Flexural Behavior of Corroded High-Speed Railway Simply Supported Prestressed Concrete Box Girder

: Simply supported prestressed concrete (PC) box girders have been widely adopted in high-speed railway bridges. In complex climatic environments, the corrosion of the prestressing strands always occurs and deteriorates the ﬂexural behavior of PC box girders. In the present study, six T-shaped scaled beams were designed and fabricated according to the speciﬁcations for a high-speed railway PC box girder. The corrosion process of the prestressing strand in scaled beams was experimentally simulated by using the constant current accelerated corrosion method. The ﬂexural behavior of corroded high-speed railway simply supported PC box girders was then investigated through four-point bending tests and theoretical investigation. The experimental results showed that strand corrosion signiﬁcantly decreased the ﬂexural behavior of the test beams. When the mass loss was 12.30%, the cracking load, ultimate load, and ductility decreased by 27.8%, 29.9%, and 11.5%, respectively. The effect of strand corrosion on ﬂexural stiffness displayed a difference before and after concrete cracking. The failure mode changed when strand mass loss was above a critical value (7%). The ﬂexural bearing capacity degradation law of corroded PC beams could be divided into two distinct stages. A strand mass loss of less than 7% could lead to a linear degradation law with a relatively slight reduction. As mass loss increased, it exhibited an exponential and sharp declining trend. An analytical model including the effects of strand cross-section reduction, strand property deterioration, and concrete cracking was also proposed to predict the ﬂexural behavior of corroded PC beams. By comparison with the experimental data, it was found that the model could predict the cracking moment, ﬂexural bearing capacity, and failure mode well.


Introduction
Recently, the high-speed railway industry has been developing rapidly worldwide [1].In China, the mileage of the Chinese high-speed railway reached 40,000 km by 2021, ranking first among all countries and regions.Simply supported prestressed concrete (PC) box girders are used extensively in high-speed railway bridges due to their high strength, excellent resistance to cracking, and outstanding seepage resistance [2].Complex climatic environments increase the chances of chlorides reaching the surface of the prestressing strands and causing them to corrode [3].Owing to a high stress level in prestressing strands, the collapse of corroded PC beams usually exhibits a brittle failure without apparent signs [4].The effect of strand corrosion on flexural behavior deserves in-depth investigation to ensure the safety of corroded PC beams.
The corrosion characteristics and property degradation of corroded prestressing strands are essential for predicting the flexural behavior of PC beams.In general, the degree of strand corrosion is assessed by comparing the mass loss before and after corrosion.However, Li et al. [5] observed that prestressing strands mainly exhibited inhomogeneous corrosion in a chloride salt environment and usually suffered fracture in the most corroded areas.In references [6][7][8], a nonuniform chloride-induced corrosion model was used to estimate the residual area of steel bars, which in turn provided new insights into the seismic response and seismic fragility estimates of aging highway concrete structures.Jeon et al. [9,10] acquired corroded strands from actual structures, and they classified the corrosion pit configuration into three types.By analyzing the reduction in tensile strength and ductility of the corroded strands, they proposed an equivalent stress-strain constitutive model.Furthermore, Vecchi et al. [11] conducted a 3D laser scan of 24 strands within a 10-year-old naturally corroded prestressing beam.They found that the majority of the corrosion pits belonged to Types I and III, with only a very small number belonging to Type II.Yu et al. [12] adopted the constant current accelerated corrosion method to investigate the effect of different mass losses on the degradation of strand mechanical properties.Then, a bilinear constitutive model for corroded prestressing strands was proposed.
However, there is excellent scope for exploring flexural behavior such as the flexural stiffness, cracking moment, and ultimate bearing capacity of corroded PC beams.Strand corrosion could decrease the flexural stiffness of PC beams because it induces prestress [13][14][15].Harries et al. [16] tested the flexural bearing capacity of two concrete box girders serviced for 42 years.They found that strand corrosion could cause reduced flexural bearing capacity.Rinaldi et al. [17] experimentally investigated the flexural properties of nine pretensioned PC beams with different mass losses.They observed that when the strand mass loss was higher than 7%, the failure mode of the PC beams changed.The test beams had two different failure modes: concrete crushing and wire rupture.Through 10 specimens with different strand mass losses, Wang et al. [18] found that the degree of bond degradation and the mass loss were not simply linear.In addition, considering the cross-section reduction and mechanical property degradation of the corroded strand, some empirical models [19][20][21] were proposed to predict the flexural behavior of corroded PC beams.
The above-mentioned flexural experiments have achieved some research results, but specific investigation of the flexural behavior of corroded high-speed railway simply supported PC box girders is limited.In addition, there are fewer empirical models for cracking moment prediction, and the above empirical models cannot predict the failure mode of corroded PC beams.In the present study, six T-shaped scaled beams were designed and fabricated according to the specifications for a high-speed railway PC box girder.The corrosion process of the prestressing strands in scaled beams was simulated by using the constant current accelerated corrosion method.The flexural behavior of scaled beams was revealed through four-point bending tests.Afterward, an analytical model including the effects of strand cross-section reduction, strand property deterioration, and concrete cracking was proposed and verified by comparison with the experimental results.This study can provide some useful suggestions for assessing the flexural behavior of the highspeed railway simply supported PC box girder subjected to complex climatic environments.

Experiment Details 2.1. Design and Fabrication of Scaled Beams
The present experimental study is mainly concerned with the flexural behavior of PC box girders.As the prototype box girder was not suitable for carrying out series tests, we first designed the scaled beam according to the similarity theory [22][23][24].The standard 32 m double-line PC simply supported box girder of Chinese high-speed railway bridges (referred to as 2322A-II in Tong Bridge (2008)) was taken as the prototype structure.This prototype structure is a typical single-box, single-room PC box girder with equal height, whose length and width are 32.6 m and 12.0 m, respectively.To arrange the prestressing strands at 120-140 kN/m prestress, a total of 246 strands were required.Detailed dimensions of the PC box girder are shown in Figure 1.
strands at 120-140 kN/m prestress, a total of 246 strands were required.Detailed dimensions of the PC box girder are shown in Figure 1.Similarity theory in structural testing needs to follow both geometric similarity and physical similarity requirements.Geometric similarity means that the linear dimensions, cross-section area, section moment of inertia corresponding to the prototype structure, and scaled beam are in a certain proportion.The beam should be designed as a scaled box beam, but the web and bottom plates in this condition were too thin to be fabricated.Therefore, the test beam was designed as a T-section by matching the flexural stiffness of the scaled box beam.To arrange only one strand in the test beam for subsequent analysis of the effect of strand mass loss on the flexural properties of the test beam, the scale ratio was set as 1:15.68 according to the principle of prestressing similarity.The dimensions of the scaled beam were calculated and are shown in Figure 2. Table 1 compares some geometric parameters of the scaled beam and the prototype structure.The theoretical similarity ratio and actual similarity ratio are, respectively, the design scale ratio and measured scale ratio of the model beams.Small differences can be found between these two similar ratios due to the construction, fabrication, and concrete protective layer requirements of the specimens.However, the differences are relatively small, which indicates that the dimensions of the model beam were chosen reasonably.Similarity theory in structural testing needs to follow both geometric similarity and physical similarity requirements.Geometric similarity means that the linear dimensions, cross-section area, section moment of inertia corresponding to the prototype structure, and scaled beam are in a certain proportion.The beam should be designed as a scaled box beam, but the web and bottom plates in this condition were too thin to be fabricated.Therefore, the test beam was designed as a T-section by matching the flexural stiffness of the scaled box beam.To arrange only one strand in the test beam for subsequent analysis of the effect of strand mass loss on the flexural properties of the test beam, the scale ratio was set as 1:15.68 according to the principle of prestressing similarity.The dimensions of the scaled beam were calculated and are shown in Figure 2. Table 1 compares some geometric parameters of the scaled beam and the prototype structure.The theoretical similarity ratio and actual similarity ratio are, respectively, the design scale ratio and measured scale ratio of the model beams.Small differences can be found between these two similar ratios due to the construction, fabrication, and concrete protective layer requirements of the specimens.However, the differences are relatively small, which indicates that the dimensions of the model beam were chosen reasonably.Similarity theory in structural testing needs to follow both geometric similarity and physical similarity requirements.Geometric similarity means that the linear dimensions, cross-section area, section moment of inertia corresponding to the prototype structure, and scaled beam are in a certain proportion.The beam should be designed as a scaled box beam, but the web and bo om plates in this condition were too thin to be fabricated.Therefore, the test beam was designed as a T-section by matching the flexural stiffness of the scaled box beam.To arrange only one strand in the test beam for subsequent analysis of the effect of strand mass loss on the flexural properties of the test beam, the scale ratio was set as 1:15.68 according to the principle of prestressing similarity.The dimensions of the scaled beam were calculated and are shown in Figure 2. Table 1 compares some geometric parameters of the scaled beam and the prototype structure.The theoretical similarity ratio and actual similarity ratio are, respectively, the design scale ratio and measured scale ratio of the model beams.Small differences can be found between these two similar ratios due to the construction, fabrication, and concrete protective layer requirements of the specimens.However, the differences are relatively small, which indicates that the dimensions of the model beam were chosen reasonably.Physical similarity requires the stress, strain, stiffness, and deformation to be similar between the scaled beam and the prototype structure.All materials used for the scaled beam were therefore strictly consistent with the prototype structure.The standard value of concrete strength was designed to be 50 MPa, a grade of C50 in the Chinese code [25].To match the concrete mix ratio of the prototype structure exactly (as shown in Table 2), the laboratory self-mixing method was used to fabricate the test beam.The gelling materials included 425# low-alkali silicate cement, first-grade fly ash, and S95-grade finely ground mineral powder.Three standard cubic concrete specimens were cast simultaneously as the test beam.Due to the uneven mixture in the laboratory, the measured 28-day compressive strength was 45.24 MPa [25], which was slightly lower than the design value.The prototype girder has a total of 246 seven-wire strands.According to the similarity requirement of effective prestress, the scaled beams had only one seven-wire strand with an outer diameter of 15.2 mm and a standard ultimate strength of 1860 MPa.A metal bellow with a diameter of 50 mm was used to fix the strand position.In addition, the normal reinforcement bars of the scaled beams were calculated according to the similarity requirement of normal stress and constructional requirements of T-shaped beams.Eight HRB335 bars with a diameter of 8 mm at the top and two HRB335 bars with a diameter of 8 mm at the bottom were used as non-prestressed reinforcement.A stirrup with a diameter of 6 mm, spacing of 100 mm, and a grade of 235 MPa was used to resist external shear forces.The arrangement of reinforcement bars of the scaled specimen is shown in Figure 2. Table 3 briefly reports the uncorroded mechanical properties obtained from standard tensile tests [26].Six T-shaped scaled beams, L1, L2, L3, L4, L5, and L6, were fabricated using the posttension method.After pouring concrete and removing formwork, the beams were cured using the steam curing method for 28 days.The strand in scaled beams was prestressed by controlling both tension and elongation indexes, with an initial strand prestress of 0.72 times the yield strength.According to the similarity theory, the final tensioning control stress was set as 157 kN, and the corresponding elongation was 12 mm.The grout (425# ordinary silicate cement:grouting agent:water = 1370:152.3:487)was injected into the metal bellow with a high-pressure grouting pump.

Corrosion Scheme
Among six scaled beams, specimen L1 was used as a comparison beam, and the other five specimens were subjected to the constant current accelerated corrosion method [27].Scaled beams were semi-soaked in a 3.5% NaCl solution, and the horizontal level of the NaCl solution approached but did not exceed the position of the strand.To ensure the chloride-induced corrosion process took place, the scaled beams were drilled with five small holes [28].A 7 cm thick sponge was placed on the surface of scaled beams.The 3.5% mass concentration NaCl solution was poured on the sponge three times a day to keep the sponge moist.The strand was connected to the positive electrode of the DC power supply.It would lose electrons, thus causing mass loss.A corrosion-resistant plate was soaked in sodium chloride solution and connected to the negative electrode of the DC power supply to form an electrolytic cell.Figure 3 shows a schematic diagram of the constant current accelerated corrosion method.

Corrosion Scheme
Among six scaled beams, specimen L1 was used as a comparison beam, and the other five specimens were subjected to the constant current accelerated corrosion method [27].Scaled beams were semi-soaked in a 3.5% NaCl solution, and the horizontal level of the NaCl solution approached but did not exceed the position of the strand.To ensure the chloride-induced corrosion process took place, the scaled beams were drilled with five small holes [28].A 7 cm thick sponge was placed on the surface of scaled beams.The 3.5% mass concentration NaCl solution was poured on the sponge three times a day to keep the sponge moist.The strand was connected to the positive electrode of the DC power supply.It would lose electrons, thus causing mass loss.A corrosion-resistant plate was soaked in sodium chloride solution and connected to the negative electrode of the DC power supply to form an electrolytic cell.Figure 3 shows a schematic diagram of the constant current accelerated corrosion method.As shown in Table 4, the corrosion time required to reach each corrosion level was estimated using Faraday's second law of electrolysis [29,30].The corrosion current density applied in this test was 980 µA/cm 2 , which is approximately 100 times higher than the figure observed for steel under natural conditions.After the accelerated corrosion process, the four-point bending test was performed to reveal the flexural behavior.Then, the scaled beams were broken open and corroded strands were carefully taken out.Each strand was cut into five 40 cm long pieces.They were acid washed with 12% hydrochloric acid, dried, and weighed on an electronic balance with an accuracy of 0.01 g.The average mass losses of the five strand sections were taken as the actual mass losses (ρ) of the scaled beam, which were 0%, 2.12%, 4.06%, 6.28%, 7.84%, and 12.30%, respectively.In addition, the standard deviation of the mass loss gradually became larger as the energization time increased.As shown in Table 4, the corrosion time required to reach each corrosion level was estimated using Faraday's second law of electrolysis [29,30].The corrosion current density applied in this test was 980 µA/cm 2 , which is approximately 100 times higher than the figure observed for steel under natural conditions.After the accelerated corrosion process, the four-point bending test was performed to reveal the flexural behavior.Then, the scaled beams were broken open and corroded strands were carefully taken out.Each strand was cut into five 40 cm long pieces.They were acid washed with 12% hydrochloric acid, dried, and weighed on an electronic balance with an accuracy of 0.01 g.The average mass losses of the five strand sections were taken as the actual mass losses (ρ) of the scaled beam, which were 0%, 2.12%, 4.06%, 6.28%, 7.84%, and 12.30%, respectively.In addition, the standard deviation of the mass loss gradually became larger as the energization time increased.

Four-Point Bending Test
The static four-point bending tests of scaled beams were carried out using a hydraulic jack with a maximum capacity of 50 tons and a distribution steel beam (as shown in Figure 4).The loading process included preloading and formal loading, which were applied step by step.To check that all parts of the contact and instruments were fine, three preloads to 30 kN were carried out.The formal loading phase was divided into two stages: force control and displacement control.The load first rose approximately 10 kN at each step until the concrete cracked.After the scaled beams cracked, the loading stage was controlled using midspan deflection, with an increment of 0.5 mm per step.After each load step, the test beam was left to rest for 2 min and the data were then recorded.In this test, five dial indicators with an accuracy of 1/1000 mm and a range of 100 mm were used to measure the vertical deflection.Finally, the loading was stopped until the concrete was crushed or the strand ruptured.

Four-Point Bending Test
The static four-point bending tests of scaled beams were carried out using a hydraulic jack with a maximum capacity of 50 tons and a distribution steel beam (as shown in Figure 4).The loading process included preloading and formal loading, which were applied step by step.To check that all parts of the contact and instruments were fine, three preloads to 30 kN were carried out.The formal loading phase was divided into two stages: force control and displacement control.The load first rose approximately 10 kN at each step until the concrete cracked.After the scaled beams cracked, the loading stage was controlled using midspan deflection, with an increment of 0.5 mm per step.After each load step, the test beam was left to rest for 2 min and the data were then recorded.In this test, five dial indicators with an accuracy of 1/1000 mm and a range of 100 mm were used to measure the vertical deflection.Finally, the loading was stopped until the concrete was crushed or the strand ruptured.

Impact of Strand Corrosion
It was discovered that the effects of strand corrosion on flexural behavior mainly include the following four aspects: strand cross-section reduction, strand property degradation, concrete cracking in the tensile zone, and the deterioration of the bond between surrounding grouting and strands.As post-tensioned prestressed beams transfer force through anchorages, the deterioration of the bond has little effect on the flexural behavior of PC beams [31].Thus, the analytical model in this paper takes into account the former three aspects.

Analytical Model 3.1. Impact of Strand Corrosion
It was discovered that the effects of strand corrosion on flexural behavior mainly include the following four aspects: strand cross-section reduction, strand property degradation, concrete cracking in the tensile zone, and the deterioration of the bond between surrounding grouting and strands.As post-tensioned prestressed beams transfer force through anchorages, the deterioration of the bond has little effect on the flexural behavior of PC beams [31].Thus, the analytical model in this paper takes into account the former three aspects.

Strand Cross-Section Reduction
In the present study, the seven-wire strand was adopted.As shown in Figure 5, it is generally assumed that strand corrosion can only occur on the external surface [11,32,33].
For chloride-induced corrosion, there are many suggestions [34][35][36] to use the hemispherical corrosion pit model to calculate the remaining area; the predictions are well in agreement with the tests.In the present study, the seven-wire strand was adopted.As shown in Figure 5, it is generally assumed that strand corrosion can only occur on the external surface [11,32,33].For chloride-induced corrosion, there are many suggestions [34][35][36] to use the hemispherical corrosion pit model to calculate the remaining area; the predictions are well in agreement with the tests.The area reduced by corrosion (Ac) of an external wire is calculated as follows: where D0 is the initial nominal diameter of the external wire, which is 5.025 mm in this test; pm is the maximum corrosion depth; and A1 and A2 are the areas of the blue and pink parts of Figure 5, respectively.
The other parameters in Equation ( 1) are given as follows: According to Vecchi et al. [11], there is a ratio (R) between the maximum corrosion depth and the average corrosion depth of the corroded prestressing strand, ranging from 2.93 to 15.05.Since there is no well-developed model for the stochastic distribution of the corrosion penetration factor of prestressed strands, the parameter in this analytical model is taken similarly to reference [21]; here, it is taken as the mean value R = 9.
( ) where pa is the average corrosion depth.
To simplify the analysis, six external wires share the same corrosion pit depth.The remaining area Ap,ρ is given by Equation (6).
The area reduced by corrosion (A c ) of an external wire is calculated as follows: where D 0 is the initial nominal diameter of the external wire, which is 5.025 mm in this test; p m is the maximum corrosion depth; and A 1 and A 2 are the areas of the blue and pink parts of Figure 5, respectively.The other parameters in Equation ( 1) are given as follows: According to Vecchi et al. [11], there is a ratio (R) between the maximum corrosion depth and the average corrosion depth of the corroded prestressing strand, ranging from 2.93 to 15.05.Since there is no well-developed model for the stochastic distribution of the corrosion penetration factor of prestressed strands, the parameter in this analytical model is taken similarly to reference [21]; here, it is taken as the mean value R = 9.
where p a is the average corrosion depth.
To simplify the analysis, six external wires share the same corrosion pit depth.The remaining area A p,ρ is given by Equation (6).
where D 1 is the initial nominal diameter of the middle wire, which is 5.15 mm in this test.

Strand Property Degradation
Corrosion can deteriorate the mechanical properties, i.e., yield strength, ultimate strength, elasticity modulus, and ultimate strain, of the prestressing strand.In the present study, the degradation models proposed by Yu et al. [12] and Li [19] were utilized.
where E p,ρ and E p denote the strand elasticity modulus before and after corrosion, respectively; f ptk and f ptk,ρ are standard values of the strand ultimate strength before and after corrosion, respectively; ε pu and ε pu,ρ are the strand ultimate strain before and after corrosion, respectively; and ε py,ρ is the corroded strand yield strain.

Concrete Cracking
Corrosion of prestressing strands results in volume expansion, creating many microcracks in the surrounding concrete.The presence of internal microcracks leads to a deterioration in the concrete compressive strength in the tensile zone.It is noted that corrosion has a minor effect on the concrete strength in the compression zone.Thus, the reduced concrete strength in the tensile zone can be calculated by adopting the formula proposed by Coronelli [37]: where f ck and f ck,ρ are the standard values of the concrete axial compressive strength before and after strand corrosion, respectively; K is a coefficient related to strand diameter and roughness [38], and here has a value of 0.1; ε c is the concrete peak compressive strain (when f cu ≤ 50 MPa, ε c = 0.002) [19]; and ε 1 is the average concrete tensile strain that is perpendicular to the prestressing force direction, which can be calculated as follows: where n is the number of steel wires to be considered for corrosion; b is the section width at the position of the strand; and w cr is the width of the crack at a certain corrosion depth.Assuming that all corrosion products are incompressible, the crack width w cr equals the increase in circumference (as shown in Figure 6).Therefore, Equation ( 13) can calculate the width of the crack: where µ rs is the volume ratio of corroded oxide to the strand subject to corrosion, recommended to be 2.78 [39].After the concrete compressive strength in the tensile zone has been calculated, it is then possible to estimate the concrete tensile strength (ftk,ρ) by taking 1/10th of the compressive strength as suggested [40].

Cracking Moment
Based on the code for the design of prestressed concrete structures [41], the cracking After the concrete compressive strength in the tensile zone has been calculated, it is then possible to estimate the concrete tensile strength (f tk,ρ ) by taking 1/10th of the compressive strength as suggested [40].

Cracking Moment
Based on the code for the design of prestressed concrete structures [41], the cracking moment can be calculated if the effective prestress and section properties are known.The strand corrosion reduces its effective prestressing force, which is taken into account here with a reduction in the elastic modulus.And the volume expansion in the corrosion products degrades the strength of the concrete, causing the test beam to crack more easily.The calculation formula for the cracking moment of a corroded prestressed beam can be obtained by introducing a mass loss amendment to the code formula.
where W 0,ρ is the flexural section resistance factor of the converted section near the midspan; σ pc,ρ is the concrete precompression stress generated by the effective prestress at the section's lower edge; and γ is the plasticity coefficient of the resisting moment, i.e., γ = 1.0.

Flexural Bearing Capacity
The basic assumptions for the flexural bearing capacity of corroded PC beams are as follows: (1) for PC beams having a reliable anchorage system, the deterioration of the bond has little effect on the flexural behavior; (2) the plane section remains satisfied for average concrete strain during bending; (3) the contribution of concrete in the tensile zone can be neglected; (4) the constitutive models of corroded steel strands and concrete are in Yu [12] and the Chinese code [25], respectively; (5) all normal reinforcing bars yield in the ultimate condition.
The stress-strain constitutive model of concrete adopts a quadratic parabolic segment followed by a horizontal linear segment.The prestressing strand adopts a bilinear constitutive relationship, with the nominal yield strength set at 0.85 times the ultimate strength.When the strand mass loss is greater than a critical value, the hardened section of the stress-strain curve disappears and the curve becomes a single linear curve.The strain and stress distribution of a normal section of the test beam is shown in Figure 7.The following equation can be obtained: where ε p,ρ denotes the strain of the corroded prestressing strand; ε cu is the ultimate compressive strain (when f cu ≤ 50 MPa, ε cu = 0.0033); h p is the distance from the strand to the compressive edge; x and x c are the actual and equivalent compressed zone heights of the concrete, respectively; and β 1 denotes the height coefficient of the equivalent rectangular compression zone, which is set to 0.8 when the concrete strength is less than 50 MPa.
Considering the axial force equilibrium, the equivalent compressed zone height of concrete is calculated from Equation (16).For all test beams, it lies in the flange plate, which can be verified in the crack profiles of the test beams.
where σ p,ρ is the stress of the corroded strand calculated by Equation (17); α 1 denotes the stress coefficient of the equivalent rectangular compression zone; b f ' denotes the width of the flange plate; f yk and A s are the standard yield strength and area of a normal tensile steel bar, respectively; and f yk ' and A s ' are the standard yield strength and area of a normal compressive steel bar, respectively.Considering the axial force equilibrium, the equivalent compressed zone height of concrete is calculated from Equation (16).For all test beams, it lies in the flange plate, which can be verified in the crack profiles of the test beams.

( ) (
) where σp,ρ is the stress of the corroded strand calculated by Equation (17); α1 denotes the stress coefficient of the equivalent rectangular compression zone; bf' denotes the width of the flange plate; fyk and As are the standard yield strength and area of a normal tensile steel bar, respectively; and fyk' and As' are the standard yield strength and area of a normal compressive steel bar, respectively.
The ultimate and actual strain of the corroded strand can be calculated according to Equations ( 9) and ( 15), respectively.By comparing these two values, the failure mode can be determined.If εp,ρ ≤ εpu,ρ, the test beams fail with concrete crushing; otherwise, rupture of the corroded strand happens.
By taking the moment of the point of the equivalent compressive force, the normal section flexural bearing capacity can be calculated as follows: where hs and as' are the distance from the non-prestressed tensile steel bar and non-prestressed compressive steel bar to the compressive edge, respectively.If εp,ρ > εpu,ρ, it means that the strand's actual strain calculated based on the plane section exceeds its ultimate strain.In this case, the equivalent compression height needs to be amended as follows: The ultimate and actual strain of the corroded strand can be calculated according to Equations ( 9) and ( 15), respectively.By comparing these two values, the failure mode can be determined.If ε p,ρ ≤ ε pu,ρ , the test beams fail with concrete crushing; otherwise, rupture of the corroded strand happens.
By taking the moment of the point of the equivalent compressive force, the normal section flexural bearing capacity can be calculated as follows: where h s and a s ' are the distance from the non-prestressed tensile steel bar and nonprestressed compressive steel bar to the compressive edge, respectively.If ε p,ρ > ε pu,ρ , it means that the strand's actual strain calculated based on the plane section exceeds its ultimate strain.In this case, the equivalent compression height needs to be amended as follows: Replacing x in Equation ( 18) with x' yields a formula for calculating the flexural bearing capacity of a test beam with strand rupture.

Experimental Phenomena
During the loading process, the crack development and failure of the scaled beams were recorded and summarized as follows.Taking specimen L1 as an example, when loaded to 72 kN, small oblique cracks first appeared on the web side in the segment between the loading point and the support.As the load increased, oblique cracks developed, and vertical bending cracks appeared in the middle segment.Then, bending cracks extended from the middle span to both ends.The average crack distance was about 100 mm, which was comparable to the stirrup spacing.As the load kept increasing, oblique cracks and vertical bending cracks continued to develop, and the crack number and width increased.When the load increased to 120 kN, cracks developed on the beam flange plate.Figure 8 shows the crack profile of all the scaled beams.
During the loading process, the crack development and failure of the scaled beams were recorded and summarized as follows.Taking specimen L1 as an example, when loaded to 72 kN, small oblique cracks first appeared on the web side in the segment between the loading point and the support.As the load increased, oblique cracks developed, and vertical bending cracks appeared in the middle segment.Then, bending cracks extended from the middle span to both ends.The average crack distance was about 100 mm, which was comparable to the stirrup spacing.As the load kept increasing, oblique cracks and vertical bending cracks continued to develop, and the crack number and width increased.When the load increased to 120 kN, cracks developed on the beam flange plate.Figure 8 shows the crack profile of all the scaled beams.Beams with different mass losses exhibited different failure characteristics.For specimens L1~L4 with low mass loss, their non-prestressed tensile reinforcement first yielded and then fractured, and then the flange concrete crushed, which exhibited obvious ductile characteristics.When specimens L5 and L6 were damaged, the strand ruptured with a strong and brittle sound.The failure of these two specimens has obvious brittle characteristics, with the bearing capacity decreasing sharply.The failure mode of the scaled beams, which changes from ductile failure to brittle failure as the mass loss increases, can also be supported by reference [17].
The experiment results are summarized in Table 4 and Figure 9. Strand corrosion could reduce the flexural behavior of PC beams.When the mass loss was 12.30%, the cracking load, ultimate load, and ductility decreased by 27.8%, 29.9%, and 11.5%, respectively.The cracking load of specimen L4 was higher than that of specimen L3, which may be due to the variability in the concrete properties inside each specimen and between different specimens [42,43].There was an obvious turning point in the ultimate load reduction of the scaled beam.When the mass loss exceeded 7%, the ultimate load would degrade faster.
The experiment results are summarized in Table 4 and Figure 9. Strand corrosion could reduce the flexural behavior of PC beams.When the mass loss was 12.30%, the cracking load, ultimate load, and ductility decreased by 27.8%, 29.9%, and 11.5%, respectively.The cracking load of specimen L4 was higher than that of specimen L3, which may be due to the variability in the concrete properties inside each specimen and between different specimens [42,43].There was an obvious turning point in the ultimate load reduction of the scaled beam.When the mass loss exceeded 7%, the ultimate load would degrade faster.

Load-Deflection Curve at Midspan
Figure 10 compares the load-deflection curves for all the scaled beams.It can be found that the curves can be divided into three stages.Before the concrete in the tension zone cracked (point A), the load and midspan deflection were approximately linear and the scaled beam was in elastic working condition (stage 1).All the scaled beams exhibited a similar load-deflection behavior at stage 1, demonstrating that the strand corrosion had a negligible effect on the initial stiffness of PC beams.Stage 2 was the cracking elastic phase, where the concrete in the tension zone cracked and stopped working.This stage terminated when the non-prestressed tensile reinforcements yielded.Stage 3 was the plastic stage, where the cracks extended to the upper flange, and the flexural stiffness of the beam section decreased rapidly.This stage ended with the concrete being crushed in the compressed zone when the strand mass loss was low, or with the strand being ruptured when the mass loss was high.The inflection points in the curve were, sequentially, concrete cracking (point A), yielding of non-prestressed tensile reinforcement (point B), concrete crushing (point C), and strand rupture (point D).

Load-Deflection Curve at Midspan
Figure 10 compares the load-deflection curves for all the scaled beams.It can be found that the curves can be divided into three stages.Before the concrete in the tension zone cracked (point A), the load and midspan deflection were approximately linear and the scaled beam was in elastic working condition (stage 1).All the scaled beams exhibited a similar load-deflection behavior at stage 1, demonstrating that the strand corrosion had a negligible effect on the initial stiffness of PC beams.Stage 2 was the cracking elastic phase, where the concrete in the tension zone cracked and stopped working.This stage terminated when the non-prestressed tensile reinforcements yielded.Stage 3 was the plastic stage, where the cracks extended to the upper flange, and the flexural stiffness of the beam section decreased rapidly.This stage ended with the concrete being crushed in the compressed zone when the strand mass loss was low, or with the strand being ruptured when the mass loss was high.The inflection points in the curve were, sequentially, concrete cracking (point A), yielding of non-prestressed tensile reinforcement (point B), concrete crushing (point C), and strand rupture (point D).Under general conditions, the corroded PC beam is usually working with cracks, i.e. in an approximately elastic stage.The deflection calculations of corroded PC beams are aimed at both elastic and cracking elastic phases.Therefore, a homogeneous elastic material model was identified as the object of study.The stiffness of the purely bending seg- Under general conditions, the corroded PC beam is usually working with cracks, i.e., in an approximately elastic stage.The deflection calculations of corroded PC beams are aimed at both elastic and cracking elastic phases.Therefore, a homogeneous elastic material model was identified as the object of study.The stiffness of the purely bending segment was taken for the study based on the "principle of minimum stiffness" [40].Given the short period of the test loading, the short-term stiffness of the corroded PC beam was studied.
For the four-point bending test, deflection can be obtained using the principle of superposition.The flexural stiffness of scaled beams is obtained as follows: where Q is the external load applied at one loading point; δ is the midspan deflection, which is taken as a positive value when moving downwards; a is the distance from the loading point to the support; and L is the calculated span of the scaled beam.
Figure 11 compares the load-flexural stiffness curves calculated using Equation ( 20).It can be found that the general trend of relative stiffness variation was decreasing, with two obvious stages: the stiffness remained unchanged during the period from loading to cracking, and decreased sharply after the test beam crack.The load-flexural stiffness curves of the scaled beams still satisfied the bifold variation law.After cracking, the slope of the second stage increased as the mass loss increased; that is, the flexural stiffness decreased more and more sharply.This implies that strand corrosion has a significant effect on flexural stiffness after cracking.Under general conditions, the corroded PC beam is usually working with cracks, i.e. in an approximately elastic stage.The deflection calculations of corroded PC beams are aimed at both elastic and cracking elastic phases.Therefore, a homogeneous elastic mate rial model was identified as the object of study.The stiffness of the purely bending segment was taken for the study based on the "principle of minimum stiffness" [40].Given the short period of the test loading, the short-term stiffness of the corroded PC beam was studied.
For the four-point bending test, deflection can be obtained using the principle of su perposition.The flexural stiffness of scaled beams is obtained as follows: where Q is the external load applied at one loading point; δ is the midspan deflection which is taken as a positive value when moving downwards; a is the distance from the loading point to the support; and L is the calculated span of the scaled beam.
Figure 11 compares the load-flexural stiffness curves calculated using Equation ( 20) It can be found that the general trend of relative stiffness variation was decreasing, with two obvious stages: the stiffness remained unchanged during the period from loading to cracking, and decreased sharply after the test beam crack.The load-flexural stiffness curves of the scaled beams still satisfied the bifold variation law.After cracking, the slope of the second stage increased as the mass loss increased; that is, the flexural stiffness de creased more and more sharply.This implies that strand corrosion has a significant effec on flexural stiffness after cracking.

Cracking Moment
The experimental and theoretical values of the cracking moment for the scaled beams are listed in Table 4.As previously mentioned, the cracking moment of specimen L4 showed a large deviation, and these data were excluded from the analysis of the effect of strand mass loss on the cracking moment to ensure data reliability.It can be found that the analytical cracking model was valid, with a maximum relative error between the experimental and theoretical cracking moment of 3.14%.Strand corrosion reduced the strand prestress and concrete strength, which in turn caused the test beam to crack more easily.Figure 12 compares the variation in cracking moment ratios, from which it can be seen that the deterioration trends of the cracking moment obtained from the experiment and theory are highly close.
the analytical cracking model was valid, with a maximum relative error between the experimental and theoretical cracking moment of 3.14%.Strand corrosion reduced the strand prestress and concrete strength, which in turn caused the test beam to crack more easily.Figure 12 compares the variation in cracking moment ratios, from which it can be seen that the deterioration trends of the cracking moment obtained from the experiment and theory are highly close.

Flexural Bearing Capacity
Table 4 compares the flexural bearing capacity and failure mode obtained from the experimental and analytical methods, and the comparison reveals a good agreement between them.The test results show that high mass losses significantly reduced the flexural bearing capacity and led to strand rupture failure.By contrast, for low mass losses, the flexural bearing capacity was only slightly reduced with concrete being crushed.Similar to those of the cracking moment, the experimental and theoretical values of the flexural bearing capacity decreased as the strand mass loss increased.A critical mass loss (a range of 6~8%) between these two stages seemed to exist.The critical mass loss for failure mode changes was highlighted by Rinaldi et al. [11] who observed extremely brittle behavior in specimens with an average mass loss of 7%.Similar results exist, such as the 5.06% and 5% mass losses proposed, respectively, by Benenato et al. [44] and Menouf [45].
An in-depth analysis of the results of this test and those of other researchers led to the determination of a critical mass loss of 7%. Figure 13 compares the variation in flexural bearing capacity ratios.The flexural bearing capacity of corroded prestressed beams depended on the strand mass loss.A strand mass loss of less than 7% could lead to a linear degradation pattern with a relatively slight reduction.As mass loss increased, the flexural bearing capacity exhibited an exponential and sharp declining trend.

Flexural Bearing Capacity
Table 4 compares the flexural bearing capacity and failure mode obtained from the experimental and analytical methods, and the comparison reveals a good agreement between them.The test results show that high mass losses significantly reduced the flexural bearing capacity and led to strand rupture failure.By contrast, for low mass losses, the flexural bearing capacity was only slightly reduced with concrete being crushed.Similar to those of the cracking moment, the experimental and theoretical values of the flexural bearing capacity decreased as the strand mass loss increased.A critical mass loss (a range of 6~8%) between these two stages seemed to exist.The critical mass loss for failure mode changes was highlighted by Rinaldi et al. [11] who observed extremely brittle behavior in specimens with an average mass loss of 7%.Similar results exist, such as the 5.06% and 5% mass losses proposed, respectively, by Benenato et al. [44] and Menouf [45].
An in-depth analysis of the results of this test and those of other researchers led to the determination of a critical mass loss of 7%. Figure 13 compares the variation in flexural bearing capacity ratios.The flexural bearing capacity of corroded prestressed beams depended on the strand mass loss.A strand mass loss of less than 7% could lead to a linear degradation pattern with a relatively slight reduction.As mass loss increased, the flexural bearing capacity exhibited an exponential and sharp declining trend.

Displacement Ductility Factor
Components with better ductility can absorb a certain amount of energy after reaching the yield or maximum bearing capacity state.The ductility of a member can be characterized by the displacement ductility factor [46,47], which is the ratio of ultimate displacement to yield displacement.Among the points, the yield point can be obtained by

Displacement Ductility Factor
Components with better ductility can absorb a certain amount of energy after reaching the yield or maximum bearing capacity state.The ductility of a member can be characterized by the displacement ductility factor [46,47], which is the ratio of ultimate displacement to yield displacement.Among the points, the yield point can be obtained by graphing the load-deflection curve using the isoenergetic method [48].And the limit point is taken to be the point corresponding to the maximum value of the load-deflection curve or the point corresponding to 85% of the peak load.
The ductility factors are listed in Table 4 and shown in Figure 14.The displacement ductility factors of the corroded beams were smaller than the figure of the comparison specimen.When strand corrosion was 12.30%, the reduction rate of the displacement ductility factor was 11.5%.There was also a relationship between the failure mode and ductility.The displacement ductility factor of specimens L5 and L6 decreased to below three, which corresponded to brittle failure with strand rupture in the experiments.

Displacement Ductility Factor
Components with better ductility can absorb a certain amount of energy after reaching the yield or maximum bearing capacity state.The ductility of a member can be characterized by the displacement ductility factor [46,47], which is the ratio of ultimate displacement to yield displacement.Among the points, the yield point can be obtained by graphing the load-deflection curve using the isoenergetic method [48].And the limit point is taken to be the point corresponding to the maximum value of the load-deflection curve or the point corresponding to 85% of the peak load.
The ductility factors are listed in Table 4 and shown in Figure 14.The displacement ductility factors of the corroded beams were smaller than the figure of the comparison specimen.When strand corrosion was 12.30%, the reduction rate of the displacement ductility factor was 11.5%.There was also a relationship between the failure mode and ductility.The displacement ductility factor of specimens L5 and L6 decreased to below three, which corresponded to brittle failure with strand rupture in the experiments.

Conclusions
To address the lack of research on the flexural behavior of corroded high-speed railway simply supported PC box girders, the failure mode, flexural stiffness, cracking moment, flexural bearing capacity, and displacement ductility factor of corroded scaled beams with different mass losses were experimentally and theoretically investigated.The main conclusions are as follows:

Conclusions
To address the lack of research on the flexural behavior of corroded high-speed railway simply supported PC box girders, the failure mode, flexural stiffness, cracking moment, flexural bearing capacity, and displacement ductility of corroded scaled beams with different mass losses were experimentally and theoretically investigated.The main conclusions are as follows: (1) Strand corrosion significantly reduced its mechanical properties, which in turn degraded the flexural behavior of the PC beams.In this test, when the mass loss was 12.30%, the cracking load, ultimate load, and ductility decreased by 27.8%, 29.9%, and 11.5%, respectively.(2) There existed a critical strand mass loss (7% in this study) for the reduction in the flexural bearing capacity of a corroded PC beam.A strand mass loss of less than 7% could lead to a linear degradation law with concrete crushing failure.As mass loss increased, the flexural bearing capacity exhibited an exponentially decreasing trend with strand rupture failure.(3) Strand corrosion had a negligible effect on initial flexural stiffness before concrete cracking.But after concrete cracking, the higher the mass loss was, the faster the flexural stiffness decreased.(4) The analytical model considering the effects of strand cross-section reduction, strand property deterioration, and concrete cracking can predict the cracking moment, flexural bearing capacity, and failure mode well.

Figure 1 .
Figure 1.Tong Bridge (2008) 2322A-II standard box girder dimensions (units: mm); the left crosssection is near the midspan, and the right cross-section is near the support.

Figure 1 .
Figure 1.Tong Bridge (2008) 2322A-II standard box girder dimensions (units: mm); the left crosssection is near the midspan, and the right cross-section is near the support.
Appl.Sci.2023, 13, x FOR PEER REVIEW 3 of 19 strands at 120-140 kN/m prestress, a total of 246 strands were required.Detailed dimensions of the PC box girder are shown in Figure 1.

Figure 1 .
Figure 1.Tong Bridge (2008) 2322A-II standard box girder dimensions (units: mm); the left crosssection is near the midspan, and the right cross-section is near the support.

Figure 2 .
Figure 2. Dimensions and section reinforcement diagram of scale beam (units: mm): (a) cross-section near the support; (b) cross-section near the midspan; (c) section reinforcement diagram near the support; (d) section reinforcement diagram near the midspan; (e) side view; (f) top view.

Figure 3 .
Figure 3. Schematic diagram of constant current accelerated corrosion method.(a) on-site photo; (b) schematic diagram.

Figure 3 .
Figure 3. Schematic diagram of constant current accelerated corrosion method.(a) on-site photo; (b) schematic diagram.

19 Figure 6 .
Figure 6.Corrosion crack width wcr calculation diagram: (a) corrosion product accumulation around a seven-wire strand; (b) the increase in one external wire's circumference.

Figure 6 .
Figure 6.Corrosion crack width w cr calculation diagram: (a) corrosion product accumulation around a seven-wire strand; (b) the increase in one external wire's circumference.

Figure 9 .
Figure 9. Cracking and ultimate loads vs. mass loss.

Figure 9 .
Figure 9. Cracking and ultimate loads vs. mass loss.

Figure 10 .
Figure 10.Comparison of the load-deflection curve.

Figure 10 .
Figure 10.Comparison of the load-deflection curve.

Figure 11 .
Figure 11.Comparison of the load-stiffness curve.

n
Number of steel wires to be considered for corrosion w cr Width of crack at a certain corrosion depth µ rs Volume ratio of corroded oxide to strand subject to corrosion b Section width at the position of strand b f ' Section width of flange plate W 0,ρ Flexural section resistance factor of converted section near the midspan σ pc,ρ Concrete precompression stress generated by the effective prestress at the section's lower edge γ Plasticity coefficient of the resisting moment h p Distance from strand to the compressive edge h s Distance from non-prestressed tensile steel bar to the compressive edge a s ' Distance from non-prestressed compressive steel bar to the compressive edge x (x c ) Actual and equivalent compressed zone heights of concrete α 1 Stress coefficient of equivalent rectangular compression zone β 1 Height coefficient of equivalent rectangular compression zone M cr,ρ Calculated cracking moment M u,ρ Calculated flexural bearing capacity Q External load applied at one loading point δ Midspan deflection a Distance from the loading point to the support L Calculated span of the scaled beam

Table 1 .
Comparison of geometric parameters between prototype structure and scaled beam.