Propagation Effect Analysis of Existing Cracks in Box Girder Bridges Based on the Criterion of Compound Crack Propagation

Abstract

Cracking in concrete box girder bridges will have a significant impact on the safety and durability of the structure, and many box girder bridges which are in service have undergone varying degrees of cracking. Currently, the safety design of actual bridge projects place an emphasis on the stress or the load value of a cross section at the limit value specified in the code for safety control. This design method assumes that the member itself is of uniform and continuous material and is internally undamaged. However, the bridge structure is more or less cracked to varying degrees during the period from casting to construction to operation of the concrete members. In this paper, a finite element computational model of a three-span prestressed concrete box girder bridge with existing cracks is established based on the fracture mechanics theory, and the critical parameters of crack extension are introduced to evaluate the extension state of cracks. At the same time, the extended stability of the existing cracks of the box girder bridge is analyzed by considering the temperature effect, vehicle loading, and prestressing loss, and the sensitivity of crack extension under each working condition is investigated. The results show that, with the increase in crack length and depth, the crack expansion is promoted, but the effect is relatively small, and the maximum stress intensity factor is only 6.48 MPa mm^1/2. Under the multi-factor coupling effect, the cracks show a composite crack expansion dominated by type I cracks, the longitudinal cracks of the existing base plate are in a stable state, the maximum value of the crack expansion critical parameter of the vertical cracks of the webs reaches 1.087, and there is a tendency to expand locally. The maximum value of the critical parameter for crack extension of the vertical crack in the web plate reaches 1.087, and there is a tendency towards local expansion. The crack extension evaluation criteria proposed in this paper have a certain reference value for crack extension research on similar concrete box girder bridges and provide a scientific basis for the optimized design of similar bridges.

1. Introduction

Prestressed concrete box girder bridges are subjected to a variety of operating conditions in the long-term operation process; there are generally different degrees of concrete cracking problems, and the existence of cracks reduces the effective cross section of the concrete structure, weakening the stiffness of the concrete box girder and making the mid-span deflection increase, thus reducing the safety and durability of the bridge [1]. In the past, the bridge collapsed before the bridge reached its standard strength and was damaged, often due to the existence of cracks in the fatigue damage caused by the expansion of instability. Therefore, it is particularly important to investigate whether the existing cracks in concrete box girder bridges further expand during operation [2].

The problem of cracking in concrete arises from the initial defects inherent in its being a non-homogeneous material, and these small cracks will gradually expand to form macroscopic cracks when subjected to different loads. The traditional strength theory assumes the concrete material to be continuous and homogeneous, which has limitations [3]. However, the application of fracture mechanics theory presupposes the existence of macroscopic cracks in the structure, which can make up for the shortcomings of the traditional strength theory. Fracture mechanics theory can more accurately describe the crack behavior and expansion law of concrete when subjected to external forces and provide a theoretical basis for the design, assessment, and maintenance of existing cracked concrete structures.

Scholars at home and abroad have carried out relevant studies on concrete cracking based on the theory of fracture mechanics. Walsh [4] investigated precast notched concrete beams using LEFM, focusing on the effect of concrete beam dimensions on the fracture parameters, and pointed out that the size of the member has a significant effect on the extrapolation of the actual structural fracture parameters, i.e., there is a size effect. Li et al. [5] used resistance strain gauges to measure the initiation loads of three-point bending beams with different heights and deduced the initiation fracture toughness and the destabilization fracture toughness under the double K model based on the crack opening displacement curves. Xu Shilong collected experimental data on the fracture toughness of concrete three-point bending specimens and conducted a probabilistic modeling study [6]. The results showed that the fracture toughness of concrete conformed to the Weibull distribution, revealing the phenomenon that the fracture toughness of concrete is affected by size effect. Combined with the reliability theory, an expression for the statistical distribution of the fracture toughness of concrete specimens was proposed and the relative failure probability of a structural fracture was calculated. Feng Bolin studied the fracture mechanics analysis method of arch dam cracks under temperature loading. He concluded that it is reasonable to apply the theory and method of fracture mechanics to the analysis and calculation of crack extension in dams as well as to practical engineering [7]. Yanhua Zhao proposed the double G criterion for concrete crack extension from the perspective of energy, combining the linear elastic fracture mechanics and the viscous cohesion distribution in the virtual crack zone with the energy release rate as a parameter [8]. Wu Zhimin proposed the corresponding crack extension criterion to calculate the whole process of I–II composite crack extension based on the experimentally measured critical K_I–K_II curves of I–II composite cracks in concrete materials [9]. Fan Guogang used the interaction integral method to study the fracture parameters of different crack patterns in concrete dams and used the cohesive crack model to analyze the effect of crack patterns on the bearing capacity of concrete structures [10]. Chen Xingda used a variable stiffness crack model to investigate the dynamic characteristics of an existing cracked T-beam under vehicular loading and evaluated the crack extension state using a three-dimensional dynamic stress intensity factor [11]. Mi Zhengxiang proposed the concept of fracture strength by combining the traditional strength theory with fracture mechanics theory for the problem of concrete cracking in dams, which brought about new ideas for the design of concrete structures [12]. Li-Qiong Ren used the cohesive crack model to study the subcritical crack expansion stage of concrete before complete failure and used fracture mechanics-related parameters to assess the mechanism of subcritical crack expansion -n concrete [13].

Based on fracture mechanics, most of the cracking problems of concrete by the above scholars focus on the test members, and there is a lack of research on the crack propagation of practical projects. In this paper, the finite element modeling of a prestressed concrete box beam bridge with existing cracks is carried out and the composite crack expansion evaluation criteria based on fracture mechanics is introduced. Under the comprehensive consideration of temperature effect, vehicle load, prestress loss, and other factors, the expansion stability of existing cracks is analyzed, which provides references for the design of and research on similar bridge structures.

2. Basic Theory of Fracture Mechanics and Extended Evaluation Criteria

In fracture mechanics, according to the stress type and displacement characteristics of cracks, the fracture morphology is divided into three types: open (type I) cracks, in-plane shear (type II) cracks, and anti-plane shear (type III) cracks, as shown in Figure 1 [14].

In practical engineering, because of the action form of loads, the different characteristics of materials and the development direction of cracks, cracks are often presented in a composite form, and people are concerned about the expansion conditions and directions of cracks, that is, the critical state and angle of composite crack expansion [15]. Therefore, it is necessary to seek a reasonable and reliable crack propagation criterion for judgment. Currently, the common composite crack propagation criteria include the maximum circumferential tensile stress intensity factor criterion, the maximum energy release rate criterion, and the K judgment criterion based on composite critical curves. The first two types of criteria are mostly used for metal materials. Due to the particularity of concrete materials, it has certain limitations, so this paper adopts the K judgment criterion of composite critical curve, which is quoted in the data of fracture test conducted on concrete I–II composite cracks in Yu Xiaozhong [16] and gives the curve expression of the empirical fracture criteria of I–II composite cracks as follows:

$${\left({\displaystyle \frac{K_I}{K_{IC}}}\right)}^2+4.2{\left({\displaystyle \frac{K_{II}}{K_{IC}}}\right)}^2=1$$

(1)

According to the above expression, the critical parameter of crack propagation $\phi$ is introduced in this paper to describe whether existing cracks enter the expanding state under the multi-factor coupling action of concrete box beams, as shown in Equation (2):

$$\phi ={\displaystyle \frac{K_I^2+4.2K_{II}^2}{K_{IC}^2}}$$

(2)

According to Equations (1) and (2), when the critical parameter of crack propagation is $\phi =1$, the crack is in a critical state of propagation. $\phi >1$, the cracks were in a spreading state; $\phi <1$, the cracks were stable. However, it should be emphasized that, although $\phi <1$ may indicate that the fracture is stable at the macro level, in some cases the fracture propagation trend is still affected by the stress intensity factor at the crack tip. Therefore, in addition to the evaluation $\phi$, it is necessary to comprehensively consider the stress field of the crack tip under the action of different factors and the influence on the stress intensity factor to judge the potential expansion behavior of the existing stable crack.

In the determination of the fracture toughness of concrete structures ${\mathrm{K}}_{\mathrm{I}\mathrm{c}}$, it has been observed that the values measured by small size specimens are affected by the size effect and are not applicable to large-size concrete structures. Through the fracture testing of large-size concrete specimens, Wu Zhimin [17] found that, when the height of the specimen was above 800 mm, the measured value ${\mathrm{K}}_{\mathrm{I}\mathrm{c}}$ had nothing to do with the size of the specimen. However, for concrete box girder bridges, it is very difficult to carry out large-scale fracture tests of the same size or scale as the structure. In view of this, based on the method proposed by Mai Jiaxuan [18] and Xu Shi-lang [6], this paper deduces the fracture toughness suitable for large-size specimens from the measured values ${\mathrm{K}}_{\mathrm{I}\mathrm{c}}$ of small-size specimens. The corresponding size effect equation is as follows:

$$K_{IC}=K_{IC\mathrm{s}}\sqrt{{\displaystyle \frac{h}{h_s}}}{\left({\displaystyle \frac{V_s}{V}}\right)}^{{\displaystyle \frac{1}{\alpha }}}$$

(3)

where, ${\mathrm{K}}_{\mathrm{ICs}}{, \mathrm{h}}_{\mathrm{s}}{, \mathrm{V}}_{\mathrm{s}}$ are corresponding to the fracture toughness, height, and volume of small-size specimens; ${\mathrm{K}}_{\mathrm{IC}}, \mathrm{h}, \mathrm{V}$ are derived the fracture toughness, height, and volume of large-size specimens, wherein h ≤ 800 mm; and $\alpha$ is the Weibull modulus, which is related to the coefficient of variation in the experimental value of fracture toughness ${\mathrm{K}}_{\mathrm{I}\mathrm{c}}$ measured, with the specific value being referred to in the literature.

The above equation is used to calculate the fracture toughness of specimens of different sizes in the literature; the relative error between the results and the test values is less than 10% and the agreement is high. Therefore, this paper refers to the wedge splitting test conducted by Gao Hongbo [19]; the specimen size is 300 mm × 300 mm × 200 mm and the concrete strength grade is C50. The test results show that the cubic compressive strength, tensile strength, and elastic modulus of the concrete specimen are 50.2 MPa, 3.95 Mpa, and 34.8 Gpa, and the average fracture toughness of the concrete specimen is 1.147 $\mathrm{Mpa}\sqrt{\mathrm{m}}$. Based on the small size specimen, the fracture toughness of the prestressed concrete continuous box girder bridge is 0.8781 $\mathrm{Mpa}\sqrt{\mathrm{m}}$ by Equation (3). This result is consistent with the interval range of size-independent fracture toughness of C50 concrete measured by Xu Shi-lang [20].

3. Finite Element Model

This paper takes a prestressed concrete continuous box girder bridge as the engineering background. The total length of the box girder bridge is 869 m with nine connections. Among them, the crack phenomenon of the third joint is the most prominent, so this paper takes the third joint box girder bridge as the research object. The span arrangement of the joint is 35 m + 35 m + 35 m. The prestressed concrete box girder structure with a single box and four chambers is adopted, and the beam height is 1.8 m. The thickness of the top plate and bottom plate of the box girder is 0.25 m, while the top plate thickness of the cantilever part is 0.2 m and the length of the cantilever is 2.5 m. The horizontal width of the inclined web is 0.433 m. The width of the middle beam is 2 m and the width of the end beam is 1.5 m. In addition, the width of the box girder top plate is 20 m, and the width of the bottom plate is 15 m. The general layout of the bridge and the section layout of the box girder are shown in Figure 2 and Figure 3, respectively.

In this paper, the three-dimensional finite element model is established by ANSYS Workbench 16.0 and the concrete and rebar units are simulated by SOLID186 and BEAM188 units, respectively. The hexahedral grid division method is used to grid the concrete solid model, and the rebar is meshed separately. The initial stress is applied to the prestressed steel bar by the cooling method, several nodes of the concrete unit are selected by CEINTF command, and the constraint equation is established with the nodes of the steel bar in the tolerance range to realize the coupling between the prestressed steel bar and the concrete node. According to the actual situation of the bridge, the boundary conditions are set and the external load is simulated. The finite element model of prestressed concrete continuous box girder bridge has a total of 520,460 nodes and 939,448 units. The overall wire frame and grid division are shown in Figure 4 and Figure 5.

The main objective of the bridge static load test is to assess whether the actual working condition of the bridge structure is consistent with the design expectation by measuring the deformation, stress, and strain responses of the bridge structure under static loading. The test not only directly and effectively verifies the mechanical properties of the bridge structure but also provides key data to verify that the finite element model accurately simulates the initial stress state of the bridge.

(1)

Test control section

The static loading test takes into account the distribution characteristics of bridge cracks, structural characteristics, and the actual situation on site, and it selects the maximum positive bending moment cross section of the third span and the maximum positive bending moment cross section of the second span of the link as the test control cross sections.

(2)

Loading method

According to the current situation of the bridge structure, this static load test adopts the automobile-loading method.

(3)

Layout of test points

The selected test control sections include the A-A section of the side span, the B-B section subjected to negative moments, and the C-C section of the center span. The layout of the test sections is shown in Figure 6, and the arrangement of the measurement points for each section is shown in Figure 7.

In order to enable the solid finite element model to accurately reflect the stress state of the prestressed concrete continuous box girder bridge under the action of multiple factors, the static load test data of the bridge with the maximum positive bending moment control section at the mid-span and side spans were compared and analyzed. The related measured and calculated values of deflection and strain are detailed in

Table 1. Side span maximum positive moment control section measured point values.

Measurement Point	$\mathbf{Deflection}/\mathbf{m}\mathbf{m}$		Relative Error	$\mathbf{Strain}/\mathsf{\mu }\mathsf{\epsilon }$		Relative Error
	Measured Value	Calculated Value		Measured Value	Calculated Value
A1	1.75	2.74	56.57%	11	17	54.55%
A2	2.45	3.37	37.56%	9	28	211.11%
A3	3.92	4.70	19.90%	30	41	36.67%
A4	5.93	6.97	17.54%	47	58	23.40%
A5	7.30	8.86	21.37%	22	83	277.27%

and

Table 2. Mid-span maximum positive moment control section measured point values.

Measurement Point	$\mathbf{Deflection}/\mathbf{m}\mathbf{m}$		Relative Error	$\mathbf{Strain}/\mathsf{\mu }\mathsf{\epsilon }$		Relative Error
	Measured Value	Calculated Value		Measured Value	Calculated Value
C1	1.65	2.11	27.88%	9	14	55.56%
C2	2.38	2.65	11.34%	12	26	116.67%
C3	3.46	3.86	11.56%	25	39	56%
C4	5.04	5.85	16.07%	86	61	29.07%
C5	5.85	7.09	21.19%	55	79	43.64%

, respectively.

By comparing the finite element simulation results in the above table with the static load test data of the bridge, the deflection and strain values of the corresponding measurement points are more consistent with the measured data, and the rule of change is consistent. However, some measurement points still show large errors [21,22]. These errors are mainly attributed to the existence of new cracks in the actual structure, resulting in the distortion of the strain values at some measurement points. In addition, the discrepancy between the inhomogeneity of the actual concrete material and the ideal material properties assumed in the model, coupled with the deviation of the eccentric or locally concentrated force of the actual bridge load test from the model setting and the actual complexity of the bearing and boundary conditions not accurately reflected in the model, may lead to the discrepancy between the actual stress state and the simulation results [23]. Although the model shows some errors due to the above factors, the overall error is still within the acceptable range, indicating that the model can effectively simulate the initial static load-bearing behavior of the bridge under the basic working conditions and can be used for the subsequent analysis of the expansion behavior of existing cracks in box girders [24].

4. Expansion Effect Analysis of Box Girder Bridges with Existing Cracks

According to the spatial stress distribution of the box girder and the location where the actual cracks are concentrated, the cracks are selected based on the longitudinal cracks in the bottom plate of the middle span and the vertical cracks in the web plate of the side span. According to the actual bridge test, the most unfavorable loading arrangement is selected for the vehicle load and the most unfavorable loading is arranged for the middle and side spans; the specific crack parameters and arrangement are shown in

Table 3. Set basic crack parameters.

Fracture Morphology	Length (mm)	Depth (mm)
Longitudinal cracks in mid-span floor	$800~1600 (\mathsf{\Delta }200)$	$10~50 (\mathsf{\Delta }10$)
Vertical crack of side span web	$300~500(\mathsf{\Delta }50)$

and Figure 8.

4.1. Analysis of Influencing Factors of Box Girder Crack Propagation

In order to investigate the effects of different influencing factors on the stress and extension behavior of longitudinal cracks in the bottom plate and vertical cracks in the web plate of the box girder [25], the effects of crack characteristics, temperature change, and prestress loss on the stress distribution and crack extension pattern of the crack tips are investigated under the combined action of the box girder’s self-weight, the second-phase constant load, and the prestressing stress as the basic working condition [26,27,28].

The calculation working condition is set up in combination with the field bridge static load test in order to more accurately respond to the stress state of the prestressed concrete continuous box girder bridge; the finite element model of the load is applied in accordance with the relevant provisions of the JTG D60-2015 [29] general specification for the design of highway bridges to load, and the analysis of the overall spatial cracking effect of the box girder is mainly considered for the following influencing factors:

(1)

Phase I dead load: Box girder bridge dead weight.

(2)

Phase II dead load: Bridge deck pavement layer, anti-collision guardrail and other facilities, a total of 129.5 kN/m.

(3)

Vehicle load: The bridge road grade is a double-width eight-lane first-level highway. According to the regulations, the lateral reduction coefficient should be considered when the multi-lane loading is carried out in the overall analysis and calculation of the bridge. The model in this paper is a single four-lane road with a reduction coefficient of 0.67. qk = 10.5 kN/m, Pk = 360 kN. The loading schematic for single-lane loading is shown in Figure 9.

(4)

Temperature gradient: According to the provisions of the general specification for highway bridge design, the vertical temperature gradient is selected for loading, as shown in Figure 10. The specifications specify T1 and T2 of different paving types. The paving layer thickness of the box girder bridge in this paper is 10 cm asphalt concrete, so the vertical positive temperature gradient T1 = 14 °C and T2 = 5.5 °C. The vertical negative temperature gradient of the concrete superstructure is a positive temperature gradient multiplied by −0.5, i.e., negative temperature gradient T1 = −7 °C, T2 = −2.75 °C.

4.1.1. Effect of Fracture Length on Fracture Propagation

(1)

Longitudinal crack of floor

Taking the mid-span longitudinal crack of the bottom plate with a crack depth of 30 mm as the reference, five working conditions with crack lengths of 800 mm, 1000 mm, 1200 mm, 1400 mm, and 1600 mm were considered, respectively, to extract the maximum stress intensity factor of the crack tip and draw the integrated path value into a graph, as shown in

Table 4. Maximum value of stress intensity factor in longitudinal crack length of different floor.

Longitudinal Crack Length (mm)	800	1000	1200	1400	1600
${\mathrm{K}}_{\mathrm{I}} (\mathrm{MPa}·{\mathrm{mm}}^{1/2})$	−0.919	−0.235	−0.889	0.209	1.227
${\mathrm{K}}_{\mathrm{II}} (\mathrm{MPa}·{\mathrm{mm}}^{1/2})$	0.508	3.01	2.21	3.22	6.48
${\mathrm{K}}_{\mathrm{III} }(\mathrm{MPa}·{\mathrm{mm}}^{1/2})$	0.377	1.03	−1.38	−1.73	−3.52

and Figure 11.

It can be seen from Table 4 and Figure 11 that, under the basic working conditions, the stress intensity factor changes in the longitudinal crack tip of the bottom plate in a similar way, increasing first and then decreasing from the two ends of the crack to the middle. This trend becomes more obvious with the increase in fracture length, and the peak value tends to appear in the middle of the fracture, while the value at both ends of the fracture is relatively stable and does not change significantly with the increase in fracture length. The maximum value of the stress intensity factor increases with the increase in crack length, and the maximum value is 6.48 $\mathrm{MPa}·{\mathrm{mm}}^{1/2}$. K_II is larger than K_I and K_III, and, with the increase in fracture length, the mixture of the types of fractures, mainly type II, is the main factor affecting the longitudinal fracture propagation of the floor.

(2)

Vertical cracks in web

Taking the vertical crack of the side span web with a crack depth of 30 mm as the basis, five working conditions of crack lengths of 300 mm, 350 mm, 400 mm, 450 mm, and 500 mm were considered, respectively, the extreme value of the stress intensity factor of the crack tip under each working condition was extracted, and the integral path value was plotted as a graph, as shown in

Table 5. The maximum value of stress intensity factor for different vertical crack lengths of web.

Vertical Crack Length of Web (mm)	300	350	400	450	500
${\mathrm{K}}_{\mathrm{I}} (\mathrm{MPa}·{\mathrm{mm}}^{1/2})$	−79.157	−80.282	−81.152	−81.832	−82.659
${\mathrm{K}}_{\mathrm{II}} (\mathrm{MPa}·{\mathrm{mm}}^{1/2})$	−0.532	0.513	0.566	0.613	0.727
${\mathrm{K}}_{\mathrm{III}} (\mathrm{MPa}·{\mathrm{mm}}^{1/2})$	−1.67	−1.63	−1.654	−1.824	−1.803

and Figure 12.

As can be seen from Table 5 and Figure 12, with the increase in the vertical crack length of the web, the K_I value increases somewhat at both ends of the crack, but the increase is small; meanwhile, the extreme value in the middle of the crack decreases continuously, showing a large negative value as a whole, with the maximum negative value being −82.659 $\mathrm{MPa}·{\mathrm{mm}}^{1/2}$. It is shown that the presence of large prestress in the longitudinal direction of the web has a significant inhibitory effect on the expansion of type I cracks. For K_II and K_III, the maximum value increases with the increase in crack length, and the fracture end away from the floor side shows a higher value. The results show that the propagation of type I cracks is inhibited under the basic working conditions, and type II and type III cracks are the main factors affecting the vertical crack propagation of the web.

4.1.2. Effect of Fracture Depth on Fracture Propagation

(1)

Longitudinal crack of floor

Taking the mid-span longitudinal crack of the bottom plate with the crack length of 1200 mm as the basis, five working conditions with crack depths of 10 mm, 20 mm, 30 mm, 40 mm, and 50 mm were considered, respectively. The maximum stress intensity factor of the crack tip under each working condition was extracted, and the integrated path value was plotted as a graph, as shown in

Table 6. The maximum value of stress intensity factor at different longitudinal crack depth of floor.

Longitudinal Crack Depth (mm)	10	20	30	40	50
${\mathrm{K}}_{\mathrm{I}} (\mathrm{MPa}·{\mathrm{mm}}^{1/2})$	−0.984	−1.744	−1.995	−2.438	−2.612
${\mathrm{K}}_{\mathrm{II} }(\mathrm{MPa}·{\mathrm{mm}}^{1/2})$	−0.676	−0.704	−2.217	−1.392	−1.436
${\mathrm{K}}_{\mathrm{III} }(\mathrm{MPa}·{\mathrm{mm}}^{1/2})$	−0.607	−0.993	−1.381	−1.573	−1.992

and Figure 13.

As can be seen from Table 6 and Figure 13, with the increase in crack depth, the change in stress intensity factor presents a trend in first increasing, then decreasing, and then increasing. A peak value of −2.217 $\mathrm{MPa}·{\mathrm{mm}}^{1/2}$ occurs at the crack tip when the depth is 30 mm and stabilizes with the increasing crack depth. The distribution of K_I and K_III at the fracture tip is stable, and their maximum values appear in the middle of the fracture and increase with the increase in fracture depth. All K_I values are negative, which indicates that the influence of typeIcrack propagation decreases with the increase in crack depth, and type II and III cracks are the main factors affecting the longitudinal crack propagation of the floor.

(2)

Vertical cracks in web

Taking the vertical crack of the side span web with a crack length of 400 mm as the basis, five working conditions with crack depths of 10 mm, 20 mm, 30 mm, 40 mm, and 50 mm were considered, respectively, and the maximum stress intensity factor of the crack tip under each working condition was extracted, while the integral path value was plotted as a graph, as shown in

Table 7. The maximum value of stress intensity factor at different vertical crack depths of web.

Vertical Crack Depth of Web (mm)	10	20	30	40	50
${\mathrm{K}}_{\mathrm{I}} (\mathrm{MPa}·{\mathrm{mm}}^{1/2})$	−21.556	−65.692	−81.152	−91.443	−98.956
${\mathrm{K}}_{\mathrm{II} }(\mathrm{MPa}·{\mathrm{mm}}^{1/2})$	−0.816	−0.388	0.541	0.503	1.187
${\mathrm{K}}_{\mathrm{III}} (\mathrm{MPa}·{\mathrm{mm}}^{1/2})$	−0.324	−1.552	−1.568	−1.891	−3.169

and Figure 14.

As can be seen from Table 7 and Figure 14, with the increase in vertical crack depth, the maximum value of K_I continues to increase with a large range of changes, all of which are negative. When the depth reaches 30 mm, the change tends to be stable. K_II showed a trend in first decreasing and then increasing, and its overall distribution had no obvious rule, while the maximum value mainly appeared at the crack tip far away from the bottom plate. When the crack depth is 50 mm, the stress intensity factor can reach 1.187 $\mathrm{MPa}·{\mathrm{mm}}^{1/2}$. K_III increases with depth, but the overall change is relatively gentle. The value of K_I is much higher than that of K_II and K_III and is sensitive to the change in fracture depth. The results show that, with the increase in fracture depth, the propagation of type I fractures is inhibited to a large extent, while the effect on the propagation of type II and type III fractures is small.

4.2. Effect of Prestress Loss on Crack Propagation

The loss of prestress is unavoidable during bridge operation, and, with the loss of prestress, the crack resistance of the bridge decreases. In order to investigate the effect of prestress loss on the crack extension performance of box girders, the effects of longitudinal cracks in the bottom plate and vertical crack extension in the web plate were analyzed under five working conditions, with no prestress loss, 10%, 20%, 30%, and 40% prestress loss being considered, respectively.

(1)

Longitudinal crack of floor

Taking the longitudinal crack length of 1200 mm and depth of 30 mm in the base plate as the reference, the maximum value of the stress intensity factor at the crack tip under different degrees of prestress loss is extracted and the integral path value is plotted as a graph; the results are shown in

Table 8. Maximum value of stress intensity factor of longitudinal crack in bottom plate under different prestress losses.

Loss of Prestress	0	10%	20%	30%	40%
${\mathrm{K}}_{\mathrm{I}} (\mathrm{MPa}·{\mathrm{mm}}^{1/2})$	−0.889	−0.703	−0.520	−0.338	−0.156
${\mathrm{K}}_{\mathrm{II}} (\mathrm{MPa}·{\mathrm{mm}}^{1/2})$	−2.205	−1.917	−1.629	−1.340	−1.052
${\mathrm{K}}_{\mathrm{III}} (\mathrm{MPa}·{\mathrm{mm}}^{1/2})$	−1.401	−1.225	−1.052	−0.879	−0.699

and Figure 15.

From Table 8 and Figure 15, it can be seen that the maximum values of the stress intensity factor K_I for longitudinal cracks in the base plate all tend to increase with the increase in prestress loss, while the maximum values of K_II and K_III show a decreasing trend. When the prestress loss reaches 40%, the increase in K_I maxima is as high as 82.45%. The K_II maxima appear at the crack ends and the K_III maxima appear in the middle of the cracks, which show overall negative values. In the case of larger prestress loss, it promotes the expansion of type I cracks and inhibits the expansion of type II and III cracks to some extent.

(2)

Vertical cracks in web

Taking a web vertical crack length of 400 mm and a depth of 30 mm as the reference, the maximum value of the stress intensity factor at the crack tip under different degrees of prestress loss is extracted and the integral path value is plotted as a graph; the results are shown in

Table 9. Maximum stress intensity factors of vertical cracks of web under different prestress losses.

Loss of Prestress	0	10%	20%	30%	40%
${\mathrm{K}}_{\mathrm{I}}(\mathrm{MPa}·{\mathrm{mm}}^{1/2})$	−60.049	−51.246	−42.443	−33.641	−24.838
${\mathrm{K}}_{\mathrm{II}}(\mathrm{MPa}·{\mathrm{mm}}^{1/2})$	0.574	0.559	0.544	0.529	0.515
${\mathrm{K}}_{\mathrm{III}}(\mathrm{MPa}·{\mathrm{mm}}^{1/2})$	−1.649	−1.751	−1.853	−1.956	−2.035

and Figure 16.

As can be seen from Table 9 and Figure 16, the expansion of type I cracks was suppressed due to the presence of prestressing, which resulted in a large negative value of K_I as a whole. However, with the increase in prestress loss, the maximum value of the stress intensity factor K_I of the vertical cracks in the web increases more significantly, and it increases by 35.21 when the prestress loss reaches 40%. In contrast, the overall values of K_II and K_III do not change much, but the trend in change is the opposite. Under the influence of prestress loss, it mainly promotes the expansion of type I and III cracks and has less influence on the expansion of type II cracks.

4.3. Stability Analysis of Longitudinal Cracks in Existing Floor

In order to investigate the influence of multi-factors on the expansion performance and force state of existing longitudinal cracks in the base plate under the operation stage, combined with the results of the analysis of the factors affecting the expansion of the cracks in the box girder, the longitudinal cracks in the base plate with a length of 1200 mm and a depth of 30 mm are selected as the cracking conditions, the most unfavorable bias loading arrangement of the spanning span in the automobile load is taken into account, and the combined effect of the factors is considered comprehensively, while the specific conditions are set up as follows (

Table 10. Working condition setting.

Combination of Operating Conditions
A	Dead weight of box girder + second stage dead load + no prestressed loss + automobile load + temperature gradient
B	Dead weight of box girder + second phase dead load + no prestress loss + 1.5 times automobile load + temperature gradient
C	Dead weight of box girder + second stage dead load + prestress loss 20% + car load + temperature gradient
D	Dead weight of box girder + second stage dead load + prestress loss 20% + 1.5 times automobile load + temperature gradient

):

In order to study the stability of longitudinal cracks in the floor during operation, the combined loads that may produce the most adverse effects are first analyzed to determine the distribution trend and maximum value of the stress intensity factors so as to judge the type of crack expansion. At the same time, based on the empirical fracture criterion of Equation (2), whether the existing fracture is in the expanding state is judged.

The maximum value of the stress intensity factor of the crack tip under various working conditions and the integral path value are drawn into a curve, as shown in Figure 17.

As can be seen from Figure 13, under the action of automobile load, the maximum value of the stress intensity factor at the crack tip is positive and K_I is much larger than K_II and K_III. According to the distribution of the curve diagram, the distribution trend in K values for different types of cracks is basically the same, with the larger value of K_I mainly appearing in the middle of the crack, and the value of the two ends of the crack is relatively small. Comparing the working conditions, it can be seen that the effect of automobile load is obviously greater than the loss of prestressing force, and the influence on K_I is the largest, with an overall increase of more than 50% in the case of overloading, while the change in value is more significant. In contrast, although the overall value of K_II shows an upward trend, the increase is only about 30% and the overall value of K_III does not change much. The above analysis shows that the automobile load is the main factor affecting the stability of longitudinal crack expansion in the base plate during the operation stage, which mainly shows the expansion of mixed type I–II cracks, with type I cracks as the main type, and it tends to expand to the middle part of the cracks instead of both ends of the cracks.

In order to further explore the stability of longitudinal crack growth in the floor, the maximum critical parameter of crack growth under various working conditions is extracted and the curve of critical parameter of crack growth is drawn according to the empirical fracture criteria of composite cracks, as shown in

Table 11. The maximum value of critical parameter $\phi$ of crack propagation under different working conditions.

Working Condition	$\mathbf{A}$	$\mathbf{B}$	$\mathbf{C}$	D
$\phi$	0.240	0.572	0.469	0.607

and Figure 18:

It can be seen from Table 11 and Figure 18 that prestress loss and overload are the main factors affecting the increase in critical parameter $\phi$ for crack expansion. When prestress loss reaches 20%, the maximum value of the corresponding parameter increases nearly twice, while, under overload, the maximum value increases more than twice. Under different working conditions, the maximum value of critical parameter $\phi$ of crack propagation is less than 1, indicating that the longitudinal crack of the floor is in a stable state and has not entered the expansion state.

4.4. Stability Analysis of Vertical Cracks of Existing Web

As shown in Section 4.3, the maximum stress intensity factor of the crack tip under working conditions is plotted as a curve, as shown in Figure 19.

It can be seen from Figure 19 that the maximum value of K_I changes most significantly in various working conditions, especially when prestressing loss and overload are considered. When only automobile load is considered, the value of K_I is negative, which indicates that the tensile stress at the crack tip is small and the crack propagation is inhibited. However, under the influence of prestress loss and overload, the increase of K_I is large and positive, and large values appear at both ends of the crack. The maximum values of K_II and K_III do not change much overall, and K_I is much larger than K_II and K_III. The above analysis shows that prestress loss and overload have a great influence on the expansion of type I cracks in the operation stage, the influence of the two is not a simple linear superposition, and their combined effect will be far more than the sum of the single influencing factors. During operation, the vertical crack propagation of the web may show a trend in regard to compound cracks, mainly type I crack propagation along both ends of the crack.

In order to further explore the stability of the vertical crack propagation of the web, the maximum critical parameter of crack propagation under various working conditions is extracted and the curve $\phi$ of critical parameter of crack propagation is drawn, as shown in Section 4.3. In this case, the stress intensity factor at the crack tip is basically negative under working condition A, the crack propagation is inhibited, and the $\phi$ value is 0, as shown in

Table 12. The maximum value of critical parameters of crack propagation under different working conditions.

Working Condition	$\mathbf{A}$	$\mathbf{B}$	$\mathbf{C}$	D
$\mathsf{\phi }$	0	0.186	0.171	1.087

and Figure 20.

It can be seen from Table 12 and Figure 20 that the critical parameter $\phi$ of crack propagation is less than 1 only when the vehicle load and prestress loss act separately, which indicates that the vertical cracks of the web are in a stable state and have not yet entered the expanding state. However, under the combined influence of prestress loss and overload, the critical parameter $\phi$ of crack propagation appears to be greater than 1 on the side of the crack tip near the box girder bottom plate, which means that the vertical cracks of the web plate may be in an unstable state and have a tendency to expand towards the box girder bottom plate.

5. Conclusions

In this paper, based on the crack distribution characteristics of actual prestressed concrete continuous box girder bridges, the longitudinal cracks in the bottom slab of the mid-span and the vertical cracks in the webs of the side spans are selected as the research objects and the effects of different crack lengths, depths, temperature gradients, prestress losses, and vehicle loads on the crack extension of box girders are investigated. The crack extension critical parameters are introduced to assess the crack extension stability, the crack extension performance is quantitatively analyzed, and the following conclusions are obtained:

• Under the action of basic working conditions, the results of the analysis of the stress intensity factor at the tips of cracks of different lengths and depths show that the longitudinal cracks of the bottom plate mainly show the expansion of composite cracks dominated by type II cracks as the lengths of the cracks increase. Meanwhile, for the increase in the vertical crack length of the web plate, it mainly shows a type II and III crack extension dominated by the type II and III cracks. When the crack depth increases, the common characteristic of both is to promote the expansion of II and III type cracks while restricting the development of I type cracks.
• Under the effect of prestress, the expansion of longitudinal cracks in the bottom plate and vertical cracks in the web plate showed obvious differences. The reduction in prestress promoted the expansion of longitudinal cracks of type I in the bottom plate and inhibited the expansion of type II and III cracks to some extent. In contrast, the expansion of vertical cracks in the web is more sensitive to the prestress reduction, which promotes the expansion of type I, II, and III cracks but mainly affects the expansion of type I cracks.
• The analysis results of the extension stability of the existing longitudinal cracks on the base plate show that the automobile load is the main factor affecting the extension stability of the longitudinal cracks on the base plate, which mainly shows the composite crack extension dominated by the type I cracks.
• The results of the extended stability analysis of the existing longitudinal cracks in the web plate show that the loss of prestress and overloading significantly affect the expansion of the type I cracks. In the case of no overloading, the vertical cracks of the web plate will not be extended under the action of prestressing, while, under the joint influence of the loss of prestressing and overloading, the critical parameter φ of the crack extension locally appear to be greater than 1, which indicates that the vertical cracks of the web plate are in an unstable state and that the possibility of destabilization and expansion is present.

The causes of cracks in continuous prestressed concrete box girder bridges, as well as the crack patterns themselves, are complex and diverse, and there are still many deficiencies in this paper that deserve further study.

• The analysis of existing crack extensions in box girders in this paper only considers the longitudinal cracks of the bottom plate and the vertical cracks of the web plate, and it is a single crack, while the actual box girder needs to be further analyzed for the combination of different crack patterns and multiple cracks.
• This paper is limited to only analysing crack extensions under static loads [30,31], and stability analyses of existing crack extensions under dynamic loads should be considered in later studies.