# Reliability Design of Soft Soil Foundation Preloading Method for Airport Runway

Authors: Qingkun Yu; Liangcai Cai; Xiang Shi
Journal: Advances in Civil Engineering
 (2019)

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## Abstract

Focusing on the rapid assessment of the reliability of airport runway soft soil foundation, taking the soft soil foundation preloading method as the research object, we established the corresponding settlement calculation model and studied the soil parameters and how they affect the settlement values, and the reliability of the soft soil foundation preloading method was obtained by the checking point method (JC). In order to realize the fast calculation of reliability, Visual Studio 2010 is used to design the visual reliability calculation program of the airport runway soft soil foundation preloading method. The analysis results showed that the soil index with the greatest influence on the final consolidation settlement was the compression coefficient, followed by the soil layer thickness, the compression modulus, the void ratio, and the consolidation coefficient. On the basis of the limit state equation, we got the final failure probability formula. Combined with an airport project example, the reliability calculation results showed that the longer the preloading time was, the more reliable the treatment plan was, and the less the probability of failure was; the larger the preloading pressure was, the more reliable the treatment plan was, and the less the probability of failure was. In the engineering example, it can be found that the treatment plan with the preloading of 100 kPa and the stacking time of 150 days is the best solution.


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## Body

## 1. Introduction

With the rapid development of the country’s economic construction, the role of air transport is becoming more and more prominent. In particular, in recent years, more and more airports have been rebuilt and expanded. In fact, the airport occupies a large area, but the urban land is limited, so most of the airports are located in poor soil and wasteland, and the contradiction is particularly prominent in the coastal areas. Then, how to effectively evaluate the reliability of airport construction in soft soil area has become an important issue for airport design and engineering personnel.Some scholars have carried out fruitful work on the reliability of ground base. Cherubini [1] obtained the general probability distribution of shallow foundation and calculated the cumulative distribution curve. Based on Cherubiniʼs research results, Easa [2] got the rough probability distribution of shallow foundation, obtained the cumulative distribution curve by calculation, and compared with those results obtained by the Monte Carlo method. Qidong and Gao [3] carried out relatively systematically reliability research on the bearing capacity of natural foundation according to the bearing capacity calculation method proposed by Hansen and conducted sensitivity analysis. Ding et al. [4] established a random field model under the condition of inclined additional stress and studied the reliability sensitivity of the soil index according to the actual engineering conditions. The first-order second-moment method is employed to calculate the reliability of bearing capacity of foundation under inclined additional stress condition, combined with the bearing capacity calculation method proposed by Hansen [5]. Taking into account the uncertainties of the calculation model of bearing capacity and the load, with the reliability analysis method and the theory of probability and statistics, Bian et al. obtained respectively the calculation formulas of reliability indices for ultimate limit state (ULS) and serviceability limit state (SLS). Meanwhile, the linear relationship between reliability indices for ULS and SLS was presented. Finally, the impact of stochastic behaviors of limiting tolerable settlement of pile head (slt) on reliability analysis for SLS was studied. The following conclusions could be drawn from this study. First, the influences of soil type and pile type on the model factor for SLS were slight. Second, the reliability index for SLS decreased with an increase of the uncertainties from the bearing capacity calculation model and the load, although the decrease rate gradually diminished. And the total variation in the reliability index for SLS was limited, which means that it is reasonable to ignore the influences of the uncertainties from the bearing capacity calculation model and the load on the reliability analysis for SLS in engineering practice. Third, the stochastic behaviors of slt had a significant effect on reliability analysis for SLS [6]. The reliability of the bearing capacity of the CFG pile composite foundation was studied by Zhang and Zheng, and the sensitivity analysis of the influence factors was also carried out [7]. In Zhang et al. [8], based on the uncertainty analysis to calculation model of settlement, the formula of reliability index of foundation pile was derived; based on this formula, the influence of different parameters on reliability was analyzed; the results indicated that the high-reliability index could be obtained by increasing the safety coefficient; reliability index would be reduced with increasing of the mean value of the calculation model coefficient and coefficient of variation of the permissible limit of settlement; the reliability index would not always monotonically increase or decrease with increasing of coefficient of variation of the calculated settlement and coefficient of variation of the calculation model coefficient. The reliability calculation method of multipile composite foundation was proposed by Ding et al., and the reliability was solved by the checking point method [9].Based on the aforementioned research, we find that there is no research on the reliability of airport soft soil foundation. In this paper, we focus on the rapid evaluation of the reliability of the airport runway soft soil foundation, taking the treatment technology of preloading method soft soil foundation as the research object; combined with an airport construction example, a corresponding settlement calculation model is established, and the sensitivity of the soil parameters affecting the settlement calculation model is studied, and then the JC method is used to get the reliability of the soft soil foundation preloading method treatment plan. In order to realize the fast calculation of the reliability, Visual Studio 2010 is used to design the visual program interface, and the reliability calculation program of the airport runway soft soil foundation preloading method treatment plan is written, which provides a simple and convenient way for the airport design and construction.


## 2. Reliability Theory

### 2.1. Limit State Equation

If the engineering structure can achieve the desired function, we call it a reliable state. If the engineering structure cannot achieve the expected function, we call it a failure state. In the middle state of reliability and failure, we call it the limit state, expressed as follows:(1)Z=gX1,X2,…,Xn.


### 2.2. Reliability Index

LetX1,X2,…,Xn be a random variable in the limit state equation, and fxx1,x2,…,xn is a corresponding probability density function. Then, the failure probability of the engineering structure can be expressed as follows:(2)Pf=PZ<0=∬Z<0⋯∫fXx1,x2,…,xndx1dx2⋯dxn.Suppose thatZ is related to the two random variables R and S, both R and S satisfy the normal distribution, the mean value is μR, μS, and the standard deviation is σR, σS. Then, the function can be expressed as Z=R−S by R and S and also satisfies the normal distribution, with the mean value Z=R−S and standard deviation σZ=σR2+σS2.As shown in Figure1, Z<0 means the probability of failure and can be written as Pf=PZ<0, that is, the shaded area shown in Figure 1. The distance from 0 to the mean value μZ can be expressed as μZ=βσZ. It is not difficult to find that when β is smaller, Pf is larger, when β is larger, Pf is smaller. Since there is such a negative correlation between the probability of failure Pf and the value β, we can refer to β as the reliability index. Then, the failure probability can be expressed as follows:(3)Pf=PZ<0=FZ0=∫−∞012πσZexp−z−μZ22σZ2dZ.Figure 1
Probability density curve of performance function.To normalize equation (3) (i.e., μt=0, σt=1), then(4)Pf=∫−∞−μz/σZ12πexp−t22dt=Φ−β.The reliability indexβ can be expressed as follows:(5)β=μZσZ=μR−μSσR2+σS2.


### 2.3. Calculation of Reliability Index by JC Method

Before solving the reliability indexβ, we must standardize the data Xi first, namely,(6)Xi¯=Xi−μXiσXi,i=1,2,…,n,where μXi is the mean value and σXi is the standard deviation. Then, in the coordinate space O¯X1¯X2¯⋯Xn¯, the following equation can be used to express the limit state:(7)Z=gX1¯σX1+μX1,X2¯σX2+μX2,…,Xn¯σXn+μXn=0.Equation (7) can be understood as the shortest distance from the limit state surface in the standard normal space coordinate system O¯X1¯X2¯⋯Xn¯ to the origin O¯, which is regarded as the reliability index β. The limit state surface of the three normal random variables is shown in Figure 2, and the perpendicular point P∗ of the shortest point-to-surface distance is the “design checking point.”Figure 2
The curved surface of limit state.As shown in Figure2, the direction cosine of the normal line O¯P∗ relative to the coordinate vector is(8)cosθxi¯=cosθxi=−∂g/∂Xip8σXi∑i=1n∂g/∂Xip∗σXi21/2,where ∂g/∂XiP∗ is the partial derivative at the point P∗.By the definition of the direction cosine, we can see(9)Xi∗¯=O¯P∗¯cosθXi¯=O¯P∗¯cosθXi=βcosθXi,i=1,2,…,n.It can be obtained by Formula (6):(10)Xi∗¯=Xi∗−μXiσXi=βcosθXi.So, in the coordinate spaceO¯X1¯X2¯⋯Xn¯, the coordinates of P∗ are(11)Xi∗=μXi+βσXicosθXi,i=1,2,…,n.SinceP∗ is on the limit state surface, there must be the following equation:(12)gX1∗,X2∗,…,Xn∗=0.From equations (9) to (11) to form an equation system, we can solve β and Xi∗i=1,2,…,n.After the random variables are equivalently normalized, the reliability indexβ can be solved by equations (9)–(11). However, μXi′ and σXi′ are done by the checking point xi∗, but the value of the checking point is unknown. It can be seen that the conditions of “equivalent normalization” and equations (9)–(11) are related to each other, and the iterative method is usually used to solve β.


## 2.1. Limit State Equation

If the engineering structure can achieve the desired function, we call it a reliable state. If the engineering structure cannot achieve the expected function, we call it a failure state. In the middle state of reliability and failure, we call it the limit state, expressed as follows:(1)Z=gX1,X2,…,Xn.


## 2.2. Reliability Index

LetX1,X2,…,Xn be a random variable in the limit state equation, and fxx1,x2,…,xn is a corresponding probability density function. Then, the failure probability of the engineering structure can be expressed as follows:(2)Pf=PZ<0=∬Z<0⋯∫fXx1,x2,…,xndx1dx2⋯dxn.Suppose thatZ is related to the two random variables R and S, both R and S satisfy the normal distribution, the mean value is μR, μS, and the standard deviation is σR, σS. Then, the function can be expressed as Z=R−S by R and S and also satisfies the normal distribution, with the mean value Z=R−S and standard deviation σZ=σR2+σS2.As shown in Figure1, Z<0 means the probability of failure and can be written as Pf=PZ<0, that is, the shaded area shown in Figure 1. The distance from 0 to the mean value μZ can be expressed as μZ=βσZ. It is not difficult to find that when β is smaller, Pf is larger, when β is larger, Pf is smaller. Since there is such a negative correlation between the probability of failure Pf and the value β, we can refer to β as the reliability index. Then, the failure probability can be expressed as follows:(3)Pf=PZ<0=FZ0=∫−∞012πσZexp−z−μZ22σZ2dZ.Figure 1
Probability density curve of performance function.To normalize equation (3) (i.e., μt=0, σt=1), then(4)Pf=∫−∞−μz/σZ12πexp−t22dt=Φ−β.The reliability indexβ can be expressed as follows:(5)β=μZσZ=μR−μSσR2+σS2.


## 2.3. Calculation of Reliability Index by JC Method

Before solving the reliability indexβ, we must standardize the data Xi first, namely,(6)Xi¯=Xi−μXiσXi,i=1,2,…,n,where μXi is the mean value and σXi is the standard deviation. Then, in the coordinate space O¯X1¯X2¯⋯Xn¯, the following equation can be used to express the limit state:(7)Z=gX1¯σX1+μX1,X2¯σX2+μX2,…,Xn¯σXn+μXn=0.Equation (7) can be understood as the shortest distance from the limit state surface in the standard normal space coordinate system O¯X1¯X2¯⋯Xn¯ to the origin O¯, which is regarded as the reliability index β. The limit state surface of the three normal random variables is shown in Figure 2, and the perpendicular point P∗ of the shortest point-to-surface distance is the “design checking point.”Figure 2
The curved surface of limit state.As shown in Figure2, the direction cosine of the normal line O¯P∗ relative to the coordinate vector is(8)cosθxi¯=cosθxi=−∂g/∂Xip8σXi∑i=1n∂g/∂Xip∗σXi21/2,where ∂g/∂XiP∗ is the partial derivative at the point P∗.By the definition of the direction cosine, we can see(9)Xi∗¯=O¯P∗¯cosθXi¯=O¯P∗¯cosθXi=βcosθXi,i=1,2,…,n.It can be obtained by Formula (6):(10)Xi∗¯=Xi∗−μXiσXi=βcosθXi.So, in the coordinate spaceO¯X1¯X2¯⋯Xn¯, the coordinates of P∗ are(11)Xi∗=μXi+βσXicosθXi,i=1,2,…,n.SinceP∗ is on the limit state surface, there must be the following equation:(12)gX1∗,X2∗,…,Xn∗=0.From equations (9) to (11) to form an equation system, we can solve β and Xi∗i=1,2,…,n.After the random variables are equivalently normalized, the reliability indexβ can be solved by equations (9)–(11). However, μXi′ and σXi′ are done by the checking point xi∗, but the value of the checking point is unknown. It can be seen that the conditions of “equivalent normalization” and equations (9)–(11) are related to each other, and the iterative method is usually used to solve β.


## 3. Research on the Reliability of Preloading Method Treatment Plan

In this paper, the foundation settlement is taken as an index to study the reliability of the soft soil foundation treatment plan. According to the reliability theory, the limit state is designed and the limit state equation is constructed. Due to the large area of the airport, when the preloading method is used, a large area will be covered. Under these circumstances, the final vertical deformation of the runway foundation can be obtained by the stratified summation method.


### 3.1. Preloading Method Foundation Settlement Calculation Model and Method

According to the stratified summation method proposed by Terzaghi [10], under the condition of preloading, the final vertical deformation can be obtained by the following equation:(13)St=SUt=N∑i=1navi1+eoiΔPhi1−8π2e−8Ch/Fnde2+π2Cv/4H2t=N∑i=1n1EsiΔPhi1−8π2e−8Ch/Fnde2+π2Cv/4H2t,where St is the vertical deformation after treatment; S is the foundation ultimate vertical deformation; Ut is the average consolidation degree of sand well drainage (excluding the influence of well resistance and smearing); N is the correction coefficient; avi is the compressibility coefficient of soil; eoi is the void ratio of the soil layer; Cv is the vertical drainage consolidation coefficient; Ch is the horizontal drainage consolidation coefficient; avi is the compressive modulus of the soil layer; eoi is the void ratio of soil layer; Cv is the vertical drainage consolidation coefficient; Ch is the horizontal drainage consolidation coefficient; Esi is the compressive modulus of the soil layer; ΔP is the additional stress, which is set as the constant; hi is the thickness of the ith compression layer; H is the sand well longest drainage path length; de is the effective drainage diameter of the sand well; n is the well diameter ratio; Fn is a function related to the well diameter ratio.


### 3.2. Research on the Sensitivity of Soil Index

For the settlement calculation model of preloading method foundation treatment plan, the final consolidation settlement of foundation is affected by many soil indexes [11], and the degree of influence is still different. In order to clarify the degree of influence of each index on the final consolidation settlement, so as to simplify the actual calculation model, it is necessary to conduct a sensitivity study of the soil index. According to the preloading method settlement calculation model, the final consolidation settlement is affected by the void ratio e, the consolidation coefficient Cv, the compression modulus Es, the compression coefficient av, and the soil thickness h.According to the engineering geological investigation report of an airport, silt clay is taken as the research object to study the degree of influence of each soil index on the final consolidation settlement. The results of the sensitivity study are as follows:(14)St=SUt=av1+eΔPh1−8π2e−8Ch/Fnde2+π2Cv/4H2t=1EsΔPh1−8π2e−8Ch/Fnde2+π2Cv/4H2t,where St is the vertical deformation after the treatment; S is the final foundation vertical deformation; and Ut is the sand drainage average consolidation degree (excluding well resistance and smear effects).Among them, the preloading loadΔP is assumed as 100 kPa, the well diameter ratio n=de/dw=1.47/0.07=21, Fn=n2/n2−1lnn−3n2−1/4n2=2.3, the sand well effective drainage cylinder diameter de = 1.05l = 1.05 × 0.4 = 1.47 m, the maximum drainage path length of sand well H=15m, and the stacking time t is 150 d. The soil index values are shown in Table 1.(1)
Calculation of the slope of the final consolidation settlement by the compression coefficient.av∼Φμav,σav2, μav=0.941MPa−1, σav=0.081, the changing range of av is 0.770∼1.190 MPa−1, the slope is l1, l1=Stmax−Stmin/avmax−avmin/μav=0.090, as shown in Figure 3.(2)
Calculation of the slope of the final consolidation settlement by the void ratio.e∼Φμe,σe2, μe=1.317, σe=0.737, the changing range of e is 1.162∼1.450, the slope is l2, l2=Stmax−Stmin/emax−emin/μe=0.052, as shown in Figure 4.(3)
Calculation of the slope of the final consolidation settlement by the compression modulus.Es∼ΦμEs,σEs2, μEs=2.593MPa, σEs=0.392, the changing range of Es is 1.860∼4.330 MPa, the slope is l3, l3=Stmax−Stmin/Esmax−Esmin/Es=0.071, as shown in Figure 5.(4)
Calculation of the slope of the final consolidation settlement by the soil thickness.h∼Φμh,σh2, μh=3.9, σh=0.792, the changing range of h is 3.6∼4.3 m, the slope is l4, l4=Stmax−Stmin/hmax−hmin/μh=0.078, as shown in Figure 6.(5)
Calculation of the slope of the final consolidation settlement by the consolidation coefficient.Cv∼ΦμCv,σCv2, μCv=2.5×10−4cm/s2, σCv=1.95×10−5, the changing range of Cv is 0.0002∼0.0003cm/s2, the slope is l5, l5=Stmax−Stmin/Cvmax−Cvmin/μCv=0.040, as shown in Figure 7.Table 1
The statistics of soil parameter.
Silt clay
e
a
v
E
s
C
v
h
Mean value
1.317
0.941
2.593
2.51E-04
3.900
Standard deviation
0.737
0.081
0.392
1.95E-05
0.792Figure 3
Effect of compression coefficient on final consolidation settlement.Figure 4
Effect of void ratio on final consolidation settlement.Figure 5
Effect of compressive modulus on final consolidation settlement.Figure 6
Effect of soil thickness on the final consolidation settlement.Figure 7
Effect of consolidation coefficient on the final consolidation settlement.As Figures3–7 show, l1>l4>l3>l2>l5. The results show that the soil index which has the most influence on the final consolidation settlement is the compression coefficient, and then the soil thickness, compression modulus, void ratio, and consolidation coefficient.


### 3.3. Reliability Calculation of Preloading Method Soft Soil Foundation Treatment Plan

The limit state is analyzed [12], and the limit state equation is determined as follows:(15)gx=N∑i=1n1EsiΔPhi1−8π2e−8Ch/Fnde2+π2Cv/4H2t−S′=0,where S′ refers to the foundation settlement in the construction period.As for the judgment of the compressive soil layer, for the soft clay, it is judged by the principle that the prestress at a certain depth is equal to 0.1 times of the soil self-weight, that is, when∑i=1Nγhi≥10ΔP, hi is the i-layer compressive soil thickness. And the effect of consolidation coefficient on foundation final consolidation settlement is very small, so it can be assumed that the consolidation coefficients of all the soil layers are the same, and the consolidation coefficients are the same in the transverse and vertical direction.AssumingCv=Ch, the statistics of Esi, hi, Cv are shown in Table 2.Table 2
The statistics ofEsi, hi, Cv.
Random variables
Mean value
Standard derivation
Variation coefficient
E
s
i
μ
E
s
i
σ
E
s
i
δ
E
s
i
h
i
μ
h
i
σ
h
i
δ
h
i
C
v
μ
C
v
σ
C
v
δ
C
vThen, you can use the iterative method to calculate the reliabilityβ and failure probability Pf; the calculation steps are as follows [13]:(1)
Calculate the derivative of each variable.(16)∂g∂Esi∗=∂g∂EsiEsi∗σEsi=−N1Esi∗2ΔPhi∗σEsi1−8π2e−8Ch∗/Fnde2+π2Cv∗/4H2t,∂g∂hi∗=∂g∂hihi∗σhi=N1Esi∗ΔPσhi1−8π2e−8Ch∗/Fnde2+π2Cv∗/4H2t,∂g∂Cv∗=∂g∂CvCv∗σCv=N∑i=1n1Esi∗ΔPhi∗σCv8π28tFnde2+π2t4H2e−8Ch∗/Fnde2+π2Cv∗/4H2t.(2)
Iterative method to solve the result: takeEsi∗=μEsi, hi∗=μhi, and Cv∗=μCv, and the iterative solution is shown in Table 3.Table 3
Iterative solution.
Iteration number
x
j
x
j
∗
∂
g
/
∂
x
j
∗
cos
θ
j
x
j
′
∗
1
E
s
i
μ
E
s
i
∂
g
/
∂
x
j
x
j
∗
σ
x
j
−
∂
g
/
∂
x
j
∗
/
∑
∂
g
/
∂
x
j
∗
2
x
j
∗
+
β
σ
x
j
cos
θ
j
h
i
μ
h
i
C
v
μ
C
vThen, we put the new checking point back into the limit equationgx1∗,x2∗,…,xj∗=0 and repeat the calculation. Finally, the value of β can be obtained by the iteration method. And the probability of failure can be written as follows [14]:(17)Pf=Φ−β=1−Φβ.


### 3.4. Engineering Example Analysis

According to the engineering geological investigation report of an airport, in accordance with the preloading method soft soil foundation treatment plan requirements, the settlement generated in the construction time isS′. Assuming that sand wells’ effective drainage cylinder diameter de = 1.47 m, well diameter ratio n=de/dw=1.47/0.07=21, Fn=n2/n2−1lnn−3n2−1/4n2=2.3, the longest drainage path length in a sand well H=66m, and take μCv=0.001, σCv=0.001.In this engineering example, the engineering technicians have designed three sets of plans:(1)
Preloading pressure is 80 kPa, stacking time is 150 d(2)
Preloading pressure is 80 kPa, stacking time is 180 d(3)
Preloading pressure is 100 kPa, stacking time is 150 dBy the iterative method, solve the reliabilityβ of different plans:(1)
Preloading pressure is 80 kPa, stacking time is 150 d.Determine the thickness of soft soil layer. Because∑i=1nγi′hi<10ΔP, the thickness of the compressive layer is equal to the entire thickness of the soft soil layer.Solve the value ofS′ (N value is 1.0) by equations (13) and (14):(18)St=SUt=N∑i=1n1EsiΔPhi1−8π2e−8Ch/Fnde2+π2Cv/4H2t=0.623m.From equation (14), make S′=0.6m, the result obtained by the JC method is as follows:(19)Pf=1−Φβ=1−Φ0.125=0.450.(2)
Preloading pressure is 80 kPa, stacking time is 180 days.TakeS′=0.6m, the result obtained by the JC method is as follows:(20)Pf=1−Φβ=1−Φ0.273=0.392.(3)
Preloading pressure is 100 kPa, stacking time is 150 days.TakeS′=St=0.78m, the result obtained by the JC method is as follows:(21)Pf=1−Φβ=1−Φ0.535=0.294.It is not difficult to find that the degree of reliability index affected by different parameters is not the same: the longer the stacking time, the more reliable the design plan is, the less likely the failure; the greater the preloading pressure, the more reliable the design plan is, the less likely the failure. In the engineering example, we adopted the third solution of which the failure probability is the least.


## 3.1. Preloading Method Foundation Settlement Calculation Model and Method

According to the stratified summation method proposed by Terzaghi [10], under the condition of preloading, the final vertical deformation can be obtained by the following equation:(13)St=SUt=N∑i=1navi1+eoiΔPhi1−8π2e−8Ch/Fnde2+π2Cv/4H2t=N∑i=1n1EsiΔPhi1−8π2e−8Ch/Fnde2+π2Cv/4H2t,where St is the vertical deformation after treatment; S is the foundation ultimate vertical deformation; Ut is the average consolidation degree of sand well drainage (excluding the influence of well resistance and smearing); N is the correction coefficient; avi is the compressibility coefficient of soil; eoi is the void ratio of the soil layer; Cv is the vertical drainage consolidation coefficient; Ch is the horizontal drainage consolidation coefficient; avi is the compressive modulus of the soil layer; eoi is the void ratio of soil layer; Cv is the vertical drainage consolidation coefficient; Ch is the horizontal drainage consolidation coefficient; Esi is the compressive modulus of the soil layer; ΔP is the additional stress, which is set as the constant; hi is the thickness of the ith compression layer; H is the sand well longest drainage path length; de is the effective drainage diameter of the sand well; n is the well diameter ratio; Fn is a function related to the well diameter ratio.


## 3.2. Research on the Sensitivity of Soil Index

For the settlement calculation model of preloading method foundation treatment plan, the final consolidation settlement of foundation is affected by many soil indexes [11], and the degree of influence is still different. In order to clarify the degree of influence of each index on the final consolidation settlement, so as to simplify the actual calculation model, it is necessary to conduct a sensitivity study of the soil index. According to the preloading method settlement calculation model, the final consolidation settlement is affected by the void ratio e, the consolidation coefficient Cv, the compression modulus Es, the compression coefficient av, and the soil thickness h.According to the engineering geological investigation report of an airport, silt clay is taken as the research object to study the degree of influence of each soil index on the final consolidation settlement. The results of the sensitivity study are as follows:(14)St=SUt=av1+eΔPh1−8π2e−8Ch/Fnde2+π2Cv/4H2t=1EsΔPh1−8π2e−8Ch/Fnde2+π2Cv/4H2t,where St is the vertical deformation after the treatment; S is the final foundation vertical deformation; and Ut is the sand drainage average consolidation degree (excluding well resistance and smear effects).Among them, the preloading loadΔP is assumed as 100 kPa, the well diameter ratio n=de/dw=1.47/0.07=21, Fn=n2/n2−1lnn−3n2−1/4n2=2.3, the sand well effective drainage cylinder diameter de = 1.05l = 1.05 × 0.4 = 1.47 m, the maximum drainage path length of sand well H=15m, and the stacking time t is 150 d. The soil index values are shown in Table 1.(1)
Calculation of the slope of the final consolidation settlement by the compression coefficient.av∼Φμav,σav2, μav=0.941MPa−1, σav=0.081, the changing range of av is 0.770∼1.190 MPa−1, the slope is l1, l1=Stmax−Stmin/avmax−avmin/μav=0.090, as shown in Figure 3.(2)
Calculation of the slope of the final consolidation settlement by the void ratio.e∼Φμe,σe2, μe=1.317, σe=0.737, the changing range of e is 1.162∼1.450, the slope is l2, l2=Stmax−Stmin/emax−emin/μe=0.052, as shown in Figure 4.(3)
Calculation of the slope of the final consolidation settlement by the compression modulus.Es∼ΦμEs,σEs2, μEs=2.593MPa, σEs=0.392, the changing range of Es is 1.860∼4.330 MPa, the slope is l3, l3=Stmax−Stmin/Esmax−Esmin/Es=0.071, as shown in Figure 5.(4)
Calculation of the slope of the final consolidation settlement by the soil thickness.h∼Φμh,σh2, μh=3.9, σh=0.792, the changing range of h is 3.6∼4.3 m, the slope is l4, l4=Stmax−Stmin/hmax−hmin/μh=0.078, as shown in Figure 6.(5)
Calculation of the slope of the final consolidation settlement by the consolidation coefficient.Cv∼ΦμCv,σCv2, μCv=2.5×10−4cm/s2, σCv=1.95×10−5, the changing range of Cv is 0.0002∼0.0003cm/s2, the slope is l5, l5=Stmax−Stmin/Cvmax−Cvmin/μCv=0.040, as shown in Figure 7.Table 1
The statistics of soil parameter.
Silt clay
e
a
v
E
s
C
v
h
Mean value
1.317
0.941
2.593
2.51E-04
3.900
Standard deviation
0.737
0.081
0.392
1.95E-05
0.792Figure 3
Effect of compression coefficient on final consolidation settlement.Figure 4
Effect of void ratio on final consolidation settlement.Figure 5
Effect of compressive modulus on final consolidation settlement.Figure 6
Effect of soil thickness on the final consolidation settlement.Figure 7
Effect of consolidation coefficient on the final consolidation settlement.As Figures3–7 show, l1>l4>l3>l2>l5. The results show that the soil index which has the most influence on the final consolidation settlement is the compression coefficient, and then the soil thickness, compression modulus, void ratio, and consolidation coefficient.


## 3.3. Reliability Calculation of Preloading Method Soft Soil Foundation Treatment Plan

The limit state is analyzed [12], and the limit state equation is determined as follows:(15)gx=N∑i=1n1EsiΔPhi1−8π2e−8Ch/Fnde2+π2Cv/4H2t−S′=0,where S′ refers to the foundation settlement in the construction period.As for the judgment of the compressive soil layer, for the soft clay, it is judged by the principle that the prestress at a certain depth is equal to 0.1 times of the soil self-weight, that is, when∑i=1Nγhi≥10ΔP, hi is the i-layer compressive soil thickness. And the effect of consolidation coefficient on foundation final consolidation settlement is very small, so it can be assumed that the consolidation coefficients of all the soil layers are the same, and the consolidation coefficients are the same in the transverse and vertical direction.AssumingCv=Ch, the statistics of Esi, hi, Cv are shown in Table 2.Table 2
The statistics ofEsi, hi, Cv.
Random variables
Mean value
Standard derivation
Variation coefficient
E
s
i
μ
E
s
i
σ
E
s
i
δ
E
s
i
h
i
μ
h
i
σ
h
i
δ
h
i
C
v
μ
C
v
σ
C
v
δ
C
vThen, you can use the iterative method to calculate the reliabilityβ and failure probability Pf; the calculation steps are as follows [13]:(1)
Calculate the derivative of each variable.(16)∂g∂Esi∗=∂g∂EsiEsi∗σEsi=−N1Esi∗2ΔPhi∗σEsi1−8π2e−8Ch∗/Fnde2+π2Cv∗/4H2t,∂g∂hi∗=∂g∂hihi∗σhi=N1Esi∗ΔPσhi1−8π2e−8Ch∗/Fnde2+π2Cv∗/4H2t,∂g∂Cv∗=∂g∂CvCv∗σCv=N∑i=1n1Esi∗ΔPhi∗σCv8π28tFnde2+π2t4H2e−8Ch∗/Fnde2+π2Cv∗/4H2t.(2)
Iterative method to solve the result: takeEsi∗=μEsi, hi∗=μhi, and Cv∗=μCv, and the iterative solution is shown in Table 3.Table 3
Iterative solution.
Iteration number
x
j
x
j
∗
∂
g
/
∂
x
j
∗
cos
θ
j
x
j
′
∗
1
E
s
i
μ
E
s
i
∂
g
/
∂
x
j
x
j
∗
σ
x
j
−
∂
g
/
∂
x
j
∗
/
∑
∂
g
/
∂
x
j
∗
2
x
j
∗
+
β
σ
x
j
cos
θ
j
h
i
μ
h
i
C
v
μ
C
vThen, we put the new checking point back into the limit equationgx1∗,x2∗,…,xj∗=0 and repeat the calculation. Finally, the value of β can be obtained by the iteration method. And the probability of failure can be written as follows [14]:(17)Pf=Φ−β=1−Φβ.


## 3.4. Engineering Example Analysis

According to the engineering geological investigation report of an airport, in accordance with the preloading method soft soil foundation treatment plan requirements, the settlement generated in the construction time isS′. Assuming that sand wells’ effective drainage cylinder diameter de = 1.47 m, well diameter ratio n=de/dw=1.47/0.07=21, Fn=n2/n2−1lnn−3n2−1/4n2=2.3, the longest drainage path length in a sand well H=66m, and take μCv=0.001, σCv=0.001.In this engineering example, the engineering technicians have designed three sets of plans:(1)
Preloading pressure is 80 kPa, stacking time is 150 d(2)
Preloading pressure is 80 kPa, stacking time is 180 d(3)
Preloading pressure is 100 kPa, stacking time is 150 dBy the iterative method, solve the reliabilityβ of different plans:(1)
Preloading pressure is 80 kPa, stacking time is 150 d.Determine the thickness of soft soil layer. Because∑i=1nγi′hi<10ΔP, the thickness of the compressive layer is equal to the entire thickness of the soft soil layer.Solve the value ofS′ (N value is 1.0) by equations (13) and (14):(18)St=SUt=N∑i=1n1EsiΔPhi1−8π2e−8Ch/Fnde2+π2Cv/4H2t=0.623m.From equation (14), make S′=0.6m, the result obtained by the JC method is as follows:(19)Pf=1−Φβ=1−Φ0.125=0.450.(2)
Preloading pressure is 80 kPa, stacking time is 180 days.TakeS′=0.6m, the result obtained by the JC method is as follows:(20)Pf=1−Φβ=1−Φ0.273=0.392.(3)
Preloading pressure is 100 kPa, stacking time is 150 days.TakeS′=St=0.78m, the result obtained by the JC method is as follows:(21)Pf=1−Φβ=1−Φ0.535=0.294.It is not difficult to find that the degree of reliability index affected by different parameters is not the same: the longer the stacking time, the more reliable the design plan is, the less likely the failure; the greater the preloading pressure, the more reliable the design plan is, the less likely the failure. In the engineering example, we adopted the third solution of which the failure probability is the least.


## 4. Reliability Calculation Program Design of Soft Soil Foundation Preloading Method Treatment

### 4.1. Interface Design

This program uses C # to realize the fast reliability calculation of the soft ground treatment plan, builds the development environment by using Visual Studio 2010, and creates the project file of “Reliability Calculation.” The design of the login interface is shown in Figure8. After completing the setting and making of the login form, we need to implement the login form selection processing technology and link to the interface of the corresponding processing technology. Then, the processing technology interface of the preloading method is designed, as shown in Figure 9. Finally, the code of the reliability calculation program is implemented according to the algorithm of reliability calculation.Figure 8
The login form.Figure 9
Preloading method reliability calculation interface.


### 4.2. Reliability Algorithm Programming

#### 4.2.1. Reliability Algorithm

(1)
Input each random variable and constant
Constant:n, de, Fn, H, N, ΔP, S′, t:
Variable:Xjj=1,…,n,n+1,…,2n,2n+1 means the average of each soil index; Yjj=1,…,n,n+1,…,2n,2n+1 means the standard deviation of each soil index.
(2)
Calculate∂g/∂xj∗
LetXj∗=Xj,
Whenj=1,…,n, Zj=−N1/Xj∗2ΔPXj+n∗Yj1−8/π2e−8/Fnde2+π2/4H2X2n+1∗t
Whenj=n+1,…,2n, Zj=N1/Xj−n∗ΔPYj1−8/π2e−8/Fnde2+π2/4H2X2n+1∗t
Whenj=2n+1, Zj=N∑i=1n1/Xi∗ΔPXi+n∗Yj8/π28/Fnde2+π2/4H2e−8/Fnde2+π2/4H2Xj∗t
(3)
Calculatecosθj(22)cosθj=−Zj∑Zj2,j=1,…,n,n+1,…,2n,2n+1.
(4)
Calculate the new checking pointXj′∗(23)Xj′∗=Xj+βYjWj,j=1,…,n,n+1,…,2n,2n+1.
(5)
TakeXj′∗ into the equation gXj′∗=0, and solve β.(24)Letβ=0gXj′∗=N∑j=1n1Xj′∗ΔPXj+n′∗1−8π2e−8/Fnde2+π2/4H2X2n+1′∗t−S′.Determine whether gXj′∗<ε, if yes, then export β1=β; if no, then let β=β+Δβ, Δβ=10−3, replace β in gXj′∗=0, iterate until gXj′∗<ε, and export β1=β.
(6)
Takeβ=β1, Xj∗=Xj′∗, repeat Step 2 to Step 5, and then export β2=β.
(7)
Iterate untilβi+1−βi≤10−3 and then export β=βi+1.


#### 4.2.2. Program Implementation

After the program design is completed, the program needs to be tested so as to observe whether the program can successfully realize its proper function. In the engineering example in this paper, the preloading method test results are shown in Figure10.Figure 10
The test result of the preloading method reliability calculation.The result of the program is correct, the fast calculation is realized, and the efficiency of work is improved.


## 4.1. Interface Design

This program uses C # to realize the fast reliability calculation of the soft ground treatment plan, builds the development environment by using Visual Studio 2010, and creates the project file of “Reliability Calculation.” The design of the login interface is shown in Figure8. After completing the setting and making of the login form, we need to implement the login form selection processing technology and link to the interface of the corresponding processing technology. Then, the processing technology interface of the preloading method is designed, as shown in Figure 9. Finally, the code of the reliability calculation program is implemented according to the algorithm of reliability calculation.Figure 8
The login form.Figure 9
Preloading method reliability calculation interface.


## 4.2. Reliability Algorithm Programming

### 4.2.1. Reliability Algorithm

(1)
Input each random variable and constant
Constant:n, de, Fn, H, N, ΔP, S′, t:
Variable:Xjj=1,…,n,n+1,…,2n,2n+1 means the average of each soil index; Yjj=1,…,n,n+1,…,2n,2n+1 means the standard deviation of each soil index.
(2)
Calculate∂g/∂xj∗
LetXj∗=Xj,
Whenj=1,…,n, Zj=−N1/Xj∗2ΔPXj+n∗Yj1−8/π2e−8/Fnde2+π2/4H2X2n+1∗t
Whenj=n+1,…,2n, Zj=N1/Xj−n∗ΔPYj1−8/π2e−8/Fnde2+π2/4H2X2n+1∗t
Whenj=2n+1, Zj=N∑i=1n1/Xi∗ΔPXi+n∗Yj8/π28/Fnde2+π2/4H2e−8/Fnde2+π2/4H2Xj∗t
(3)
Calculatecosθj(22)cosθj=−Zj∑Zj2,j=1,…,n,n+1,…,2n,2n+1.
(4)
Calculate the new checking pointXj′∗(23)Xj′∗=Xj+βYjWj,j=1,…,n,n+1,…,2n,2n+1.
(5)
TakeXj′∗ into the equation gXj′∗=0, and solve β.(24)Letβ=0gXj′∗=N∑j=1n1Xj′∗ΔPXj+n′∗1−8π2e−8/Fnde2+π2/4H2X2n+1′∗t−S′.Determine whether gXj′∗<ε, if yes, then export β1=β; if no, then let β=β+Δβ, Δβ=10−3, replace β in gXj′∗=0, iterate until gXj′∗<ε, and export β1=β.
(6)
Takeβ=β1, Xj∗=Xj′∗, repeat Step 2 to Step 5, and then export β2=β.
(7)
Iterate untilβi+1−βi≤10−3 and then export β=βi+1.


### 4.2.2. Program Implementation

After the program design is completed, the program needs to be tested so as to observe whether the program can successfully realize its proper function. In the engineering example in this paper, the preloading method test results are shown in Figure10.Figure 10
The test result of the preloading method reliability calculation.The result of the program is correct, the fast calculation is realized, and the efficiency of work is improved.


## 4.2.1. Reliability Algorithm

(1)
Input each random variable and constant
Constant:n, de, Fn, H, N, ΔP, S′, t:
Variable:Xjj=1,…,n,n+1,…,2n,2n+1 means the average of each soil index; Yjj=1,…,n,n+1,…,2n,2n+1 means the standard deviation of each soil index.
(2)
Calculate∂g/∂xj∗
LetXj∗=Xj,
Whenj=1,…,n, Zj=−N1/Xj∗2ΔPXj+n∗Yj1−8/π2e−8/Fnde2+π2/4H2X2n+1∗t
Whenj=n+1,…,2n, Zj=N1/Xj−n∗ΔPYj1−8/π2e−8/Fnde2+π2/4H2X2n+1∗t
Whenj=2n+1, Zj=N∑i=1n1/Xi∗ΔPXi+n∗Yj8/π28/Fnde2+π2/4H2e−8/Fnde2+π2/4H2Xj∗t
(3)
Calculatecosθj(22)cosθj=−Zj∑Zj2,j=1,…,n,n+1,…,2n,2n+1.
(4)
Calculate the new checking pointXj′∗(23)Xj′∗=Xj+βYjWj,j=1,…,n,n+1,…,2n,2n+1.
(5)
TakeXj′∗ into the equation gXj′∗=0, and solve β.(24)Letβ=0gXj′∗=N∑j=1n1Xj′∗ΔPXj+n′∗1−8π2e−8/Fnde2+π2/4H2X2n+1′∗t−S′.Determine whether gXj′∗<ε, if yes, then export β1=β; if no, then let β=β+Δβ, Δβ=10−3, replace β in gXj′∗=0, iterate until gXj′∗<ε, and export β1=β.
(6)
Takeβ=β1, Xj∗=Xj′∗, repeat Step 2 to Step 5, and then export β2=β.
(7)
Iterate untilβi+1−βi≤10−3 and then export β=βi+1.


## 4.2.2. Program Implementation

After the program design is completed, the program needs to be tested so as to observe whether the program can successfully realize its proper function. In the engineering example in this paper, the preloading method test results are shown in Figure10.Figure 10
The test result of the preloading method reliability calculation.The result of the program is correct, the fast calculation is realized, and the efficiency of work is improved.


## 5. Conclusions

In this paper, we take the airport runway soft soil foundation as the research object and research on the reliability of preloading method soft ground plans by means of theoretical analysis, numerical simulation, and engineering example analysis and program design. According to the settlement calculation model, the influence degree of each soil parameter on the final consolidation settlement is analyzed. Moreover, we find the relationship between the stacking time and the reliability, and the preloading pressure and the reliability that the longer the stacking time is, the more reliable the design plan is, and the less likely the failure possibility is; the larger the preloading pressure is, the more reliable the design plan is, and the less likely the failure possibility is. Finally, we solve the complicated problem of reliability calculation and write the calculation program. The reliability calculation program is much faster than the manual calculation, which can save time and reduce the workload. According to the engineering example of an airport runway soft soil foundation, we carry out the reliability calculation; the theoretical calculation result is consistent with the practical application, which shows the feasibility of the reliability calculation method. In the next step, we can consider other foundation treatment technology reliability calculation programss and gradually form a set of reliability research system.



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*Source: 5204372-2019-03-03.xml*