Study on the influence of expansive soil slope on bridge foundation under excavation and rainfall conditions

Expansive soil is a kind of special soil that mainly composes of hydrophilic minerals and has significant deformation characteristics of water absorption expansion and water loss shrinkage. The characteristics often bring great harms to engineering construction. Relying on the bridge foundation on the expansive soil slope of the Yangtze‐to‐Huaihe Water Diversion Project, this study simulates the humidity field with temperature field for expansive soil using the numerical simulation method when short‐term or long‐term rainfall occurs during slope excavation. The horizontal deformation of bridge foundation on expansive soil slope considering four factors such as self‐weight of slope soil, excavation unloading rebound, hygroscopic expansion of expansive soil, and decrease of soil strength caused by change of water content was studied herein. The research results show that during the excavation process, the bridge foundation produces a horizontal displacement towards the top of the slope, and the rainfall after excavation introduces a decrease to the horizontal displacement of the bridge foundation and gradually develops in the opposite direction. During the excavation process, the rebound of slope excavation is the main factor affecting the horizontal displacement of the bridge foundation. Under short‐term rainfall conditions, the hygroscopic expansion of the slope soil becomes the main influencing factor. Under long‐term rainfall conditions, the main influencing factors of horizontal deformation of bridge foundation are the self‐weight of slope soil and the decrease of soil's strength.

commonly carried out. The results showed that the macro mechanical properties were closely related to the absorption and release of water from soils. 3 Most importantly, the cohesion, internal friction angle, and shear strength of expansive soil decreased with the number of dry-wet cycles. [4][5][6][7] Both the characteristics of swelling shrinkage and the reduction of hygroscopic strength of expansive soil were attributed to its complex microstructure. As same as other clays, the deformation of expansion and contraction can also be divided into both the macro-structure part and the micro-structure part. 8 The mechanism of soil swelling induced by absorbing water was the formation and the increase of thickness of the water film, which generates a "wedge" force between clay particles, increasing the particle spacing and enlarging the pores. Therefore, the strength of expansive soil is finally determined by the distribution of pores in the soil. [9][10][11] As expansive soil is very sensitive to water variation, it is hygroscopic and expansive during rainfall, and the bridge foundation on the expansive soil slope will also be affected by the slope deformation.
When the bridge foundation is located on the slope, the situation becomes very complex, and many scholars have carried out researches on it. Martin investigated the influence of slope horizontal displacement on bridge pile foundations. 12 Frank F studied the displacement, bending moment, and reaction force of the pile foundation on the unstable slope. 13 Fu studied the lateral characteristics of the pile foundation and stable pile on the high and steep slope. 14 Andrea studied the vertical and lateral response of a scoured bridge pier founded on a cylindrical caisson foundation embedded in a layer of dense sand. 15 Although many scholars have done a lot of researches on the bridge foundation on the slope, few bridge foundations are built on the expansive soil slope due to the special properties of expansive soil. Therefore, there are little researches in this area. The excavation rebound has attracted attention in the Terzaghi Era, and many scholars have studied the rebound deformation of excavation, such as foundation pit, [16][17][18] channel, 19 and slope. 20 The excavation of the foundation pit will generally lead to the vertical rebound at the bottom of the pit, while in the process of slope excavation; there is not only a vertical rebound but also a horizontal rebound on the slope. Hence, when the slope is excavated, the bridge foundation on the slope will deform not only vertically but also horizontally.
Therefore, when rainfall occurs during the excavation of expansive soil slope, the deformation of expansive soil slope is affected by the self-weight of slope soil, excavation unloading rebound, hygroscopic expansion of soil, and decrease of soil strength caused by change of water content. For the bridge foundation on the expansive soil slope, affected by rainfall and excavation, the deformation of the expansive soil slope will also act on the bridge foundation, which may lead to the large displacement of the bridge foundation, and ultimately affect the safety and use of the bridge. In this paper, the influence law and mechanism of the bridge foundation on the slope of the Yangtze-to-Huaihe Water Diversion Project in China under the conditions of excavation and rainfall are studied.

THEORY OF TEMPERATURE FIELD SIMULATION OF UNSATURATED SOIL SEEPAGE FIELD
Moisture content has a significant impact on the swelling properties of expansive soil. Therefore, the study first analyze the change and distribution of the moisture content of the soil. The temperature field control equation is similar to the seepage field and humidity field control equations. 21 The changes of water content in regional soil can be expressed by transient seepage analysis based on Darcy's law, and the changes in the soil temperature can be expressed by heat conduction analysis. 22 The two physical phenomena have the same mathematical expression.
The thermal energy balance equation without internal heat sources in the heat conduction problem is: x where: is the material density; C v is the specific heat capacity of the material (J / (kg ⋅ • C)); T is the transient temperature of the object; is the thermal conductivity (w / (kg ⋅ • C)). Rainfall and water level rise and fall infiltration are unsaturated seepage problems. The traditional three-dimensional unsaturated seepage problem has a continuous volume equation: where: w is the volume moisture content of soil unit; w is the pore water density; d is the dry density of soil mass; w is the moisture content; k is the permeability coefficient; h is the total head. The change of volumetric water content per unit time caused by the change of net normal stress v − u a and matric suction u a − u w can be determined by the following formula 23 : where: v is the volume stress; u a is the pore air pressure; u w is the matric suction of soil, and m w 2k are the water volume coefficients corresponding to changes in normal stress and matric suction. In the process of unsaturated seepage, it is generally assumed that the total stress keeps a constant with time, that is, v ∕ t = 0; Assuming that the pore air pressure is the same as the atmospheric pressure or changes little with time, then u a ∕ t = 0. The water head is the sum of the gravity head and matric suction head: h = z − u w ∕ w g. Then equation can be obtained: When the water content in the soil is not close to the saturated water content, the matric suction head changes significantly with the water content. In this case, the last item on the left of Equation (4) is negligible compared with the second item in the bracket on the left, 24 k y ∕ y ≈ 0. The matric suction head is set as u = u w ∕ ( w g), so the differential equation of the unsaturated seepage problem is simplified as follows: where: C w = w gm w 2k , it is the specific water capacity (1 / M), that is, the amount of water that can be released or absorbed by the soil when the matric potential of the unit volume of soil decreases or increases by one unit head.
The change of water head (moisture content) is regarded as the change of temperature, which k can be replaced by and C w can be replaced by C v so that the temperature field can be used to simulate the seepage field.

The equivalent condition between hygroscopic expansion deformation and temperature rise expansion deformation
The two theories have a common form of linear expansion, and the strain caused by temperature change and the strain caused by water content change can be expressed as: where: and are linear expansion coefficients of temperature and humidity respectively ΔT and Δw are the changes of temperature and moisture content per unit volume of soil respectively. When the two strains are equal, the strain and stress of the humidity stress field can be calculated equivalently by using the temperature field stress theory.
Through experiments, Zhang 25 deduced that: where: H is the unloaded expansion rate obtained from the test; Δw is the change of moisture content (because the unit of Δw is %, so = 0.01 ; According to the definition of the expansion coefficient, the expansion coefficient is 1/100 of the slope of the immersion expansion curve, so = 0.01 H Δw ).
The field expansive soil was taken for the tests, and the relationship between unloaded expansion rate and water content is shown in Figure 1. The initial stage of the immersion curve is approximately linear. 26 The linear expansion was used to obtain the hygroscopic expansion coefficient and the coefficient of temperature expansion, which is shown in Table 1.

The equivalent condition between the permeability coefficient and the thermal conductivity coefficient
According to the test results of Wang Zhao 27 and Liu Xiaowen, 28 the relationship between matric suction and water content of unsaturated soil is as follows: According to the definition of m w 2k , the specific water capacity is the negative value of the partial derivative of the volumetric water content to the matric suction u a − u w : Without considering the change of pore pressure, replace Equation (8) with Equation (9) and express it by gravitational water content: where: a and b are test constants. Equation (1) is equivalent to Equation (4), and the temperature field is used to simulate the humidity field, assuming k x = k y = k z and x = y = z , replace with k, and the coefficients on the right of the two equations shall also be equivalent, that is:

The equivalent condition between the thermal expansion force and the hygroscopic expansion force
The change of expansion force caused by temperature variation can be expressed as: where: K is the bulk modulus.
The swelling force of expansive soil with different moisture contents can be obtained from the moisture absorption tests. Then the relationship between water content and swelling force can be established by curve fitting: When the temperature expansion force is equal to the humidity expansion force, replace Equation (13) with Equation (12): When using the temperature field to simulate the humidity field, the temperature expansion coefficient of Equation (7) instead of the humidity expansion coefficient, the heat conductivity coefficient of Equation (11) instead of the permeability coefficient, and the temperature expansion force of Equation (14) instead of the humidity expansion force can be used to simulate the humidity field equivalently through the temperature field. The mechanical indexes and heat conduction indexes of expansive soil with a dry density of 1.45 g/cm 3 and initial moisture content of 15% (saturated moisture content of 35%) under different subsequent moisture content are shown in Table 2.
In order to facilitate simulation, based on Mohr-Coulomb elastic-plastic constitutive model, the expansion deformation caused by humidity change is added to Mohr-Coulomb constitutive model by using the method of literature 29 : The elastic modulus E ij , Poisson's ratio ij , cohesion c, and internal friction angle of the soil change with the water content.

DEFORMATION MECHANISM OF EXPANSIVE SOIL SLOPE UNDER EXCAVATION AND RAINFALL
When rainfall occurs during the excavation of expansive soil slope, the slope is mainly affected by four effects (as shown in Figure 2): excavation rebound, expansion deformation, self-weight deformation, and strength reduction. If the slope is a rigid body, after the excavation the slope will not appear any deformation, and actually, the soil is usually the elastic-plastic body, on the one hand, the slope soil rebounds after excavation and unloading, on the other hand, the unloading surface deforms under the action of self-weight. For expansive soil, the slope soil will expand and reduce its strength under the condition of rainfall. When the quality of the soil is good (such as hard clay) and there is no rainfall, springback plays a dominant role in the process of slope excavation, that is, the deformation of slope soil caused by springback is greater than that caused by excavation unloading. When the soil quality is poor (such as soft clay), excavation unloading plays a leading role, that is, the deformation of slope soil caused by excavation unloading is greater than that caused by excavation rebound. In the process of rainfall, when the rainfall time is short, the slope soil will expand, and when the rainfall time is long, the slope soil strength will decrease greatly, and the reduction of strength becomes the dominant factor of slope soil deformation. The soil is regarded as a simple elastic-plastic body with no self-weight and no expansion. After excavation and unloading, the slope springs back. The horizontal deformation of the slope soil is shown in Figure 2B (the displacement to the right is positive in the figure). After the soil on the slope is removed, the soil will rebound vertically upward. Due to the coordination of the overall deformation, the vertical deformation of the soil will also affect its horizontal deformation. Near the slope foot, the soil will produce horizontal displacement to the left, and the slope top will produce horizontal displacement to the right. When there is a structure on the slope, the structure will rotate towards the top of the slope under the influence of slope soil rebound deformation, and the translation of the structure may be towards the top of the slope.
When the slope soil is regarded as an expansion body without self-weight, and the excavation rebound is not considered (increasing the elastic modulus and expansion coefficient of soil, the ratio of resilience to expansion can be neglected), the slope expands hygroscopically after rainfall, and the horizontal deformation of slope soil is shown in Figure 2C (the positive displacement in the figure is to the right). Under the action of expansion load, the slope produces expansion deformation perpendicular to the slope line, the horizontal displacement at the toe and top of the slope is small, and the horizontal displacement at the middle of the slope is large. When there is a structure on the slope, the structure will shift and rotate towards the bottom of the slope under the action of soil expansion and deformation.
Without considering rebound and expansion, the horizontal deformation of slope under the action of self-weight and soil strength reduction is shown in Figure 2D. If the sliding surface is circular, the slope line rotates around the center of the circle.
When rainfall occurs during the excavation of expansive soil slope, the slope is jointly affected by the above four actions, and its deformation is also the coupling of the slope soil deformation under the four actions. The final deformation state of slope soil is related to many factors such as soil strength, elastic modulus (rebound modulus), density, hygroscopicity, and expansibility. If there are structures on the slope, the deformation of structure and soil is also mutually affected.

Modeling
The Anhui section of the Yangtze-to-Huaihe Water Diversion Project is 293.6 km, and 68 bridges will be built, many of which are built on the expansive soil slope. Affected by the channel navigation function, the piers of channel crossing bridges can only be distributed on the slope. The channel excavation and bridge foundation are carried out alternately. When the bridge foundation is constructed first and then the channel excavation is carried out, under the condition of rainfall, the characteristics of swelling deformation and strength reduction of expansive soil and the rebound characteristics of slope excavation are bound to have a certain effect on the bridge foundation. Therefore, this paper selects several typical construction conditions for analysis to study the action mode and influence mechanism of expansive soil slope on bridge foundation under the comprehensive action of excavation and rainfall. A three-dimensional numerical calculation model for determining the effect of channel slope excavation on bridge foundation under six working conditions is established, as shown in Table 3 and Figure 3. There are two kinds of channel slopes: 24 m high slope (working conditions 1 ∼ 3 ) and 42 m high slope (working conditions 4 ∼ 6 ). After the channel slope is excavated to a certain depth for H, the bridge or bridge foundation construction shall be carried out, and then the remaining channel slope excavation shall be continued (the excavation thickness of each layer is the same). Taking the early slope excavation and the completion of bridge or bridge foundation construction as the initial conditions, the interaction between expansive soil slope and bridge foundation under the conditions of subsequent residual slope excavation and rainfall during excavation is studied.
The 24 m high side slope is divided into four levels, each level is 6 m high, and the slope ratio is 1:3. The 42 m high side slope is divided into seven levels, each level is 6 m high, the lower four level slope ratio is 1:2, and the upper three level slope ratio is 1:3. Due to the channel navigation requirements, the bridge pile foundation is located on the slope, and the dimensions of the bridge foundation (including pier and pile foundation) are shown in Table 4.
Equivalent hygroscopic expansion by heating expansion, and coupling Mohr-Coulomb constitutive model (as described in Sections 2 and 3), the expansive soil can be simulated. At the same time, Mohr-Coulomb constitutive model is adopted for the sandy soil layer, with a density of 2 g/cm 3 , elastic modulus of 60 MPa, internal friction angle of 29 • , and cohesion of 0 kPa; The lowest layer is 8 m thick foundation rock, which adopts linear elastic constitutive model,  with an elastic modulus of 100 MPa and a density of 2.2 g/cm 3 . For the convenience of calculation, the three-dimensional solid element coupled with temperature and displacement was also used for sand and bedrock, but the value of the temperature expansion coefficient is very small compared with expansive soil, and its expansion can be ignored. A linear elastic constitutive model was adopted for the bridge and foundation, with an elastic modulus of 30 GPa and a density of 2.45 g/cm 3 . Hard contact is established between the pier, bearing platform, and pile foundation embedded in the soil, the friction coefficient between the expansive soil and the structure is 0.2, and the friction coefficient between the sand and bedrock and the structure is 0.24.
The structure (including bridge, pile foundation, pier, bearing platform, abutment, and abutment pile foundation) adopts three-dimensional eight-node elastic solid element, and the slope soil adopts three-dimensional eight-node elastic-plastic solid element coupled with temperature and displacement.
The corresponding normal displacement constraints are applied to the side of the soil model and the middle axle section of the bridge (take the center line of the channel as the axis of symmetry, and take half of the channel and bridge for modeling and analyzing), the displacement in three directions is constrained by the bottom of the soil model, and the rotation displacement is constrained by the side of the soil at the slope bottom (the middle axle surface of the whole channel) and the middle axle section of the bridge. The gravity acceleration of 10 m/s 2 is applied to the whole model, the initial temperature field is applied to the whole soil model (when the initial moisture content of the soil is 15%, the corresponding temperature is 15 • C, as shown in Table 2) and the surface temperature is applied to the slope surface (when the final moisture content of the soil after rainfall is 60%, the corresponding temperature is 131.33 • C, as shown in Table 2).
During the simulation, the in-situ stress balance shall be carried out after the early slope excavation and the bridge or bridge foundation construction, and then the subsequent construction and rainfall simulation shall be carried out.

Impact of excavation rebound on bridge foundation (no rainfall)
In the excavation stage, the horizontal displacement of the bridge foundation is shown in Figure 4 (the displacement towards the channel center is positive). In the excavation stage, due to the rebound effect (the rebound effect is greater than the self-weight effect), the horizontal displacement of the bridge foundation develops towards the slope top. With the increase of excavation depth, the horizontal displacement of the bridge foundation increases continuously. For the 24 m slope, when there is a bridge (upper bridge span structure, the same below), the bridge can restrict the development of horizontal displacement of the foundation, and the horizontal displacement of the bridge foundation is the smallest. Under the condition of surcharge, the horizontal displacement of the bridge foundation decreases. When there is a bridge, in the first and second steps of excavation, the change of bridge foundation displacement is smaller, in the third and fourth steps of excavation, the rebound effect becomes larger and larger, and the displacement of the bridge foundation increases gradually. When there is no bridge, the situation is similar to that with bridge, but the displacement is much larger.
For the 42 m high slope, the horizontal displacement change of the bridge foundation at the lower part of the slope is the smallest (the soil at the slope bottom is mainly vertical deformation), and the horizontal displacement change of the bridge foundation at the middle and upper part of the slope is larger. When the bridge foundation is located at the lower part of the slope, the displacement change of the bridge foundation is relatively smaller after the first and second excavation steps, and slightly larger after the third and fourth excavation steps. When the bridge foundation is located at the middle and upper parts of the slope, the horizontal displacement of the bridge foundation changes relatively larger after the first to the third step of excavation, and smaller after the fourth to sixth steps of excavation. So, the displacement change of the bridge foundation during each excavation is related to the relative position relationship between bridge foundation and newly excavated slope.
The displacement vector of the slope after excavation is shown in Figure 5. For the 24 m slope, after excavation, the slope moves clockwise around the bridge foundation, but the soil displacement is smaller on the upper side of the foundation, larger on the lower side of the foundation, and reaches the maximum at the slope bottom. The displacement mode of the slope also causes the foundation to rotate clockwise. When there is a bridge, the rotation of the foundation displacement is limited. When there is surcharge on the top of the slope, the soil displacement on the upper side of the foundation is downward along the slope, thus offsetting part of the rebound deformation. For the 42 m slope, the excavated slope will also produce clockwise displacement around the foundation. When the foundation is located at the lower part of the slope, the rebound of the slope is smaller due to the relatively smaller excavation depth. When the foundation is displaced in the middle of the slope, the rebound of the slope is larger, and the foundation is most affected by the slope. When the foundation is located at the upper part of the slope, on the one hand, the rebound at the upper part of the slope is smaller, on the other hand, the foundation is far from the slope bottom, and the effect of the foundation on the rebound is not as great as that in the middle of the slope.

Influence of short-term rainfall after excavation on bridge foundation
Under the condition of short-term rainfall (continuous rainfall for 1 day) after excavation, the horizontal displacement of the foundation is shown in Figure 6. After short-term rainfall, on the one hand, the slope soil will absorb moisture and expand; on the other hand, its strength will also decrease to a certain extent. Under the combined action of these two factors, the horizontal displacement of the bridge foundation is reduced or even reversed (24 m slope). For the 24 m slope, since the horizontal displacement of the foundation is smaller after excavation, it is significantly affected by the change of slope soil after short-term rainfall. The horizontal displacement of the foundation decreases quickly and gradually develops in the opposite direction. When there is bridge, the change rate of horizontal displacement of the bridge foundation is slower, while when there is no bridge and surcharge on the slope top, the change rate of horizontal displacement is the fastest.
For 42 m slope, the horizontal displacement of the foundation is larger due to excavation. Although the horizontal displacement of the foundation has recovered after the short-term rainfall, the displacement direction has not changed. When the foundation is located in the middle of the slope, its horizontal displacement changes the slowest.
The displacement vector of the slope after the short-term rainfall is shown in Figure 7. Rainfall will reduce the strength of slope soil, which will also lead to the weakening of rebound, and the displacement of the bridge foundation caused by excavation will also be reduced to a certain extent. At the same time, in the process of rainfall, the slope soil will absorb moisture and expand, (the displacement vector is perpendicular to the soil surface), and the slope soil produces a displacement vector opposite to the rebound effect. In addition, when the strength decreases, the displacement of the slope along the slope surface will also increase to a certain extent under the action of self-weight.

5.2.3
Influence of long-term rainfall after excavation on bridge foundation The horizontal displacement of the foundation under long-term rainfall (continuous rainfall for 10 days) is shown in Figure 8. After the long-term rainfall, the horizontal displacement of the foundation develops greatly, and the displacement direction changes totally. For the 24 m slope, due to the thick expansive soil layer, the slope landslide occurs after the long-term rainfall. During the whole rainfall process, the horizontal displacement of the foundation changes consistently under different conditions, and the displacement is not much different. Under the condition of no bridge and no surcharge, the foundation is slightly affected by the rainfall.
For the 42 m slope, the bridge foundation located in the middle and lower part of the slope is not directly affected by the expansive soil layer, and though the bridge foundation at the upper part of the slope is located in the expansive soil layer, the expansive soil layer is thin after excavation. Although the long-term rainfall will lead to the landslide of the upper expansive soil layer, the strength of the lower sandy soil layer shows no decreases (the decrease of sandy soil moisture absorption strength is not considered in the calculation). During the whole rainfall process, the horizontal displacement of the foundation changes linearly, but the horizontal displacement of the foundation located on the upper part of the slope develops slightly faster.
Under the condition of the long-term rainfall, the displacement vector of slope soil is shown in Figure 9. Under the condition of the long-term rainfall, the strength of expansive soil decreases greatly, its expansion basically reaches the limit, and the slope slides. After landslide, the displacement vector of slope soil is quite different from that after the short-term rainfall, and the displacement vector is no longer perpendicular to the soil surface, which indicates that the deformation of the slope is mainly controlled by its self-weight and strength.

CONCLUSION
This paper simulates humidity field with temperature field for expansive soil using the numerical simulation method, when rainfall (short-term and long-term) occurs during slope excavation, the horizontal deformation of bridge foundation on expansive soil slope under four actions (self-weight of slope soil, excavation unloading rebound, hygroscopic expansion of expansive soil and decrease of soil strength caused by change of water content) was studied, and the conclusions are as follows: 1. The rebound of slope soil during excavation has a certain effect on the horizontal displacement of the bridge foundation, which is related to the excavation depth of the slope and the position of the bridge foundation. The greater the excavation depth of the slope is, the greater the horizontal deformation of the bridge foundation is. When the excavation surface crosses the bridge foundation, the excavation increases the displacement of the bridge foundation. When the bridge foundation is located in the middle of the slope, the excavation leads to the maximum horizontal displacement of the bridge foundation, and when it is located in the lower part of the slope, the excavation leads to the minimum horizontal displacement of the bridge foundation. 2. During rainfall after excavation, the soil displacement of slope and the horizontal displacement of the bridge foundation gradually decrease and develop in the opposite direction (compared to the rebound). In the 24 m slope, when the foundation is not constrained by the bridge, the displacement development rate is slightly faster, but the development law is basically the same (compared to the bridge is constructed). In the 42 m slope, when the bridge foundation is located within the expansive soil layer, the displacement of the bridge foundation develops slightly faster, but the displacement development law is also basically the same (compared to the foundation located at other place). 3. During the excavation process, the rebound of slope excavation is the main factor affecting the horizontal displacement of the bridge foundation. Under the condition of the short-term rainfall, the hygroscopic expansion of expansive soil is the main factor affecting the horizontal displacement of the bridge foundation. Under the condition of the long-term rainfall, the self-weight and strength reduction of slope soil are the main factors affecting the horizontal displacement of the bridge foundation.

DATA AVAILABILITY STATEMENT
The data used to support the findings of this study are included within the article.