# Digital Evaluation of Vertical Compressive Bearing Capacity for Jet Grouting Pile-Mini Steel Pipe Pile Composite Foundation

Authors: Xiaojun Wang; Zhipeng Mao; Wendi Wang
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

The jet grouting pile-mini steel pipe pile in soft soil area is composed of the jet grouting pile and the mini steel pipe pile. Through the analysis of the stress mode of the composite foundation under the overall compression of the pile top, the hypothetical diagram and calculation equation of the interfacial friction resistance distribution of the composite foundation are put forward. Thus, the calculation equation of the axial force of the mini steel pipe pile and the jet grouting pile is deduced, and the axial force distribution diagram of the mini steel pipe pile and the jet grouting pile in the composite foundation is defined. The vertical compression composite foundation in the field was selected for bearing capacity test and calculation and comparative analysis, and the relative error was 3%. Therefore, the stress mode and calculation method of the jet grouting pile-mini steel pipe pile proposed in this paper can meet the needs of engineering design and popularization and application of the composite foundation.


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## Body

## 1. Introduction

The jet grouting pile-mini steel pipe pile composite foundation refers to the combined system formed by the jet grouting pile and the mini steel pipe pile to share external load [1]. In the soft soil layer, the jet grouting pile with a certain depth and diameter shall be constructed first. Before the initial solidification of the jet grouting pile [2], the mini steel pipe pile with diameter less than Φ300 mm and length less than the length of the jet grouting pile is vertically pushed into the jet grouting pile along the central part by the improved microinserting and pressing mechanical equipment. After the strength of the jet grouting pile reaches 28 days, the composite foundation is formed [3]. In this combined system, the pile top load is mainly transmitted to the jet grouting pile through the friction around the mini steel pipe pile [4], and then the load is transmitted to the soil around the pile mainly by the friction force around the jet grouting pile [5] so as to complete the load transmission [6]. This system can take maximum advantage of the “high strength” characteristics of mini steel pipe pile to bear the load [7] and use the reinforcement effect of jet grouting pile on soft soil and large pile side surface area to provide sufficient side friction resistance [8]. Although the load plate test is more appropriate to discuss the bearing capacity of the composite foundation [9], its deformation and load transfer law cannot be directly extended to the actual engineering design. Therefore [10], it is necessary to study and give a reasonable and simple calculation method of the bearing capacity of the composite foundation [11] so as to make the composite foundation better popularized in engineering application [12], which is the main scientific and technological problem to be solved in this paper. The composite foundation is used in the foundation reinforcement project with the engineering geological condition of soft soils. For the convenience of description, the jet grouting pile-mini steel pipe pile composite foundation is abbreviated as the composite foundation.


## 2. Stress Mode Analysis of the Composite Foundation under Vertical Compressive Load

It is assumed that in the composite foundation, the uniformity of the jet grouting pile is good [13], and the diameter of the pile body remains unchanged in the depth direction. The mini steel pipe pile and the composite foundation are regarded as pure friction piles, and the bottom axial force is considered as 0. Figure 1 is a schematic diagram of the force transfer mechanism of the composite foundation under vertical compressive load. According to the vertical force transfer characteristics of the composite foundation shown in Figure 1, the composite foundation can be divided into upper and lower parts of “inserted mini steel pipe pile section” and “pure jet grouting pile section” along the depth direction. Due to the small rigidity of the jet grouting pile, when the length of the pure jet grouting pile section is large, the side friction resistance of the “pure jet grouting pile section” mainly acts on the “effective section of the pure jet grouting pile” with the length of Lc. The side friction resistance between the noneffective section at the bottom of the jet grouting pile and soil around the pile is very small, which can be ignored. Therefore, in the calculation of bearing capacity, the effective length of the “pure jet grouting pile section” in the composite foundation should be guaranteed as far as possible so as to reduce the length of the mini steel pipe pile. It is generally believed that the effective pile length of the jet grouting pile shall not exceed 10 m. Combined with the concept of effective pile length of jet grouting pile, the value of the “effective section of pure jet grouting pile” Lc in the composite foundation can be obtained by the following equation:(1)Lc=Lj−Ls≤minβDj,10.Figure 1
The schematic diagram of the force transfer mechanism of the composite foundation under vertical compressive load.In the formula,Lc is the length of the “effective section of the pure jet grouting pile”; Lj is the effective length of the jet grouting pile; Ls is the length of the mini steel pipe pile. According to the strength of the jet grouting pile body, β=10∼18. Dj is the diameter of the jet grouting pile.Under the overall compression mode of the mini steel pipe pile and jet grouting pile jointly bearing the upper load, the bearing performance of the mini steel pipe pile and the jet grouting pile should be considered at the same time [14, 15].The ultimate compressive bearing capacity of the mini steel pipe pile at the pile top shall meet the following equation:(2)Q0s≤Qsu=σsuAs.The load shared by the top of the jet grouting pile can be calculated according to the following formula:(3)Qju≥Q0j=QsuAjAsnp,(4)Aj=Dj2−Ds2π40<Z<Ls,Dj2π4Ls≤Z..In the equation,Q0s is the compressive load shared by the pile top of the mini steel pipe pile; Q0j is the compressive load shared by the top of the jet grouting pile; Qsu is the vertical ultimate compressive bearing capacity of the mini steel pipe pile controlled by yield strength; Qju is the vertical ultimate compressive bearing capacity of the jet grouting pile without confined compressive strength control; σsu is the yield strength of the mini steel pipe pile; As is the cross-sectional area of the mini steel pipe pile, As=πTsDs−Ts2, where Ts is the wall thickness of the mini steel pipe pile, and Ds is the outer diameter of the mini steel pipe pile; Dj is the diameter of the jet grouting pile; np=σ0s/σ0j is the stress sharing ratio of the mini steel pipe pile and the jet grouting pile at the pile top; Aj is the cross-sectional area of the jet grouting pile, determined by (4); Z is the depth; Ls is the length of the mini steel pipe pile; Lj is the effective length of the jet grouting pile.In addition, to coordinate the ultimate side friction at the interface between the mini steel pipe pile and the jet grouting pile with the ultimate side friction at the interface between the jet grouting pile and soil, it should also be noted that there is no shear failure between the mini steel pipe pile and the jet grouting pile under the condition of ensuring the material strength of the pile body [16–18]. Therefore, the bonding strength (ultimate side friction) at the interface between the mini steel pipe pile and the grouting pile should meet the following conditions:(5)Q0s≤fsu,(6)fsu=Us∫0LsqsZdz0≤Z≤Ls,The composite foundation is used as a friction pile, so the pile top load shall not be greater than the ultimate side friction resistance of the jet grouting pile-soil interface.(7)Q0≤fju,(8)Q0=Q0s+Q0j,(9)fju=Uj∫0LjqjZdz0≤Z≤Lj,In the equation,qsZ is the side friction resistance at the interface between the mini steel pipe pile and the jet grouting pile at depth Z; qjZ is the side friction resistance at the interface between the jet grouting pile and soil at depth Z; fsu is the ultimate side friction resistance at the interface between the mini steel pipe pile and the jet grouting pile; fju is the ultimate side friction resistance at the interface between the jet grouting pile and soil; Us is the perimeter of the mini steel pipe pile; Uj is the perimeter of the jet grouting pile; Z is the depth; Ls is the length of the mini steel pipe pile; Lj is the effective length of the jet grouting pile; Q0 is the pile top load.According to the analysis results of the numerical model [19, 20], under the vertical compressive load, the maximum axial force of the jet grouting pile appears at the depth of 0 m and the bottom of the mini steel pipe pile (Z=Ls). Therefore, when calculating the compressive bearing capacity of the composite foundation, the strength of the jet grouting pile needs to be checked according to equations (10) to (12).(10)Q0j≤Qju=σjuAj,(11)QjLs≤R2,(12)R2=γcQju.In the equation,σju is the vertical compressive strength of the jet grouting pile (taking unconfined compressive strength); QjLs is the axial force of the jet grouting pile at depth Ls; R2 is the maximum bearing capacity controlled by the pile body material of the “pure jet grouting pile section”; γc is the reduction coefficient of vertical compressive strength of the jet grouting pile. In case of lack of test data, γc=1.


## 3. Assumption of the Interfacial Friction Distribution of the Composite Foundation under Vertical Compressive Load

According to relevant research results [20–22], the distribution law of side friction resistance between the mini steel pipe pile and the jet grouting pile and side friction resistance between the jet grouting pile and the soil around pile can be simply assumed, as shown in Figures 2 and 3, so as to facilitate design and calculation.Figure 2
Distribution of interfacial friction resistance between the mini steel pipe pile and the jet grouting pile under overall compression mode.Figure 3
Distribution of interfacial friction resistance between the jet grouting pile and soil under overall compression mode.


### 3.1. Assumption of Interfacial Friction Resistance Distribution between the Mini Steel Pipe Pile and the Jet Grouting Pile under Vertical Compressive Load

As can be seen from Figure2, after simplifying the hypothesis, the side friction resistance between the mini steel pipe pile and the jet grouting pile presents a three-step distribution of small in the upper and large in the lower. The depth range of the first step is 0∼L1, and the average side friction resistance is τ1; the depth range of the second step is L1∼L2, and the average side friction is τ2; the depth range of the third step is L2∼Ls, and the average side friction resistance is τ3. The distribution of side friction resistance qsZ at the interface between the mini steel pipe pile and the jet grouting pile can be expressed by as(13)qsZ=τ10≤Z<L1τ2L1≤Z<L2τ3L2≤Z<Ls,(14)τ2=2τ1=ξτ3,(15)τ3=τu=κσju,(16)L1=αDs,(17)L2=Ls−L1.In the equation,qsZ is the side friction resistance between the mini steel pipe pile and the jet grouting pile; Z is the depth; τ1, τ2, and τ3 are the average side friction resistance between the mini steel pipe pile and the jet grouting pile in different depth ranges. The value of ξ is based on the extent of the bonding force in the middle section of the mini steel pipe pile, and its proposed value is 0.05 according to the numerical simulation results. τu is the ultimate side friction resistance at the interface between the mini steel pipe pile and the jet grouting pile, which is related to the compressive strength of the jet grouting pile and the smoothness of the mini steel pipe pile κ, κ=0.15∼0.4; σju is the compressive strength of the jet grouting pile; Ls is the length of the mini steel pipe pile; Ds is the outer diameter of the mini steel pipe pile; the value of α is taken according to the range of bonding force between the mini steel pipe pile and the jet grouting pile. When the whole composite foundation is compressed, α=15.


### 3.2. Assumption of the Distribution of Interficial Friction Resistance between the Jet Grouting Pile and Soil under Vertical Compressive Load

As can be seen from Figure3, the side friction resistance at the interface between the jet grouting pile and the soil after the simplified assumption is divided into the upper and lower parts of “inserted mini steel pipe pile section” and “pure jet grouting pile section.” The properties of the upper soil layer are often poor, and the ultimate side friction resistance at the upper part of the jet grouting pile-soil interface will be less than that at the lower part, which generally presents a two-step distribution. Therefore, the distribution of side friction resistance at the interface between the jet grouting pile and soil is(18)qjZ=τ40≤Z<Lsτ5Ls≤Z<Lj0Lj≤Z≤L..In the equation,qjZ is the side friction resistance between the jet grouting pile and the soil around the pile; Z is the depth; τ4 and τ5 are the average ultimate side friction resistance of the jet grouting pile-soil interface in different depth ranges, which can be obtained according to the geological exploration report; Ls is the length of the mini steel pipe pile; Lj is the effective length of the jet grouting pile, and the value is shown in equation (1); L is the pile length of the jet grouting pile.


## 3.1. Assumption of Interfacial Friction Resistance Distribution between the Mini Steel Pipe Pile and the Jet Grouting Pile under Vertical Compressive Load

As can be seen from Figure2, after simplifying the hypothesis, the side friction resistance between the mini steel pipe pile and the jet grouting pile presents a three-step distribution of small in the upper and large in the lower. The depth range of the first step is 0∼L1, and the average side friction resistance is τ1; the depth range of the second step is L1∼L2, and the average side friction is τ2; the depth range of the third step is L2∼Ls, and the average side friction resistance is τ3. The distribution of side friction resistance qsZ at the interface between the mini steel pipe pile and the jet grouting pile can be expressed by as(13)qsZ=τ10≤Z<L1τ2L1≤Z<L2τ3L2≤Z<Ls,(14)τ2=2τ1=ξτ3,(15)τ3=τu=κσju,(16)L1=αDs,(17)L2=Ls−L1.In the equation,qsZ is the side friction resistance between the mini steel pipe pile and the jet grouting pile; Z is the depth; τ1, τ2, and τ3 are the average side friction resistance between the mini steel pipe pile and the jet grouting pile in different depth ranges. The value of ξ is based on the extent of the bonding force in the middle section of the mini steel pipe pile, and its proposed value is 0.05 according to the numerical simulation results. τu is the ultimate side friction resistance at the interface between the mini steel pipe pile and the jet grouting pile, which is related to the compressive strength of the jet grouting pile and the smoothness of the mini steel pipe pile κ, κ=0.15∼0.4; σju is the compressive strength of the jet grouting pile; Ls is the length of the mini steel pipe pile; Ds is the outer diameter of the mini steel pipe pile; the value of α is taken according to the range of bonding force between the mini steel pipe pile and the jet grouting pile. When the whole composite foundation is compressed, α=15.


## 3.2. Assumption of the Distribution of Interficial Friction Resistance between the Jet Grouting Pile and Soil under Vertical Compressive Load

As can be seen from Figure3, the side friction resistance at the interface between the jet grouting pile and the soil after the simplified assumption is divided into the upper and lower parts of “inserted mini steel pipe pile section” and “pure jet grouting pile section.” The properties of the upper soil layer are often poor, and the ultimate side friction resistance at the upper part of the jet grouting pile-soil interface will be less than that at the lower part, which generally presents a two-step distribution. Therefore, the distribution of side friction resistance at the interface between the jet grouting pile and soil is(18)qjZ=τ40≤Z<Lsτ5Ls≤Z<Lj0Lj≤Z≤L..In the equation,qjZ is the side friction resistance between the jet grouting pile and the soil around the pile; Z is the depth; τ4 and τ5 are the average ultimate side friction resistance of the jet grouting pile-soil interface in different depth ranges, which can be obtained according to the geological exploration report; Ls is the length of the mini steel pipe pile; Lj is the effective length of the jet grouting pile, and the value is shown in equation (1); L is the pile length of the jet grouting pile.


## 4. Axial Force Calculation of the Composite Foundation under Vertical Compressive Load

The above assumption of interfacial friction resistance distribution can be applied to the calculation of axial force of the composite foundation. According to the numerical simulation results, the mini steel pipe pile and the composite foundation can be regarded as the pure friction pile, and the bottom axial force can be regarded as 0. According to the force transmission mechanism of the composite foundation, the calculation equation of axial force of the composite foundation is established.The cumulative sum of the side friction resistance of the mini steel pipe pile is equal to the upper load shared by its pile top. The axial force of the mini steel pipe pile can be expressed as(19)QsZ=Q0s−Us∫0ZqsZdZ.Substituting equation (13) into equation (19), we obtain the following equation:(20)QsZ=Q0s−Usτ1Z0≤Z<L1,Q0s−Usτ1L1+τ2ZL1≤Z<L2,τ3Ls−ZL2≤Z<Ls.In the equation,QsZ is the axial force of the mini steel pipe pile at different depths; τ1, τ2, and τ3 are the average side friction resistance at the interface between the mini steel pipe pile and the jet grouting pile in different depth ranges; Us is the perimeter of the mini steel pipe pile; Z is the depth; Ls is the length of the mini steel pipe pile; the values of L1 and L2 refer to equations (16) and (17), respectively.The axial force of the jet grouting pile in the “inserted mini steel pipe pile section” is composed of the upper load shared by the top of the jet grouting pile, the side friction resistance of the mini steel pipe pile-jet grouting pile interface, and the side friction resistance of the jet grouting pile-soil interface, which can be expressed by equation (21).The axial force distribution of the jet grouting pile is(21)QjZ=Q0j+Us∫0ZqsZdZ−Uj∫0ZqjZdZ.Substituting equations (13) and (18) into equation (21), equation (22) can be obtained as follows:(22)QjZ=Q0j+Usτ1−Ujτ4Z0≤Z<L1,Q0j+Usτ1−Ujτ4L1+Usτ2−Ujτ4ZL1≤Z<L2,Q0j+Usτ1−Ujτ4L1+Usτ2−Ujτ4L2+Usτ3−Ujτ4ZL2≤Z<L3,UjLj−Zτ5Ls≤Z<Lj..In the equation,QjZ is the axial force of the jet grouting pile; Q0j is the load shared on the top of the jet grouting pile; τ1, τ2, and τ3 are the average side friction resistance at the interface between the mini steel pipe pile and the jet grouting pile in different depth ranges; Us is the perimeter of the mini steel pipe pile; Uj is the perimeter of the jet grouting pile; τ4 and τ5 are the average side friction resistance of the jet grouting pile-soil interface in different depth ranges, which can be obtained according to the geological exploration report; Ls is the length of the mini steel pipe pile; Lj is the effective length of the jet grouting pile.According to equations (20) and (22), under the overall compression mode of the composite foundation, the axial force distribution of the mini steel pipe pile and the jet grouting pile is shown in Figures 4 and 5, respectively.Figure 4
Axial force distribution of the mini steel pipe pile in the composite foundation under overall compression mode.Figure 5
Axial force distribution of the jet grouting pile in the composite foundation under overall compression mode.


## 5. Calculation of Bearing Capacity of the Composite Foundation under Vertical Compressive Load

The field vertical compression composite foundation in soil layers shown in Table1 was selected for bearing capacity test and calculation comparative analysis. The calculation of the vertical compressive bearing capacity of the composite foundation is divided into the calculation of the ultimate bearing capacity controlled by the interface strength of the composite foundation and the calculation of the ultimate bearing capacity controlled by the material strength of the pile body of the composite foundation, and the smaller of the two is taken as the ultimate compressive load of the composite foundation. The parameters of the jet grouting pile and the mini steel pipe pile of the test pile are shown in Tables 2 and 3, respectively. The ultimate side friction resistance of each soil layer in the test site is shown in Table 1. The stratum distribution and composite profile at the test site is shown in Table 4.Table 1
Recommended value of ultimate side friction resistance of each soil layer.
NumberName of soilThe state of the soilRecommended value of ultimate side friction resistance/kPa1Silty clayPlastic state, medium to high compressibility302Silty claySoft plastic state, medium to high compressibility233Mucky clayHigh water content, high compressibility, flow plastic state184Silty clay with sandMedium compressibility, slightly dense∼moderately dense state405Silty claySoft plastic state, high compressibility29Table 2
Calculation parameters of the jet grouting pile.
Type of pilePile diameterDjPile lengthLjUnconfined compressive strengthTensile strengthJet grouting pile0.6 m20 m8 MPa0.6 MPaTable 3
Calculation parameters of the mini steel pipe pile.
Type of pileOuter diameterDsPile lengthLsWall thicknessTsCompressive yield strengthTensile strengthMini steel pipe pile0.108 m10 m8 mm358 MPa410 MPaTable 4
Comparison of measured and calculated bearing capacity of the composite foundation.
Load modeMeasured value of bearing capacityRa/kNCalculated value of bearing capacityRc/kNAbsolute error/kNRelative error/%Overall compressive strength of the composite foundation10831051323According to the concept of the effective pile length,Dj is substituted into (1), and β=18. It is calculated that the “effective section of pure jet grouting pile” Lc=10m, and the calculation process is shown in equations (23) and (24), so there is no “invalid section of the jet grouting pile” in the composite foundation.(23)Lc=Lj−Ls=10m,(24)minβDj,10=min18×0.6,10=10m≥Lc.


### 5.1. Maximum Bearing Capacity Controlled by Iinterface Strength

According to Table1 and Figure 6, the ultimate side friction resistance of the jet grouting pile-soil interface calculated in combination with equation (9) is(25)fju=Uj∫0LjqjZdz=0.6π×1×30+2×23+6×18+5×40+6×29=1051kN.Figure 6
Stratum distribution and composite foundation profile at the test site.Combined with equation (6), equation (13), and Figure 2, the ultimate side friction resistance at the interface between the mini steel pipe pile and the jet grouting pile is calculated. Substituting the data in Table 3 into equations (16) and (17), respectively, L1=1.62m and L2=8.38m can be obtained, where α=15. According to Table 2, σju=8MPa=8000kPa. κ=0.15∼0.4, so when taking κ=0.15+0.4/2=0.275 and substituting it into equation (15), τ3=2200kPa can be obtained; substituting ξ=0.05 and τ3 into equation (14), τ2=110kPa and τ1=55kPa can be obtained. Therefore, from equations (6) and (13), we obtain(26)fsu=Us∫0LsqsZdz=UsL1τ1+L2−L1τ2+L1τ3=0.108π×1.62×55+8.38−1.62×110+1.62×2200=1492kN.Therefore, according to equations (25) and (26), fju<fsu, so the maximum bearing capacity controlled by the interface strength of the composite foundation under the overall compression mode is(27)Ri=fsu=1051kN.


### 5.2. The Maximum Bearing Capacity of the Composite Pile Top Controlled by Pile Material Strength

The maximum bearing capacity of the mini steel pipe pile top under the control of pile material strength calculated by equation (2) and Table 3 is(28)Q0s≤Qsu=σsuAs=σsuTsDs−Ts2π=358×103×0.008×0.108−0.0082π=900kN.The cross-sectional area of the jet grouting pile at the pile top calculated by equation (4) and Table 2 is(29)Aj=Dj2−Ds2π4=0.62−0.1082π4=0.274m2.The maximum bearing capacity of the jet grouting pile top under the control of pile material strength is calculated by substituting equation (10):(30)Qju=σjuAj=8×103×0.274=2192kN.According to the numerical simulation analysis under the overall compression mode of the composite foundation, the pile top stress sharing ratio of the mini steel pipe pile and the jet grouting pile isnp = 140 [7]. The load Q0j shared by the top of the jet grouting pile calculated by equation (3) is(31)Q0j=QsuAjAsnp=900×0.2740.002513×140=700kN<Qju=2192kN.Therefore, the maximum bearing capacity of the pile top of the composite foundation under the overall compression mode is(32)Rt=Qsu+Q0j=900+700=1600kN.


### 5.3. Maximum Bearing Capacity of the “Pure Jet Grouting Pile Section”

Using equation (22) and Table 1, the maximum bearing capacity of the “pure jet grouting pile section” under the control of ultimate side friction resistance of the jet grouting pile is(33)R1=QjLc=Uj∫LsLjqjZdz=0.6π×4×40+6×29=630kN.Using equation (10), Qju of the “pure jet grouting pile section” can be obtained as follows:(34)Qju=σjuAj=σjuDj2π4=8×103×0.62×π4=2260kN.We substitute equation (12), and γc=1.0. The maximum bearing capacity of the jet grouting pile under the control of material strength is(35)R2=γcQju=1×2260=2260kN.Therefore, according to equations (33) and (35), R1<R2, so the “pure jet grouting pile” section of the composite foundation will not be damaged under the overall compression mode.Based on the above (1) to (3), Ri=1051kN<Rt=1600kN, so the ultimate bearing capacity of the composite foundation under the overall compression mode is(36)Rc=Ri=1051kN.


### 5.4. Comparative Analysis of the Experimental Calculation of Vertical Compressive Bearing Capacity of the Composite Foundation

The comparison of measured and calculated bearing capacity of the composite foundation is shown in Table4.As can be seen from the table, under the overall compression mode, the measured test value of the bearing capacity of the composite foundation is 1083 kN. The calculated value determined by the calculation formula deduced under the assumed force distribution mode in this paper is 1051 kN, the absolute error between them is 32 kN, and the relative error is 3%. Therefore, both absolute error and relative error are very small. Therefore, the stress mode and calculation method of the jet grouting pile-mini steel pipe pile composite foundation proposed in this paper can meet the needs of engineering design and popularization and application of the composite foundation.


## 5.1. Maximum Bearing Capacity Controlled by Iinterface Strength

According to Table1 and Figure 6, the ultimate side friction resistance of the jet grouting pile-soil interface calculated in combination with equation (9) is(25)fju=Uj∫0LjqjZdz=0.6π×1×30+2×23+6×18+5×40+6×29=1051kN.Figure 6
Stratum distribution and composite foundation profile at the test site.Combined with equation (6), equation (13), and Figure 2, the ultimate side friction resistance at the interface between the mini steel pipe pile and the jet grouting pile is calculated. Substituting the data in Table 3 into equations (16) and (17), respectively, L1=1.62m and L2=8.38m can be obtained, where α=15. According to Table 2, σju=8MPa=8000kPa. κ=0.15∼0.4, so when taking κ=0.15+0.4/2=0.275 and substituting it into equation (15), τ3=2200kPa can be obtained; substituting ξ=0.05 and τ3 into equation (14), τ2=110kPa and τ1=55kPa can be obtained. Therefore, from equations (6) and (13), we obtain(26)fsu=Us∫0LsqsZdz=UsL1τ1+L2−L1τ2+L1τ3=0.108π×1.62×55+8.38−1.62×110+1.62×2200=1492kN.Therefore, according to equations (25) and (26), fju<fsu, so the maximum bearing capacity controlled by the interface strength of the composite foundation under the overall compression mode is(27)Ri=fsu=1051kN.


## 5.2. The Maximum Bearing Capacity of the Composite Pile Top Controlled by Pile Material Strength

The maximum bearing capacity of the mini steel pipe pile top under the control of pile material strength calculated by equation (2) and Table 3 is(28)Q0s≤Qsu=σsuAs=σsuTsDs−Ts2π=358×103×0.008×0.108−0.0082π=900kN.The cross-sectional area of the jet grouting pile at the pile top calculated by equation (4) and Table 2 is(29)Aj=Dj2−Ds2π4=0.62−0.1082π4=0.274m2.The maximum bearing capacity of the jet grouting pile top under the control of pile material strength is calculated by substituting equation (10):(30)Qju=σjuAj=8×103×0.274=2192kN.According to the numerical simulation analysis under the overall compression mode of the composite foundation, the pile top stress sharing ratio of the mini steel pipe pile and the jet grouting pile isnp = 140 [7]. The load Q0j shared by the top of the jet grouting pile calculated by equation (3) is(31)Q0j=QsuAjAsnp=900×0.2740.002513×140=700kN<Qju=2192kN.Therefore, the maximum bearing capacity of the pile top of the composite foundation under the overall compression mode is(32)Rt=Qsu+Q0j=900+700=1600kN.


## 5.3. Maximum Bearing Capacity of the “Pure Jet Grouting Pile Section”

Using equation (22) and Table 1, the maximum bearing capacity of the “pure jet grouting pile section” under the control of ultimate side friction resistance of the jet grouting pile is(33)R1=QjLc=Uj∫LsLjqjZdz=0.6π×4×40+6×29=630kN.Using equation (10), Qju of the “pure jet grouting pile section” can be obtained as follows:(34)Qju=σjuAj=σjuDj2π4=8×103×0.62×π4=2260kN.We substitute equation (12), and γc=1.0. The maximum bearing capacity of the jet grouting pile under the control of material strength is(35)R2=γcQju=1×2260=2260kN.Therefore, according to equations (33) and (35), R1<R2, so the “pure jet grouting pile” section of the composite foundation will not be damaged under the overall compression mode.Based on the above (1) to (3), Ri=1051kN<Rt=1600kN, so the ultimate bearing capacity of the composite foundation under the overall compression mode is(36)Rc=Ri=1051kN.


## 5.4. Comparative Analysis of the Experimental Calculation of Vertical Compressive Bearing Capacity of the Composite Foundation

The comparison of measured and calculated bearing capacity of the composite foundation is shown in Table4.As can be seen from the table, under the overall compression mode, the measured test value of the bearing capacity of the composite foundation is 1083 kN. The calculated value determined by the calculation formula deduced under the assumed force distribution mode in this paper is 1051 kN, the absolute error between them is 32 kN, and the relative error is 3%. Therefore, both absolute error and relative error are very small. Therefore, the stress mode and calculation method of the jet grouting pile-mini steel pipe pile composite foundation proposed in this paper can meet the needs of engineering design and popularization and application of the composite foundation.


## 6. Conclusions

(1)
Under the overall compression mode of the pile top of the composite foundation, the mini steel pipe pile and jet grouting pile jointly bear the upper load. According to the vertical force transmission characteristics of the composite foundation, the composite foundation can be divided into “inserted mini steel pipe pile section” and “pure jet grouting pile section” along the depth direction, and the jet grouting pile has “effective length.” In the checking calculation of bearing capacity, the bearing performance of the mini steel pipe pile and jet grouting pile should be considered. Accordingly, the coordination effect of ultimate side friction resistance between the mini steel pipe pile and the jet grouting pile and ultimate side friction resistance between the jet grouting pile and soil should also be considered.(2)
The simplified hypothetical distribution diagram and calculation formula of interfacial friction resistance between the mini steel pipe pile and the jet grouting pile and interfacial friction resistance between the jet grouting pile and soil are put forward. The interfacial friction resistance between the mini steel pipe pile and the jet grouting pile presents a three-step distribution of small in the upper and large in the lower. The interfacial friction resistance between the jet grouting pile and soil generally presents a two-step distribution, and the ultimate side friction resistance at the upper part is less than that at the lower part.(3)
The mini steel pipe pile and the composite foundation can be regarded as pure friction piles, and the bottom axial force can be regarded as 0. According to the force transmission mechanism of the composite foundation, the calculation equation of the axial force of the composite foundation is established, the calculation formula of the axial force of the mini steel pipe pile and the jet grouting pile under the overall compression mode is deduced, and the axial force distribution diagram of the mini steel pipe pile and the jet grouting pile in the composite foundation is put forward.(4)
The calculation of the vertical compressive bearing capacity of the composite foundation is divided into the calculation of the ultimate bearing capacity controlled by the interface strength of the composite foundation and the calculation of the ultimate bearing capacity controlled by the material strength of the pile body of the composite foundation. The smaller of the two is taken as the ultimate compressive load of the composite foundation.(5)
The conclusions of this paper need to be further verified and improved by more engineering design examples.



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*Source: 8499597-2022-05-11.xml*