IntroductionA simple welding connection system for the replacement of damaged/fractured longitudinal rebars in severely damaged reinforced concrete (RC) columns with new rebar segments, which have reduced diameters in their intermediate parts (machined part) compared with the original rebars, is theoretically and experimentally investigated in this paper. The proposed rehabilitation procedure includes column repair and shear strength upgrading to obtain compliance with structural performance requirement of CEN (2020), whereas flexural strength upgrading is not considered, as in the case of many existing bridge columns that present only shear strength seismic deficiencies due to construction error or old (not seismic design) code.The present procedure explores the possibility of connecting new rebar segments to the original weldable undamaged rebar ends by a connection system composed of a part from an off-the-shelf steel equal angle; a welding connection is adopted to simplify the on-site process, resolve geometrical problems due to nonalignment of the original rebars, limit the demolishment of the core concrete to the damaged parts only, retain the positions of the plastic hinge thanks to the containment of the welding length, and reduce the uncertainties regarding the bridge capacity after rehabilitation.That the bars are weldable has a huge connotation for the application of this technique. It is therefore strongly recommended that a chemical analysis of the bars to be replaced is carried out to check their weldability.Provided that the original rebars are weldable, the rehabilitation process solves the following two crucial aspects of plastic hinge rehabilitation: (1) it develops a simple technique for replacing a damaged/fractured longitudinal rebar even in complicated site conditions; and (2) it preserves the connections to the original rebars from yielding and allows the location of the plastic hinge above the foundation to remain unchanged if not slightly reduced. An additional positive aspect worth mentioning is that the shear demand remains unchanged or is slightly reduced. Finally, to simplify the concrete placement with a positive impact on execution, the hoops are reduced, which is not discussed in this paper.The paper includes an analytical determination of the geometries of the new rebar segments and of the connections to the original weldable rebars. Also presented, to confirm the design procedure, are experimental tests on three 1∶6-scale specimens representative of the column of a prototype bridge before and after rehabilitation with rebar segments of different machined lengths and diameters. Finally, the main conclusions are drawn, and open issues for future research are presented.Literature ReviewThe typical damages corresponding to different damage states that RC bridge columns experience following earthquakes are as follows: concrete cracking; concrete cover spalling; and yielding, buckling, or fracturing of hoops and/or longitudinal rebars (Vosooghi and Saiidi 2010a, b). Substandard columns (not designed for modern seismic codes) and sometimes standard columns (designed for modern seismic codes) may experience buckling, rebar fracturing, and shear failure (He et al. 2015).He et al. (2015) classified bridge column rehabilitation as follows: (1) rehabilitation of RC bridge columns without fractured longitudinal rebars; and (2) rehabilitation of RC bridge columns with fractured longitudinal rebars.The rehabilitation of RC bridge columns without fractured longitudinal rebars usually refers to bridge columns with small to moderate damage; rehabilitation results in adequate strength capacity without external reinforcement systems but obtains reduced stiffness (French et al. 1990; Lehman et al. 2001) or has adequate external reinforcement systems, such as RC jackets (Bett et al. 1988; Kunwar et al. 2021), steel jackets (Chai et al. 1991; Zhang et al. 2021), fiber-reinforced polymer (FRP) jackets (Chang et al. 2004; Elsouri and Harajli 2011; He et al. 2013b, 2014; Li and Sung 2003; Priestley and Seible 1993; Saadatmanesh et al. 1997; Sheikh and Yau 2002; Vosooghi et al. 2008), near-surface mounting (NSM) of carbon fiber–reinforced polymer (CFRP) laminate systems or hybrid CFRP systems (Chellapandian and Prakash 2018), grooving and corner strip-batten reinforcement (Saljoughian and Mostofinejad 2020), and shape memory alloys (SMAs) (Shin and Andrawes 2011), which recover the bridge column strength and often the stiffness as well. Rehabilitation with external RC jackets may require a relatively long time to cure, as well as considerable labor. For the others, reduced stiffness could occur, which is a limitation.Rehabilitation of RC bridge columns with fractured longitudinal rebars was proposed by Albanesi et al. (2009), Cheng et al. (2003, 2004), Hwang et al. (2021), He et al. (2013a, b, 2016), Lavorato and Nuti (2015), Lavorato et al. (2015, 2017), Lehman et al. (2001), Parks et al. (2016), Rodrigues et al. (2017, 2018), Rutledge et al. (2014), Saiidi and Cheng (2004), Shin and Andrawes (2011), Vosooghi and Saiidi (2013), Wu and Pantelides (2017), Xue et al. (2018), and Yang et al. (2015a, b). These procedures involve the replacement of fractured rebars with other rebars or with external reinforcements in FRP, with or without new external jackets. Some methods include relocation of the plastic hinge. Such rehabilitation procedures can require substantial labor and be time-consuming, with the exception of those that consist of external FRP applications only, which require additional studies to achieve effective strength, ductility, and stiffness.The closure of one or more bridges can paralyze an entire region (Briseghella et al. 2019; Nuti and Vanzi 2003; Nuti et al. 2010); therefore, there has been growth in new rehabilitation technique proposals for reopening bridges to traffic in a short time (He et al. 2013a, b; Sun et al. 2017; Wu and Pantelides 2018), even with reduced performance (for example, limited to sustaining emergency traffic).Considering this, an appealing roadmap is the optimization of some rehabilitation procedures of RC bridge columns with fractured longitudinal rebars, such as those proposed by Cheng et al. (2003, 2004), Lehman et al. (2001), Shin and Andrawes (2011), and Yang et al. (2015a), which include fractured rebar replacement and could be grouped with that of Cheng and Mander (1997), which implies the connection of the new rebar segments with the original rebars through mechanical connectors.Rehabilitation has some important open issues. First, because of the dimensions and stiffness of the mechanical coupler, the plastic hinge is relocated to a position above or below the connection. This may reduce the length of the plastic hinge and increase the shear demand, with unfavorable effects on the displacement capacity of the bridge. Additionally, connecting the new rebar segments with the original rebar in the bridge column may be a complex and difficult process in different real cases (e.g., the operating space to apply the coupler in the case of congested rebar may be insufficient, or rehabilitation could be difficult in the case of an irregular arrangement of the rebars); therefore, alternative connection systems should be evaluated (Kheyroddin et al. 2021). Moreover, with the exception of dog-bone solutions (Cheng and Mander 1997), there could be the risk of unwanted plastic hinge relocation or connection failure due to segment overcapacity.The present proposal, following those by Cheng and Mander (1997) and Cheng et al. (2003, 2004), localizes the new plastic hinge on the new rebar segments only. The following simplifications are discussed: the segment replacement is realized using a machined rebar, and assembly is realized by means of a connector composed of an off-the-shelf steel equal angle, which is also applicable in the case of congested and/or irregular rebars, with the use of welding.Rehabilitation ProcedureAn RC bridge column is considered with adequate longitudinal rebars but inadequate hoops for confinement and shear (Lavorato et al. 2015) that is damaged by strong seismic action. A method is discussed involving the replacement of damaged/fractured longitudinal rebars with new rebar segments and the use of CFRP wrapping to enhance the shear strength, to provide continuum bracing (which can keep the new machined rebars from buckling) and to allow large hoop spacing in the rehabilitation zone. Alternative transverse reinforcement solutions have been presented by Lavorato et al. (2017) and Xue et al. (2018).New rebar segments are commercially available. The new rebar segments are connected with the original weldable undamaged rebars by means of an off-the-shelf steel equal angle, to which the ends of the original rebars and new rebar segments are spliced by fillet and butt welding (Fig. 1).The selected steel equal angle permits the original column geometry to remain unchanged while simplifying the alignment between the original rebars and new rebar segments. The rebars can be welded to the steel equal angle from the exterior of the column. According to the authors’ experience, the rehabilitation operation can be completed in a few days (3 days for the lab specimen). The rehabilitation operation on site could take more time owing to careful design and preparations, including stabilizing the column if needed, taking samples, and conducting mechanical and metallurgical tests. Moreover, it is recommended to wait for concrete to harden before applying CFRP wrapping and opening the bridge to traffic. The large spacing of new hoops and self-compacting concrete (Lavorato and Nuti 2015) simplifies the placement of rehabilitation concrete. Other researchers (He et al. 2013a, b; Wu and Pantelides 2017) have proposed rapid rehabilitation techniques, according to ATC-18 (ATC 1997), with 3–5 days of rehabilitation time. Yang et al. (2015a) used mechanical bar couplers, which imply a reduced cover and require good alignment.A typical example of severe damage in RC bridge columns damaged by strong seismic action is shown in Fig. 2(a). The rehabilitation technique can be divided into five phases. The damaged longitudinal rebar, hoops, and concrete cover were removed [Fig. 2(b)]. The new rebar segments were connected to the original weldable undamaged rebars with off-the-shelf steel equal angles by fillet and butt welding [Fig. 2(c)]. The new hoops were placed [Fig. 2(d)]. After the application of self-compacting concrete (SCC) [Fig. 2(e)], unidirectional CFRP jacketing was applied [Fig. 2(f)].There was no axial force during rehabilitation. Even if this is an approximation with respect to possible real cases, it is considered acceptable, as the applied axial force would be smaller than 15% of the residual core column strength. According to the literature (Ferrotto et al. 2018a, b, c), for small preload values, as in the case analyzed, the capacity of the confined column is similar to that obtained without considering the preload. The connection system design and the geometry of new rebar segments are presented in the following sections.The rehabilitation details for rebar replacement design of the rehabilitated columns (RRs) should be designed on the basis of those of the original columns (ORs). The basis of the procedure is the assumption of equal total top horizontal displacement (δ) before and after rehabilitation, as follows: δtot (for the OR) = δtot′ (for the RR). The design of the new rebar segment consists of defining the geometry of the connections to the cut original weldable rebars and the diameter (dt) and length (Lt) of the machined part of the new rebar segment.The connector proposed in this paper consists of a steel equal angle to which the original weldable rebars and new rebar segments are connected by fillet welding. The ends of the original weldable rebars and new rebar segments are also butt-welded to reduce the length of the fillet weld (Fig. 1). Therefore, the total length of connector (Cc) is as follows: (1) where db = original longitudinal rebar diameter in the column; and Cw = gap needed for butt-welding in accordance with rebar welding standards (AWS 2011; MOH 2012). Usually, Cw should be less than db; for simplicity, it can be assumed that Cw is equal to db.The selection of dt and Lt can be obtained by considering the following four issues: (a) dt should be determined considering that the maximum force transmitted by the machined rebar should be smaller than the yield force in the connector and in the original anchoring underneath; (b) the effective machined length (Lte) where the new rebar segment undergoes plasticity should be shorter than Lt; (c) the expected plastic curvature demand (ϕp′) in the plastic hinge after rehabilitation; and (d) the relation between ϕp′, dt, and db.Fig. 1 shows that the total length of the new rebar segment (Ltot) usually depends on the dimensions of the damaged part of the column in the plastic hinge zone. The new rebar segment consists of a central machined part and two nonmachined parts on the two sides, as follows: (2) where Lnt = total length of the new rebar segment with no diameter reduction (nonmachined parts). To allow welding the new rebar segment to the off-the-shelf steel equal angles, the shortest Lnt is as follows: (3) where C1 = gap between the connector end and the machined start position, which ranges from 5 to 15 mm (Fig. 1).The total length between the column bottom end and the machined start position (C) is as follows: (4) C=Cc+C1=4·db+Cw+C1≈5.5·dbSometimes part of the connector could enter the foundation. The length of the removed concrete part (Ldem) will include, in addition to the substitute new rebar segment length (Ltot), the portions of original rebars in the columns to weld the steel equal angles (top and bottom), i.e., 5·db, and in the upper part only an additional gap length (LG), which is usually 10  mm (Ldem=Ltot+5·db+LG).The dimensions of the machined part, dt and Lt, determine the behavior of the column after rehabilitation. This will be correlated with the behavior of the original column. To guarantee that the plastic excursion develops in the new rebar segment, the maximum force of the new rebar segments must be smaller than the resistance of the connectors and of the original rebars.The relation between ϕp′, dt, and db can be found by means of capacity design strategy, as follows: (5) min(fsyc·Ac;fsy·(db2·π/4))≥γRd2·fsy′·(dt2·π/4)where fsyc and fsy = yield stress of the connector with the cross-sectional area (Ac) and of the original rebar (MPa), respectively; fsy′ = yield stress of the new rebar segment (MPa); and γRd>1 [squared in Eq. (5) to be applied to dt2] takes into account the uncertainties regarding the original and new steel material characteristics (in the rebar and steel equal angle) (Forte et al. 2018) and the hardening of the new rebar segment after yielding. For the assumed present application, γRd=1.2.If fsyc·Ac≫fsy·(db2·π/4), which is easily guaranteed (the connector must be stiff with a large Ac), then the second term in the parenthesis of min() is the smallest, from Eq. (6), and dt can be obtained: (6) The curvature ductility demand of the RR can be formulated on the basis of the curvature ductility demand of the OR. A model for δtot evaluation is proposed, which holds true for the OR and RR. δtot can be subdivided into four contributions: (7) δtot=δy+δp=δflex,y+δslip,y+δflex,p+δslip,pwhere δy = yield displacement of the original column; and δp = displacement beyond yielding of the original column. Both displacements are contributed by a part due to the column internal deformations, δflex,y and δflex,p, and a second part due to the slip into the foundation of the column, δslip,y and δslip,p. The two latter variables are determined by the two concentrated rotations at the base, θslip,y at yielding and θslip,p beyond yielding.For the OR, δflex,y, δslip,y, δflex,p, and δslip,p are as follows (see Fig. 3 detail i): (8) δflex,y=ϕy·H2/3δslip,y=H·θslip,yδflex,p=θp·(H−Lflex,p/2)=ϕp·Lflex,p·(H−Lflex,p/2)δslip,p=H·θslip,pwhere ϕy = yield curvature before rehabilitation; H = column height from the base to the zero-moment section; θp = plastic rotation due to the column internal plastic deformation before rehabilitation; Lflex,p = section of plastic hinge length along which there are internal plastic deformations before rehabilitation; and ϕp = plastic curvature before rehabilitation.Therefore, the contributions for δtot can be given, and ϕp in Lflex,p can be derived: (9) δtot=δy+δp=δflex,y+δslip,y+δflex,p+δslip,p=ϕy·H2/3+H·θslip,y+ϕp·Lflex,p·(H−Lflex,p/2)+H·θslip,pϕp=δtot−(ϕy·H2/3+H·θslip,y+H·θslip,p)Lflex,p·(H−Lflex,p/2)The ϕy for a circular column (Priestley 1997), as suggested in the code (CEN 2004), is as follows: (10) where εsy = yield strain of the rebar; and D = column diameter.θslip,y is as follows (Sezen and Setzler 2008): (11) θslip,y=sy/(D−0.4·yc)=0.5·εsy·lr(εsy)/(0.6·D)=εsy·fsy·db/(6.1·fc′·0.6·D)where sy = rebar slip at yielding, for a concrete-steel bond stress of 0.76·fc′; fc′ = concrete cylinder compressive strength (MPa); lr(εsy) = anchorage length at yielding, lr(εsy)=1/4·db/(0.76·fc′)·fsy; and the term (D−0.4·yc) is approximated to 0.6·D, where yc = height of the compressed part of the cross section.θslip,p with a similar formulation is as follows: (12) θslip,p=sp/(D−0.4·yc)=0.5·(εs+εsy)·lr(εs)/(0.6·D)=(εs+εsy)·[fs(εs)−fsy]·db/(cred,bond·6.1·fc′·0.6·D)where sp = additional rebar slip to sy when the actual strain of the original rebar (εs)>εsy; fs(εs) = stress of the original rebar (MPa) at εs>εsy; and lr(εs) = increment of the rebar anchorage length with respect to lr(εsy) when fs(εs)>fsy, lr(εs)=1/4·db/(cred,bond·0.76·fc′)·[fs(εs)−fsy]. Note that along lr(εs), a residual concrete-steel bond stress (cred,bond)<1 of the maximum available stress (0.76·fc′) is assumed.From Eqs. (9)–(12), ϕp can be obtained: (13) ϕp=13.02·δtot/H2−10.42·εsy/D−3.56·εsy·r·db/(H·D)−3.56·εsy·hs·r·{[μϕ(ϕ)/α]2−1}·db/(cred,bond·H·D)where r=fsy/fc′; μϕ = section curvature ductility for the OR; α = coefficient to obtain the section curvature ductility from the rebar ductility [see Eqs. (26) and (27)]; hs=Es/E is the strain hardening of the original rebar; Es = elastic modulus of the original rebar after yielding (MPa); E = elastic modulus of the original rebar (MPa); and Lflex,p=0.08·H [see Eq. (14)].The general presentation above is suitable for treatment before and after rehabilitation (see Fig. 3 detail i). Indeed, for the OR, as an alternative to the general model presented, various (sometimes simpler) empirical models based on different expressions of the plastic hinge length (Lp) have been proposed by different researchers (Bae and Bayrak 2008; Berry et al. 2008; Feng et al. 2021; Lu et al. 2005; Ning and Li 2016; Panagiotakos and Fardis 2001; Paulay and Priestley 1992; Priestley and Park 1987). The empirical model of Lp [Eq. (14)] (Paulay and Priestley 1992) suggested in the code (MOT 2008) is as follows (see Fig. 3 detail ii): (14) Lp=0.08·H+0.022·fsy·db=Lflex,p+Lslip,pwhere Lflex,p=0.08·H is the part of the plastic hinge along which the longitudinal rebars can plasticize within the column; Lslip,p = increment of the plastic hinge length to account, in a simplified way, for the contribution to the column top displacement associated with the rebar slip in the foundation, including both elastic and plastic slips. Therefore, instead of Eq. (9), δtot can be expressed as follows: (15) δtot=δy+θp·(H−Lp/2)=ϕy·H2/3+ϕp·Lp·(H−Lp/2)Comparing Eq. (9) with Eq. (15), it can be found that δy=ϕy·H2/3 in the latter equation coincides with δflex,y in the former equation, and the term ϕp·Lp·(H−Lp/2) in the latter equation coincides with the contributions of the three terms (δslip,y, δflex,p, δslip,p) given by Eq. (8): (16) ϕp·Lp·(H−Lp/2)=H·θslip,y+H·θslip,p+θflex,p·(H−Lflex,p/2)=H·θslip,y+H·θslip,p+ϕp·Lflex,p·(H−Lflex,p/2)According to Eq. (16), the simplified Eq. (15) loses part of the clear physical representation, concentrating the effects in a single plastic hinge of length Lp whose midpoint is at a distance (H–Lp/2) from the top of the column. To reproduce the results of the more general model of Eq. (9), Lp should depend on the ductility demand. In fact, the expression of the contribution to the hinge length, i.e., 0.022·fsy·db, is valid in the case of large ductility demands for typical column geometries.Given δtot, the ϕp evaluation through the inversion of Eq. (13) requires an iterative procedure, whereas for large plastic displacements, the use of Eq. (15) is very simple and straightforward, and the use of Eq. (10) leads to curvature ductility. The adopted simplification is useful for the design procedure: (17) ϕp=(δtot−ϕy·H2/3)/[Lp·(H−Lp/2)]μϕ=[δtot·D/(2.4·εsy)−H2/3]/[Lp·(H−Lp/2)]+1After rehabilitation, i.e., in the RR, only the flexural part of the plastic hinge within the column, above the connection, i.e., Lflex,p′=0.08·(H−C), can be found (Fig. 3), as the slip in the foundation is due to the anchorage, which remains in the elastic range (δslip,p′=H·θslip,p′=H·0=0). It can be assumed that rebar anchorages cannot yield because of the machining of rebars.The maximum possible length of the substituted segment that yields is Lte along the machined part Lt included in Lflex,p (see Fig. 1). The part of the machined rebar that eventually extends beyond the top of Lflex,p′ will remain elastic and is called the useless machined length (Ltu); therefore, we have the following: (18) Lte=Lflex,p′;if Lt>Lflex,p′;Ltu=Lt−Lflex,p′Lte=Lt;if  Lt≤Lflex,p′To evaluate ϕp′, a vertical cantilever in Fig. 3 with a horizontal force applied on the top is considered. The curvature distributions at yielding are shown in Figs. 3(a and b). The plastic range (Lp in the OR and Lp′ in the RR), which is within the Lt before and after rehabilitation, is shown in Figs. 3(c and d) detail i.For the RR, δ at yielding (δy′) is, with a small approximation, as follows: (19) δy′=δflex,y′+δslip,y′=(1/γRd2)·ϕy·[H/(H−C)]·(H2/3)+(1−1/γRd2)·ϕy·Lt·(H−C−Lt/2)+δslip,y′where δflex,y′ = yield δ of the rehabilitated column due to the column internal deformations; and δslip,y′ = yield δ of the rehabilitated column due to the slip into the foundation of the column, contributed from the rotation (θslip,y′) at the base, due to the strain penetration [from Eq. (11)] at yielding, as follows: (20) δslip,y′=δslip,y/γRd2=H·θslip,y/γRd2=H·θslip,y′For the RR, δtot′, which is analogous to Eq. (7) of the OR, is as follows: (21) δtot′=δy′+δp′=δflex,y′+δslip,y′+δflex,p′+δslip,p′=(1/γRd2)·ϕy·[H/(H−C)]·(H2/3)+(1−1/γRd2)·ϕy·Lt·(H−C−Lt/2)+(1/γRd2)·δslip,y+ϕp′·Lte·(H−C−Lte/2)+0ϕp′=(δtot′−δy′)[Lte·(H−C−Lte/2)]=(δtot′−{(1/γRd2)·ϕy·[H/(H−C)]·(H2/3)+(1−1/γRd2)·ϕy·Lt·(H−C−Lt/2)+(1/γRd2)·δslip,y})[Lte·(H−C−Lte/2)]where δp′=δ beyond yielding of the rehabilitated column; and δflex,p′ = plastic δ of the rehabilitated column due to the column internal deformations.The proper design of the machined rebar must take into account the increase of the section plastic demand with respect to the RR for effect of the imposed plastic strain along the machined rebar part only, i.e., Lte, while δslip,p′ is null; as for adequate diameter reduction, hardening of the machined part cannot induce yielding of the foundation, and the anchorages remain elastic.The designer can adopt a reduced Lte with respect to the maximum effective value of 0.08 · (H−C). However, this reduction is limited by the available section curvature ductility after rehabilitation (μϕ′). The prediction of μϕ′ can be obtained by comparing the responses of the OR and RR columns.It can be assumed that under earthquake action, δtot′=δtot. Therefore, from Eqs. (9) and (21), Eq. (22) can be obtained: (22) (1/γRd2)·ϕy·[H/(H−C)]·(H2/3)+(1−1/γRd2)·ϕy·Lt·(H−C−Lt/2)+(1/γRd2)·δslip,y+ϕp′·Lte·(H−C−Lte/2)+0=ϕy·H2/3+δslip,y+ϕp·Lflex,p·(H−Lflex,p/2)+δslip,pThe plastic and elastic contributions are rearranged as follows: (23) ϕp′·Lte·(H−C−Lte/2)−ϕp·Lflex,p·(H−Lflex,p/2)−δslip,p=ϕy·[H2/3−(1/γRd2)·H/(H−C)·(H2/3)−(1−1/γRd2)·Lt·(H−C−Lt/2)]+(1−1/γRd2)·δslip,yFrom Eq. (23), μϕ′ can be calculated as a function of μϕ with the following assumptions: H−C≈H, rounding to 1 some ratios (H(H−Lte2))≈H−Lt2H−Lte2≈H−Lflex,p2H−Lte2≈1and Lte=η·Lflex,p=η·0.08·H: (24) μϕ′≈(1−1/γRd2)·[1/η·4.2−Lt/(η·Lflex,p)+θslip,y/(ϕy·η·Lflex,p)]+(μϕ−1)·1/η+θslip,p/(ϕy·η·Lflex,p)+1Usually, Lt/Lflex,p≥1; however, it can be assumed that Lt/Lflex,p≈1: (25) μϕ′≈(1−1/γRd2)·1/η·[3.2+θslip,y/(ϕy·Lflex,p)]+1/η·[(μϕ−1)+θslip,p/(ϕy·Lflex,p)]+1Eq. (25) can be further simplified by expressing θslip,p as a function of μϕ (elastoplastic model with hardening), as follows: (26) fs(εs)−fsy=hs·fsy·[μs(εs)−1]εs=εsy+[fs(εs)−fsy]/(hs·E)εs=εsy+εsy·[μs(εs)−1]where μs(εs)=1+(εs–εsy)/εsy is the ductility of the rebar calculated for εs>εsy.It can be assumed that the ductility of the rebar calculated for the strain εs>εsy, [μs(εs)=1+(εs–εsy)/εsy], is as follows: (27) μs(εs)=μϕ(ϕ)/α=(1+ϕp/ϕy)/αfs(εs)−fsy=hs·fsy·[μϕ(ϕ)/α−1]where α depends on the vertical axial load and, for a circular column, varies from 0.8 (in flexure) to 1.2 (for high axial loads) [see Fig. 7(a) and (b)].From Eqs. (12), (26), and (27), θslip,p can be calculated as follows: (28) θslip,p=0.27·εsy·hs·r·{[μϕ(ϕ)/α]2−1}·db/(cred,bond·D)By substituting Eqs. (10), (11), and (28) into Eq. (25), μϕ′ can be obtained: (29) μϕ′≈3.2  (1−1γRd2)η+(1−1γRd2)·1.41·r·dbH·η+(μϕ−1)η+{[μϕ(ϕ)α]2−1}·1.41·hs·r·dbcred,bond·H·η+1Note that the first term, (1−1/γRd2)/η·3.2, depends on the elastic flexural displacement at yielding of the RR column obtained from that of the OR through the term (1–1/γRd2); the second term, (1−1/γRd2)·1.41·r·db/H/η, depends on the slip at yielding of the RR and OR; the third term, (μϕ−1)/η, depends on the flexural plastic demand of the OR; and the fourth term ((μϕ(ϕ)/α)2−1)·1.41·hs·r·db/(cred,bond·H)/η, depends on the slip after yielding of the RR. In addition, all four terms depend on the reduction of flexural plastic hinge length in the RR with respect to the OR, as they are divided by η=Lte/Lflex,p.In Eq. (29), the first term is usually small and hardly overtakes 1; the second term is even smaller than half of the first term; the part depending on (−1) in the fourth term can be neglected for large ductility (μϕ>7). Therefore, Eq. (29) can be simplified for large ductility demands (μϕ>7), including a coefficient of 1.3, as follows: (30) μϕ′≈1.3·{(μϕ−1)η+[(μϕ(ϕ)α]2·1.41·hs·r·dbcred,bond·H·η+1}As pointed out earlier, the ductility demand in the RR depends on η and hs/(η·H). The latter expression accounts for the fact that the RR column has no contribution from the plastic slip of the OR column.If Lte decreases, the ductility demand increases as expected. An increase in the ductility demand is inevitable after rehabilitation, as all the demand is concentrated in the column, whereas in the OR column, part of the demand extends into the foundation. It can be observed that the plastic demand concentration in the column above the foundation guarantees the resilience of the RR column where all the plastic zones are visible. The comparisons among the experimental results and estimations based on “accurate” [Eq. (29)] or “simplified” [Eq. (30)] ductility demand expressions are provided next.The relation among Lte, μϕ′, and the possible limitations for (dt/db) can be found through enacting the following procedures. Given the increase in μϕ′ and the ductility of the machined rebar (με′) with respect to the corresponding parameters of the OR columns, according to Eqs. (29) and (30) or Eq. (17), the adopted diameter reduction (dt/db)ad can be derived, which is sufficient to ensure that the total force transmitted by the new rebar segment excludes yielding in the connections (having a larger area of the connected rebars) or in the original rebars. (31) fsy·(db2·π/4)>fs′(εs′)·(dt2·π/4)where fs′(εs′)=fsy′+hs′·E′·εsp′>fsy′ is the stress of the machined rebar (MPa) for the strain at maximum δ (εs′); εsp′=εs′–εsy′ is the plastic strain of the machined rebar; εsy′ is the yield strain of the machined rebar; hs′=Es′/E′ is the strain hardening of the machined rebar; Es′ = elastic modulus of the machined rebar after yielding (MPa); and E′ = elastic modulus of the machined rebar (MPa).From Eq. (31), considering Eqs. (26) and (27), Eq. (32) can be obtained. 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