Low cyclic loading tests and fatigue damage models of concrete bridge piers reinforced with high-strength rebar

Abstract

Reinforced concrete bridge piers are extremely vulnerable to damage during long-duration ground excitations or main shock-aftershock type earthquakes due to accumulated damage caused by a great number of reversed excursions in elastic-plastic range. However, few studies on fatigue damage of piers can be found in literature. Low-cyclic loading tests of four identical RC bridge piers with high-strength rebar HRB600E (yield strength about 600 MPa) were carried out in this study. One of specimens was taken as the benchmark and was subjected to a conventional load protocol, and the rest was subjected to one, two and three times yield displacement, respectively. The research results showed that the fatigue strength of RC bridge piers tended to drop drastically at about ten cycles and then slowed down gently. It was found that strength degradation rate increased significantly with the displacement amplitude for fatigue tests while the fatigue life decreased dramatically with the displacement amplitude. In particular, when the cyclic loading displacement exceeded 2 times the yield displacement, the fatigue life dropped dramatically. Based on the experimental data, an exponential-type damage model was proposed with the peak lateral force at the first cycle as the coefficient, the cycle count as the base and factor of the loading displacement amplitude as the exponent, which could accurately predict the degraded lateral force of the bridge piers at different constant drifts. An accumulative fatigue damage index was established to evaluate the damage level of bridge piers.

Keywords

high strength rebar bridge pier low-cyclic fatigue fatigue damage model experimental study

Introduction

In recent years, the world has been greatly astonished by the devastating damage caused by several long-duration major earthquakes (Liu, 2009; Mohammed et al., 2017), such as Wenchuan Earthquake (2008, M_w 8.0), Chile Maule Earthquake (2010, M_w 8.8), Eastern Japan Earthquake (2011, M_w 9.0). For example, the recorded ground motions lasted for 50-270 seconds in the 2011 Japan Tohoku earthquake, and 20 to 90 s in the 2010 Chile Maule Earthquake. Such long duration records could subject structures to the plastic regime at increased reversals, which can result in accumulative fatigue damage and even collapse.

In view of the significant damage during the long-duration earthquakes (Buckle et al., 2012; Kawashima et al., 2011), a number of researchers have approached this research topic through extensive numerical and experimental studies in recent years. For example, Ou et al. (2013) conducted pseudo-static tests on the two flexural dominated reinforced concrete columns under a long-duration loading protocol and a baseline loading protocol with one cycle for each drift loading. It was found that strength degradation is related to maximum displacement and energy dissipation. Chandramohan et al. (2016) performed nonlinear numerical study on a reinforced concrete bridge pier using assembling sets of ‘spectrally equivalent’ long and short duration records. They concluded that the collapse capacity of the concrete bridge pier is 17% lower when subjecting to the set of long-duration records. Mohammed (2016) and Mohammed et al. (2017) conducted a shake table experiment on a series of identical bridge columns under ‘spectrally equivalent’ long and short records. It was concluded that bridge columns had 25% smaller displacement capacity when subjected to long-duration records. On the other hand, a number of researchers have performed low-cyclic fatigue tests on reinforced concrete columns to improve their seismic design in past decades (Ei-Bahy et al., 1999; Ge et al., 2013; Hindi and Sexsmith, 2001; Li et al., 1998; Liu et al., 1996; Park and Ang, 1985; Ranf et al., 2006; Tsuno and Park, 2004; Zhang et al., 2008). All of these studies indicate that the collapse capacity of bridge columns will be reduced significantly when subjected to lager number of reversal displacements. However, all of these studies mainly focus on bridge columns with conventional reinforcing steel.

In addition, high-strength reinforcement has attracted considerable attention to researchers and practice engineers for construction of bridge columns to increase the design strength. Some researchers have performed pseudo-static tests on reinforced concrete columns with high-strength rebar, which show that it is beneficial to the seismic performance of bridge columns (Barbosa et al., 2016; Kelly et al., 2014; Ou et al., 2010; Ousalem et al., 2009; Rautenberg et al., 2012; Restrepo et al., 2006; Rong et al., 2015; Su et al., 2015, 2019). However, these experiments are only limited to conventional loading protocol of few cycles of the reversal displacements. Lack of fatigue test data on bridge columns with high-strength rebar hinders the application of this type of bridge columns in areas where future long-duration earthquakes could occur, like the Pacific Northwest Coast of the United States.

In consideration of the potential advantages of the application of high-strength rebars in medium to high intensity seismic regions (Zhuo et al., 2018, 2019), it is necessary to investigate low-cyclic fatigue behavior of bridge piers reinforced with high-strength rebar. In this paper, fatigue tests of several large-scale pier specimens reinforced with high strength rebar of HRB600E (yield strength about 600 MPa) have been carried out to fill the knowledge gap in literature for the design of this type of bridge columns. To begin, four large-scale identical bridge columns were designed. One of the specimens was taken as the benchmark, which was subjected to a conventional loading protocol. Yield displacement of the benchmark specimen was computed from the test data based on the Park’s method, which was taken as a reference for the input to the rest three specimens. To be specific, the rest three specimens were loaded to failure by reversed cycles of constant displacement of one, two, and three times yield displacement of the benchmark specimen, respectively. The strength degradation of the specimens was investigated by the test data, on which a model was proposed to predict the strength degradation with the number of loading cycles. In addition, an accumulative fatigue damage index was defined and computed from the proposed strength degradation model to evaluate the damage level of bridge piers.

Low cyclic fatigue test program

Reinforced concrete pier specimens

To investigate the fatigue life of concrete piers under different constant displacement amplitudes, three fatigue specimens and one benchmark specimen with identical geometry and reinforcement details were designed and constructed, which were named as FA0, FA1, FA2 and FA3, respectively. Each specimen consisted of a footing (1500 mm × 1500 mm × 600 mm), a circular column (600 mm in diameter and 2350 mm high) and a loading block (800 mm × 800 mm × 600 mm). The scale was about 1:3–1:2 to the practical pier columns. The reinforcement arrangement in the column consisted of 14 longitudinal bars with 20 mm-diameter of HRB600E, the spiral stirrup with spacing of 50 mm and 8 mm-diameter of HRB400. The designed specimen is shown in Figure 1.

Figure 1.

Specimen size and reinforcement: (a) CIP pier; (b) cross section.

The concrete strength at the test day was measured by six concrete cubes of 150 × 150 × 150 mm, which were simultaneously cast and cured under the same condition as pier specimens. The average compressive strength of 44.56 MPa and the specified strength of 33.87 MPa was determined according to the Chinese Standard for Test Method of Concrete Structures (GB/T50152, 2012). The mechanical properties of the rebar were tested and tabulated in Table 1. Note that the rebar elongation percentage after fracture was measured in a 100 mm gauge length. It is seen in this table that the yield strength of the longitudinal rebar is 627 MPa, which is 57% larger than that of conventional normal strength rebar HRB400 (yield strength = 400 MPa).

Table 1.

Sample test results of HRB600E and HRB400 rebar.

Steel type	Yield strength(MPa)	Ultimatestrength (MPa)	Modulus ofelasticity (MPa)	Elongation percentageafter fracture (%)
HRB600E	627	825	1.99 × 10⁵	20
HRB400	434	578	1.95 × 10⁵	28

Experimental setup

Each specimen was connected to the lab floor through four slip-critical bolts to avoid relative slippage of the specimens to the floor. All specimens were loaded cyclically in the lateral direction by a MTS servo-hydraulic horizontal actuator (100 t) under a constant axial compression load by two vertical hydraulic jacks. One end of the horizontal actuator was connected to the center of the loading block while other end to a strong reaction wall. As a result, the effective loading height was 2650 mm. The applied horizontal load and displacement were measured by the embedded sensors inside the actuator. The axial force was applied by a reaction beam system, including a steel beam, two threaded bars and two 100 t hydraulic jacks. The axial force of 766 kN was applied, which provided axial compression ratio of 8% to emulate the dead load in regular highway bridges. The test setup was shown in Figure 2.

Figure 2.

Experimental setup.

Loading protocol

During the tests, the vertical hydraulic jakes were set to apply the prescribed constant axial load of 766 kN. For all specimens, the horizontal actuator was loaded in a displacement-control algorithm and the lateral displacement of the column at the center axis of horizontal actuator was taken as the control variable.

For the benchmark specimen FA0, the loading scheme consisted of three stages: (1) initial stag: loading from 0 to 10 mm with increments of 2 mm, (2) stage of 10 to 50 mm with increments of 5 mm, and (3) beyond 50 mm with increments of 10 mm to failure. Each load grade had 3 cycles. At the end of each grade, one cycle of the previous load grade was applied to simulate the main shock-aftershock earthquake. The load protocol of specimen FA0 is shown in Figure 3. Specimens FA1 to F3 were loaded cyclically to failure under a constant displacement amplitude of 1, 2 and 3 times equivalent yield displacement ( $Δ_{y}$ ) of specimen FA0, respectively. Note that $Δ_{y}$ of specimen FA0 was determined by Park’s method (Park, 1989), which is described in ‘Backbone curve of specimen FA0’. The loading to all specimens was terminated when their strength dropped to 85% of the maximum strength.

Figure 3.

Load protocol of specimen FA0.

Experimental results

Damage observations

During the tests, photos were taken to the column bottom at the end of each load grade to document the evolution of damage for all specimens. Figure 4(a) to (h) show the typical damage of all specimens at the end of final load grade.

Figure 4.

Damage photos at the pier bottom of all specimens at the end of the tests: (a)(c)(e)(g) close to the actuator, (b)(d)(f)(h) opposite side of the actuator.

Specimen FA0 started to experience transverse flexural cracking at displacement amplitude of 8 mm. When the displacement was increased to 45 mm, spalling of cover concrete started to occur. Extensive spalling was observed at 110 mm. At loading of 160 mm, longitudinal rebar was fractured and the lateral strength was decreased to 85% of maximum strength. The specimen was identified as failure and the test was terminated. Figure 4(a) and (b) shows the column bottom of specimen FA0 at the end of the specimen. As seen in these figures, specimen FA0 experienced extensive concrete spalling and longitudinal rebar fracture and buckling at end of the tests.

Specimen FA1 started to experience hairline cracks at the pier bottom on both sides of the pier after the first cycle of loading of $Δ_{y}$ . During the subsequent cycles of loading, the cracking zone was extended from the pier bottom to the height of around 1.2 m, with an average cracking spacing being about 150 mm, as shown in Figure 4(c) and (d). The test was terminated at the 501st cycle as the lateral force dropped to 85% of the maximum value. No significant concrete spalling was observed at the end of the test as seen in Figure 4(c) and (d). Sound of bar fracture was observed during the final cycle. These two observations indicate that specimen failed due to fatigue fracture of longitudinal rebar. The observations of no-obvious concrete spalling and longitudinal rebar fracture will pose a significant challenge for post-earthquake evaluation and repair of the bridge columns after subjecting to long-duration records.

Specimen FA2 experienced fully developed cracks at the pier bottom at the end of first cycle of loading of 2 $Δ_{y}$ and the average crack spacing was about 150 mm. These cracks were oriented horizontally on both sides perpendicular to the loading direction, while at an inclined angle of 45° on the pier sides. At about the 5th cycle, the concrete cover began to spall. The test was terminated at the 160th cycle as the lateral force dropped to 85% of the maximum value. At the end of the test, extensive cover concrete spalling was observed at the pier bottom with an approximate range of 400 mm × 200 mm close to the actuator and 500 mm × 200 mm at the opposite side of actuator, as seen in Figure 4(e) and (f). In addition, the cracks were extended to a height of 500 mm from the pier bottom and some spirals were exposed.

Specimen FA3 experienced fully developed cracks at the pier bottom at the end of first cycle of loading of 3 $Δ_{y}$ and the average crack spacing was about 100 mm. These cracks were oriented horizontally on both sides perpendicular to the loading direction, while at an inclined angle of 45° on the pier sides. At the subsequent loading, vertical cracks were developed on both sides perpendicular to the loading direction. At about the 4th cycle, one of longitudinal rebar was fractured. The test was terminated at the 8th cycle as the lateral force dropped to 85% of the maximum value. At the end of the test, extensive cover concrete spalling with was observed with an approximate range of 610 mm × 640 mm close to the actuator and 690 mm × 530 mm at the opposite side of actuator, as seen in Figure 4(g) and (h). In addition, the spirals and some longitudinal rebar were exposed and the core concrete at the pier bottom was locally crushed with one longitudinal rebar fractured. Compared with specimens FA1 and FA2, concrete spalling area in specimen FA3 was much wider.

Hysteretic loops

During the tests, the applied horizontal load and displacements at the column head were measured by the embedded sensors inside the horizontal actuator and these values were used to plot the hysteretic curves. Figure 5(a) to (d) show the hysteretic curves of specimens FA0 to FA3, respectively. Also shown in Figure 5(a) are the typical damage points of specimen FA0 during the experiment, like first yield of longitudinal rebar, cover concrete spalling and rebar fracture.

Figure 5.

Hysteretic loops of the specimens: (a) FA0, (b) FA1, (c) FA2, (d) FA3.

As seen in Figure 5(a), when the displacement is low, the force-displacement relationship of FA0 is linear. When further increasing the load, the hysteretic loops are opened and the entrapped area increases rapidly, indicating good energy dissipation capacity. These loops are approximately anti-symmetric during the loading and unloading stages. The maximum lateral force of specimen FA0 was 282 kN, and the corresponding displacement was 50 mm.

As seen in Figure 5(b), under repeated cycles of $Δ_{y}$ , the force-displacement relationships of FA1 is approximately linear and the loops are narrow. In addition, strength degradation is clearly observed when increasing the number of cycles. The peak lateral force dropped significantly within around 10 cycle counts and then slowed down gently. However, it is difficult to see the detailed evolution of strength degradation from the hysteretic loops in Figure 5(b) since the specimen FA1 failed at the 501st cycle and the loops almost overlapped with each other. The evolution of strength degradation is described in detail in ‘Fatigue damage models’. The maximum lateral strength occurred at the first cycle and the average of maximum forces in the positive and negative loading directions was 222.5 kN.

It is seen in Figure 5(c), under repeated cycles of 2 $Δ_{y}$ , the force-displacement relationship of FA2 has some degrees of nonlinearity and the loops are opened up, compared with those of the specimen FA1 in Figure 5(b). In addition, strength degradation is also observed when increasing the number of cycles. It is still difficult to see the detailed evolution of strength degradation from the hysteretic loops in Figure 5(b) since the specimen FA1 failed at the 160th cycle and the loops almost overlapped with each other. The evolution of strength degradation is described in detail in ‘Fatigue damage models’. The maximum lateral strength occurred at the first cycle and the average of maximum forces in the positive and negative loading directions was 275.6 kN.

As seen in Figure 5(d), under repeated cycles of 3 $Δ_{y}$ , the force-displacement relationship of FA3 has obvious nonlinearity and the entrapped area is larger, comparing with those of the specimens FA1 and FA2. In addition, strength degradation is also observed when increasing the number of cycles. The lateral strength deteriorates faster and the specimen failed at the 8th cycle. The average of maximum lateral strength at the first cycle in the positive and negative loading directions was 268.2 kN.

Backbone curve of specimen FA0

The backbone curve of the specimen FA0 was plotted by connecting the peak lateral strength at each load grade, as shown in Figure 6(a). It is noted that each load grade was repeated at three cycles. As mentioned in ‘Low cyclic fatigue test program’, the specimen FA0 was taken as a benchmark from which the loading displacements of the specimens FA1 to FA3 were determined. For this purpose, the equivalent yield displacement ( $Δ_{y}$ ) of specimen FA0 was computed based on the Park’s method. To be specific, the average backbone curve of specimen FA0 in the positive and negative directions was first computed and the results are plotted in the first quartile, as shown in Figure 6(b). The points of maximum lateral force and 75% maximum lateral force were determined from the plot, which were points A and B, respectively. Based on the definition of the Park’s method, an inclined line was drawn from the origin through point B, and a horizontal line was drawn through point A. The intersection of these two lines (point C in Figure 6(b)) gave the point for $Δ_{y}$ . Based on this method, the $Δ_{y}$ of specimen FA0 was computed as 32 mm. Then the ductility capacity $μ$ of specimen FA0 was computed as $Δ_{u}$ / $Δ_{y}$ =5.0 based on Figure 6(a). This indicates the specimen FA0 had good ductility capacity.

Figure 6.

Envelope of the hysteretic curves of specimen FA0: (a) Complete backbone curve, (b) Park’s method to estimate equivalent yield displacement $Δ_{y}$ .

Comparison of fatigue tests

A comparison of the fatigue test results of specimens FA1, FA2 and FA3 is summarized in Table 2, including the displacement load amplitude, maximum lateral force, and the number of cycles when various typical damage stages were initiated. It can be seen from this table that with the increase of the loading displacement from $Δ_{y}$ to 3 $Δ_{y}$ , the total number of cycles leading to the failure of specimens is decreased dramatically from 501 to 9. In addition, when the displacement load was equal to $Δ_{y}$ , no significant exterior damage to the cover concrete was observed while extensive cover concrete spalling and core concrete crushing were observed when the displacement load was equal to 3 $Δ_{y}$ . These observations indicate that strength degradation rate will increase with the displacement load amplitude for fatigue tests and the fatigue life will decrease dramatically with the loading displacement amplitude.

Table 2.

Summary of fatigue tests for specimens FA1 to FA3.

Specimen	Displacement amplitude (mm)	Lateral drift	F_max (kN)	Cycle counts
Specimen	Displacement amplitude (mm)	Lateral drift	F_max (kN)	Initial cracking	Spalling	Significant spalling	0.85F_max
FA1	32 ( $Δ_{y}$ )	1.21%	222.5	1	–	–	501
FA2	64 (2 $Δ_{y}$ )	2.42%	275.6	1	5	72	160
FA3	96 (3 $Δ_{y}$ )	3.62%	268.2	1	1	1	9

Fatigue damage models

The accumulative damage of concrete columns is visibly characterized by the degradation of horizontal bearing capacity (Liu et al., 1996). In this section, the collected data of specimens FA1 to FA3 is presented and interpreted to explore the degradation of lateral load capacity of reinforced concrete bridge piers with high-strength rebar with the loading cycles. To be specific, the peak lateral force at each cycle was output and plotted against the number of cycles. It is note that the peak lateral force at each cycle is the average of peak lateral forces in the positive and negative directions. Then, a strength degradation model is proposed to capture the degradation of the lateral force with the number of loading cycles at different displacement amplitudes. The proposed model is then compared against the test data to validate its accuracy.

Fatigue strength degradation model

Specimen FA1

Figure 7 shows the peak lateral load versus the number of cycles of specimen FA1. It is seen in this figure that the peak lateral force drops significantly at the first 10 cycles. With the increase of loading cycles, the degradation rate slows down. The lateral force dropped to 85% of the maximum value at the 501^st cycle. The peak lateral force at the first cycle was 222.5 kN for specimen FA1. It is noted that there is a jump in the recorded lateral force at around 170th cycles. This is mainly because during the tests, the axial force was compensated by the hydraulic jacks after significant decrease, which resulted in a slight increase of the lateral force.

Figure 7.

Comparison of peak lateral force versus cycles of displacement load between test data and the proposed model for specimen FA1.

Based on the test data presented in Figure 7, a fatigue model of the peak lateral force $F (N_{1}, d_{1})$ was proposed as an exponential function for the specimen FA1.

F (N_{1}, d_{1}) = F_{1, d_{1}} N_{1}^{β_{1}}

(1)

where $d_{1}$ is the drift ratio of the column of specimen FA1 and $d_{1}$ = $Δ_{y}$ /H = 1.21%; $N_{1}$ is cycle count at $d_{1}$ ; $F_{1, d_{1}}$ is the peak lateral force at the first cycle and $β_{1}$ is an attenuation exponent.

According to the cyclic loading result of specimen FA1 (see Figure 5(b)), the peak lateral force of specimen FA1 was $F_{1, d_{1}}$ = 222.50 kN. By numerical curve fitting, the attenuation exponent of $β_{1}$ = −0.027 was obtained. With these parameters determined, the peak lateral force versus loading cycles of specimens can be estimated by equation (1) and the result is also plotted in Figure 7 for a comparison. As seen in Figure 7, the proposed model for strength degradation of specimen FA1 yields comparable results with the test data. For example, compared with the experimental lateral force, the predicted value based on the proposed model had a maximum deviation of less than 1.4%. Therefore, the proposed model in equation (1) can accurately capture the strength degradation of specimen FA1 with increasing number of cycles of displacement load at $Δ_{y}$ .

Specimen FA2

Figure 8 shows the peak lateral load versus the number of cycles of specimen FA2. It is seen in this figure that the peak lateral force drops significantly at the first several cycles. With the increase of loading cycles, the degradation rate slows down. The lateral force dropped to 85% of the maximum value at the 160th cycle. The peak lateral force at the first cycle was 275.6 kN for specimen FA2 which was 23.9% larger than that of specimen FA1. It is noted that there is a jump in the recorded lateral force at around 60th cycles. This is mainly because during the tests, the axial force was compensated by the hydraulic jacks after significant decrease, which resulted in a slight increase of the lateral force.

Figure 8.

Comparison of peak lateral force versus cycles of displacement load between test data and the proposed model for specimen FA2.

Similarly, a fatigue model of the peak lateral force $F (N_{2}, d_{2})$ was proposed as an exponential function for the specimen FA2.

F (N_{2}, d_{2}) = F_{1, d_{2}} N_{2}^{β_{2}}

(2)

where $d_{2}$ is the drift ratio of the column of specimen FA2 and $d_{2}$ =2 $Δ_{y}$ /H = 2.42%; $N_{2}$ is cycle count at $d_{2}$ ; $F_{1, d_{2}}$ the peak lateral force at the first cycle and $β_{2}$ is an attenuation exponent. Based on the test data in Figure 8, the peak lateral force of specimen FA2 at the first cycle $F_{1, d_{2}}$ =275.61kN. Based on the regression analysis, $β_{2}$ was determined as $β_{2}$ = −0.033. Substituting these values into equation (2), the strength degradation model of specimen FA2 is computed and the result is also plotted in Figure 8 for a comparison.

As seen in Figure 8, similarly, the predicted peak lateral forces are close to those of test data for the specimen FA2, with a maximum discrepancy of 0.9%. Therefore, the proposed model in equation (2) can accurately capture the strength degradation of specimen FA2 with increasing number of cycles of displacement load at 2 $Δ_{y}$ .

Specimen FA3

Figure 9 shows the peak lateral load versus the number of cycles of specimen FA3. It is seen in this figure that the peak lateral force was deteriorated quickly with the increase of loading cycles. The lateral force dropped to 85% of the maximum value at the 9th cycle. The peak lateral force at the first cycle was 268.2 kN for specimen FA3 which was comparable with that of specimen FA2 but was 20.5% larger than that of specimen FA1. It is noted that there is a jump in the recorded lateral force at around 3rd cycles. This is mainly because during the tests, the axial force was compensated by the hydraulic jacks after significant decrease, which resulted in a slight increase of the lateral force.

Figure 9.

Comparison of peak lateral force versus cycles of displacement load between test data and the proposed model for specimen FA3.

Based on the characteristics of the test data in Figure 9, a similar fatigue model of the peak lateral force $F (N_{3}, d_{3})$ was proposed as an exponential function for the specimen FA3.

F (N_{3}, d_{3}) = F_{1, d_{3}} N_{3}^{β_{3}}

(3)

where $d_{3}$ is the drift ratio of the column of specimen FA3 and $d_{3}$ = 3 $Δ_{y}$ /H = 3.62%; $N_{3}$ is cycle count at $d_{3}$ ; $F_{1, d_{3}}$ is the peak lateral force at the first cycle corresponding to displacement amplitude of $d_{3}$ , $F_{1, d_{3}}$ = 268.24kN; and $β_{3}$ is an attenuation exponent. Based on the test data and regression analysis, $F_{1, d_{3}}$ = 268.24kN and $β_{3}$ = −0.078. Substituting these two parameters into equation (3), the strength degradation model of specimen FA3 is computed and the result is also plotted in Figure 9 for a comparison.

As seen in Figure 9, similarly, the predicted peak lateral forces are close to those of test data for the specimen FA3, with a maximum discrepancy of 2.4%. Therefore, the proposed model in equation (3) can accurately capture the strength degradation of specimen FA3 with increasing number of cycles of displacement load at 3 $Δ_{y}$

Based on the above analyses, the strength degradation models of specimens FA1 to FA3 in equations (1) to (3) have the same format, which can be written as follows:

F (N, d_{r}) = F_{1, d_{r}} N^{β}

(4)

Where $d_{r}$ is a constant drift at which the specimens are loaded during the fatigue test; $N$ is cycle count at $d_{r}$ ; $F_{1, d_{r}}$ is the peak lateral force at the first cycle and $β$ is an attenuation exponent. The attenuation exponent $β$ and constant drift $d_{r}$ for specimens FA1 to FA3 are listed in Table 3.

Table 3.

Summary of attenuation exponent $β$ and constant drift $d_{r}$ for specimens FA1 to FA3.

Specimen	FA1	FA2	FA3
Constant drift $d_{r}$	1.21%	2.42%	3.62%
Attenuation exponent $β$	−0.027	−0.033	−0.078

As seen in Table 3, it is clear that the attenuation exponent $β$ is related to the applied constant drift $d_{r}$ . Based on the data in Table 3, the relationship between the attenuation exponent $β$ and the constant drift $d_{r}$ can be established in the form of a quadratic equation.

β = A {d_{r}}^{2} + B d_{r} + C

(5)

Where A, B and C are constant respectively, here A = −133.729, B = 4.348, C = −0.060.

It can be seen in Figures 7 to 9 that the fatigue damage model in equation (4) is in good agreement with the test results, which can be expressed that the lateral force at the first loop degrades drastically at initial several cycles and then slows down gently.

In fact, the model proposed can also be validated by the specimen FA0. Knowing from Figures 3 and 5(a), during the three cycles at the displacement of 35mm, the three peak lateral forces tested were 259.19 kN, 251.95 kN and 251.86 kN, respectively. As comparison for this case, equations (5) and (4) can yield a set of results by $β$ = −0.02591, $F_{2, d_{r}}$ = 254.58 kN, and $F_{3, d_{r}}$ = 251.92 kN. Compared with the measured values, the maximum difference of lateral forces is 1.04%.

Accumulative fatigue damage model

To evaluate the degree of damage at a certain cycle, an accumulative fatigue damage index $D_{N}$ is defined to quantify the lateral strength loss rate.

D_{N} = \frac{F_{1, d_{r}} - F (N, d_{r})}{α F_{1, d_{r}}}

(6)

Where $α$ is the reduction factor of lateral strength at failure and it is usually assumed that the specimen will fail when the lateral force drops by $α F_{1, d_{r}}$ . It generally varies from 0.15 to 0.20 and here it is taken 0.15. N is the cycle count, and N≥ 2. Figure 10 illustrates the defined accumulate fatigue damage index $D_{N}$ .

Figure 10.

Illustration of fatigue damage index. $D_{N}$

By substituting $F (N, d_{r})$ in equation (4) into equation (6), the accumulative fatigue damage index $D_{N}$ can be simplified as:

D_{N} = \frac{F_{1, d_{r}} - F_{1, d_{r}} N^{β}}{α F_{1, d_{r}}} = \frac{1 - N^{β}}{α}

(7)

It is in equation (7) that the $D_{N}$ varies exponentially with the number of cycles as the base and the attenuation exponent $β$ which is in turn related to the constant loading drift ratio of the fatigue test as the exponent. In addition, the $D_{N}$ will increase the the number of loading cycles. Taking the specimen FA2 as an example, the fatigue damage index $D_{N}$ was calculated by equation (7) for each load cycle. The results are summarized in Table 4 and plotted in Figure 11.

Table 4.

Summary of lateral forces of specimen FA2 at various cycle counts.

Cyclecount	Fatiguedamageindex $D_{N}$	Cycle count	Fatiguedamageindex $D_{N}$
2	0.146	50	0.784
3	0.230	60	0.819
4	0.289	70	0.847
5	0.335	80	0.872
6	0.371	90	0.894
7	0.402	100	0.913
8	0.429	110	0.931
9	0.453	120	0.947
10	0.474	130	0.962
20	0.609	140	0.975
30	0.687	150	0.988
40	0.742	160	0.999

Figure 11.

Fatigue damage index versus cycle count for specimen FA2.

To correlate the observation of damage with the computed accumulated damage index $D_{N}$ , the typical damage of the specimen FA2 is listed along with the number of cycles in Table 5. For example, the specimen FA2 experienced yielding at the first cycle, spalling of cover concrete at the 12th cycle, and failure at the 160th cycle.

Table 5.

Observed damage for specimen FA2.

Cycle count	Displacement (mm)	Damage description
1	64	Yielding, crack width exceeded 2.0 mm, cracks were fully developed
2	64	Crack width exceeded 2.5 mm
5	64	Minor spalling, crack stability
12	64	Spalling of concrete cover
50	64	Some loose concrete was picked off
90	64	Significant cracking and concrete spalling, Spalling progressed to 200 mm from the base
150	64	Spalling progressed to 340 mm from the base
160	64	Failure

Based on these observations and the corresponding cycles, the $D_{N}$ was correlated to the damage level of the specimen by combining the Tables 4 and 5. The results are shown in Table 6.

Table 6.

Fatigue damage index versus damage level for specimen FA2.

Fatigue damage index $D_{N}$	Damage level	Description
[0, 0.1)	No damage	No visible hair cracking
[0.1, 0.25)	Minor damage	Visible cracking throughout
[0.25, 0.5)	Moderate damage	Severe cracking, localized spalling
[0.5, 0.90)	Severe damage	Concrete crushing, rebar exposed
[0.90,+)	Failure or collapse	The strength of the pier specimen drops to the maximum strength of 85%

As seen in Table 6, five different damage levels are presented along with the fatigue damage index and description of damage. For example, when the $D_{N}$ varies from 0 to 0.1, the specimen FA2 is determined as no damage. When the $D_{N}$ varies from 0.25 to 0.5, the specimen FA2 is determined as moderate damage with severe cracking and localized spalling. When the $D_{N}$ is larger than 0.9, the specimen FA2 is approaching to failure.

Conclusion

Fatigue damage of reinforced concrete bridge piers can be critical in long duration earthquake. However, little research has been done for reinforced concrete bridge piers with high-strength rebar. This study focuses on fatigue tests on four identical bridge specimens with high-strength rebar of about 600 MPa for yield strength. One of the specimens named FA0 was taken as benchmark and was loaded under a conventional load protocol, while the rest specimens named FA1 to FA3 were loaded using a fatigue load protocol with one, two, and three times the equivalent yield displacement of the benchmark specimen, respectively. Detailed test program is introduced. Observations of damage and hysteretic loops are presented and interpreted to gain insight into the performance of the piers under fatigue loading. The strength degradation due to fatigue loading is explored based on the test data. An empirical-based strength degradation model is proposed for the test specimens. Based on this model, a fatigue damage index is defined and computed, which is then correlated with the observed damages. The main observations and findings are as follows.

The hysteresis curve of the specimens at the first loop under a constant drift followed the path of monotonically pushover skeleton curve and gradually stabilized after the first 10 loops.

With the increase of constant loading displacement from one to three times equivalent yield displacement of the benchmark specimen, the number of cycles required to load to failure decreased dramatically from 501 to 9. This indicates that strength degradation rate will increase with the displacement load amplitude for fatigue tests and the fatigue life will also decrease dramatically with the loading displacement amplitude.

The strength degradation of the specimens followed an approximately exponential function of number of cycles as the base and factor of the loading displacement amplitude as the exponent.

The proposed strength degradation model can accurately estimate the evolution of strength deterioration of the specimens with the increasing number of loading cycles under different displacement amplitudes.

Fatigue damage index ranges were given along with typical damage levels.

It is worth noting that the empirical-based strength degradation model for reinforced concrete piers with high-strength rebar are only based on the test data of specimens FA1 to FA3. Even good agreement has been obtained between the test data and proposed model for the test specimens in this study, it does not necessarily mean it can be applied to other piers with high-strength rebar due to the complexity of nonlinear material properties (concrete and steel) and physical interaction between the reinforcement and concrete. The model is a first attempt of its kind to approach the strength degradation of reinforced concrete piers with high-strength rebar. Due to the limit test data of this type of bridge piers, further study with more experimental data and rigorous numerical analysis is needed to check the accuracy. The research conclusions may be biased, but the idea and the proposed formulas can provide additional input for subsequent experimental research.

Footnotes

Acknowledgements

The high-strength rebar (HRB600E) were donated by Jiangsu Tianshun Group, China. Special thanks to Dr. Junsheng Su for generously providing the test data of Figure 5(a).

Declaration of Conflicting Interest

The author(s) declare that there is no conflict of interest.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The project was sponsored by the National Key R&D Program of China (Grant No. 2017YFC0806009), and the Scientific Research Foundation for the High-level Personnel of Nanjing Institute of Technology (Grant No. YKJ201985).

ORCID iD

Zhao Liu

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