### Damage observations

The CCC (control column without steel strands) exhibited flexural cracks at a nominal drift ratio of 0.25%. The lateral force of the column was maximum at a nominal drift ratio of 2.0% in the positive and negative directions. According to the orientation of the column during testing, west to east was defined as the positive direction (compression on the east side), and east to west was defined as the negative direction (compression on the west side). At a nominal drift ratio of 3.0%, spalling of the concrete cover was observed near the base of the column. At a nominal drift ratio of 7.0%, the longitudinal steel deformed bars fractured.

CSC40 (with steel strands and concrete cover thickness of 40 mm) exhibited flexural cracks at a nominal drift ratio of 0.25%. The lateral force of the column reached a maximum value at a nominal drift ratio of 2% in the negative direction. At a nominal drift ratio of 4.0%, spalling of the concrete cover was observed near the base of the column. The lateral force of the column reached a maximum value at a nominal drift ratio of 5.0% in the positive direction. At a nominal drift ratio of 6.0%, the compressed steel strands began to bulge (Fig. 3(a)). At a nominal drift ratio of 7.0%, the longitudinal steel deformed bars fractured, as displayed in Fig. 3(b).

CSC30 (with steel strands and concrete cover thickness of 30 mm) exhibited flexural cracks at a nominal drift ratio of 0.25%. At a nominal drift ratio of 4.0%, spalling of the concrete cover was observed near the base of the column. The lateral force of the column reached a maximum value at a nominal drift ratio of 5.0% in the positive and negative directions. At a nominal drift ratio of 6.0%, the compressed steel strands began to bulge (Fig. 3(c)). At a nominal drift ratio of 7.0%, the longitudinal steel deformed bars fractured, as depicted in Fig. 3(d).

CSC20 (with steel strands and concrete cover thickness of 20 mm) exhibited flexural cracks at a nominal drift ratio of 0.25%. The lateral force of the column reached a maximum value at a nominal drift ratio of 5.0% in the positive and negative directions. At a nominal drift ratio of 6.0%, the compressed steel strands began to bulge, and the concrete cover was severely spalled (Fig. 3(e)). At a nominal drift ratio of 7.0%, the longitudinal steel deformed bars fractured, as illustrated in Fig. 3(f).

The test results revealed that the maximum lateral force of the columns with steel strands was higher than that of the column without steel strands. Moreover, the maximum lateral force was observed at a higher drift ratio when using steel strands than when not using steel strands. These results indicate that using steel strands is beneficial for developing and maintaining the post-yield stiffness ratio. However, the drift ratio for the maximum force in the positive direction was 5.0% for CSC40, whereas that in the negative direction was substantially reduced to 2.0%. This result was likely attributed to the non-uniform distribution of actual concrete cover thicknesses of the column. This phenomenon is further discussed in subsequent sections.

### Hysteretic response

The lateral force–displacement response of each column was measured and recorded. The envelope response was obtained from the peak value of the first cycle for each drift ratio in the lateral force–displacement response, and the envelope response of each column was idealized using a bilinear relationship in accordance with Federal Emergency Management Agency (FEMA) standard 356 (2000). The first line of the bilinear relationship passed through the envelope response at approximately 0.6 (P_{y}), where (P_{y}) is the force of the idealized yield point. The first line ended at the idealized yield point. The second line began from the idealized yield point and ended at the ultimate drift on the envelope response. The yield point was adjusted such that the area below the idealized bilinear relationship approximated that below the envelope response. The process of obtaining the idealized bilinear relationship is illustrated in Fig. 4. In Fig. 4, (D_{y}) is the idealized yield drift; (P_{y}) is the idealized yield force; (D_{u}) is the ultimate drift; (P_{u}) is the lateral force at (D_{u}); (k_{i}) is the initial stiffness; (k_{{2}}) is the post-yield stiffness; and (alpha) is the post-yield stiffness ratio. The post-yield stiffness ratio *α* is defined as follows:

$$alpha = frac{{k_{2} }}{{k_{i} }}$$

(2)

The ultimate drift ((D_{u})) of each column was defined in accordance with the damage condition described in Sect. 3.1. For CCC, the drift of the ultimate point was defined as the drift level immediately before the longitudinal steel bars fractured. For CSC40, CSC30, and CSC20, the ultimate drift was defined as the drift immediately before the steel strands bulged because the lateral force declined after this point. The positive post-yield stiffness could not be maintained after strand bulging. The ultimate force ((P_{u})) was the force corresponding to the ultimate drift in the envelope response of each column.

The lateral force–displacement response, the envelope response, and the idealized response of each column are illustrated in Fig. 5. The lateral force presented in Fig. 5 was modified to remove the P-Delta effect of the axial load system (Ou et al. 2015). The drift ratios shown in Fig. 5 were measured using an optical system and differed from the nominal drift ratios.

Table 3 reveals that compared with the conventional column (CCC), the three columns with steel strands (CSC40, CSC30, and CSC20) had higher ultimate strength and post-yield stiffness ratios; thus, these three columns have better self-centering ability after an earthquake. The test results revealed that CSC40, which had the largest specified concrete cover ratio of 7.5%, had the smallest average post-yield stiffness ratio (2.1%), followed by CSC30 and CSC20 (specified concrete cover ratios of 5.5 and 3.6%, respectively, and post-yield stiffness ratios of 3.9 and 6.4%, respectively). Because the concrete cover was not confined and could spall easily, a higher concrete cover ratio resulted in a higher loss of concrete in the compression zone. The loss of the compression zone resulted in the shortening of the distance between the resultant tension and compression forces acting on the critical column cross-section (column base) and decreased the post-yield stiffness. This explanation is further supported by other test observations presented in subsequent sections.

In Table 3, (D_{p}) is the peak drift; and (P_{p}) is the peak lateral force.

### Energy dissipation

The energy dissipation of columns was evaluated using the equivalent viscous damping ratio calculated as follows:

$$beta_{eq} = frac{1}{2pi } times left( {frac{{E_{D} }}{{K_{eff} D^{2} }}} right)$$

(3)

$$K_{eff} = frac{{left| {F^{ + } } right| + left| {F^{ – } } right|}}{{left| {Delta^{ + } } right| + left| {Delta^{ – } } right|}}$$

(4)

where (beta_{eq}) is the equivalent viscous damping ratio; (E_{D}) is the energy dissipation of a hysteretic loop; *D* is the maximum displacement of a hysteretic loop; (K_{eff}) is the effective stiffness; Δ^{+} and Δ^{−} are the maximum positive and negative displacement of a hysteretic hoop, respectively; and *F*^{+} and *F*^{−} are the forces corresponding to Δ^{+} and Δ^{−}, respectively.

Figure 6 presents the equivalent damping ratio of each column (the average value of two cycles) at each nominal drift ratio. The lateral strength of the columns with steel strands was partly contributed by the steel strands. However, because the strands maintained predominantly elastic to enable self-centering, as evident from the strain data presented in the next section, the strands could not provide hysteretic energy dissipation through yielding. Figure 6 reveals that at a drift ratio higher than 2.0%, the equivalent damping ratio of the conventional column (CCC) was higher than those of the columns with steel strands (CSC40, CSC30, and CSC20). The effect of different concrete cover thickness ratios on the equivalent damping ratio was insignificant.

### Tensile strain responses

Strain gauges were installed on the longitudinal reinforcement (deformed bars and steel strands) to observe the strain responses under loading. Figure 7 presents the strain distribution of the longitudinal deformed bars of the columns with strands (CSC40, CSC30, and CSC20), and Fig. 8 shows the strain distribution of the steel strands of these columns. The y axis of Figs. 7 and 8 represents the elevation of the strain gauge from the bottom of the column.

Figures 7 and 8 reveal that the maximum strains of the longitudinal deformed bars exceeded the yield strain at a drift ratio of 1.0%. At a drift ratio of 4.0%, the maximum strain exceeded 0.02. The strain distribution of the steel strands was substantially different from that of the longitudinal deformed bars. The strains were similar across the unbonded length (from y = 0 to 2300 mm). Wrapping the strands with tape effectively prevented bonding between the steel strand and the concrete; thus, the strain values did not vary. Because of the spreading of the strain across the strands, their maximum strain was considerably smaller than that of the longitudinal deformed bars. At a drift ratio of 4.0%, the maximum strain was approximately 0.004, which was considerably smaller than the yield strain of the strands (i.e., 0.0089). The use of partially unbonded strands effectively reduced the maximum strain and delayed yielding. Comparing the strain responses between the three columns reveal that the differences in the peak value and distribution of strains were not significant for the same drift ratio. This indicates that the effect of the concrete cover thickness ratio on the strain responses was not significant.

### Analysis of the concrete cover thickness ratio

As described in Sects. 3.1 and 3.2, the post-yield stiffness ratios differed significantly between positive and negative loading directions for columns with steel strands (CSC40, CSC30, and CSC20). After completing a cyclic loading test, the crushed and spalled concrete of the column was removed, and the concrete cover thickness around the column was measured (Fig. 9(a)). Figure 9(b)–(e) presents the measurement results for the four columns tested. Moreover, Table 4 compares the measured concrete cover thickness and post-yield stiffness ratio of the columns. The offset of the core concrete centroid was calculated based on the measured concrete cover thickness.

According to the data in Table 4, CCC and CSC20 had similar concrete cover thickness on their east and west sides. The specified concrete cover thickness of CSC40 was 40 mm; however, the measurement results revealed that the thicknesses of its east and west sides were 33 and 50 mm, respectively. The core concrete centroid was offset to the east by approximately 8.5 mm. The specified concrete cover thickness of CSC30 was 30 mm; however, the measurement results revealed that the thickness of its east and west sides were 48 and 20 mm, respectively. The core concrete centroid of this column was offset to the west by approximately 14.0 mm. The variation of the measured concrete cover thickness around the columns was due to the construction error. However, this variation also provided more test data between the cover thickness and the post-yield stiffness ratio.

In Table 4, (c_{c,A}) is the measured concrete cover thickness.

For further exploring the influence of the loss of concrete cover on the post-yield stiffness ratio, the measured concrete cover thickness ((c_{c,A})) was divided by the measured effective depth ((d_{t,A})) to obtain the measured concrete cover ratio ((c_{L,A})). (d_{t,A}) is the measured distance from the extreme compression fiber of the section to the centroid of the farthest layer of longitudinal tension reinforcement.

$$c_{L,A} = frac{{c_{c,A} }}{{d_{t,A} }}$$

(5)

Table 5 and Fig. 10 show that the post-yield stiffness ratio tended to be inversely proportional to the measured concrete cover thickness ratio. This was because the distance of the resultant tension and compression forces (moment arm) acting on the critical column cross-section was positively related to the effective depth of the cross-section. As the concrete cover thickness ratio increased, the moment arm tended to decrease by a higher percentage after the loss of concrete cover. The reduction in the moment arm likely decreased the increase rate of the column lateral strength, leading to a smaller post-yield stiffness ratio. In the positive direction of CSC40 (CSC40 +) and for CSC40 − and CSC30 + , the concrete cover thickness ratio on the compression side was large, which resulted in a considerable loss of the moment arm after spalling of the concrete cover. Thus, the post-yield stiffness ratios were small. Smaller concrete cover thickness ratios were observed on the compression side for CSC30 − , CSC20 + , and CSC20 − than for CSC40 + , CSC40 − , and CSC30 + ; thus, for CSC30 − , CSC20 + , and CSC20 − , the reduction in the moment arm was smaller after the concrete cover spalled. The post-yield stiffness ratios for CSC30 − , CSC20 + , and CSC20 − were higher than those for CSC40 + , CSC40 − , and CSC30 + , which revealed the influence of the loss of concrete cover thickness ratio on the post-yield stiffness ratio.

A linear regression was used to derive a formula to relate the measured concrete cover ratio ((c_{L,A})) with the post-yield stiffness ratio (*α*). The derived linear regression formula is as follows:

$$alpha = – 1.0{1}c_{L,A} + 10.16$$

(6)

Note that the effective depth of the cross-section ((d_{t,A})) was used to represent the effect of the moment arm. It is known that the moment arm depends on design parameters such as the concrete strength, the axial load, and the amount and distribution of longitudinal reinforcement. Since bridge columns usually have a low axial load and the moment arm is not sensitive to the concrete strength and the design of longitudinal reinforcement for typical bridge columns, the use of the effective depth provides a good approximation of the effect of the moment arm. Moreover, the use of the effective depth simplifies the calculation. Hence, Eq. (6) can be used in the preliminary design when most of the design details are still unknown to determine the feasibility of using the proposed column to reduce the residual displacement. Further research can be done to refine Eq. (6) to include the effects of design parameters more directly. Note that the post-yield stiffness ratio depends on the amount, distribution, and strength of the steel strands relative to those of the deformed bars. Thus, care should be taken if Eq. (6) is used for columns with the proportion of the steel strands relative to the deformed bas significantly different than that examined in this research.

Kawashima et al. (1998) reported that if the post-yield stiffness ratio is 5%, the residual displacement of a bridge column can be effectively controlled. In accordance with this result, Eq. (6) reveals that the concrete cover thickness ratio should not exceed 5.1%. However, the concrete cover is used to protect reinforcements from corrosion. Based on ACI 318–19 (ACI 2019), the concrete cover thickness for columns exposed to weather or in contact with the ground should not be less than 50 mm. This is a typical condition for bridge columns. For the case of a concrete cover thickness of 50 mm, the upper limit of 5.1% concrete cover thickness ratio suggests that the effective depth of the cross-section of the proposed self-centering column be not less than 980 mm to ensure a post-yield stiffness ratio of at least 5%.

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