# FEM Analysis and Simplified Approach for a Single Energy Pile Subjected to Thermomechanical Loads

Authors: Zhong-jin Wang; Peng-fei Fang; Ri-hong Zhang; Kui-hua Wang; Xin-yu Xie
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

The distribution of temperature in sand soils was measured through laboratory tests, and the temperature influence on friction resistances at the concrete-soil interface was analyzed. Based on the results of laboratory tests, the finite element model was established using the sequential thermal coupling method. The influences of temperature on the bearing characteristics of energy pile were analyzed. The analysis results show that the cyclic temperature will cause additional displacement along pile depth. It is pointed out that if applied vertical loads at energy pile head exceed the value from which nonlinear settlements would be initiated, irrecoverable additional settlement will occur at pile head. Based on the analysis results, a simplified approach was proposed to estimate the zero point of additional displacement along pile shaft and the additional axial pile force. The comparison between the calculated results obtained by the proposed method and that of ABAQUS on single energy pile was given to verify the accuracy of the proposed method. It is shown that reasonable predictions can be obtained without expensive and time-consuming analyses by the proposed method in this paper.


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## Body

## 1. Introduction

Over the past decades, many experimental and theoretical researches have been carried out to analyze the bearing characteristics of pile foundations, because of their extensive application in many infrastructures and structures [1–3]. In recent years, pile foundations have been increasingly used in an innovative form of energy piles, which couples the structural role of pile foundations to that of heat exchangers to exploit the large thermal storage capabilities of the ground, particularly when these energy piles are coupled to heat pumps [4–6]. Due to their economic benefit and environment-friendly advantages, energy piles were widely used around the world recently. Although having been widely used in the world, various aspects of the technology are relatively unknown.As carrier of heat exchanger, piles will expand and contract while enduring heating and cooling, resulting in thermomechanical phenomena. In the past years, some efforts had been made to study the mechanisms of thermomechanical soil-structure interaction. In situ tests were carried out to investigate the changes of pile characteristics due to temperature, and the results shown that additional thermal stresses would be mobilized in the pile during the heating and cooling. The change of energy pile bearing characteristic was subjected to the restraint conditions of the tested piles, and it was also found that the use of energy piles causes significant thermally induced additional deformation in the pile itself [7–10]. Model tests were also carried out to examine the heat transfer performance and bearing characteristics of piles with embedded tubes under normal working conditions over repeated temperature cycling, and the results show that the thermal stresses were superimposed with the mechanical stresses [11–14]. Centrifuge modeling of soil-structure interaction in energy foundations was also carried out to measure the transient thermomechanical response of end-bearing energy pile during heating-cooling cycles, and the result shown that the model pile was affected by the heating and cooling cycles [15, 16], being consistent with the conclusion that the effect of temperature on the shear strength of sand, clay, and the clay-concrete interface is negligible [17, 18].To obtain the characteristics of energy pile foundations during heating-cooling cycles, numerical method was also used to analyze the variations of pile stresses and settlements caused by temperature. Scholars also have carried out many researches to analyze the characteristics of piles during the heating and cooling processes [19–22], and the finite element method was still one of the most extensive methods used to simulate energy piles subjected to a constant mechanical load and a seasonally cyclic thermal load over several years. The results from the above analysis show that the finite element method is a very effective mean to simulate the change of energy pile characteristics.As discussed above, it is conclude that both the thermal-induced displacements and stresses must be taken into account in the geotechnical design of energy piles despite being acceptable under normal working conditions. The additional axial force and pile shaft friction resistance change are complex, being closely related to the site engineering geological conditions. In this paper, the conduction characteristics of temperature on concrete-soil interface were tested through laboratory tests, and a finite element model was established based on the results from the above laboratory tests. The influences of temperature load on bearing characteristics of a single energy pile were analyzed, and a simplified method was proposed to estimate the zero point of additional displacement and the additional pile axial force was calculated by the proposed method. The comparisons between the calculated results obtained by the proposed method and that of ABAQUS were given to verify the accuracy of the proposed method. It is shown that reasonable predictions can be obtained without expensive and time-consuming analyses by the proposed method in this paper.


## 2. Laboratory Test

To study the changes of bearing behavior caused by temperature, it is necessary to study the change of temperature with time at the concrete-soil interface and its distribution around the concrete and surrounding soil. A self-developed friction resistance testing device, which can be considering the influence of temperature as shown in Figure1, was used to measure the temperature conduction and friction change on the concrete-soil interaction surface. The heat pipes were embedded in the prefabricated concrete plate, the distance from the lower surface is 1mm. The temperature sensors were buried in the soil at 1mm from the concrete-soil interface, which is regarded as the pile-soil interface. Test material parameters were shown in Table 1.Table 1
Parameters of model tests.
Sand parameters
Water content
C
u
C
c
Size (mm)
5%
3.1
0.9
300x300x75
Concrete parameters
Size (mm)
500x350x100Figure 1
Temperature conduction and friction testing device.The device consists of upper and lower shear boxes with unequal sizes. The upper shear box is fixed on the reaction force rack, and the lower shear box is connected to the reaction force rack through the guide rails, sliding along the guide rails during tests. The upper shear box has a vertical compression plate that can apply normal stress to the soil sample. The loading device is fixed at one side of the lower shear box, and horizontal loads can be applied, and a displacement sensor is provided testing horizontal displacement. The above device was also described by the WANG et al. [13].The constant temperature heating system is mainly composed of a heating pool, an electric heater, delivery tubes, a thermal fluid, a delivery pump, a temperature probe, and an intelligent thermostat. The temperature probe measures the thermal fluid temperature in the heating pool, and the signal output of the temperature probe is connected with the signal receiving end of the intelligent controller. When the temperature of the thermal fluid in the heating pool reaches the set value, the intelligent temperature controller controls the electric heater to stop working. The delivery pump is connected with the catheter, and the catheter is embedded in the preparation process of the concrete. After the temperature of the thermal fluid reaches a set value, the constant-temperature thermal fluid passes through the inside of the concrete by the delivery tubes. By heat conduction, the temperature of test block can be to a constant value, and the soil is heated through the block. The change of temperature in the soil is measured through the embedded temperature sensor.11 flexible tubes are embedded in the test concrete block at equal distance, the distance between the hoses and top of the test concrete block is about 10mm, and the length of the hoses is 3 m. When the concrete reaches the certain strength, the wood models were removed and maintained. The surface of the test concrete block is polished, as shown in Figure2.Figure 2
Molding process of the test concrete block.
(a)
(b)When the temperature of heating fluid is stable, the temperature sensor values were begun recording. The tested results were shown in Figure3. It can be seen that the changes of temperature with time at the concrete soil interface were nonlinear, and the temperature were tending to a constant value after heating for a period of time. From Figure 3, it also could be seen that the frictional resistance with the relative displacement at the concrete-soil interface has no significant changes as the temperature changing. The above results are consistent with existing research conclusions by Yavari et al. [17].Figure 3
Experimented results of temperature conduction and friction.
(a)
Temperature changes at pile-soil interface with time
(b)
Curves of interface friction resistance-displacement


## 3. Finite Element Analyses for Energy Pile due Temperature

### 3.1. Finite Element Model

According to the above laboratory test results, it can be seen that the temperature at the concrete-soil interface increases with time, and tends to be constant with time. The temperature has no significant effect on the contact friction between concrete and soils. Based on the results mentioned above, a finite element model was established, as shown in Figure4. The length of the pile is 40m, and the diameter of the energy pile is 1000mm. The soil depth is 80m in the direction of the pile depth, and the width value is 40m.Figure 4
The simplified diagram for calculation model.
(a)
(b)As for the related parameters, Alessandro et al. [6] suggested that the ratio of piles expansion coefficient to that of soils is taken as 0.25~4.0. The values of parameters in the model adopt that recommended by “Thermal design code for civil building” [23] and “Code for Design of Concrete Structures” [24]. The heating surfaces were assumed to be on the spiral-tube surface, ignoring the effects of fluid flow on the change of temperature. The calculation parameters of the model are shown in Table 2.Table 2
Parameters of finite element model.
Parameter
Value
Parameter
Value
Elastic modulus of pile GPa
30
Pile unit weight kN/m3
25
Soil elastic modulus MPa
20
Soil unit weight kN/m3
20
Coefficient of linear thermal expansion for pile °C−1
1x10−5
Friction angle of soil °
30
Coefficient of linear thermal expansion for soil °C−1
2x10−6
Soil Poisson’s ratio
0.35
Pile conductivity W/m/K
1.74
Soil cohesion kPa
5
Specific heat capacity of soil J/kg/K
1305
Pile Poisson's ratio
0.15
Specific heat capacity of pile J/kg/K
1706
Soil conductivity W/m/K
1.16The axial symmetry finite element model was adopted to analyze the influence of temperature on the pile. The soil adopts More-Coulomb model, and the pile was deemed to be elastic. The effect of the liquid flow on the temperature distribution in the heating pipe was neglected. Based on the measurements in laboratory tests, the influence of temperature change on the mechanical properties of the contact interface was not considered. The finite element analysis adopted sequential thermomechanical coupling method. The implication of this method is that stress does not affect temperature distribution, but temperature causes stress changes. In the process of heat conduction analysis, the heat conduction elements were adopted and the three-dimensional stress elements were adopted in the process of coupled thermomechanical analysis.According to relevant experimental test data [19, 25] (Li et al., 2013), after the energy pile was heated for a certain period of time, the overall temperature of the pile is increased by about 20°C, and the measured soil temperature at a certain depth was 17.5-22.5°C below the ground (Li et al., 2013) [26]. The initial temperature of the adopted model above was regarded as 20°C. After heating, the increase of temperature was set to be 20°. The temperature load was applied according to the actual increase from the laboratory test as shown in Figure 2(a). The finite element model adopted the sequential thermocoupling analysis method. In the thermal analysis process, the heat transfer element was adopted and three-dimensional stress element is used in the coupling analysis process. The bottom and the sides of the model were fixed, and the top is unconstrained.


### 3.2. Analysis Results

The load-settlement curves from the calculated model were shown in Figure5, and it could be seen that while pile top load exceeded 4700kN, the settlement began to drop steeply. The influences of temperature on pile top displacements and pile end settlements of energy piles were shown in Figure 6. From Figure 6, it could be seen that the additional displacements and end settlements caused by temperature have no significant difference when the pile top has no load and the applied load was equal to 1000kN and 3000kN. While the pile top load increases more than about 4700kN, the pile top displacements and end settlements caused by cycle heating and cooling were becoming obvious. As the pile top load increases, the pile top displacements caused by heating were decreased, but the pile end settlements become lager. With the progress of cooling, the additional pile top settlements caused by heating increased with increase of the pile top load. Through the above analyses, it could be concluded that the level of pile top load has significant influence on the displacements and end settlements caused by temperature. The displacements along the pile depth caused by cycle heating and cooling were shown in Figure 6. From Figure 6, it could be seen that the displacements along pile depth changed linearly. The lines for changes of temperature-induced displacements along pile depth were parallel, and the temperature-induced displacements along pile depth increase as the pile top loads increasing.Figure 5
Curves of load and displacements of pile.Figure 6
Displacements of pile top and end caused by temperature.It could be seen from Figure8 that the loads at pile top had great influences on changes of pile axial forces due to temperature. The pile top load increases to the value, P
a, from which the applied pile heat load could lead to pile end settlement suddenly increasing, the relative displacement of the pile and surrounding soils increasing, the side friction resistance fully developing, and the axial force caused by temperature significantly being reduced. As the pile top loads increasing, the maximum change of pile axial force caused by the temperature reduced. While the load at pile top is less than P
a, the maximum change of pile axial force caused by heating is approximately 25m below pile top. When the loads at the top of the pile were larger than the value of P
a, the maximum changes of energy pile axial force due to heating were reduced with increase of pile top loads and the corresponding positions of the maximum changes of axial force rise up along the pile depth with increase of pile top load.The displacement differences between pile top and end caused by temperature were shown in Figure9. It can be seen from Figure 9 that the displacement differences of energy pile caused by heating and cooling were approximately consistent at different load levels.As discussed above, the conclusion that heating and cooling would lead to additional displacement of energy pile with pile head load ofP
a from which nonlinear settlements begun to occur and from safety view of engineering, loads applied at energy pile top should not be heavier than the value of P
a. Within the load value of P
a mentioned above, the displacement of energy pile caused by heating could be eliminating with the process of cooling, as shown in Figure 9.


## 3.1. Finite Element Model

According to the above laboratory test results, it can be seen that the temperature at the concrete-soil interface increases with time, and tends to be constant with time. The temperature has no significant effect on the contact friction between concrete and soils. Based on the results mentioned above, a finite element model was established, as shown in Figure4. The length of the pile is 40m, and the diameter of the energy pile is 1000mm. The soil depth is 80m in the direction of the pile depth, and the width value is 40m.Figure 4
The simplified diagram for calculation model.
(a)
(b)As for the related parameters, Alessandro et al. [6] suggested that the ratio of piles expansion coefficient to that of soils is taken as 0.25~4.0. The values of parameters in the model adopt that recommended by “Thermal design code for civil building” [23] and “Code for Design of Concrete Structures” [24]. The heating surfaces were assumed to be on the spiral-tube surface, ignoring the effects of fluid flow on the change of temperature. The calculation parameters of the model are shown in Table 2.Table 2
Parameters of finite element model.
Parameter
Value
Parameter
Value
Elastic modulus of pile GPa
30
Pile unit weight kN/m3
25
Soil elastic modulus MPa
20
Soil unit weight kN/m3
20
Coefficient of linear thermal expansion for pile °C−1
1x10−5
Friction angle of soil °
30
Coefficient of linear thermal expansion for soil °C−1
2x10−6
Soil Poisson’s ratio
0.35
Pile conductivity W/m/K
1.74
Soil cohesion kPa
5
Specific heat capacity of soil J/kg/K
1305
Pile Poisson's ratio
0.15
Specific heat capacity of pile J/kg/K
1706
Soil conductivity W/m/K
1.16The axial symmetry finite element model was adopted to analyze the influence of temperature on the pile. The soil adopts More-Coulomb model, and the pile was deemed to be elastic. The effect of the liquid flow on the temperature distribution in the heating pipe was neglected. Based on the measurements in laboratory tests, the influence of temperature change on the mechanical properties of the contact interface was not considered. The finite element analysis adopted sequential thermomechanical coupling method. The implication of this method is that stress does not affect temperature distribution, but temperature causes stress changes. In the process of heat conduction analysis, the heat conduction elements were adopted and the three-dimensional stress elements were adopted in the process of coupled thermomechanical analysis.According to relevant experimental test data [19, 25] (Li et al., 2013), after the energy pile was heated for a certain period of time, the overall temperature of the pile is increased by about 20°C, and the measured soil temperature at a certain depth was 17.5-22.5°C below the ground (Li et al., 2013) [26]. The initial temperature of the adopted model above was regarded as 20°C. After heating, the increase of temperature was set to be 20°. The temperature load was applied according to the actual increase from the laboratory test as shown in Figure 2(a). The finite element model adopted the sequential thermocoupling analysis method. In the thermal analysis process, the heat transfer element was adopted and three-dimensional stress element is used in the coupling analysis process. The bottom and the sides of the model were fixed, and the top is unconstrained.


## 3.2. Analysis Results

The load-settlement curves from the calculated model were shown in Figure5, and it could be seen that while pile top load exceeded 4700kN, the settlement began to drop steeply. The influences of temperature on pile top displacements and pile end settlements of energy piles were shown in Figure 6. From Figure 6, it could be seen that the additional displacements and end settlements caused by temperature have no significant difference when the pile top has no load and the applied load was equal to 1000kN and 3000kN. While the pile top load increases more than about 4700kN, the pile top displacements and end settlements caused by cycle heating and cooling were becoming obvious. As the pile top load increases, the pile top displacements caused by heating were decreased, but the pile end settlements become lager. With the progress of cooling, the additional pile top settlements caused by heating increased with increase of the pile top load. Through the above analyses, it could be concluded that the level of pile top load has significant influence on the displacements and end settlements caused by temperature. The displacements along the pile depth caused by cycle heating and cooling were shown in Figure 6. From Figure 6, it could be seen that the displacements along pile depth changed linearly. The lines for changes of temperature-induced displacements along pile depth were parallel, and the temperature-induced displacements along pile depth increase as the pile top loads increasing.Figure 5
Curves of load and displacements of pile.Figure 6
Displacements of pile top and end caused by temperature.It could be seen from Figure8 that the loads at pile top had great influences on changes of pile axial forces due to temperature. The pile top load increases to the value, P
a, from which the applied pile heat load could lead to pile end settlement suddenly increasing, the relative displacement of the pile and surrounding soils increasing, the side friction resistance fully developing, and the axial force caused by temperature significantly being reduced. As the pile top loads increasing, the maximum change of pile axial force caused by the temperature reduced. While the load at pile top is less than P
a, the maximum change of pile axial force caused by heating is approximately 25m below pile top. When the loads at the top of the pile were larger than the value of P
a, the maximum changes of energy pile axial force due to heating were reduced with increase of pile top loads and the corresponding positions of the maximum changes of axial force rise up along the pile depth with increase of pile top load.The displacement differences between pile top and end caused by temperature were shown in Figure9. It can be seen from Figure 9 that the displacement differences of energy pile caused by heating and cooling were approximately consistent at different load levels.As discussed above, the conclusion that heating and cooling would lead to additional displacement of energy pile with pile head load ofP
a from which nonlinear settlements begun to occur and from safety view of engineering, loads applied at energy pile top should not be heavier than the value of P
a. Within the load value of P
a mentioned above, the displacement of energy pile caused by heating could be eliminating with the process of cooling, as shown in Figure 9.


## 4. Simplified Calculation Method

### 4.1. Analysis of Pile under Thermomechanical Load

Even though many existing methods about the settlement of pile foundation and load transfer mechanism analysis of single pile, these methods for the energy pile are not very applicable due to the temperature load.The behavior between the pile shaft and the surrounding soils in this paper is described by a simple linear model. For the linear relationship, the pile side friction is increasing linearly with gradually increasing relative displacement between pile shaft and surrounding soils. The relationship between the pile end loads and the settlements also can be expressed by the linear model based on the existing research results. The relationships can be expressed in the following:(1)
τ
z
=
k
s
Δ
s
z
τ
b
=
k
b
s
bwhere τ
(
z
) is the shaft shear stress at a given depthz, k
s is the initial stiffness of the soil surrounding pile shaft, Δ
s
z is the pile-soil relative displacement developed in the pile-soil interface at a given depthz. τ
b is the pile end resistance, k
b is initial soil stiffness at the pile base, and s
b is the pile end settlements. The coefficients k
s and k
b can be obtained by the proposed method in the references [1–3, 27].As discussed above, it can be seen that there was a zero point of displacement change at which the additional displacement caused by temperature was zero, while the pile top load was not heavier than the value,P
a, that leading to nonlinear settlement of pile top, as shown in Figures 6 and 7. Position of the point can be obtained by the condition of mechanical equilibrium due to heating, as shown in Figure 10. Assuming that the temperature caused expansions of soil surrounding pile end and pile shaft were negligible, only the expansion of the pile is considered. The behavior between the pile shaft and the surrounding soils and the relationship between pile end load and settlement were all expressed by linear models.Figure 7
Temperature-induced displacements along pile depth.Figure 8
Variations of pile axial force caused by temperature.Figure 9
Settlement difference caused by heating 30d and cooling 60d.Figure 10
Zero point of displacement change.The zero point of displacement change is considered as origin of coordinate, and the length from zero point to the pile head has the value ofl
1. l
1 can be written by the equilibrium expression:(2)
∫
0
l
1
2
π
r
0
k
s
λ
p
Δ
T
l
d
l
=
∫
0
L
-
l
1
2
π
r
0
k
s
λ
p
Δ
T
l
d
l
+
π
r
0
2
k
b
λ
p
Δ
T
L
-
l
1where λ
p is the coefficient of linear expansion for pile and Δ
T is the temperature increase of pile body. Thus, the zero point of displacement change l
1 can be obtained in (3)
l
1
=
k
s
L
2
+
r
0
k
b
L
2
k
s
L
+
r
0
k
bThe additional axial pile shaft force from the pile head to the zero point can be obtained by(4)
Δ
p
z
=
∫
0
z
2
π
r
0
k
s
1
Δ
T
λ
p
l
1
-
l
d
lThe analytic solutions to (4) can be expressed in (5)
Δ
p
z
=
2
π
r
0
k
s
1
Δ
T
λ
p
l
1
z
-
π
r
0
k
s
1
Δ
T
λ
p
z
2The additional axial pile shaft force from the zero point to the pile end can be obtained by(6)
Δ
p
z
=
π
r
o
2
k
b
Δ
T
l
2
λ
p
+
∫
l
1
z
2
π
r
0
k
s
Δ
T
λ
p
L
-
z
d
zThe analytic solutions to (6) can be expressed in(7)
Δ
p
z
=
π
r
o
2
k
b
Δ
T
l
2
λ
p
+
2
π
r
0
k
s
Δ
T
λ
p
L
z
-
l
1
-
π
r
0
k
s
Δ
T
λ
p
z
2
-
l
1
2In multilayered soils, assume the zero point of displacement change is in theith layer and the length from theith layer soil heat to the zero point isl
1, as shown in Figure 11. The zero point of displacement change can be obtained by (8)
∑
j
=
0
i
-
1
∫
l
j
l
j
+
1
2
π
r
0
k
s
j
λ
p
Δ
T
l
d
l
+
∫
0
l
1
2
π
r
0
k
s
i
λ
p
Δ
T
l
d
l
=
∑
j
=
i
+
1
n
∫
l
j
l
j
+
1
2
π
r
0
k
s
j
λ
p
Δ
T
l
d
l
+
∫
0
L
i
-
l
1
2
π
r
0
k
s
i
λ
p
Δ
T
l
d
l
+
π
r
0
2
L
-
l
1
-
∑
j
=
1
i
-
1
L
j
Δ
T
λ
p
k
bFigure 11
Calculation model for zero point in multilayered soils.All the parameters in the above equation are shown in Figure11. The additional axial pile shaft force from the pile head to the zero point can be obtained by the following, assuming that the relative displacements between the pile shaft and the surrounding soils are in linear relationship in all the multilayered soils:(9)
Δ
p
z
=
∑
j
=
0
i
-
1
∫
∑
m
=
0
j
l
m
∑
m
=
0
j
+
1
l
m
2
π
r
0
k
s
j
λ
p
Δ
T
∑
j
=
0
I
-
1
L
i
+
l
1
-
z
d
z
+
∫
∑
m
=
0
i
-
1
l
m
z
2
π
r
0
k
s
i
λ
p
Δ
T
∑
j
=
0
I
-
1
L
i
+
l
1
-
z
d
zThe additional axial pile shaft force from the pile top to the zero point can be obtained by(10)
Δ
p
z
=
π
r
o
2
k
b
Δ
T
L
-
∑
j
=
0
I
-
1
L
j
-
l
1
λ
p
+
∑
j
=
I
i
∫
∑
m
=
I
j
l
m
∑
m
=
I
j
+
1
l
m
2
π
r
0
k
s
j
λ
p
Δ
T
∑
j
=
I
n
-
I
L
j
-
l
1
-
z
d
z
+
∫
∑
m
=
I
n
-
i
+
1
l
m
z
2
π
r
0
k
s
i
λ
p
Δ
T
∑
j
=
I
n
-
i
+
1
L
i
-
l
1
-
z
d
zIn order to determine the specific position of layers in (10), the procedure can be adopted to calculate(1) Confirm the values ofk
si in theith soil layer.(2) Assume that the zero point is in the lowest layer and calculate to obtain the value ofl
1, as shown in Figure 11.(3) If the solution ofl
1 is negative, then assuming the zero point is in the (n-1)th as shown in Figure 11. Continue to calculate the solution to (8) until obtaining the positive value ofl
1.(4) Calculate the additional axial pile shaft force using (9) and (10) along pile shaft.


## 4.1. Analysis of Pile under Thermomechanical Load

Even though many existing methods about the settlement of pile foundation and load transfer mechanism analysis of single pile, these methods for the energy pile are not very applicable due to the temperature load.The behavior between the pile shaft and the surrounding soils in this paper is described by a simple linear model. For the linear relationship, the pile side friction is increasing linearly with gradually increasing relative displacement between pile shaft and surrounding soils. The relationship between the pile end loads and the settlements also can be expressed by the linear model based on the existing research results. The relationships can be expressed in the following:(1)
τ
z
=
k
s
Δ
s
z
τ
b
=
k
b
s
bwhere τ
(
z
) is the shaft shear stress at a given depthz, k
s is the initial stiffness of the soil surrounding pile shaft, Δ
s
z is the pile-soil relative displacement developed in the pile-soil interface at a given depthz. τ
b is the pile end resistance, k
b is initial soil stiffness at the pile base, and s
b is the pile end settlements. The coefficients k
s and k
b can be obtained by the proposed method in the references [1–3, 27].As discussed above, it can be seen that there was a zero point of displacement change at which the additional displacement caused by temperature was zero, while the pile top load was not heavier than the value,P
a, that leading to nonlinear settlement of pile top, as shown in Figures 6 and 7. Position of the point can be obtained by the condition of mechanical equilibrium due to heating, as shown in Figure 10. Assuming that the temperature caused expansions of soil surrounding pile end and pile shaft were negligible, only the expansion of the pile is considered. The behavior between the pile shaft and the surrounding soils and the relationship between pile end load and settlement were all expressed by linear models.Figure 7
Temperature-induced displacements along pile depth.Figure 8
Variations of pile axial force caused by temperature.Figure 9
Settlement difference caused by heating 30d and cooling 60d.Figure 10
Zero point of displacement change.The zero point of displacement change is considered as origin of coordinate, and the length from zero point to the pile head has the value ofl
1. l
1 can be written by the equilibrium expression:(2)
∫
0
l
1
2
π
r
0
k
s
λ
p
Δ
T
l
d
l
=
∫
0
L
-
l
1
2
π
r
0
k
s
λ
p
Δ
T
l
d
l
+
π
r
0
2
k
b
λ
p
Δ
T
L
-
l
1where λ
p is the coefficient of linear expansion for pile and Δ
T is the temperature increase of pile body. Thus, the zero point of displacement change l
1 can be obtained in (3)
l
1
=
k
s
L
2
+
r
0
k
b
L
2
k
s
L
+
r
0
k
bThe additional axial pile shaft force from the pile head to the zero point can be obtained by(4)
Δ
p
z
=
∫
0
z
2
π
r
0
k
s
1
Δ
T
λ
p
l
1
-
l
d
lThe analytic solutions to (4) can be expressed in (5)
Δ
p
z
=
2
π
r
0
k
s
1
Δ
T
λ
p
l
1
z
-
π
r
0
k
s
1
Δ
T
λ
p
z
2The additional axial pile shaft force from the zero point to the pile end can be obtained by(6)
Δ
p
z
=
π
r
o
2
k
b
Δ
T
l
2
λ
p
+
∫
l
1
z
2
π
r
0
k
s
Δ
T
λ
p
L
-
z
d
zThe analytic solutions to (6) can be expressed in(7)
Δ
p
z
=
π
r
o
2
k
b
Δ
T
l
2
λ
p
+
2
π
r
0
k
s
Δ
T
λ
p
L
z
-
l
1
-
π
r
0
k
s
Δ
T
λ
p
z
2
-
l
1
2In multilayered soils, assume the zero point of displacement change is in theith layer and the length from theith layer soil heat to the zero point isl
1, as shown in Figure 11. The zero point of displacement change can be obtained by (8)
∑
j
=
0
i
-
1
∫
l
j
l
j
+
1
2
π
r
0
k
s
j
λ
p
Δ
T
l
d
l
+
∫
0
l
1
2
π
r
0
k
s
i
λ
p
Δ
T
l
d
l
=
∑
j
=
i
+
1
n
∫
l
j
l
j
+
1
2
π
r
0
k
s
j
λ
p
Δ
T
l
d
l
+
∫
0
L
i
-
l
1
2
π
r
0
k
s
i
λ
p
Δ
T
l
d
l
+
π
r
0
2
L
-
l
1
-
∑
j
=
1
i
-
1
L
j
Δ
T
λ
p
k
bFigure 11
Calculation model for zero point in multilayered soils.All the parameters in the above equation are shown in Figure11. The additional axial pile shaft force from the pile head to the zero point can be obtained by the following, assuming that the relative displacements between the pile shaft and the surrounding soils are in linear relationship in all the multilayered soils:(9)
Δ
p
z
=
∑
j
=
0
i
-
1
∫
∑
m
=
0
j
l
m
∑
m
=
0
j
+
1
l
m
2
π
r
0
k
s
j
λ
p
Δ
T
∑
j
=
0
I
-
1
L
i
+
l
1
-
z
d
z
+
∫
∑
m
=
0
i
-
1
l
m
z
2
π
r
0
k
s
i
λ
p
Δ
T
∑
j
=
0
I
-
1
L
i
+
l
1
-
z
d
zThe additional axial pile shaft force from the pile top to the zero point can be obtained by(10)
Δ
p
z
=
π
r
o
2
k
b
Δ
T
L
-
∑
j
=
0
I
-
1
L
j
-
l
1
λ
p
+
∑
j
=
I
i
∫
∑
m
=
I
j
l
m
∑
m
=
I
j
+
1
l
m
2
π
r
0
k
s
j
λ
p
Δ
T
∑
j
=
I
n
-
I
L
j
-
l
1
-
z
d
z
+
∫
∑
m
=
I
n
-
i
+
1
l
m
z
2
π
r
0
k
s
i
λ
p
Δ
T
∑
j
=
I
n
-
i
+
1
L
i
-
l
1
-
z
d
zIn order to determine the specific position of layers in (10), the procedure can be adopted to calculate(1) Confirm the values ofk
si in theith soil layer.(2) Assume that the zero point is in the lowest layer and calculate to obtain the value ofl
1, as shown in Figure 11.(3) If the solution ofl
1 is negative, then assuming the zero point is in the (n-1)th as shown in Figure 11. Continue to calculate the solution to (8) until obtaining the positive value ofl
1.(4) Calculate the additional axial pile shaft force using (9) and (10) along pile shaft.


## 5. Case Study

The proposed method is an approximate calculation method, and the existing tests mainly focused on heat transfer analysis of pile to study the responses of heat conduction, and the bearing characteristics of field tested pile were also carried out by many scholars. But the bearing characteristics were influenced by many factors such as the stratums and groundwater flow. Therefore, the finite element method is adopted to compare with the results obtained by the above proposed method. The parameters in the analysis adopted the values from the Table3.Table 3
Parameters of case study.
Layer of soils
Thickness of soil layer (m)
Parameters
Values(kPa/mm)
Layer 1
5
k
s1
1.094
Layer 2
5
k
s1
1.276
Layer 3
5
k
s1
2.161
Layer 4
5
k
s1
2.719
Layer 5
5
k
s1
2.771
Layer 6
5
k
s1
2.875
Layer 7
5
k
s1
3.066
Layer 8
5
k
s1
3.241
l
1=26.31 m (Obtained by the proposed method)
l
1= 24.65 m (Obtained by ABAQUS)
r
pile=0.5m, H
pile=40m, λ
pile  °C−1=1x10−5, k
b1=28.91kPa/mmComparisons between the results form ABAQUS and those computed by the proposed approach are shown in Figure12. Comparison between the curves computed by the proposed approach and that of ABAUQS is shown in Figure 12. Figure 12 shows that the additional displacements along pile shaft caused by heating calculated by the proposed method are generally consistent with the finite method results.Figure 12
Result comparisons between the proposed method and finite method (ABAQUS).


## 6. Conclusions

As discussed above, the following conclusions can be obtained:(1)
The frictional resistance at the concrete-soil interface has no obvious relationship to the variation of temperature. The cycle temperature would lead to unrecoverable additional displacement at energy pile heat with heavy loads from which nonlinear settlement occurred on the load-settlement curve.
(2)
The position of the zero point of displacement change has no obvious relationship to temperature change of pile body. The main factors of influence on the point are the initial soil stiffness at the pile base and shear stiffness of soils surrounding pile and the coefficients of linear expansion for pile.
(3)
The proposed method can be used to estimate the zero point of additional displacement and additional pile axial force along pile length.



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*Source: 2098350-2019-02-13.xml*