Numerical Investigation of Residual Displacement of Rocking Self-Centering Columns Under Cyclic Loading

3 Well - designed rocking self - centering (RSC) columns are capable of achieving small 4 residual displacement. However, few studies conducted the quantitative analysis for the 5 residual displacement of RSC columns. The residual displacement is the product of the 6 struggle between the self - centering (SC) capacity and the energy dissipation (ED) 7 capacity. In this study, a SC factor and an ED parameter were defined to reflect the SC 8 and ED capacity of the RSC column, respectively. The influence of eight common 9 design parameters on the SC factor and the ED parameter was explored using factorial 10 analysis. Parametric analysis was performed to investigate the tendency of the SC factor 11 and the ED parameter with the increase of maximum drift. According to the results of 12 the parametric analysis, the effect of the SC factor and the ED parameter on the 13 distribution of the residual drift was researched statistically. A simplified formula was 14 proposed to calculate the upper limit of the residual drift. What is more, a set of 15 predictive regression formulas was established to estimate the actual residual drift, these 16 regression formulas have an applicable condition that the ED parameter should be larger 17 than 0.75. When the ED parameter was less than 0.75, the residual drift is approximate 18 to zero. 19


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Past devastating earthquakes have demonstrated that the reinforced concrete (RC) 23 bridge structures are highly vulnerable to earthquakes with extensive damage 24 concentrated in plastic region of bridge pier (Han et al. 2009; Shi et al. 2020). It is likely 25 to cause large residual displacement as a result of serious damage. The bridge structure 26 with significant permanent displacement is hard to return back to the initial position, 27 which causes tremendous difficulties to the post-earthquake recovery and huge 28 economic losses (Kawashima et al. 1998). For example, about 100 RC bridge piers 29 were eventually demolished due to a residual drift ratio of 1.75% after the 1995 Kobe 30 earthquake in Japan (Kawashima et al. 1998). Therefore, the residual displacement is 31 regarded as a main index to evaluate the earthquake resilience and the self-centering 32 capacity of piers become an important design consideration (Uma et al. 2010;Palermo 33 and Mashal. 2012). 34 Based on accelerated bridge construction philosophy, the rocking self-centering 35 (RSC) column was proposed, and the post-tensioning has been considered to be an 36 efficient way to drastically reduce the residual displacement (Marsh et al. 2011). To data, . It can be concluded that there is a trade-off between the SC and ED capacity, the 55 residual displacement is the product of their mutual struggle. 56 The harm of residual displacement has been realized gradually, and residual 57 displacement has been regarded as an important consideration in seismic design. 58 According to earthquake damage investigation results, the Japan Road Association code 59 first proposed a reparability limit of 1% for RC columns (Japan Road Association,  The purpose of this study is exploring the distribution of residual drift of RSC 80 columns under cyclic loading and achieving its prediction. Two governing parameters 81 including a SC factor and an ED parameter were introduced to describe the SC capacity 82 and ED capacity of piers. The contribution of eight common design parameters to the 83 6 SC factor and ED parameter was compared. Then the effect of the SC factor and ED 84 parameter on the residual drift was investigated, a set of predictive formulas for the 85 residual drift was obtained from regressive analyses and an application of it was given. To capture the hysteretic behavior of the RSC columns, a numerical model is 89 established using OpenSees. Fig. 1 shows the schematic of the analytical model. The 90 displacement of the RSC columns is dominated by rigid rotation, the bending 91 deformation occurs before the column lifts up and its contribution is very small, so the 92 elastic beam-column elements can be used to model the column, this simplified 93 simulation strategy was also adopted in previous studies ( The total unbonded length of ED bars in numerical model, Lub, consists of two 111 parts. One is the designed unbonded length (L0) which can prevent the early fracture of 112 ED bars, the other is the equivalent unbonded length (Leu) caused by the strain 113 penetration. The Lub can be expressed as: If Leu=0, the value of Lub is minimum, so the stress of ED bars is largest at the same 116 lateral displacement. As a result, the strength of an RSC column will be overestimated.

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The value of Leu is significant, but it is difficult to determine its value precisely because ( 2 ) where  is elongation of an ED bar, due to the opening of the joint; y  is yield strain  ff where fs is stress of ED bars; fg is compression strength of grout.    results. For specimen PT1 as shown in Fig. 2(a), the simulated initial stiffness is slightly 161 larger than the test result and the simulated strength was slightly lower. Besides, 162 because the elastic beam-column element is adopted to model the column, the simulated 163 hysteretic curve is unable to reflect the little energy dissipation during the cyclic loading.

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On the whole, the FE result matches well with the test result, which suggests that the 165 simulation techniques except for ED bars are effective. Due to the reinforcing steel in specimen HBD2 has a fuse length of 50 mm, strain 172 will be concentrated on the fuse segment, so ub L is set to 50 mm. The hysteretic curve 173 based on FEA is plotted in Fig. 2 7).

Definition and calculation of SC factor
The self-centering and energy dissipation contribution can be derived from the 212 monolithic beam analogy procedure ). However, the iterative 213 calculation process is still relatively complicated. What's more, SC  is obtained from 214 theoretical calculation, while the residual displacement is determined by simulation. In 215 order to reduce error, it is better that using simulation method to measure the 216 self-centering performance index. For this reason, the SC factor SC  used in this study 217 is defined as follows: where SC F is the self-centering contribution provided by gravity load and PT tendons, 220 and ED F is the energy dissipation contribution provided by ED bars. As shown in Fig. 4, 221 the calculation of SC  requires the following two steps:

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(1) Establish the numerical model of RSC-ED columns and conduct a pushover analysis, 223 then the force-displacement curve can be obtained. The force FRSC-ED includes the 224 self-centering and energy dissipation contribution.

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(2) Delete the ED bar elements from the existing numerical model of the RSC-ED 226 column and perform a pushover analysis again, then the force-displacement curve which 227 only contain the self-centering contribution is obtained. Therefore, the energy 228 dissipation contribution FED can be derived as:  The self-centering model is so ideal that it is unable to capture the residual 242 displacement in fact. Implementing two rotational springs in parallel with appropriate 243 hysteretic models is another simplified simulation method. The RSC column without 244 ED bars is modelled by a bilinear elastic model, as shown in Fig.5 (b). The ED bar is 245 simulated using a bilinear elasto-plastic model, as shown in Fig.5 (c). With the coaction 246 of the two springs, the RSC column presents a trilinear skeleton curve to be exact, there 247 is a stiffness reduction after the column is lifted up. Since the stiffness reduction is very 248 slight, the skeleton curve can be approximate to a bilinear one.

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To extract the ED parameter, an equivalent linearization method is used. The ED 250 parameter can be calculated as:     Table 2. For example, in order to ensure the ED bars have enough 281 deformation capacity at larger loading drift, the value of L0 of the specimen RS is set to 282 300 mm rather than 150 mm. The prototype pier of specimen HBD1 has a total mass of 283 180 t, but the gravity force was not applied in the test due to the limitation of the 284 experimental prestressing, the value of G P of the specimen RS is 1800 kN. Meanwhile, 285 G P will enhance the self-centering capacity of the specimen RS, which may lead to no 286 residual displacement. Therefore, the value of PT P is set to 600 kN rather than 1800 287 kN. 288

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(a) Cross-section (b) hysteretic curve 291 Fig. 6 Cross-section and cyclic behavior of specimen RS (Unit: mm) 292   Table 3. To prevent the PT tendon yields under cyclic loading, the 307 value of Pd corresponding to its high level is set to 0.4, and the maximum loading drift 308 is 4%.  Table 4). The other part includes three generators 319 (the last three columns in Table 4) which determines the factors that are not included in 320 the basic design. -and + are the symbols of low level and high level, respectively. The 321 levels of the last three factors are determined by using generating relations.

Table 3 Two levels of the considered factors 323
Level Note: Pd is the ratio between the initial stress of a PT tendon and its yield stress. 324  The effect of each key parameter on the SC factor at 2%, 3% and 4% drift is 338 compared as shown in Fig. 7 with a linear regression model which can be expressed as follows:  Table 5.

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The purpose of this part is exploring the effect of the SC factor and ED parameter   379 In order to obtain enough data to analyze the distribution of residual drift, a 380 parametric study is first carried out. Table 6 shows the all the specimens used in the 381 parametric study, the specimens are designed based on the specimen RS. Each specimen 382 has a special denotation which corresponding to a value of the parameter. For example, 383 for the aspect ratio λ, five specimens are designed and tagged with A1, A2, A3, A4 and 384 A5. Among these tags, A1 corresponds to a value of 3.6.

Parametric study
385 Table 6 Denotation of specimens in parametric study (g) ED bar ratio 403 Fig. 10 Influence of each parameter on ED parameter 404 Fig. 9 shows the influence of each parameter on the SC factor, the uncertainty of 405 Lub in the simulation process is considered. It can be concluded that the value of the SC 406 factor corresponding to minimum Lub is always smaller than that corresponding to 407 maximum Lub, which indicates that the self-centering capacity of bridge columns will be 408 underestimated if the strain penetration is not considered. The uncertainty of Lub has 409 little influence on the change tendency of the SC factor, the SC factor presents a 410 downward trend as the drift increases.

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As shown in Fig. 9(a), the five specimens (A1-A5) have a similar SC factor at 1% 412 drift. With the increase of drift, the SC factor of the specimen with a larger aspect ratio 413 descends more rapidly because of the P-Δ effect. Specimen A1 is different from other 414 specimens, its SC factor has a slight growth when the drift exceeds a certain value.

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Increasing d P , PT  , G  and 0 L are all benefit to improve the self-centering force, so 416 their increase will enhance the SC factor as shown in Fig. 9(b), (c), (d) and (e). The 417 improvement effect of d P and G  on the SC factor is very similar and dependent on 418 the drift as shown in Fig. 9(b) and (d), a larger d P and G  can generate a larger SC 419 factor generally, but the self-centering capacity is not stable, which decreases sharply at 420 the drift increases. As shown in Fig.9 (c), it can be concluded that increasing the PT 421 tendon ratio is an effective method to strength the stability of the self-centering capacity.

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Compared to d P , PT  and G  , the effect of 0 L on the SC factor is very limited as 423 shown in Fig.9 (e). The value of 0 L increase from 300 mm to 700 mm, while the 424 increase of SC factor does not exceed 0.3. Increasing y f or ED  is helpful to improve 425 the energy dissipation capacity, so their increase will reduce the SC factor as shown in 426 Fig. 9(f) and (g). There is a greatest reduction of the SC factor when ED  increases 427 from 0.46% to 0.69% as shown in Fig. 9(g).

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Similarly, Fig. 10 shows the influence of each parameter on the ED parameter, it 429 can be observed that the ED parameter corresponding to the minimum Lub is larger than  drift. On the other hand, it can be concluded that the residual drift is close to zero when 475 the ED parameter is less than 0.75, which is not related to the maximum drift.

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As shown in Fig.13, the data in Fig.11  L5 are given as an example, as shown in Fig.14. In fact, a too long unbonded length will 498 not be designed generally due to it may lead to the buckling of ED bars, so it is 499 unnecessary to pay much attention on the effect of L0.  Table 7.   Based on the analyses, the following conclusions can be drawn:

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(1) Ignoring the strain penetration of ED bars will underestimate the self-centering 522 capacity of RSC columns, and conservative residual displacement will be obtained.

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(2) The total contribution of the eight parameters to the SC factor and ED parameter are 524 stable and maintain at 84% and 95% around, respectively. No significant interaction 525 between these factors is observed. Due to the P-Δ effect, the contribution of the gravity 526 loading ratio to the SC factor will decreases rapidly as the drift increase, and the effect 527 of the aspect ratio will grow.

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(3) The SC factor, ED parameter, and maximum drift dominate the distribution of

Conflict of interest
The authors declare that they have no known competing financial 540 interests or personal relationships that could have appeared to influence the work 541 reported in this paper.

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Availability of data and material Not applicable.

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Code availability Not applicable.