Introduction

Green, low-carbon and high-efficiency are the trend of modern civil engineering technology1,2. The utilisation of prefabrication technology has been identified as a key factor in enhancing the economic, technical and environmental performance of construction projects. With the development of prefabrication technology, prefabricated culverts in highways have increasingly been performed. The prefabrication of culverts or channels can be categorized into integral prefabrication and modular prefabrication3. Integral prefabrication, which involves large cross-sections, requires high precision in lifting equipment, transport, and installation. Compared to cast-in-place construction, both prefabrication methods can significantly improve the quality and progress of a project. However, the larger sections and weights of precast components place much greater demands on lifting equipment, lifting accuracy, and operator control. Modular precast culverts with delicate sections and appropriate weights allowing for flexibility in the size of individual modules, facilitating convenient lifting and transportation are more adaptable. Modular precast culverts, designed with appropriate sections and optimized weights, offer flexible sizing options for individual modules. This design enhances their adaptability and ensures ease of lifting and transportation. As a result, prefabricated modular culverts have been rapidly developed and applied to Chinese highways4,5,6,7,8, achieving excellent economic and social performance9,10,11,12.

The prefabricated box culverts currently used in China are mainly constructed with hinged connections in a single section to realize the modularity of prefabricated culverts. Unfortunately, suitable design standards are not available for this innovative culvert design. The primary reference is the monolithic box culvert load determination method. However, the difference in the culvert–soil interaction relationship between prefabricated box culvert and the monolithic box culvert would cause the difference between prefabricated box culvert and load, and its earth pressure distribution law has been a critical problem that needs to be addressed urgently. However, the different interactions between the soil and the prefabricated box culvert compared to the monolithic box culvert result in different loading conditions. This discrepancy requires urgent attention to the critical issue of earth pressure distribution. Presently, the calculation of the vertical earth pressure concentration coefficient on the top slab and the design codes for culverts in several nations continue to adhere to the “load-structure” design approach13,14,15. Structural modifications to the culvert can impact the interaction between the culvert and the soil. Nevertheless, culverts are influenced by geology, topography, the construction method, culvert structural type, and various other aspects. Excessive earth pressure beneath culverts is a common issue that can lead to structural damage and affect both stability and regular usage16,17,18. Culvert design codes across different countries generally rely on the vertical earth pressure concentration coefficient on the top slab to determine the allowable top load19,20,21, factoring in the interaction between the culvert and the soil22,23,24. Unlike monolithic culverts, prefabricated ones often incorporate specific connection designs, such as groove25 and hinge26 connections within each section, which significantly influence their structural behaviour and performance.

In addition, the bolted connection can be used in the longitudinal direction. This new type of culvert with a connection structure adopts a prefabricated process, which is frequently applied to highway and municipal projects. According to the structural moment distribution law, the prefabricated box culvert members in China generally perform modular segmentation and usually set the hinge position where the moment of the sidewall is zero. For the hinged prefabricated box culvert (HPBC), the hinged structure would impact the culvert–soil interaction; currently, research on this aspect is limited. Moreover, the longitudinal connection performance is enhanced by fitting longitudinal flexural bolts between the HPBC sections. This novel culvert integrates both intra-segmental and longitudinal connection designs, featuring an innovative concept. In China, prefabricated box culvert components are usually divided into modules according to the principles of structural moment distribution. Hinge points are normally positioned at areas where the sidewall moment is zero. The presence of hinges in hinged prefabricated box culverts (HPBC) has a considerable impact on the interaction between the culvert and the surrounding soil. Nevertheless, there is currently a lack of study investigating this particular element.

Researchers have conducted several investigations on the earth pressure characteristics of prefabricated box culverts. Li et al.25 analyzed the damage modes, bearing capacity, and deformation characteristics of jointed heart peduncles and structures of precast concrete box culverts with U-shaped sections by full-scale static tests and numerical simulation methods. Gong et al.26 used an improved fuzzy clustering method and a detection method for acoustic emission characteristics on two precast box culverts. They conducted an analysis on the deterioration process of the box culverts using the traditional fuzzy C-mean (FCM) method. Garg et al.27 used laboratory model testing to perform shear tests on precast box culverts measuring 1.22 m in length, width, and height. They simulated the impact of AASHTO 2005 and HS20 vehicle loads on the structural strain, deflection, and fracture patterns. Ramadan et al.28 examined the structural behaviour and distribution patterns of earth pressure in reinforced concrete precast culverts with spans of 7.3, 10.4, and 13.5 m. They conducted a finite element simulation and compared the results with field observations. Bian et al.29 studied the earth pressure distribution law of top slab, vault deformation, and soil settlement law of prefabricated arch culverts by full-size model test and numerical simulation. Zenagebriel et al.30 used the finite element method to simulate the settlement of adjacent longitudinal precast box culverts. The settlement of adjacent longitudinal precast box culverts, bottom slab reaction forces, and soil on both sides under different loading conditions were investigated. Gong et al.31 performed a comparative examination of stress, strain, displacement, and damage laws of different types of culvert structures (monolithic, flat-joint, round-hinged, and tongue-and-groove) using laboratory testing and finite element methods. Mário et al.32 conducted field experiments and numerical simulations to analyze the earth pressure, consequent deformation law, and damage mechanism of the perimeter of the container culvert of the enterprise port connection under a backfill height of 9.5 m. Ma et al.33 conducted field tests to investigate the earth pressure distribution and structural deformation around prefabricated arch culverts with sizes ranging from 1 to 7 m. Kang et al.34 formulated the performance function of prefabricated box culvert components, constructed the generalized probability density evolution equation of the function, and subsequently calculated the resulting probability density function and the failure probability of the components. Du et al.35 used the Drucker-Prager Cap model in the finite element software ADINA to analyze the hysteresis curves of precast reinforced concrete box culverts and compared them with the results of laboratory model tests. Abuhajar et al.36 used laboratory centrifugal model test to study the interaction between box culvert and sand, and analyzed the earth pressure and structural moment distribution law around the culvert.

The literature examines the mechanical and deformation properties of monolithic prefabricated box culverts. There is a scarcity of research conducted on hinged prefabricated box culverts (HPBC). However, if the monolithic culvert approach is employed, the estimation of the earth pressure concentration coefficient on the top slab would be unreliable due to the alteration in structural form and mechanical mode. The distribution law of vertical earth pressure on the top slab is a crucial component that impacts the mechanical and deformation characteristics of a culvert. Nevertheless, there has been limited investigation about the distribution pattern of vertical earth pressure on the top of the hinged prefabricated box culvert. Consequently, this paper presented the results of a field experiment conducted to test the distribution law of vertical earth pressure on the top slab of the Xiyu Expressway project at different filling stages. Furthermore, FLAC3D numerical software was employed to examine the impact of backfill height, foundation modulus and backfill modulus on the distribution law of vertical earth pressure on the top slab. A modified method to calculate the vertical earth pressure on the top slab was proposed, and the obtained results were subsequently compared with those of the AASHTO and Chinese culvert codes. Additionally, a sensitivity analysis of the influencing factors was conducted by combining the present method with the principle of the orthogonal integer test. This study can serve as a reference for the design of other precast culvert projects.

Analysis of field test programs and results

Project overview

In China, the Xiyu Expressway constructed several hinged prefabricated box culverts (HPBC) with a span of 4.0 m and a rise of 4.0 m. The structure was modularized into a prefabricated top slab, prefabricated side walls, and a bottom slab using the cast-in-place method. Both the precast elements and the cast-in-place bottom slab were composed of C40 concrete. The prefabrication process performed centrally within the prefabrication facility, resulting in standardized lengths of 3.0 m for each individual section. Adjacent 3-m sections were jointed using M10 mortar, and the joints were sealed with asphalt and asphalt-soaked wadding. As illustrated in Figs. 1 and 2, the hinged prefabricated box culverts comprised a top slab, two side walls, and a bottom slab casted in place. The top slab and side walls were connected via a hinge mechanism to allow for articulation. Additionally, the prefabricated box culvert was designed with a 10 cm thick C30 concrete foundation as the cushion.

Figure 1
figure 1

Design diagram of hinged prefabricated box culverts (a) Structural dimensions; (b) side view of standard culvert section; and (c) structure diagram of hinge joint (unit: cm); (d) vertical arrangement diagram; (e) bend bolt connection diagram of the top slab.

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Figure 2
figure 2

On-site hinged prefabricated box culverts (HPBC).

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Field test scheme

In this paper, field tests of vertical earth pressure on the top slab were carried out using the HPBC of the Xiyu Expressway as a research object. Figure 3 illustrates the schematic diagram of the location of the field test section. Tests were conducted under the traffic lane, approximately 460 cm from the roadbed centerline. Firstly, it should be noted that this section is relatively vulnerable, due to the loss of stiffness in the joint of the HPBC, in comparison to the rigid joint of the box. On the other hand, it is necessary to investigate the earth pressure distribution pattern around the joint, as the load distribution pattern at the joint is crucial to the structure of the culvert. Compaction tests were carried out during the filling stage. All tests achieved compaction of more than 96%. In the process of culvert filling, the earth pressure distribution law on the top of HPBC with backfill heights of 0.3, 0.9, and 2.0 m was tested. Figure 4 shows the layout scheme of gauges.

Figure 3
figure 3

Schematic diagram of the location of the field test section.

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Figure 4
figure 4

Schematic diagram of vertical earth pressure gauges on site on the top slab.

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Field test element placement

The earth pressure gauge using YT-ZX-0300 type dual-films earth pressure gauge, the size of φ120 × 28 mm, as shown in Fig. 5, the specific parameters of the earth pressure gauge are shown in Table 1.

Figure 5
figure 5

YT-ZX-0300 double-films earth pressure gauge.

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Table 1 YT-ZX-0300 double-films earth pressure gauge technical parameters.
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Figure 6 shows the layout process. Before installing the double-films earth pressure gauges, the sensor and data cable line should be tested to confirm if they work properly. The pressure box used was a double-films earth pressure box. Installation involved placing the stressed film (pressure-bearing layer) facing upwards, levelling the base with fine-grained soil or fine sand, and compacting the fine-grained soil covering around the earth pressure box. Note that within 1 m of the earth pressure gauge, manual bulldozing and small machinery should be used to compact the earth or gravel, and large machinery bulldozing and compacting should be avoided to prevent damage to the earth pressure gauges. In addition, the number of sensors and the data line length shall be checked to confirm if they are correct. Measure according to the site conditions, determine the installation position of the gauges and extend the earth pressure gauge data cable ports to the culvert cavity to access data.

Figure 6
figure 6

The process of burial of double-films earth pressure gauges on the top slab.

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Field test results and analysis of earth pressure distribution law on the top slab

The distribution law of vertical earth pressure on the top slab in the field test is shown in Figs. 7 and 8.

Figure 7
figure 7

The vertical earth pressure distribution law on the top slab in field test.

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Figure 8
figure 8

Comparison of vertical earth pressure at the characteristic point of culvert top (field test and linear earth pressure calculation method).

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Based on Figs. 7 and 8, it was observed that the vertical earth pressure value on the top slab significantly increased compared to the middle of the span. As the backfill height on the top slab increased, the vertical earth pressure value also increased. When the backfill height was merely 0.3 m, the vertical earth pressure exhibited horizontal distribution. As the backfill height increased, the earth pressure distribution on the top slab demonstrated a non-linear pattern. increased from 0.3 to 2.0 m. The center of the culvert top had earth pressure values of 5.2 kPa, 15.1 kPa, and 27.5 kPa, corresponding to each backfill height. Meanwhile, the vertical earth pressure values at the end of the culvert top were 6.0 kPa, 27.2 kPa, and 77.1 kPa.

The earth pressure in the span of the top slab at each backfill height was consistent with the calculated value of linear earth pressure. However, the earth pressure on the top slab non-linearly increased, and the changing trend with the height of the filling on the top was consistent with the calculated value of 2.1 times linear earth pressure. This was due to the soil overlaid the culvert (interior soil column), and after compaction, there were different settlement laws outside the culvert (exterior soil column). This is due to differential settlement of the backfill above the culvert and the surrounding backfill. The term “interior soil column” pertains to the backfill located directly above the culvert, while the “exterior soil column” designates the backfill situated outside the culvert’s boundaries. These two columns exhibit a notable difference in settlement deformation, with the external soil column generally undergoing more significant settlement than its internal counterpart. This disparity leads to friction between the two columns, which in turn results in the redistribution of vertical loads. As a consequence, the interior soil column experiences greater vertical loads, demonstrating a concentrated effect of vertical earth pressure. Thus, the distribution of vertical earth pressure is characterized by a non-linear pattern.

Moreover, the end of the culvert had greater stiffness than the middle span. Therefore, the vertical deformation on the end of top slab could be approximated to zero. Conversely, the middle of the span had some deformation under the load of backfill. The soil column on the top slab had the same frictional resistance transferred to the soil column on the end of the top slab. Thus, the deformation in the span of the top slab was lower, so the earth pressure in the span was slightly closer than the linear earth pressure value.

Numerical simulation scheme and analysis of results

Model building

The numerical simulation was conducted by FLAC3D finite difference simulation software, as shown in Fig. 9. To enhance the accuracy of the model calculation, the thickness of the ground was selected as 15 times the span of the culvert. The calculated width of the model was taken as ten times the span. The bottom and sides of the model were normal to the fixed constraints. The corresponding model can be established according to the assigned backfill heights. Interfaces were configured for both intra-segmental (hinge) and longitudinal connections in order to replicate the mechanical properties of the HPBC connection (Fig. 10). The backfill units can be added or subtracted from a numerical model to simulate varied backfill heights.

Figure 9
figure 9

Numerical model diagram.

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Figure 10
figure 10

Numerical simulation detail diagram.

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Calculation parameters

This research utilizes the equivalent modulus of elasticity method to determine the modulus of elasticity of reinforced concrete materials. Equation (1) calculates the equivalent modulus of elasticity of the section by translating the modulus of elasticity of the reinforcement to the modulus of elasticity of concrete with the same cross-sectional dimensions.

$$E = \frac{{E_{s} A_{s} + E_{c} A_{c} }}{{A_{s} + A_{c} }}.$$
(1)

As shown in the Fig. 11, the section reinforcement of a culvert section with a longitudinal unit length of 1.00 m was set as a standard section.

Figure 11
figure 11

Longitudinal section reinforcement diagram of culvert structure (unit: cm).

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The concrete modulus of elasticity Ec of 30,000 MPa. The steel diameter was 20 mm HPB400 reinforcement, and the modulus of elasticity of steel ES was 200 GPa. Substitution into the Eq. (1) can be obtained. The equivalent modulus E of reinforced concrete was 34,187 MPa. Interface parameters, according to the literature37,38, it is known that the normal stiffness kn and tangential stiffness ks of the peripheral unit body are calculated as Eqs. (2)–(4).

$$k_{n} = k_{s} = 10\max \left[ {\frac{{K + {{4G} \mathord{\left/ {\vphantom {{4G} 3}} \right. \kern-0pt} 3}}}{{\Delta Z_{{{\text{min}}}} }}} \right],$$
(2)
$$K = \frac{E}{3(1 - 2\nu )},$$
(3)
$$G = \frac{E}{2(1 + \nu )},$$
(4)

where K is the bulk modulus, G is the shear modulus, \(\Delta Z_{\min }\) is the normal minimum width of the surrounding unit body, and v is the Poisson’s ratio.

The elastic model was selected for culvert structure and cushion, while backfill and ground were used in the Mohr–Coulomb model. The parameters of backfill and ground were set concerning the survey and design data, and the specific parameters were shown in Table 2 and Table 3. The foundation stiffness for this simulation is assumed to be 20 MPa according to the culvert fill modulus study by Xie et al.39, Feng et al.40, and Ma et al.41. The foundation soil needs to meet the foundation-bearing capacity requirements for compaction treatment. According to the reference values of rock and soil parameters42, the elasticity modulus of the hard plastic powdery clay was in the range of 6–50 MPa. In this paper, the elasticity modulus of the foundation was selected as 50 MPa, which was considered an acceptable state of compaction. Considering that the bedding layer applied in the actual project was 4% cement soil, the C30 concrete foundation was set, and the thickness was only 10 cm, in order to improve the calculation efficiency and accuracy, the two materials were combined into the cushion, and the value of its modulus of elasticity was discounted compared with that of the concrete. The value of the modulus of elasticity was taken as 20,000 MPa.

Table 2 Material parameters.
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Table 3 Mechanical parameters of interface.
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Connections were configured both intra-segmental and longitudinally, as shown in Fig. 10. The interface parameters assume the interface was smooth, and the unfavorable working conditions ignoring the hinge joint friction can be obtained. For the longitudinal interface, the calculation is as follows.

The modulus of elasticity of the culvert structure was 34,187 MPa with Poisson’s ratio of 0.15, and the minimum normal width of the perimeter unit cell was 0.221, so it can be calculated:

$$k_{n} = k_{s} = 10\max \left[ {\frac{K + 4G/3}{{\Delta Z_{{{\text{min}}}} }}} \right] = 10 \times \frac{16,280 + 14,864 \times 4/3}{{0.221}} \times 10^{6} = 1627.95 \times 10^{9} \;{\text{Pa}} = 1627.95\;{\text{GPa}}{.}$$

The longitudinal interface was converted to equivalent cohesive force by converting the total strength of the flexural bolts to equivalent cohesive force according to the strength of the flexural bolt reinforcement and the reinforcement area.

The strength parameters of different geotechnical materials have a nearly linear relationship with the bulk density of the material. The equation that converts the parameters can be derived as follows:

$$c = c_{0} + \left( {\frac{{f_{{{\text{sd}}}} A_{{\text{s}}} + f_{{\text{M}}} A_{{\text{c}}} }}{{A_{{\text{c}}} }}} \right)c_{0} ,$$
(5)

where c is the equivalent cohesion, c0 is the mortar cohesion, fsd is the design value of the tensile strength, AS is the total cross-sectional area of the flexural reinforcement, fcu is the compressive strength of the mortar, Ac is the cross-sectional area of the longitudinal joint.

Yang et al.43, Li44, Ruan et al.45, studied the selection of interface parameters. The cohesive force c0 and the angle of internal friction φ of the cement mortar can be assumed to be 1.2 MPa and 50°, respectively. According to the Chinese Code for the Design of Concrete Structures (GB 50010-2010)46 it can be determined that the design value of the tensile strength of the HRB400 hot-rolled ribbed steel bars fsd is 330 MPa, and the compressive strength of the concrete cube fM is 10 MPa. The cross-sectional area of the bent bolt As is 0.00196 m2, and the overall area of the longitudinal contact surface A is 5.64 m2. Calculate the parameters by substituting them into Eq. (5).

$$c = c_{0} + \left( {\frac{{f_{{{\text{sd}}}} A_{{\text{s}}} + f_{{\text{M}}} A_{{\text{c}}} }}{{A_{{\text{c}}} }}} \right)c_{0} = 1.2 + \frac{330 \times 0.00196 + 10 \times 5.64}{{5.64}} \times 1.2 = 11.31\;{\text{MPa}}{.}$$

Analysis of working conditions

Prefabricated box culverts are commonly used in regions with mostly plain land, considering the actual site topography, construction conditions, and the convenience of transport the culvert. The current standard design drawings for HPBC used in specific provinces in China specify a maximum backfill height of 6 m. In the numerical simulation, the selected backfill height included the critical backfill heights of 0.5 m for the open culvert and the concealed culvert. In addition, the HPBC with a 1 m backfill height was more common in the Xiyu Expressway, which was also a typical design backfill height. Backfill heights of 3, 6, and 9 m were set to analyze the distribution pattern of HPBC at larger backfill heights. Therefore, 0.5 m, 1 m, 3 m, 6 m, and 9 m are selected for the numerical simulation to provide a reference for further analysis of the earth pressure distribution.

Numerical model validation analysis

To verify the accuracy of the numerical simulation, the field test 0.9 m was used as a case for verifying the simulation. The comparison of vertical earth pressure distribution law on the top slab is shown in Fig. 12.

Figure 12
figure 12

Comparison between numerical simulation and field test of vertical earth pressure distribution law on the top slab.

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As shown in Fig. 12, the vertical earth pressure distribution law of the culvert perimeter of the numerical simulation is consistent with the field test results. The numerical model accurately represents the real force characteristics of the culvert in the field. The distribution law of vertical earth pressure on the top slab was that the earth pressure value at both ends increases significantly. The numerical model tends to slightly overestimate compared to field tests, due to its simplified assumptions about material consistency and solving process. Actual conditions, with their inherent complexities and environmental variability, are better captured in field tests. The minor discrepancies observed serve to reinforce the reliability of the numerical model.

Analysis of numerical simulation results of earth pressure distribution law on the top slab

The vertical earth pressure distribution law of the culvert top of the HPBC under different backfill heights is shown in Figs. 13 and 14.

Figure 13
figure 13

Vertical earth pressure distribution law on the top slab with different backfill heights.

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Figure 14
figure 14

Earth pressure variation law at the characteristic point on the top slab.

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From Figs. 13 and 14, it can be seen that the vertical earth pressure distribution law on the top of the HPBC under each backfill height tended to be stable in the range of about 1.5 m from the center line of the top slab. The earth pressure concentration phenomenon existed on the top slab. The vertical earth pressures in the central region of the top slab were observed to be 8.9, 18.1, 55.0, 110.7 and 163.1 kPa, respectively, at backfill heights ranging from 0.5 to 9.0 m. In contrast, the vertical earth pressures at the end were recorded at 16.5, 34.9, 97.3, 222.2 and 325.8 kPa, respectively. When the backfill height was 0.5 m and 1.0 m, the vertical earth pressure distribution law on the top slab was nearly linearly distributed. The difference between earth pressure values at the end of the end and the span was insignificant. When the backfill height was 3 m and above, the vertical earth pressure on the top slab increased significantly. The overall vertical earth pressure value on the top slab in the span was close to the calculated value of linear earth pressure. The overall vertical earth pressure value on the top slab at the end was relative to the computed value of 2 times linear earth pressure. It indicates that the earth pressure concentration on the top slab will be more prominent when the backfill height is higher.

Influence of foundation modulus on vertical earth pressure on the top slab of the HPBC

The modulus of the culvert foundation has a significant impact on both the overall stability of the culvert and the concentration of earth pressure exerted on it. This study examined different module values for the culvert foundation, represented as Ef, which included 30, 40, 50, 60, and 80 MPa. The foundation's width extended 0.5 m beyond each side of the culvert base, and its depth was 1.5 m. It might be considered a traditional method of treating replacement fill as a foundation. Figures 15 and 16 illustrate the fluctuations in vertical earth pressure on the top slab as the foundation modulus undergoes alterations.

Figure 15
figure 15

Vertical earth pressure on the top slab with different modulus of foundation.

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Figure 16
figure 16

Vertical earth pressure variation law at the characteristic point on the top slab with different modulus of foundation.

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Figure 15 demonstrates that changes in foundation modulus have a minimal effect on the vertical earth pressure applied to the top slab of an HPBC construction. More specifically, changes in the foundation modulus mostly increased the earth pressure at the far end of the top slab, while the pressure at the middle section remained almost unchanged.

Figure 16 provided more evidence that an increase in foundation modulus results in an elevation of vertical earth pressure at the culvert’s endpoint. Conversely, a slight decline was seen at the midpoint. This phenomenon was caused by the decreased settling of the interior earth column, which in turn increased the earth pressure at the end of the culvert. The subsequent rise in earth pressure at the edge of the top slab affected the overall deformation pattern, leading to a tiny increase in vertical deformation near the middle of the top slab and a corresponding small decrease in vertical earth pressure at this point.

Influence of backfill modulus on vertical earth pressure on the top slab of the HPBC

Variations in vertical earth pressure on the top slab versus modulus of backfill are presented in Figs. 17 and 18. This study investigates the influence of backfill modulus (Eb) on the distribution of vertical earth pressure exerted on the top slab of a culvert. To comprehensively analyze this relationship, a range of backfill moduli was considered, specifically 20, 25, 30, 35, 40, and 50 MPa. The variations in vertical earth pressure corresponding to these different backfill moduli are graphically represented in Figs. 17 and 18. These results may provide insights into how the stiffness of the backfill material affects the load distribution over the culvert structure.

Figure 17
figure 17

Vertical earth pressure on the top slab with different modulus of backfill.

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Figure 18
figure 18

Vertical earth pressure variation law at the characteristic point on the top slab with different modulus of backfill.

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As depicted in Fig. 17, the modulus of backfill significantly influenced the vertical earth pressure over the HPBC structure. Specifically, an increase in backfill modulus resulted in a notable decrease in vertical earth pressure at the end regions of the top slab, while showing an increase in the central area. This phenomenon can be attributed to the enhanced stress dissipation capability of the backfill material with a higher modulus. For example, areas susceptible to stress concentration on the top slab end experienced a substantial mitigating effect upon increasing the backfill modulus. Simultaneously, the central region of the top slab exhibited a distinct upward trend in earth pressure, indicating more effective load distribution across the backfill.

Further analysis in Fig. 18 reveals a considerably higher increase in stress at the end of the top slab compared to the mid-span region. This observation highlighted the improved uniformity in load distribution over the top slab, which in turn led to a significant reduction in the deformation at the mid-span of the top slab. These findings collectively demonstrate that increasing the modulus of backfill not only reduces the deformation of the HPBC’s top slab but also promotes a more homogeneous distribution of loads.

Difference in earth pressure distribution law between the hinged prefabricated box culvert and monolithic box culvert

In order to investigate the difference of earth pressure distribution law between HPBC and monolithic box culvert (MBC), based on the above numerical model, the hinged top and bottom were set as a continuum for analysis. The distribution law of vertical earth pressure on the top of the culvert under the backfill height 3 m is shown in Fig. 19. What’s more, Fig. 20 illuminates the deformation difference between the of the two types of culverts.

Figure 19
figure 19

Comparison of earth pressure distribution law at different parts of HPBC and MBC.

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Figure 20
figure 20

Comparison of deformation of the top slab between the HPBC and and MBC.

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As shown in Fig. 19, the distribution of earth pressure around the HPBC culvert exhibited notable differences compared to that of the monolithic culvert. Mainly, the vertical earth pressure on the top of the span of the HPBC was lower than that of the MBC, which was about 93.7% of the MBC. In comparison, the earth pressure on the top slab was relatively larger, which is about 105.2% of the MBC.

In order to ascertain the cause of the discrepancy, it can be seen in Fig. 20. In this paper, the HPBC has a hinged structure on the side wall, which relieved the side wall from transferring the bending moment, so the top slab deformation increased. It has been demonstrated that the vertical earth pressure exerted on the top slab of both the HPBC and MBC was directly proportional to the deformation of the top slab. The top slab of the HPBC presented significantly more deformation than that of the MBC. This vertical deformation influenced how the backfill load was distributed among the interior soil columns situated above the top slab. The most substantial vertical deformation takes place near the center of the culvert span, leading the interior soil columns above to transfer loads to adjacent interior soil columns. In the HPBC culvert span, the majority of vertical earth pressure was concentrated at the ends. Consequently, the HPBC exhibited a more pronounced non-linear distribution pattern of earth pressure than the MBC.

The modified method of vertical earth pressure on the top slab of the HPBC

In general, the above analysis presented that the vertical earth pressure relationship on the top slab in the span of the HPBC tends to be a linear distribution law, and the vertical earth pressure relationship on the top end of its top slab could have a linear earth pressure of 2 times. It is essentially the height of the backfill that will dominate the magnitude of the load, with the modulus of the foundation and the modulus of the backfill controlling only the distribution pattern. Compared with the monolithic culvert, the vertical earth pressure on the top of the span range of the HPBC was lower than that of the monolithic one under the same backfill height, about 90% of that of the monolithic culvert. In contrast, the earth pressure value on the top slab was relatively higher, about 105% of that of the monolithic culvert. Overall, the size of the vertical earth pressure on the top slab of the HPBC can be calculated by the linear earth pressure equation, and two times the linear earth pressure can calculate the stress at the end. Conventionally, the calculation of the vertical earth pressure concentration coefficient has not taken into account the non-linear nature of the earth pressure distribution on the top slab. Instead, an approximate method has been used, considering the average distribution of the culvert top earth pressure as the design load for the culvert.

However, the actual vertical earth pressure distribution law on the top slab was non-linear due to the difference in stiffness between the culvert structure and the surrounding soil, which contributes to the non-linear load distribution on the top slab. Hence, it is necessary to rectify the calculations in order to ascertain the vertical earth pressure exerted on the top slab, thereby enhancing both the specification and design methodology.

This paper presented a method for calculating the vertical earth pressure correction on the top of a culvert. The method was based on determining the distribution density function φ(x) by analyzing the actual distribution law. The distribution function ψ(x) of the total load on the top of the culvert was obtained by integrating φ(x) along the length of the culvert. Finally, the correction of vertical earth pressure η on the top of the culvert was obtained by dividing ψ(x) by the calculated width. Refer to Fig. 21 for a visual representation. Subsequently, the interconnection between each function was formed by Eqs. (6), (7), and (8).

$$\varphi \left( x \right) = f(x),$$
(6)
$$\psi (x) = \int {\varphi (x)dx} ,$$
(7)
$$\eta = \psi (x)/B^{\prime}.$$
(8)
Figure 21
figure 21

Schematic diagram of the modified calculation method of vertical earth pressure on the top of the HPBC.

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Equations (9), (10), (11), (12), and (13) could be derived by fitting the vertical earth pressure on the top slab under the backfill height of 0.5, 1, 3, 6, and 9 m.

$$\varphi_{0.5} \left( x \right) = 8.80343 + 0.47026x^{2} + 0.19639x^{4} ,\;R^{2} = 0.97894,$$
(9)
$$\varphi_{1} \left( x \right) = 18.12176 - 0.90544x^{2} + 0.78518x^{4} ,\;R^{2} = 0.98704,$$
(10)
$$\varphi_{3} \left( x \right) = 54.74712 - 0.83405x^{2} + 1.728x^{4} ,\;R^{2} = 0.98493,$$
(11)
$$\varphi_{6} \left( x \right) = 112.53305 - 6.95381x^{2} + 5.28547x^{4} ,\;R^{2} = 0.99374,$$
(12)
$$\varphi_{9} \left( x \right) = 165.63487 - 9.55981x^{2} + 7.61483x^{4} ,\;R^{2} = 0.99497.$$
(13)

Integrating Eq. (9) leads to Eq. (14). Equations (15), (16), (17), and (18) can also be obtained by integration.

$$\psi_{0.5} (B^{\prime}) = \int\limits_{{ - 0.5B^{\prime}}}^{{0.5B^{\prime}}} {\varphi_{0.5} (x)} dx = (8.80343x + 0.15675x^{3} + 0.039278x^{5} )|_{{ - 0.5B^{\prime}}}^{{0.5B^{\prime}}} = 49.37\;{\text{kN}} ,$$
(14)
$$\psi_{1} (B^{\prime}) = \int\limits_{{ - 0.5B^{\prime}}}^{{0.5B^{\prime}}} {\varphi_{1} (x)} dx = (18.12176x + 0.30181x^{3} + 0.15704x^{5} )|_{{ - 0.5B^{\prime}}}^{{0.5B^{\prime}}} = 110.92\;{\text{kN}} ,$$
(15)
$$\psi_{3} (B^{\prime}) = \int\limits_{{ - 0.5B^{\prime}}}^{{0.5B^{\prime}}} {\varphi_{3} (x)} dx = (54.74712x - 0.27802x^{3} + 0.34560x^{5} )|_{{ - 0.5B^{\prime}}}^{{0.5B^{\prime}}} = 289.56\;{\text{kN}} ,$$
(16)
$$\psi_{6} (B^{\prime}) = \int\limits_{{ - 0.5B^{\prime}}}^{{0.5B^{\prime}}} {\varphi_{6} (x)} dx = (112.53305x - 2.31794x^{3} + 1.05709x^{5} )|_{{ - 0.5B^{\prime}}}^{{0.5B^{\prime}}} = 597.32\;{\text{kN}} ,$$
(17)
$$\psi_{9} (B^{\prime}) = \int\limits_{{ - 0.5B^{\prime}}}^{{0.5B^{\prime}}} {\varphi_{9} (x)} dx = (165.63487x - 3.18657x^{3} + 1.52297x^{5} )|_{{ - 0.5B^{\prime}}}^{{0.5B^{\prime}}} = 880.43\;{\text{kN}} .$$
(18)

Thus, the modified vertical earth pressure on the top slab can be determined by considering different backfill heights of 0.5, 1, 3, 6, and 9 m, as well as the corresponding vertical earth pressure concentration coefficient on the top slab (Ks). Comparisons were made with AASHTO culvert design specifications and Chinese culvert design specifications to further characterize the revised calculation method.

The vertical earth pressure concentration coefficient of the AASHTO specification13 increases as the backfill height rises, as shown in Eq. (19).

$$F_{\text{e}} = 1 + 0.2\frac{H}{{B_{c} }},$$
(19)

where H = height of the backfill above the top slab; Bc = width of the section of the culvert.

The Chinese culvert design specification15 considers the influence of different culvert types, valley widths, valley slopes and backfill heights on the vertical earth pressure concentration coefficient on the top of the culvert, and provides the method of taking the value, and the box culvert takes the value as shown in Table 4.

Table 4 Vertical earth pressure concentration coefficient of box culvert of specifications for design of highway culverts of China.
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In Table 5, D is the outer width of the culvert (m), Bg is the width of the valley, α is the slope of the valley, and H is the height of the fill on the top of the culvert (m).

Table 5 Summary of vertical earth pressure values on the top slab obtained by the modified method.
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Table 5 and Fig. 22 show a comparison of the results obtained by the modified calculation method in this paper for representative working conditions with normative values.

Figure 22
figure 22

Comparison of the calculated vertical earth pressure value on the top of the HPBC with the design specification.

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Based on Table 5 and Fig. 22, the specification requirements were higher than the numerical simulation value, indicating that the specification value is relatively conservative. In the AASHTO method, the vertical earth pressure concentration coefficient on the top of culverts increases linearly with the backfill height and could not take into account the effects of differential settlement on the earth pressure on the top of high-fill culverts, sufficiently. Chinese culvert design specifications consider the non-linear variation of the vertical earth pressure concentration coefficient on the top of the culvert, whereas low-fill culverts tend to be more conservative. The numerical simulation results were lower than the specification values because the load structure design method had a higher safety factor. Compared with the culvert specifications, the modified method proposed in this paper verified the reasonableness of the specification value. When the backfill height was relatively low, and the correction formula calculation value was greater than the specification value. However, after the backfill height was higher than 4 m, the specification value was more than the correction formula calculation value.

As a result, correcting the vertical earth pressure on the top of the HPBC required consideration of nonlinear distribution characteristics. It is suggested that in the design of prefabricated box culvert, the value of vertical earth pressure concentration coefficient Ks on the top slab can be taken concerning the calculation method and specification in this paper. In the actual project, the design of the Ks should be higher than the modified value in this paper due to the construction and various impact factors.

Sensitivity analyses of key parameters of theHPBC

Orthogonal array experimental design

The pressure of the culvert is related to the external load and the interaction between the culvert and the soil. The backfill height directly determined the amount of load, and the stiffness of the foundation and the backfill further affected the distribution law of the vertical soil pressure on the top of the culvert. Combined with the orthogonal array test principle, this paper selected the backfill height, foundation modulus, and backfill modulus as the factors, respectively, for each factor to set five levels, as shown in Table 6.

Table 6 Levels of orthogonal factors.
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Sensitivity analysis results

In accordance with the values of each factor at varying levels, as presented in Table 6, a three-factor, five-level orthogonal test was conducted, and the resulting orthogonal table is displayed in Table 7. In this table, factors A, B, and C represent the factors above, respectively. Each influencing factor is respectively combined with working conditions according to the levels in Table 7, with the values of the corresponding levels selected from Table 6. A total of 25 distinct combinations of conditions were established using finite element models, with the vertical earth pressure concentration coefficient on the top slab (Ks) calculated for each. The results of these calculations are presented in Table 7.

Table 7 Calculated results of vertical earth pressure concentration factor on the top of the HPBC.
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Table 8 presents the mean values of the earth pressure coefficients at the culvert top at varying levels for each factor, along with the maximum deviation (D) of the mean values of the earth pressure coefficients at the culvert top at different levels for each factor.

Table 8 Calculation of the value of each level and the extreme difference R.
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In accordance with the tenets of extreme difference analysis, it can be observed that the magnitude of the extreme difference is directly proportional to the influence exerted by the factors in question on the soil pressure coefficient at the culvert’s apex. As evidenced in Table 8, the extreme difference corresponding to the backfill height was 0.246, the extreme difference corresponding to the foundation modulus was 0.045, and the extreme difference corresponding to the backfill modulus was 0.086. The order of influence was as follows: backfill height > backfill modulus > foundation modulus. The backfill height directly defined the load magnitude, and thus, it was inevitable that it exhibited the highest sensitivity. With regard to the backfill and foundation modulus, the two factors in question were the changes in the interaction relationship caused by the difference in the modulus of the culvert and the surrounding soil mass. The effect of the backfill modulus directly influenced the settlement difference between the interior and exterior soil columns. An increase in the backfill modulus results in a reduction in the settlement between the interior and exterior soil columns, thereby mitigating the concentrated effect of the culvert earth pressure. The objective of the culvert foundation design is to determine the foundation bearing capacity and, consequently, the foundation treatment based on the design conditions. An increase in foundation modulus will result in a reduction in foundation settlement, which will indirectly lead to a reduction in the settlement of the interior soil column. This will have the effect of increasing the upper soil pressure exerted by the culvert. In the context of the actual project, it is important to ensure that the foundation stiffness of the culvert is maintained to a reasonable degree, taking into account the bearing capacity of the foundation. With regard to the backfill modulus, it is essential to ensure that it is fully compacted in order to minimize the later post-work settlement, which will, in turn, reduce the earth pressure concentration effect of the culvert. The backfill height is associated with design factors. Culverts with large backfill heights can benefit from flexible materials or low-density backfill in order to reduce the impact of the upper loading.

Conclusion

  1. (1)

    The distribution pattern of vertical earth pressure on the top slab of HPBC was found to be significantly affected by the backfill height through field testing. When the filling height was 0.3 m, the pressure value was observed to be uniformly distributed. However, when the filling height was 0.9 m and 2.0 m, the pressure exhibited a non-linear distribution. As the backfill height increased from 0.3 to 2.0 m, the vertical earth pressure value in the culvert top span became linearly proportional to the backfill height. Meanwhile, the earth pressure on the top slab increased non-linearly, and the changing trend with the backfill height of the culvert top was consistent with a calculated value of 2.1 times the linear earth pressure.

  2. (2)

    According to the vertical earth pressure distribution law of HPBC, the earth pressure value at both ends of the culvert top increased significantly while the earth pressure value in the span remained relatively low. This was due to the overlying backfill load of the culvert which produced vertical deformation on the top slab in the span under the effect of weight. As the weight of the soil column in the span accumulated gradually to the soil column on the end of the top slab through the action of frictional resistance between the soil columns, the earth pressure values at the two ends of the top slab were larger. Similarly, the culvert structure acted as a foundation, resulting in smaller settlement values for the culvert top compared to the backfill on both sides.

  3. (3)

    Numerical simulation shows that the vertical earth pressure value of the top slab of HPBC is not only affected by the settlement difference between the interior and exterior soil columns, but also the vertical deformation of its top slab will further affect the load distribution between the interior soil columns. The earth pressure distribution around the culvert of the HPBC was different from that of the monolithic culvert. Specifically, the vertical earth pressure on the top of the span of the HPBC was smaller, approximately 93.7% of the monolithic culvert. Conversely, the earth pressure on the top slab was relatively larger, approximately 105.2% of the monolithic culvert. This can be attributed to the comparatively minor deformation of the top slab of the monolithic box culvert in comparison to the HPBC, which was responsible for redistributing the earth pressure through differential settlement within the interior soil column.

  4. (4)

    This paper proposed the vertical earth pressure correction method for the HPBC top, which verified the reasonableness of the specification taking value. Additionally, for backfill heights less than 9 m, the vertical earth pressure concentration coefficient of the culvert top of a monolithic box culvert could be adopted for the HPBC. Therefore, it is suggested that the value of the vertical earth pressure concentration coefficient Ks on the top of the culvert be taken with reference to the calculation method and specification in this paper for the design of prefabricated box culvert. It will enable a more reasonable determination of the load capacity.

  5. (5)

    The sensitivity of backfill height, foundation modulus, and backfill modulus to the vertical earth pressure concentration coefficient on the top slab (Ks) was analysed by orthogonal array tests. The results demonstrated that backfill height was the most significant factor, followed by backfill modulus and foundation modulus.