Simulation Research on Local Settlement of Kiln Foundation Based on Winkler Theory

The masonry structure is one of the main types of modern industrial buildings in China because of its local materials and simple economy. However, these buildings often suffer damage and cracks due to uneven settlement of the foundation and disrepair, which affects the safe and normal use of the structure. It is urgent to maintain and strengthen the structure to restore the safety of the structure. This paper takes Baoji Shenxin Yarn Factory - Kiln Factory as the actual engineering background, and use the finite element software ABAQUS based on Winkler’s theory to establish a finite element calculation model that can more conveniently respond to the effects of different foundation settlements. Through the simulation analysis, the influence law of the local settlement of the kiln foundation on the overall structural force state was obtained, which can provide an important reference basis for the maintenance and reinforcement of the structure.


Introduction
Located in Jintai District, Baoji City, Shaanxi Province, Baoji Shenxin Yarn Factory is the most complete surviving anti-war industrial site in China.Due to the relatively long time and the local area in the wet-submerged loess area, the foundation of the kiln factory has been subject to uneven settlement, thus causing the formation of multiple cracks in the cave body with misaligned cracks in the arch ring, localized damage such as small slump and water seepage in the wall surface, as shown in Figure 1.Due to the diversity of foundation models, the linear elastic model is more frequently used for its study, among which the Winkler elastic foundation model is the simplest and most convenient foundation model.Zhang B et al. [1] based on Winkler's elastic foundation beam theory and arranged horizontal and rotational springs around the shaft to model; Zhang C R et al. [2][3] linked the Winkler foundation modulus to the earth pressure free field by selecting a suitable pipe -soil interaction; Kurian N P et al. [4] obtained the actual nonlinear response of the characterized soil to the applied load by fitting the load-settlement data nonlinearly through the Winkler model.Okeagu B J et al. [5] found that the soil reaction coefficients around the pipe varied widely and the magnitude was related to the span of the pipe, the depth of the soil and the direction of action.Nogami T K et al. [6] found that the constants of springs and dampers of viscoelastic soils can be derived from the continuum equations under plane strain conditions.Li J G et al. [7] summarized the test methods for the three moduli of foundation soils and the relationships between them, and gave the methods for taking the values of different moduli in finite element calculations.Hu J [8] selected the appropriate values according to the geology by comparing the bed coefficients obtained by different methods.It can be found that the introduction of Winkler foundation model to simulate the uneven settlement of foundation is simple and convenient with few parameters, which is easy to apply in practical engineering.If the bed coefficient k is chosen reasonably, then its spring model will be more consistent with the actual foundation conditions and more satisfactory results can be obtained.

Winkler Foundation Model and Parameters
In 1867, Winkler, a Czech engineer, proposed an elastic foundation model that divides the soil into several vertical soil columns, each of which can be replaced by an independent spring whose pressure on each spring is proportional to the deformation of that spring, and the shape of the vertical displacement at the base of the foundation is similar to that of the base reaction graph, as shown in Equation (1).

𝑝(𝑥, 𝑦)=𝑘𝑠(𝑥, 𝑦)
(1) in the formula: --pressure intensity per unit area, Unit: / 2 ; --foundation Settlement, Unit: ; --Substrate coefficient, Unit: / 3 .Because the foundation of the kiln is a strip foundation and because the foundation only bears the self-weight of the kiln, which is equivalent to the uniform load acting on the foundation, the settlement result when subjected to the uniform load  0 is shown in Equation (2).The value of the bed coefficient  in equation (1) can be derived by the elasticity theory calculation method, as shown in equation (3).
in the formula: -coefficient of influence of settlement at corner points; --width of the load;   -modulus of deformation of foundation soil; --poisson's ratio of foundation soil, take 0.3.According to the geotechnical investigation report and field test of the kiln plant, mortar strength grade (M7.5) and sintered common brick (MU20) type, referring to"the Code for Design of Masonry Structures" (GB 50003-2011), the design value of tensile strength of masonry structure is taken as   =0.29MPa, and the design value of compressive strength is taken as ==-2.39MPa.structural material parameters are shown in

Comparative Study of Two Foundation Models
The finite element simulation models of the structure were established using the actual foundation model (Scheme 1) and the Winkler spring foundation model (Scheme 2), respectively.The structure of the kiln is simulated by three-dimensional shell unit, and the foundation is simulated by spring unit and solid unit respectively, and the finite element model is shown in Figure 2. A spring unit is set every 800 mm along the length of each kiln, and the bed coefficient is calculated according to Equation ( 3), and the self-weight action is applied to the kiln structure to obtain the numerical simulation results of the structure under the two foundation models.And the maximum vertical displacements of six representative kiln holes with two types of foundation models are shown in Table 2.As seen in Table 2, the maximum difference in vertical displacement between two foundation solutions for kiln No.①is 0.566mm; the maximum difference in vertical displacement between two foundation solutions for kiln No.② is 0.359mm; the maximum difference in vertical displacement between two foundation solutions for kiln No.⑥ is 0.235mm; the maximum difference in vertical displacement between two foundation solutions for kiln No.⑦ is 0.353mm; the maximum difference in vertical displacement between two foundation solutions for kiln No.⑧ is 0.416mm; the maximum difference in vertical displacement between two foundation solutions for kiln No.⑫ is 0.361mm.The average error of vertical displacement of the top of each kiln under both schemes is less than 10%, which indicates that it is feasible to use the Winkler foundation model instead of the actual foundation model, and also provides a theoretical basis for calculating the uneven settlement of the kiln foundation in the next step.

Effect of Local Settlement on the Stress State of Kiln Structure
After the field survey, ② kiln cave and ⑬ kiln cave junction there is 3.6 meters length range of foundation settlement occurred 8 mm.According to Section 2, the Winkler spring foundation model can be used instead of the actual foundation model, so vertical displacements are applied to the spring units in this range to simulate the uneven settlement of the foundation.In order to analyze the effect of uneven settlement on the overall kiln stress state, four scenarios of 0mm, 8mm, 10mm and 12mm were calculated.In order to understand the stress state of the overall structure of the kiln, a total of eight stress paths were defined, as shown in Figure 3. Figure 4 shows the various parts of the kiln schematic diagram.The variation of the main tensile stress on each path under the four calculation schemes was collated, as shown in Figure 5.   (unit: MPa) As shown by path1 and 2, when no settlement occurs, the vault of the kiln is in a tensile state, with maximum values of 0.036 MPa and 0.199 MPa, respectively, where the bottom of the wall of path 1 is also in a tensile state, with a maximum value of 0.268 MPa, and at this time, the maximum principal tensile stress in each part is less than the design value of the tensile strength of the material.With the increase of settlement, all parts of path1 and 2 are in tensile state and the tensile stress increases, which may cause structural damage to the kiln and must be paid attention to, and reinforcement measures must be taken in a timely manner, and at the same time, it is shown that the further away from the location of settlement, the smaller the impact of local settlement.
As can be seen from path 3 and 4, the stresses in the vault of kiln ② were relatively uniform when no settlement occurred at the top of kiln ②, but with the increase of settlement, the main tensile stress at the connection between kiln ② and kiln ⑬ changed dramatically, increasing from 0.214 MPa to 2.408 MPa.With the increase of distance, the magnitude of stress change decreases, and the effect of the local settlement of kiln ② on the stress of the arch ring of kiln ⑬ connected to it is basically on the connected side, and has no effect on the stress of the other side.
From path5 and 6, it can be seen that path5 is affected by the local settlement of ② kiln hole, the maximum main tensile stress at the left arch foot and left arch waist position changes a lot, and the tensile stress of the left arch ring has a significant increase, and the stress on the path6 has a slight change, but the magnitude of the change is very small, which indicates that with the increase of distance, the effect of the local settlement on the structure of the state of stress in the gradual reduction.As can be seen from path7 and 8,after the local settlement of No. ② kiln cave, the maximum main tensile stress in the arch of No. ⑬ kiln cave gradually decreases with the increase of distance, and the influence range is basically within 1.5 times of the settlement area, and the maximum main tensile stress in the arch of No. ⑧ kiln cave basically does not change, which is mainly due to the role of the arch ring of No. ⑬ kiln cave connected with it.

Conclusion
In this paper, Winkler theory is introduced to simulate the kiln factory foundations of Changle loess industrial site, and numerical simulation analysis of the local settlement of kiln foundations is carried out to obtain the influence law of the local settlement of kiln foundations on the overall structural stress state, and the main conclusions are as follows: (1) Numerical simulation models of the kiln plant were established using the actual foundation model and the Winkler spring foundation model, respectively.The results show that the vertical displacements of the kiln structure are basically comparable under the two different foundation models, and the vertical displacement distribution pattern of each kiln is basically the same.The average error of vertical displacement at the top of each kiln under both schemes is less than 10%, which indicates that it is feasible to use the Winkler foundation model instead of the actual foundation model; (2) When settlement does not occur, the vault and arch waist of No. ② kiln are in tension, with the increase in settlement, the main tensile stress in various parts of the increase will lead to the overall destruction of the kiln structure, must pay attention to, and take timely reinforcement measures;

Figure 2 .
Figure 2. Finite element model diagram.Table 2. Comparison of the displacement of the top of each kiln under the two schemes.(unit: mm)

Figure 5 .
Figure 5. Path maximum principal tensile stress variation graph.(unit: MPa)As shown by path1 and 2, when no settlement occurs, the vault of the kiln is in a tensile state, with maximum values of 0.036 MPa and 0.199 MPa, respectively, where the bottom of the wall of path 1 is also in a tensile state, with a maximum value of 0.268 MPa, and at this time, the maximum principal tensile stress in each part is less than the design value of the tensile strength of the material.With the increase of settlement, all parts of path1 and 2 are in tensile state and the tensile stress increases, which may cause structural damage to the kiln and must be paid attention to, and reinforcement measures must be taken in a timely manner, and at the same time, it is shown that the further away from the location of settlement, the smaller the impact of local settlement.As can be seen from path 3 and 4, the stresses in the vault of kiln ② were relatively uniform when no settlement occurred at the top of kiln ②, but with the increase of settlement, the main tensile stress at the connection between kiln ② and kiln ⑬ changed dramatically, increasing from 0.214 MPa to 2.408 MPa.With the increase of distance, the magnitude of stress change decreases, and the effect of the local settlement of kiln ② on the stress of the arch ring of kiln ⑬ connected to it is basically on the connected side, and has no effect on the stress of the other side.From path5 and 6, it can be seen that path5 is affected by the local settlement of ② kiln hole, the maximum main tensile stress at the left arch foot and left arch waist position changes a lot, and the tensile stress of the left arch ring has a significant increase, and the stress on the path6 has a slight change, but the magnitude of the change is very small, which indicates that with the increase of distance, the effect of the local settlement on the structure of the state of stress in the gradual reduction.As can be seen from path7 and 8,after the local settlement of No. ② kiln cave, the maximum main tensile stress in the arch of No. ⑬ kiln cave gradually decreases with the increase of distance, and the influence range is basically within 1.5 times of the settlement area, and the maximum main tensile stress in the arch of No. ⑧ kiln cave basically does not change, which is mainly due to the role of the arch ring of No. ⑬ kiln cave connected with it.
Table 1 is shown.Table 1.Structural material parameters.

Table 2 .
Comparison of the displacement of the top of each kiln under the two schemes.(unit: mm)