Parametric sensitivity analysis of cellular diaphragm wall

Abstract

The deformation law of the cellular diaphragm wall in deep foundation pits was studied through numerical simulation. Based on the example of the dock wall in engineering, the full three-dimensional finite element model was used to simulate the excavation of the foundation pit. Interaction between the cellular diaphragm wall and the soil was also taken into account in the calculation. The results indicated that the maximum lateral displacement, which is the evaluation index of sensitivity analysis, appeared on the top of the interior longitudinal wall with an excavation depth of 10 m. The centrifuge model test was carried out to study the deformation regulation for a cellular diaphragm wall. The most sensitive factor was found by adjusting the length of the partition wall, the spacing of the partition wall and the thickness of the wall. In the end, a suggestion was proposed to optimize the cellular diaphragm by adjusting the length of the partition wall.

Keywords

cellular diaphragm wall / sensitivity analysis / optimization / centrifuge model test

Introduction

As a type of permanent self-standing structure, a cellular diaphragm wall is widely used in excavation and harbor engineering [1]. However, fundamental studies of this type of structure are relatively limited and infrequently reported in the literature. Wang [2] studied the stability of cellular diaphragm walls and analyzed monitoring data. A three-dimensional numerical simulation taking into account the interaction between the soil and the cellular diaphragm wall was carried out to study the space effect of the special structure in Li’s paper [3]. Fang [4] introduced the construction technology in the dock extension project. Xu [5,6] studied the interaction between cellular diaphragm wall and soil and derived the distribution law of soil pressure.

The analysis method of the traditional retaining structure is insufficient for this type of structure. Due to the lack of theoretical research, caution must be used during design and construction. Therefore, optimization analysis for cellular diaphragm walls is necessary. The centrifuge model test is carried out in this paper to study the deformation regulation for a cellular diaphragm wall. Then, an optimization analysis is provided to study the size sensitivity to maximum lateral displacement.

Numerical simulation for a cellular diaphragm wall

A 3D nonlinear finite element method was applied in ABAQUS to simulate the excavation of the foundation pit.

Description of the project

The dock extension project has a length of 180 m, a width of 76 m and a depth of 10 m. The top elevation is+ 0.0 m, the bottom elevation of the interior longitudinal wall is -25 m, the bottom elevation of the exterior longitudinal wall is -23 m, and the bottom elevation of the partition wall is -21 m (the interior longitudinal wall faces the pit, and the exterior longitudinal wall backs against the earth). The sectional dimensions of the cellular diaphragm wall are shown in Table 1.

3D finite element simulation model

Calculation scale

The length of the foundation pit was assumed to be infinite because the length is much longer than the width for a real-world pit. The model was formed by four pieces of a grid and required half the width of the foundation pit because of its asymmetry. The distance from the pit edge to the model boundary was determined to be six times the excavation depth, and the distance from the excavation bottom to the bottom of the boundary was the same. The width of the model was five times the partition spacing according to previous literature [7,8]. Finally, the calculation range was set as 110 m by 40 m by 70 m.

The finite element calculation model of the cellular diaphragm wall is shown in Fig. 1.

Boundary condition

The boundary conditions were as follows:

1) Restrict horizontal displacement of the two sides in the corresponding direction.

2) Restrict the horizontal and vertical displacement of the underside.

3) Set the boundary at the top surface as a free boundary.

Selection of the material parameters

The structure material of the concrete diaphragm wall and the lining was assumed to be elastic. The modulus of elasticity and the Poisson ratio of the diaphragm wall were assumed to be that of C35 concrete. The parameters for the soil layer were assigned using the soil parameter table in Shanghai summarized by Xu [9]. The physical and mechanical parameters for each soil layer are shown in Table 2.

The cam-clay model was chosen as the constitutive relation for soil in the simulation [10-15].

Contact algorithm between the diaphragm wall and soil

The materials were all simulated using hexahedron elements [9]. The interaction was simulated by setting contact elements between structural elements and geological materials. The tangential friction coefficient of the contact surface was 0.3. The embedded deformation was not considered to be normal. That is, normal stiffness was taken to be infinite.

Work conditions in the simulation

The work conditions [10] are shown in Table 3.

Analysis of simulation results

Simulation results of displacement for the cellular diaphragm wall are shown in Table 4.

As shown in Table 4, the lateral displacement of the interior longitudinal wall was much larger than that of the exterior longitudinal wall. The maximum displacement of each wall appeared in the top. The displacement decreased as the wall’s depth increased, and it increased greatly as the excavation depth increased. Therefore, the optimization analysis used the lateral displacement of the interior longitudinal wall with an excavation depth of 10 m as the evaluation index.

Centrifuge model test for the cellular diaphragm wall

Model design

Experimental principle

The centrifuge test is a scale model test. Because the applied test model has a centrifugal acceleration much larger than the acceleration due to gravity in the centrifugal field, the loss of gravitational stress due to reduced dimensions is compensated for by the centrifuge. Therefore, the stress of soil in the centrifuge test could become the same as the stress on the undisturbed soil. This test could simulate the real-world construction conditions. The similarity relationship between the centrifugal model and the prototype is shown in Table 5.

Model design

The centrifuge test was performed using the TJL150 centrifuge machine. The maximum capacity of the machine is 150 gt, the maximum acceleration is 200 g and the effective rotation radius is 3 m. The model ratio was fixed as 100, and the model box is 400 mm × 500 mm × 600 mm (length × width × height) according to test objective and laboratory arrangement. Aluminum alloy was used as a substitute for concrete. The dimensions for the model were fixed according to the principle of equivalent flexural rigidity. The flexure (w) equation under a uniformly distributed load (q) was as follows:

(1){\displaystyle \frac{{\partial }^{\mathrm{4}}w}{\partial x^{\mathrm{4}}}}+\mathrm{2}{\displaystyle \frac{{\partial }^{\mathrm{4}}w}{\partial x^{\mathrm{2}}\partial y^{\mathrm{2}}}}+{\displaystyle \frac{{\partial }^{\mathrm{4}}w}{\partial y^{\mathrm{4}}}}={\displaystyle \frac{q^{\mathrm{4}}}{k}}\cdot w,

(2)k={\displaystyle \frac{E{h}^{\mathrm{3}}}{\mathrm{12}(\mathrm{1}-{\mu }^{\mathrm{2}})}},

where k denotes flexural rigidity, E denotes modulus of elasticity, and μrepresents Poisson’s ratio. According to the principle of equivalent flexural rigidity,

(3){\displaystyle \frac{E_{\mathrm{m}\mathrm{1}}·{\delta }_{\mathrm{m}\mathrm{1}}^3}{\mathrm{12}(\mathrm{1}-{\mu }_{\mathrm{m}\mathrm{1}}^2)}}={\displaystyle \frac{E_{\mathrm{m}\mathrm{2}}·{\delta }_{\mathrm{m}\mathrm{2}}^2}{\mathrm{12}(\mathrm{1}-{\mu }_{\mathrm{m}\mathrm{2}}^2)}},

where {\delta }_{\mathrm{m}\mathrm{1}}={\displaystyle \frac{{\delta }_{\mathrm{p}}}{n}}, E_p = E_m1, μ_p = μ_m1, n denotes model ratio, subscript p represents prototype, subscript m1 is centrifugal model made of concrete, subscript m2 denotes centrifugal model made of aluminum alloy. So,

(4){\delta }_{\mathrm{m}\mathrm{2}}={\displaystyle \frac{{\delta }_{\mathrm{p}}}{\mathit{n}}}{\left[\begin{array}{ll}{\displaystyle \frac{(\mathrm{1}-{\mu }_{\mathrm{m}\mathrm{2}}^{\mathrm{2}})}{1-{\mu }_{\mathrm{m}\mathrm{2}}^{\mathrm{2}}}}& {\displaystyle \frac{E_{\mathrm{p}}}{E_{\mathrm{m}2}}}\end{array}\right]}^{\mathrm{1}/\mathrm{3}}.

Adjust the length and spacing of partition wall (shown in Table 6) to realize the deformation law for the diaphragm wall, respectively. The size in Table 6 is the prototype size. The centrifugal size can be calculated by Eq. (4).

Soil parameters

Clay in layer ⑤₁ of Shanghai was used in this centrifuge model test. The physico-mechanical parameters are shown in Table 7.

Data acquisition

An Eddy displacement sensor was placed on top of the cellular diaphragm wall to measure the displacement. The installation condition is shown in Fig. 3.

Analysis of the test results

Lateral displacement of the top of the interior longitudinal wall is shown in Tables 7 and 8. The displacement of the centrifugal model has been converted to the prototype.

The lateral displacement of the top of the interior longitudinal wall decreases as the length of the partition wall increases and increases as the spacing of the partition wall increases.

Sensitivity analysis

A numerical simulation was carried out to obtain more data for sensitivity analysis because the experiment provided limited data.

Sensitivity analysis theory

The optimization analysis for a cellular diaphragm wall was performed through a sensitivity analysis of the sectional dimensions. The parameters for numerical simulation referred to the second section.

Three sectional dimensions of the cellular diaphragm wall were taken as the analysis factors: the length of the partition wall, the spacing of the partition wall and the thickness of the wall. The lateral displacement of the interior longitudinal wall with an excavation depth of 10 m was used as the evaluation index. The single-factor sensitivity coefficient was calculated and compared to obtain the most affected factor. The formula of the single-factor sensitivity coefficient is as follows:

(5)E={\displaystyle \frac{\mathrm{\Delta }A}{\mathrm{\Delta }F}},

where E is the sensitivity coefficient, \mathrm{\Delta }A is change ratio of maximum displacement, \mathrm{\Delta }F is change ratio of uncertain factor. The analysis was based on the prototype size as shown in Table 1.

Analysis of the calculation results

Sensitivity analysis for the length of the partition wall

As shown in Table 9, there is a negative correlation between the displacement of the diaphragm wall top and the length of the partition wall. The displacement decreases greatly with an increase in the length of the partition wall. It follows that the displacement of the diaphragm wall is fairly sensitive to the length of the partition wall.

Sensitivity analysis for the spacing of the partition wall

As shown in Table 10, there is a positive correlation between the displacement of the diaphragm wall top and the spacing of the partition wall. The displacement increases with an increase in the spacing of the partition wall. It shows that the displacement varies with the spacing of the partition wall in a smaller range than with the length of the partition wall.

Sensitivity analysis for the thickness of the wall

As shown in Table 11, there is a negative correlation between the displacement of the diaphragm wall top and the thickness of the wall. The displacement decreases with an increase in the thickness of the wall in a very small range. It follows that the thickness of the wall has little effect on the displacement of the top of the diaphragm wall. The thickness of the wall varies from 0.8 m to 1.2 m normally in Shanghai, so its effect could be ignored.

Conclusions

The centrifuge model test was carried out in this paper to study the regulation of deformation for a cellular diaphragm wall. The optimization analysis for a cellular diaphragm wall was conducted through sensitivity analysis of the sectional dimensions. Three sectional dimensions of the cellular diaphragm wall were used as the analysis factors: the length of the partition wall, the spacing of the partition wall and the thickness of the wall. The lateral displacement of the interior longitudinal wall with an excavation depth of 10 m was considered as the evaluation index. A single-factor sensitivity coefficient was calculated and compared to obtain the most affected factor in the end. The following four research findings and conclusions were found:

1) The lateral displacement of the top of the cellular diaphragm wall decreases with an increase in the length of the partition wall, increases with an increase in the spacing of the partition wall, and decreases with an increase in the thickness of wall.

2) The length of the partition wall has a much greater effect on the lateral displacement of the top of the wall than the spacing of the partition wall. The thickness of the wall has little effect on the lateral displacement of the top of the wall.

3) Based on the overall balance and deformation, adjusting the length of the partition wall is the best way to optimize the cellular diaphragm.

4) The sensitivity analysis of the cellular diaphragm wall only calculates a single factor. Multivariate analysis can be carried out in future research. Further, economic factors can also be considered.

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