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Plasticity solution for the limit vertical pressure of a single rigid pile with a pile cap in soft soil
Computers and Geotechnics 117 (2020) 103260
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Plasticity solution for the limit vertical pressure of a single rigid pile with a pile cap in soft soil
T
Hang Zhou , Hanlong Liu, Xueyuan Li, Xuanming Ding ⁎
Key Laboratory of New Technology for Construction of Cities in Mountain Area, College of Civil Engineering, Chongqing University, Chongqing 400045, China
ARTICLE INFO
ABSTRACT
Keywords: Finite element limit analysis Limit lateral resistance XCC pile group Undrained soil
This paper presents an analysis of the limit vertical capacity of a pile with a cap in soft soil and evaluates the influence of the cap on the limit vertical pressure (LVP) through finite element limit analysis methods based on OptumG2 and OptumG3. Rigorous upper and lower bound plastic solutions for the LVP are given. Subsequently, the effect of the pile-soil interface adhesion factor α, the ratio between the pile length and diameter L/d and the ratio between the pile cap diameter and pile length D/d on the LVP are investigated through parametric analysis. Moreover, a simple empirical equation for LVP is proposed by conducting nonlinear regression analysis with the least squares method. The presented equation for LVP can be used to construct a simple load-displacement relation for a single pile with a cap in the future.
1. Introduction The rigid pile composite foundation (RPCF) technique has been widely used to address soft soil improvement in China in recent decades. It is a ground improvement method that the mechanism is between the pile foundation and natural ground. Concrete piles and mixing piles are often adopted as the rigid piles in the composite foundation. RPCF can improve the ground capacity and reduce settlement. A rigid pile of a RPCF often produces upward or downward penetration deformation when the external load is relatively large. Therefore, an additional cap is placed on top of the rigid pile to reduce the penetration deformation. The cap changes the vertical rigid pile capacity and the behaviour of the RPCF. Butterfield and Banerjee conducted an elastic analysis of the load displacement behaviour of the pile group-pile cap system and the load distribution between the piles in the group and the cap based on Mindlin’s elastic solution [1]. Kim et al. studied the influence of the lateral capacity of the caps of pile groups by conducting full-scale model tests [2]. Desai et al. presented a numerical analysis of structural systems composed of a pile, cap and soil and identified the influence of relative stiffness on the distribution of loads and displacements in the cap-pile-soil system [3]. Joen and Park experimentally investigated the seismic characteristics of a prestressed concrete pile-pile cap system, and the test results showed that the pile-pile cap system could undergo large post-elastic deformation without an obvious loss in strength under severe seismic loading [4]. Chow and Teh numerically checked the
⁎
behaviour of vertically loaded pile groups embedded in a nonhomogeneous soil with the pile caps in contact with the ground [5]. Liu and Novak compared the behaviour of axially loaded single piles without caps and single piles with caps subjected to monotonic loading by combining the use of finite and infinite element methods in which the pile and the near-field soil are modelled by finite elements, while the far-field soil is modelled by mapped infinite elements [6]. Shen et al. used a variational approach to analyse the behaviour of a pile group–pile cap system and gave the finite series solutions for the deformations and shear stresses of group piles by using the principle of minimum potential energy [7]. Won et al. presented a numerical method that considers the coupling between the rigidities of the piles, the cap, and the column for investigating the response of columns supported by pile groups [8]. Chen et al. theoretically investigated the stress transfer mechanisms of a pile-supported embankment system, by considering the embankment fill, the piles and caps, and the foundation soils, and derived a closed-form solution to capture the soil arching effect [9]. Zheng et al. conducted a three-dimensional nonlinear finite element analysis of a composite foundation formed by CFG–lime piles to investigate the stress transfer and settlement behaviour [10]. In addition, the pile cap-soil interaction problem can be investigated through the classical model of a disk embedded in a semi-infinite domain. This fundamental problem was extensively studied by Ahmadi et al. who presented a series of dynamic and static analytical solutions for a disk embedded in a semi-infinite domain through Green’s functions, displacement potential functions, Hankel transforms, and some
Corresponding author. E-mail address: [email protected] (H. Zhou).
https://doi.org/10.1016/j.compgeo.2019.103260 Received 21 August 2019; Received in revised form 15 September 2019; Accepted 15 September 2019 0266-352X/ © 2019 Elsevier Ltd. All rights reserved.
Computers and Geotechnics 117 (2020) 103260
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area of the pile cap. The presented equation for the LVP can be used in future applications to construct a simple load-displacement relation for a single pile with a cap.
p q
q t
Undrained clay
Rigid pile cap
2. Problem statement
Rigid pile
Fig. 1 shows that the rigid single pile with a rigid pile cap is embedded in undrained soft clay. The pile length is defined as L, and the diameter of the pile is d. Two types of pile caps, namely, caps with a square (pile cap A) or circular (pile cap B) cross sections, are discussed in this paper. For a pile cap with a circular cross section, the problem reduces to an axisymmetric problem, while it reduces to a three-dimensional problem for a pile cap with a square cross section. The thickness of the pile cap is t, and D is used to describe the circular crosssectional diameter or square cross-sectional side length. In addition, the undrained strength of the soft soil is defined as su0. The top of the pile cap is subjected to a pressure p, while a pressure q is applied to the top of surrounding clay. To obtain the LVP of the pile-cap system pu, an axisymmetric model is used for the pile-circular cap system, and a three-dimensional model is chosen for the pile-square cap system. The difference in the LVP results between the two systems will be discussed. The surrounding soft clay is assumed to be a rigid, perfectly plastic Tresca material with an associated flow rule. Undrained conditions are considered in the computation. The shear strength of the pile-soil interface is described by = su (0 1) , where is the interface adhesion factor and ranges from 0 (representing a smooth interface) to 1 (representing a rough interface). A normalized LVP, Np = (pu q) su0 , is used in the following analysis. pu is equal to Fu A c , where Fu is the limit vertical bearing capacity of the single pile with a pile cap. The normalized LVP (hereafter referred to as the LVP) depends on the geometric parameters of the pile and pile cap (L, D, d), the undrained strength of the soil (su) and the pile-soil interface adhesion factor . Therefore, these parameters are normalized, and the LVP can be expressed as a function of these normalized parameters as follows:
L Rigid pile cap A
d Rigid pile cap B
D
z
z
Fig. 1. Computation model for pile-cap system.
LVP = (pu
(1)
q) su0 = f ( , , )
where = L d and culated as follows:
LVC = (su × LVP
= D d . In addition, the value of LVC can be cal(2)
q) A c D2 4
where Ac = for a square pile cap and Ac = for a circular pile cap. The side length of the square pile cap in cross section is assumed to be equal to the diameter of the circular pile cap in cross section. Then, the problem reduces to find the detailed form of the function on the right side of Eq. (1) through a series of parametric studies using FELA based on OptumG2 and G3.
D2
Fig. 2. Numerical model of the pile-circular cap system using OptumG2.
3. Numerical model
mathematical techniques [11–16]. More recently, Phutthananon et al. investigated the effect of pile cap size, soft layer thickness and pile strength on the load transfer and settlement mechanisms of pile-supported embankments [17]. However, a review of the previous studies shows that little attention has been paid to the effect of the pile cap on the limit vertical capacity of a pile. The bearing capacity of a composite foundation depends on the vertical capacity of a single pile, and the presented design method does not consider the pile cap effect when calculating the bearing capacity of a single pile. Therefore, the aim of this paper is to investigate the limit vertical capacity of a pile with a cap in soft soil and, in particular, to evaluate the influence of the cap on the limit vertical capacity (LVC). Two-dimensional and three-dimensional finite element limit analysis methods based on OptumG2 and OptumG3 are used to derive a simple expression for calculating the limit vertical pressure (LVP) of a single pile with a cap through a series of parametric studies. Then, the LVC can be obtained by multiplying the LVP and the cross-sectional
The analysis described in this paper was conducted using OptumG2 and G3, commercially available finite element and finite element limit analysis software (Optum CE 2018). OptumG2 can address two-dimensional plane strain and axisymmetric problems, while OptumG3 can solve three-dimensional problems. OptumG2 and OptumG3 incorporate the adaptive remeshing technique, which allows the failure mechanisms to be automatically optimized in terms of the size, position and orientation of the mesh elements. Therefore, these programs can provide highly accurate upper and lower bound solutions for limit analysis problems. OptumG2 is selected to model the pile-circular cap system due to its axisymmetric characteristics, while OptumG3 is adopted to simulate the pile-square cap system. To compare the 2D solution from OptumG2 and 3D solution from OptumG3, the pile-circular cap system is also simulated in OptumG3. The numerical models are shown in Figs. 2 and 3. The surrounding soil is described as a rigid-plastic Tresca model that 2
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Fig. 3. Numerical model of the pile-circular cap system and pile-square cap system using OptumG3.
70 65
D/d=2
60 55 50
(b)
50 45
UB(D/d=2) LB(D/d=2) UB(D/d=3) LB(D/d=3) UB(D/d=4) LB(D/d=4) UB(D/d=5) LB(D/d=5)
L/d=40
40
45
(pu q)/su
(pu q)/su
55
UB(L/d=10) LB(L/d=10) UB(L/d=20) LB(L/d=20) UB(L/d=30) LB(L/d=30) UB(L/d=40) LB(L/d=40)
(a)
40 35
35 30 25
30 25
20
20 15
15 10
0
2500
5000
7500
10000
12500
15000
17500
20000
Grid numbers
10
0
2500
5000
7500
10000
12500
15000
17500
20000
Grid numbers
Fig. 4. Variation in LVP for the pile-circular cap system with mesh numbers for different (a) L/d and (b) D/d values using OptumG2.
incorporates the associate flow rule. The pile and the pile cap are assumed to be rigid bodies. su0 is set to 10 kPa to represent a soft soil. β ranges from 10 to 40, and δ ranges from 1 to 5, which covers a relatively large range of commonly used sizes in practice. The adhesion factor α varies between 0 (smooth pile-soil interface) and 1 (perfectly rough interface). The adhesion factor α is selected as 1 for the validation analysis. The model size is selected as 50 m × 50 m (50 m × 50 m × 50 m for a quarter of the three-dimensional model) to avoid boundary effects. The pile diameter d is fixed at 0.5 m, while the other parameters change with the normalized parameters β and δ. The pressure q on the top surface of the surrounding soil is asset to 5 kPa. To obtain a plasticity solution with sufficient accuracy, the effect of mesh numbers on the calculated LVP is investigated by letting the mesh range from 500 to 20,000 elements. The mesh adaptivity provided in the Optum software is used with five adaptive iterations. The
corresponding LVP is then plotted against the mesh number to determine the convergence condition. The calculated parameters are given in the previous section. The results of OptumG2 and OptumG3 are shown in Figs. 4–6, respectively. It can be found that the value of LVP tends to be stable when the mesh number reaches 20,000 elements; therefore, a total of 20,000 elements will be used in the next parametric analysis. Furthermore, it can be seen from Table 1 that the difference in the LVP between the results of the circular pile cap case predicted by the 2D and 3D numerical models is insignificant. This proves the validity of the 3D numerical model. The calculated LVCs for the pilecircular cap and pile-square cap systems are also compared in Table 2. Although the values of the LVP for the two systems are different, the difference between the LVCs of the two systems is insignificant. This indicates that the influence of the pile cap shape on the LVC is limited. Therefore, this paper only considers the circular pile cap case using 3
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75
65
D/d=2
60
75
UB(L/d=10) LB(L/d=10) UB(L/d=20) LB(L/d=20) UB(L/d=30) LB(L/d=30) UB(L/d=40) LB(L/d=40)
70 (a) circular pile cap
55 50
65
55 50 45
(pu q)/su
(pu q)/su
D/d=2
60
45 40 35
40 35
30
30
25
25
20
20
15
15
10
10
5
5
0
UB(L/d=10) LB(L/d=10) UB(L/d=20) LB(L/d=20) UB(L/d=30) LB(L/d=30) UB(L/d=40) LB(L/d=40)
(b) square pile cap
70
0
2500
5000
7500
10000
12500
15000
17500
0
20000
0
2500
5000
7500
Grid numbers
10000
12500
15000
17500
20000
Grid numbers
Fig. 5. Variation in LVP for pile-circular cap and pile-square cap systems with mesh numbers for different L/d values using OptumG3.
85 75 70
L/d=20
65 60 55
60 55
L/d=20
50 45
50 45 40 35
40 35 30
30
25
25
20
20
15
15
10
10
5
5 0
UB(D/d=2) LB(D/d=2) UB(D/d=3) LB(D/d=3) UB(D/d=4) LB(D/d=4) UB(D/d=5) LB(D/d=5)
(b) square pile cap
65
(pu q)/su
(pu q)/su
70
UB(D/d=2) LB(D/d=2) UB(D/d=3) LB(D/d=3) UB(D/d=4) LB(D/d=4) UB(D/d=5) LB(D/d=5)
80 (a) circular pile cap
0
2500
5000
7500
10000
12500
15000
17500
20000
0
0
2500
5000
7500
Grid numbers
10000
12500
15000
17500
20000
Grid numbers
Fig. 6. Variation in LVP for pile-circular cap and pile-square cap systems with mesh numbers for different D/d values using OptumG3. Table 2 Comparison of LVC for pile-circular cap and pile-square cap systems.
Table 1 Comparison of LVPs for pile-circular cap and pile-square cap systems. LVP
L/d = 10, L/d = 20, L/d = 30, L/d = 40,
Circular
D/d = 2 D/d = 2 D/d = 2 D/d = 2
LVC (kN)
Square
2D_UB
2D_LB
3D_UB
3D_LB
UB
LB
L/d = 10, D/d = 2
17.826 28.443 38.817 49.09
17.684 28.305 38.674 48.93
17.18 28.92 37.168 46.284
16.536 27.96 36.324 44.308
14.392 23.724 30.172 37.832
13.784 23.204 29.448 36.692
L/d = 20, D/d = 2 L/d = 30, D/d = 2 L/d = 40, D/d = 2
4
UB LB UB LB UB LB UB LB
Circular
Square
Difference (%)
13.99 13.88 22.33 22.33 30.47 30.36 38.54 38.41
14.39 13.78 23.72 23.20 30.17 29.45 37.83 36.69
−2.77 0.71 −5.87 −3.76 0.99 3.09 1.86 4.68
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D/d=2 =0
D/d=3 =0
D/d=4 =0
D/d=2 = 0.6
D/d=3 = 0.6
D/d=4 = 0.6
D/d=2 =1
D/d=3 =1
D/d=4 =1
D/d=5 =0
D/d=5 = 0.6
D/d=5 =1
Fig. 7. Failure mode of the soil surrounding the pile cap.
220
200
(a) Lower bound solution D/d=1 D/d=2 d = 0.25m D/d=3 q = 5kPa D/d=4 su=10kPa D/d=5 Fitting line
140
(pu-q)/su
120
160 140
100 80
120 100 80
60
60
40
40
20
20
0
D/d=1 D/d=2 d = 0.25m D/d=3 q = 5kPa D/d=4 su=10kPa D/d=5 Fitting line
180
(pu-q)/su
160
(b) Upper bound solution
200
180
0
5
10
15
20
25
30
35
40
45
5
10
15
20
25
30
35
40
45
L/d
L/d
Fig. 8. (a) Lower bound and (b) upper bound solutions for the variation in LVP with L/d for different values of D/d.
OptumG2 in the following parametric analysis.
with decreasing adhesion factor α. Moreover, increasing D/d increases the failure zone size, while the shape of the zone is not changed. Fig. 8 plots the plasticity solutions for the variation in LVP with L/d for different values of D/d. It can be found that an increase in L/d leads to a linear increase in LVP; thus, a linear function is used to fit the relation between L/d and LVP. Furthermore, the slope of the L/d-LVP relation decreases as D/d increases. Fig. 9 shows the variation in LVP with L/d for different values of α. The slope of the L/d-LVP relation increases with increasing adhesion factor α. Fig. 10 gives the relation between LVP and D/d for different values of L/d, showing that the value of LVP decreases with increasing D/d. It should be clarified here that
4. Parametric analysis Fig. 7 plots the failure mode (displacement pattern) of the soil surrounding the pile cap for the pile-circular pile cap system under vertical loading conditions. The parameter L/d = 20 is used, and the other parameters used are shown in the figure. It can be seen that the failure mode is similar to that of the strip footing, in that a rigid inclusion zone appears beneath the pile for the rough interface case (α = 1). The zone is aligned with the rigid pile and gradually decreases 5
Computers and Geotechnics 117 (2020) 103260
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60
60
(b) Upper bound solution
(a) Lower bound solution =0 =0.2 =0.4 =0.7 =1 Fitting line
(pu-q)/su
40
40
30
20
10
10
0
5
10
15
20
25
30
35
40
D/d=2 d=0.25m q=5kPa su=10kPa
30
20
0
=0 =0.2 =0.4 =0.7 =1 Fitting line
50
D/d=2 d=0.25m q=5kPa su=10kPa (pu-q)/su
50
45
5
10
15
20
25
30
35
40
45
L/d
L/d
Fig. 9. (a) Lower bound and (b) upper bound solutions for the variation in LVP with L/d for different values of α.
200
200
(b) Upper bound solution
(a) Lower bound solution 180
180
160
160
d = 0.25m q = 5kPa su=10kPa
(pu-q)/su
120 100
140 120
(pu-q)/su
140
L/d=10 L/d=20 L/d=30 L/d=35 L/d=40 Fitting line
100
80
80
60
60
40
40
20
20
0
0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
L/d=10 L/d=20 L/d=30 L/d=35 L/d=40 Fitting line
d = 0.25m q = 5kPa su=10kPa
0
1
2
3
4
5
6
D/d
D/d
Fig. 10. (a) Lower bound and (b) upper bound solutions for the variation in LVP with D/d for different values of L/d.
this finding is not contradictory to the physical intuition that the bearing capacity should increase when increasing the size of the pile cap. It is noted that LVP is the limit pressure applied at the pile cap. The increase in D/d leads to an increase in the area of the cross section of the pile and, clearly, an increase in the vertical force LVC. In fact, the bearing capacity of a single pile with cap is the limit vertical force LVC described in Eq. (2) rather than the limit pressure. If the limit pressure
is transformed to a limit vertical force, as in Eq. (2), the relation describing how the LVC increases as D/d increases will be obtained (but is not presented here). Fig. 11 presents the variation in LVP with D/d for different values of α. A power function, as shown in Figs. 10 and 11, is selected to fit the LVP-D/d relation. In addition, Figs. 12 and 13 show the variations in LVP with α for different values of L/d and D/d, respectively. An increase in adhesion factor α results in an increase in
6
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120
120
(a) Lower bound solution
100
(pu-q)/su
80 70
100
=0 =0.2 =0.4 =0.7 =1 Fitting line
L/d d = 0.25m q = 5kPa su=10kPa
90
60
90
70 60 50
40
40
30
30
20
20
10
10 0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
=0 =0.2 =0.4 =0.7 =1 Fitting line
L/d d = 0.25m q = 5kPa su=10kPa
80
50
0
(b) Upper bound solution
110
(pu-q)/su
110
0
6.0
1
2
3
4
5
6
D/d
D/d
Fig. 11. (a) Lower bound and (b) upper bound solutions for the variation in LVP with D/d for different values of α.
60
45
(pu-q)/su
40 35
65
D/d d = 0.25m q = 5kPa su=10kPa
L/d=10 L/d=20 L/d=30 L/d=35 L/d=40 Fitting line
60 55 50
30 25
45 40
D/d d = 0.25m q = 5kPa su=10kPa
35 30
20
25 20
15
15
10
10
5 0
(b) Upper bound solution
70
L/d=10 L/d=20 L/d=30 L/d=35 L/d=40 Fitting line
50
75
(pu-q)/su
55
80
(a) Lower bound solution
5
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Fig. 12. (a) Lower bound and (b) upper bound solutions for the variation in LVP with α for different values of L/d.
LVP. Moreover, an increase in L/d or D/d leads to an increase in the slope of the LVP-α relation. All the detain informations about the fitting lines see Tables 3–14.
5. Empirical lower bound equation for the LVC in uniform soil From the above results, a simple general form for computing LVP can be obtained:
LVP = (A1 + A2 )(B1 + B2 )
7
C1
(3)
Computers and Geotechnics 117 (2020) 103260
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120
100
D/d=1 D/d=2 D/d=3 D/d=4 D/d=5 Fitting line
90 80
(pu-q)/su
(b) Upper bound solution
70
100
L/d d = 0.25m q = 5kPa su=10kPa
D/d=1 D/d=2 D/d=3 D/d=4 D/d=5 Fitting line
80
(pu-q)/su
110
120
(a) Lower bound solution
60 50
L/d d = 0.25m q = 5kPa su=10kPa
60
40
40 30
20
20 10 0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Fig. 13. (a) Lower bound and (b) upper bound solutions for the variation in LVP with α for different values of D/d. Table 3 Information about fitting lines in Fig. 8(a). Equation
y = a + bx
Plot
D/d = 1
D/d = 2
D/d = 3
D/d = 4
D/d = 5
Intercept Slope Residual Sum of Squares Pearson's r R-Square (COD) Adj. R-Square
14.95925 ± 0.36769 4.15749 ± 0.01366 0.65267 0.99997 0.99995 0.99994
7.45636 ± 0.09302 1.03876 ± 0.00345 0.04178 0.99997 0.99994 0.99993
6.43564 ± 0.041 0.46184 ± 0.00152 0.00812 0.99997 0.99995 0.99993
6.15343 ± 0.02246 0.25977 ± 8.3407E-4 0.00243 0.99997 0.99995 0.99994
5.02511 ± 0.51146 0.19741 ± 0.019 1.26286 0.97763 0.95575 0.9469
Table 4 Information about fitting lines in Fig. 8(b). Equation
y = a + bx
Plot
D/d = 1
D/d = 2
D/d = 3
D/d = 4
D/d = 5
Intercept Slope Residual Sum of Squares Pearson's r R-Square (COD) Adj. R-Square
15.19064 ± 0.4514 4.15661 ± 0.01676 0.98368 0.99996 0.99992 0.99999
7.54096 ± 0.09348 1.04098 ± 0.00347 0.04219 0.99997 0.99994 0.99993
6.5045 ± 0.0409 0.46259 ± 0.00152 0.00807 0.99997 0.99995 0.99994
6.20796 ± 0.02235 0.26012 ± 8.30091E-4 0.00241 0.99997 0.99995 0.99994
6.06929 ± 0.02844 0.167 ± 0.00106 0.0039 0.9999 0.9998 0.99976
Table 5 Information about fitting lines in Fig. 9(a). Equation
y = a + bx
Plot
α=0
α = 0.2
α = 0.4
α = 0.7
α=1
Intercept Slope Residual Sum of Squares Pearson's r R-Square (COD) Adj. R-Square
7.82268 ± 0.73905 0.02841 ± 0.02745 2.63677 0.9995 0.999 0.99993
7.857 ± 0.09172 0.24123 ± 0.00341 0.04061 0.9995 0.999 0.99993
7.91275 ± 0.09174 0.44114 ± 0.00341 0.04063 0.99985 0.9997 0.99994
7.70089 ± 0.09223 0.84094 ± 0.00343 0.04106 0.99996 0.9992 0.99994
7.45057 ± 0.08974 1.03917 ± 0.00333 0.03888 0.99997 0.9995 0.99976
8
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Table 6 Information about fitting lines in Fig. 9(b). Equation
y = a + bx
Plot
α=0
α = 0.2
α = 0.4
α = 0.7
α=1
Intercept Slope Residual Sum of Squares Pearson's r R-Square (COD) Adj. R-Square
7.85218 ± 0.09116 0.04088 ± 0.00339 0.04012 0.98328 0.996684 0.96021
7.9505 ± 0.09236 0.2413 ± 0.00343 0.04119 0.9995 0.99899 0.99879
7.91275 ± 0.09174 0.44114 ± 0.00341 0.04063 0.99985 0.9997 0.99964
7.70089 ± 0.09223 0.84094 ± 0.00343 0.04106 0.99996 0.9992 0.9999
7.54043 ± 0.09198 1.04103 ± 0.00342 0.04084 0.99997 0.9995 0.99994
Table 7 Information about fitting lines in Fig. 10(a). Equation
y = axb
Plot
L/d = 10
L/d = 20
L/d = 30
L/d = 35
L/d = 40
a b Reduced Chi-Sqr R-Square (COD) Adj. R-Square
55.69932 ± 2.14206 −1.46076 ± 0.10239 4.67384 0.9916 0.9888
98.413 ± 1.65226 −1.70378 ± 0.056 2.76065 0.99857 0.9981
139.47429 ± 2.40998 −1.76059 ± 0.06073 5.86589 0.9985 0.99847
160.02632 ± 2.43406 −1.78942 ± 0.05489 5.9802 0.99885 0.9999
180.50224 ± 2.44653 −1.81163 ± 0.04992 6.039 0.9991 0.9988
Table 8 Information about fitting lines in Fig. 10(b). Equation
y = axb
Plot
L/d = 10
L/d = 20
L/d = 30
L/d = 35
L/d = 40
a b Reduced Chi-Sqr R-Square (COD) Adj. R-Square
55.69932 ± 2.14206 −1.46076 ± 0.10239 4.67384 0.9916 0.9888
98.413 ± 1.65226 −1.70378 ± 0.056 2.76065 0.99857 0.9981
139.77543 ± 2.42401 −1.75822 ± 0.06082 5.93466 0.99849 0.99799
160.36622 ± 2.44794 −1.7871 ± 0.05497 6.04886 0.99884 0.99846
180.53316 ± 2.44937 −1.80754 ± 0.04978 6.05354 0.99909 0.99879
Table 9 Information about fitting lines in Fig. 11(a). Equation
y = axb
Plot
α=0
α = 0.2
α = 0.4
α = 0.7
α=1
a b Reduced Chi-Sqr R-Square (COD) Adj. R-Square
17.71843 ± 1.19082 −0.79773 ± 0.09911 1.51586 0.9564 0.94186
33.793 ± 1.74371 −1.1861 ± 0.10664 3.13903 0.98231 0.97641
49.89232 ± 2.02686 −1.38924 ± 0.10121 4.19667 0.9903 0.98706
82.20908 ± 1.61148 −1.64079 ± 0.06168 2.63016 0.99801 0.997434
98.17849 ± 1.65092 −1.70648 ± 0.05622 2.75597 0.99857 0.99809
Table 10 Information about fitting lines in Fig. 11(b). Equation
y = axb
Plot
α=0
α = 0.2
α = 0.4
α = 0.7
α=1
a b Reduced Chi-Sqr R-Square (COD) Adj. R-Square
17.86163 ± 1.17186 −0.79822 ± 0.09679 1.46787 0.95846 0.94461
33.92459 ± 1.72578 −1.18332 ± 0.10487 3.07536 0.98278 0.97704
50.01786 ± 2.01246 −1.38612 ± 0.09995 4.13778 0.99047 0.98729
82.33222 ± 1.61429 −1.63802 ± 0.06154 2.63952 0.998 0.99734
98.413 ± 1.65226 −1.70378 ± 0.056 2.76065 0.99857 0.9981
9
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Table 11 Information about fitting lines in Fig. 12(a). Equation
y = a + bx
Plot
L/d = 10
L/d = 20
L/d = 30
L/d = 35
L/d = 40
Intercept Slope Residual Sum of Squares R-Square (COD) Adj. R-Square
8.27119 ± 0.36129 10.00655 ± 0.62143 0.73219 0.98856 0.98475
8.92013 ± 0.71236 20.3819 ± 1.22529 2.84653 0.98927 0.9857
8.26054 ± 1.24859 32.04316 ± 2.14764 8.74504 0.9867 0.98227
9.48574 ± 1.24342 35.94231 ± 2.13874 8.67271 0.98949 0.98599
9.62671 ± 1.41981 41.13063 ± 2.44215 11.30791 0.98953 0.98605
Table 12 Information about fitting lines in Fig. 12(b). Equation
y = a + bx
Plot
L/d = 10
L/d = 20
L/d = 30
L/d = 35
L/d = 40
Intercept Slope Residual Sum of Squares R-Square (COD) Adj. R-Square
8.27119 ± 0.36129 10.00655 ± 0.62143 0.73219 0.98856 0.98475
8.92013 ± 0.71236 20.3819 ± 1.22529 2.84653 0.98927 0.9857
9.27406 ± 1.06992 30.82465 ± 1.84032 6.42134 0.98942 0.98589
9.40549 ± 1.25531 35.88111 ± 2.15921 8.83947 0.98925 0.98567
9.54234 ± 1.43374 41.07883 ± 2.46612 11.53096 0.9893 0.98574
Table 13 Information about fitting lines in Fig. 13(a). Equation
y = a + bx
Plot
D/d = 1
D/d = 2
D/d = 3
D/d = 4
D/d = 5
Intercept Slope Residual Sum of Squares R-Square (COD) Adj. R-Square
18.47967 ± 2.87855 83.24636 ± 4.95126 46.48033 0.9895 0.986
8.8355 ± 0.72311 20.345 ± 1.24379 2.93314 0.98891 0.98522
7.10817 ± 0.32854 9.02136 ± 0.5651 0.60547 0.98837 0.98449
6.51725 ± 0.19779 5.1112 ± 0.34022 0.21946 0.98688 0.98251
6.79153 ± 0.53824 −0.41073 ± 0.9258 1.62506 0.98514 0.98112
Table 14 Information about fitting lines in Fig. 13(b). Equation
y = a + bx
Plot
D/d = 1
D/d = 2
D/d = 3
D/d = 4
D/d = 5
Intercept Slope Residual Sum of Squares R-Square (COD) Adj. R-Square
18.59122 ± 2.85548 83.32823 ± 4.91157 45.7382 0.98968 0.98625
8.92013 ± 0.71236 20.3819 ± 1.22529 2.84653 0.98927 0.9857
7.18046 ± 0.32547 9.02813 ± 0.55983 0.59423 0.9886 0.98479
6.57248 ± 0.1954 5.11851 ± 0.33609 0.21417 0.98723 0.98297
6.78894 ± 0.5377 −0.40465 ± 0.92487 1.6218 0.98623 0.98199
6. Conclusion
Table 15 Optimal value of the constant coefficients. A1
A2
B1
B2
C1
3.89
8.73
2.08
0.26
−1.67
This paper provides a finite element limit analysis of the LVP of a single pile with a cap in soft soil by using OptumG2 and OptumG3. The influence of the pile-soil interface adhesion factor α, the ratio between the pile length and diameter L/d and the ratio between the pile cap diameter and pile length D/d on the LVP are investigated through parametric analysis. The results indicate that LVP linearly increases as L/d and the adhesion factor α increase, while LVP non-linearly decreases as D/d increases. Moreover, a simple empirical equation for LVP is proposed by conducting nonlinear regression analysis with the least squares method.
where A1, A2, B1, B2, and C1 are constant coefficients. A nonlinear regression analysis with the least squares method as described in Zhou et al. [18–20] is conducted to derive the five constant coefficients. The constant coefficients are shown in Table 15. To validate the derived equation, the FELA results and theoretical calculations using Eq. (3) are compared in Table 16. The largest differences between the FELA results and theoretical calculations are less than 7%.
10
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Table 16 Comparison of the FELA results and theoretical predictions using the derived empirical lower bound equation. α
β
δ
FELA
Prediction
Error (%)
α
β
δ
FELA
Prediction
Error (%)
α
β
δ
FELA
Prediction
Error (%)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
8.92 9.939 10.958 11.977 12.996 14.015 15.034 16.053 17.072 18.091 19.11 20.129 21.148 22.167 23.186 24.205 25.224 26.243 27.262 28.281 29.3
8.96 8.96 9.97 10.98 11.99 12.99 14.00 15.01 16.02 17.02 18.03 19.04 20.04 21.05 22.06 23.07 24.07 25.08 26.09 27.10 28.10
0.48 0.48 0.32 0.18 0.07 −0.02 −0.10 −0.18 −0.24 −0.29 −0.34 −0.38 −0.42 −0.46 −0.49 −0.52 −0.54 −0.57 −0.59 −0.61 −0.63
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
10 11.5 13 14.5 16 17.5 19 20.5 22 23.5 25 26.5 28 29.5 31 32.5 34 35.5 37 38.5 40
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
17.84 19.40 20.96 22.52 24.08 25.63 27.19 28.75 30.31 31.87 33.43 34.98 36.54 38.10 39.66 41.22 42.77 44.33 45.89 47.45 49.01
18.69 20.25 21.81 23.38 24.94 26.51 28.07 29.63 31.20 32.76 34.32 35.89 37.45 39.01 40.58 42.14 43.70 45.27 46.83 48.39 49.96
4.51 4.19 3.92 3.68 3.47 3.28 3.12 2.97 2.84 2.72 2.61 2.52 2.43 2.34 2.27 2.19 2.13 2.07 2.01 1.95 1.90
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20
1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5
98.41 72.13 55.47 44.19 36.15 30.21 25.68 22.14 19.32 17.03 15.14 13.56 12.23 11.10 10.12 9.27 8.53 7.88 7.31 6.80 6.34
92.45 68.22 52.76 42.23 34.70 29.11 24.83 21.48 18.80 16.61 14.81 13.30 12.02 10.93 9.99 9.17 8.45 7.82 7.26 6.76 6.32
−6.46 −5.74 −5.15 −4.63 −4.18 −3.78 −3.42 −3.09 −2.78 −2.50 −2.24 −2.00 −1.77 −1.56 −1.36 −1.17 −0.99 −0.81 −0.65 −0.49 −0.34
Acknowledgement [9]
The work is supported by the National Natural Science Foundation of China, Grant/Award Number: 51978105 and 51708063; Chongqing Research Program of Basic Research and Frontier Technology, Grant/ Award Number: cstc2017jcyjAX0261; and the Fundamental Research Funds for the Central Universities, Grant/Award Number: 2018CDQYTM0045.
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References
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[1] Butterfield R, Banerjee PK. The problem of pile group–pile cap interaction. Geotechnique 1971;21(2):135–42. [2] Kim JB, Brungraber RJ, Singh LP. Pile cap soil interaction from full-scale lateral load tests. J Geotechnical Eng Div 1979;105(5):643–53. [3] Desai CS, Alameddine AR, Kuppusamy T. Pile cap-pile group-soil interaction. J Struct Div 1981;107(5):817–34. [4] Joen PH, Park R. Simulated seismic load tests on prestressed concrete piles and pile-pile cap connections. PCI Journal 1990:35(6). [5] Chow YK, Teh CI. Pile-cap-pile-group interaction in nonhomogeneous soil. J Geotechnical Eng 1991;117(11):1655–68. [6] Liu W, Novak M. Soil–pile–cap static interaction analysis by finite and infinite elements. Can Geotech J 1991;28(6):771–83. [7] Shen WY, Chow YK, Yong KY. A variational approach for the analysis of pile group–pile cap interaction. Geotechnique 2000;50(4):349–57. [8] Won J, Ahn SY, Jeong S, et al. Nonlinear three-dimensional analysis of pile group
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supported columns considering pile cap flexibility. Comput Geotech 2006;33(6–7):355–70. Chen RP, Chen YM, Han J, et al. A theoretical solution for pile-supported embankments on soft soils under one-dimensional compression. Can Geotech J 2008;45(5):611–23. Zheng JJ, Abusharar SW, Wang XZ. Three-dimensional nonlinear finite element modeling of composite foundation formed by CFG–lime piles. Comput Geotech 2008;35:637–43. Ahmadi SF, Eskandari M. Vibration analysis of a rigid circular disk embedded in a transversely isotropic solid. J Eng Mech 2013;140(7):04014048. Ahmadi SF, Eskandari M. Axisymmetric circular indentation of a half-space reinforced by a buried elastic thin film. Mathematics Mech Solids 2014;19(6):703–12. Shodja HM, Ahmadi SF, Eskandari M. Boussinesq indentation of a transversely isotropic half-space reinforced by a buried inextensible membrane. Appl Math Model 2014;38(7–8):2163–72. Ahmadi SF, Samea P, Eskandari M. Axisymmetric response of a bi-material full-space reinforced by an interfacial thin film. Int J Solids Struct 2016;90:251–60. Robertson IA. Forced vertical vibration of a rigid circular disc on a semi-infinite elastic solid. Mathematical proceedings of the Cambridge philosophical society. Cambridge University Press; 1966. p. 547–53. Pak RYS, Gobert AT. Forced vertical vibration of rigid discs with arbitrary embedment. J Eng Mech 1991;117(11):2527–48. Phutthananon C, Jongpradist P, Jamsawang P. Influence of cap size and strength on settlements of TDM-piled embankments over soft ground. Mar Georesour Geotechnol 2019:1–20. Zhou H, Liu H, Yin F, et al. Upper and lower bound solutions for pressure-controlled cylindrical and spherical cavity expansion in semi-infinite soil. Comput Geotech 2018;103:93–102. Zhou H, Liu H, Wang L, et al. Finite element limit analysis of ultimate lateral pressure of XCC pile in undrained clay. Comput Geotech 2018;95:240–6. Zhou H, Liu H, Li Y, et al. Limit lateral resistance of XCC pile group in undrained soil. Acta Geotech 2019:1–11.
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Technical Communication
Plasticity solution for the limit vertical pressure of a single rigid pile with a pile cap in soft soil
T
Hang Zhou , Hanlong Liu, Xueyuan Li, Xuanming Ding ⁎
Key Laboratory of New Technology for Construction of Cities in Mountain Area, College of Civil Engineering, Chongqing University, Chongqing 400045, China
ARTICLE INFO
ABSTRACT
Keywords: Finite element limit analysis Limit lateral resistance XCC pile group Undrained soil
This paper presents an analysis of the limit vertical capacity of a pile with a cap in soft soil and evaluates the influence of the cap on the limit vertical pressure (LVP) through finite element limit analysis methods based on OptumG2 and OptumG3. Rigorous upper and lower bound plastic solutions for the LVP are given. Subsequently, the effect of the pile-soil interface adhesion factor α, the ratio between the pile length and diameter L/d and the ratio between the pile cap diameter and pile length D/d on the LVP are investigated through parametric analysis. Moreover, a simple empirical equation for LVP is proposed by conducting nonlinear regression analysis with the least squares method. The presented equation for LVP can be used to construct a simple load-displacement relation for a single pile with a cap in the future.
1. Introduction The rigid pile composite foundation (RPCF) technique has been widely used to address soft soil improvement in China in recent decades. It is a ground improvement method that the mechanism is between the pile foundation and natural ground. Concrete piles and mixing piles are often adopted as the rigid piles in the composite foundation. RPCF can improve the ground capacity and reduce settlement. A rigid pile of a RPCF often produces upward or downward penetration deformation when the external load is relatively large. Therefore, an additional cap is placed on top of the rigid pile to reduce the penetration deformation. The cap changes the vertical rigid pile capacity and the behaviour of the RPCF. Butterfield and Banerjee conducted an elastic analysis of the load displacement behaviour of the pile group-pile cap system and the load distribution between the piles in the group and the cap based on Mindlin’s elastic solution [1]. Kim et al. studied the influence of the lateral capacity of the caps of pile groups by conducting full-scale model tests [2]. Desai et al. presented a numerical analysis of structural systems composed of a pile, cap and soil and identified the influence of relative stiffness on the distribution of loads and displacements in the cap-pile-soil system [3]. Joen and Park experimentally investigated the seismic characteristics of a prestressed concrete pile-pile cap system, and the test results showed that the pile-pile cap system could undergo large post-elastic deformation without an obvious loss in strength under severe seismic loading [4]. Chow and Teh numerically checked the
⁎
behaviour of vertically loaded pile groups embedded in a nonhomogeneous soil with the pile caps in contact with the ground [5]. Liu and Novak compared the behaviour of axially loaded single piles without caps and single piles with caps subjected to monotonic loading by combining the use of finite and infinite element methods in which the pile and the near-field soil are modelled by finite elements, while the far-field soil is modelled by mapped infinite elements [6]. Shen et al. used a variational approach to analyse the behaviour of a pile group–pile cap system and gave the finite series solutions for the deformations and shear stresses of group piles by using the principle of minimum potential energy [7]. Won et al. presented a numerical method that considers the coupling between the rigidities of the piles, the cap, and the column for investigating the response of columns supported by pile groups [8]. Chen et al. theoretically investigated the stress transfer mechanisms of a pile-supported embankment system, by considering the embankment fill, the piles and caps, and the foundation soils, and derived a closed-form solution to capture the soil arching effect [9]. Zheng et al. conducted a three-dimensional nonlinear finite element analysis of a composite foundation formed by CFG–lime piles to investigate the stress transfer and settlement behaviour [10]. In addition, the pile cap-soil interaction problem can be investigated through the classical model of a disk embedded in a semi-infinite domain. This fundamental problem was extensively studied by Ahmadi et al. who presented a series of dynamic and static analytical solutions for a disk embedded in a semi-infinite domain through Green’s functions, displacement potential functions, Hankel transforms, and some
Corresponding author. E-mail address: [email protected] (H. Zhou).
https://doi.org/10.1016/j.compgeo.2019.103260 Received 21 August 2019; Received in revised form 15 September 2019; Accepted 15 September 2019 0266-352X/ © 2019 Elsevier Ltd. All rights reserved.
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area of the pile cap. The presented equation for the LVP can be used in future applications to construct a simple load-displacement relation for a single pile with a cap.
p q
q t
Undrained clay
Rigid pile cap
2. Problem statement
Rigid pile
Fig. 1 shows that the rigid single pile with a rigid pile cap is embedded in undrained soft clay. The pile length is defined as L, and the diameter of the pile is d. Two types of pile caps, namely, caps with a square (pile cap A) or circular (pile cap B) cross sections, are discussed in this paper. For a pile cap with a circular cross section, the problem reduces to an axisymmetric problem, while it reduces to a three-dimensional problem for a pile cap with a square cross section. The thickness of the pile cap is t, and D is used to describe the circular crosssectional diameter or square cross-sectional side length. In addition, the undrained strength of the soft soil is defined as su0. The top of the pile cap is subjected to a pressure p, while a pressure q is applied to the top of surrounding clay. To obtain the LVP of the pile-cap system pu, an axisymmetric model is used for the pile-circular cap system, and a three-dimensional model is chosen for the pile-square cap system. The difference in the LVP results between the two systems will be discussed. The surrounding soft clay is assumed to be a rigid, perfectly plastic Tresca material with an associated flow rule. Undrained conditions are considered in the computation. The shear strength of the pile-soil interface is described by = su (0 1) , where is the interface adhesion factor and ranges from 0 (representing a smooth interface) to 1 (representing a rough interface). A normalized LVP, Np = (pu q) su0 , is used in the following analysis. pu is equal to Fu A c , where Fu is the limit vertical bearing capacity of the single pile with a pile cap. The normalized LVP (hereafter referred to as the LVP) depends on the geometric parameters of the pile and pile cap (L, D, d), the undrained strength of the soil (su) and the pile-soil interface adhesion factor . Therefore, these parameters are normalized, and the LVP can be expressed as a function of these normalized parameters as follows:
L Rigid pile cap A
d Rigid pile cap B
D
z
z
Fig. 1. Computation model for pile-cap system.
LVP = (pu
(1)
q) su0 = f ( , , )
where = L d and culated as follows:
LVC = (su × LVP
= D d . In addition, the value of LVC can be cal(2)
q) A c D2 4
where Ac = for a square pile cap and Ac = for a circular pile cap. The side length of the square pile cap in cross section is assumed to be equal to the diameter of the circular pile cap in cross section. Then, the problem reduces to find the detailed form of the function on the right side of Eq. (1) through a series of parametric studies using FELA based on OptumG2 and G3.
D2
Fig. 2. Numerical model of the pile-circular cap system using OptumG2.
3. Numerical model
mathematical techniques [11–16]. More recently, Phutthananon et al. investigated the effect of pile cap size, soft layer thickness and pile strength on the load transfer and settlement mechanisms of pile-supported embankments [17]. However, a review of the previous studies shows that little attention has been paid to the effect of the pile cap on the limit vertical capacity of a pile. The bearing capacity of a composite foundation depends on the vertical capacity of a single pile, and the presented design method does not consider the pile cap effect when calculating the bearing capacity of a single pile. Therefore, the aim of this paper is to investigate the limit vertical capacity of a pile with a cap in soft soil and, in particular, to evaluate the influence of the cap on the limit vertical capacity (LVC). Two-dimensional and three-dimensional finite element limit analysis methods based on OptumG2 and OptumG3 are used to derive a simple expression for calculating the limit vertical pressure (LVP) of a single pile with a cap through a series of parametric studies. Then, the LVC can be obtained by multiplying the LVP and the cross-sectional
The analysis described in this paper was conducted using OptumG2 and G3, commercially available finite element and finite element limit analysis software (Optum CE 2018). OptumG2 can address two-dimensional plane strain and axisymmetric problems, while OptumG3 can solve three-dimensional problems. OptumG2 and OptumG3 incorporate the adaptive remeshing technique, which allows the failure mechanisms to be automatically optimized in terms of the size, position and orientation of the mesh elements. Therefore, these programs can provide highly accurate upper and lower bound solutions for limit analysis problems. OptumG2 is selected to model the pile-circular cap system due to its axisymmetric characteristics, while OptumG3 is adopted to simulate the pile-square cap system. To compare the 2D solution from OptumG2 and 3D solution from OptumG3, the pile-circular cap system is also simulated in OptumG3. The numerical models are shown in Figs. 2 and 3. The surrounding soil is described as a rigid-plastic Tresca model that 2
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Fig. 3. Numerical model of the pile-circular cap system and pile-square cap system using OptumG3.
70 65
D/d=2
60 55 50
(b)
50 45
UB(D/d=2) LB(D/d=2) UB(D/d=3) LB(D/d=3) UB(D/d=4) LB(D/d=4) UB(D/d=5) LB(D/d=5)
L/d=40
40
45
(pu q)/su
(pu q)/su
55
UB(L/d=10) LB(L/d=10) UB(L/d=20) LB(L/d=20) UB(L/d=30) LB(L/d=30) UB(L/d=40) LB(L/d=40)
(a)
40 35
35 30 25
30 25
20
20 15
15 10
0
2500
5000
7500
10000
12500
15000
17500
20000
Grid numbers
10
0
2500
5000
7500
10000
12500
15000
17500
20000
Grid numbers
Fig. 4. Variation in LVP for the pile-circular cap system with mesh numbers for different (a) L/d and (b) D/d values using OptumG2.
incorporates the associate flow rule. The pile and the pile cap are assumed to be rigid bodies. su0 is set to 10 kPa to represent a soft soil. β ranges from 10 to 40, and δ ranges from 1 to 5, which covers a relatively large range of commonly used sizes in practice. The adhesion factor α varies between 0 (smooth pile-soil interface) and 1 (perfectly rough interface). The adhesion factor α is selected as 1 for the validation analysis. The model size is selected as 50 m × 50 m (50 m × 50 m × 50 m for a quarter of the three-dimensional model) to avoid boundary effects. The pile diameter d is fixed at 0.5 m, while the other parameters change with the normalized parameters β and δ. The pressure q on the top surface of the surrounding soil is asset to 5 kPa. To obtain a plasticity solution with sufficient accuracy, the effect of mesh numbers on the calculated LVP is investigated by letting the mesh range from 500 to 20,000 elements. The mesh adaptivity provided in the Optum software is used with five adaptive iterations. The
corresponding LVP is then plotted against the mesh number to determine the convergence condition. The calculated parameters are given in the previous section. The results of OptumG2 and OptumG3 are shown in Figs. 4–6, respectively. It can be found that the value of LVP tends to be stable when the mesh number reaches 20,000 elements; therefore, a total of 20,000 elements will be used in the next parametric analysis. Furthermore, it can be seen from Table 1 that the difference in the LVP between the results of the circular pile cap case predicted by the 2D and 3D numerical models is insignificant. This proves the validity of the 3D numerical model. The calculated LVCs for the pilecircular cap and pile-square cap systems are also compared in Table 2. Although the values of the LVP for the two systems are different, the difference between the LVCs of the two systems is insignificant. This indicates that the influence of the pile cap shape on the LVC is limited. Therefore, this paper only considers the circular pile cap case using 3
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75
65
D/d=2
60
75
UB(L/d=10) LB(L/d=10) UB(L/d=20) LB(L/d=20) UB(L/d=30) LB(L/d=30) UB(L/d=40) LB(L/d=40)
70 (a) circular pile cap
55 50
65
55 50 45
(pu q)/su
(pu q)/su
D/d=2
60
45 40 35
40 35
30
30
25
25
20
20
15
15
10
10
5
5
0
UB(L/d=10) LB(L/d=10) UB(L/d=20) LB(L/d=20) UB(L/d=30) LB(L/d=30) UB(L/d=40) LB(L/d=40)
(b) square pile cap
70
0
2500
5000
7500
10000
12500
15000
17500
0
20000
0
2500
5000
7500
Grid numbers
10000
12500
15000
17500
20000
Grid numbers
Fig. 5. Variation in LVP for pile-circular cap and pile-square cap systems with mesh numbers for different L/d values using OptumG3.
85 75 70
L/d=20
65 60 55
60 55
L/d=20
50 45
50 45 40 35
40 35 30
30
25
25
20
20
15
15
10
10
5
5 0
UB(D/d=2) LB(D/d=2) UB(D/d=3) LB(D/d=3) UB(D/d=4) LB(D/d=4) UB(D/d=5) LB(D/d=5)
(b) square pile cap
65
(pu q)/su
(pu q)/su
70
UB(D/d=2) LB(D/d=2) UB(D/d=3) LB(D/d=3) UB(D/d=4) LB(D/d=4) UB(D/d=5) LB(D/d=5)
80 (a) circular pile cap
0
2500
5000
7500
10000
12500
15000
17500
20000
0
0
2500
5000
7500
Grid numbers
10000
12500
15000
17500
20000
Grid numbers
Fig. 6. Variation in LVP for pile-circular cap and pile-square cap systems with mesh numbers for different D/d values using OptumG3. Table 2 Comparison of LVC for pile-circular cap and pile-square cap systems.
Table 1 Comparison of LVPs for pile-circular cap and pile-square cap systems. LVP
L/d = 10, L/d = 20, L/d = 30, L/d = 40,
Circular
D/d = 2 D/d = 2 D/d = 2 D/d = 2
LVC (kN)
Square
2D_UB
2D_LB
3D_UB
3D_LB
UB
LB
L/d = 10, D/d = 2
17.826 28.443 38.817 49.09
17.684 28.305 38.674 48.93
17.18 28.92 37.168 46.284
16.536 27.96 36.324 44.308
14.392 23.724 30.172 37.832
13.784 23.204 29.448 36.692
L/d = 20, D/d = 2 L/d = 30, D/d = 2 L/d = 40, D/d = 2
4
UB LB UB LB UB LB UB LB
Circular
Square
Difference (%)
13.99 13.88 22.33 22.33 30.47 30.36 38.54 38.41
14.39 13.78 23.72 23.20 30.17 29.45 37.83 36.69
−2.77 0.71 −5.87 −3.76 0.99 3.09 1.86 4.68
Computers and Geotechnics 117 (2020) 103260
H. Zhou, et al.
D/d=2 =0
D/d=3 =0
D/d=4 =0
D/d=2 = 0.6
D/d=3 = 0.6
D/d=4 = 0.6
D/d=2 =1
D/d=3 =1
D/d=4 =1
D/d=5 =0
D/d=5 = 0.6
D/d=5 =1
Fig. 7. Failure mode of the soil surrounding the pile cap.
220
200
(a) Lower bound solution D/d=1 D/d=2 d = 0.25m D/d=3 q = 5kPa D/d=4 su=10kPa D/d=5 Fitting line
140
(pu-q)/su
120
160 140
100 80
120 100 80
60
60
40
40
20
20
0
D/d=1 D/d=2 d = 0.25m D/d=3 q = 5kPa D/d=4 su=10kPa D/d=5 Fitting line
180
(pu-q)/su
160
(b) Upper bound solution
200
180
0
5
10
15
20
25
30
35
40
45
5
10
15
20
25
30
35
40
45
L/d
L/d
Fig. 8. (a) Lower bound and (b) upper bound solutions for the variation in LVP with L/d for different values of D/d.
OptumG2 in the following parametric analysis.
with decreasing adhesion factor α. Moreover, increasing D/d increases the failure zone size, while the shape of the zone is not changed. Fig. 8 plots the plasticity solutions for the variation in LVP with L/d for different values of D/d. It can be found that an increase in L/d leads to a linear increase in LVP; thus, a linear function is used to fit the relation between L/d and LVP. Furthermore, the slope of the L/d-LVP relation decreases as D/d increases. Fig. 9 shows the variation in LVP with L/d for different values of α. The slope of the L/d-LVP relation increases with increasing adhesion factor α. Fig. 10 gives the relation between LVP and D/d for different values of L/d, showing that the value of LVP decreases with increasing D/d. It should be clarified here that
4. Parametric analysis Fig. 7 plots the failure mode (displacement pattern) of the soil surrounding the pile cap for the pile-circular pile cap system under vertical loading conditions. The parameter L/d = 20 is used, and the other parameters used are shown in the figure. It can be seen that the failure mode is similar to that of the strip footing, in that a rigid inclusion zone appears beneath the pile for the rough interface case (α = 1). The zone is aligned with the rigid pile and gradually decreases 5
Computers and Geotechnics 117 (2020) 103260
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60
60
(b) Upper bound solution
(a) Lower bound solution =0 =0.2 =0.4 =0.7 =1 Fitting line
(pu-q)/su
40
40
30
20
10
10
0
5
10
15
20
25
30
35
40
D/d=2 d=0.25m q=5kPa su=10kPa
30
20
0
=0 =0.2 =0.4 =0.7 =1 Fitting line
50
D/d=2 d=0.25m q=5kPa su=10kPa (pu-q)/su
50
45
5
10
15
20
25
30
35
40
45
L/d
L/d
Fig. 9. (a) Lower bound and (b) upper bound solutions for the variation in LVP with L/d for different values of α.
200
200
(b) Upper bound solution
(a) Lower bound solution 180
180
160
160
d = 0.25m q = 5kPa su=10kPa
(pu-q)/su
120 100
140 120
(pu-q)/su
140
L/d=10 L/d=20 L/d=30 L/d=35 L/d=40 Fitting line
100
80
80
60
60
40
40
20
20
0
0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
L/d=10 L/d=20 L/d=30 L/d=35 L/d=40 Fitting line
d = 0.25m q = 5kPa su=10kPa
0
1
2
3
4
5
6
D/d
D/d
Fig. 10. (a) Lower bound and (b) upper bound solutions for the variation in LVP with D/d for different values of L/d.
this finding is not contradictory to the physical intuition that the bearing capacity should increase when increasing the size of the pile cap. It is noted that LVP is the limit pressure applied at the pile cap. The increase in D/d leads to an increase in the area of the cross section of the pile and, clearly, an increase in the vertical force LVC. In fact, the bearing capacity of a single pile with cap is the limit vertical force LVC described in Eq. (2) rather than the limit pressure. If the limit pressure
is transformed to a limit vertical force, as in Eq. (2), the relation describing how the LVC increases as D/d increases will be obtained (but is not presented here). Fig. 11 presents the variation in LVP with D/d for different values of α. A power function, as shown in Figs. 10 and 11, is selected to fit the LVP-D/d relation. In addition, Figs. 12 and 13 show the variations in LVP with α for different values of L/d and D/d, respectively. An increase in adhesion factor α results in an increase in
6
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120
120
(a) Lower bound solution
100
(pu-q)/su
80 70
100
=0 =0.2 =0.4 =0.7 =1 Fitting line
L/d d = 0.25m q = 5kPa su=10kPa
90
60
90
70 60 50
40
40
30
30
20
20
10
10 0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
=0 =0.2 =0.4 =0.7 =1 Fitting line
L/d d = 0.25m q = 5kPa su=10kPa
80
50
0
(b) Upper bound solution
110
(pu-q)/su
110
0
6.0
1
2
3
4
5
6
D/d
D/d
Fig. 11. (a) Lower bound and (b) upper bound solutions for the variation in LVP with D/d for different values of α.
60
45
(pu-q)/su
40 35
65
D/d d = 0.25m q = 5kPa su=10kPa
L/d=10 L/d=20 L/d=30 L/d=35 L/d=40 Fitting line
60 55 50
30 25
45 40
D/d d = 0.25m q = 5kPa su=10kPa
35 30
20
25 20
15
15
10
10
5 0
(b) Upper bound solution
70
L/d=10 L/d=20 L/d=30 L/d=35 L/d=40 Fitting line
50
75
(pu-q)/su
55
80
(a) Lower bound solution
5
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Fig. 12. (a) Lower bound and (b) upper bound solutions for the variation in LVP with α for different values of L/d.
LVP. Moreover, an increase in L/d or D/d leads to an increase in the slope of the LVP-α relation. All the detain informations about the fitting lines see Tables 3–14.
5. Empirical lower bound equation for the LVC in uniform soil From the above results, a simple general form for computing LVP can be obtained:
LVP = (A1 + A2 )(B1 + B2 )
7
C1
(3)
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120
100
D/d=1 D/d=2 D/d=3 D/d=4 D/d=5 Fitting line
90 80
(pu-q)/su
(b) Upper bound solution
70
100
L/d d = 0.25m q = 5kPa su=10kPa
D/d=1 D/d=2 D/d=3 D/d=4 D/d=5 Fitting line
80
(pu-q)/su
110
120
(a) Lower bound solution
60 50
L/d d = 0.25m q = 5kPa su=10kPa
60
40
40 30
20
20 10 0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Fig. 13. (a) Lower bound and (b) upper bound solutions for the variation in LVP with α for different values of D/d. Table 3 Information about fitting lines in Fig. 8(a). Equation
y = a + bx
Plot
D/d = 1
D/d = 2
D/d = 3
D/d = 4
D/d = 5
Intercept Slope Residual Sum of Squares Pearson's r R-Square (COD) Adj. R-Square
14.95925 ± 0.36769 4.15749 ± 0.01366 0.65267 0.99997 0.99995 0.99994
7.45636 ± 0.09302 1.03876 ± 0.00345 0.04178 0.99997 0.99994 0.99993
6.43564 ± 0.041 0.46184 ± 0.00152 0.00812 0.99997 0.99995 0.99993
6.15343 ± 0.02246 0.25977 ± 8.3407E-4 0.00243 0.99997 0.99995 0.99994
5.02511 ± 0.51146 0.19741 ± 0.019 1.26286 0.97763 0.95575 0.9469
Table 4 Information about fitting lines in Fig. 8(b). Equation
y = a + bx
Plot
D/d = 1
D/d = 2
D/d = 3
D/d = 4
D/d = 5
Intercept Slope Residual Sum of Squares Pearson's r R-Square (COD) Adj. R-Square
15.19064 ± 0.4514 4.15661 ± 0.01676 0.98368 0.99996 0.99992 0.99999
7.54096 ± 0.09348 1.04098 ± 0.00347 0.04219 0.99997 0.99994 0.99993
6.5045 ± 0.0409 0.46259 ± 0.00152 0.00807 0.99997 0.99995 0.99994
6.20796 ± 0.02235 0.26012 ± 8.30091E-4 0.00241 0.99997 0.99995 0.99994
6.06929 ± 0.02844 0.167 ± 0.00106 0.0039 0.9999 0.9998 0.99976
Table 5 Information about fitting lines in Fig. 9(a). Equation
y = a + bx
Plot
α=0
α = 0.2
α = 0.4
α = 0.7
α=1
Intercept Slope Residual Sum of Squares Pearson's r R-Square (COD) Adj. R-Square
7.82268 ± 0.73905 0.02841 ± 0.02745 2.63677 0.9995 0.999 0.99993
7.857 ± 0.09172 0.24123 ± 0.00341 0.04061 0.9995 0.999 0.99993
7.91275 ± 0.09174 0.44114 ± 0.00341 0.04063 0.99985 0.9997 0.99994
7.70089 ± 0.09223 0.84094 ± 0.00343 0.04106 0.99996 0.9992 0.99994
7.45057 ± 0.08974 1.03917 ± 0.00333 0.03888 0.99997 0.9995 0.99976
8
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Table 6 Information about fitting lines in Fig. 9(b). Equation
y = a + bx
Plot
α=0
α = 0.2
α = 0.4
α = 0.7
α=1
Intercept Slope Residual Sum of Squares Pearson's r R-Square (COD) Adj. R-Square
7.85218 ± 0.09116 0.04088 ± 0.00339 0.04012 0.98328 0.996684 0.96021
7.9505 ± 0.09236 0.2413 ± 0.00343 0.04119 0.9995 0.99899 0.99879
7.91275 ± 0.09174 0.44114 ± 0.00341 0.04063 0.99985 0.9997 0.99964
7.70089 ± 0.09223 0.84094 ± 0.00343 0.04106 0.99996 0.9992 0.9999
7.54043 ± 0.09198 1.04103 ± 0.00342 0.04084 0.99997 0.9995 0.99994
Table 7 Information about fitting lines in Fig. 10(a). Equation
y = axb
Plot
L/d = 10
L/d = 20
L/d = 30
L/d = 35
L/d = 40
a b Reduced Chi-Sqr R-Square (COD) Adj. R-Square
55.69932 ± 2.14206 −1.46076 ± 0.10239 4.67384 0.9916 0.9888
98.413 ± 1.65226 −1.70378 ± 0.056 2.76065 0.99857 0.9981
139.47429 ± 2.40998 −1.76059 ± 0.06073 5.86589 0.9985 0.99847
160.02632 ± 2.43406 −1.78942 ± 0.05489 5.9802 0.99885 0.9999
180.50224 ± 2.44653 −1.81163 ± 0.04992 6.039 0.9991 0.9988
Table 8 Information about fitting lines in Fig. 10(b). Equation
y = axb
Plot
L/d = 10
L/d = 20
L/d = 30
L/d = 35
L/d = 40
a b Reduced Chi-Sqr R-Square (COD) Adj. R-Square
55.69932 ± 2.14206 −1.46076 ± 0.10239 4.67384 0.9916 0.9888
98.413 ± 1.65226 −1.70378 ± 0.056 2.76065 0.99857 0.9981
139.77543 ± 2.42401 −1.75822 ± 0.06082 5.93466 0.99849 0.99799
160.36622 ± 2.44794 −1.7871 ± 0.05497 6.04886 0.99884 0.99846
180.53316 ± 2.44937 −1.80754 ± 0.04978 6.05354 0.99909 0.99879
Table 9 Information about fitting lines in Fig. 11(a). Equation
y = axb
Plot
α=0
α = 0.2
α = 0.4
α = 0.7
α=1
a b Reduced Chi-Sqr R-Square (COD) Adj. R-Square
17.71843 ± 1.19082 −0.79773 ± 0.09911 1.51586 0.9564 0.94186
33.793 ± 1.74371 −1.1861 ± 0.10664 3.13903 0.98231 0.97641
49.89232 ± 2.02686 −1.38924 ± 0.10121 4.19667 0.9903 0.98706
82.20908 ± 1.61148 −1.64079 ± 0.06168 2.63016 0.99801 0.997434
98.17849 ± 1.65092 −1.70648 ± 0.05622 2.75597 0.99857 0.99809
Table 10 Information about fitting lines in Fig. 11(b). Equation
y = axb
Plot
α=0
α = 0.2
α = 0.4
α = 0.7
α=1
a b Reduced Chi-Sqr R-Square (COD) Adj. R-Square
17.86163 ± 1.17186 −0.79822 ± 0.09679 1.46787 0.95846 0.94461
33.92459 ± 1.72578 −1.18332 ± 0.10487 3.07536 0.98278 0.97704
50.01786 ± 2.01246 −1.38612 ± 0.09995 4.13778 0.99047 0.98729
82.33222 ± 1.61429 −1.63802 ± 0.06154 2.63952 0.998 0.99734
98.413 ± 1.65226 −1.70378 ± 0.056 2.76065 0.99857 0.9981
9
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Table 11 Information about fitting lines in Fig. 12(a). Equation
y = a + bx
Plot
L/d = 10
L/d = 20
L/d = 30
L/d = 35
L/d = 40
Intercept Slope Residual Sum of Squares R-Square (COD) Adj. R-Square
8.27119 ± 0.36129 10.00655 ± 0.62143 0.73219 0.98856 0.98475
8.92013 ± 0.71236 20.3819 ± 1.22529 2.84653 0.98927 0.9857
8.26054 ± 1.24859 32.04316 ± 2.14764 8.74504 0.9867 0.98227
9.48574 ± 1.24342 35.94231 ± 2.13874 8.67271 0.98949 0.98599
9.62671 ± 1.41981 41.13063 ± 2.44215 11.30791 0.98953 0.98605
Table 12 Information about fitting lines in Fig. 12(b). Equation
y = a + bx
Plot
L/d = 10
L/d = 20
L/d = 30
L/d = 35
L/d = 40
Intercept Slope Residual Sum of Squares R-Square (COD) Adj. R-Square
8.27119 ± 0.36129 10.00655 ± 0.62143 0.73219 0.98856 0.98475
8.92013 ± 0.71236 20.3819 ± 1.22529 2.84653 0.98927 0.9857
9.27406 ± 1.06992 30.82465 ± 1.84032 6.42134 0.98942 0.98589
9.40549 ± 1.25531 35.88111 ± 2.15921 8.83947 0.98925 0.98567
9.54234 ± 1.43374 41.07883 ± 2.46612 11.53096 0.9893 0.98574
Table 13 Information about fitting lines in Fig. 13(a). Equation
y = a + bx
Plot
D/d = 1
D/d = 2
D/d = 3
D/d = 4
D/d = 5
Intercept Slope Residual Sum of Squares R-Square (COD) Adj. R-Square
18.47967 ± 2.87855 83.24636 ± 4.95126 46.48033 0.9895 0.986
8.8355 ± 0.72311 20.345 ± 1.24379 2.93314 0.98891 0.98522
7.10817 ± 0.32854 9.02136 ± 0.5651 0.60547 0.98837 0.98449
6.51725 ± 0.19779 5.1112 ± 0.34022 0.21946 0.98688 0.98251
6.79153 ± 0.53824 −0.41073 ± 0.9258 1.62506 0.98514 0.98112
Table 14 Information about fitting lines in Fig. 13(b). Equation
y = a + bx
Plot
D/d = 1
D/d = 2
D/d = 3
D/d = 4
D/d = 5
Intercept Slope Residual Sum of Squares R-Square (COD) Adj. R-Square
18.59122 ± 2.85548 83.32823 ± 4.91157 45.7382 0.98968 0.98625
8.92013 ± 0.71236 20.3819 ± 1.22529 2.84653 0.98927 0.9857
7.18046 ± 0.32547 9.02813 ± 0.55983 0.59423 0.9886 0.98479
6.57248 ± 0.1954 5.11851 ± 0.33609 0.21417 0.98723 0.98297
6.78894 ± 0.5377 −0.40465 ± 0.92487 1.6218 0.98623 0.98199
6. Conclusion
Table 15 Optimal value of the constant coefficients. A1
A2
B1
B2
C1
3.89
8.73
2.08
0.26
−1.67
This paper provides a finite element limit analysis of the LVP of a single pile with a cap in soft soil by using OptumG2 and OptumG3. The influence of the pile-soil interface adhesion factor α, the ratio between the pile length and diameter L/d and the ratio between the pile cap diameter and pile length D/d on the LVP are investigated through parametric analysis. The results indicate that LVP linearly increases as L/d and the adhesion factor α increase, while LVP non-linearly decreases as D/d increases. Moreover, a simple empirical equation for LVP is proposed by conducting nonlinear regression analysis with the least squares method.
where A1, A2, B1, B2, and C1 are constant coefficients. A nonlinear regression analysis with the least squares method as described in Zhou et al. [18–20] is conducted to derive the five constant coefficients. The constant coefficients are shown in Table 15. To validate the derived equation, the FELA results and theoretical calculations using Eq. (3) are compared in Table 16. The largest differences between the FELA results and theoretical calculations are less than 7%.
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Computers and Geotechnics 117 (2020) 103260
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Table 16 Comparison of the FELA results and theoretical predictions using the derived empirical lower bound equation. α
β
δ
FELA
Prediction
Error (%)
α
β
δ
FELA
Prediction
Error (%)
α
β
δ
FELA
Prediction
Error (%)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
8.92 9.939 10.958 11.977 12.996 14.015 15.034 16.053 17.072 18.091 19.11 20.129 21.148 22.167 23.186 24.205 25.224 26.243 27.262 28.281 29.3
8.96 8.96 9.97 10.98 11.99 12.99 14.00 15.01 16.02 17.02 18.03 19.04 20.04 21.05 22.06 23.07 24.07 25.08 26.09 27.10 28.10
0.48 0.48 0.32 0.18 0.07 −0.02 −0.10 −0.18 −0.24 −0.29 −0.34 −0.38 −0.42 −0.46 −0.49 −0.52 −0.54 −0.57 −0.59 −0.61 −0.63
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
10 11.5 13 14.5 16 17.5 19 20.5 22 23.5 25 26.5 28 29.5 31 32.5 34 35.5 37 38.5 40
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
17.84 19.40 20.96 22.52 24.08 25.63 27.19 28.75 30.31 31.87 33.43 34.98 36.54 38.10 39.66 41.22 42.77 44.33 45.89 47.45 49.01
18.69 20.25 21.81 23.38 24.94 26.51 28.07 29.63 31.20 32.76 34.32 35.89 37.45 39.01 40.58 42.14 43.70 45.27 46.83 48.39 49.96
4.51 4.19 3.92 3.68 3.47 3.28 3.12 2.97 2.84 2.72 2.61 2.52 2.43 2.34 2.27 2.19 2.13 2.07 2.01 1.95 1.90
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20
1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5
98.41 72.13 55.47 44.19 36.15 30.21 25.68 22.14 19.32 17.03 15.14 13.56 12.23 11.10 10.12 9.27 8.53 7.88 7.31 6.80 6.34
92.45 68.22 52.76 42.23 34.70 29.11 24.83 21.48 18.80 16.61 14.81 13.30 12.02 10.93 9.99 9.17 8.45 7.82 7.26 6.76 6.32
−6.46 −5.74 −5.15 −4.63 −4.18 −3.78 −3.42 −3.09 −2.78 −2.50 −2.24 −2.00 −1.77 −1.56 −1.36 −1.17 −0.99 −0.81 −0.65 −0.49 −0.34
Acknowledgement [9]
The work is supported by the National Natural Science Foundation of China, Grant/Award Number: 51978105 and 51708063; Chongqing Research Program of Basic Research and Frontier Technology, Grant/ Award Number: cstc2017jcyjAX0261; and the Fundamental Research Funds for the Central Universities, Grant/Award Number: 2018CDQYTM0045.
[10] [11] [12] [13]
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