Optimization design of anti-seismic engineering measures for intake tower based on non-dominated sorting genetic algorithm-II

Abstract

High-rise intake towers in high-intensity seismic areas are prone to structural safety problems under vibration. Therefore, effective and low-cost anti-seismic engineering measures must be designed for protection. An intake tower in northwest China was considered the research object, and its natural vibration characteristics and dynamic response were first analyzed using the mode decomposition response spectrum method based on a three-dimensional finite element model. The non-dominated sorting genetic algorithm-II (NSGA-II) was adopted to optimize the anti-seismic scheme combination by comprehensively considering the dynamic tower response and variable project cost. Finally, the rationality of the original intake tower antiseismic design scheme was evaluated according to the obtained optimal solution set, and recommendations for improvement were proposed. The method adopted in this study may provide significant references for designing anti-seismic measures for high-rise structures such as intake towers located in high-intensity earthquake areas.

Graphical abstract

Keywords

intake tower / NSGA-II / mode decomposition response spectrum method / anti-seismic engineering measures / optimization design / variable project cost

1 Introduction

The intake tower is located in front of the water conveyance system and is mainly used for water diversion and power generation [1]. In recent years, China’s large-scale water conservation and hydropower projects have mainly been built on rivers rich in hydropower resources in the southwest. The height of the intake tower tends to increase with the height of water-retaining structures. However, numerous high-intensity earthquake areas in southwest China can seriously affect the structural safety and normal operation of hydraulic structures in a strong earthquake. In addition, as a high-rise structure, the sectional area of an isolated intake tower is generally small, as well as its bending and torsional stiffnesses, therefore, it may easily lose stability under dynamic action. Therefore, analyzing the anti-seismic stability of intake towers is important [2].

Many scholars have utilized various dynamic methods for earthquake conditions to study the seismic performance of intake towers, including the pseudostatic approach, time-history analysis, and the mode decomposition response spectrum method [3–7]. Among these, the mode decomposition response spectrum method is one of the most widely used dynamic analysis methods for the seismic design of hydraulic structures [8–11]. Based on the analysis of the natural vibration characteristics of a structure, the displacement and stress of the structure under seismic conditions can be conveniently obtained by linear or nonlinear superposition. Thus, the response spectrum method is suitable for structural anti-seismic design and safety review of intake towers.

For the anti-seismic design of intake towers, engineering measures such as heightening the backfill and foundation consolidation grouting are usually adopted to further improve stability by ensuring a sufficient structural stiffness of the tower. For example, the standard for the seismic design of hydraulic structures [12] proposes that for shore-type intake towers, the gap between the tower body and excavated rock mass should be backfilled with concrete [13,14]. The tower foundation is on a rock mass with sufficient bearing capacity; hence, in some cases, consolidation grouting is required for foundation treatment [15].

However, there is no unified standard for designing anti-seismic engineering measures. If the adopted engineering measures cannot meet the seismic requirements of a structure, the structure may encounter safety problems during its operation period. In contrast, if the scales of anti-seismic measures are too large, they lead to enormous excavation and backfilling, high engineering costs, and long construction periods. Consequently, searching for an optimal combination scheme of antiseismic measures for the intake tower and minimizing project costs while ensuring safety are essential [16,17].

In summary, the optimal design of anti-seismic measures for intake towers is a multiobjective decision-making problem involving two targets: structural safety and economic feasibility. Intelligent algorithms and data mining technologies are widely applied in various industries to optimise engineering structures [18–24]. Among them, with the advantages of autonomous learning and random optimization, the non-dominated sorting genetic algorithm-II (NSGA-II) is particularly suitable for multi-objective decision-making [25–27]. The essence of multi-objective optimization is determining the optimal Pareto solution [28,29]. After further development, the improved NSGA-II, which is based on the Pareto-optimal concept, can evenly select individuals at the initial stage of the algorithm to prevent local convergence. In addition, it can improve computational efficiency and accelerate convergence speed.

This study analyzed the natural vibration characteristics and dynamic response of an intake tower in northwest China under preliminary design conditions using a three-dimensional finite element model. On this basis, the NSGA-II was adopted to optimize the anti-seismic scheme combination by comprehensively considering the tower dynamic response and variable project cost. Finally, the rationality of the original intake tower antiseismic design scheme was evaluated according to the obtained optimal solution set, and improvement recommendations were proposed. The proposed method and corresponding research conclusions provide significant references for the anti-seismic design and reinforcement of high-rise structures in high-intensity earthquake areas.

2 Three-dimensional finite element model

2.1 Project overview

An isolated shore-type high-rise intake tower is the most important diversion construction in water-supply projects in northwest China. The maximum height of the intake tower was 56.0 m, and the width of the tower in the direction vertical to the inflow and along the inflow was 14.00 and 18.35 m, respectively. The safety grade of the intake tower was Class III. The intake tower was reinforced concrete with grade C25 concrete, and the backfill concrete around the tower was made of C20 concrete. The size of the stairwell was 5.5 m × 5.5 m × 45.48 m, which was arranged on the downstream side, and the stair form was double-running. The inclined plates of the staircase were in the inflow direction, and the included angle between the plate and horizontal direction was approximately 34°. The normal storage level of the intake tower was 1432.00 m. Typical sections of the intake tower are shown in Fig.1.

The natural slope angle near the water inlet was approximately 35°, and the bedrock was composed of mixed schist, mainly medium-hard rock. There was no large-scale structural fracture zone in the rock mass. Therefore, the bedrock is of good quality and has no adverse impact on building stability. According to site seismic safety evaluation and the seismic ground motion parameter zone map of China [30], under the risk level of 10% exceedance probability during the 50-year reference period in the project site, the horizontal peak acceleration of bedrock is 0.20g. Besides, the characteristic period of the seismic response spectrum is 0.45 s, and the basic seismic intensity is VIII grade.

2.2 Finite element method model description

To ensure that the boundary constraints do not affect the real stress state of the building, the finite element model must include a certain range of foundations and surrounding rocks. Thus, the upstream and downstream boundaries of the model are intercepted at 40.00 and 39.65 m in front of and behind the intake tower, respectively. The bottom elevation of the model was 1267.00 m, and the two sides of the vertical inflow direction extend 30.00 m.

The commercial software ABAQUS was used for finite element analysis. The finite element model was established according to the design size of the intake tower after ignoring detailed structures such as parapets, piezometric tubes, and diversion pipelines. Based on the principle of the variational method, the calculation domain was discretized, and a three-dimensional finite element mesh was formed using automatic mesh generation and densification technology. Fig.2 displays the direction of the coordinate system and the generated three-dimensional finite element mesh with 123113 elements and 150687 nodes. The solid element was a hexahedral eight-node element (C3D8). The positive direction of the x-axis is the same as the inflow direction. The detailed diagrams of the intake tower and backfill concrete are shown in Fig.2(b).

Because the stairs inside the intake tower form a support structure similar to a triangle, the impact of the stairs on the seismic performance of the intake tower must be considered. The structure of the stairs is illustrated in Fig.2(c), and its inner steel bars are modeled, as shown in Fig.2(d). In an actual project, there are dowel bars between the stairs and inner walls of the intake tower; therefore, the connection between them is relatively firm. The two parts are assumed not to be separated under static or dynamic conditions. Therefore, a tie constraint connected the intake tower to its stairs. A perfect bond was used to connect the stairs and their internal bars. Fig.2(e) shows the detailed arrangement of the concrete backfill. For the finite element model, three-direction constraints were set on the bottom surface, and X- and Y-direction constraints were set on the truncated surface perpendicular to the X- and Y-axis, respectively.

3 Seismic response analysis method for intake tower

3.1 Mode-superposition response spectrum analysis

The mode decomposition response spectrum method was utilized for the dynamic analysis of the intake tower. The response spectrum theory refers to transforming a structure into a multi-degree-of-freedom system, and its seismic response can be simplified by decomposing the structure into a combination of a series of single-degree-of-freedom systems according to the mode shapes. The response spectrum is expressed by Eq. (1), as follows:

(1)

where {\beta }_{max} is the maximum value of standard design response spectrum, which is generally taken as 2.25 for intake tower; T_{\mathrm{g}} is the characteristic period, which is 0.45 s in this case.

The natural vibration characteristics of the tower were determined using the designed response spectrum to calculate the seismic response of the intake tower. Because the seismic energy of an earthquake is primarily concentrated in the frequency band below 20 Hz, the corresponding vibration frequency of the building structure response is low. Thus, using the response-spectrum method, reasonable results can be obtained by considering only a few vibration modes.

According to standard [12], the seismic effect of each mode is generally combined using the square root of the sum of squares method (SRSS), as shown in Eq. (1). However, if the ratio of the absolute frequency difference between two successive modes to a smaller frequency is less than 0.1, the seismic action effect is combined using the complete quadric combination method (CQC), as expressed by Eqs. (2) and (3), as follows:

(2)

with

(3){\rho }_{ij}=\frac{8\sqrt{{\zeta }_i{\zeta }_j}\left({\zeta }_i+{\gamma }_{\text{ω}}{\zeta }_j\right){\gamma }_{\text{ω}}^{3/2}}{{\left(1-{\gamma }_{\text{ω}}^2\right)}^2+4{\zeta }_i{\zeta }_j{\gamma }_{\text{ω}}\left(1+{\gamma }_{\text{ω}}^2\right)+4\left({\zeta }_i^2+{\zeta }_j^2\right){\gamma }_{\text{ω}}^2} ,

where S_{\mathrm{E}} is the effect of seismic actions; m is the number of mode shapes; S_i and S_j are the seismic action effects of modes i and j, respectively; {\rho }_{ij} is the correlation coefficient of modes i and j; {\zeta }_i and {\zeta }_j are the damping ratios of modes i and j, respectively; {\gamma }_{\text{ω}} is the angular frequency ratio; and {\gamma }_{\text{ω}} ={\omega }_j/{\omega }_i, with {\omega }_i and {\omega }_j representing the angular frequencies of modes i and j, respectively.

3.2 Load calculation and combination method

Because the project site of the intake tower is located in a high-intensity earthquake area, and the tower is an isolated high-rise building, its load is complex. Based on the standard for the seismic design of hydraulic structures [12] and specifications for the load design of hydraulic structures [31], the loads exerted on the intake tower during earthquakes include the self-weight, hydrostatic pressure, uplift pressure, hydrodynamic pressure, and earthquake acceleration.

The static water pressure acting vertically on the tower was applied at a gradient of 9.81 kN/m³, and the water pressure on the reservoir surface was considered zero. The uplift pressure of the tower was the total head of the corresponding reservoir water level multiplied by the gravity of the water, which was applied according to a rectangular distribution on the bottom surface of the intake tower. However, based on a related empirical formula [12,32,33], if the seismic effect imposed on the intake tower is calculated using the dynamic method, the hydrodynamic pressure inside and outside the tower can be considered by the additional mass acting on the inner and outer surfaces of the tower, which is expressed as follows:

(4)m_{\mathrm{w}}(h)={\varphi }_{\mathrm{m}}(h) {\rho }_{\mathrm{w}}{\eta }_{\mathrm{w}}A {\left(\frac{a}{2H_0}\right)}^{-0.2},

where m_{\mathrm{w}}(h) represents the added mass of hydrodynamic pressure per unit height at water depth of h, {\varphi }_{\mathrm{m}}(h) is the additional mass distribution coefficient related to water depth for hydrodynamic pressure on the outer tower surface and taken as 0.72 for hydrodynamic pressure inside the tower, {\rho }_{\mathrm{w}} is the mass density of water, {\eta }_{\mathrm{w}} is the shape coefficient, A is the envelope area of intersection between the average section of the tower and water, a is the average value of the maximum width of the upstream surface of the tower perpendicular to the direction of seismic action, and H₀ is the total water depth.

In addition, the intake tower was a non-blocking water building, and the site design seismic intensity was VIII. Therefore, the corresponding horizontal seismic peak acceleration of the bedrock was determined as 0.20g. According to the standard [12], because the intake tower is a reinforced concrete structure, its effect reduction factor of seismic impact should be taken as 0.35, when the dynamic method is utilized to calculate the seismic effect [34]. The displacement and stress of the intake tower during an earthquake were determined by the superposition of static and dynamic results.

4 Multi-objective decision making based on NSGA-II

The optimized design of anti-seismic engineering measures for intake towers is a multi-objective decision-making problem; that is, the most reasonable combination of anti-seismic engineering measures must be determined for relatively low project costs based on safety requirements. The NSGA-II was used for optimization in this study. The NSGA-II process includes fast nondominated sorting, crowding distance calculation, selection, and the same crossover and mutation operations as the traditional genetic algorithm [35–37]. The following section introduces the principles of the algorithm based on the calculation process.

4.1 Fast non-dominated sorting approach

First, the parent population P(t) is generated by combining different situations of seismic engineering measures. Then, P(t) is sorted into different non-dominated fronts. A non-dominated individual P\ast (t) of the population is only determined by the relationship expressed by Eq. (5):

(5)\begin{array}{c}\mathrm{\exists }{P}_i\left(t\right)\ne P\ast \left(t\right),\left(i=1,2,...,N\right)\in P\left(t\right):\\ \begin{array}{c}\mathrm{\forall }j\in \left\{1,2,...,m\right\}\left\{R_j\left[P_i\left(t\right)\right]⩽R_j\left[P\ast \left(t\right)\right]\right\},\end{array}\\ \begin{array}{c}\wedge \mathrm{\exists }k\in \left\{1,2,...,m\right\}\left\{R_k\left[P_i\left(t\right)\right]⩽R_k\left[P\ast \left(t\right)\right]\right\},\end{array}\end{array}

where N is the size of the parent population, m is the number of objective functions, and R_j() is the values of the jth objective function. P(t) can be sorted using Eq. (4) and all individuals belonging to the first non-dominated front can be found, and their ranks are set to 1. The remaining individuals in P(t), except those with rank 1, continue to be sorted using the same procedure. Individuals of the second non-dominated front were subsequently selected, and their ranks were set to 2. Similarly, the sorting process continues until all types are defined.

4.2 Crowding distance calculation

After sorting the population, individuals of the same rank are arranged in ascending order based on their objective values. The crowding distance is calculated using Eq. (6):

(6){\delta }_r(i )=\sum _{k=1}^m\frac{\left|R_k(i+1)-R_k(i-1)\right|}{R_k^{max}-R_k^{min}},i \in \{2 ,3,.. .,n-1\},

where r is the rank of the ith individual; n is the number of individuals with ranks of r; {\delta }_r(i ) represents the crowding distance of the ith individual; and R_k^{max} and R_k^{min} denote the maximum and minimum values of the kth objective function, respectively.

4.3 Selection operation

By comparing the rank and crowding distance of each individual, the tournament selection method is adopted for the filter of the population P(t). In detail, two individuals are selected randomly from the population every time. If they have different ranks, the one with the smaller rank is selected. If their ranks are the same, the individual with the larger crowding distance is selected. Based on this method, N individuals are chosen for crossover operation.

4.4 Crossover, mutation and elitism operations

The double-point crossover operation is performed for two random individuals using cross-probability p_{\mathrm{c}}, and the mutation operation is performed for each allele of each individual using mutation probability p_{\mathrm{m}}. After the crossover and mutation, an offspring population is generated. Then, an elitism operation is performed. First, the parent and offspring populations are combined to create a temporary population with the size of 2N, which is sorted again using the same procedure. Subsequently, the best N individuals are filtered using the same criterion as in Subsection 4.3 to be the next parent population.

A flowchart of the NSGA-II is shown in Fig.3.

4.5 Multi-objective decision making method

The optimal solution obtained by the NSGA-II is a Pareto-optimal set instead of a unique solution [24]. Therefore, the technique for order performance by similarity to the ideal solution (TOPSIS) is utilized to select the best unbiased optimal solution. First, it is assumed that there is a set of Pareto-optimal solutions of size l, that is, {\mathit{P}}_1,{\mathit{P}}_2,.. .,{\mathit{P}}_l. On this basis, the decision matrix for the solution set can be constructed as shown in Eq. (7):

(7)\mathit{R}=\left[\begin{array}{cccc}{R}_1\left({\mathit{P}}_1\right)& R_2\left({\mathit{P}}_1\right)& \dots & R_m\left({\mathit{P}}_1\right)\\ R_1\left({\mathit{P}}_2\right)& R_2\left({\mathit{P}}_2\right)& \dots & R_m\left({\mathit{P}}_2\right)\\ ⋮& ⋮& & ⋮\\ R_1\left({\mathit{P}}_l\right)& R_2\left({\mathit{P}}_l\right)& \dots & R_m\left({\mathit{P}}_l\right)\end{array}\right].

Subsequently, Eq. (8) is adopted for normalization to convert the objective function values into dimensionless values, and the decision matrix R is transformed into {\mathit{R}}^{\prime }. Subsequently, the Euclidean distances between all solutions and the coordinate origin are calculated using Eq. (9) for comparison.

(8)\begin{array}{rl}{\mathit{R}}_i^{\prime }\left({\mathit{P}}_j\right)=\left[R_i\left({\mathit{P}}_j\right)-min{R}_i\right]/& \left(max{R}_i-min{R}_i\right),\\ & \overline{i}=1,2,...,\lambda ,\end{array}

(9){\delta }_j=\sqrt{\sum _{i=1}^m{R}_i^{\prime }{\left({\mathit{P}}_j\right)}^2}, j=1,2 ,..., l.

The smaller the value of {\delta }_j, the better the Pareto-optimal solution. It is rational to consider the solution set corresponding to the minimum {\delta }_j as the best unbiased optimal solution, which is also the most reasonable combination of antiseismic engineering measures.

5 Dynamic analysis of intake tower under design condition

First, the design scheme of the intake tower was considered for the finite element analysis, and the calculation results were referred to for comparison, based on which, the following seismic design optimization was performed.

5.1 Designed mechanical parameters of materials

Because the response spectrum theory can only calculate the linear response of the structure and most of the intake tower materials are in the linear deformation stage under normal operation, all materials are considered as linear elastic models. The tower foundation surface is set in the middle and lower parts of the slightly and weakly weathered rock, and the mechanical parameters of the surrounding rocks at different weathered degrees are listed in Tab.1.

In the static calculation, the density of the surrounding rocks was determined, as listed in Tab.1. In the dynamic calculation, the foundation massless method was adopted; that is, the density of the foundation material was adjusted to 1 × 10⁻⁶ kg/m³, and the concrete density of the intake tower and backfill was based on the actual values in Tab.1. In contrast, according to relevant regulations, under the action of dynamic load, the dynamic compressive strength of concrete with the grades of C20 and C25 are 22.2 and 26.9 MPa, respectively [29]. The dynamic tensile strength was considered to be 10% of the dynamic compressive strength. In addition, the standard value of the dynamic elastic modulus of concrete can be increased by 50% compared to the static elastic modulus [12]. The mechanical parameters of concrete are listed in Tab.2.

5.2 Analysis of natural vibration characteristics

Before analyzing the response spectrum, the natural vibration of the intake tower was analyzed to obtain the natural frequency and corresponding vibration mode. Simultaneously, considering that the added mass of hydrodynamic pressure affects the natural vibration characteristics of the intake tower, two conditions, including empty and full reservoirs, were calculated. Because earthquake energy is mainly concentrated in lower-frequency bands, reasonable results can be obtained by combining the first 20 modes. The first five natural frequencies and vibration characteristics of the intake tower are summarized in Tab.3, and the first five vibration diagrams and corresponding frequencies for the full reservoir are shown in Fig.4.

Subsequently, two load cases were considered for the superposition of the static and dynamic results, as listed in Tab.4. The results were utilized to analyze tower displacement and stress.

5.3 Displacement analysis of the tower

First, the static state of the intake tower under the action of self-weight and water pressure was calculated, and a dynamic calculation under seismic action was performed according to the mode decomposition response spectrum method. The solution technology for the finite element static analysis is based on the full Newton method. In addition, the convergence options in the static and dynamic analyses were set to default values, and the total solution time was approximately 31 s, indicating that the calculation efficiency was quite high.

Using the superposition method of the static and dynamic results, the contour maps of the tower displacement in the two load cases are illustrated in Fig.5 and Fig.6, respectively. As detailed in the figures, the maximum displacement along the flow direction is −4.628 mm, which appears at the top of the tower; the maximum displacement perpendicular to the flow direction is 8.091 mm, which appears on the upstream side of the top plate; and the maximum settlement is 7.135 mm at the top of the tower. Therefore, the deformation of the intake tower was small and did not cause any damage.

5.4 Stress analysis of the tower

Similarly, the contour maps of the tower stress for the two load cases are shown in Fig.7 and Fig.8. As detailed in the figures, the maximum tensile stress, maximum compressive stress, and vertical normal stress are 2.659, −5.235, and −4.925 MPa, respectively, all of which appear on the joint between the right bank of the intake tower and concrete backfill. Compared to the dynamic stress standard values, the stress of the intake tower did not exceed the allowable value in the design case of an earthquake. The joint between the right bank of the tower and backfill is the most crucial position for stress control.

The calculated displacement and stress are summarized in Tab.5.

6 Optimization of anti-seismic measures for the intake tower

6.1 Sensitivity analysis of anti-seismic measures

Because the intake tower is fixed with concrete backfill, the displacement is small during an earthquake. In addition, the tower is located on slightly or weakly weathered rocks; therefore, its foundation can bear large nonuniform stress. Consequently, there is generally no stability problem that must be emphasized in the anti-seismic design of intake towers on rock foundations. In summary, the intake tower was not damaged by deformation or instability during vibrations. However, the maximum tensile stress of the tower (2.659 MPa) is close to the standard dynamic tension (2.69 MPa) in the current design scheme, and there is a certain risk of tensile damage; therefore, the design and optimization of anti-seismic measures for the intake tower should be controlled by the maximum tensile stress at the crucial position.

First, four anti-seismic measures were considered, and four values for each factor were proposed for sensitivity analysis. Compared to the original design scheme, only one factor was changed each time to study its influence on the maximum tensile stress (2.659 MPa) of the intake tower. The values of each factor in the single-factor sensitivity analysis are presented in Tab.6.

On the one hand, the elastic modulus of backfill concrete and bottom consolidation grouting area in the finite element model shown in Fig.2 was changed for sensitivity analysis, respectively. The elastic modulus of the backfill concrete was determined by its grade, which was calculated using Eq. (10). By adjusting the grade and elastic modulus of the backfill concrete, its effect on the maximum tensile stress of the tower was analyzed. The relationship curves between the grade of the backfill concrete and maximum tensile stress multiple of the design scheme are shown in Fig.9(a). The relationship curves were obtained using polynomial fitting.

(10)E_{\mathrm{c}}=\frac{{10}^5}{2.2+{\displaystyle \frac{34.7}{f_{\mathrm{c}\mathrm{u}}}}} ,

where E_{\mathrm{c}} is the elastic modulus in MPa and f_{\mathrm{c}\mathrm{u}} is the compressive strength of concrete.

Similarly, the elastic modulus of the bottom consolidation grouting is increased by 0.3, 0.6, 0.9, and 1.2 times respectively based on the foundation elastic modulus (3.45 GPa), and the relation curves and corresponding fitting polynomial between the elastic modulus of the bottom consolidation grouting and the maximum tensile stress multiple of the design scheme are shown in Fig.9(b).

On the other hand, the finite element model was adjusted for the analysis of the influence of the backfill height at the tower back and bottom consolidation grouting depth on the maximum tensile stress. The sensitivity analysis schemes for the concrete backfill heights are listed in Tab.7. Through numerical calculations, the relationship between the backfill height and maximum tensile stress multiple and relationship between the bottom consolidation grouting depth and maximum tensile stress multiple are summarized and shown in Fig.9. Similarly, relationship curves were obtained by fitting.

The calculation results show that the relationship between each anti-seismic measure and the maximum tensile stress of the intake tower can be determined by fitting quadratic or cubic polynomials. According to the relationship curves, the tensile stress of the tower can be reduced by strengthening various anti-seismic measures; however, the reduction efficiency of strengthening anti-seismic measure on the maximum tensile stress gradually decreases.

6.2 Establishment of multi-objective function

In the calculation within the range of each factor in the above sensitivity analysis, the maximum tensile stress of the intake tower appears in the joint between the right bank of the tower and backfill and does not change because of the change in the values of the factors. Therefore, optimizing the tensile stress at this crucial position is reasonable.

By combining the fitting polynomial of the relationship between each factor and maximum tensile stress multiple, the fitting formula for the maximum tensile stress of the intake tower related to the four factors was obtained, as shown in Eq. (11). In addition, the maximum tensile stress should not exceed the dynamic compressive standard of C25 concrete; therefore, a restrictive condition was set, as expressed in Eq. (12):

(11)\begin{array}{rl}\delta & \left({\mu }_1{\mu }_2,{\mu }_3,{\mu }_4\right)=2.659k_1{k}_2{k}_3{k}_4=2.659\\ & \times (-0.00000752{\mu }_1^3+0.000676946{\mu }_1^2\\ & -0.02072{\mu }_1+1.20384)(0.00137{\mu }_2^2-0.02207{\mu }_2\\ & +1.08599)(-0.0000859591{\mu }_3^3+0.00417{\mu }_3^2\\ & -0.11229{\mu }_3+1.78857)(0.000041874{\mu }_4^2\\ & -0.00101{\mu }_4+1.00402),\end{array}

(12)

where f_{\mathrm{t}} is the dynamic compressive standard value and f_{\mathrm{t}} is 2.69 MPa in this case.

However, the variable project cost will generally increase to strengthen various anti-seismic measures. Therefore, the cost of anti-seismic measures was analyzed and calculated as follows: First, according to the general cost of similar projects, the unit price for each component of the anti-seismic engineering measures is summarized and listed in Tab.8.

Second, the engineering quantities for each part of the anti-seismic measures were estimated. In this study, the considered variable costs mainly included the labor and material costs for excavation and concrete backfilling of the mountain at the back of the tower, the cost of anchor arms for connecting the concrete backfill with the intake tower or mountain, and the cost of consolidation grouting for the tower foundation. The concrete backfill at the back of the tower was regarded as a triangular prism; hence, its volume and contact areas with both sides at different backfill heights could be calculated. The density of the anchor arms was uniformly taken as 3.6 m⁻². The volume of cement mortar used for foundation consolidation grouting was estimated using the weighted average method of the volume-to-foundation elastic modulus within the grouting range, and it was assumed that the elastic modulus of the mortar after solidification was 45 GPa. Thus, the calculation formula for the variable engineering cost is expressed as Eq. (13):

(13)\begin{array}{rl}\varpi \left({\mu }_1,{\mu }_2,{\mu }_3,{\mu }_4\right)=& 1.575{\mu }_3^2\left[\left(4{\mu }_1+180\right)+100.0\right]\\ & +160\times 3.6\times 21.46{\mu }_3\\ & +{\displaystyle \frac{\left({\mu }_2-3.45\right)\left(256.9{\mu }_4\right)}{45}}\times 3.15\times 350.00\end{array}.

It is worth noting that the price calculated in this section was only the variable cost of the engineering measures taken to improve the seismic performance of the intake tower, instead of the total engineering cost. The constraint conditions for each variable were determined according to a reasonable range of realistic construction measures, as listed in Tab.6. In summary, the multiobjective function can be expressed by Eq. (14):

(14)

6.3 Optimization of anti-seismic measures based on multi-objective decision making

By inputting any set of the anti-seismic measure parameters, the corresponding maximum tensile stress of the intake tower and variable engineering cost can be calculated via the formulas shown by Eqs. (10) and (12). The NSGA-II was applied for the iterative optimization of the objective functions shown in Eq. (13).

Through multiple trial calculations, the parameters of the NSGA-II are set as follows: the size of parent population N is 50; maximum iteration is 200; crossover p_{\mathrm{c}} and mutation probabilities p_{\mathrm{m}} are 0.8 and 0.05, respectively. After the iterative operation, the top five solutions in the Pareto-optimal solution set were sorted by calculating their Euclidean distances, and the position of the first solution set was searched using the TOPSIS method proposed in Subsection 4.5, as summarized in Tab.9 and Fig.10. Adjusting the initial value and number of iterations several times showed that the best solution set always approached the same position after 100 iterations, indicating that the global optimum can be obtained using this method, and the result is repeatable.

The calculation results show that the optimization of anti-seismic engineering measures for the intake tower is a multi-objective Pareto optimal solution problem; that is, the two objective functions cannot simultaneously meet the minimum objective. The first solution set in Tab.9 was selected as the best unbiased optional solution, which was considered the optimal combination scheme for anti-seismic measures.

A comparison of the optimal result and original design scheme shows that the two parameters of the backfill concrete grade and the elastic modulus of the foundation consolidation grouting area are relatively close. In contrast, the two parameters of the consolidation grouting depth and backfill height of the tower back obtained by the NSGA-II were significantly different from the initial design.

Recommendations for the improvement of the antiseismic measures were proposed. First, the grade and height of the backfill concrete were appropriately increased, and the corresponding relative height of the backfill at the most suitable height was 0.62. In addition, to ensure the elastic modulus of the foundation grouting area, the consolidation depth can be appropriately reduced to reduce costs. Finally, the steel ratio at the crucial position (the joint between the right bank of the tower and backfill) can be increased to prevent local tensile damage and cracking.

6.4 Implications

This study analyzes the dynamic response of an intake tower under an earthquake based on the mode decomposition response spectrum method and optimize its anti-seismic measures using the NSGA-II. The adopted methods and relative conclusions of this research can provide experience for the design of similar projects. However, according to the design request of “no damage in small earthquake, no collapsing in strong earthquake,” the intake tower concrete can be theoretically allowed to enter the plastic phase in an actual operation state. However, because the response spectrum theory can only consider the linear response, obtaining the real displacement and stress values of the structure is difficult; thus, the strength of concrete after entering the plastic phase is ignored, which can easily lead to wastage in the design of seismic measures. In addition, the response spectrum theory ignores the interaction between the structure and foundation during ground motion. Considering these shortcomings, research on dynamic numerical methods must be conducted in the future.

7 Conclusions

This study discusses the dynamic response and anti-seismic measures of an intake tower based on the response spectrum theory and NSGA-II, and the main conclusions are as follows.

1) According to the dynamic FEM calculation results, the structure of the intake tower is safe for earthquakes at their initial states. The maximum tensile stress generally occurs at the joint between the tower and side of the higher backfill, which can be regarded as the most crucial position that affects structural safety. Antiseismic measures, including heightening the concrete backfill and foundation consolidation grouting, can reduce the maximum tensile stress; however, the reduction efficiency gradually weakens.

2) It is suggested that the grade of backfill concrete and depth of foundation consolidation grouting can be appropriately increased and reduced, respectively. The most reasonable relative height of the concrete backfill at the back of the tower was 0.62, indicating that the backfill height must be properly increased to ensure the safety of the project.

3) For intake towers on a rock foundation, on the premise of ensuring stability and deformation safety, the Pareto-optimal problem can be constructed with the goals of controlling the maximum tensile stress at the crucial position and reducing the cost of seismic engineering measures. A multi-objective optimization algorithm should be used to find the optimal solution. The objective functions and constraint conditions established in the optimization are reasonable and can be applied to the design optimization of seismic measures for similar projects.

References

[1] Zhang H Y, Li T C, Li Z K. Modeling in SolidWorks and analysis of temperature and thermal stress during construction of intake tower. Water Science and Engineering, 2009, 2(1): 95–102

[2] GongC LLiu HZhangJ. Study on dynamic properties of the intake tower with finite element method. Applied Mechanics & Materials, 2014, 501–504: 1888–1891

[3] Wang Y, Lin Z T, Song C, Dong B Y. Seismic analysis of a water release integrated structure part I: Response spectrum analysis of intake tower. Applied Mechanics and Materials, 2014, 470: 938–941

[4] YanY ZXiong C XWenX HLiW H. Structure seismic analysis on intake tower of spillway tunnel. Advanced Materials Research, 2015, 1065–1069: 1427–1432

[5] Alembagheri M. Earthquake response of solitary slender freestanding intake towers. Soil Dynamics and Earthquake Engineering, 2016, 90: 1–14

[6] Zhang H Y, Zhang L J. Tuned mass damper system of high-rise intake towers optimized by improved harmony search algorithm. Engineering Structures, 2017, 138: 270–282

[7] Bartoli G, Betti M, Galano L, Zini G. Numerical insights on the seismic risk of confined masonry towers. Engineering Structures, 2019, 180: 713–727

[8] Zou X K, Chan C M. An optimal resizing technique for seismic drift design of concrete buildings subjected to response spectrum and time history loadings. Computers & Structures, 2005, 83(19): 1689–1704

[9] Ftima M, Léger P. Seismic stability of cracked concrete dams using rigid block models. Computers & Structures, 2006, 84(28): 1802–1814

[10] Gorai S, Mait D. Seismic response of concrete gravity dams under near field and far field ground motions. Engineering Structures, 2019, 196: 109292

[11] Aldemir A. Prediction equation for the fundamental vibration period of concrete gravity dams with impounded water. Earthquake Spectra, 2021, 37(3): 1710–1725

[12] GB51247-2018. Standard for Seismic Design of Hydraulic Structures. Beijing: China Planning Press, 2018 (in Chinese)

[13] ZhangYLi S YXiaKGuoJ JHeG J LiM. Seismic study on backfill height of tower-back for high-rise intake tower structure. Water Resources and Hydropower Engineering, 2018, 49(11): 62−67 (in Chinese)

[14] YangGLi S YLiLXiaoYZhangZ X YangY. Effect of backfill height on seismic performance of high intake tower. Journal of Water Resources & Water Engineering, 2019, 30: 6 (in Chinese)

[15] ZhangW JLu W BChenMYanPZhouC B. Analysis of consolidation grouting effect of rock mass based on comparison of wave velocity before and after grouting. Chinese Journal of Rock Mechanics & Engineering, 2012, 31(3): 469–478 (in Chinese)

[16] Ang H, Lee J. Cost optimal design of R/C buildings. Reliability Engineering & System Safety, 2001, 73(3): 233–238

[17] Esteva L, Díaz-López O, García-Pérez J. Life-cycle optimization in the establishment of performance-acceptance parameters for seismic design. Structural Safety, 2002, 24: 187–204

[18] Zheng P, Hobbs B, Koonce J. Optimizing multiple dam removals under multiple objectives: Linking tributary habitat and the Lake Erie ecosystem. Water Resources Research, 2009, 45(12): 1699–1702

[19] Lozano M, Ramos J, Serra L. Cost optimization of the design of CHCP (combined heat, cooling and power) systems under legal constraints. Energy, 2010, 35(2): 794–805

[20] ZhongD HLi ZWuB PHuWLüP. Time-quality-cost tradeoff optimization of rockfill dam construction based on pareto solution. Journal of Tianjin University (Science and Technology), 2016, 49(10): 1001−1007

[21] BuK ZZhao YZhengX C. Optimization design for foundation pit above metro tunnel based on NSGA2 genetic algorithm. Journal of Railway Science and Engineering, 2021, 18(2): 459−467 (in Chinese)

[22] Zhang W B, Shi D D, Shen Z Z, Wang X H, Gan L, Shao W, Tang P, Zhang H W, Yu S Y. Effect of calcium leaching on the fracture properties of concrete. Construction & Building Materials, 2023, 365: 130018

[23] Zhang W B, Shi D D, Shen Z Z, Shao W, Gan L, Yuan Y, Tang P, Zhao S, Chen Y S. Reduction of the calcium leaching effect on the physical and mechanical properties of concrete by adding chopped basalt fibers. Construction & Building Materials, 2023, 365: 130080

[24] Chao Z M, Dang Y B, Pan Y, Wang F Y, Wang M, Zhang J, Yang C X. Prediction of the shale gas permeability: A data mining approach. Geomechanics for Energy and the Environment, 2023, 33: 100435

[25] Chang L, Chang F. Multi-objective evolutionary algorithm for operating parallel reservoir system. Journal of Hydrology, 2009, 377(1−2): 12–20

[26] Zare O, Saghafian B, Shamsai A, Nazif S. Multi-objective optimization using evolutionary algorithms for qualitative and quantitative control of urban runoff. Hydrology and Earth System Sciences Discussions, 2012, 9: 777–817

[27] Wang S P, Zhao D M, Yuan J Z, Li H J, Gao Y. Application of NSGA-II algorithm for fault diagnosis in power system. Electric Power Systems Research, 2019, 175: 105893

[28] Ghasemi H, Rafiee R, Zhuang X, Muthu J, Rabczuk T. Uncertainties propagation in metamodel-based probabilistic optimization of CNT/polymer composite structure using stochastic multi-scale modeling. Computational Materials Science, 2014, 85: 295–305

[29] Ghasemi H, Brighenti R, Zhuang X, Muthu J, Rabczuk T. Optimal fiber content and distribution in fiber-reinforced solids using a reliability and NURBS based sequential optimization approach. Structural and Multidisciplinary Optimization, 2015, 51(1): 99–112

[30] GB18306-2015. Seismic Ground Motion Parameters Zonation Map of China. Beijing: China Earthquake Administration, 2015 (in Chinese)

[31] SL744-2016. Specification for Load Design of Hydraulic Structures. Beijing: China Water Power Press, 2016 (in Chinese)

[32] Goyal A, Chopra A K. Hydrodynamic and foundation interaction effects in dynamics of intake towers: Earthquake responses. Journal of Structural Engineering, 1989, 115(6): 1386–1395

[33] Goyal A, Chopra A K. Earthquake analysis of intake-outlet towers including tower-water-foundation-soil interaction. Earthquake Engineering & Structural Dynamics, 1989, 18(3): 325–344

[34] Miranda E, Bertero V V. Evaluation of strength reduction factors for earthquake-resistant design. Earthquake Spectra, 1994, 10(2): 357–379

[35] Deb K. Multi-objective genetic algorithms: Problem difficulties and construction of test problems. Evolutionary Computation, 1999, 7(3): 205–230

[36] Ye C J, Huang M X. Multi-objective optimal power flow considering transient stability based on parallel NSGA-II. IEEE Transactions on Power Systems, 2015, 30(2): 857–866

[37] Liu D, Huang Q, Yang Y Y, Liu D F, Wei X T. Bi-objective algorithm based on NSGA-II framework to optimize reservoirs operation. Journal of Hydrology, 2020, 585: 124830

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