To obtain the optimal load-reduction scheme and calculation method of earth pressure on the crown of the pipe, the load-reduction efficiency of rigid pipe and flexible pipe with different thicknesses and layers of expanded polystyrene (EPS) is investigated by model tests, and the law of load reduction is obtained by analyzing the earth pressure and the displacements of the filling around the pipe. The test results show that the earth pressure is obviously reduced with EPS laying on the crown of pipe, and the load-reduction efficiency is increased to be constant with increasing of EPS thickness. In the case that the summation thickness of EPS is constant, the load-reduction efficiency of EPS with two layers spread on the pipe is higher than that of one layer only. Compared with the rigid pipe, the load-reducing effect of flexible pipe is more significant. Based on the data obtained from the model tests, the nonlinear earth pressure calculation formula obtained from regression analysis is adopted, and the results from it are compared with the existing formula consequences of the earth pressure on the buried pipe. The results show that the earth pressure calculated by nonlinear earth pressure theory is on the brink of that tested in the field. The research results can provide references for selection of load-reduction measures and calculation of earth pressure on the crown of the pipe.

With the fast development and expansion of modern cities, buried pipes have been extensively applied for transmission of various media such as water, gas, and petroleum. The underground pipe project is concealed, and its investment is huge; once the pipe is affected, it may cause huge economic loss and even the crisis of public safety. In recent years, the damage phenomenon of urban road surface of underground pipe is serious, especially the ruin of shallow buried pipeline, which greatly affects the smoothness and safety of urban traffic. In order to ensure the normal operation of pipelines, many scholars have studied the stress, deformation, settlement, and cracking of underground pipe [

In order to change the stress concentration phenomenon of earth pressure on crown of pipe and reduce earth pressure, artificial measures should be adopted. Firstly, the distribution of earth pressure along pipe can be adjusted by laying relieving slab to improve the overall stress state of the structure [

Marston first proposed lying compressible flexible material on crown of the culvert (pipe) for earth pressure load reduction and verified the feasibility of the method of pipe structure load reduction [

Expanded polystyrene (EPS) is a kind of foam plastic, because of its light weight, corrosion resistance, and good compression deformation performance, in the research subject has been favored by many scholars and has been widely used in practical engineering. Bartlett et al. [

Obviously, EPS is a very useful load-reducing material. However, at present, the research on the earth pressure reduction at the crown of the pipe mostly stays on the parameters of EPS. For example, Zhang et al. [

Therefore, some scholars put forward the method of laying multilayer polymer or combined reinforcement on pipe roof to reduce the earth pressure. Zhao et al. [

Laying EPS on the crown of pipe can reduce the settlement difference between central and lateral soil prism, and reducing the concentration of earth pressure on the crown of pipe is a feasible and effective load-reduction scheme. However, there are many papers on load reduction of single-layer EPS on culvert crown, which does not consider the load-reduction effect of multilayer EPS and seldom use it in flexible pipe. In this paper, the load-reduction effects of shallow buried rigid pipe and flexible pipe under different thickness and layers of EPS are analyzed by laboratory model tests. Considering the arch effect on the crown of pipe, the nonlinear earth pressure calculation formula is obtained by regression analysis. And the reliability of the formula is verified by comparing with the current six classical calculation methods, which provides a reference for the design of pipe load reduction and the calculation of earth pressure.

Regardless of the impact of the boundary of the earth pressure of the buried pipe, the pipe under such burying conditions is a buried pipe. Therefore, the left and right baffle of the test model box is 4 times the diameter of the pipe away from the pipe axis, which is used to simulate the trench slope far away from the pipe axis, as the boundary condition of this test.

This test mainly focuses on the variation of earth pressure around the pipe. Flexible pipe and rigid pipe are selected according to their mechanical properties. Usually, the deformation of pipe is larger than the settlement of soil above the pipe. The soil on both sides of the pipe will produce upward friction on the soil above it, which reduces the phenomenon of earth pressure concentration on the crown of the pipe. This kind of pipeline is called flexible pipe; conversely, it is called rigid pipe. Therefore, when choosing pipe material, the flexible pipe is simulated by the PVC pipe with larger deformation, and the rigid pipe is simulated by the steel pipe with smaller deformation. And the specifications are DN110 (wall thickness is 3.2 mm). The filling material is ISO standard sand, and its particle size distribution is given in Figure

Gradation curve.

Compressible stress-strain curve of EPS.

Parameters of sand.

Maximum dry density (g·cm^{−3}) |
Minimum dry density (g·cm^{−3}) |
Internal friction angle (°) | Degree of compaction (%) | Water content (%) |
---|---|---|---|---|

1.70 | 1.65 | 30 | 95 | 8 |

Parameters of EPS.

Density (kg·m^{−3}) |
Modulus of proportion (MPa) | Modulus of elasticity (MPa) | Yield modulus (MPa) |
---|---|---|---|

18 | 2.22 | 1.40 | 0.04 |

Static load is applied by means of stress control and step compression. The concentrated load is transformed into a uniformly distributed load and transmitted to the filling soil through the bearing plate (steel plate with a length of 790 mm × width of 400 mm × thickness of 14 mm). The loading force of the model is determined according to the height of the simulated fill, and the height of 50 cm above is simulated through the stacking weight on the bearing plate. The weight has three specifications of 5 kg, 10 kg, and 20 kg, and the weight to be loaded to simulate different filling height is shown in Table

The applied load on the model filling height.

Load weight (kg) | 0 | 55 | 110 | 165 | 275 |

Height of backfill (cm) | 50 | 60 | 70 | 80 | 90 |

Figure

Model test box (unit: mm).

Figure

Table

Test programs.

Scheme | Pipe material | Load-reducing material | Thickness of EPS, |
Layer spacing of EPS, |
Number of EPS layers, |
EPS width, |
---|---|---|---|---|---|---|

1 | Steel pipe | — | — | — | 0 | — |

2 | Steel pipe | EPS | 1 | — | 1 | 16.5 |

3 | Steel pipe | EPS | 2 | — | 1 | 16.5 |

4 | Steel pipe | EPS | 3 | — | 1 | 16.5 |

5 | Steel pipe | EPS | 4 | — | 1 | 16.5 |

6 | Steel pipe | EPS | 2 | 5 | 2 | 16.5 |

7 | PVC pipe | — | — | — | 0 | — |

8 | PVC pipe | EPS | 1 | — | 1 | 16.5 |

9 | PVC pipe | EPS | 2 | — | 1 | 16.5 |

10 | PVC pipe | EPS | 3 | — | 1 | 16.5 |

11 | PVC pipe | EPS | 4 | — | 1 | 16.5 |

Figure

Pressure on crown of pipe vs. height change of filling soil.

Load-reduction rate of EPS in different soil cover depths.

Figure

Figure

According to the above analysis, the earth pressure on the crown of the pipe presents a nonlinear increasing trend with the increase of filling height, and all of the monitored pressure is greater than the linear theoretical value. On the one hand, the gravity stress on the crown of the pipe increases with the increase of the height of the fill (the gravity stress is the intergranular stress transmitted between the soil particles at the contact points, neglecting the effect of friction and cohesion between the particles, and the existence of the frictional force and cohesion is the reason for the nonlinear growth of the pressure of the crown of the pipe with the height of the backfill). On the other hand, with the increase of the height of the backfill, the differential settlement will be produced between the central soil prism and the lateral on the crown of the pipe. The stiffness of the rigid pipe is greater than the surrounding soil, and the settlement of the pipe side backfill is greater than that of the crown of the pipe. That is, the settlement difference between the central and lateral soil prisms is Δ_{1} < 0 (as shown in Figure

By laying EPS to increase the settlement of backfill on the crown of pipe and reduce the differential settlement between the central and lateral soil prisms, the concentrated stress on the crown of the pipe can be transferred. Figure

Distribution of vertical earth pressure above the pipe.

Pressure variation of No. 1–5 pressure cells with filling height.

Figure

Figure

The test results demonstrate that the earth pressure on the crown of the pipe decreases and the earth pressure on the side of the pipe increases with the increase of the thickness of EPS. The main reason is that the EPS of a certain great thickness has a large compressible deformation, which reduces the differential settlement between the central and lateral soil prisms (as shown in Figure

Figure

Pressure on crown of pipe vs. filling height.

Load-reduction rate of different layers of EPS vs. filling height.

Figure

It can be seen from Figure

The above test results show that when the backfill is high, under the condition that the total thickness of EPS is the same, the load-reduction efficiency of two layers of EPS is greater than that of one layer. The probable reason is that the soil particles contact each other to transfer stress, and the stress is spread layer by layer. When the crown of the pipe is overlaid two layers of EPS, the upper layer of EPS increases the stress diffusion angle (as shown in Figure

Figure

Distribution of vertical earth pressure above the pipe.

Pressure of No. 3 earth pressure cell with filling height.

It can be seen from Figure

Figure

The above test results show that the load-reduction effect of two layers of EPS on the crown of pipe is better than that of single-layer, but at the same time, the increase of horizontal earth pressure measured in the pipe is more obvious. On the one hand, the deformation of the two-layer EPS is more sufficient than that of the single-layer EPS, thus the differential settlement is better reduced by two-layer EPS, (as shown in Figure

The earth pressure on the crown of the pipe measured by earth pressure cell No. 8 in scheme 1 and schemes 7 to 11 is shown in Figure

Crown earth pressure vs. the filling height.

Crown earth pressure vs. filling height.

Figure

It can be seen from Figure

The analysis shows that the flexible pipe has a certain deformation capacity, which can reduce the differential settlement between central and lateral soil prisms and the stress concentration on the crown of the pipe. Therefore, compared with rigid pipe, the deformation of flexible pipe contributes a certain load-reduction capacity. The settlement of the soil prism on crown of flexible pipe can be further increased through the compression deformation of EPS with different thickness. As shown in Figure _{2} > 0, and the earth pressure on pipe crown is smaller than the weight of the overburden. On the other hand, EPS installed can not only increase the bearing capacity of flexible pipe, but also protect the pipe from stress concentration and excessive deformation.

The variation of earth pressure measured by the earth pressure cells of No. 6 to No. 10 with the filling height of 90 cm on the crown of the pipe is illustrated in Figure

Distribution of vertical earth pressure above the pipe.

Pressure of No. 3 load cells with filling height.

Figure

As saw from Figure

In conclusion, the variation of horizontal earth pressure on the side of flexible pipe is similar to that of the rigid pipe, which increases with the increase of filling height and load-reduction efficiency. From Figure

_{1} is the width of the loose zone.

_{e} of the filling on the crown of pipe. That is to say, a plane of equal settlement is formed. The calculation formula based on the plane of equal settlement is as follows.

When _{e},

When _{e},_{e} is equal level height;

_{h} is the deformation modulus of the backfill in height of _{c} is a coefficient related to the ratio of the length to width of a rigid pipe; and

_{c} is the outer diameter of the pipe and _{e} is the interaction coefficient between soil and structure of buried pipe.

According to the model tests results, the pressure on the crown of the pipe presents a nonlinear change with the increase of the filling height, which is caused by the load transition inside the backfill on the crown of the pipe. Considering the influence of load transition on pipe, the following nonlinear earth pressure calculation formula of pipe is proposed, and the undetermined coefficients in the formula are determined by the nonlinear regression analysis of the data measured by model tests [

According to the tested results of scheme 1, the formula of regression earth pressure is

According to the tested results of scheme 1 to scheme 5, the formula of regression earth pressure of rigid pipe is_{1} and _{2} are coefficients related to the thickness of EPS, whose values can be approximated to _{1} = −405.14^{2} − 3.8303_{2} = −311.93^{2} + 14.206

According to the tested results of scheme 7 to scheme 11, the formula of regression earth pressure on flexible pipe is_{5} = −140.17^{2} − 24.307_{6} = −226.79^{2} + −4.5176

The correlation index is as follows:

Figure

Comparison of earth pressure on crown of the pipe.

According to the comparison, when the backfill on a buried pipe reaches a certain height, the earth pressure on the crown of the pipe is obviously less than the measured earth pressure, which is calculated by Terzaghi’s method, Gu Anquan formula, and CGCDHBC and CFCDRBC methods. Among them, Terzaghi’s underground cavern theory underestimated the pressure and coefficient as compared with the field data. When the filling reaches a certain height, the pressure will no longer be transmitted downward, and the calculated earth pressure is minimal. Gu Anquan formula is based on the elastic theory, assuming that the pipe is a bar foundation, and the earth pressure is calculated by the settlement of the fill, but the calculation results are only closer to the measured value only when the height of the fill is large. The experience coefficient method is adopted in the CFCDRBC methods in China and the AASHTO methods in the United States. But the pressure of the pipe crown soil is obviously less than the measured earth pressure value calculated by the CFCDRBC methods, it is unsafe. And the pressure of the pipe crown soil of the AASHTO methods will increase with the height of the fill, it is not economic. Marston’s formula needs to take into account the problem of the plane of equal settlement, and when the filling height is greater than that of it, the pressure on the crown of the pipe will be much larger than the measured value as the filling height increases. Here in, the nonlinear earth pressure calculation formula obtained by regression analysis of the test data are closest to the measured values of the earth pressure on the crown of pipe, which can better reflect the change process of the pressure of the crown of the pipe.

In this paper, model tests and theoretical analysis were employed to study the earth pressure on rigid pipe and flexible pipe underneath EPS, and the following conclusions can be drawn:

The instrument of EPS on the crown of the pipe can play a significant role in reducing load, and with the increase of the thickness of EPS, the efficiency of load reduction firstly increases and then tends to be constant. The load-reduction effect of two layers EPS on the crown of the pipe is better than that of single-layer EPS with the same total thickness.

Compared with rigid pipe, flexible pipes themselves play a role in reducing load, so when the filling height is low, the use of flexible pipe can play a valuable role in load reduction. When the height of the fill is high, placing the EPS on the flexible pipe can reduce the pressure on the crown of the pipe and increase the bearing capacity of the pipe effectively, therefore guarantee the safety of the flexible pipe in practical engineering.

The settlement difference between central and lateral soil prisms can be reduced by laying flexible filler on the crown of the pipe or replacing the flexible pipe, so that the concentrated stress on the crown of the pipe can be transferred to the adjacent soil mass, and the vertical earth pressure on the side of the pipe can be increased, and the horizontal earth pressure on the pipe can also be increased. Therefore, while considering the load reduction of the vertical soil pressure on the crown of the pipe, the effect on the horizontal soil pressure of the pipe cannot be ignored.

According to the model tests data, the nonlinear earth pressure formula is fitted by regression analysis and compared with the contemporary earth pressure calculation formulae of pipe. It is indicated that the calculation formula of nonlinear earth pressure is closer to the measured value.

Finally, the formula for calculating the nonlinear earth pressure proposed in this paper is simple in form and highly correlated with engineering practice. And it has an important reference value for calculating the earth pressure of pipe under different conditions. In addition, Chen et al. [

The data used to support the findings of this study are available from the corresponding author upon request, and the readers can access the data supporting the conclusions of the study.

The authors declare that they have no conflicts of interest.

This work was supported by grants from the National Natural Science Foundation of China (NSFC) (nos. 51678223 and 51608181) and Green Industrial Project of Hubei University of Technology (YXQN2017001). The authors would like to express their appreciation to these financial assistances.