Experimental and Theoretical Study on the Crack Defect Effect on the Bearing Capacity of a Rectangular Culvert

Abstract

Rectangular culverts are extensively employed in various municipal projects. Due to their frequent placement in weak strata within urban areas, these structures are susceptible to defects like cracks, which can significantly reduce their bearing capacity and even result in collapse or other failures. This study investigates the mechanical behavior of cracked rectangular culverts through laboratory tests, analyzing both the failure mechanisms and the impact of crack depth on the ultimate bearing capacity (UBC) of the culverts. Focusing solely on the effect of crack depth on the UBC of rectangular culverts with defects, this paper establishes fundamental assumptions and proposes a calculation method for the residual bearing capacity (RBC) of rectangular culverts under crack-induced conditions. The theoretical calculations closely align with the experimental results. The findings indicate that (1) the failure process of a rectangular culvert under trilateral normal external load can be categorized into the initial stage with no visible cracks, the crack progression stage, and the final failure stage; (2) the increase in crack depth leads to a decrease in the UBC of the rectangular culvert, indicating that cracks have a negative impact on its bearing capacity; and (3) when the crack depth reaches the protective layer’s thickness, the UBC of the rectangular culvert reduces by over 30%, providing further evidence that an increase in crack depth significantly diminishes its bearing capacity.

1. Introduction

With the expansion of cities and the growth of the population, the original under-ground drainage systems, especially the culvert facilities, are facing a series of problems, such as an insufficient bearing capacity, serious aging and damage, and poor drainage. These problems not only threaten the safe and stable operation of urban infrastructure, but also pose severe challenges to urban water environment management, flood control and drainage capacity improvement, and the improvement of residents’ quality of life [1].

Rectangular culverts are frequently situated in weak urban strata with complex geological conditions, and their structures are often subject to defects such as corrosion and cracking. Cracks are the primary form of failure in concrete structures, which can reduce the load-bearing capacity of the structure, leading to structural failure and disaster [2]. Experts and scholars have thoroughly examined the underlying factors leading to surface cracks in reinforced concrete and their progression patterns, as well as the impact of cracking on the structural bearing capacity [3,4,5,6,7]. With the passage of time, there is a risk of the deterioration of the material of the rectangular culvert itself, which has certain safety hazards. Consequently, research on the mechanical properties of existing rectangular culvert structures can offer a scientific foundation for quantitatively assessing the health status of reinforced concrete structures and propose a theoretical method for future maintenance and repair efforts [8]. In comparison to conventional reinforced concrete drainage pipelines, rectangular drainage culverts typically have a rectangular cross-section, providing benefits such as high spatial efficiency, strong stability, and favorable foundation adaptability. However, the difference in the cross-sectional shape results in variations in stress patterns, failure modes, and methods of analyzing the bearing performance between drainage pipelines and rectangular drainage culverts in the stratum [9].

Recently, to investigate the external load on rectangular culverts in practical projects—specifically, the distribution of or variation in earth pressure—extensive research has been carried out by scholars globally. Based on the Duncan hyperbolic model, elastic model, force balance, and deformation compatibility conditions, Li Yonggang et al. [10] derived a calculation method of earth pressure on rectangular culverts. Abuhajar et al. [11] investigated the soil arching effect of rectangular culverts in physical engineering through experiments and analyzed the factors influencing this effect. With the improvement in the calculation theory of earth pressure, scholars at home and abroad have gradually begun to pay attention to the study of the mechanical properties of rectangular culvert structures. Xie Yongli et al. [12] conducted preliminary research on the mechanical properties of high-fill rectangular culvert structures and discussed the applicable conditions of different structural types of culverts. Chen Baoguo [13] and Yang Xiwu [14] introduced the structural design and construction points of a new load-reducing rigid culvert in detail and carried out field tests based on the physical project. Dasgupta et al. [15] calculated the deformation, bending moment, and shear force of the rectangular culvert structure, and compared the calculated data with the experimental data.

In summary, the existing research is mostly aimed at the failure mode and bearing capacity of the intact rectangular culvert structure after being subjected to an external load in the actual project. However, there is a lack of research on the overall mechanical properties of rectangular culverts with preset single defects (such as cracks) after being subjected to external loads. Based on structural mechanics, this paper puts forward a calculation formula for the RBC of the rectangular culvert under the action of crack defects. It carries out an external pressure test of a rectangular culvert with crack defects to verify the correctness and applicability of the RBC theory of a rectangular culvert with crack defects.

2. Experimental Study on External Pressure of Rectangular Culvert with Crack Defects

2.1. Experiment Ideas

To explore the ultimate bearing capacity (UBC) and influencing factors of a rectangular culvert with crack defects, the variation law of the UBC of rectangular culverts with crack defects at different depths was obtained. In this study, an external pressure load test of a rectangular culvert with crack defects was carried out, and a trilateral normal external load test of a rectangular culvert with different crack depths was carried out [16].

Displacement meters were placed vertically and horizontally to obtain the load–displacement curve of the rectangular culvert. By analyzing the test results, the compression failure law of rectangular culverts with crack defects was obtained. The mechanical performance test of rectangular culverts with crack defects primarily examined the impact of crack depth on the UBC. Four test groups were designed based on the inner protective layer’s thickness. Key parameters were assessed for defective culverts with crack depths of 1/3, 2/3, and the full inner protective layer’s thickness, along with a non-defective culvert for comparison.

2.2. Experiment Method and Procedure

2.2.1. Specimen Preparation

The rectangular culverts used in this experiment were all manually cast. The specimen’s section dimensions were a × b = 500 mm × 400 mm, with a length of 100 mm, a wall thickness of 60 mm, and a concrete protective layer thickness of 7.5 mm. All rectangular culvert samples were produced using molds. Cement and aggregate were mixed thoroughly, and then water was gradually added while continuing to stir. A water–cement ratio of 0.38 and a sand–cement ratio of 1.2 were used. Details of the pouring material are provided in

Table 1. Rectangular culvert specimen pouring material table.

Testing Material	Material Type and Model
Cement	Normal Portland cement (P.I.42.5)
Sand	Manufactured sand (60–80 mesh)
Longitudinal bar	Hot rolled plain steel bars (5 mm)
Stirrup	Hot rolled plain steel bars (5 mm)
Compressive strength of concrete	23.4 MPa
Stirrup tensile strength	650 MPa

. The prepared concrete mixture was quickly and evenly poured into molds containing pre-positioned steel cages, and a vibrating machine (HZJ-A concrete shaking table; Shengke Test Instrument Co., Ltd., Cangzhou, China) was used to remove internal air bubbles. After casting and demolding, the specimens were cured naturally for 28 days (28 ± 2 °C, spraying tap water on time).

After the specimen was cured, the specimen was pretreated. The group A1 rectangular culvert did not need to be treated, and the group C1~C3 culvert needed to be prefabricated. Three groups of rectangular culvert specimens with crack defects were obtained using the concrete cutting device. The physical representation of the specimens and the locations of crack defects are illustrated in Figure 1. The location of each group of defects is described in

Table 2. A1, C1, C2, and C3 groups of rectangular culvert cracks.

Groups	Influencing Factors of Defects	Defection Evaluation	Quantity
A1	Undamaged	Zero defect	1
C1	Crack depth	Two cracks are positioned on the inner side of the upper and lower surfaces (ULSs) of the rectangular culvert, and two cracks are arranged on the outer side of the left and right surfaces. The depth is 1/3 of the protective layer thickness, with a width of 0.2 mm.	1
C2	Crack depth	Two cracks are positioned on the inner side of the ULSs of the rectangular culvert, and two cracks are arranged on the outer side of the left and right surfaces. The depth is 2/3 of the protective layer thickness, with a width of 0.2 mm.	1
C3	Crack depth	Two cracks are positioned on the inner side of the ULSs of the rectangular culvert, and two cracks are arranged on the outer side of the left and right surfaces. The depth is the protective layer thickness, with a width of 0.2 mm.	1

. There were two reasons for using manual cutting: (1) Manual cutting can accurately control the length, depth, and width of cracks, so as to quantitatively analyze the effect of a certain factor on the UBC of a rectangular culvert. If the fracturing method is used, the crack progression becomes uncontrollable, affecting the bond between the steel bar and concrete, thus failing to meet the single-variable assumption. (2) The artificial cutting method offers benefits such as being low-cost, easy to operate, and less time-consuming.

2.2.2. Stress and Displacement Measurement

The strain–displacement measuring device was installed in the mid-span section of the rectangular culvert to measure strain distribution in critical areas such as the inner and outer sides of the top and bottom roofs, the inner and outer sides at the midpoints of the side walls, and the roof’s right angle. Two LVDT sensors monitored horizontal and vertical displacement variations in the rectangular culvert. Sensors were mounted using a universal magnetic clamp with a 30 mm range. Sensor positions are depicted in Figure 2, and displacement data were gathered in real time via a static strain acquisition system.

2.2.3. Loading

In this work, the UBC of the rectangular culvert specimen was determined through a trilateral external load test (TEB), with the stress model depicted in Figure 3. A ring stiffness machine was utilized to apply graded loading to the specimen by controlling the displacement. The loading rate was initially set at 3 mm/min for 0~5 kN, then reduced to 1 mm/min, stopping once the specimen failed.

2.3. Test Results and Analysis

2.3.1. Specimen Failure Mode

This section evaluates specimen failure modes under various conditions. Figure 4 and Figure 5 show the load–displacement curve and crack progression of the non-defective rectangular culvert (Group A1). The load–displacement curves for the four groups with varying crack depths exhibit similar trends. The deformation process of the rectangular culvert can be divided into three stages: an initial stage with no visible cracks, a crack progression stage, and a final failure stage [17]. The concrete and internal steel bars’ stress characteristics at each stage were examined, and the factors contributing to key point curve changes were summarized.

Point a is where the ring stiffness machine first touches the A1 specimen’s roof, with its displacement showing the culvert roof’s mid-span movement. The machine’s pressure on the culvert is shown in Figure 5a. From a to b, the structure is intact before any cracks, and the load increases linearly with the machine’s displacement, acting like an elastic beam. The concrete’s compression zone stays elastic, but the tensile edge strain rises quickly, showing plastic behavior. Additionally, owing to the strong bond between the concrete and steel bars, the strain in the steel bars matches that in the concrete at the same level. The initial stage with no visible cracks in the rectangular culvert exhibits the following features: (1) the specimen has not yet cracked, and deflection is minimal; and (2) the stress curve in the compression area remains linear, and the stress curve in the tensile area is initially linear and later curves as the load increases.

Point b in Figure 5b marks the crack onset in a rectangular culvert. Cracks first appear in critical areas like the roof’s mid-span and the bottom plate’s stress concentration. These cracks develop from the tensile zone to the compression zone, gradually forming through-cracks with small widths. Point b’s bearing capacity underpins the crack resistance calculations. From point b to point c, the structure enters crack progression, where cracks expand rapidly due to concrete’s brittleness. However, part of the tensile force initially carried by the concrete is transferred to the internal steel reinforcement, leading to increased stress in the steel. The load carried by the rectangular culvert, as seen in Figure 5, continues to rise with displacement, but the curve’s slope decreases compared to the initial stage. This may be attributed to plastic deformation in the compression zone, leading to a faster strain growth rate in the concrete. The crack progression stage exhibits the following features: (1) concrete in the tension zone at the crack no longer participates in load bearing and exhibits solely compressive behavior; (2) the tensile force at the crack is carried by the steel bars, which do not reach the yield point; and (3) concrete in the compression zone undergoes plastic deformation, though not to a significant extent.

Point c denotes the normal stress state of the rectangular culvert, which is crucial for deformation checks under regular use (Figure 5c). The shift from point c to point d signifies the culvert’s ultimate failure. During this stage, the steel bar yields, leading to significant deformation. The structure’s deformation leads to increased roof deflection, mid-span curvature, and crack width, while the compression area of the roof diminishes. The load increase from point c to point e is attributed to the accumulation of elastic strain energy during initial compression, which surpasses the energy needed for crack progression, thus accelerating crack growth. Additionally, the steel bar has not yet reached its yield limit, allowing the tensile stress and internal specimen stress to increase. At point e, the culvert’s load reaches its peak at 8091.35 N with a displacement of 9.65 mm. Following point e, the load on the rectangular culvert drops abruptly, leading to brittle failure. Cracks appear in the mid-span roof and the stress concentration areas of the bottom plate, causing the loss of bearing capacity. After reaching point e, the load enters a decline stage, and a nearly horizontal curve appears due to the steel reaching its yield strength, while the tensile force remains constant. The concrete continues to be crushed, and the UBC is mainly supported by the steel bars. The deformation of the culvert continues to increase with no significant change in load. At point d, as the crack widens, the strain in the steel bar continues to increase until it eventually breaks, leading to complete structural failure, as shown in Figure 5d.

Figure 5d shows that the cracks were developed at the support and loading points. Support and loading points are critical load-bearing areas in structures, often subjected to significant loads and stresses. When these stresses exceed the tensile strength of concrete, cracks may form near the support or loading points. Simultaneously, the constraint imposed by these critical locations restricts the free development of cracks within concrete members. Specifically, crack formation near support or loading points leads to increased stress concentrations due to this constraint. This stress concentration not only accelerates crack propagation but also triggers additional crack formation. As cracks expand, the carrying capacity of support points gradually decreases, posing a threat to the overall structural stability and safety. Moreover, excessive or unevenly distributed loading at these points can induce additional stresses and deformations near support areas, further exacerbating crack initiation and progression.

The load–displacement and crack progression of the C1–C3 group of cracked rectangular drainage culverts are similar to those of the A1 group. It is observed that most cracks develop in the middle of the roof and floor spans, indicating this area as the stress concentration zone. The failure mode can still be divided into three stages: the initial stage with no visible cracks, the crack progression stage, and the final failure stage.

2.3.2. Influence of Crack Depth on Rectangular Culvert UBC

The rectangular culvert with crack defects was analyzed to determine the impact of crack depth on its UBC. Figure 6 shows the load–displacement curves of rectangular culverts with varying crack depths. e0, e1, e2, and e3 represent the ultimate bearing capacities of the A1 non-defective rectangular culvert and the C1, C2, and C3 crack defect rectangular culverts, respectively. The detailed data are presented in

Table 3. UBC of intact and crack defect rectangular culverts.

Groups	Displacement at UBC (mm)	UBC (N)
A1	9.65	7768.24
C1	8.42	6738.31
C2	16.98	6389.86
C3	15.57	5218.53

. The results indicate that the UBC decreases as the crack depth increases, with no clear pattern in the displacement corresponding to the UBC. When the crack depth equals the protective layer’s thickness, the rectangular culvert’s UBC is reduced by 32.83%, demonstrating the significant effect of crack depth on its bearing capacity.

3. Calculation Model of UBC of Rectangular Culvert with Crack Defects

3.1. UBC Calculation Model of Non-Defective Rectangular Culvert

To facilitate the calculation, the influence of the wall thickness was ignored in the process of the force analysis of rectangular reinforced concrete structures, and the rectangular culvert was simplified as a frame structure, as evidenced by the force distribution diagram in Figure 7. The simplified force model of the rectangular culvert is an axisymmetric model, so a half component was taken for calculation. Because there was only vertical displacement in the middle of the rectangular culvert floor and roof during the test, it was regarded as a sliding bearing. The distance from the rectangular culvert’s bottom plate to the midpoint of the span was 50 mm, which is less than the average length b of the culvert. As shown in Figure 7, the model may be optimized for calculation purposes.

According to the simplified model, the displacement method was used to analyze the structure to obtain the bending moment of each point, as shown in Formula (1):

$$\begin{array}{cc}\begin{array}{c}{\mathrm{M}}_{\mathrm{AM}}={\mathrm{i}}_1{\mathsf{\theta }}_{\mathrm{A}}-{\displaystyle \frac{\mathrm{Fa}}{8}}\\ {\mathrm{M}}_{\mathrm{MA}}=-{\mathrm{i}}_1{\mathsf{\theta }}_{\mathrm{A}}-{\displaystyle \frac{\mathrm{Fa}}{8}}\\ {\mathrm{M}}_{\mathrm{AC}}={2\mathrm{i}}_2{\mathsf{\theta }}_{\mathrm{A}}+{4\mathrm{i}}_2{\mathsf{\theta }}_{\mathrm{C}}\\ {\mathrm{M}}_{\mathrm{CA}}={ 4\mathrm{i}}_2{\mathsf{\theta }}_{\mathrm{A}}+{2\mathrm{i}}_2{\mathsf{\theta }}_{\mathrm{C}}\\ {\mathrm{M}}_{\mathrm{CN}}{=\mathrm{i}}_1{\mathsf{\theta }}_{\mathrm{C}}-{\displaystyle \frac{\mathrm{Fa}}{8}}\\ {\mathrm{M}}_{\mathrm{NC}}=-{\mathrm{i}}_1{\mathsf{\theta }}_{\mathrm{C}}-{\displaystyle \frac{\mathrm{Fa}}{8}}\end{array}& ,\end{array}$$

(1)

$$\left\{\begin{array}{c}{\mathrm{i}}_1={\displaystyle \frac{2\mathrm{EI}}{\mathrm{a}}}\\ {\mathrm{i}}_2={\displaystyle \frac{\mathrm{EI}}{\mathrm{b}}}\end{array}\right.$$

(2)

where,

• θ_A—Point A’s angular displacement;
• θ_C—Point C’s angular displacement;
• a—Average length of the rectangular culvert section, m;
• b—Average width of the rectangular culvert section, m;
• c—Longitudinal length of the rectangular culvert, m;
• i₁—AM and CN rod line stiffness; see Formula (2);
• i₂—AC rod line stiffness; see Formula (2);
• M_AC—A-end bending moment of the AC rod;
• M_CA—C-end bending moment of the AC rod;
• M_AM—A-end bending moment of the AM rod;
• M_MA—M-end bending moment of the AM rod;
• M_NC—N-end bending moment of the CN rod;
• M_CN—C-end bending moment of the CN rod;
• F—M, N point load, N.

According to the displacement method, the following equations can be listed. Simultaneously, Formulae (1)–(3), θ_A, and θ_C were obtained, as witnessed in Formula (4). The obtained θ_A and θ_C were brought into Formula (1) to obtain the bending moment of each point, such as in Formula (5). From the bending moment of each point, it was inferred that the bending moment of the M and N points is the largest, suggesting that the roof and floor span of the rectangular culvert are dangerous points. The M point bending moment was taken for analysis:

$$\left\{\begin{array}{c}{\mathrm{M}}_{\mathrm{A}}={\mathrm{M}}_{\mathrm{AM}}+{\mathrm{M}}_{\mathrm{AC}}=0\\ {\mathrm{M}}_{\mathrm{C}}={\mathrm{M}}_{\mathrm{CN}}+{\mathrm{M}}_{\mathrm{CA}}=0\end{array}\right.,$$

(3)

$${\mathsf{\theta }}_{\mathrm{A}}={\mathsf{\theta }}_{\mathrm{C}}={\displaystyle \frac{\mathrm{Fa}}{48{\mathrm{i}}_2+8{\mathrm{i}}_1}},$$

(4)

$$\left\{\begin{array}{c}{\mathrm{M}}_{\mathrm{MA}}={\mathrm{M}}_{\mathrm{NC}}=-{\displaystyle \frac{3\mathrm{F}{\mathrm{a}}^2+2\mathrm{Fab}}{24\mathrm{a}+\mathrm{b}}}\\ {\mathrm{M}}_{\mathrm{AM}}={\mathrm{M}}_{\mathrm{CN}}={\displaystyle \frac{3\mathrm{F}{\mathrm{a}}^2}{24\mathrm{a}+\mathrm{b}}}\\ {\mathrm{M}}_{\mathrm{AC}}={\mathrm{M}}_{\mathrm{CA}}=-{\displaystyle \frac{3\mathrm{F}{\mathrm{a}}^2}{24\mathrm{a}+\mathrm{b}}}\end{array}\right..$$

(5)

From the bending moment of point M, the relationship between the load borne by the rectangular culvert and the mid-span bending moment of the rectangular culvert roof under the condition of a trilateral normal external load on the rectangular culvert could be obtained as follows:

$$\mathrm{F}={\displaystyle \frac{\left(24\mathrm{a}+\mathrm{b}\right)\mathrm{M}}{3{\mathrm{a}}^2+2\mathrm{ab}}}.$$

(6)

Hence, the UBC of the rectangular culvert subjected to trilateral normal external load could be determined from the maximum bending moment at the midpoint of the roof span of the culvert, which is

$${\mathrm{F}}_{\max }={\displaystyle \frac{\left(24\mathrm{a}+\mathrm{b}\right){\mathrm{M}}_{\max }}{3{\mathrm{a}}^2+2\mathrm{ab}}}.$$

(7)

In this paper, the rectangular culvert specimen was a rectangular reinforced concrete culvert with double-layer reinforcement. Taking a small area near the mid-span line load of the double-layer-reinforced rectangular culvert roof, the section could be treated as a double-layer-reinforced rectangular section flexural member under normal section bending, as depicted in Figure 8. The bending moment at the cross-section could be derived by analyzing both the front and side views of the isolator.

According to the equilibrium condition of force, the following equation can be listed:

$${\mathrm{\alpha }}_1{\mathrm{f}}_{\mathrm{cc}}\mathrm{cx}+{\mathrm{f}}_{\mathrm{rc}}{\mathrm{A}}_{\mathrm{s}1}={\mathrm{f}}_{\mathrm{rt}}{\mathrm{A}}_{\mathrm{s}}$$

(8)

where,

• A_s—The combined cross-sectional area of stirrups within the section’s tensile zone, mm²;
• f_cc—Compressive strength of concrete, MPa;
• f_rt—Stirrup tensile strength, MPa;
• f_rc—Stirrup compressive strength, MPa;
• x—Compression zone height, mm;
• α₁—The equivalent rectangular stress coefficient in the concrete compression zone is 1.0 for concrete grades of C50 or lower.

The height of the compression zone is

$$\mathrm{x}={\displaystyle \frac{{\mathrm{f}}_{\mathrm{rt}}{\mathrm{A}}_{\mathrm{s}}-{\mathrm{f}}_{\mathrm{rc}}{\mathrm{A}}_{\mathrm{s}1}}{{\mathrm{\alpha }}_1{\mathrm{f}}_{\mathrm{cc}}\mathrm{c}}}.$$

(9)

The moment of the point of action of the resultant force on the compressive steel bar was obtained:

$${\mathrm{M}}_{\mathrm{u}}={\mathrm{\alpha }}_1{\mathrm{f}}_{\mathrm{cc}}\mathrm{cx}\left({\mathrm{h}}_0-{\displaystyle \frac{\mathrm{x}}{2}}\right)+{\mathrm{f}}_{\mathrm{rc}}{\mathrm{A}}_{\mathrm{s}1}({\mathrm{h}}_0-{\mathrm{a}}_{\mathrm{s}1}).$$

(10)

Substituting x into Formula (10), we could obtain

$${\mathrm{M}}_{\mathrm{u}}=\left({\mathrm{f}}_{\mathrm{rt}}{\mathrm{A}}_{\mathrm{s}}-{\mathrm{f}}_{\mathrm{rc}}{\mathrm{A}}_{\mathrm{s}1}\right)\left({\mathrm{h}}_0-{\displaystyle \frac{{\mathrm{f}}_{\mathrm{rt}}{\mathrm{A}}_{\mathrm{s}}-{\mathrm{f}}_{\mathrm{rc}}{\mathrm{A}}_{\mathrm{s}1}}{2{\mathrm{\alpha }}_1{\mathrm{f}}_{\mathrm{cc}}\mathrm{c}}}\right)+{\mathrm{f}}_{\mathrm{rc}}{\mathrm{A}}_{\mathrm{s}1}({\mathrm{h}}_0-{\mathrm{a}}_{\mathrm{s}1}),$$

(11)

$${\mathrm{h}}_0=\mathrm{h}-{\mathrm{a}}_{\mathrm{s}}$$

(12)

where,

• M_u—Section under the bending moment, N·m;
• A_s1—Total stirrup cross-sectional area in the section’s compression zone, mm²;
• h₀—Section’s effective depth, mm;
• h—Sectional height, mm;
• a_s—Thickness of the inner protective layer of the rectangular culvert, mm;
• a_s1—Thickness of the outer protective layer of the rectangular culvert, mm.

When the rectangular culvert is damaged under the action of trilateral normal external load

$${\mathrm{M}}_{\mathrm{u}}={\mathrm{M}}_{\max }.$$

(13)

By combining Formulae (7), (11) and (13), the maximum load that the double-layer-reinforced rectangular culvert can bear under the condition of three-side normal external load could be obtained.

$${\mathrm{F}}_{\max }={\displaystyle \frac{\left(\left({\mathrm{f}}_{\mathrm{rt}}{\mathrm{A}}_{\mathrm{s}}-{\mathrm{f}}_{\mathrm{rc}}{\mathrm{A}}_{\mathrm{s}1}\right)\left({\mathrm{h}}_0-\frac{{\mathrm{f}}_{\mathrm{rt}}{\mathrm{A}}_{\mathrm{s}}-{\mathrm{f}}_{\mathrm{rc}}{\mathrm{A}}_{\mathrm{s}1}}{2{\mathrm{\alpha }}_1{\mathrm{f}}_{\mathrm{cc}}\mathrm{c}}\right)+{ \mathrm{f}}_{\mathrm{rc}}{\mathrm{A}}_{\mathrm{s}1}({\mathrm{h}}_0-{\mathrm{a}}_{\mathrm{s}1})\right)\left(24\mathrm{a}+\mathrm{b}\right)}{3{\mathrm{a}}^2+2\mathrm{ab}}}$$

(14)

where,

• F_max—The maximum load of the rectangular culvert, N;
• M_max—Maximum bending moment of the cross-section, N·m.

3.2. UBC Calculation Model for Cracked Rectangular Culvert

This study primarily focused on the effect of crack depth on the UBC of rectangular culverts. It considered only the variation in geometric parameters, ignoring material deterioration and the reduction in material interaction forces. The tension zone is usually considered as a dangerous section in reinforced concrete structures. For both non-defective and crack defect rectangular culverts, the tensile zones are concentrated in the top plate, the interior of the bottom plate, the outer wall of the side plates, and the top and bottom corners of the rectangular culvert. The mid-span of the roof is considered a critical section where cracks are positioned on the inner wall of the mid-span, as illustrated in Figure 9.

The stress analysis of the crack defect rectangular culvert was carried out, and the following assumptions were made: (1) Crack defects are often accompanied by other defects, such as corrosion. This section only considers the influence of single crack defects on the mechanical properties of rectangular culverts. (2) The mechanical parameters of concrete and steel bar materials in the rectangular culvert with crack defects will change. The strength reduction coefficient kc of concrete and the strength reduction coefficient k_r of the steel bar material are introduced to represent the influence of concrete and steel bar deterioration on strength, respectively. k_c and k_r can be measured using relevant laboratory tests. (3) Cracks significantly affect the bonds between reinforced concrete. A reduction coefficient k_cr is introduced to describe the change in bonding behavior between concrete and the steel reinforcement in the presence of cracks in the defective pipeline. k_cr can be measured by the direct shear test. The value of k_cr is related to crack depth, the crack progression degree of the reinforced concrete bonding section, and other factors. (4) The effect of crack depth on the mechanical properties of the rectangular culvert can be regarded as the change in component thickness. With the increase in crack depth, the thickness of reinforced concrete members becomes smaller. Considering steel bar corrosion and concrete bonding, the impact of crack depth on rectangular culvert members can be categorized into two cases. First, when the crack depth is less than the protective layer thickness, only the compressive strength of concrete is compromised, while the size and bonding effect remains unchanged, and internal steel bars are unaffected. Second, when the crack depth exceeds the protective layer thickness, both the tensile strength of steel bars and their bonding with concrete are weakened, along with a reduction in the concrete compressive strength.

Based on the above assumptions, the maximum bending moment of the double-reinforced rectangular section after a crack defect was determined by comparing the crack depth with the protective layer thickness of the reinforced concrete member:

$${\mathrm{M}}_{\mathrm{u}1}=\left\{\begin{array}{c}{\mathrm{k}}_{\mathrm{cr}}\left({\mathrm{\alpha }}_1{{\mathrm{k}}_{\mathrm{c}}\mathrm{f}}_{\mathrm{cc}}\mathrm{c}{\mathrm{x}}_1\left({\mathrm{h}}_0-{\displaystyle \frac{{\mathrm{x}}_1}{2}}\right)+{\mathrm{f}}_{\mathrm{rc}}{\mathrm{A}}_{\mathrm{s}1}({\mathrm{h}}_0-{\mathrm{a}}_{\mathrm{s}1})\right), \mathrm{t}\le {\mathrm{a}}_{\mathrm{s}}\\ {\mathrm{k}}_{\mathrm{cr}}\left({\mathrm{\alpha }}_1{{\mathrm{k}}_{\mathrm{c}}\mathrm{f}}_{\mathrm{cc}}\mathrm{c}{\mathrm{x}}_1\left(\mathrm{h}-{\displaystyle \frac{{\mathrm{x}}_1}{2}}-\mathrm{t}\right)+{{\mathrm{k}}_{\mathrm{r}}\mathrm{f}}_{\mathrm{rc}}{\mathrm{A}}_{\mathrm{s}1}(\mathrm{h}-{\mathrm{a}}_{\mathrm{s}1}-\mathrm{t})\right), \mathrm{t}>{\mathrm{a}}_{\mathrm{s}}\end{array}\right.,$$

(15)

$${\mathrm{x}}_1={\mathrm{k}}_{\mathrm{r}}{\displaystyle \frac{{\mathrm{f}}_{\mathrm{rt}}{\mathrm{A}}_{\mathrm{s}}-{\mathrm{f}}_{\mathrm{rc}}{\mathrm{A}}_{\mathrm{s}1}}{{\mathrm{\alpha }}_1{\mathrm{f}}_{\mathrm{cc}}\mathrm{c}}}={\mathrm{k}}_{\mathrm{r}}\mathrm{x}$$

(16)

where,

• M_u1—The maximum bending moment of the rectangular culvert section with crack defects, N·m;
• k_c—Reduction coefficient of the concrete compressive strength;
• k_r—Compressive strength reduction factor of the concrete/steel bar (the reduction in the tensile strength of the steel bar is roughly equivalent to that of the compressive strength);
• k_cr—The bond strength reduction coefficient between the concrete and steel bar;
• t—Fracture depth, mm;
• x₁—Crack defect section concrete compression zone height, mm.

While

$${\mathrm{M}}_{\mathrm{u}1}={\mathrm{M}}_{\max },$$

(17)

the maximum load calculation formula of a double-layer-reinforced crack defect rectangular culvert under the condition of a three-side normal external load can be obtained:

$${\mathit{F}}_{\max 1}=\left\{\begin{array}{c}{\displaystyle \frac{{\mathit{k}}_{\mathit{cr}1}\left(24\mathit{a}+\mathit{b}\right)\left({\mathit{\alpha }}_1{{\mathit{k}}_{\mathit{c}1}\mathit{f}}_{\mathit{cc}}\mathit{c}{\mathit{x}}_1\left({\mathit{h}}_{0 }-\frac{{\mathit{x}}_1}{2}\right)+{\mathit{f}}_{\mathit{rc}}{\mathit{A}}_{\mathit{s}1}({\mathit{h}}_0-{\mathit{a}}_{\mathit{s}1})\right)}{3{\mathit{a}}^2+2\mathit{ab}}}, \mathit{t}\le {\mathit{a}}_{\mathit{s}}\\ {\displaystyle \frac{{\mathit{k}}_{\mathit{cr}1}\left(24\mathit{a}+\mathit{b}\right)\left({\mathit{\alpha }}_1{{\mathit{k}}_{\mathit{c}1}\mathit{f}}_{\mathit{cc}}\mathit{c}{\mathit{x}}_1\left(\mathit{h} -\frac{{\mathit{x}}_1}{2}-\mathit{t}\right)+{{\mathit{k}}_{\mathit{r}1}\mathit{f}}_{\mathit{rc}}{\mathit{A}}_{\mathit{s}1}(\mathit{h}-{\mathit{a}}_{\mathit{s}1}-\mathit{t})\right)}{3{\mathit{a}}^2+2\mathit{ab}}}, \mathit{t}>{ \mathit{a}}_{\mathit{s}}\end{array}\right..$$

(18)

The data in Table 2 were substituted into Formula (18) for calculating the UBC of rectangular culverts with varying crack depths. In the test, cracks at different depths were prefabricated by the manual cutting method, and the pouring materials and curing conditions of each group of rectangular culverts were the same as those of the A1 group of non-defective rectangular culverts. Therefore, the strength reduction coefficient k_r of the steel bar and the strength reduction coefficient kc of concrete in the model of the UBC of rectangular culverts with cracks at different depths were taken as one. The reduction coefficient k_cr value of the bond strength of cracked reinforced concrete and the theoretical calculation value of the UBC of the rectangular culvert are presented in

Table 4. Bond strength reduction factor and UBC of reinforced concrete of rectangular culvert with crack defects.

Groups	Bond Strength Reduction Factor of Reinforced Concrete	The Theoretical Calculation Value of the UBC (N)
A1	1	7176.17
C1	0.9	6418.77
C2	0.8	5670.65
C3	0.7	4931.63

. The test value, theoretical calculation value, and error of the UBC of rectangular culverts with different depth crack defects are shown in Figure 10.

Based on Figure 10, the calculated UBC of the non-defective rectangular culvert in group A1 is 7176.17 N, while the value measured through the trilateral normal outward load test is 7768.24 N, resulting in an error of 8.25%, which is within acceptable limits (10%). As the crack depth increases, both theoretical and experimental values for the rectangular culvert’s UBC decline. Consequently, the theoretical capacity value decreases steadily with an increasing crack depth, resulting in a linear trend. When the crack depth of the inner wall of the rectangular culvert roof is 2.5 mm, the C1 group’s UBC is 6418.77 N, a 10.49% reduction compared to the non-defective rectangular culvert, with an error of 4.99% compared to the experimental value. For a crack depth of 5 mm, the calculated capacity for the C2 group is 5670.65 N, showing a 20.99% reduction compared to the non-defective culvert and an error of 12.68% compared to the experimental value. When the crack depth reaches 7.5 mm, equal to the inner protective layer’s thickness, the calculated ultimate capacity for the C3 group is 4931.63 N, a reduction of 31.28% with an error of 5.82% compared to the experimental results. The results from the four experimental groups indicate that when the crack depth matches the protective layer’s thickness, the rectangular culvert’s bearing capacity is reduced by approximately 30%. As the crack depth increases, both the theoretical and test values of the UBC of the rectangular culvert with crack defects decrease. Additionally, the discrepancy between the theoretical and experimental values is minimal, with the theoretical values being lower, suggesting that the theoretical calculations provide a more conservative estimate. The UBC of rectangular culverts with different depths of crack defects can be estimated using Formula (18).

According to the fracture propagation mechanism, when a crack appears in a rectangular box culvert, the stress concentration will occur at the crack tip. This means that the stress level near the tip of the crack will be much higher than in the region away from the crack. With the increase in crack depth, the deflection of the roof of the rectangular box culvert increases, and the stress concentration effect at the crack tip will be more significant, leading to a further increase in the stress level in this area. Therefore, the originally uniformly distributed stress will be redistributed due to the existence of cracks. Cracks weaken the overall strength and stiffness of the structure, making it easier for the structure to reach the limit state when subjected to the same external force.

4. Conclusions

In this study, the mechanical properties of rectangular culverts with crack defects were examined. The key findings are summarized as follows:

• In the trilateral external load test, the failure process of the rectangular culvert can be categorized into three distinct phases: the initial stage with no visible cracks, the crack progression stage, and the final failure stage. During the intact stage, the load applied to the rectangular culvert increased proportionally with displacement. In the crack progression phase, the load continued to grow as the displacement increased, with a decreasing curvature of the load–displacement curve, and cracks gradually appeared and expanded along the culvert wall. In the final failure stage, the rectangular culvert’s structural integrity was compromised, with the load first increasing and then declining.
• Because the change in crack depth will change the bond strength between reinforced concrete, it is necessary to discuss the reduction coefficient k_cr of the bond strength between reinforced concrete. Based on the comparison between the test results and the theoretical calculation of the rectangular culvert’s UBC, it can be inferred that the reduction coefficient kc for the bond strength of reinforced concrete is linked to the ratio of the crack depth to the protective layer thickness. The error between the theoretical calculation correction value and the test value obtained by this calculation method is about 10%, so the calculation formula is suitable for the calculation model of the UBC of the crack defect pipeline considering only the single influence factor of the crack depth.
• The effect of crack depth on the bearing capacity of rectangular culverts can be considered as a reduction in the wall thickness. When the crack depth is less than or equal to the thickness of the concrete protective layer, the UBC of the rectangular culvert reduces as the crack depth increases. When the crack depth equals the protective layer’s thickness, both the test value and calculated value of the UBC decrease by 31%, with an error of less than 10%. This study demonstrates that an increasing crack depth leads to a substantial reduction in the bearing capacity, with implications for the design and maintenance of urban drainage structures. In the follow-up monitoring of components, if cracks are found, surface repair, injection repair, replacement repair, paste repair, and other technologies can be used to treat the cracks according to their different causes and characteristics.