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Geostatic stress field procedure is the first and the most important step for the numerical simulation of geotechnical engineering, which greatly influences the simulation results. Traditional methods often fail when the model is complex. In this paper, based on finite element method (FEM) software ABAQUS, failure reasons of four commonly used methods for the geostatic stress field are studied. According to the analysis results, a new set of methods, which can provide reasonable displacement and stress field results under complex conditions, is proposed. The proposed methods follow the principle that the stress of different materials should be obtained separately to avoid stress distortion. Then, the accuracy and applicability of the proposed method are verified through a comparison study and a specific application. This study provides a theoretical basis for the method of geostatic stress field procedure under complex condition and can serve as a reference for relevant studies.

Geostatic stress (geo-stress) is the natural stress existing in rock and soil structure, also known as the initial stress, absolute stress, or original rock stress of rock and soil mass [

ABAQUS is a general finite element software. Because of its excellent nonlinear computing capability, ABAQUS has been widely used in the field of geotechnical engineering [

There are four basic methods for geostatic stress field procedure in ABAQUS: A: automatic method, B: direct defining method, importing methods (which can be further divided into two different methods, i.e., C: importing stress results from output files (odb files of ABAQUS), and D: importing nodal force from output files [

When method A is invoked, the software automatically computes the equilibrium corresponding to the initial loads and the initial configuration, allowing only small displacements within specified tolerances. The procedure is available with a limited number of elements and materials and is intended to be used in analyses in which the material response is primarily elastic.

Method B is used when the initial stress state is approximately known and rather simple so that the stress can be directly given with explicit equations. If the stresses given as initial conditions are far from equilibrium under the geostatic loading and boundary conditions, the method will fail.

For a rather complex model, that is, whether the model has a complex geometry and boundary conditions or it is a nonlinear constitutive model, methods C and D should be applied. Methods C and D involve two steps: (a) run a regular simulation rather than a geostatic stress field procedure and (b) run the geostatic stress field procedure by importing the stress or nodal results of step a.

To study the flexibility of the four methods under different conditions, three numerical models representing three different complexities are established.

Model 1. Simple model.

Model 2. Model with tie constraint.

Model 3. Model with contact interaction.

The first condition (simple model) represents a model with simple stress state and geometric characteristics, while the other two models represent a model with a complex stress state or geometric characteristics of two different types. The second condition (tie constraint) represents the model with two different parts with no relative displacement at the interface. The third condition (contact interaction) represents the model with two different parts allowing relatively displacement at the interface.

As shown in Figure

Dimensions of model 1.

Parameters of the materials in model 1.

Material | Elastic modulus (kPa) | Poisson’s ratio | Cohesion (kPa) | Friction angle (°) | Density (kg/m^{3}) |
---|---|---|---|---|---|

Clay | 1.8 × 10^{4} | 0.35 | 32 | 14 | 2.0 |

FEA model of Model 1.

Applying four methods to model 1, vertical deformation and vertical stress (_{z}) contours are obtained as shown in Figure

Vertical deformation results for model 1 (unit: m). (a) Method A, (b) method (B), (c) method C, and (d) method D.

Stress results for model 1 (unit: kPa). (a) Method A, (b) method B, (c) method C, and (d) method D.

Table of the simulation results.

Method | Max stress (kPa) | Max displacement (m) | Conclusion |
---|---|---|---|

Method A | −9.706 × 10^{2} | 2.828 × 10^{−16} | Equilibrium acquired |

Method B | −9.706 × 10^{2} | 7.076 × 10^{−16} | Equilibrium acquired |

Method C | −9.706 × 10^{2} | 9.370 × 10^{−09} | Equilibrium acquired |

Method D | −9.706 × 10^{2} | 9.289 × 10^{−8} | Equilibrium acquired |

The ground displacement caused by the process of foundation pit excavation or tunnel excavation is mostly bigger than 10^{−3} m, and displacement smaller than 10^{−4} m is usually ignored in engineering practice. Therefore, a 10^{−4} m level of displacement can be a reasonable upper bound for the results of the geostatic stress field procedure. As shown in Figures ^{−4} m.

In addition, it can be seen from Figures

In ABAQUS, to establish a tie constraint between two surfaces is to constrain each of the nodes on one surface to have the same value of displacement as the nodes on the other surface that it contacts. In geotechnical engineering simulation, tie constraint is often used to simulate the relationship between artificial structure and the soil when relative slip can be ignored, such as the interaction between the lining and the surrounding soil in the shield tunnel excavation simulation, the interaction between the foundation and the soil, and the connection between the retaining pile and the soil in the simulation of foundation pit supporting system. As shown in Figure

Dimensions of model 2.

Parameters of the materials in Model 2.

Material | Elastic modulus (kPa) | Poisson’s ratio | Cohesion (kPa) | Friction angle (°) | Density (kg/m^{3}) |
---|---|---|---|---|---|

Soil | 1.8 × 10^{4} | 0.35 | 32 | 14 | 20 |

Concrete | 3.15 × 10^{7} | 0.2 | — | — | 25 |

FEA model of model 2.

Only methods C and D reach convergence, and the calculation results are obtained (shown in Figures

Simulation results of method C. (a) Vertical displacement results (unit: m) and (b) vertical stress results (unit: kPa).

Simulation results of method D. (a) Vertical displacement results (unit: m) and (b) vertical stress results (unit: kPa).

It can be seen from Figures ^{−4} m). Furthermore, stress near the tie surface is distorted. This is because the stress results in the first step of methods C and D are distorted because of the abrupt change of stiffness at the tied surface.

Considering the stress distortion results of methods C and

In the numerical simulation such as pile-soil interaction simulation, friction contact is often used to simulate the interaction between the pile side surface and soil side surface. Infinite element analysis, contact interaction is a nonlinear boundary condition, which often causes convergence difficulty. The behavior of the four methods was discussed under such conditions.

Settings of model 3 are almost the same as model 2 except that the tie constraint between soil and concrete is replaced by a surface contact interaction (Figure

Dimensions of model 3.

Similar to the situation of model 2, only methods C and D converged, and the calculation results are shown in Figures

Simulation results of method C. (a) Displacement results (unit: m) and (b) vertical stress results (unit: kPa).

Simulation results of method D. (a) Displacement results (unit: m) and (b) vertical stress results (unit: kPa).

It can be seen from Figures ^{−2} m, which is far bigger than that requiring 10^{−4} m standard [

Vertical deformation contours of concrete (unit: m).

Vertical deformation contours of soil (unit: m).

Therefore, the four basic methods cannot produce a reasonable result for a model with contact interaction either.

In summary, four basic methods can come up with reasonable results for a rather simple finite element model with uncomplicated geometry, single material property, and no nonlinear boundary conditions. Since the automatic method is the simplest and produces the most ideal results, it should be used when possible. However, when nonlinear boundary conditions such as tie constraint or contact interaction were involved, which is quite common in the numerical simulation in foundation pit excavation or tunneling, all four methods are unable to function properly.

In the numerical simulation of geotechnical engineering problems, the interaction of soil layers and artificial structure in a model can be pretty complex, such as the simulation of the influence of the excavation of foundation pit on existing structures, or the excavation simulation or slope stability analysis considering the retaining pile. For these sorts of problems, the geostatic stress field procedure will be difficult because of the poor performance of the commonly used methods. In this study, a new method for geostatic stress field procedure is proposed to deal with these problems.

According to the analysis in the above section, the reasons for the failure of the basic methods lie in the huge stiffness difference between soil and artificial structure material such as steel and concrete. Because of the stiffness difference, relative displacement occurs at the interface of the soil and artificial structure, which will lead to a redundant stress increment that makes the acquired stress field inappropriate, and thus the displacement cannot meet the requirement. Since the redundant stress near the interface is the main cause of the failure of the geostatic stress field procedure, avoiding the redundant stress during the geostatic stress field procedure may be a practicable approach.

Therefore, the following geostatic stress field procedure was proposed and verified by applying to model 2 and model 3. The proposed method follows the principle that the stress of different materials should be obtained separately to avoid stress distortion.

The proposed method contains two similar forms to deal with the tie constraint and contact interaction conditions.

For tie constrained:

Remove the tie constraint that connects soil and artificial structures in the model, and apply appropriate displacement boundary conditions to the corresponding surface of the model (for the above example, the applied displacement boundary condition for model 2 should be zero normal displacement on both sides of soil and concrete).

Run a static analysis of the model, and import the output file to the geostatic stress field procedure.

Restore the tie constraints and run the geostatic stress field procedure.

For contact interaction:

Remove the contact interaction that connects soil and artificial structures in the model, and apply appropriate displacement boundary conditions to the corresponding surface of the model, or set the friction coefficient to zero so that the contact interaction can act as a zero normal displacement boundary condition.

Run a static analysis of the model, and import the output file to the geostatic stress field procedure.

Restore the contact interaction and run the geostatic stress field procedure, or restore the friction coefficient and run the geostatic stress field procedure.

If each part of the model is simple enough for the automatic method, then the method can be simplified as follows:

Deactivate the contact interaction (or set the friction coefficient to zero) at the first step of the simulation, and use the automatic method to run the geostatic stress field procedure for each part of the model.

Activate the contact interaction (or restore the friction coefficient) at the second step and run a static analysis.

The proposed method is applied to model 2 and model 3 to verify the validity and accuracy of the method.

Model 2

Displacement and stress results of the proposed method applying to the tie constraint are shown in Figures

Model 3

Displacement and stress results of the proposed method applying to the contact interaction model are shown in Figures

Vertical deformation contours of model 2 (unit: m).

Stress contours of model 2 (unit: kPa).

Vertical deformation contours of model 3 (unit: m).

Stress contours of model 3 (unit: kPa).

According to Figures ^{−5} m. The produced stress field is not distorted on the joint area. Therefore, the proposed method is adequate for complex models. Although only ABAQUS is used in this paper to demonstrate the procedure, the proposed method is also available for other FEM software such as ANSYS and MIDAS.

The accuracy and applicability of the proposed method are further verified through the simulation of two practical engineering applications.

The first practical engineering case is the excavation of a deep foundation pit of a subway station. The depth and width of the foundation pit are 18 m and 19.2 m, respectively. The foundation pit adopts the combined support system composed of supporting pile, top beam, middle beam, and supporting steel pipe. The initial state of the model is shown in Figure

Model before and after excavation. (a) Before excavation. (b) After excavation.

Vertical stress contours (unit: kPa).

Vertical deformation contours (unit: m).

According to Figures

The second practical engineering case is the bearing capacity test of a friction pile. The diameter and length of the pile are 2 m and 55 m, respectively. Axisymmetric modeling is used for the model (see Figure

Model of the pile-soil interaction.

Stress contours of pile-soil interaction.

Vertical deformation contours of pile-soil interaction.

Comparison of the simulated deformation results and monitoring data of the model.

From the displacement and stress results in Figures

In this study, four commonly used methods for the geostatic stress field procedure were applied to three models with different complexities to investigate the flexibility of the methods. When the model involves tie constraints or contact interactions, none of the four methods can produce a reasonable result. The reason for the failure of the methods is the distorted stress distribution near the adjacent area caused by stiffness difference between materials such as soil and concrete.

According to the analysis of the failure of the commonly used methods, a new method was proposed to overcome the defect of the commonly used methods. The proposed methods follow the principle that the stress of different materials should be obtained separately to avoid stress distortion.

Finally, the accuracy and applicability of the proposed methods were proved through the comparison with the commonly used methods and the application to two practical engineering applications. The proposed method is a good solution for the geostatic stress field procedure under complex conditions.

The data used to support the findings of this study are available from the first author upon request.

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

This work was supported by the Key Research and Development Project of Shaanxi Province (Grant nos. 2020SF-373 and 2021SF-523) and the Special Research Project of Shaanxi Provincial Education Department (Grant no. 19JK0381).