Over time, the uneven settlements of the structure and foundation are prominent in constructing ship lock heads on soft soil. These deformations endanger the safety of ship lock heads during construction. This research aimed to establish the ship lock head’s structural optimization procedure on soft soil, considering the time-varying effects of the structure and foundation. By comprehensively considering the linear viscoelastic creep of concrete and the elastoplastic consolidation characteristic of soft soil, a perfect time-dependent analysis method for the lock head on soft soil was proposed. Furthermore, a hybrid particle swarm optimization, enhanced whale optimization, and differential evolution (PSO-EWOA-DE) algorithm was proposed to optimize thirty-four design variables of a lock head. With the minimal volume of the lock head as the optimization objective, the finite element model was established. In the optimization process, three types of constraints were evaluated. The result showed that the optimized design could reduce 10.45% of structure volume. Through comparing and analysing the maximum principle stresses and vertical displacements of the lock head before and after optimization, some conclusions were drawn. The optimization procedure proposed in this paper provides a new perspective for the structural optimization of hydraulic structures on soft soil.

As one of the most critical navigation structures, ship locks use hydraulic power to lift ships through dams built on natural or canalized rivers [

Many algorithms have been used in structural engineering optimization tasks over the past few decades, from gradient-based algorithms to nongradient probabilistic-based algorithms [

The creep phenomenon was discovered by Hatt in 1907 [

Consolidation is the time-settlement behaviour of clay. The clay instantaneously settles under the load’s action without drainage, and then the excess pore pressure gradually dissipates [

In this article, a structural optimization method for ship lock heads on soft soil was established considering the time-varying effects of the structure and foundation. Meanwhile, by comprehensively considering the linear viscoelastic creep of concrete and the elastoplastic consolidation characteristic of soft soil, a perfect time-dependent analysis method for the lock head on soft soil was proposed. With the minimal volume of the lock head as the optimization objective, a hybrid particle swarm optimization, enhanced whale optimization, and differential evolution (PSO-EWOA-DE) algorithm was applied to obtain the optimal shape. A comparison of the results obtained by the PSO-EWOA-DE algorithm with those of PSO, WOA, DE, EWOA, and PSO-DE methods revealed that the proposed algorithm was more effective in the structural optimization of a ship lock head and had a faster convergence rate. The remainder of this paper is structured as follows. Section

For mass concrete structures, the compliance function is expressed as follows [

The creep compliance function can be written in the following Dirichlet series form [

In addition,

For viscoelastic materials, stress increment in 3D can be calculated by the following equation [

In these equations,

According to the classical elastoplastic theory of geotechnical engineering, the stress-strain relationship is shown as follows [

In addition,

For an isotropic hardening material, the yield function is given as follows [

According to the flow rule, the plastic strain increment is defined as the following equation [

In the MCC model, the volumetric modulus

In these equations,

The yield function of the MCC model is given as follows [

In the Cartesian coordinate system, the mean effective stress

According to the consistency condition, the derivatives of the yield function

Based on equations (

According to equation (

The thermal strain increment is calculated by equation (

For the saturated continuous soil, the equilibrium equation can be discretized into the following form [

Based on the theory of seepage, the pore water pressure

Equation (

In addition,

For a porous elastic medium, the continuity equation can be expressed as follows:

Considering the fluid-skeleton compatibility, the fluid flow equation is converted into the following equation [

In a time increment

In addition,

Parametric modelling is the basis for structural optimization design. The ship lock head is a symmetrical structure along the water flow direction. Therefore, half of the geometric model was established. As shown in Figure

Schematics of the spatial (a) ship lock head and (b) lock head-soft foundation-backfilled soil system.

The horizontal and vertical views of the bottom plate are shown in Figure

The basic dimensions of the bottom plate. (a) Plane figure of the bottom plate. (b) Sectional drawing of the bottom plate.

The second-stage concrete section.

The corridor layer section.

Designers save concrete materials by setting up empty boxes in the lock head. According to this method, 34 design variables (_{1}–_{15}, _{1}–_{12}, and _{1}–_{7}) were distinguished in the

Design variables. (a) Horizontal view. (b) Elevation view.

The purpose of structural optimization is to find the optimal design that meets the stability and financial requirements. In this work, the lock head volume was taken as the objective function. The optimization problem can be expressed as follows:_{1}, …, _{15}, _{1}, …, _{12}, _{1}, …, _{7}) is the updating design variables,

According to the previous research [

The PSO is a swarm intelligence-based algorithm proposed by Kennedy and Eberhart [

In addition,^{th} design variable of particle ^{th} iteration, ^{th} design variable, ^{th} iteration,

The WOA mimics the bubble-net hunting behaviour of humpback whales. The unique hunting model is shown in Figure

Unique bubble-net spiral hunting behaviour.

As shown in Figure

Exploitation phase in WOA. (a) Shrinking encircling mechanism. (b) Bubble-net spiral attacking mechanism.

In this equation,

To expand the searching area, the WOA updates each design variable’s position using a randomly selected candidate when

Flowcharts of the (a) WOA algorithm and (b) EWOA algorithm.

The WOA has an effective global search capability. In order to enhance the WOA in terms of reliability, EWOA was proposed. The EWOA maintains the simplicity of the WOA. In the exploitation phase, to maintain a balance between the diversity and intensification of search results, equation (

In the exploration phase, each candidate randomly changes the value of a design variable with a probability

For candidate ^{th} design variable, respectively. The flowchart of EWOA is presented in Figure

The DE is a simple population-based evolutionary algorithm proposed by Storn and Price [

The PSO-EWOA-DE algorithm is a population-based evolutionary algorithm. Initially, the PSO-EWOA-DE algorithm explores the search space globally by using PSO and EWOA and then utilizes DE to fine-tune the selected solution locally. This process is repeated until the stopping criterion is met. This algorithm can ensure diversity and maintain convergence accuracy. The PSO-EWOA-DE algorithm steps are summarized as follows:

Set the population size

Calculate the fitness value for each candidate.

Update the parameters

Generate the trial population

Generate the trial population

Select the trial population

where ^{th} candidate of the trial population, ^{th} candidate of ^{th} candidate of

Generate a new population for the next generation according to the DE process shown in Section

Terminate the PSO-EWOA-DE algorithm by satisfying the termination condition. If so, go to Step 9; otherwise, go to Step 2.

Output the optimal solution obtained in Step 6.

The flowchart of the PSO-EWOA-DE algorithm is presented in Figure

Flowcharts of the (a) PSO-EWOA-DE algorithm and (b) structural optimization procedure.

This section presents a structural optimization procedure of the ship lock head structure based on the PSO-EWOA-DE algorithm. The optimization procedure was implemented by using a self-developed Python script, which was worked in Abaqus 6.14. The heat of hydration was calculated by using a user subroutine (HETVAL). Furthermore, a user subroutine (UMAT) was developed to implement the MCC model and calculate the thermal creep stress.

The structural optimization process of a ship lock head on soft soil, considering the time-varying effect of the structure and foundation, can be summarized as follows:

The population size

The FEM model for a ship lock head was generated. Furthermore, the initial temperature, convection conditions, construction steps, and the heat of hydration were applied to calculate the temperature field.

The element type was automatically modified to calculate the initial geostress field.

The pore water pressure boundaries, displacement boundaries, and design loads were applied to calculate the stress field by calling Abaqus.

Three types of constraints shown in

A new population was generated using the PSO-EWOA-DE algorithm shown in

The stopping criterion was checked, and Steps 2 to 6 were repeated when

The optimum solution with the highest fitness obtained in Step 6 was output.

The flowchart of the structural optimization procedure is shown in Figure

The structural optimization of the proposed ship lock head during the construction period considered only the static load. The design loads of the lock head-soft foundation-backfilled soil system were gravity loads, active earth pressure, live loads, and temperature. A lock head with a width of 53.8 m and a height of 12.2 m was presented here. Table

Range of design variables.

Design variable | Lower limits (m) | Upper limits (m) | Initial values (m) | Optimum values (m) |
---|---|---|---|---|

_{1} | 3.29 | 4.59 | 4.59 | 4.3 |

_{2} | 0.69 | 1.29 | 1.29 | 0.69 |

_{3} | 0.06 | 0.24 | 0.24 | 0.102 |

_{4} | 3.71 | 3.91 | 3.71 | 3.91 |

_{5} | 0.06 | 0.71 | 0.71 | 0.506 |

_{6} | 0.06 | 0.69 | 0.69 | 0.6 |

_{7} | −1.31 | −0.51 | −0.51 | −0.78 |

_{8} | 17.92 | 18.19 | 17.92 | 18.139 |

_{9} | 0.165 | 0.19 | 0.165 | 0.19 |

_{10} | 3.75 | 3.89 | 3.75 | 3.89 |

_{11} | 3.79 | 3.9 | 3.79 | 3.88 |

_{12} | 22.9 | 23.39 | 22.9 | 23.39 |

_{13} | 0.06 | 0.69 | 0.69 | 0.69 |

_{14} | 23.29 | 23.49 | 23.29 | 23.30 |

_{15} | 0.06 | 0.39 | 0.39 | 0.37 |

_{1} | 0.06 | 0.69 | 0.69 | 0.456 |

_{2} | 0.06 | 0.59 | 0.59 | 0.201 |

_{3} | 0.49 | 0.89 | 0.89 | 0.506 |

_{4} | 0.06 | 1.81 | 1.81 | 0.348 |

_{5} | 0.11 | 1.16 | 1.16 | 0.152 |

_{6} | 11.59 | 11.9 | 11.9 | 11.6 |

_{7} | 0.06 | 1.51 | 1.51 | 0.605 |

_{8} | 0.06 | 0.69 | 0.69 | 0.62 |

_{9} | 0.06 | 0.61 | 0.61 | 0.45 |

_{10} | 0.39 | 0.49 | 0.49 | 0.42 |

_{11} | 0.06 | 0.71 | 0.71 | 0.6 |

_{12} | 0.06 | 1.89 | 1.89 | 1.15 |

_{1} | 0.06 | 1.60 | 1.60 | 0.3 |

_{2} | 0.06 | 0.29 | 0.29 | 0.06 |

_{3} | 0.06 | 0.23 | 0.23 | 0.06 |

_{4} | 0.43 | 1.02 | 1.02 | 0.49 |

_{5} | 0.06 | 0.52 | 0.52 | 0.3 |

_{6} | 0.06 | 0.31 | 0.31 | 0.07 |

_{7} | 0.06 | 0.57 | 0.57 | 0.06 |

The lock head density is 2400 kg/m^{3}, and Poisson’s ratio is 0.167. According to the previous research [

Table

Main material properties of the soil foundation and backfilled soil.

Definition | Soft foundation | Backfilled soil |
---|---|---|

Mass density (kg/m^{3}) | 1900 | 1900 |

Poisson’s ratio | 0.3 | 0.3 |

Initial void ratio | 0.837 | 0.837 |

Virgin compression index | 0.0853 | 0.082 |

Unloading-reloading index | 0.00714 | 0.00697 |

Permeability coefficient | 0.776 | 0.003 |

Critical-state parameter | 0.933 | 1.502 |

Preconsolidation pressure (KPa) | 450.587 | 191.361 |

The entire construction process of the ship lock head was 386 days. As shown in Table

Ship lock construction process.

Construction sequence | Construction process | Construction duration (day) | Construction intermissions (day) |
---|---|---|---|

C1 | Pouring bottom plate | 2 | 8 |

C2 | Removing bottom plate formwork | 1 | 35 |

C3 | Backfilling the first layer of soil | 20 | 96 |

C4 | Pouring the corridor layer | 1 | 10 |

C5 | Removing corridor layer formwork | 1 | 25 |

C6 | Backfilling the second layer of soil | 19 | 26 |

C7 | Pouring the empty-box layer | 1 | 9 |

C8 | Removing empty-box layer formwork | 1 | 53 |

C9 | Pouring the second-stage concrete | 1 | 8 |

C10 | Removing the second-stage concrete formwork | 1 | 20 |

C11 | Backfilling the third layer of soil | 18 | 30 |

In this section, the previously described procedure was used to optimize the ship lock head. The optimization work met the requirements of China. For 60 runs of PSO [

Evolutionary processes of PSO, WOA, and EWOA algorithms.

Table

Results of optimization.

Volume (m^{3}) | PSO | WOA | DE | EWOA | PSO-DE | PSO-EWOA-DE |
---|---|---|---|---|---|---|

Initial design | 3338.04 | 3338.04 | 3338.04 | 3338.04 | 3338.04 | 3338.04 |

Optimal design | 3031.81 | 3019.49 | 3011.39 | 2994.07 | 2992.55 | 2989.30 |

Decreasing ratio | 9.17% | 9.54% | 9.79% | 10.30% | 10.35% | 10.45% |

The antisliding safety factors and the anti-overturning safety factors for the optimal results of the PSO, WOA, DE, EWOA, PSO-DE, and PSO-EWOA-DE are summarized in Figure

Comparisons of antisliding safety factors and anti-overturning safety factors.

Figure ^{3}. When the population sizes were 10, 20, and 30, the population sizes were too small to find the optimum solution. When the population sizes were 40 and 50, the PSO-EWOA-DE converged at the 13^{th} and 11^{th} iteration, respectively. The number of numerical simulations of the latter was 30 runs more than that of the former. As shown in Figure ^{th} and 18^{th} iteration, respectively. The number of numerical simulations of the latter was 140 runs more than that of the former. It could be concluded that population size 40 was more suitable for optimizing the lock head.

Effects of population size

Figure

Locations of four feature points.

It can be found from Figure

Comparisons of maximum (max.) principle stress before and after optimization at (a) point

Figure

The selected feature points’ time-history curves of the vertical displacements before and after optimization are shown in Figure

Comparisons of vertical displacements before and after optimization at (a) point

Figure

The structural optimization procedure for ship lock heads on soft soil, considering the time-varying effects of the structure and foundation, was established in this paper. By comprehensively considering the linear viscoelastic creep of concrete and the elastoplastic consolidation characteristic of soft soil, a perfect time-dependent analysis method for the lock head on soft soil was proposed. The MCC model was applied in the consolidation calculation of the soft foundation. To obtain the minimum volume, the PSO-EWOA-DE algorithm was proposed to optimize thirty-four design variables of a lock head. Among all the compared algorithms, the PSO-EWOA-DE algorithm showed promising results and had a faster convergence rate. The following conclusions are drawn by analysing the maximum principle stresses and vertical displacements of the lock head:

The temperature load and the backfilling soil’s gravity are the main reasons for increasing the ship lock head’s tensile stress on the soft foundation. The lock head should be protected against cracking when pouring the concrete and backfilling the soil.

The lock head settled instantaneously when pouring the concrete. Subsequently, as the soft foundation’s pore water pressure dissipated, the lock head’s vertical displacements changed slowly. The maximum settlement occurred at the side bottom plate.

In general, structural optimization could noticeably reduce the lock head’s settlement during construction, which was beneficial to engineering safety. The analysis method that comprehensively considered the concrete creep and soft soil consolidation could reflect the lock head’s stress field and displacement field’s time-varying characteristics.

This structural optimization method of the ship lock head considered the time-varying effects of the structure and foundation. Future work should consider the optimization of the ship lock head’s construction sequence.

The data used to support the findings of this study are included within the article.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China (no. 51579089).