Numerical analysis of uneven settlement of highway subgrade based on nonlinear algorithm

2.1 Soil settlement theory and constitutive model

Roadbed deformation on highways in service is caused by self-weight and vehicle-in-vehicle effects, and foundation deformation is caused by a change in foundation stress. In conducting soil research, there is always an important issue, that is, the problem of soil settlement. According to the settlement consolidation theory, the simulation calculation of the settlement is the main research content of the calculation of soil settling deformation. From the perspective of the overall evaluation of road ground foundation safety stability, considering the direction of the highway transportation service level, the size of the ground base of the ground, and the difference in the decline of the new and old base sinks is the important basis of the above. The three most common soil changes that cause foundation sinking are described below. Heavy rain and flooding also have an influence on soil settling when wet clay soil holds moisture and becomes extremely elastic. As a result, the home may slide or “sink” into the weak soft soil. In some ways, it is similar to how your foot “squishes” into the ground when walking through mud. Furthermore, if water with poor drainage is permitted to pool or stand near your house, the earth will absorb the water and grow weaker once more. Soil settlement is mainly composed of three aspects: first, the fixed stress between the combined water and soil particles changes due to the inherent moisture in the pore, thereby causing the soil settling; Soil due to self-weight, vehicle-loaded load, resulting in the phenomenon of pore moisture and air in the roadber geotorus. This can be obtained by calculating the settlement amount, as shown in Eq. (1):

where S ( t ) is the total settlement of the soil when T, S d ( t ) is the initial settlement of the soil when T, S s ( t ) is the secondary consolidation settlement of the local body when T, S c ( t ) is the consolidation settlement of the soil; the overall settlement amount produced by the road base to complete this whole period of time is completed, which is the key consideration to expand whether the project is stable. The consolidation theory is how the soil settling is the theory of time variation, and the main consolidation theory has two terzaghi consolidation theories and Bio (Biot) consolidated theory. Wave propagation in a pores-saturated media, also known as a fluid-saturated solid matrix medium, is described by Biot’s theory. The hydrostatic pressure filling the pores of a porous material resists stress when it is applied, but Biot overlooks the microscopic scale and claims that continuum thermodynamics may be applied to quantifiable macroscopic parameters. Among them, the Austrian sugar junction is to describe the consolidation process of the multi-dimensional potential, and the taiski theory is a simple one-dimensional solid knocked theory. Therefore, it is more complicated than the Austrian theory, which needs to be used: the score transformation and the number of methods, despite this. Bio theory can only meet the solve demand for solving a simple problem than a small number of boundary values. However, with the continuous development of modern computer science and technology, a variety of finite element software based on the analysis of the Austrian solids theory, so that the nonlinear and non-homogeneous materials, complex stress–strain in actual engineering problems and irregular boundary conditions can be completely simulated and calculated.

2.2 Widening the finite element model

2.2.1 Basic assumptions and sizes

When using the finite element entity to model the road engineering, in order to better fit the boundary load, the following assumptions are made: (i) the length of the road engineering model in the direction is much larger than the road surface width, in line with plane strain the basic hypothesis of the problem is therefore carried out in accordance with the above type; (ii) roadbed soil, including new filling and ground soil, is regarded as an amoelastic material of the same-tolerability; (iii) double-sided widened construction the default left and right construction progress is always consistent; some of the data show that the two-side settlement difference caused by the two-side broadness method is smaller than the settlement deformation of the single side, and therefore, the text is selected for both sides. Construction is carried out in a way out. The size of the model is shown in Figure 1.

Figure 1

The subgrade widening structure model.

2.3 Numerical calculation method of settlement analysis

In recent years, due to the continuous development and maturity of computer technology, many complex geotext has got a certain degree of solving through the computer. Since the commonly used parsing method is difficult to consider complex boundary conditions and initial conditions when calculating the ground base setting, the calculation error is largely due to the complexity and nonlinearity of the foundation soil. The development of finite elements studied various application units and high precision units and introduced numerical calculation methods in the calculation such that the nonlinear stress–strain relationship between complex boundary conditions and anisotropic soil becomes possible. The numerical method currently used for soft land base deposition analysis is mainly differential method, limited element method, and boundary element, and its development trend is a finite element method and differential method or combined with boundary meta-way solutions, in order to play a variety of methods.

1. The differential difference method is to discrete the study area with differential grids, and the consolidated differential equation is simultaneously divided into differential equations for each node, and then combined with the initial conditions and boundary conditions, the linear equation group obtains a numerical solution. Taking the planar problem as an example, the difference method can obtain the pore pressure cloth at each time in the study plane, so the initial sink value can be derived, and the total settling S or the total settlement amount of any time T is S c . Due to the vertical strain of the soil, as shown in formula (2):

   (2) ε z = 1 + μ E [ ( 1 − μ ) ( σ z − μ ) − μ ( σ x − μ ) ] .

   The settlement of a lead hammer line along the foundation is shown in Eq. (3):

   (3) ∫ 0 H 1 + μ E [ ( 1 − μ ) σ z − μ σ x ] d H .

   In the formula, H is the effective compression layer thickness. The total settlement is shown in formula (4):

   (4) ∫ 0 H 1 + μ E [ ( 1 − μ ) ( σ z − μ ) ( σ x − μ ) ] d H .

   In the formula, E and μ vary with the effector, so the above formula can be a total settlement at any time. Any engraved fixed settlement is S c = S − S t . When calculating the final settlement, the pore pressure μ = 0 in the formula is calculated. The disadvantage of the differential method is that the derivative of the approximate solution is inaccurate, which is difficult to introduce boundary conditions along the nonlinear boundary, and the complex domain is difficult to accurately express and not suitable for non-uniform and non-rectangular grids. In particular, when encountering the complex boundary conditions of geometric shapes, the accuracy of finite differences is often limited, and even solving is impossible.

2. Finite element method finite element is the foundation and structure as an overall analysis, dividing the grid, forming a dispersion structure, and forming a limited number of regional units, which are only strong at the node. The stress of the material can be expressed as the formula (5):

   (5) { σ } = [ D ] { ε } .

   The relationship between the nodes and nodes displacement of the unit body can be established by the virtual displacement principle, and the total balance equation is written, such as the formula (6):

   (6) [ K ] { δ } = { R } .

   In the formula, [ K ] , { δ } , and { R } are a stiffness matrix, a node displacement matrix, and a node load matrix. Then, they are combined with the initial and boundary conditions to solve the linear equations, calculate the displacement and stress of each point of the foundation and the structure at any time, and obtain a numerical solution to the problem. The finite element method can be considered a two-dimensional or even three-dimensional problem, reflecting the effects of lateral deformation.

3. Boundary elemental boundary element method is to convert the integral of the field belonging to the area boundary to the regional boundary and uses discrete technology for the numerical solution of the boundary integration equation; because the factor matrix of boundary elements is full, resulting in calculation storage, the amount is large, and it is not necessary to save computation time, and the incapacity of nonlinear issues, etc., causes the current boundary dollar in processing the consolidation issue.

3.1 Analysis of the impact of the step height and width ratio on the subgrade settlement

Analysis of step fixed height and width effect (Figure 2).

Figure 2

Soil settlement curve when the step height is unchanged: (a) settlement of the subgrade top surface; (b) foundation top surface settlement.

As analyzed in Figure 2, we can conclude that when the height of the control steps is unchanged, the width of the steps is increased, then the settlement of both the subgrade top surface, and the foundation top surface will be reduced. Compared with the average settlement of the subgrade top surface when the step width was 1 m, there are the average settlement of about 3.5% at 1.2 m and the average settlement of about 10.1% at the width of 1.6 m. These results show that increasing the step width during excavation is beneficial to reduce soil settlement and maintain soil stability.

According to Figure 3, when the height of the excavation steps remains unchanged, the horizontal displacement of the foundation soil directly below the slope foot also increases with the width of the widening embankment. And the increase decreases with the width of the steps. Analysis of step width and height is as follows:

Figure 3

Step elevation is unchanged in the horizontal displacement curve of the new subgrade slope toe foundation.

When the step height is 0.6, 1.0, 1.25, and 2.0 m, respectively, the deformation values of the soil are as follows (Table 1):

According to Table 1, when the height of the steps is unchanged, the settlement and horizontal displacement of the subgrade soil will increase. After comparing the simulation results of the two cases, the height ratio of the steps is minimized. In the actual construction, the height ratio of the steps cannot be infinitely reduced, and the smaller the width ratio of the steps, the more the steps need to excavate. After many years of vehicle action, the upper edge of the original subgrade slope is more solid than the lower soil, and the slope foot is more prone to water erosion and need to clear more surface loose soil [13]. Therefore, for most expressways with original slopes of 1:1.5, the height ratio of the steps should be controlled at 1:1.2–1:1.5 according to the above research results and relevant specifications. That is, when the width is controlled at about 1.2 m, the height of the steps is kept at 0.8–1.0 m.

3.2 Geogrid contact model principle

Since the tensile stress under the interaction between the geogrid and the soil is much less than the allowable stress of the tensile strength, the stress–strain of the grilles is always considered to maintain the line elastic relationship [14]. In the analysis and calculation of the finite element model, the Yang modulus of the grille material is much larger than that of the soil. Young’s modulus is a measurement of either a material’s ability to withstand changes in length when compressed or tensioned along its length. The modulus of elasticity, commonly known as Young’s modulus, is computed by dividing the longitudinal stress by the strain. Stress and strain may well be explained for a metal bar under tension as follows. The orientation of materials has a large influence on Young’s modulus. Isotropic materials have mechanical properties that are constant in all directions. Two examples are ceramics and pure metals. Grain architectures can be generated by treating a material or by adding impurities that direct the mechanical properties of the material [15,16]. After the two are combined into the reinforcement body under the same force, the strain value difference is large, resulting in relative sliding and even cracks of the two [17]. Therefore, the reinforcement body should not be analyzed directly when establishing the finite element model, but both are defined as two materials after the contact, set as embedded constraints, namely the embedded contact [18]. During construction, the original subgrade excavated steps and the newly laid subgrade as the platform, and the laying length of the steps shall play the anchor role to strengthen the laid part of the steps [19]. The vertical deformation of the subgrade after the disposal of geotechnical reinforcement and the foundation top surface decreases, but the decrease is very small, which will almost have no impact on the consolidation settlement of the foundation [20]. It can be concluded that the grille reinforcement disposal of the soil is very small for the change of the soil consolidation settlement compared with the total settlement increment, so the reinforcement treatment should be combined with other effective methods to reduce the difference settlement between old and new roadbeds.