Numerical Analysis and Load-Carrying Capacity Estimation of Reinforced Concrete Slab Culvert Rehabilitated with a Grouted Corrugated Steel Plate

Two laboratory tests were conducted to investigate the mechanical properties and develop a load-carrying capacity estimation method for reinforced concrete (RC) slab culverts rehabilitated with a grouted corrugated steel plate (CSP). Subsequently, 216 numerical models of RC slab culverts rehabilitated with different shapes of CSPs and grout strengths were established to investigate the influence of these parameters and the arch effect on the rehabilitated system. A mechanical model was proposed based on the elastic center method, and a load-carrying capacity estimation method of RC slab culverts rehabilitated with grouted CSPs was established and verified. It was concluded that the load-carrying capacity of the rehabilitated system increased with a decrease in the radius of the side walls and crown at a constant radius of the CSP haunch. At a constant radius of the side walls and CSP crown, the load-carrying capacity of the rehabilitated system increased with an increase in the haunch radius. The most effective way to improve the load-carrying capacity of the rehabilitated system was to increase the radius of the haunches and reduce the radius of the arch crown and side wall. The arch effect of the grout was related to the load type. The load-carrying capacity of the rehabilitated system was the highest when the CSP was similar to or the same as the arch axes of the grout. The most important function of the grout was to provide strong lateral restraint for the CSP, reducing the required span and improving the load-carrying capacity of the CSP. In addition, the shear strength of the grout contributed to improving the load-carrying capacity by exerting an arch effect or experiencing shear failure. The proposed load-carrying capacity estimation method is applicable to rehabilitated systems with a box or arch-type CSP. Our findings provide guidance for engineers to design similar rehabilitated systems.


Introduction
A corrugated steel plate (CSP) arch shell is a thin-walled, highly flexible structure that can be shaped into circular, pear-shaped, arch, and box structures (AASHTO, 2010;CHBDC, 2006). The rehabilitation of small bridges and culverts with grouted CSPs is a convenient method that only requires lining existing bridges and culverts with a CSP matching the structure's cross-sectional shape (ASTM, 2013;SnapTite, 2013). It has been increasingly used to repair small bridges and culverts that are difficult to dismantle or where traffic cannot be interrupted. A grouted CSP can improve the load-carrying capacity of old small bridges and culverts without requiring demolition or traffic detours while reducing environmental impacts and engineering costs (Chen, 2016;Shang, 2017;Wang, 2012). However, this technology poses new challenges to structural analysis (Li et al., 2021). Many factors affect the mechanical properties of rehabilitated systems, such as the structural form, the size of small bridges and culverts, the strength and thickness of the grout, and the shape of the CSP (Garcia & Moore, 2015;Kovalchuk et al., 2021;Li et al., 2020aLi et al., , 2020bLi et al., , 2020cSmith et al., 2015;Tetreault et al., 2018Tetreault et al., , 2020. The structural form of small bridges and culverts differs significantly from that of pipes; thus, the mechanical properties of small bridges 1 3 and culverts rehabilitated with grouted CSP will likely differ from those of slip-lined pipes. The above conclusions only apply to small bridges and culverts rehabilitated with grouted CSPs but cannot guide their design or internal force analysis. If the mechanical properties and load-carrying capacity calculation methods of small bridges and culverts rehabilitated with grouted CSP are not sufficiently understood, over-rehabilitation or under-rehabilitation may occur in engineering applications. Assessments of the rehabilitation performance of using CSPs for small bridges and culverts in practical engineering have shown that CSPs improve the load-carrying capacity of old small bridges and culverts (He et al., 2019;Vaslestad et al., 2002Vaslestad et al., , 2004. However, there is no load-carrying capacity estimation method of small bridges and culverts rehabilitated with grouted CSP up to now, which limits the application of CSP in rehabilitation engineering. Small bridges and culverts have many structural forms, making it impossible to analyze all. Therefore, a reinforced concrete (RC) slab culvert rehabilitated with a grouted CSP was investigated. The RC slab, grout, and CSP are in contact and exhibit slip (McAlpine, 2006;Zhao & Daigle, 2001), and grout may or may not create an arch effect.
Thus, the material nonlinearity of concrete and the contact nonlinearity between different material layers were considered in the numerical analysis. The influence of the CSP shape and grout strength on the load-carrying capacity of the RC slab culvert rehabilitated with a grouted CSP is investigated, and whether an arch effect can occure in the grout or not and its determination method are also investigated. Furthermore, the load-carrying capacity estimation method is proposed. The research results will provide reference and guidance for engineers to design such rehabilitation engineering.

Laboratory Test
Laboratory tests were performed to verify the numerical results. The dimensions and parameters of the test specimen and the schematic diagram of the loading frame are shown in Fig. 1. The strength grade of the RC slab and integral foundation was C40, with a compressive strength of 40.375 MPa and an elastic modulus of 32.5 GPa. The strength grade of the steel bars was HRB400m with a minimum yield strength of 400 MPa, a tensile strength of 575 MPa, and an elastic modulus of 210 GPa.
The corrugation amplitude of the CSP was 55 mm with a period of 200 mm. The CSP had a wall thickness of 3 mm. The type of CSPs was Q235. It had a minimum yield strength of 235 MPa, a minimum tensile strength of 370 MPa, and an elastic modulus of 210 GPa. C20 concrete was used as the grout; its thickness at the base and the crown was 95 mm. The C20 concrete had a compressive strength of 21 ± 2.00 MPa and an elastic modulus of 25.5 GPa.
Two midspan single-point loading experiments of the RC slabs rehabilitated with arch-and box-type CSPs were conducted, and the contact area between the loading device and the RC slab is 150 mm × 500 mm. String potentiometers with an accuracy of 0.1 mm and strain gauges with a resistance of 120 Ω ± 0.3% and a gauge factor of 2.11 ± 1% were used to monitor the displacements and strains of the CSP's crown and haunches.
The purpose of these two experiments was to verify the accuracy of the numerical models. Subsequently, numerical models were established to investigate the influence of the CSP shape and grout strength on the load-carrying capacity of the rehabilitated system. Finally, a load-carrying capacity estimation method was proposed based on the numerical results.

Stress-Strain Relationship of Concrete
The stress-strain relationship recommended by Ref. (Ministry of Housing & Urban-Rural Development, 2011) was adopted for the numerical analysis. It is expressed as follows: Uniaxial tensile stress-strain relationship: Uniaxial compressive stress-strain relationship: where α t is a parameter of the descending section of the uniaxial tensile stress-strain curve of concrete; α c is a parameter of the descending section of the uniaxial compressive stress-strain curve of concrete; f t,r is a characteristic value of the concrete's uniaxial tensile strength; f c,r is a characteristic value of the concrete's uniaxial compressive strength; ε t,r is the peak tensile strain corresponding to the uniaxial tensile

Fig. 1
Dimensions and parameters of the test specimen and schematic diagram of the loading frame strength f t,r ; ε c,r is the peak compressive strain corresponding to the uniaxial compressive strength f c,r ; E c is the elastic modulus of concrete; ε is the nominal strain of concrete; σ is the nominal stress of concrete; y is the ratio of the concrete's strain to its peak strain; ρ t , ρ c , and n are parameters during the calculation.
This stress-strain relationship is a nominal relationship that must be converted into a stress-strain relationship for input into a numerical model (ABAQUS, 2011;Liu, et al., 2022;Ministry of Housing & Urban-Rural Development, 2011). The conversion expression is as follows: where ε true is the actual strain; σ true is the actual stress; ε is the nominal strain of concrete.; σ is the nominal stress of concrete.
The inelastic strain can be converted as follows: where ε in is the inelastic strain; ε true is the actual strain; σ true is the actual stress. The true stress-strain relationships of the C20 and C40 concrete obtained by Eqs. (1)-(4) are shown in Figs. 2 and 3.

Contact Settings
Due to different construction sequences of RC slab culverts rehabilitated with grouted CSPs, not all materials are completely bonded. During construction, the grout forms contact surfaces with the RC slab, foundation, and CSP. We simplified the contact settings to establish the numerical model, as shown in Fig. 4. The surface-to-surface contact was simulated; its normal behavior was defined as a hard contact, and the tangential behavior was defined as Coulomb friction contact. The friction coefficients between the grout and the RC slab and between the grout and the foundation were 0.6; that between the grout and CSP was 0.2 (ACI, 2019).

Numerical Model
ABAQUS was used to establish the numerical model of the specimens (Fig. 1). The C3D8R element was used to simulate the concrete, the T3D2 element was used to simulate the steel bars, and the S4R element was used to simulate the CSP.
The translational degrees of freedom in the three directions at the bottom, left, and right surfaces were restricted, as shown in Fig. 4. Displacement loading was adopted in the middle of the RC slab, and the value was − 20 mm   (vertically downward). Since the numerical model contains many nonlinear factors, an explicit algorithm was used to solve it, with a duration of 2 s and a time increment of 0.003 s. The large deformation effect was also considered. The numerical models are shown in Fig. 5.

Comparison of Experimental and Numerical Results
The crack distributions obtained from the numerical analysis and the experiment are shown in Fig. 6. The distributions and types of cracks generated in the arch-and box-type rehabilitated systems were significantly different.
In the arch-type rehabilitated system, bending and shear cracks occurred in the middle span of the RC slab, whereas only shear oblique cracks were generated in the grout. The   shear cracks generated in the RC slab and grout were parallel and angled at 45° downward. In addition, oblique shear cracks in the grout intersected with the haunches of the CSP arch. In the box-type rehabilitated system, bending and shear cracks also occurred in the middle span of the RC slab, whereas only bending cracks were generated in the grout. In both rehabilitated systems, the cracks in the RC slab and grout were not connected. Furthermore, the two ends of the RC slab warped upward, and the RC slab and grout became concave in the middle of the span. In general, the numerical simulation results and experimental results were consistent. The load-displacement curves at the crown of the CSP are shown in Fig. 7. The numerical results were highly consistent with the experimental results when the test load was less than the ultimate load-carrying capacity because the concrete damage plastic model was considered in the numerical model. The values obtained from the numerical analysis of the arch type for a load of 1851.19 kN (14.38 mm) were 0.11% greater than those derived from the experiment for a load of 1830.55 kN (11.72 mm). Similarly, the values derived from the numerical analysis of the box type for a load of 526.17 kN (16 mm) were 13% greater than those obtained from the experiment for a load of 465.77 kN (17.81 mm).
The comparison of the strains at the CSP haunch and crown is shown in Fig. 8. The trends of the numerical results and the test results are the same, and the values tend to be consistent. Therefore, the numerical model is reasonable and effective and can be used to analyze the influence of the CSP shape and grout strength on the load-carrying capacity of the rehabilitated system. Moreover, the load-carrying capacity estimation method is based on these numerical models.

Arch Effect
Previous studies have shown that even without a CPS, the grout contributes the most to the load-carrying capacity of the rehabilitated system in some cases (Li et al., 2021) due to the grout's arch effect.
The arch axis or the arch's compressive stress line under external loads changes with the rise-to-span ratio due to the arch effect of the grout, i.e., different rise-to-span ratios results in different arch axes (Long et al., 2019). In this study, the external loads were distributed to the RC slab and grout with 45° angle, and the edge of the load distribution range was the path of load diffusion. The arch axis is a parabola within the load distribution range and a straight line outside of the diffusion range. The axes are tangent at the intersection. A CSP is typically placed symmetrically in the middle of the grout. Therefore, the arch axes were drawn with the midpoint at the top of the grout as the reference point. The span ranged from 900 to 2300 mm with increments of 200 mm, and the rise ranged from 100 to 900 mm with increments of 100 mm. The arch axes with different rise-to-span ratios are shown in Fig. 9.
Arch axes also exist in the internal area of the grout, as shown in Fig. 9. They are derived by moving the existing arch axes downward. This part of the arch axes is not shown to avoid confusion. In addition, the position of the blank area is generally occupied by the CSP; thus, it is unnecessary to draw these arch axes.
If the CSP shape is similar to or the same as these arch axes, an arch effect is produced in the grout. Conversely, there is no arch effect if the CSP shape is dissimilar. As shown in Fig. 9, the box type cuts off most of the arch axes in the grout, and the crown of the CSP is very flat. Therefore, the arch effect is less pronounced, and the box arch resists external load due to its flexural load-carrying capacity. In contrast, the arch type has a similar shape as the arch axes; thus, an arch effect occurs, enhancing the load-carrying capacity of the rehabilitated system. A very simple method can be used to determine whether an arch effect is produced in the grout. If the radius of the haunch and side wall of the CSP is less than 0.2 times that of the crown, no arch effect is produced in the grout; whereas greater than or equal to 0.2 times that of the crown, arch effect is produced in the grout (CHBDC, 2006).

Mechanical Model
The box-type CSP is also called a multicentered or fivecentered arch structure. It consists of circular arcs of a crown (qs), two haunches (pq and st), and two sidewalls (Ap and tE) as shown in Fig. 10. These arcs are tangent to each other. The CSP shape changes as the arc radius changes; thus, the selection of the CSP shape is relatively flexible. An equivalent rigid frame is typically used by engineers to calculate the internal force of multicentered arches (Cao, 2007). Simplifying a multicentered arch into a rigid frame requires five key points: two feet (A, E), two haunch midpoints (B, D), and the crown (C). These five key points are connected to form a rigid frame (ABCDE), whose flexural and compressive stiffness values are the same as those of the CSP (Fig. 10). Under vertical loads, the beams BC and CD bend downward, whereas the columns AB and DE bend outward. Due to the strong constraints provided by the grout and foundation (areas filled with grids), horizontal outward movement rarely occurs (or the value is very small; the experimental displacements of points B and D were 0-0.2 mm). Thus, it can be assumed that these two points (B and D) are fixed, and the only parts that can deform are the beams BC and CD, corresponding to the arc BqsD in the original CSP. Beams BC and CD are further simplified to a skene arch BCD with the same span and rise as the folded beam to ensure that the beams BC and CD are similar to the original arc BqsD. As a result, the control points of the internal force calculation are B, C, and D.
As mentioned before, if an arch effect is produced in the grout, shear failure will occur, and the failure path lines are mB and nD. Slight bending cracks may occur at lines aB and bD. If no arch effect is produced in the grout, flexural failure will occur in the grout above arc BqsD of the CSP. No cracks will occur in other areas filled with grids.
It is challenging to calculate the load-carrying capacity of the rehabilitated system based on the CSP shape because it deviates from the original structural design. Therefore, only the internal forces of the key points B, C, and D need to be calculated by simplifying arc BqsD to the skene arch BCD, significantly reducing the calculation difficulty. Arch axes with different rise-to-span ratios Flexural failure or shear failure only affects the grout, and the failure mode determines how much strength it contributes. Since no bond exists between different materials, the total load-carrying capacity of the rehabilitated system is equal to the sum of the load-carrying capacity of the RC slab, grout, and CSP (Eq. (5)). The occurrence of flexural or shear failure in the grout depends on the whether an arch effect is produced in the grout.
where F 1 is the flexural capacity of the RC slab; F 2 is the shear capacity of the grout; F 3 is the flexural capacity of the CSP; P is the total load carried by the rehabilitated system.

Flexural Capacity of RC Slab
The load-carrying capacity of the RC slab was calculated by assuming a doubly-reinforced beam was present in the rectangular section (Ministry of Housing & Urban-Rural Development, 2011).
where α 1 is a coefficient related to the concrete grade; this value is 1.0 when the concrete grade does not exceed C50 and 0.94 when the concrete strength grade is C80. It is determined by linear interpolation when the concrete grade is between C50 and C80; f c is the design compressive strength value of the concrete; b is the width of a simply-supported RC slab; h 0 is the effective thickness of a simply-supported RC slab; x is the height of the compression zone; f y and f y ʹ are the design tensile and compressive strengths of the steel bars, respectively; A s and A s ʹ are the tensile and compressive areas of the steel bars, respectively; l is the effective span (the length between point m and n) of a simply-supported RC slab, which equals L-2(d + f).

Shear Capacity of Grout
The shear capacity of the grout can only be considered when an arch effect is produced in the grout. It can be calculated as follows (Ministry of Housing & Urban-Rural Development, 2011): where f t2 is the design tensile strength of the grout; A 2 is the total area of the shear surface of the grout, which is equal to 2(d + f)/cos 45° multiplied by the width b equal to that of a simply-supported RC slab.

Flexural Capacity of CSP
As mentioned before, the key points of the internal force calculation of the CSP are B, C, and D, and the calculation of these internal forces is based on simplifying the arc BqsD to the skene arch BCD. Consequently, the problem is greatly simplified. The elastic center method (Compile Group of Design Manual for Highway Bridge & Culvert, 1984) is used to calculate the internal forces in these simplified arches.
The computing model is shown in Fig. 11. The height from the elastic center to the arch crown y s , the radius R, and the half-center angle φ 0 of the simplified arch are defined as follows: where y s is the height from the elastic center of the simplified skene arch to the arch crown; R is the radius of the Fig. 10 Mechanical model of RC slab culvert rehabilitated with grouted CSP simplified skene arch; φ 0 is the half-center angle of the simplified skene arch; L is the effective span of the simplified skene arch; f is the rise of the simplified skene arch. Generally, the half-center angle φ 0 of the simplified skene arch is less than 45 degrees, and the combined stress at points B and D of this arch type is greater than at point C. Under a distributed load gʹ, points B and D yield first and produce plastic hinges (indicated by the time the bending moment reaches M u , as shown in Fig. 12①). At this time, the bending moment of point C is M c , which is less than M u , indicating that the loadcarrying capacity of the simplified skene arch has not been exhausted (Fig. 12①). Since points B and D have produced plastic hinges, they can be simplified as two hinged arches (Fig. 12②) to calculate the internal force of point C; however, the bending moments of points B and D remain unchanged at M u . When the distributed load increases to gʺ, the bending moment of point C reaches M u , the load-carrying capacity of the simplified skene arch is exhausted, and the arch is transformed into a three-hinged arch.
The internal forces of points C, B, and D under a distributed load gʹ in Fig. 12① can be calculated as follows (Compile

Group of Design Manual for Highway Bridge & Culvert, 1984):
The internal forces of points C, B, and D under a distributed load gʺ in Fig. 12② can be calculated as follows (Compile Group of Design Manual for Highway Bridge & Culvert, 1984): Fig. 11 Computing model of CSP where k 1 , k 2 , k 3 , k 4 , k 5 , k 6 , k 7 , Φ Mc ʹ, Φ Nc ʹ, Φ Mf ʹ, Φ Nf ʹ, Φ Mc ʺ, Φ Nc ʺ, and Φ Nf ʺ are the coefficients of the elastic center method for calculating the internal force of the simplified skene arch; gʹ and gʺ are the distributed loads carried by the simplified skene arch in Fig. 12① and ②, respectively; M c ʹ and N c ʹ are the bending moment and axial force of point C in Fig. 12①, respectively; M f ʹ and N f ʹ are the bending moment and axial force of points B and D in Fig. 12①, respectively; M c ʺ and N c ʺ are the bending moment and axial force of point C in Fig. 12②, respectively; N f ʺ is the axial force of points B and D in Fig. 12②. It is necessary to check whether the combined stress generated by the axial force and the bending moment of the key point exceeds the yield stress of the steel to determine whether the simplified skene arch produces a plastic hinge. It is calculated using Eq. (13) (Liu, 2004). If the yield stress of the steel is known, the maximum uniformly distributed loads gʹ and gʺ carried by the CSP can be solved by the reverse calculation of Eq. (13).
where M is the bending moment of the key point; N is the axial force of the key point; f csp is the design value of the yield strength of the CSP; d csp is the corrugation height of the CSP; I is the moment of inertia of the CSP; A is the sectional area of the CSP.
The distributed load gʹ shown in Fig. 12① can be calculated as follows (Compile Group of Design Manual for Highway Bridge & Culvert, 1984): The value of gʹ is substituted into Eq. (9) to calculate M c ʹ and N c ʹ, and the combined stress σʹ of point C is calculated by Eq. (13).
At this time, point C can bear the additional stress σʺ, which can be expressed as: gʺ can be solved by the reverse calculation of Eq. (13): The maximum uniformly distributed load g carried by the simplified skene arch of the CSP is the sum of gʹ and gʺ.
Finally, the flexural capacity of the CSP can be expressed as: and Φ Nc ʺ are the coefficients of the elastic center method for calculating the internal force of the simplified skene arch; gʹ and gʺ are the distributed loads carried by the simplified skene arch shown in Fig. 12① and ②, respectively; M c ʹ and N c ʹ are the bending moment and axial force of point C in Fig. 12①, respectively; M f ʹ and N f ʹ are the bending moment and axial force of points B and D in Fig. 12①, respectively; M c ʺ and N c ʺ are the bending moment and axial force of point C in Fig. 12②, respectively; N f ʺ is the axial force of points B and D in Fig. 12②; σʹ and σʺ are the combined stress of point C in Fig. 12① and ②, respectively; f csp is the design value of the yield strength of the CSP; d csp is the corrugation height of the CSP; I is the moment of inertia of the CSP; A is the sectional area of the CSP; g is the maximum uniformly distributed load carried by the simplified skene arch of the CSP.

Result
The CSP can have many shapes at the same rise-span ratio; engineers are more likely to choose box and arch shapes based on experience. However, many box-or arch-type CSP shapes can be obtained by changing the radius of the haunch R s , crown R c , and side wall R w (as shown in Fig. 13, R c = R w ) of the CSP. The same numerical method as before was used to investigate the influence of the CSP shape, identify the arch effect, and evaluate the applicability of the load-carrying capacity estimation method. Meanwhile, the rise of 1000 mm and the span of 2000 mm of the CSP, the material strength, and the number of steel bars in the RC slab and foundation remained the same. We investigated 216 CSP shapes; the shape number and radius parameters are listed in Table 1. Fig. 13 Shape parameters of the CSP It is observed in Fig. 14 that the load-carrying capacity of the rehabilitated system increased with a decrease in the radius of the sidewall and crown at a constant haunch radius of the CSP. The larger the haunch radius, the smaller the reduction in the load-carrying capacity. When the radius of the crown and side wall of the CSP remained constant, the load-carrying capacity of the rehabilitated system increased with an increase in the haunch radius. The larger the radius of the crown and side wall, the greater the rate of increase of the load-carrying capacity. The closer the CSP shape was to a semicircle, the higher the load-carrying capacity of the rehabilitated system. The haunch represents the fillet of the crown and the side wall. The larger the fillet radius, the faster the CSP changed from a box-type arch to a semicircle. Therefore, increasing the haunch radius substantially improved the load-carrying capacity of the rehabilitated system, more so than reducing the radius of the crown and side wall. In other words, the most effective way to improve the load-carrying capacity of the rehabilitated system is to increase the haunch radius, followed by reducing the radius of the crown and side wall.
The relationship between the load-carrying capacity of the rehabilitated system and the CSP shape is illustrated in Fig. 14. Therefore, Fig. 15 only shows the comparison between the numerical and calculation results. It is worth noting that the calculation results were obtained using the proposed load-carrying capacity estimation method. It is observed in Fig. 15 that the trends of the calculation results and numerical results are highly consistent. Two situations deserve special attention. When no arch effect is produced in the grout, the shear capacity of the grout does not have to be considered, but the grout's shear capacity must be considered. The determination method of the arch effect has been described above. When the radius of the haunch and side wall of the CSP is more than 0.2 times that of the crown, an arch effect will occur. An arch effect was observed in the rehabilitated systems with shapes No. 10,19,28,29,37,38,39,46,47,48,55,56,57,58,64,65,66,67, and 68. These rehabilitated systems had high load-carrying capacities, corresponding to the peaks in Fig. 15. It is evident that the grout strength significantly affected the load-carrying capacity of the rehabilitated system but only when an arch effect occurred. This behavior is depicted in Fig. 15; the data peaks correspond to the arch effect in the grout, which increases the grout's shear capacity. The higher the grout strength, the higher the load-carrying capacity of the rehabilitated system, i.e., the load-carrying capacity is higher for the rehabilitated system with C40 grout than for the system with C30 grout, Fig. 14 Influence of the CSP shape on the load-carrying capacity of the rehabilitated system 1 3 while C20 grout has the lowest load-carrying capacity. The rehabilitated systems not mentioned above did not produce an arch effect, and the grout strength did not affect sensitively their load-carrying capacities. The calculation results are highly similar to the numerical results and only slightly lower, which is conservative for structural design. If the arch effect was produced in the grout, the shear capacity F 2 of the grout was considered in the calculation results, and the results increased with an increase in the grout strength. However, when no arch effect was produced, the load-carrying capacities of the rehabilitated systems with different grout strengths were the same, and only F 1 and F 3 were included. The comparison of the calculation and numerical results indicate that the load-carrying capacity estimation method is reasonable for investigating these problems.

Discussion
Flexibility is an inherent characteristic of CSP structures. The CSP produces horizontal deformation under vertical loads and compresses the soil, resulting in passive earth pressure that limits the horizontal deformation of the CSP, demonstrating the vertical load-carrying capacity of the CSP (Sun et al., 2021;Wadi et al., 2021). The high flexibility of the CSP improves its load-carrying capacity, and the deformation level depends on interactions with the surrounding medium. However, if the surrounding medium is highly rigid concrete, the CSP's horizontal deformation is relatively small under a vertical load. In a rehabilitated system, the CSP cannot squeeze the sidewall of the foundation of small bridges, culverts, and grout because the soil surrounding the foundation sidewall has been consolidated. Therefore, horizontal deformation is nearly impossible because of the passive earth pressure of the consolidated and stable soil. As a result, experiments with lateral restraint were adopted in this study. In addition, we ignored the part of the CSP below the midpoint of the haunch and simplified it into a skene arch. The crown C and haunches B and D were used as control points to calculate CSP's internal forces. The calculation results showed that this approach was correct.
Midspan loading was the most unfavorable load case in evaluating the load-carrying capacity of a rehabilitated system. Generally, the worst structural damage is the loss of bearing capacity due to the plastic hinge of the simplysupported RC slab in the middle of the span. A composite structure is formed after rehabilitation. However, since the plastic hinge already exists in the RC plate, it is the weakest position of the rehabilitated system under a vertical load, subjecting the midspan of the rehabilitated system to a concentrated load due to the limited rotation of the plastic hinge. Under this condition, the arch axis is a parabola within the load distribution and a straight line outside the diffusion range. However, the arch axis is always a parabola under a uniform load.

Conclusion
We analyzed the mechanical properties of RC slab culverts rehabilitated with a grouted CSP. The most important contribution of this paper is that the CSP shape significantly affected the ultimate load-carrying capacities of the rehabilitated systems. The proposed load-carrying capacity estimation method was verified using experiments and numerical analysis, and a standardized calculation method suitable for rehabilitated systems with different CSP shapes was developed.
At a constant haunch radius of the CSP, the load-carrying capacity of the rehabilitated system increased with a decrease in the radius of the side walls and crown. When the radius of the side walls and crown of the CSP remained constant, the load-carrying capacity of the rehabilitated system increased with an increase in the haunch radius. The most effective way to improve the load-carrying capacity of the rehabilitated system was to increase the haunch radius and reduce the radius of the arch crown and side wall. In general, the closer the shape of the CSP was to a circle or arc, the higher the load-carrying capacity of the rehabilitated system was.
The most important role of the grout was to provide strong lateral restraint for the CSPs and reduce their effective span, indirectly improving the load-carrying capacity of the rehabilitated system. In addition, when an arch effect was produced in the grout, the shear strength of the grout substantially contributed to improving the load-carrying capacity of the rehabilitated system.
The proposed load-carrying capacity estimation method is suitable for rehabilitated systems with box-or arch-type CSPs and provides conservative estimates for structural design.
In future research, many numerical calculations should be performed to verify the applicability and rationality of the proposed load-carrying capacity estimation method. In addition, the influence of the material strength, the structural dimension of the RC slab culvert, and the steel ratio should be evaluated.
Appendix α t is a parameter of the descending section of the uniaxial tensile stress-strain curve of concrete.α c is a parameter of the descending section of the uniaxial compressive stress-strain curve of concrete.f t,r is a characteristic value of concrete uniaxial tensile strength.f c,r is a characteristic value of concrete uniaxial compressive strength.ε t,r is the peak tensile strain corresponding to the uniaxial tensile strength f t,r .ε c,r is the peak compressive strain corresponding to the uniaxial compressive strength f c,r . E c is the elastic modulus of concrete.ε is the nominal strain of concrete.σ is the nominal stress of concrete.y is the ratio of the concrete's strain to its peak strain.ρ t , ρ c , and n are parameters during calculations.ε true is the actual strain.σ true is the actual stress.ε in is the inelastic strain.
F 1 is the flexural capacity of the RC slab. F 2 is the shear capacity of the grout. F 3 is the flexural capacity of the CSP. P is the total load carried by the rehabilitated system.α 1 is a coefficient related to the concrete grade; this value is 1.0 when the concrete grade does not exceed C50 and 0.94 when the concrete strength grade is C80. It is determined by linear interpolation when the concrete grade is between C50 and C80.f c is the design compressive strength value of the concrete.
B is the width of a simply-supported RC slab. H 0 is the effective thickness of a simply-supported RC slab. X is the height of the compression zone. F y and f y ʹ are the design tensile and compressive strengths of the steel bars, respectively.
A s and A s ʹ are the tensile and compressive areas of the steel bars, respectively.l is the effective span (the length between point m and n) of a simply-supported RC slab, which equals L-2(d + f).f t2 is the design tensile strength of the grout.
A 2 is the total area of the shear surface of the grout, which is equal to 2(d + f)/cos45° multiplied by the width b equal to that of a simply-supported RC slab.y s is the height from the elastic center of the simplified skene arch to the arch crown.
R is the radius of the simplified skene arch; φ 0 is the halfcenter angle of the simplified skene arch.
L is the effective span of the simplified skene arch.f is the rise of the simplified skene arch.e is the distribution of the vertical load acting on the top surface of the RC slab.h is the thickness of the RC slab.d is the thickness of the grout at the crown.k 1 , k 2 , k 3 , k 4 , k 5 , k 6 , k 7 , Φ Mc ʹ, Φ Nc ʹ, Φ Mf ʹ, Φ Nf ʹ, Φ Mc ʺ, Φ Nc ʺ, and Φ Nf ʺ are the coefficients of the elastic center method for calculating the internal force of the simplified skene arch.  Fig. 12① and ②, respectively. M c ʹ and N c ʹ are the bending moment and axial force of point C in Fig. 12①, respectively. M f ʹ and N f ʹ are the bending moment and axial force of points B and D in Fig. 12①, respectively. M c ʺ and N c ʺ are the bending moment and axial force of point C in Fig. 12②, respectively. N f ʺ is the axial force of points B and D in Fig. 12②. M is the bending moment of the key point. N is the axial force of the key point.σʹ and σʺ are the combined stress of point C in Fig. 12① and ②, respectively.f csp is the design value of the yield strength of the CSP.d csp is the corrugation height of the CSP.
I is the moment of inertia of the CSP.
A is the sectional area of the CSP.g is the maximum uniformly distributed load carried by the simplified skene arch of the CSP.
Acknowledgements This work was supported by the Young Scientists Fund of China (Grant No. 52108142).

Data Availability
The data (experimental results) used to support the findings of this study are included in the article.

Conflict of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
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