1 Introduction

For many years, buried corrugated steel plate (CSP) bridges and culverts have been widely used in transport infrastructure (tunnels, culverts, underpasses, animal crossings, footbridges etc.). An increase in interest in this type of structure can be observed since the 1970s. The growing popularity of this type of buried structure is mainly due to the simplicity of execution and relatively low costs compared to reinforced concrete (RC) bridges and culverts. Recently, using CSP elements for constructing bridges and culverts allows for structures with a span of even more than 30 m (Embaby et al. 2022a).

An essential element in the design and construction of buried structures is the appropriate height of the cover in the crown of these objects (Pettersson and Sundquist 2014). This is important for two reasons: firstly, to reduce traffic impacts on the steel shell/pipe, and secondly, to allow positive arching phenomenon to occur (Beben 2020). Bridge standards, i.e. AASHTO LFRD (AASHTO 2017), CHBDC (CHBDC 2014) and design recommendations (Rowińska et al. 2004), help to predict the appropriate minimum depth of soil cover in buried structures. However, in many cases of buried bridges and culverts, the minimum depth of soil cover in the crown cannot be applied due to the connection with the designed road/railway gradeline and/or hydraulic calculations resulting in a larger cross-section of the CSP shell (and at the same time a smaller depth of the soil cover). Therefore, additional elements should be used in such cases to help transfer live loads. These elements can be divided into two groups. The first group includes elements reinforcing the steel shell/pipe (not analysed in this study), while the second contains elements reinforcing the backfill.

Embaby et al. (Embaby et al. 2022a, 2022b) analysed the use of steel circumferential angled ribs and RC collars, respectively, while Maleska and Beben (Maleska and Beben 2019) and Maleska et al. (Maleska et al. 2021) evaluated the use of additional CSP ribs (including concrete-filled ribs) and RC longitudinal beams to reinforce the shell. Sun et al. (Sun et al. 2023) analysed the CSP bridge with protection concrete to improve mechanical performance. It was found that using the elements mentioned above reduced the induced straining actions in the steel shell. However, such reinforcements require quite complicated construction work, which reduces their attractiveness and increases construction costs.

The best way to increase the load capacity of buried bridges and culverts/pipes is to strengthen the backfill. For this purpose, RC slab, reinforced soil with cement or recycled tire chips, geosynthetics, geocell, geogrid, steel mesh reinforcement, steel flat bars and expanded polystyrene (EPS) are used.

Duncan et al. (Duncan et al. 1985) analysed the effect of using the RC relieving slab on the deformations of the CSP shell. The authors found that bending moments are reduced by 40% when using the RC slab. The experimental and finite element results of soil-steel bridges with relieving slab under static loading were presented by Beben and Stryczek (Beben and Stryczek 2016).

Pearson and Milligan (Pearson and Milligan 1991) conducted laboratory tests of the CSP culvert model with strengthened soil using steel flat bars. Tests were conducted for different backfill depths and different reinforcement locations. As a result of using this reinforcement, the deformation of the culvert was reduced by approx. 50%. Embaby et al. (Embaby et al. 2022a, 2022b) evaluated the performance of large-span soil-steel bridge with steel mesh reinforcements (to strengthen the backfill).

Bathurst and Knight (Bathurst and Knight 1998) used a cellular polymer material to strengthen the soil cover above the CSP culvert. They noticed that the reinforced soil cover reduces construction costs and significantly relieves the CSP culvert. Besides, Elshesheny et al. (Elshesheny et al. 2019), Fattah et al. (Fattah et al. 2018) and Kou et al. (Kou et al. 2018) conducted small-scale laboratory tests on flexible buried pipes with backfill reinforced with geogrid, geocell and geotextile, respectively. Wysokowski (Wysokowski 2021) evaluated changes in the load-bearing capacity of buried flexible structures using geotextiles to reinforce backfill. The results showed that the vertical displacement of the culvert crown was significantly reduced after using the geotextile.

Expanded polystyrene (EPS) geofoam is commonly used as lightweight fill materials in various geotechnical engineering applications, including protecting buried small-scale culverts and pipelines (Abdollahi et al. 2021; Bartlett et al. 2015; Vaslestad and Sayd 2019). Ma et al. (Ma et al. 2019) conducted a laboratory test on small-scale rigid and flexible pipes using EPS to reduce the earth pressure and the displacements of the backfill around the pipe. The test results show that the earth pressure is reduced with EPS laying on the crown of the pipe, and the load-reduction efficiency is increased to be constant with increasing EPS thickness. Santos et al. (Santos et al. 2020) investigated the effects of trench installation methods with EPS geofoam on the behaviour of buried corrugated steel arch structure. Three configurations were investigated without EPS geofoam, imperfect trench installation and embedded trench installation. The FEM results showed that embedded trench installation models reduced the earth pressure, deflection and stress at the crown by 68%, 40% and 39% on average. Meguid et al. (Meguid et al. 2017) conducted a numerical and experimental analysis of the interaction between backfill-EPS-steel culvert. They proved that increasing EPS’s density reduces the steel culvert’s stress concentration. Maleska et al. (Maleska et al. 2019) conducted a finite element analysis of the behaviour of a CSP bridge with EPS blocks below a foundation subjected to seismic excitation and rockburst. The results showed the benefits of using EPS blocks in reducing internal forces in the steel shell and reducing backfill acceleration.

Considering the applications of EPS in buried culverts, it should be noted that so far, it has been used for small cross-section culverts/pipes. The analysed buried bridge does not meet the requirements (AASHTO LFRD, CHBDC, Polish recommendations) regarding the mini-mum cover depth in the crown. Therefore, this article presents a finite element analysis and experimental study using an RC relieving slab and, for comparison purposes, different types of EPS geofoam for a shallow-buried CSP bridge with a span of 10 m under static live load. Three static live-load schemes with a total weight of over 500 kN each were used. Five FEM models of the buried bridge were adopted for the analysis: (i) model I (without RC relieving slab), (ii) model II (with RC slab with a thickness of 0.20 m), (iii) model III with the use of expanded polystyrene EPS15, (iv) model IV–EPS29 and (v) model V–EPS46. Models III–V differ in the type of EPS that was applied instead of the RC slab. The finite element method results were compared with the experimental ones of this bridge. The experimental and numerical results were also confronted with the permissible displacement levels provided for in the standards and recommendations for this type of bridge.

2 Short Description of Buried CSP Bridge

The analysed shallow-buried bridge in cross-section has an arch structure with a span of 10 m and a height of 4.02 m (Fig. 1). The shell consists of CSP elements of 0.003 m thick, and corrugation 0.05 × 0.15 m, joined together with the high-strength bolts. The steel arch structure is supported on two RC foundations made of grade C25/30 concrete. The bridge is situated at a skew of 58.81° relative to the road’s axis. A detailed description of the analysed bridge is presented in Beben and Stryczek (2016).

Fig. 1
figure 1

Cross section of shallow-buried CSP bridge

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During the construction phase of this bridge, it was necessary to significantly reduce the planned soil cover thickness above the steel shell from 1.25 to 0.65 m. This was caused by the surveyor’s incorrect marking out of the bridge foundations. Thus, by the applicable bridge standards for the design of buried CSP bridges, the analysed bridge needs to meet the minimum depth of soil cover requirements. According to AASHTO LRFD (AASHTO 2017), the minimum soil cover for the analysed buried bridge should be 1.25 m, and CHBDC (CHBDC 2014) suggests that this cover should be at least 2.62 m. However, Polish design recommendations (Rowińska et al. 2004) say that the depth of the soil cover in buried bridges should be at least 1.66 m. Such a change in the depth of the soil cover forced the use of some element that would relieve the steel shell from the live loads. For this purpose, a 0.20-m-thick RC relieving slab made of grade C16/20 concrete was used, which was placed 0.10 m above the shell crown. The RC slab extends by about 1.50 m beyond the outline of the cross-section on both sides of the arch structure (Fig. 1), and in the longitudinal section, it covers the entire width of the roadway with the sidewalk. During the numerical analysis, various types of EPS geofoam were used instead of the RC slab.

3 Experimental Testing of the Buried CSP Bridge

During field tests, the shallow-buried CSP bridge with the RC relieving slab was loaded with two heavy trucks weighing over 500 kN. The loads on individual vehicle axles are shown in Fig. 2a. Field tests were planned in three static load scenarios (Fig. 2), i.e. different positions of vehicles on the bridge. The vehicles were positioned on the bridge in such a way as to cause the most significant possible impact, i.e. scheme A (unsymmetrical load—Fig. 3) and schemes B and C (symmetrical load to the longitudinal axis of the bridge). Such recommendations come from the bridge standard (PN-89/S-10050 1989), which requires adopting vehicle load systems that are most unfavourable for the bridge. A detailed description of static load schemes can be found in the paper (Beben and Stryczek 2016). The results of the experimental tests of the buried CSP bridge with the RC slab will be compared with the finite element analysis results and presented in the discussion of the results (Section 6).

Fig. 2
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Load schemes used in the buried CSP bridge: a longitudinal section, b cross-section

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Fig. 3
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Side view of the tested buried CSP bridge according to load scheme A

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4 Finite Element Model Description

The numerical simulations of shallow-buried CSP bridge were made using the finite element method (FEM) in the DIANA FEA program. A FEM analysis was planned for five buried bridge models:

  • I (without the RC relieving slab),

  • II with the RC slab,

  • III with the use of expanded polystyrene EPS 15,

  • IV with the use of expanded polystyrene EPS 29 and

  • V with the use of expanded polystyrene EPS 46.

The shallow-buried CSP bridge models were modelled as a 3D object (Fig. 4) with the following material parameters:

Fig. 4
figure 4

FEM model of the buried bridge made in the DIANA FEA program: a with dimensions and boundary conditions, b with density of mesh of finite elements

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  • The corrugated steel plates were modelled as curved shell elements (T15SH) using an elastic-plastic model with Young’s modulus of 210 GPa, Poisson ratio of 0.3, yield strength of steel of 235 MPa, density of 7850 kg/m3, a moment of inertia of 1145 mm4/mm, the plate thickness of 0.003 m and cross-section area of 3.76 mm2/mm,

  • The backfill was modelled as an elastic–plastic material (TE12L–solid elements) with the Mohr–Coulomb yield criterion with Young’s modulus of 100 MPa, density of 2050 kg/m3, dilation angle of 5°, angle of internal friction of 39°, cohesion of 3 kPa and initial tension equal to 0 Pa. The soil cover depth over the CSP shell was 0.65 m,

  • The RC slab with a thickness of 0.20 m was modelled as the Total Strain crack model (solid elements–T18IF) with a Young modulus of 2.8 GPa, Poisson ratio of 0.2, density of 24,000 kg/m3 and mean compressive strength of 24 MPa. The RC slab was placed 0.10 m over the CSP shell,

  • The EPS (models III–V) was modelled as a linear elastic model using solid elements (T18IF) with a Poisson ratio of 0.09 and parameters presented in Table 1. The thickness of the EPS blocks was 0.20 m and was located in the place of the RC slab,

    Table 1 The EPS parameters used for numerical analysis
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  • The interaction between the backfill and the steel shell has been considered by applying an automatic interface with the “Coulomb friction” function. In this function, the angle of internal friction of 39°, the dilation angle of 5°, the stiffness of 2000 kN/m3 and the cohesion of 3 kPa were assumed. The interaction between backfill and RC slab/EPS geofoam was modelled similarly to that between backfill and CSP shell. The difference was only in the stiffness interface (250 kN/m3).

  • The boundary conditions were modelled as pinned supports for the shallow-buried bridge model (backfill) for x, y and z directions (Fig. 4a),

  • The finite elements were automatically divided so that their maximum dimensions did not exceed 1.0 m for backfill, RC slab, EPS geofoam and 0.5 m for CSP shell. In summary, the number of finite elements was equal to 766 960. In the contact zones of the CSP shell, EPS geofoam, RC slab with backfill, or in places where maximum values were expected, the mesh of finite elements was denser (Fig. 4b). In the FEM analysis, attempts were made to eliminate the influence of the finite element mesh density on the obtained results. For this purpose, the automatic function of finite element mesh refinement was used. As a result, the smallest possible elements were obtained, eliminating the effect of finite element density (effect of too large size of finite elements).

5 FEM Results

5.1 Buried CSP Bridge With/Without RC Slab

To determine the impact of the RC slab on the level of displacements and stresses in the steel shell of the shallow-buried CSP bridge, it was decided to perform numerical simulations for the bridge models without and with the RC slab (models I and II). In addition, one should ask whether the permissible displacements for this type of underground bridge by applicable regulations are not exceeded if the RC slab is not used.

5.1.1 Vertical Displacements

The analysis of the impact of the RC relieving slab on the shallow-buried CSP bridge was carried out for three different settings of the loading trucks by the adopted program during the experimental tests (Fig. 2).

Analysing the obtained values of bridge displacements based on models I (without the RC slab) and II (with the RC slab), it can be seen that the largest displacements were obtained in model I. This is quite obvious as the live loads from the trucks directly affected the soil layers above the bridge shell. In this model, the largest displacements of the bridge were obtained for the arrangement of trucks by scheme B. They amounted to − 5.52 × 10–4 m (Fig. 5b). On the other hand, for model II with the RC relieving slab, the maximum displacements were equal to − 3.30 × 10–4 m (Fig. 5f) and were obtained for scheme C. For both models, the maximum recorded values were located near the shell crown, close to the pressure of the truck wheels. These places are best visible in model I, where the relieving slab was not used (Fig. 5a–c). It should be added that the differences in the maximum displacements from the individual load schemes (A, B and C) for the respective models I and II are very small (do not exceed 10%).

Fig. 5
figure 5

Top view of maps of maximum displacements of model #I of CSP bridge (a–c) load schemes A, B and C, respectively, and model #II (d–f) load schemes A, B and C, respectively

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In addition, it can be seen that the use of the RC relieving slab in the shallow-buried CSP bridge not only reduces the maximum displacements (in the range of 39–41%—Table 2) but also has a significant impact on the distribution of displacements in the steel shell (cf. Figure 5a–b and 5d–f). Figure 5 shows that the distribution of displacements in model II occurs over a larger area than it can be seen in model I. This should be seen as the main benefit of using the RC relieving slab in a buried CSP bridge because the displacement values, despite significant reductions resulting from using the RC slab, are generally low. In addition, it can be assumed that using the RC slab may positively affect the durability of such a bridge, as it minimises the direct impact of heavy vehicle traffic on the CSP shell of the buried bridge. This is an essential statement because these bridges in Poland are designed for 100 years, just like typical bridges.

Table 2 Comparison of displacements and stresses in CSP shell obtained from FEM models I and II
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5.1.2 Stresses

The stresses in models I and II of the buried CSP bridge are generally low due to the three different loading schemes. It can be seen that the maximum values occurred near the quarter points (above the foundations). The highest stresses in model I were − 21.3 MPa (Fig. 6b) and in model II − 18.4 MPa (Fig. 6e). These were compressive stresses in both cases. The applied loading schemes A, B and C caused similar stresses in the steel shell (differences of 0.1–0.3 MPa) (Table 2). Thus, using the RC relieving slab reduced the maximum stresses in the shell of the buried bridge in the range of about 14% (Table 2). It was also noticed that the distribution of maximum stresses at quarter points due to using the RC slab occurs over a larger area (length) than in the case of the model without the relieving slab (Fig. 6).

Fig. 6
figure 6

Maximum stresses of CSP bridge model #I (a–c) load schemes A, B and C, respectively, and for model #II (d–f) load schemes A, B and C

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The stresses in the crown area of the shell were also analysed to relate the results obtained from the numerical analysis to the experimental results discussed in detail in the next section of this work. In the vicinity of the bridge crown, the highest stresses were − 11.1 MPa for model I and − 7.58 MPa for model II. Thus, the maximum stresses in the crown area of the steel shell were much lower than in the vicinity of the quarter points of the shell by 48% and 59% for models I and II, respectively.

5.2 Buried CSP Bridge with EPS Geofoam

An alternative to the heavy and rigid RC slab relieving the shallow-buried CSP bridge may be using lighter material in the form of EPS geofoam. This solution is definitely cheaper and faster to implement. Three different types of EPS (different in density and Young’s modulus) were used to assess the effect of the EPS layer on displacements and stresses in the buried CSP bridge. The EPS layer in the III–V models was used in the same place as the RC slab and with the same thickness, i.e. 20 cm.

5.2.1 Vertical Displacements

The numerical analysis shows that the impact of the EPS geofoam on the displacement of the buried CSP bridge is comparable to using a heavy RC slab (model II). The use of different types of EPS (models III–V) to relieve the buried bridge structure resulted in maximum displacements oscillating around − 3.25 × 10−4 m for all EPS models (Fig. 7). The differences in maximum displacements between the III and V models were minimal and did not exceed 1%. However, they were smaller by about 2% compared to model II (− 3.30 × 10−4 m) and about 41% compared to model I (− 5.52 × 10−4 m—Table 3).

Fig. 7
figure 7

Top view of maps of maximum displacements of CSP bridge model #V (a–c) load schemes A, B and C, respectively

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Table 3 Comparison of displacements and stresses in CSP shell obtained from FEM calculations for model I (without EPS) and models with EPS (III, IV and V)
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The above shows that, despite using EPS with different parameters (density and Young’s modulus), it did not significantly reduce/increase the displacement of the buried CSP bridge. This trend differs from using EPS geofoam to mitigate seismic impacts in buried CSP bridges/culverts (Maleska et al. 2019). In this case, the EPS layer acted as a damper, absorbing the seismic waves. The use of EPS with a lower density resulted in a reduction of maximum displacements in the CSP shell.

The maximum displacements in models III–V were located near the crown of the steel shell, similar to model II, where the RC slab was used. The maximum displacements are distributed over a larger area of the shell than in the case of model I and in a very similar way as in model II (with the RC slab). Thus, using different types of EPS (models III–V) does not significantly affect the redistribution of displacements in the shallow-buried CSP bridge compared to model II (with RC slab).

5.2.2 Stresses

The highest stress values were recorded in the V model (EPS 46) and were equal to − 17.3 MPa. They were obtained in the vicinity of the quarter points of the shell as a result of the load set by the load scheme B (Fig. 8). For models III and IV, the stresses were very similar to model V (Table 3), and the difference between them did not exceed 1%. EPS reduced stresses in the shell of the buried CSP bridge by about 19% (Table 3). Moreover, it can be seen that EPS, regardless of type, has a more favourable effect on the stress level in the buried CSP bridge than in the case of using the RC slab (cf. Tables 2 and 3). It was noticed that the reduction of maximum stresses in models III–V is greater by ca. 5% than those obtained when using the model with the RC slab (#II).

Fig. 8
figure 8

Maximum stresses of model #V (a–c) load schemes A, B and C, respectively

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It is also worth mentioning the stresses in the crown of the bridge shell, where the maximum displacements were recorded and strain measurements were made. It was noticed that the highest stresses in the crown in the analysed models III–V are at a similar level, i.e. ca. − 7.20 MPa. However, compared to model II (with the RC plate), the stresses in the crown were lower by about 5%. The stress distribution as a result of the use of EPS (models III–V) occurs over a larger area (length) in the vicinity of the quarter points than it was observed in model I and is comparable to that in model II, in which RC slab was used (cf. Figures 6 and 8). It follows from the above that using EPS results in comparable stress reductions in the shell of the buried CSP bridge as the RC slab.

Determining how using an EPS/RC slab affects the concentration and redistribution of stresses in the backfill around the relieving elements is also interesting. For this purpose, the distribution of stresses at the border of the backfill and the relieving layers in the considered models I–V was analysed. Examples of stress distributions in the backfill from load scheme C are shown in Fig. 9. It was noticed that due to the use of relieving elements, the stresses in the backfill concentrated above these elements (models II–V) and spread beyond the outline of the CSP shell. Thereby minimising the impact of the live load on the shell of the shallow-buried CSP bridge.

Fig. 9
figure 9

Stress concentration in backfill in the CSP bridge for model a I (without relieving slab), b II (RC slab), c III (EPS 15), d IV (EPS 29) and e V (EPS 46)

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Comparing the way of stress distribution around the relieving elements (models II–V), one can see a significant similarity in the redistribution of stresses in model II (with RC slab) and models IV and V (EPS 29 and EPS 46), please see Fig. 9b, d, e. It follows that EPS 29 and EPS 46, i.e. with the higher density (and smaller pores) and Young’s modulus, behave like a rigid slab. It was also noticed that the RC slab experiences greater deformations than the EPS layer. This may indicate that RC slab, as a stiff element, cooperates less with the backfill than EPS, which is more flexible to loads and thus experiences smaller deformations.

A different behaviour was noticed in model III, which uses a soft EPS15 (larger pores). In this case, the stress distribution in the backfill, to a greater extent, extends beyond the range of the EPS blocks and partially reduces the pressure on the relieving layer. This proves that EPS absorbs part of the pressure from live loads. An increase in stresses was also noticed in the subsurface layers near the end of the EPS layer for model III (Fig. 9c). This statement is of great practical importance because EPS application is definitely easier and cheaper than RC slab. However, in model I (without relieving elements), the load distribution in the backfill is obviously more even. The largest ones occurred close to the steel shell, just above the support (Fig. 9a). There are no increased stress concentrations in the backfill, but they run from the quarter points to the road surface and on both sides of the shell. Additionally, the arching phenomenon, i.e. redistribution of loads in the backfill, is clearly visible, which is disturbed in models with relieving elements (Fig. 9b–e). On the other hand, in this case, the CSP shell is exposed to the direct impacts of service loads, which may affect its durability due to the 100 years of use of bridge structures in Poland.

6 Discussion of the FEM Simulations and Experimental Results

To verify the correctness of numerical calculations from the DIANA FEA program, the results obtained from the numerical model II (with RC slab) were compared with the results of experimental testing (Beben and Stryczek 2016). The same arrangement of load schemes (location of vehicles on the bridge) was used for experimental and numerical analysis. Additionally, the FEM results were read from the numerical model from the same places where the inductive and strain gauges were installed (during experiments). The sensors were located at the shell crown of the buried CSP bridge.

Based on the results presented in Fig. 10, it can be seen that the displacements of the shallow-buried CSP bridge shell obtained from experimental tests and numerical simulations with relieving layers (RC slab and EPS geofoam) were similar to each other. Significant differences were visible when comparing the results of models II–V with model I (without the relieving layer), proving the considerable impact of the relieving layers on the CSP bridge’s displacement level. However, the trends in the course of displacements were similar in the experiment and numerical simulations. The most significant differences in the course of displacements were observed in the load scheme B (symmetric), which may result from the initial consolidation of the soil layers under the load and the sensitivity of the CSP shell to the applied live load. In addition, it can be seen that for the load schemes (A and C), the course of the experimental and computational displacements is close to each other. This proves the high sensitivity of the bridge to the load (especially the impact of the first load—scheme A) and the internal stabilisation of the soil medium as a result of subsequent loads (schemes B and C).

Fig. 10
figure 10

Comparison of results from numerical analysis and experimental testing

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It should also be added that the differences in the calculated and measured displacements may be caused by the method of loading the bridge in experimental tests, i.e. the loading patterns used. First, an asymmetrical scheme (A) was used, and the bridge experienced significant displacements directly under the loading vehicles. In the next step, the vehicles were moved to the centre of the bridge (symmetrical scheme B), and the displacements occurring in the shell were balanced, thus spreading the displacements over a larger surface. However, this process is difficult to reproduce in numerical simulations, hence the differences obtained. In the case of load patterns A and B, the front axles of the vehicles are located outside the bridge outline. However, for scheme C (symmetrical), the largest bridge displacements occurred due to the location of all forces coming from individual vehicle axles near the bridge crown. In this case, a relative balance of displacements in the shell was achieved, which resulted in their relatively even distribution over the entire width of the bridge and a reasonably good agreement between calculated and measured values. The largest difference between the maximum displacements measured and calculated using DIANA FEA did not exceed 27% (Table 4).

Table 4 Comparison of maximum displacements and stresses obtained from measurements and FEM calculations for model II
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In the case of stresses analysis, the discrepancies in obtained values did not exceed 59% (Table 4, Fig. 10). The measured values of unit strains of corrugated steel plates (on which base the stresses were calculated) were at a very low level (in the order of reading error (Beben and Stryczek 2016)). Therefore, the stresses derived from this may only be partially reliable. Nevertheless, similar trends in the course of stresses in the CSP shell were observed (Fig. 10).

The obtained differences in the results of FEM calculations and experimental tests may result from the heterogeneity of the backfill used for the construction of the bridge, which is difficult to represent in the numerical model. In addition, some simplifications of the numerical model, e.g. the lack of bridge equipment elements, could also have influenced the discrepancies obtained. FEM simulations were carried out on idealised numerical models that did not take into account some imperfections in execution and resulting, for example, from the consolidation of soil layers.

Another important aspect of the differences obtained between experimental studies and numerical calculations may be including the arching phenomenon in the FEM model. In the case of numerical simulations, the arching phenomenon was modelled using an interface (contact layer), which is a kind of approximation that considers the actual operation of the structure and the interaction between the individual elements of the bridge. It should be added that the arching effect was modelled similarly for all analysed models (I–V). However, in the case of a real bridge, which was subjected to experimental tests, many aspects may influence the arching effect (including sliding force and shear stresses in the vicinity of the shell), as well as the effect of using relieving elements that disrupt the course of redistributive forces in the ground. This isn’t easy to take into account in the FEA model. Additionally, the density of the backfill, which changes with the use of the CSP bridge, has an impact. Also, permanent deformations in the CSP bridge may impact the arching effect, which is difficult to consider in numerical analysis.

In addition, it is essential to determine whether the analysed models of the CSP bridge do not exceed the permissible displacement levels provided for in the standards and recommendations. The maximum design displacement is determined as the allowable deflection of the buried CSP bridges/culverts, which should not exceed the value calculated based on the following conditions, namely:

  1. 1.

    5% of the span, i.e. 500 mm (AASHTO LFRD (AASHTO 2017)),

  2. 2.

    0.5% of the span, i.e. 50 mm (CHBDC (CHBDC 2014)),

  3. 3.

    2% of the span, i.e. 200 mm (CSPI (CSPI 2007)) and

  4. 4.

    10% of the rise, i.e. 402 mm (NCSPA (NCSPA 2018)).

It is clearly visible that the conditions resulting from the NCSPA (NCSPA 2018) and the AASHTO (AASHTO 2017) recommendations are less restrictive than the requirements proposed by CHBDC (CHBDC 2014) and CSPI (CSPI 2007) for the buried CSP bridges/culverts. As shown in the study, the maximum buried CSP bridge displacement (0.552 mm–FEM model #I–without EPS/RC slab) is definitely smaller than the permissible values calculated under the above four conditions. The above conditions are caused by a conservative approach to determining the limit values (not considering the soil-structure interaction phenomenon and increased safety factors).

It should be added that the CEN (CEN 2020) does not propose an estimation of the allowable deflection for buried CSP bridges/culverts. It’s strange because many bridges of this type are being built in Europe, and this issue should be regulated. On the other hand, the application of recommendations provided for, for example, vaulted or masonry bridges does not consider the specific behaviour of buried CSP bridges.

Generally, deflections of buried CSP bridges/culverts are acceptable (even up to 5% of the span), provided the shape change has stopped, the shape is suitable for the intended function and the backfill has become suitably consolidated. However, as practice shows, such large displacements are rare and are caused by factors other than the live load, e.g. incorrect execution, including poor quality of the backfill and its appropriate compaction, incorrect execution of screw joints etc.

7 Conclusions

Taking into account the obtained results of numerical analysis (DIANA FEA) and experimental tests under static live loads, it can be concluded that the use of a relieving layer (RC slab or EPS geofoam) significantly reduces displacements and stresses in the shell of the shallow-buried CSP bridge. In addition, the use of relieving elements results in a more uniform distribution of stresses and displacements in the bridge shell. In addition, the following conclusions can be drawn:

  • The use of EPS geofoam and the RC slab reduced vertical displacements in the range of 40–41% and 39–41%, respectively, in the buried CSP bridge. On the other hand, stresses in the shell of the CSP bridge were reduced by ~ 19% and ~ 14%, thanks to the use of EPS geofoam and RC slab, respectively,

  • The use of different types of EPS (in terms of density and Young’s modulus–FEM models #III–V) did not significantly reduce displacements and stresses in the buried bridge. The differences between the individual models did not exceed 1%,

  • From a practical point of view, using EPS geofoam is more beneficial than RC slab. While maintaining a similar range of deformation reduction in the buried CSP bridge as a result of using both relieving elements, time-consuming and cost-intensive works on the RC slab is avoided,

  • The distribution of stresses in the backfill in the vicinity of the relieving elements is similar, especially for RC slab (model II) and EPS 29 and EPS 46 (models IV and V). This means that high-density EPS can be successfully used instead of RC slab. However, to avoid too high-stress concentrations above the relieving element, it is recommended to use low-density EPS (with larger pores). EPS takes over part of the pressure from live loads,

  • A fairly good convergence of displacements and stresses (even 3% and 20%, respectively) calculated using the FEM method (for model II with the RC slab) and those obtained from experimental tests was observed. The shape of the experimental and calculation curves is similar. This means that the proposed computational model based on FEM is a good tool for estimating the level of effort in buried CSP bridges/culverts,

  • The reduced depth of the soil cover (from 1.25 to 0.65 m) over the buried bridge shell (model I) did not cause displacements and stresses threatening the safety of the bridge. According to various standards and recommendations, the CSP bridge’s allowable displacements have not been exceeded. Therefore, it seems that the relieving elements are not necessary for this bridge even though the condition of minimum soil cover over the shell was not ensured. However, bridge structures, including CSP bridges, are designed in Poland for a period of 100 years of service. It is currently difficult to determine whether the reduced soil cover could impact the bridge's durability in the future.

Further studies on the behaviour of shallow-buried bridges and culverts (smaller than 0.5 m) with the use of relieving or damping elements (geotextiles, geogrids, recycled tires, lightweight backfill, gravel-rubber mix etc.) under the influence of static and dynamic loads are justified. The obtained results may allow for increasing the potential of buried CSP bridges.