The Finite Element Analysis of Dislocated Lock Heads in Double-Lane Ship Lock

Abstract

In order to investigate the mechanical characteristics of dislocated lock heads, a three-dimensional finite element model of dislocated lock heads in double-lane ship lock was established, then the interaction between two middle piers was researched by the mixed finite element method, the influence of the new layout pattern on stress and deformation of lock heads was studied. The results showed this new pattern had little effect on structural stress, but it had obvious influence on subsidence and foundation reaction force of two middle piers. The anti-slide safety factor of transverse stability in design flood was smaller, this working condition was controllability in design.

1 Introduction

The layout pattern of dislocated lock heads in double-lane ship lock is a new structural type, it has some advantages of less land occupation, environmental protection, low maintenance cost and reducing investment. This pattern has unique structural features, such as an expansion joint is built between two middle piers of lock heads, the axes of lock heads are not on same line, the earth pressure acting on side pier is dissymmetric.

Aiming at the support type of a double diaphragm wall, a finite element model was constructed to study the deformation and stresses of adjacent ship lock, the results showed larger wall spacing can cause greater reduction of the ship lock displacement (Carlos et al. 2020). Two layout schemes in accordance with Xi-Cheng canal ship lock project were proposed, the pattern of dislocated lock heads was considered to a more reasonable scheme through comparison and analysis (Chen et al. 2018). By the viscoelastic boundary and the dynamic contact model, the influence of contact nonlinearity on the structural stress and deformation under the action of earthquake was studied, the opening width of structural joints under the design earthquake and its influence on water seal were analyzed in Xin-Xia ship lock, the researches will improve the seismic design of dislocated lock heads (Ding et al. 2018).

Due to few practical engineering cases and above structural features, the stress and deformation behavior of dislocated lock heads in double-lane ship lock are unknown, the promotion and application of this structural pattern are limited. In order to investigate the mechanical characteristics of dislocated lock heads, a three-dimensional finite element model of dislocated lock heads in double-lane ship lock is established, then the interaction between two middle piers is researched by the mixed finite element method, the influence of this layout pattern on stability, structural stress , subsidence of lock heads is studied, these results in this paper will provide a theoretical basis for proving the design of the new structural pattern.

2 The Mixed Finite Element Method

Analysis on the mechanical mechanism of dislocated lock heads in double-lane ship lock is a key problem in the design, the mixed finite element method is used to solve contact problems between lock heads in order to research the interaction between two adjacent lock heads. Based on the characteristic of local nonlinearity, the forces acting on the contact bodies are divided into two parts: external forces and contact forces. The displacement of contact bodies is regarded as the basic variable, the nodal contact force in possible contact area is regarded as the iterative variable, so that the nonlinear iteration process was only limited in the possible contact surface. The complex contact nonlinearity is transformed into the variation of the contact forces, so the iterative procedure became easily to be carried out and highly efficient (Zhao et al. 2006).

The adjacent lock heads are denoted as ${\it \Omega }_{1}$ and ${\it \Omega }_{2}$ in Fig. 1, $u$ and $f$ are displacement and contact force of contact point pair on contact surface, respectively. $\xi$, $\eta,\, \zeta$ are the local coordinate system defined on the contact surface, respectively. The normal gap value between contact points can be expressed as:

$$ w = \left( {u - u} \right)\vec{\xi } + w_{0} $$

(1)

where $w$ denotes gap value of contact points, ${w}_{0}$ denotes initial gap value of contact points.

When the contact process is regarded as a mechanical model, three contact conditions should be satisfied normal impenetrability condition, normal contact force is compressive stress condition or tangential friction condition. Three contact states can be summarized as follows:

$$ {\text{open}}\,{\text{state}}\!:\quad \quad \quad f_{1} = f_{2} = 0,\quad w > 0 $$

(2)

$$ {\text{close}}\,{\text{state}}\!:\quad \quad \quad w_{0} = 0,\quad f_{1} = - f_{2} ,\quad \sqrt {f_{\zeta }^{2} + f_{\eta }^{2} }\,<\,Ac - \mu f_{\xi } ,\quad f_{\xi } < A\sigma_{t} $$

(3)

$$ {\text{slip}}\,{\text{state}}\!:\quad \quad \quad w = 0,\quad \sqrt {f_{\zeta }^{2} + f_{\eta }^{2} } = Ac - \mu f_{\xi } ,\quad f_{\xi } < A\sigma_{t} $$

(4)

where ${f}_{\zeta }$, ${f}_{\eta }$ denote component of contact force along tangential directions, $\mu$ is friction coefficient, A is area controlled by contact points, c denotes cohesion, ${\sigma }_{t}$ denotes tensile strength of contact surface, ${f}_{\xi }$ denotes normal contact force component.

Under static conditions, the finite element equilibrium equation at the n + 1 load increment step can be expressed as follows:

$$ K{\it \Delta} u_{n} = (F_{n + 1} + f_{n} - \smallint B^{T} \sigma_{n} d\Omega ) + \Delta f_{n} $$

(5)

where $K$ is global stiffness matrix, $\Delta {u}_{n}$ is displacement increment matrix, ${F}_{n+1}$ is external load vector, ${f}_{n}$ is the contact force vector at the previous one step, $\Delta {f}_{n}$ is the contact force increment at the present step, $B$ is strain matrix, ${\sigma }_{n}$ is total stress increment step at the previous one step.

3 Engineering Example

A double-lane ship lock locates in Wuxi city, Jiangsu Province, and it is an integral part of the Xi-Cheng canal regulation project. The downstream head of this project is 1.5 km away from the main channel of the Yangtze River, and this ship lock project is a pivot engineering in Xi-Cheng canal.

Fig. 1.

Mechanical model for contact problem

Fig. 2.

The dislocation lock heads

The effective size of each single-lane ship lock is 180 m × 23 m × 4.0 m (length of chamber × width of chamber × minimum water depth), and the design water head is 3.47 m. the design navigability is 42 million tons per year. The chamber adopts a separate structural type, and the chamber walls on both side piers apply to single-anchored steel sheet pile. The shared middle chamber wall uses a counter-pulled sheet piles structural type.

The project adopts a new pattern of dislocated lock heads in double-lane ship lock, there is 28.8 m along the flow direction between two adjacent lock head axis, the arrangement is shown in Figure 2 and 3.

Fig. 3.

The general layout of this double-lane ship locks

To account for deformation feature and mechanical mechanism of this new pattern, a three-dimensional finite element model is created. In the model, 8-node solid isoparametric and 6-node prism elements are adopted. the special contact elements are also used in the computational analysis. The whole FEM model consists of 417290 elements and 1154860 nodes, as seen in Figure 4. Dimensions of the simulations are 112.8 m (X axis) × 77.6 m (Y axis) × 30 m (Z axis), the transverse of this model is X axis and the along river direction is Y axis. In this study, we use GEIDP to calculate the deformations and stresses of this model, GEIDP is a finite element analysis software package developed by Li T.C. professor.

Fig. 4.

Finite element model of dislocated lock heads

The main loads such as self-weight, water pressure, earth pressure, uplift pressure and so on are carried on the dislocated lock heads. The load combination is given in Table 1.

Table 1. Load combination

4 Result Analysis

The maximum tensile stresses in the lock heads are shown in Table 2, the calculation proves the tensile stress is generally small in the dislocated lock heads, the maximum tensile stress is 1.69 MPa in the side pier, the maximum tensile stress is 1.73 MPa in the middle pier, the maximum tensile stress is 1.09 MPa in the lock floor, the maximum tensile stress does not exceed the concrete tensile strength for the whole structure.

The anti-slide safety factor of lock heads can reflect structural stability under various working conditions, the tangential force and normal force of each element on the sliding surface can be obtained according to the finite element results (Niu et al. 2009), the stability safety factor can be calculated by traditional formulas shown as following:

$$ K_{s} = \frac{{\sum \sigma_{i} f_{i} A_{i} + \sum c_{i} A_{i} }}{{\sum \tau_{i} A_{i} }} $$

(6)

where ${\sigma }_{i}$ is normal stress of element i, ${f}_{i}$ is friction coefficient of element i, ${c}_{i}$ is cohesion of element i, ${\tau }_{i}$ is tangential stress of element i, ${A}_{i}$ is area of element i in the sliding direction. The stability safety factor is obtained in each working condition shown in Table 3.

Table 2. The maximum tensile stress unit: MPa

From Table 3, it is obvious that each safety factor of shear fracture under four working conditions is more than 1.8 and the result satisfies the criterion, the structure can get enough safety reserve in overall stability. The anti-slide safety factor of transverse stability in design flood is smaller than other conditions, this working condition is controllability in design.

Table 3. Stability safety factor of gate head based on finite element method

Figure 5 shows the foundation reaction force distribution of lock heads under different working conditions. It is seen that the great foundation reaction force occurs in middle piers for each working condition. These forces are 74.45 kPa, 93.58 kPa, 98.04 kPa, respectively, under three working conditions. The foundation reaction force is largest under design flood condition, The foundation reaction forces of the middle-pier floors are greater due to the two middle piers interact with each other. When this ship lock is completed, the ratio of the maximum to the minimum of the foundation reaction force in the middle-pier floors is 4.13, which is slightly smaller than the allowable value 5.0 of the current ship lock design code. When the ship locks are operating, due to the action of water pressure in the lock heads, foundation reaction force can tend to be uniformly, the ratio of the maximum and minimum foundation reaction force is 2.40, which meets the requirements of the design code.

Fig. 5.

The foundation reaction force distributions of lock heads

The subsidence of lock-head floor is shown in Fig. 6, we can see that there is a large subsidence on side floor, and the maximum value is 20.3 mm. Under different working conditions, the subsidence of lock-head floor is mainly rigid displacement, and this subsidence will not result in greater stress on the lock head structure.

Fig. 6.

Horizontal distribution of lock head floor subsidence

5 Conclusions

Based on the analysis above; the following conclusions could be reached:

• (1) For the dislocated lock heads, the maximum tensile stress does not exceed the concrete tensile strength under different working conditions, this layout pattern has little effect on structural stress.

• (2) This layout pattern has obvious influence on subsidence and foundation reaction force of middle piers. The subsidence of dislocated lock heads is mainly rigid displacement and large subsidence occurs in middle piers. The foundation reaction force of middle piers is quite large under design flood.

• (3) For the dislocated lock heads, the anti-slide safety factor of transverse stability in design flood is smaller, this working condition is controllability in design.