Model test of the group piles foundation of a high-speed railway bridge in mined-out area

Abstract

The research on the mechanism of pile-soil-cap-goaf interaction and settlement of high-speed railway bridge located in mined-out area is still relatively rare. By taking the pile group of Guanshandi bridge foundation in Hefei-Fuzhou high-speed railway as the prototype, a model test is carried out. According to the similarity theory, the similar constant is derived and the similar model material is determined. Meanwhile, three types of data including the bearing behavior of piles, and the settlement law, and soil among piles are investigated. It can be found that: the influence of goaf on the bearing capacity of pile is inversely to the loading degree, the larger of loading degree, the smaller impact of goaf on the bearing capacity. There is no negative side friction can been found in pile body and the degree of downward tendency for the barycenter of side friction layout is obvious for piles in goaf. Although the bearing ratio of soil resistance under cap is relatively large, the cap effect is suggested be ignored considering the characteristic of goaf. There is a maximum critical value for the uneven settlement of pile group in goaf, and when the value is reached, the uneven settlement stop growing anymore. In addition, the formula for calculating bearing capacity and settlement of pile group in goaf based on test results, theory analysis and related standard is established.

Keywords

high-speed railway / mined-out areas / goaf / group piles foundation / physical model test / bearing capacity / settlement

Introduction

Hefei-Fuzhou High-speed railway is built as a design speed of 350 km/h in China and adopts the ballastless track technology. In this railway line, Guanshandi Bridge, a section of mined-out areas in Shangrao City, Jiangxi Province, China, adopts group piles foundation to control the deformation of the roof of goaf roadway. The bridge is of multi-span simply supported beam structure and the test prototype in this paper is the group piles foundation of the intermediate pier. The bridge adopts bored piles, whose cap along the railway is 9 m long, 11 m wide and 2.5 m high, and the foundation is composed of 9 piles with the length of 32.5 m. The angle of the extension direction of roadway and the railway direction is 60°, the roof of roadway is 13 m thick and the roadway is 4.22 m wide and 2 m high. Figure 1 illustrates the foundation layout and the stratigraphic distribution in detail. The stratum layers from top to bottom are: (1) plain fill stratum; (2) strong-weathered carbonaceous shale; (3) weak-weathered carbonaceous shale.

Model dimension and material selection

Model dimension

The main relevant parameters in this physical model test include: stress, strain, internal friction angle, cohesive force, unit weight, elastic modulus, Poisson’s ratio, unit distributed load, length and displacement. According to resemblance theories, the related parameter expression for basic physical model of pile group of bridge in goaf of High-speed Railway is listed as follows:

(1) f(\sigma ,\epsilon ,E,\nu ,\gamma ,f,c,q,l,\delta )=\mathrm{0.}

The total number of parameters n = 10 and the fundamental dimension m= 2 (for statics problem, the fundamental dimensions will be F and L. According to π theorem, there are 8 independent π items, and its π function can be expressed as follows:

(2) \phi ({\pi }_{\mathrm{1}},{\pi }_{\mathrm{2}},{\pi }_{\mathrm{3}},{\pi }_{\mathrm{4}},{\pi }_{\mathrm{5}},{\pi }_{\mathrm{6}},{\pi }_{\mathrm{7}},{\pi }_{\mathrm{8}})=\mathrm{0.}

By assuming the geometric similarity constant (C_l) is 25 and the weight similarity constant (C_g) is 1.5, the similarity constants of other physical quantity can be deduced according to the p theorem, as listed in Table 1.

Compared with the prototype, the most significant feature of the model test lies in the fact that its test process and result will be subject to the boundary conditions. By drawing upon the related model test from both China and foreign countries [ 1– 3] and considering the boundary conditions for the model test of pile group foundation in this paper, this model test species the size of model tank after the elimination of boundary effect, which is listed as follows: the length and width are three times larger than that of the boundary of the cap, meaning that the length is 11 × 3/25= 1.32 m and width is 9 × 3/25= 1.08 m. The influence range of height is made 10 times larger than the pile diameter, that is to say, the height is 1.3+ 0.1+ 0.5= 1.9 m (pile length+ the height of bearing platform and boundary effect). In light of the convenient manufacturing of the model tank, the final size of model tank is length × width × height of 1.4 m × 1.1 m × 2 m. Meanwhile, to further reduce the influence of model tank on the boundary, the lubricating oil is applied to the side wall of the model tank.

The geometric parameter of the test model and the model tank dimension are shown in Fig. 2.

Model material

In the model test, the elasticity modulus of piles should be mainly considered. The elasticity modulus of C35 concrete is 3.15 × 10⁴ MPa and that of model piles is 3.15 × 10⁴/37.5= 0.84 × 10³ MPa. By comparing the test of multiple materials and considering the model making process, PPR pipe, with an outer diameter of 32 mm, inner diameter of 23 mm and elasticity module of 0.962 × 10³ MPa, is selected as the model material of bored concrete pile. The concrete cap is simulated with cast-in situ concrete in order to intensify the integrity of pile and cap in the model. The simulation of plain fill in the model test mainly takes unit weight and internal friction angle as the main control standards. The river sand with an internal friction angle of 30° is adopted in the model to simulate the plain fill.

The experimental result of the elastic modulus of PPR pipe is shown in Table 2.

The weakly weathered shale and strongly weathered shale take the elastic modulus and compressive strength as the control standards in the model, meaning that other physical indexes could be appropriately relaxed. Based on the simulation research results [ 4– 6] of similar materials and the prototype material characteristics of this test, this test uses medium-coarse sand as aggregate, takes gypsum and cement as concrete and then prepares samples by taking six groups of prepared materials. Make three samples for each preparation ratio. After material being evenly mixed, add PVC pipe with a diameter of 5cm and a length of 10cm to be firmly tamped to make test block. After 14 days of maintenance, paste strain foil on the middle surface of the test block; after that, compress it on a press machine so as to measure its elastic modulus and compressive strength.

The strongly weathered charcoal shale in the prototype has the elastic modulus of (0.8–1.0) × 10⁴ MPa and uniaxial compressive strength of 10–12 MPa; the weakly weathered charcoal shale has the elastic modulus of (1–1.2) × 10⁴ MPa and uniaxial compressive strength of 15–18 MPa. The resemblance theory shows that the corresponding strongly weathered charcoal shale in the model has the elastic modulus of (0.8–1.0) × 10⁴/37.5= 213–267 MPa and uniaxial compressive strength of (10–12)/37.5= 0.27–0.32 MPa; the weakly weathered charcoal shale has the elastic modulus of (1–1.2) × 10⁴/37.5= 267–320 MPa and uniaxial compressive strength of (15–18)/37.5= 0.4–0.48 MPa. The specimen of No.1 preparation ratio has the elastic modulus of 283 MPa and uniaxial compressive strength of 0.46 MPa, which basically meet the mechanical index of the weakly weathered charcoal shale; however, the specimen of No. 3 preparation ratio has the elastic modulus of 217 MPa and uniaxial compressive strength of 0.29 MPa, which basically meet the mechanical index of the strongly weathered charcoal shale.

Model load

The load in this test is divided into 7 classes [ 7]. In it, the construction stage is given to 3 classes, the self-weight of structures plus the load of train is regarded as only one class and the additional load beyond the design load is divided into 3 classes, see Table 4 for specific details. For the convenience of loading, the actual test load is slightly different from the calculated model load.

Layout of monitoring instruments

The layout of monitoring point aims to obtain 3 kinds of data: soil stress; settlement of cap, soil; internal force of pile. The resistance value of resistance strain foil used in this model test is 119.9±0.1 W and the sensitivity coefficient K is 2.08±1%. The soil between piles and pile tip of this test uses a micro-soil pressure cell with a measuring range of 50 kPa; the pile top adopts a micro-soil pressure cell with a measuring range of 400 kPa. Moreover, monitor settlement with an observation pole and a dial gauge (measuring range: 0–5 mm; precision: 0.001 mm). The detailed component layout and monitoring contents are as follows:

1) Monitoring points of the soil stress between piles: t1–t4;

2) No. 3, 5 and 7 piles(see Fig. 2 to figure out the pile number) are closely attached with strain gages and micro earth pressure cells which are installed on pile top and pile tip (number of No.3: t9, t10; number of No.5: t7, t8; number of No.7: t5, t6), respectively;

3) Monitoring points of surface settlement of cap: cj1–cj4;

4) Monitoring points of soil settlement between piles: cj5–cj8;

5) Monitoring points of roof settlement at the bottom of roof: cj9 and cj10.

See Fig. 4 for the layout of specific monitoring points.

Test results and analysis

Axial force of piles

See Fig. 5 for the axial force of No. 3, No. 5 and No. 7 piles. The changing trend of the axial force of No. 3 and No. 5 is basically the same. Under the same load, the axial force of No. 5 is the highest, followed by No. 3 and then No. 7 at 0–2 cm. No. 3 and No. 7 are symmetric and the axial force on the top of No. 3 is larger than that of No.7 because the roadway weakens the bearing capacity of piles to a certain degree.

Relative values are used to compare the force situation at the top of three piles, No. 3, No. 5 and No. 7. Assuming the axial force at the top of No. 3 under the effect of each load is 1, the axial force at the top of No. 5 under the effect of the same load is expressed with the ratio of measured values of the axial forces at top of No. 5 and No. 3. For instance, under the effect of load class 4, the axial force at the top of No. 3 is 43.5232 N and that of No. 5 is 46.5802 N. Assuming the axial force at the top of No. 3 is 1, so that of No. 5 is 46.5802/43.5232≈1.07. The axial force at the top of No. 7 is handled in the same way. After conversion, the relative values of load at pile top are listed in Table 5.

No.3 and No.7 are located symmetrically to the cap. If there is no goaf in the formation of No.7, the axial force at the top of No.3 and No.7 should be identical under the effect of load. The presence of goaf brings about evident difference to the two axial forces. It is shown in Fig. 5 that, owing to the goaf, the axial force of No.7 is lower than that of No.3. When the load is small, they have a large difference, but as the load increases, the two axial forces tend to become the same. The reason is, when the load increases, the side friction and tip resistance of middle pile in the floor of goaf is further developed, the proportion of side friction provided by top of goaf in the whole resistance reduces. From the perspective of bearing capacity, the larger the load, the smaller the goaf’s influence on the bearing capacity of piles is.

The measured data of project indicates that, with general formation condition and design load, for the vast majority of pile group foundations, the axial force at the top of middle pile is larger than those of side piles [ 8]. In this experiment, the goaf changes the distribution of internal force among the piles in the group. The internal force of No.5 is slightly larger than that of side piles, and its relative proportion to normal side pile No.3 is unchanged with the change of load, but stays at 10% approximately.

Side friction of piles

It can be seen from Fig. 6 that the side friction of three piles increases and then decreases. However, under different load, the turning point of side friction from large to small is different. For example, under the load of class 1–4, the turning point is at 74cm below the pile top; under the load of class 5–7, the turning point is at 98cm below the pile top, indicating that the big load makes the pile give full play to its bearing ability in the deep location. Due to the cap is a low cap, the load can directly transfer to the soil between piles under the cap which reduces the relative pile-soil displacement. Therefore, the upper side friction of piles exerts insufficiency and the barycenter of the side friction distribution diagram of piles moves downward [ 9, 10], and the downward moving depth increases with the load. For example, the barycenter of the side friction distribution diagram of No.7 is at 45–70cm below the pile top under the load of class 1 and at 65–95 cm below the pile top under the load of class 7; the side friction of No.7 at 42.8–50.8 cm of the roadway is zero. By comparing the side friction of symmetricNo.3 and 7, the downward moving degree of the barycenter of the side friction distribution diagram of No.7 is obviously larger than that of No.3 under the load of class 2 and class 3. For example, under the load of class 2, the side friction of No.3 at 26 cm, 50 cm, 74 cm and 98 cm below the pile top is 75.7 Pa, 149.9 Pa, 201.4 Pa and 185.3 Pa respectively, and that of No.7 at 26 cm, 50.8 cm, 74 cm and 98 cm (42.8–50.8 cm is the roadway) below the pile top is 69.3 Pa, 172.4 Pa, 338.4 Pa and 112.8 Pa respectively. This is because the deformation of the roof of the roadway of No.7 is larger than that of other pile, and the settlement of the roof of roadway is relatively large under downward tension and earth pressure between soils which restrains the development of the side friction of piles. The barycenter of the side friction distribution diagram of the two piles is inclined to be consistent under the load of above class 3, which is obvious when they are under the load of class 7.

Pile and soil stress and load sharing ratio

According to Fig. 7, both of the soil stresses between 2 piles and between 4 piles increase with the load. Taking the soil stress between 4 piles at t3 as an example, when the load increases from 500 to 1850 N, the soil stress at t3 increases stably; when the load increases from 1850 to 2450 N, the increase amplitude of soil stress at t3 decreases from 28.4% to 3.7% and the increase of soil stress is obviously slow, indicating that the load borne by piles increases greatly and the load borne by both piles and soil gradually shifts to the piles. Generally, the change law of soil stress between piles at different points is basically the same. The soil stress at t1 and t2 above the roadway is slightly less than that at t3 and t4 (no roadway below) because the soil settlement of the upper part is larger than that of other parts due to settlement deformation of the roof of roadway, thus causing low soil stress below the cap.

From Fig. 8, the stress of pile top and the stress of pile tip increase with the load and the changes of the stress on pile top are obvious because the side friction of piles bears the most part of loads, thus the influence of load change on the resistance of pile tip is not so obvious compared with pile top. When the load increases from 1850 to 2450 N, the stress of pile top increases rapidly, corresponding to the change law of soil stress between piles. As the elasticity modulus of piles is larger than that of soil, the stress of pile top is larger under the same strain. The stress of pile top and pile tip under the maximum load is shown as Table 6.

It can be seen from Fig. 9, the load sharing ratio of pile increases with the load, and changes with gradually slow growth amplitude mainly because: (1) the compression modulus of soil between piles increases gradually during loading process; (2) the exertion of side friction results in a relative displacement of piles to the surrounding soil. The load borne by piles is only 56% under the load of class 4 and only 67% under the load of class 7.

Settlement analysis

It is clear from Fig. 10 that the settlement of cap and the soil between piles increases with the load. The curves in the figure change from steep to slow because the loose soil layer is compressed after loading, its compression modulus is improved gradually and the settlement amplitude reduces, therefore, the settlement curve of cap is also slow.

The settlement deformation at cj6 is less than that at cj5. under the maximum load of 2450 N, the settlement at cj5 is 0.252mm, which is larger than that at cj6 of 0.201 mm. The cj5 is above the roadway and cj6 is above the non-roadway. It is clear that the soil settlement above the roadway is larger than that above non-roadway. By comparing the settlement at cj7 and cj8, the settlement deformation at cj7 is less than that at cj8. For example, the settlement at cj7 and cj8 is 0.174 and 0.190 mm respectively under the maximum load and the settlement at cj8 is larger than that at cj7 and cj8 which is above the roadway, again indicating that the soil settlement above the roadway is larger than that above the non-roadway.

The settlement at cj1-4 at the corner of cap is 0.241, 0.230, 0.181 and 0.210 mm respectively under the maximum load, in which the distance of cj1 to the roadway is the smallest, followed by cj2 and cj4 and then cj3. The smaller the distance of monitoring point to the minded-out area is, the larger the settlement will be. Under the load of other classes, the deformation law of cap is basically the same, with slight numerical difference, indicating that the existence of roadway does affect the bridge foundation and the deformation of foundation above the roadway is larger. From the figure, the differential settlement at cj1-cj4 does not increase with the load of class 3–7, that is, the differential settlement maintains in a relatively stable range, indicating that the existence of mined-out area below does impose impacts on the settlement of cap and such impact is of a critical load. Beyond this load, the differential settlement between the roof soil layer of mined-out area and the normal soil layer under the foundation will not increase anymore and the uneven settlement is inclined to be stable. During foundation design, this law can be used to control the post-construction settlement and uneven settlement of the bridge foundation in mined-out areas.

It can be seen from Fig. 11 that the settlement of the two monitoring points increases with the load, indicating that the larger the roof load of roadway is, the greater the settlement will be. The settlement at cj10 is less than that at cj9. For example, the settlement at cj10 is 0.081 mm and that at cj9 is 0.100 mm under the maximum load. The difference between cj9 and cj10 is that the former is located at the big spacing between piles and the latter is located at the small spacing between piles. This means that the spacing between piles imposes impacts on the roof settlement of roadway, that is, for the piles going through the roadway, the larger the spacing between piles in the mined-out areas is, the greater the load borne by the soil between piles and the greater the settlement deformation at the bottom of the roof of roadway will be.

The computational formula for the bearing capacity of single pile in the goaf

The bearing capacity of single pile includes the pile side friction and the pile tip resistance. The major difference between the bearing capacity of single pile in the goaf and that of ordinary single pile lies in the side friction rather than the tip resistance. The influence of goaf on the side friction could be divided into two parts: one is that the goaf itself fails to provide the pile side friction and the other is that the side friction provided by the goaf roof is weakened under the influence of the goaf. In light of the consistency between the mechanical mechanism of the bearing capacity of pile foundation in the goaf and that in the ordinary formation, the computational formula for the bearing capacity of single pile in the goaf will be deviated on the basis of the existing formulas in this paper. Code for design on subsoil and foundation of railway bridge and culvert (TD 10002.5-2005) stipulates that the computational formula for the allowable bearing capacity of bored pile is:

(3) [P]={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}U\sum f_i{l}_i+m_{\mathrm{0}}A[\sigma ].

In Eq. (3), [P] - the allowable bearing capacity of pile; U -the perimeter of the pile shaft section; f_i -limit frictional resistance of each rock and soil layer; l_i -the thickness of each rock layer; m_{\mathrm{0}} -the reduction coefficient of pile bottom supporting force; A -the pile tip supporting area; [\sigma ] -the allowable bearing capacity of pile tip foundation soil.

In Eq. (3), the first item in the right is the calculation of the pile side friction and the second is the calculation of pile tip resistance. Due to the slight influence of the goaf on the single pile bearing capacity and pile tip resistance, the single pile bearing capacity in the goaf could be deviated by correcting the calculation of the first pile side friction. The deviated formula is:

(4) [P]={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}U\sum f_i{l}_i+m_{\mathrm{0}}A[\sigma ]-{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}U{\eta }_{out}{f}_{out}{l}_{out}.

In Eq. (4), {\eta }_{out} -the influence coefficient of the bearing capacity in the goaf; f_{out} -the limit side frictional resistance of the original rock and soil layer in the goaf; l_{out} -the pile length in the goaf section. Refer to Eq. (3) for other parameters.

{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}U{f}_{out}{\eta }_{out}{l}_{out} , the third item in the right of Eq. (4), refers to the influence value of goaf on the single pile bearing capacity. f_{out} refers to the limit side frictional resistance of the original rock and soil layer in the goaf before excavation; l_{out} is the pile length in the goaf section, or the height of the goaf; {\eta }_{out} is the influence coefficient of bearing capacity in the goaf. Considering the fact that the goaf fails to provide the pile side frictional resistance and the side friction provided by the goaf roof weakens two parts of factors under the influence of goaf, the value of over 1 is assigned. The model experimental data reveals that {\eta }_{out} , the value of the influence coefficient of the goaf bearing capacity of pile side friction, is assigned as 1.25. The basis for the assigning of value is that if the class-4 load is taken as the example, Table 5 shows that the relative value of the No.3 pile side frictional resistance in the total bearing capacity is 1-0.036= 0.964(1 refers to the relative value of the pile bearing capacity and 0.036 means that the pile tip resistance takes up a proportion of 3.6%). The relative value of No. 7 pile side frictional resistance in the total pile bearing capacity is 0.91-0.029= 0.881 with the relative difference between their pile side frictional resistance of 0.964-0.881= 0.083. It is shown in Fig. 6 that the side friction provided by the rock and soil layer of elevation section in the goaf accounts for 6.7% of that of No.3 pile; 0.083/0.067= 1.238, in which, 1.238 refers to the influence coefficient of the bearing capacity in the goaf.

In the computational formula for the allowable bearing capacity of railway bridge foundation standard bored pile, divide by the security coefficient so as to calculate the pile side frictional resistance. In light of the fact that the reduction has been carried out when the calculation of the allowable bearing capacity of pile base foundation soil is adopted for the calculation of the pile tip bearing capacity, the goaf will not exert too much influence on the single pile bearing capacity, especially under the circumstances of large pile tip bearing capacity. In actual work, the single pile bearing capacity in important and special engineering foundation is determined through test pile; it is feasible to apply the single pile bearing capacity in goaf to the estimation of the single pile bearing capacity under preliminary design; however, the accurate result is to be determined through pile test. It can be seen from the test of the bearing capacity composition that the pile tip bearing capacity is not subject to the influence of goaf; as the pile side friction carries most of the load and the pile tip bearing capacity is reliable and safe, it is of prime importance to pay attention to the pile tip bearing capacity in design.

It is the first time to adopt {\eta }_{out} , the influence coefficient of the bearing capacity in goaf. The value assigning of {\eta }_{out} in different regions and under different working conditions necessitates supplement and improvement of abundant engineering practices. Code for Design of Building Foundation (GB50007-2011), Technical Code for Building Pile Foundation (TJ94-2008) and Code for Design of Ground Base and Foundation of Highway Bridges and Culverts (JTG D63-2007) offer almost the same computational principle for pile bearing capacity but slightly different formula. Likewise, the computational method for {\eta }_{out} , the influence coefficient of bearing capacity in goaf, could also be introduced.

BHFelleniue [ 11] believed that the shallow soil around pile will undergo the settlement (apart from the particular conditions, such as expansive soil and frost heaving); in other words, the cap will unavoidably be detached from the soil; for this reason, it is unrealistic to rely on the theory that the soil under the cap and pile could jointly bear the upper loads. The American IBC2006 standards clearly specify that the load-bearing functions of the soil under the cap should not be taken into consideration; the Euricode 7:2004 lists the reasons causing the negative frictional resistance in details; CP4:2003 completely accepts this theory and incorporates it into the standards. The domestic Code for design on subsoil and foundation of railway bridge and culvert (TD 10002.5–2005) and Code for Design of Ground Base and Foundation of Highway Bridges and Culverts (JTG D63–2007) all assume that the upper loads are all borne by the piles; the Code for Design of Building Foundation (GB50007–2011) and the Technical Code for Building Pile Foundation (TJ94–2008) enumerate a few situations that take into consideration the functions of the pile cap. Although the experimental results show that the pile cap has borne large proportion of load, it is still advisable that the cap effect should be neglected in the calculation of the bearing capacity of the pile foundation because: 1. The deformation of goaf is a complex issue. As the creep deformation of the roof requires a long period of time, there are many cases showing the continuing deformation even after hundreds years of mining explorations. Therefore, the bearing capacity of the soil under the pile cap cannot be guaranteed; 2. Under long-term dynamic loading effect, the pile group foundation of railway and highway will destabilize the rock and soil layer under the cap.

The computational formula for the settlement of pile groups in the goaf

The computational formula for the surface deformation is introduced before the discussion about the computational formula for the settlement of pile groups in the goaf. The probability integral method based on stochastic medium theory, which is established by Liu Baochen, is adopted for the calculation of the surface deformation in domestic large-scale goaf. The key issue in adopting this method is the selection of prediction parameters that contain 2 methods: one is the inversion of the prediction parameter through actual measurement of ground movement data; the other is to refer to the prediction parameter empirical value in the nearby diggings or procedures when there is no measured data for reference. Currently, the prediction parameter of probability integral method is the parameter of mathematical significance with weak relationship between the parameter and geology mining condition; for this reason, it is not suggested to select the prediction parameter according to the rationality of the mining status [ 12]. The deformation of the small-scale goaf is displayed by the abrupt roof carving and collapse and surface cracking, showing no rules to follow. The settlement of the pile group foundation in the goaf is calculated to obtain the settlement after the load application is finished, which has no direct relationship with the settlement in the process of mining. Therefore, it is impracticable to adopt the computational method for the surface deformation in the goaf for the calculation of the settlement of the pile group foundation in the goaf.

Under general formation condition, the domestic standards adopt the equivalent pier method to calculate the settlement of the pile group foundation; for example, according to the Design Criterion for Railway Bridge-Culvert Base and Foundation, the pile group foundation with the center distance no less than 6 times larger than the pile diameter is regarded as the block deep foundation model. The Boussinesq stress is adopted to calculate the additional stress and layerwise summation method to calculate the settlement with the computational formula listed as:

(5) S=m_s{\displaystyle \sum _{i=\mathrm{1}}^n\Delta S_i}.

In Eq. (5), S -the total fundamental settlement; m_s - empirical correction coefficient of settlement; \Delta S_i -settlement value of rock and soil layer with the thickness of \Delta z above the depth of Zn; n-the layering number of the rock and soil layer divided according to the compression modulus within the calculation depth of foundation settlement under the base.

Boussinesq is the rational answer to stress and strain under the vertical concentrated force of the elastic semi-infinite body on its boundary, which is based on the continuum theory. The equivalent pier method takes the rock and soil layer and structure above the pile group base, including the goaf, as the solid foundation, which also meets the solving conditions of Boussinesq. The calculation of the settlement of the pile group foundation in the goaf could be divided into two parts: one is that the pile group foundation adopts the corrected formula to obtain the average settlement value of the pile group foundation when the foundation is taken as the general formation conditions; the other is to add the influence of the goaf on part of the pile tip settlement to the average settlement value so as to obtain the uneven settlement difference. The total settlement amount on the top of the goaf foundation is obtained on the basis of the two above-mentioned settlement.

(6) S_{out}={\phi }_{out}{m}_s{\displaystyle \sum _{i=\mathrm{1}}^n\Delta S_i}.

In Eq. (6), S_{out} -the total settlement amount on the top of the goaf foundation; {\phi }_{out} -the influence coefficient of the settlement of goaf with the value of over 1; refer to Eq. (5) for the meaning of other parameters.

The experimental results show that under the effect of load class-4, the settlement of cj1-cj4 on the four angular points of the pile cap is 0.162, 0.154, 0.103 and 0.134 mm respectively. Suppose the cap is absolutely rigid, the settlement value of the cap center is the average settlement of the four angular points, namely, 0.138 mm. Under such load, the value of {\phi }_{out} is assigned as 0.162/0.138= 1.174, because the prototype serves as the pile group foundation of the high-speed railway that has high requirements for settlement control. Considering the small average settlement amount of the model, the uneven settlement in the goaf brings quite less impact.

The highway specifications adopt almost the same method of solving as the railway specifications do; specifically, the block deep foundation Boussinesq solution that does not take into account the shearing strength of the pile group side rock-soil mass is utilized to calculate the settlement. By integrating the influence of the additional stress and pile group geometrical parameter when calculating the foundation settlement according to Mindlin analytical solution of soil displacement. Theoretically, some issues do exist in the calculation of pile group settlement in the goaf on the basis of the code for building pile foundations because the rock-soil mass should conform to the continuum assumption as the pile group side rock-soil mass shearing strength is taken into consideration. However, considering that the existence of goaf makes the continuum assumption invalid, this contradiction must be overcome and deserve in-depth studies when Mindlin analytical solution of soil displacement is adopted.

In terms of specific application, corrected Eq. (5) is applied to the calculation of the average settlement amount of the pile groups in the goaf while Eq. (6) to the calculation of the total settlement amount at the top of the goaf foundation.

Conclusion

Through the small-structure model test, this paper offers an analysis of the mechanism for the pile-soil-cap-goaf and the settling character and comes to the following conclusions:

1) The influence of goaf on pile bearing capacity is inversely proportional to the load; the higher load is accompanied by less influence of goaf on the bearing capacity. The pile side negative frictional resistance does not occur to any of the piles. The downward shift of the center of gravity of the side frictional resistance in piles crossing the goaf is more obvious than that of the normal status. Considering the large proportion of load borne by the soil among piles under the pile cap and the particularity of the goaf, it is advisable not to take into account the cap effect.

2) There is a critical load value in the uneven settlement of pile groups in the goaf; if this value is exceeded, the differential settlement between the rock and soil layer of the goaf roof and the normal rock and soil layer under the foundation won’t increase any longer.

3) Based on the experimental results and theoretical deviation, the computational formula for the single pile bearing capacity and the pile group settlement in goaf that are based on the existing codes are established.

4) This model test is a small-structure one with the target of studying the mechanism for the stress and settlement of the pile group. The experimental data obtained cannot be applied to the prototype in proportion.

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