3.1. Test Phenomena
Under cyclic loading, brittle fractures occurred along main cracks near Beam-2 and Beam-3 but there was no obvious indication that failure was about to occur. During the failure process of Beam-2 and Beam-3, the amount of cracks and their width and height along the length of the beams experienced three stages: the early rapid development stage, the middle stability stage, and the final failure stage. Initially, when the loading of Beam-2 had reached the 100,000th cycle, all of the cracks in the specimen were half the height of the beam height. The cracks were distributed symmetrically along the span direction and on both sides of the cross section. Both the number of cracks and their width remained constant when the number of loading cycles increased from 100,000 times to 600,000 times, with the beam continuing to remain in the crack stability stage. Finally, a dominant crack in the beam span suddenly extended and widened when the 650,147th cycle was achieved, resulting in a fracture in the beam specimen. At this point, the beam underwent a sudden, major deformation and lost its bearing capability due to the ruptures of the five rebar steels in the bottom of the beam. The concrete in the compression zone was raised, accompanied by shedding.
Similar behaviors occurred when Beam-3 reached the 10,000th load cycle but in this case the height of the cracks that developed along the side faces were larger. Beam-3 reached the crack stability stage between loadings of 100,000 times to 300,000 times. However, when the load cycles reached 377,240 an abrupt fracture occurred due to the rupture of the rebar in the bottom of the specimen. This rupture occurred without obvious precursors. Easily identifiable fatigue failure characteristics can usually be found on the fracture surface of the rebar, but in this case no necking phenomenon was observed. The limit failure modes of Beam-2 and Beam-3 are shown in
Figure 7. Excluding the main cracks, there was little variation in the width of the additional cracks in Beam-2 and Beam-3. Before the overall fracture of the beam specimens occurred, the maximum residual crack width was about 0.08 mm.
No fractures occurred in Beam-1 after 2,000,000 loading cycles with a maximum loading force of 77 kN. At the base of the beam, the penetrating cracks were distributed uniformly along the length of the beam. The cracks also extended up to half of the height of the web, with a spacing of 20 cm. The cracks tended to close up following the unloading stage, with a maximum crack width of 0.1 mm. When adjacent to the middle part of the beam, residual crack widths were generally less than 0.08 mm. Throughout the cyclic loading process, the crack development of Beam-1 could be divided into the rapid development stage, the slow development stage, and the stable stage. The final condition of Beam-1 and its crack distribution is shown in
Figure 8. In this instance, all of the cracks occurred when the load cycle had been performed 100,000 times. Beam-1 displayed more cracks than Beam-2 and Beam-3, with most of the cracks located 10 cm below the beam height. A new crack occurred at the end of the beam, on both sides, when the load cycling had been performed 1,000,000 times. The crack developed gradually, spreading up the beam and reaching about half the overall height of the beam. The number, height, and width of the cracks did not increase when the number of loadings increased from 1,000,000 times to 2,000,000 times.
A destructive test was conducted on Beam-1 after it had been subjected to 2,000,000 cyclic loadings. Finally, an abrupt fracture occurred in the Beam-1 along the cracks in the middle span with a bearing force P of 268 kN, which was defined as the residual bearing capacity of the test beam. This demonstrates that the difference between the theoretically calculated value of the ultimate bearing capacity of the beam and the residual bearing capacity was very small. Additionally, according to the test phenomena, a linear relationship exists between the force and the deflection of the mid span before the force exceeded 260 kN, indicating that the beam was basically in an elastic stage. A displacement-control loading scheme was used when the force was loaded to 260 kN, and the bearing capacity of the beam was able to remain stable for a long time during the increase in deflection that occurred when the ultimate load was reached. The ductile failure of the beam, with an ultimate deflection in the mid-span of 58 mm, was caused by the tensile yield of the bottom rebar. The residual bearing capacity of the beam was more than 90% of the theoretical ultimate bearing capacity, indicating that the ultimate bearing capacity of the reinforced concrete beams was largely provided by the tensile strength of the rebar. Under 2,000,000 standard load cycles, there was no obvious decrease in the bearing capacity of reinforced concrete beams, indicating that the residual bearing capacity of Beam-1 meets bearing capacity design requirements. However, the service life of Beam-2 and Beam-3 decreased rapidly when subjected to the overloaded cycles.
3.2. Analysis of Deflection and Stiffness Degradation
The deflection curves of the different measuring points along the span of Beam-1, Beam-2, and Beam-3 under different cyclic loads of varying amplitudes are shown in
Figure 9. For the test of Beam-1, the maximum mid-span deflection was 9.56 mm after 2,000,000 loading cycles, and no obvious change in vertical deflection was detected along the span as the loading cycles increased. The maximum mid-span deflection was only 1.5% higher than the deflection caused by the initial static load, indicating that the beam was basically in an elastic stage. For Beam-2, the vertical deflection that occurred at different positions along the beam span under peak static load was significantly higher than the deflection that occurred under the initial static load. For Beam-2, the mid-span deflection increased by 13.7% after 10,000 loading cycles. From this point the deflection increased slowly, and vertical deflection increased by 20.4% compared with the initial static load after 300,000 cyclic loading cycles. A brittle fracture failure occurred in the mid-span of Beam-2 when the loading cycle reached 650,000. When Beam-2 had experienced 600,000 loading cycles, the mid-span deflection under the peak load was 14.3 mm, 24.5% higher than the deflection that occurred under the initial static load. The deflection–span ratio was about 1/420.
Beam-3 displayed similar variation in vertical deflection to that of Beam-2. The mid-span deflection under peak loading was 13.6% higher than the deflection under the initial static load when the loading cycle had been repeated 10,000 times. The vertical deflection of the mid-span increased suddenly and obviously after 50,000 cyclic loadings, and the maximum deflection of the mid-span increased by about 23.8% compared with the initial value. The mid-span deflection was 19.75 mm, which is 30.8% higher than the initial static load after 300,000 loading cycles, and the deflection–span ratio of the beam was about 1/300. Finally, a brittle fracture failure occurred in the span of Beam-3 after to 370,000 loading cycles. During the entire cyclic loading process, the deflection of Beam-2 and Beam-3 increased nonlinearly with the increase in loading cycles, indicating that the damage to the beam accumulated gradually under repeated vehicle loads, while the stiffness of the members lessened.
The vertical deflection of the different beams’ mid-span under a range of loading cycles is shown in
Figure 10. When the overload effect is not considered, Beam-1 was able to survive 2,000,000 cyclic loads. The deflection of Beam-1 was essentially stable as the cycles increased, and there was no serious or irrecoverable deformation. For the destructive tests of Beam-2 and Beam-3, the vertical deflection of each section underwent the process of increasing–stabilizing–brittle failure with the increase in cycles. The deflection stability stage was the same as the fatigue damage process of the tensile rebar at the bottom of the beams. In addition, the deflection development of Beam-3 was significantly faster than that of Beam-2, indicating that a more serious cumulative damage can be achieved by increasing the amplitude of the cyclic loadings. Therefore, the repeated actions of overloaded vehicles can increase the deformation of concrete beams, and accelerate the development of internal damage. The higher the proportion of overload is, the more obvious this trend.
Bending stiffness is one of the most important indicators used characterize the performance of reinforced concrete beams. The deflection analysis shows that the stiffness of the test beams decreased gradually with the increase of load cycles. In the test, the bending stiffness of the beam can be reflected by the load–deflection ratio of the mid-span section under different loading cycles, as shown in
Figure 11. For the test of Beam-1, the load–deflection ratio curve decreases slowly, indicating that the bending stiffness of the beam decreased slightly with the increasing cyclic loading, and the bending stiffness of the beam decreased by 4.4% compared with the maximum value after 2,000,000 cycles. For Beam-2, the load–deflection ratio decreased rapidly after 10,000 cycles, and then decreased gradually until the final failure stage, indicating that the bending stiffness of the beam decreased gradually with the increase in loading cycles; after 60,000 cycles the stiffness was 48.7% lower than the initial stiffness. For the test of Beam-3, the load–deflection ratio maintained a relatively fast descent rate during 50,000 cycles, and the descent rate slightly decreased during 50,000 to 300,000 cycles but was still higher than that of Beam-2. The ultimate bending stiffness of the Beam-3 following 300,000 cycles was 55.4% lower than its bending stiffness under the static load.
Our results indicate that the bending stiffness of the beam did not experience a gradual, stable degradation process under the cyclic vehicle loads; its change trend was related to the values of vehicle loads. Under the repeated action of a standard load, the stiffness of the beams did not degenerate significantly, and the members remained in an elastic stage. However, under 1.6 times the standard load, the bending stiffness of the beam decreased significantly with an increase of loading cycles, indicating that the beam was irreversibly damaged. Therefore, the ratio of overloaded vehicles and the number of passages should be strictly controlled in order to improve the mechanical performance of beams and extend the service life of the overall structure.
3.3. Analysis of the Residual Deflection
The development of the internal damage done to the beams directly corresponds with residual deflection. The residual deflection curves along the key sections of Beam-1, Beam-2, and Beam-3 after different loading cycles are shown in
Figure 12. For the test of Beam-1, under a vertical load amplitude of 77 kN, the residual deflection increased linearly and slowly along the different positions of the span with the increase in the number of loading cycles, and the residual deflection decreased gradually from the mid-span to both ends. At 2,000,000 loading cycles, the maximum residual deflection of section D-4 was 1.8 mm, and the maximum residual deflection was 19.5% of the absolute deflection. For the test of Beam-2, under a vertical load amplitude of 97 kN, the vertical residual deflection along the different positions of the span increased significantly with the increase in loading cycles. After 300,000 loading cycles, the maximum residual deflection of the mid span section was 3.0 mm, which accounted for 21.6% of the absolute vertical deflection. After 600,000 loading cycles the maximum residual deflection of the mid span section was 4.8 mm, which accounted for 30.75% of the absolute vertical deflection. For the test of Beam-3 with a vertical load amplitude of 121 kN, the variation trend of the vertical residual deflection was similar to that of Beam-2. The residual deflection increased rapidly with the increase in the number of loading cycles. After 300,000 loading cycles, the maximum residual deflection in the middle span was 3.3 mm, which accounts for 18.5% of the absolute vertical deflection.
The comparison of mid-span maximum residual deflection for the different test beams under a range of loading cycles is shown in
Figure 13. Our results indicate that residual deflection increased with the increase of the vertical load amplitude over the same amount of cycles, and that the overload ratio influences the beam’s degree of cumulative damage.
According to our analysis of deflection and residual deflection, the absolute vertical deflection of each section along the span of Beam-1 changed slightly with the increase in cyclic loading times, and the corresponding residual deflection also experienced a slow increase. The deflection of Beam-1 was basically stable after 1,000,000 cycles, and the stiffness of the beam ceased to degenerate. For Beam-2 and Beam-3 the vertical absolute deflection of different sections increased with the increase in cyclic loading times, and the residual deflection also experienced a significant increase. The residual deformation of Beam-3 was notably larger than that of Beam-2 after experiencing the same number of loading cycles. This indicates that the plastic cumulative damage done to the beams increases gradually with an increase in repeated vehicle loads.
3.4. Strain and Stress Analysis
During the cyclic loading process, the average strain on the rebar and concrete in a section of the beam was taken to be representative of the corresponding strain in that section. The variation trend displayed by the average longitudinal reinforcement tensile strain and the average concrete compressive strain along the different sections of the beam span are shown in
Figure 14 and
Figure 15. The maximum tensile strain of the longitudinal reinforcement in Beam-1 was 855.1 με, and the strain on the rebar did not alter significantly with an increase of loading cycles. The rebar in Beam-1 did not reach yield strain throughout the entire testing process. The maximum tensile strain of the longitudinal reinforcement in Beam-2 was 1765.3 με, which underwent several stages of development, including sudden increase–stable–accelerated growth–stable strain development with the increase of loading cycles. The maximum tensile strain of the longitudinal reinforcement at mid-span in Beam-3 was 1858.7 με, which underwent sudden strain increase and stable strain development with the increase of loading cycles.
According to our observations, the sudden increase in the strain on the bottom tensile rebar corresponds with the generation and rapid development of the cracks in the concrete beam. The stress is gradually transferred to the rebar as the concrete slowly develops cracks. The rebar was forced to bear the cyclic loads when the cracks developed completely. At this stage, the strain on the rebar entered the stable development stage under static load with a stable peak load. The strain on the rebar underwent a smaller increase under the same load because in this case the damage to the rebar had not yet developed. Until the fatigue fracture of the rebar occurred, the maximum strain in the rebar did not reach yield strain.
Our results indicate that during the cyclic loading, when the simulated vehicle loads were 1.6 times larger than the standard load, the rebar was constantly in a high-stress state. This resulted in a rapid increase in fatigue and cumulative damage to the plasticity of the rebar. The ultimate failure of the beam was caused by the rebar’s development of a fatigue crack, not due to the yield failure of the rebar itself.
Figure 15 shows the variation in the maximum compressive strain on the concrete at the compressive edge of the top of the test beam. The maximum strain was the average strain at the compressive edge of the concrete when the static load reached its maximum value after a specific number of cycles.
During the initial loading stage, when the number of cycles was below 50,000, the concrete’s strain grows rapidly. As the cracks develop it gradually enters the stable stage. In the second stage, when the number of cycles was between 50,000 and 90% of the total number of loading cycles, the compressive strain growth of the concrete was small and the cracks also entered the stable development stage. In the third stage, when the number of cycles was more than 90% of the total number of loading cycles, the compressive strain of the concrete entered an ascending stage, ultimately leading to the failure of the specimens. The strain of the concrete at the compressive edge of the top of the beam did not reach yield strain during the entire loading process. The mechanical behavior of the concrete beams under cyclic loading is largely determined by the presence of tensile steel bars.
According to our strain analysis of the rebar and concrete, the compressive strain of concrete and the tensile strain of the rebar fluctuated slightly within a certain range but generally the strain curves developed smoothly. The maximum tensile strain on the rebar and the maximum compressive strain on the concrete did not reach their ultimate strain values. Even though fractures occurred during the testing of Beam-2 and Beam-3, the rebar still did not reach yielding strain. Our results indicate that the failure of the beams was caused by the fatigue fracture of the tensile rebar, which is in accordance with the fatigue failure characteristics of reinforced concrete beams.
In order to obtain the law of material strain distribution in the critical stress section, the top concrete compressive strain, web concrete compressive strain, tensile strain, and bottom longitudinal reinforcement tensile strain in the same section were measured, and their distribution along the section height of the beams is shown in
Figure 16,
Figure 17 and
Figure 18.
It can be observed that the strain distribution of the concrete across different sections of the test beams possessed linear characteristics, which conforms to the plane section assumption. After 10,000 loading cycles the position of the neutral axis increased compared with the initial static load, and the position of the neutral axis was seen to increase slowly as the number of cycles rose. Because Beam-1 was in an elastic state during the loading process, the rebar’s strain under different cyclic cycles remained stable. However, due to a gradual accumulation of residual strain in the rebar during the testing process, the performance of Beam-2 and Beam-3 were gradually degraded, and the strain on the rebar increased gradually during different loading cycles. Our results indicate that the fatigue damage experienced by the steel bars was small and the functional performance of the beams remained stable under the repeated action of standard vehicle loads. However, when subjected to the repeated action of overloaded vehicles the damage to the rebar accumulated quickly and their performances degenerated significantly, leading to the early fatigue failure of the beams.
Based on our analysis of the fatigue test results for reinforced concrete beams, data with similar conditions were selected. The fatigue life data of different specimens under different stress levels as stated in reference [
3,
18,
19,
20] are shown in
Figure 19. The stress amplitude–number of cycles (S–N) curve of ordinary reinforced concrete beams can be obtained by fitting these data, and the stress amplitude (
S) can be calculated as,
where
N is the number of cycles. Therefore, the relationship between the vehicle loads and the number of cycles can also be deduced from the results of this study. For example, when the vehicle load was 1.25–1.6 times the standard load, the number of cyclic loads experienced by the concrete beams was between 200,000–750,000.