {KPa} \right)$$;

λ_1,2 - respectively, the toe of the wall, the heel of the wall foundation bearing capacity improvement coefficient, general retaining wall: the toe of the wall to improve the coefficient, the default is 1.2; the heel of the wall to improve the coefficient, the default is 1.3;

[σ] - permissible bearing capacity of foundation of retaining wall (KPa)$$\left( {KPa} \right)$$;

B - wall bottom section width (m)$$\left( m \right)$$. 2)

When e > B/6:

When e > B/6, tensile stress occurs at the base of the foundation, without considering the foundation to bear tensile force, then the foundation stress redistribution, calculated according to the following formula: 10σmax=2Wa113Zn$${\sigma_{\max }} = \frac{{2{W_{a11}}}}{{3Zn}}$$ 11Zn=Ma11/Wa1$${Z_n} = {M_{a11}}/{W_{a1}}$$

σ_max - Maximum foundation compressive stress after foundation stress redistribution (kPa);

Z_n - Distance from the point of maximum foundation reaction force (m)$$\left( m \right)$$.

Retaining wall spacing L = 10m, seepage depth zw = 1m, retaining wall cross-section size z = 1m, h = 2m, the above initial conditions remain unchanged, only to change the value of slope inclination angle α, calculations to analyze the impact of slope inclination angle on the stability of retaining wall. The relationship between slope angle and Ea1 is shown in Figure 5. As can be seen from the figure, when the slope increases to about 26.14°, the downward sliding force of the shallow soil body of the slope begins to be greater than the slip resistance (Ea1>0), and only then will the shallow soil body of the slope produce a tendency of downward sliding. The influence of slope angle on the stability of retaining wall is shown in Figure 6. The slope angle is in the range of 25°~60°, the stability coefficient of retaining wall shows a rapid decrease with the increase of slope angle. When the slope height and the spacing of the retaining wall are certain, the increase of the slope inclination will make the gravity downward component of the sliding soil bar increase, and the stability coefficient of the retaining wall will appear a very small value at about 60°, and thereafter, with the decrease of the force Ea1, the stability coefficient of the retaining wall will gradually increase.

Figure 5.

The relationship between slope Angle and Ea1

Figure 6.

The effect of slope Angle on the stability of retaining wall

Slope inclination angle α = 30 °, seepage depth zw = 1 m, retaining wall cross-section size z = 1 m, h = 2 m, the above initial conditions remain unchanged, only to change the value of retaining wall spacing L, calculate and analyze the effect of retaining wall spacing on the stability of the retaining wall. The relationship between retaining wall spacing and Ea1 is shown in Fig. 7. Ea1 increases linearly with the increase of retaining wall spacing. Figure 8 shows the effect of retaining wall spacing on retaining wall stability. The stability coefficient of the retaining wall decreases with the increase of the retaining wall spacing. The increase of the retaining wall spacing increases the length of the sliding earth bar behind the retaining wall wall, and the force Ea1 applied to the retaining wall increases, which makes the stability coefficient of the retaining wall decreases with the increase of the wall spacing.

Figure 7.

The relationship between the spacing of the wall and Ea1

Figure 8.

The effect of spacing of retaining wall on the stability of retaining wall

Slope inclination angle α = 30 °, seepage depth zw = 1 m, retaining wall spacing L = 10 m, the above initial conditions remain unchanged, only change the bottom width of the retaining wall cross-section z, height h, calculations to analyze the effect of the retaining wall cross-section dimensions of the retaining wall stability. The influence of retaining wall cross-section parameters on the stability of retaining wall is shown in Fig. 9; (a) is the influence of z on the stability of retaining wall when h=2m, (b) is the influence of h on the stability of retaining wall when z=2m. From the figure, it can be seen that when h=2m, the slip resistance stability coefficient ks and the overturning resistance stability coefficient kt both increase linearly with the increase of z, in which ks increases faster than kt, which indicates that the slip resistance stability of the retaining wall is more changed by the influence of z.

Figure 9.

The effect of the section parameter on the stability of the wall

When z=2m, ks shows a gentle linear increase with the increase of h. When h increases from 0.75m to 4.45m, the slip resistance stability coefficient ks increases from 0.78 to 4.24, while the overturning resistance stability coefficient kt decreases from 17.45 to 0.75, which indicates that the change of h has a greater influence on kt. When h increases from 0.75m to 2.87m, kt decreases by 14.58, and as h continues to increase, the decrease in kt slows down significantly.

Suppose that there is a population consisting of m particles within a certain D-dimensional target search range. Let x_i be a D-dimensional vector of the i th particle: 12xi=(xi1,xi2,⋯,xiD),i=1,2,⋯m$${x_i} = \left( {{x_{i1}},{x_{i2}}, \cdots ,{x_{iD}}} \right),i = 1,2, \cdots m$$

The adaptation value of x_i is calculated based on the function to be solved, and this value is employed to evaluate the merit of the current position of the particle. The flight speed of particle i is also set as a D-dimensional vector: 13vi=(vt1,vt2,⋯,viD),i=1,2,⋯m$${v_i} = \left( {{v_{t1}},{v_{t2}}, \cdots ,{v_{iD}}} \right),i = 1,2, \cdots m$$

The optimal position searched so far by particle i is: 14pb=(pi1,pi2,⋯,ptD),i=1,2,⋯m$${p_b} = \left( {{p_{i1}},{p_{i2}}, \cdots ,{p_{tD}}} \right),i = 1,2, \cdots m$$

The entire group is currently searching for the optimal location: 15gb=(gi,gi2,⋯,giD),i=1,2,⋯m$${g_b} = \left( {{g_i},{g_{i2}}, \cdots ,{g_{iD}}} \right),i = 1,2, \cdots m$$

Once the local optimal position and the overall optimal position are found, the velocity and position of the particles are constantly updated in subsequent iterations according to the following two equations: 16vmk+1=wvmk+c1r1(pm−xmk)+c2r2(gin−xink)$$v_m^{k + 1} = wv_m^k + {c_1}{r_1}\left( {{p_m} - x_m^k} \right) + {c_2}{r_2}\left( {{g_{in}} - x_{in}^k} \right)$$ 17xmk+1=xmk+vmnk+1$$x_m^{k + 1} = x_m^k + v_{mn}^{k + 1}$$

where n = 1,2,⋯,D, k are the number of iterations, w is the inertia weight, c₁ and c₂ are the learning factors, and r₁ and r₂ are random numbers in the range [0,1]. The inertia weights act as a trade-off between local and global optimization.

Based on the basic principles and application principles of the particle swarm optimization algorithm [29] mentioned above, the algorithm is applied to the optimization design of gravity retaining wall. To solve the cross-section optimization problem of gravity retaining wall, a single-objective multi-constraint nonlinear model is used, and the optimization model building process can be divided into the following steps: 1)

Establish the objective function and the adaptive value function.

The project cost of gravity retaining wall can be calculated according to the following formula: 18T=S×L×P$$T = S \times L \times P$$

Where T - total cost of gravity retaining wall;

S - the cross-sectional area of the retaining wall;

L - the design length of the retaining wall;

P - unit price of construction materials.

The objective function calculation formula for this optimization design is as follows: 19S=Fmin(x1,x2,⋯,xn)$$S = {F_{\min }}\left( {{x_1},{x_2}, \cdots ,{x_n}} \right)$$

Where x - the dimensions of the parts that make up the area;

F_min - the minimum combined sectional area.

The adaptation value function in the particle swarm algorithm can be calculated according to the following formula: 20f(x)=S$$f(x) = S$$

The optimal solution calculated by particle swarm algorithm is the optimal individual that satisfies the requirements of the objective function. 2)

Selection of optimization design variables

In this design, six design variables will be selected to complete the gravity retaining wall cross-section optimization design, including: retaining wall top width x₁, wall face slope ratio x₂, toe width x₃, back slope ratio x₄, base slope ratio x₅, and toe height x₆.

The initialization range of each dimensional particle is set, while the amount of the range change of each dimensional particle is taken as the maximum velocity of that dimensional particle variable V_max.

In this paper, according to the design requirements of gravity retaining wall, particle swarm algorithm is used to optimize its design. The main workflow of this optimization algorithm is as follows: 1)

A structure size variable value in this optimization design is a particle, in the application of the algorithm should firstly initialize the size (m)$$\left( m \right)$$ of the particle swarm, the position (xt)$$\left( {{x_t}} \right)$$ and velocity (vt)$$\left( {{v_t}} \right)$$ of each particle;

In this paper, some parameters in the algorithm are optimized and set by particle swarm optimization algorithm. 1)

The number of swarm particles m: the number of particles in this optimization design is selected in the range between.

Where ω_min, ω_max is the lower and upper limit of the weight value, Iter_max is the maximum number of iterations, i indicates the i th iteration, and ω_i is the weight value of the i th iteration. In the selection of inertia weights should be selected according to the needs of their own design. According to the optimization design of this project, two ranges of [0,0.5] and [0.5,1] will be selected for comparison and analysis.

On the above designed particle swarm optimization algorithm based gravity retaining wall pressure calculation and cross-section optimization method, the following will design a group of comparative experiments, select the two most commonly used methods as the control object, in order to facilitate the subsequent experimental statement, the following two methods were expressed in the control group 1, the control group 2, will be designed in this paper as an experimental method for the experimental group, selecting a mountainous area of gas pipeline project as the background, the The original design of gravity retaining wall is 3.24m in height, 2.55m in width, 0.75m in thickness, the wall material is slurry masonry block, the weight of the retaining wall is 25.79kN/m³, the fill of the wall thickness is not easy to weather the stone masonry, the natural weight is 18.24kN/m³, the allowable bearing capacity of the foundation of the gravity retaining wall is 1550kPa, and the coefficient of friction of the basement is 0.47, and the design method of this paper is as follows. According to the above process of gravity retaining wall pressure calculation, randomly selected a retaining wall, gravity retaining wall pressure calculation results are shown in Figure 10. Using particle swarm optimization algorithm to optimize the objective function of the retaining wall cross-section solution, the optimized retaining wall cross-section area of 56.46m², the upper wall wall slope ratio of 1.27 lower wall wall slope ratio of 1.05, the width of the top of the wall is 2.6m, the width of the weighing platform is 0.85m, and the eccentricity distance is 0.23m.

Figure 10.

Pressure calculation results of gravity retaining wall

For gravity type retaining wall pressure calculation accuracy, select the error as an indicator, the difference between the calculated value and the actual retaining wall pressure value, to find out the retaining wall pressure calculation error, the use of spreadsheets on the experimental data records, three methods of retaining wall pressure calculation error is shown in Figure 11. The three methods in the retaining wall pressure calculation error shows obvious differences, the average calculation error of the design method is 0.028kPa, the average value of the control group 1 for the retaining wall pressure calculation error is 0.666kPa, the average value of the control group 2 for the retaining wall pressure calculation error is 0.457kPa, the control group calculation error is much larger than the experimental group, so it is proved that the design method is superior to the traditional method in the area of the gas pipeline retaining wall pressure calculation accuracy. Therefore, it is proved that the design method is better than the traditional method in terms of the accuracy of pressure calculation of district gas pipeline retaining wall.

Figure 11.

Three methods to prevent the pressure calculation error of the wall

For the optimization effect of gravity retaining wall section, the coefficient of safety of slip resistance stability of retaining wall is chosen as the index, and the higher the coefficient of safety of slip resistance stability is, the better the stability of gravity retaining wall is, eight gravity retaining walls are randomly selected in the experiment, and the coefficients of safety of slip resistance stability of the retaining walls are compared with the optimized retaining walls of the three methods, and the specific data are shown in Fig. 12. Under the application of design method, the coefficient of safety of slip-resistant stability of retaining wall is at a high level, and the average coefficient of safety of slip-resistant stability is 0.953, which can be controlled above 0.9, proving that the slip-resistant performance of optimized gravity retaining wall is significantly improved, and the stability of slip-resistant stability is high. In comparison, the coefficient of safety of slip resistance and stability of retaining wall under the application of design method is higher than that of control group 1 by 0.435, and higher than that of control group 2 by 0.487, which has a good optimization effect. Therefore, it can be proved by the above experimental data and experimental results that the design method has obvious advantages in the accuracy of retaining wall pressure calculation and in the optimization effect of retaining wall cross-section, and it is more suitable for the pressure calculation and cross-section optimization design of gravity retaining wall for district gas pipeline compared with the two traditional methods.

Figure 12.

Gravity retaining wall anti-slip stability safety coefficient

In this paper, the algorithm calculates the minimum cost of backfill for effective internal friction angles φ of 30°, 35°, 40° and 45° and finds that the number of internal cycles of the algorithm is around 1000. Table 1 shows the optimization results for different internal friction angles φ. Analyzing from the changes of each design variable, the basic dimensions of the four dimensional parameters of “toe plate, bottom of the vertical wall, heel plate and thickness of the bottom plate” are basically the same, and their parameters are around 0.307, 0.307, 1.485 and 0.099, respectively, while the parameter of the top of the vertical wall changes greatly (0.099) at 20°-50°, and the parameter of the top of the vertical wall varies greatly (0.099) at 20°-40°. 50° varies more (between 0.101-0.275), and no relevant pattern can be found for the variation of the top of the standing wall, which proves that the algorithm is better dispersed in terms of the values taken. The main change in the parameters is the change in the cross-sectional area of the reinforcement, which in each structure at an internal friction angle φ of 20° is about double the cross-sectional area of the reinforcement at an internal friction angle φ of 50°, and the ratio of the lengths of the non-passage bars to the passage bars is maintained in the range of 0.232 to 0.244.

The variation of minimum cost with internal friction angle φ is shown in Fig. 13. From the figure, it can be seen that the minimum cost of cantilever retaining wall with the change of internal friction angle φ is basically straight line, with the increase of the internal friction angle φ, the minimum cost of cantilever retaining wall is decreasing, but the minimum cost of retaining wall per meter decreases within 40 yuan. From the above calculation results, it can be seen that the value is negative, that is, the minimum cost decreases with the increase of the internal friction angle φ. For every 1° increase of the internal friction angle φ, the minimum cost per meter decreases by 1.23317 yuan.

Figure 13.

Minimum cost with internal friction Angle

In this paper, the algorithm calculates the minimum cost of backfill with fill weight γs of 10 kg/m³, 15 kg/m³, 20 kg/m³, and 25 kg/m³ respectively, and the number of internal cycles of the algorithm is between 1000 and 1200. Table 2 shows the optimization results for different fill weights γs. The five dimensional parameters are basically the same when analyzed in terms of the variation of each design variable. The main changes in the parameters are the changes in the cross-sectional area of the reinforcement, and the ratio of the length of the non-through-length reinforcement to the through-length reinforcement is maintained between 0.2307 and 0.2438, with a relatively large interval.

The variation of minimum cost with fill weight γs is shown in Fig. 14. The minimum cost of cantilever retaining wall with the change of fill weight γs is basically a straight line, with the increase of fill weight γs, the minimum cost of cantilever retaining wall in the increase, but the minimum cost of retaining wall per meter of the decrease in 25 yuan, fill weight γs on the minimum cost of cantilever retaining wall is not much influence. From the above calculations, it can be seen that the value is positive, i.e., the minimum cost increases with the increase of fill weight γs, and for every increase of 1 kg/m³ in fill weight γs, the minimum cost per meter increases by 1.481067 yuan.

Figure 14.

The minimum cost is accompanied by the change in the heavy γs

The sensitivity of each parameter is shown in Table 3. For every 1° increase in the angle of internal friction φ, the minimum construction cost per meter decreases by 1.23317 yuan. For every 1 kg/m³ increase in fill weight γs, the minimum construction cost per meter increases by 1.481067 yuan. For every 1 kN/m² increase in surface load Pk, the minimum cost per meter increases by 1.154368 yuan. Every increase of 0.1 in the base friction coefficient f, the minimum cost per meter decreases by 0.896541 yuan. Through the table, it is easy to get the sensitivity of the cost of cantilever retaining wall to each condition parameter, surface load pk >fill weight γs > angle of internal friction φ > coefficient of basal friction f.