ACI

STRUCTURAL J O U R N A L

A JOURNAL OF THE AMERICAN CONCRETE INSTITUTE

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final manuscript is tentatively scheduled
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The DOI for this paper is 10.14359/51745466

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 1 Modified Analytical Model for the Shear Capacity of Corroded Columns

2 Benjamin Matthews1*, Alessandro Palermo2, Allan Scott3

1
3 Ph.D. Candidate, Department of Civil and Natural Resources Engineering, University of Canterbury, Christchurch,
*
4 New Zealand. Corresponding Author.
2
5 Professor, Department of Structural Engineering, University of California San Diego, San Diego, USA. 3Associate

6 Professor, Department of Civil and Natural Resources Engineering, University of Canterbury, Christchurch, New

7 Zealand.

8 AUTHOR BIOGRAPHIES

9 Benjamin Matthews is a postdoctoral researcher at Eindhoven University of Technology, the

10 Netherlands. Ben received his PhD in civil engineering from the University of Canterbury, New

11 Zealand, in 2023. His PhD research focused on the assessment of reinforcement corrosion in aged

12 structures, predictive machine learning applications, and the seismic shear performance of reinforced

13 concrete bridge piers. Ben’s current research lies in the reuse of existing precast concrete elements

14 and their applications in new designs, as part of the ReCreate project for circular construction.

15 Alessandro Palermo is a Professor at the University of California, San Diego, formerly at the

16 University of Canterbury New Zealand. He received his BS, MS, and PhD in civil engineering from

17 the Technical University of Milan, 1997, 1999, 2004. He is author of over 400 papers, three patents

18 and recipient of the 2020 Alfred Noble ASCE award. His research interests are focused on seismic

19 behavior of RC structures including novel materials, long-term seismic resilience and corrosion

20 effects, and non-metal reinforcement.

21 Allan Scott is an Associate Professor in Civil Engineering at the University of Canterbury,

22 Christchurch, New Zealand. He received his B.Eng from McMaster University, Canada and Masters

23 and PhD from the University of Cape Town, South Africa in 1994, 1997 and 2004. His primary

24 research interests include: the development of sustainable construction materials; assessment of the

1
 1 residual capacity of corroded and seismically damaged reinforced concrete structures; and in situ

2 resource utilization options for off-earth civil engineering construction applications.

3 ABSTRACT

4 This paper proposes a series of empirical modifications to an existing three-step analytical model

5 used to derive the cyclic shear capacity of circular RC columns considering corrosive conditions. The

6 results of sixteen shear-critical RC columns, artificially corroded to various degrees and tested under

7 quasi-static reversed cyclic loading, are used for model verification. The final model is proposed in a

8 stepwise damage-state format relative to the measured damage of the steel reinforcement. New

9 empirical decay coefficients are derived to determine the degraded material properties based on an

10 extensive database of over 1,380 corroded tensile tests. An additional database of 44 corroded RC

11 circular piers is collected to assist in the modification of ductility-based parameters. Compared to the

12 shear-critical test specimens, the model results indicate that the peak shear capacity can be predicted

13 well across a range of deterioration severities (0 – 58.5 % average transverse mass loss), with a mean

14 predictive ratio of ± 8.60 %. As damage increases, the distribution of the corrosion relative to the

15 location of the shear plane becomes a critical performance consideration, increasing predictive

16 variance.

17 Keywords: corrosion; reinforced concrete; circular columns; shear; cyclic; analytical model.

18 INTRODUCTION

19 Aging reinforced concrete (RC) infrastructure is at risk of continual degradation throughout its service

20 life due to aggressive environmental factors and construction/design deficiencies [1, 2]. Chlorides

21 permeating through the protective concrete cover eventually depassivate the embedded steel

22 reinforcement, generating expansive rust products [3]. As the steel oxidizes locally, internal radial

23 stresses are introduced. Once expansive pressures exceed the pore pressure and concrete tensile

24 splitting capacity, micro-cracking proliferates throughout the porous material. The presence of micro-

2
 1 cracks accelerates the diffusion of chloride ions and oxygen from the concrete surface toward the

2 steel interface [1, 4]. As corrosion advances, cracking develops into visible macro-cracks, which

3 detrimentally affects the mechanical response of the material through a redistribution of the

4 constitutive stress-strain pathways [1]. At severe degradation, cracks develop into the confined

5 concrete core, compromising the structural reliability of the member to withstand seismic loads [5].

6 Transverse reinforcement possesses the most significant risk of chloride-induced deterioration due to

7 the conventional geometric and physical design of RC members. The transverse steel is typically

8 detailed as the outer-facing reinforcement, which leads to a smaller thickness of concrete cover

9 protection from chloride ingress [6, 7]. Compared to the larger longitudinal bars, the smaller

10 transverse diameter also oxidates proportionately faster. Thus, transverse steel is often observed to

11 deteriorate preferentially, with ratios of transverse-longitudinal corrosion ranging between 1.5 and

12 4.5 [8, 9]. An accelerated loss of transverse steel will lead to a disproportional degradation of shear

13 and confinement-related mechanical properties. Understanding the disproportionate reduction of

14 mechanical properties is crucial in structural reliability assessments of aging RC bridge structures,

15 particularly those lacking adequate seismic resistance. Many aging bridge assets were designed to

16 obsolete seismic guidelines, providing inadequate seismic design before the additional effects of long-

17 term corrosion are even considered [10, 11]. Thus, the degradation of the seismic shear capacity of

18 RC columns is a critical concern to be addressed across the aging infrastructure stock.

19 LITERATURE REVIEW

20 The ability of an RC column to resist shear stresses imposed on the system by seismic loading is

21 provided by the constitutive relationships between the concrete and steel materials. The concrete

22 shear resistance is provided by the transfer of stress along the crack interface via the interlocking of

23 aggregate protruding from the crack [12]. Transverse steel reinforcement intercepting the crack will

24 mobilize and redistribute the tensile strain throughout the confining cage. Dowel action transfers

25 stress from orthogonal cracks to the longitudinal reinforcement, increasing shear stiffness [12]. Axial

3
 1 compression amplifies the shear resistance of a column by increasing the effectiveness of aggregate

2 interlock. Ang [13] and Priestley et al. [14] experimentally verified the direct correlation between

3 axial load ratio and nominal shear capacity. ACI 318-19 [15] provisions consider the contribution of

4 the axial load to the shear resistance as an integral component of the concrete shear resistance. Other

5 models, such as the Kowalsky and Priestley [16] model, developed at the University of California

6 San Diego (termed the UCSD model herein), disassociate axial and concrete contributions into

7 individual terms. The concrete shear resistance is known to degrade in RC columns under substantial

8 deflection due to the progressive loss of aggregate interlock transfer after excessive crack dilation

9 and smoothening along the crack interface from continual abrasion [17 - 19]. Fig. 1a illustrates the

10 original conceptual model for shear capacity in RC column sections, proposed by the ASCE/ACI

11 Joint Task Committee [17] and adapted by Ang et al. [18].

12 Ang et al. [18] and Wong et al. [19] derived relationships between the degradation in concrete shear

13 resistance and the increase in displacement ductility. Priestley et al. [14] then proposed a displacement

14 ductility factor, 𝛾, by integrating the Ang and Wong ductility models to account for degradation in

15 concrete contribution at high displacement ductility. Fig. 1b presents the 𝛾-curve for the displacement

16 ductility modified by Kowalsky and Priestley [16] for RC columns with circular cross-sections. Other

17 models have proposed ductility-related factors to a similar effect, including Aschheim and Moehle

18 [20] (UCB Model), California Department of Transportation [21] (Caltrans Model), and Applied

19 Technology Council [22] (ATC Model). For circular cross-sections, seismic design codes such as

20 ACI 318-19 suggest indirectly deriving an equivalent square or rectangular section to calculate the

21 nominal shear capacity. NZS-3101 [23] directly evaluates the nominal shear capacity using the actual

22 circular section, replacing the total gross cross-section with an effective shear area component. Cai et

23 al. [24] evaluated the previously mentioned models and proposed a new approach, correlating the

24 seismic shear capacity with the column drift ratio, to alleviate the ambiguity around ductility-based

25 design. The proposed model effectively calculated the seismic shear capacity of circular columns

26 without requiring transformation into an equivalent square or rectangular section.

4
 1 Under ordinary uncorroded conditions, the above models provide reliable estimations of the cyclic

2 shear capacity of RC columns with various cross-sections. However, when considering the

3 debilitating effects of chloride-induced corrosion, the mechanical reliability of the degraded materials

4 is compromised and must be re-evaluated. An abundance of experimental research has been

5 conducted on corroded RC columns with square and rectangular sections, such as [7, 25 - 29]. A

6 much smaller volume of experimental data exists on corroded circular sections [6, 8, 30 - 32]. An

7 experimental program conducted by the authors investigated the rate of seismic shear loss in short

8 shear-critical RC circular columns under various levels of imposed corrosion damage [34]. Test

9 observations indicated that deflection capacity was much more heavily impacted by corrosion than

10 the peak shear capacity. A maximum reduction of 70.9 % and 37.5 % were recorded for the ultimate

11 displacement and peak shear capacity, respectively, at average mass losses of 8.87 % and 37.7 % for

12 the longitudinal and transverse reinforcement. Li et al. [32] proposed an empirical relationship

13 between the average transverse mass loss, axial load ratio, and residual shear capacity based on the

14 results of four corroded short circular piers failing in shear. Li et al. [34] extended this analysis by

15 proposing a constitutive model based on the modified compression field theory and a non-linear finite

16 element analysis. Using additional experimental results of square columns failing in shear, the

17 proposed model appeared to predict the residual shear capacity well, with prediction ratios ranging

18 between 0.78 and 1.26.

19 This paper proposes a series of modifications to the revised UCSD model used to calculate the seismic

20 shear capacity of RC circular columns under corrosive conditions. The UCSD model presented by

21 Kowalsky and Priestley [16] is selected as the basis of this investigation as a highly effective model

22 with a direct evaluation of the circular section. The changes integrate the effects of material and

23 mechanical degradation of members exposed to chloride-induced corrosion on the peak shear

24 capacity. To the best of the authors’ knowledge, the experimental tests reported in this study and Li

25 et al. [32] are the only programs that involve the corroded behavior of RC circular columns failing in

26 shear-dominated modes. Due to several essential cracking statistics not reported in Li et al. [32], the

5
 1 proposed modifications are developed primarily around the experiments presented by Matthews et al.

2 [33]. A database of 44 corroded circular columns, failing under various modes and ductilities, is

3 compiled to derive new relationships between concrete shear degradation and displacement ductility.

4 RESEARCH SIGNIFICANCE

5 The implications of material and mechanical degradation on the structural performance of chloride-

6 affected columns must be known to estimate the remaining capacity of aging structures. Shear failure

7 of short shear-critical columns is a crucial design and safety consideration due to its brittle and often

8 explosive nature. Since the transverse reinforcement typically suffers from more severe degradation

9 than the longitudinal reinforcement, ductile members designed with low flexural-to-shear capacity

10 ratios are also at substantial risk of triggering brittle shear-dominated failure modes. This paper

11 proposes a series of modifications to an existing analytical approach to predict the cyclic shear

12 capacity of corroded RC circular columns. A piece-wise damage state format for low, moderate, and

13 severe corrosion damage is suggested. The effects of corrosion on material and mechanical properties

14 are investigated separately, with two new coefficients proposed to account for 1) premature loss of

15 bond due to severe degradation and 2) an initial enhancement in shear stress transfer due to existing

16 corrosion surface cracks. The proposed model provides an enhanced predictive ability for moderately

17 to highly corroded RC columns with longitudinal mass losses greater than 7.5 %.

18 EXPERIMENTAL DATA

19 An experimental study was undertaken at the University of Canterbury, New Zealand, to investigate

20 the cyclic shear performance of sixteen artificially corroded RC short circular piers. All piers were

21 designed to trigger shear-dominated failure modes. Tests were performed under displacement-

22 controlled quasi-static fully reversed cyclic loading using a standard cyclic loading protocol [35]. The

23 study variables included corrosion current density, corrosion damage level, confinement

24 effectiveness, spiral diameter size-effect, and aspect ratio (𝐿/𝐷).

6
 1 All columns were designed with a 500 mm diameter (𝐷) and cover thickness of 25 mm. Two

2 transverse volumetric ratios were implemented to categorize different levels of confinement

3 effectiveness – under-confined (UC) and well-confined (WC) [23]. A low axial compression ratio of

4 0.02 was used to minimize the shear-resisting mechanisms enhanced by axial force [16]. Two

5 variations of the impressed-current accelerated corrosion method were implemented: cyclic wetting

6 and drying (WD) and constant saturation (CS). A wet-to-dry ratio of 4:3 was adopted for the entire

7 corrosion incubation. All UC columns were subjected to cyclic wetting and drying periods. A 200

8 μA/cm2 or 300 μA/cm2 current density was applied to each specimen.

9 Table 1 summarizes the key design and corrosion parameters for each column in this study. Where

10 𝑓𝑐′ is the concrete compressive strength at testing; 𝜌𝑠𝑝 is the transverse volumetric ratio; 𝑠 is the spiral

11 spacing; 𝑑𝑠𝑝 is the nominal spiral diameter; 𝑖𝑐𝑜𝑟𝑟 is the applied current density; 𝑡 is the duration of

12 the corrosion exposure; 𝜂𝑚,𝑙 and 𝜂𝑚,𝑠𝑝 are the average longitudinal and spiral mass loss percentages,

13 respectively. For brevity, the experimental design details, test setup, and structural analyses are not

14 discussed in this paper and can be found at [33]. Fig. 2a presents the standard design used for the

15 benchmark specimens, and Fig. 2b illustrates the cyclic test setup used in this program. Table 2

16 summarizes the structural test results for each specimen. Where 𝜃 is the inclination angle of the

17 primary diagonal strut to the column axis; ∆𝑢 is the ultimate displacement measured at 0.85 of the

18 peak load; and 𝜇∆ is the displacement ductility.

19 SHEAR MODELS

20 A comparative analysis of several analytical models with ascending complexity is presented to verify

21 the efficacy of the final proposed model. The Kowalsky and Priestley [16] revised UCSD model for

22 uncorroded columns is included to serve as a predictive benchmark (M-0). Two additional simplified

23 degradation models are selected. The first derives a simple shear capacity reduction factor relative to

24 the average longitudinal mass loss (M-1). The second considers only the degraded material properties

25 of the steel reinforcement (M-2). Finally, based on the revised UCSD model structure, the complete

7
 1 degradation model is proposed in a piece-wise format relative to different corrosion damage limit

2 states (M-3).

3 Revised UCSD Shear Model M-0

4 Kowalsky and Priestley [16] presented revisions to the analytical model for the cyclic shear capacity

5 of circular RC columns originally proposed by Priestley et al. [14], based on the work performed by

6 Ang et al. [18] and Wong et al. [19]. This study proposes modifications to the [16] revision, which

7 may serve as a useful analytical tool for the simplified estimation of the degraded shear capacity of

8 corroded circular columns. The UCSD-A model (assessment model) defines the nominal shear

9 capacity of a circular column in the form:

𝑉𝐴 = 𝑉𝑠 + 𝑉𝑝 + 𝑉𝑐 (1)

𝜋 𝐷−𝑐−𝛿 (2)
𝑉𝑠 = 𝐴𝑠𝑝 𝑓𝑦𝑡 cot (𝜃)
2 𝑠

(𝐷 − 𝑐) (3)
𝑉𝑝 = 𝑃 , 𝑓𝑜𝑟 𝑃 > 0
2𝐿

𝑉𝑐 = 𝛼𝛽𝛾√𝑓𝑐′ (0.8𝐴𝑔 ) (4)

10 The model is characterized by the shear resistance provided by the steel truss mechanism (𝑉𝑠 ), the

11 axial load component (𝑉𝑝 ), and the concrete mechanism (𝑉𝑐 ). Where 𝑓𝑦𝑡 is the nominal yield capacity

12 of the transverse reinforcement, 𝑐 is the neutral axis depth, 𝛿 represents the depth of the cover concrete

13 to the transverse reinforcement layer, and the strut inclination angle 𝜃 is assumed to be 30 degrees. 𝛼

14 is a factor accounting for column aspect ratio, 𝛽 accounts for the longitudinal steel ratio, 𝛾 represents

15 the reduction in concrete shear resistance with increasing ductility, and 𝐴𝑔 is the gross cross-sectional

16 area of the column. The original uncorroded shear model is designated herein as model M-0. Due to

17 insufficient experimental data on corroded columns failing in shear with adequate ranges of aspect

18 ratios and longitudinal steel ratios, modifications to 𝛼 and 𝛽 are considered outside of the scope of

19 this research. Eq. 5 and 6 present the original 𝛼 and 𝛽 curves proposed by Kowalsky and Priestley

20 (2000).

8
 𝑀
1 ≤ 𝛼 = 3− ≤ 1.5 (5)
𝑉𝐷

𝛽 = 0.5 + 20𝜌𝑙 ≤ 1.0 (6)

1 Where 𝑀/𝑉𝐷 is equivalent to 𝐿/𝐷, and 𝜌𝑙 is the longitudinal reinforcement ratio.

2 Simple Degradation Model M-1

3 Model M-1 includes a simple empirical degradation factor, defined by the loss of shear capacity with

4 increasing average longitudinal mass loss (𝜂𝑚,𝑙 ) relative to the uncorroded benchmark, expressed by

5 Eq. 7. The maximum shear capacity is normalized against the dependable section, 𝐴𝑔 √𝑓𝑐′ , to eliminate

6 the influence of different concrete compressive strengths. The degradation coefficient is derived

7 statistically through an ordinary least squares (OLS) evaluation, resulting in the linear relationship

8 described in Eq. 7.

𝑉𝑚𝑎𝑥,𝑐𝑜𝑟𝑟 = (1 − 0.0377𝜂𝑚,𝑙 )𝑉𝑚𝑎𝑥,0 (7)

9 Where 𝑉𝑚𝑎𝑥,0 is derived theoretically from Eq. 1 (i.e., model M-0). 𝑉𝑚𝑎𝑥,0 is equal to 470.0 kN and

10 618.7 kN for BM-UC and BM-WC, respectively. The Eq. 7 prediction provides an R2 of 0.520 and a

11 mean absolute percentage error (MAPE) of 9.80 %.

12 Simple Degradation Model M-2

13 The second degradation model (M-2) adopts the UCSD model, substituting the material properties

14 for their degraded estimates. The mean residual cross-sectional area of the transverse reinforcement

15 and the residual spiral yield capacity are considered here, given in Eq. 8 to 11.

16 Area Loss

17 As the reinforcing steel deteriorates with time, the geometric properties degrade at a proportional rate.

18 The average residual cross-sectional area of the degrading steel is equivalent to the average mass loss

19 measured gravimetrically, shown as:

𝑚0 − 𝑚𝑐𝑜𝑟𝑟 𝐴0 − 𝐴𝑐𝑜𝑟𝑟 Ø20 − Ø2𝑐𝑜𝑟𝑟

= =
𝑚0 𝐴0 Ø20

9
 1 Thus, the mean residual bar diameter and cross-sectional area of the steel reinforcement can be

2 inferred from the average mass loss percentage as:

Ø𝑟𝑒𝑠 = Ø0 − Ø0 (1 − √1 − 𝜂𝑚 ) (8)

𝐴𝑟𝑒𝑠 = 𝐴0 (1 − 𝜂𝑚 ) (9)

3 Reinforcement Yield Capacity

4 Strength-based mechanical properties have been shown throughout the literature to degrade linearly

5 with increasing average mass loss. Imperatore [36] conducted a detailed review of existing empirical

6 formulations for estimating several key mechanical properties of steel. Empirical coefficients for the

7 yield stress of a corroded bar have been found to range over [0.0016 to 0.0198] for artificially

8 corroded bars and [0.0119 to 0.0143] for naturally corroded bars [37 - 43]. The degraded yield stress

9 presented herein is considered relative to the nominal uncorroded area of the bar (engineering stress)

10 rather than a function of the residual area.

11 This study uses a database of over 1,380 monotonic tensile tests on corroded reinforcing bars to

12 update the empirical coefficient for the yield stress of the transverse reinforcement. The open-source

13 database compiled by the authors for a previous study can be found at

14 https://zenodo.org/records/8035720. Separating the data by corrosion type results in two datasets of

15 674 artificially corroded samples and 530 naturally corroded samples. Since morphological properties

16 vary widely in large-scale structural members, this study will only consider a coefficient for variation

17 in the corrosion method. Eq. 10 presents the relationship between the normalized yield stress and

18 average mass loss for different corrosion types.

𝑓𝑦,𝑟𝑒𝑠 = 𝛼𝑦 𝑓𝑦0 (10)

(1.0 − 0.0183𝜂𝑚 )𝑓𝑦0 , 0 < 𝜂𝑚 < 27.5 % (Artificial)
𝑓𝑦,𝑟𝑒𝑠 = { (1.0 − 0.0101𝜂𝑚 )𝑓𝑦0 , 0 < 𝜂𝑚 < 27.5 % (Natural)
(0.82 − 0.0086𝜂𝑚 )𝑓𝑦0 , 𝜂𝑚 ≥ 27.5 % (Combined)
19 A change in the degradation slope was observed at larger mass loss magnitudes, indicating a bilinear

20 degradation trend. For severe column damage states, specimens demonstrate transverse mass losses

21 typically between 27.7 % and 58.5 %. Therefore, the degradation slope was defined into bilinear

10
 1 ranges of [0 % < 𝜂𝑚 < 27.5 %] and [27.5 % < 𝜂𝑚 < 100 %], and the corresponding yield coefficient,

2 𝛼𝑦 , was statistically derived using an ordinary least squares (OLS) evaluation. Due to the scarce

3 available data in the higher degradation range, natural and artificial samples are combined to improve

4 the model fit. The coefficient of determination, 𝑅 2 , were calculated as 0.98, 0.984, and 0.921 for

5 earlier artificial and natural ranges, and the severe combined range, respectively. Since the degraded

6 engineering stress derived in Eq. 10 is measured relative to the nominal uncorroded cross-sectional

7 area, the appropriate modification to Eq. 2 becomes:

𝜋 𝐷−𝑐−𝛿
𝑉𝑠 = 𝛼𝑦 (𝐴𝑠𝑝,0 𝑓𝑦𝑡,0 ) cot (𝜃) (11)
2 𝑠

8 M-3 PROPOSED MODIFICATIONS

9 Model Structure

10 The proposed modifications to the UCSD model follow a damage-dependent structure as a function

11 of the average mass loss of the longitudinal steel. Longitudinal mass loss is selected as the governing

12 criterion due to more consistent experimental reliability than transverse mass loss (Table 1). From

13 the available data, four key damage states can be identified.

14 Damage State 0 (DS0)

15 Uncorroded. The model remains unchanged from Eq. 1 to Eq. 4.

16 Damage State 1 (DS1)

17 Low corrosion, pre-cracking. Bond strength is observed to increase slightly due to the additional

18 friction the rust provides [44, 45] before expansive pressures exceed the tensile capacity of the

19 concrete cover. Oh et al. [46] proposed a critical mass loss (𝜂𝑚,𝑐𝑟𝑖𝑡 ) threshold for the initiation of

20 concrete cover cracking relative to the cover depth:

𝜂𝑚,𝑐𝑟𝑖𝑡 = 0.0018𝛿 2.07 (12)

11
 1 For a cover depth (𝛿) of 25 mm, the critical mass loss threshold is approximately 1.4 %. Ou et al. [47]

2 and Cheng et al. [48] adopted a value of 1.5 % for an equivalent uncorroded condition. Thus, until

3 reaching 𝜂𝑚,𝑐𝑟𝑖𝑡 , Eq. 1 to Eq. 4 are maintained. DS1 is defined over the range:

0 < 𝜂𝑚𝑙,𝐷𝑆1 ≤ 𝜂𝑚,𝑐𝑟𝑖𝑡 (13)

4 Damage State 2 (DS2)

5 Moderate corrosion. Initial post-cracking stage with early reductions in concrete and steel material

6 properties. Characterized by changes to the material properties and mechanical structure of the shear

7 mechanisms while still maintaining a ductile failure mode. 𝜂𝑚𝑙 = 7.5 % corresponds approximately

8 to the threshold between ductile and brittle shear failures (Tables 1 and 2). Beyond this, the transverse

9 mass loss exceeds 27.5 %. DS2 is therefore defined as:

𝜂𝑚,𝑐𝑟𝑖𝑡 < 𝜂𝑚𝑙,𝐷𝑆2 ≤ 7.5 % (14)

10 Damage State 3 (DS3)

11 Severe corrosion. Significant longitudinal and transverse cracking, concrete decay (powdering),

12 cover spalling, and expected cracking into the concrete core. Due to the lattice-like interwoven crack

13 pattern, the cover is treated as a delaminated body from the rest of the rigid member and is assumed

14 to be largely ineffective at carrying stress. Brittle failures are characteristic of this damage state. It is

15 common to observe localized hotspots of 100 % spiral reduction, causing confinement to become

16 effectively reliant on the contribution of the longitudinal bars over a given region. DS3 is defined by

17 corrosion damage exceeding 7.5 % mass loss of the longitudinal steel, with an average transverse

18 mass loss greater than 27.5 %.

𝜂𝑚𝑙,𝐷𝑆3 > 7.5 % (15)

19 Material Modifications

20 Steel Elastic Modulus

21 The degradation in steel elastic modulus, measured relative to the engineering yield stress, can be

22 derived using the same aforementioned database via Eq. 16. After separating by corrosion type, 575

23 artificially corroded and 195 naturally corroded samples were available for analysis.

12
 (1.0 − 0.0105𝜂𝑚 )𝐸𝑠0 , 𝑖𝑓 𝐴𝑟𝑡𝑖𝑓𝑖𝑐𝑖𝑎𝑙
𝐸𝑠,𝑟𝑒𝑠 = { (16)
(1.0 − 0.0127𝜂𝑚 )𝐸𝑠0 , 𝑖𝑓 𝑁𝑎𝑡𝑢𝑟𝑎𝑙

1 Where 𝐸𝑠0 is the uncorroded steel elastic modulus assumed to be 200 GPa. The linear regression in

2 Eq. 16 produced an R2 of 0.646 and 0.892 for artificial and natural predictions, respectively.

3 Concrete Compressive Strength

4 Corrosion cracking affects the member on both material and mechanical levels. Campione et al. [5]

5 proposed a corroded section model for circular columns considering three critical zones: 1) the

6 cracked concrete cover, 2) a cracked portion of the concrete core, and 3) the uncracked concrete core.

7 Fig. 3 illustrates the section model adopted for this analysis. The model assumes that rust products

8 accumulate uniformly around the bar circumference, are incompressible, and thus generate expansive

9 micro-cracks bilaterally [49]. The cracked portion of the core is estimated to have a uniform depth

10 equal to the cover concrete, 𝛿 ′ ≈ 𝛿.

11 Cracked Concrete Cover: Corrosion-induced cracking proliferating throughout the concrete cover

12 will alter the constitutive stress-strain relationships between the steel, concrete, and applied loads.

13 For unconfined concrete, the softening coefficient, 𝜁𝑢𝑐 , is described as [50]:

′
𝑓𝑐,𝑐𝑜𝑟𝑟 = 𝜁𝑢𝑐 𝑓𝑐′ (17)

1.0 , 𝜀𝑐𝑟 = 0
𝜁𝑢𝑐 = { 0.9 (18)
, 𝜀𝑐𝑟 ≥ 0
√1 + 600𝜀𝑐𝑟

𝑤𝑐𝑟,𝑡𝑜𝑡
𝜀𝑐𝑟 = (19)
𝑏0

∑ 𝑤𝑐𝑟,𝑖 𝑙𝑖 (20)
𝑤𝑐𝑟,𝑡𝑜𝑡 =
𝐿
′
14 Where 𝑓𝑐,𝑐𝑜𝑟𝑟 is the corroded unconfined concrete strength, 𝜀𝑐𝑟 is the tensile strain induced by the

15 corrosion cracking, 𝑏0 is the column circumference, 𝑤𝑐𝑟,𝑡𝑜𝑡 is the total surface crack width measured

16 experimentally, 𝑤𝑐𝑟,𝑖 is the crack width increment that is assumed to be approximately constant over

17 the measured length 𝑙𝑖 , and 𝐿 is the column clear height [51]. This method geometrically accounts for

18 any subjectiveness in classifying constant crack width between experiments.

13
 1 Cracked Concrete Core: The compressive strength of the cracked zone of the confined core, 𝑓𝑐𝑐𝑐 , can

2 be estimated using a theoretical prediction of the softening coefficient. The degraded confinement

3 effectiveness will adversely affect the confined concrete strength [52].

7.94𝐾 𝐾
𝑓𝑐𝑐𝑐 = 𝜁𝑐𝑐 𝑓𝑐′ (−1.254 + 2.254√1 + − 2 )
𝑓𝑐′ 𝑓𝑐′

𝑓𝑐𝑐𝑐 = 𝜁𝑐𝑐 𝑓𝑐′ ∙ 𝛼𝐾 (21)

4 Where 𝛼𝐾 is the simplified confinement magnifier. Using Eq. 18 and 19, 𝜁𝑐𝑐 can be determined by

5 substituting a theoretical estimation of the total crack width within this zone [53]:

𝑤𝑐𝑟,𝑡ℎ = 2𝜋(𝜈𝑟𝑠 − 1)𝑃𝑎𝑣𝑒 (22)

6 Where 𝜈𝑟𝑠 is the volumetric expansion ratio between the corrosion product and base metal, taken

7 equal to 2 [53]. 𝑃𝑎𝑣𝑒 is the average penetration based on uniform corrosion, determined either

8 experimentally or estimated theoretically [54, 55]:

𝑃𝑎𝑣𝑒 = 0.0116𝑖𝑐𝑜𝑟𝑟 𝑡 (23)

9 Where time, 𝑡, is in years. 𝜁𝑐𝑐 is limited by the magnitude of the unconfined softening coefficient,

10 such that if the theoretical value is lower than the experimental value for the unconfined section, then

11 𝜁𝑐𝑐 = 𝜁𝑢𝑐 . The effective confining pressure provided by the transverse reinforcement (𝐾) can be

12 determined through the work performed by [52], updating the necessary material properties with their

13 degraded counterparts.

1
𝐾= 𝑘 𝜌 𝑓 (24)
2 𝑒 𝑠𝑝,𝑟𝑒𝑠 𝑦𝑡,𝑟𝑒𝑠

1 − 𝑠⁄2𝑑 (25)
𝑐
𝑘𝑒 =
1 − 𝜌𝑐𝑐

4𝐴𝑠𝑝,𝑟𝑒𝑠 (26)
𝜌𝑠𝑝,𝑟𝑒𝑠 =
𝑑𝑐 𝑠

14 Where 𝑘𝑒 is the confinement effectiveness coefficient, 𝜌𝑠𝑝,𝑟𝑒𝑠 is the residual transverse volumetric

15 ratio, 𝐴𝑠𝑝,𝑟𝑒𝑠 and 𝑓𝑦𝑡,𝑟𝑒𝑠 can be determined using Eq. 9 and 10, respectively, 𝑠 is the spiral spacing,

14
 1 𝑑𝑐 is the diameter of the center-to-center enclosed spiral, and 𝜌𝑐𝑐 is the ratio of the longitudinal

2 reinforcement area to the area of the confined core.

3 Uncracked Concrete Core: The compressive strength of the undamaged concrete core is affected

4 only by the reduction in confinement effectiveness due to the degrading transverse steel and can,

5 therefore, be represented as [52]:

𝑓𝑐𝑐 = 𝛼𝐾 𝑓𝑐′ (27)

6 The measured crack patterns and parameters from Eq. 17 to 27 are presented in Appendix B.

7 Mechanical Modifications

8 Shear Strut Inclination

9 In the steel truss analogy proposed by Kowalsky and Priestley [16], the angle of strut inclination to

10 the column vertical axis (𝜃) is assumed to be 30 degrees. However, the angle gradually reduces as the

11 increasing shear load causes a redistribution of imposed stresses [13]. A reliable theoretical derivation

12 of the strut angle was developed by Ang [13] through plastic analyses using the mechanical degree

13 of shear reinforcement.

𝜈−𝜓
cot(𝜃) = √ (28)
𝜓

𝜌𝑠𝑝 𝑓𝑦𝑡
𝜓= (29)
𝑓𝑐′

14 Where 𝜈 is a web effectiveness factor to account for the concrete not developing full compressive

15 strength. Recommended values of 𝜈 range between 0.7 to 0.9 [56]. Thürlimann et al. [57] presented

16 kinematic limitations to the inclination of the compressive stress field of 0.5 ≤ tan(θ) ≤ 2.0 or

17 approximately 25° ≤ 𝜃 ≤ 65°, based on the achievable plastic strain distribution in concrete. The

18 limit also ensures that the angle does not exceed a corner-to-corner failure plane of a column with an

19 aspect ratio of less than 2.0 [13].

20 It was experimentally observed that corrosion enabled exceedance of this kinematic limit, with strut

21 angles measuring as low as 20.3 degrees. Geometrically, this is achieved by the inclusion of

15
 1 intermittent vertical cracks, such that a strut can develop between the crack and neutral axis rather

2 than fully developing to the extreme fiber (Fig. 4b). As corrosion damage increased, the average strut

3 angle decreased (Table 2). Fig. 5 presents the average strut inclination for both damage states as a

4 function of mass loss. Angles presented in Table 2 and Fig. 5 were averaged over all significant

5 diagonal shear cracks forming, measured digitally prior to cover spalling. In DS2, a plateauing occurs

6 near approximately 25 degrees, while DS3 degrades slightly further, closer to 20 degrees. Neither

7 CS-S1 nor CS-S2 developed significant diagonal cracks; instead, the concrete behavior was

8 dominated by longitudinal bond damage originating from the existing corrosion damage. A

9 probabilistic determination of the location and severity of corrosion cracking is outside of the scope

10 of this study; therefore, a conservative lower-bound estimate for the strut inclination is proposed

11 based on each damage state, expressed by Eq. 30.

25 °, 𝑖𝑓 𝐷𝑆2
𝜃′ = { (30)
20 °, 𝑖𝑓 𝐷𝑆3

12 Neutral Axis

13 As the unconfined concrete layer continually softens, the depth of the dependable compression block

14 must be adjusted. When significantly pre-cracked, the idealized stress block of the concrete

15 compressive section will non-linearize, and the centroid of the compression zone will shift inward as

16 the outer extremities soften and lose stiffness. At DS2, the applied compression load should

17 sufficiently offset the cover softening, closing minor cracks. A conservative estimate of the loss of

18 section reliability can be represented as half the cover depth, resulting in a new equivalent section

19 depth equal to 𝐷 − 𝛿. At DS3, the cover is assumed to be generally unable to redistribute stresses

20 between materials, resulting in an equivalent section depth equal to the depth of the enclosed concrete

21 core (𝑑).

22 Axial Compression Performance

23 Cracking through the concrete section has been shown experimentally to degrade the axial

24 compressive capacity of corroded RC columns [5, 58, 59]. Revathy et al. [59] observed a 3 % and

16
 1 12 % decrease in the axial capacity of corroded RC circular columns for a longitudinal design mass

2 loss of 10 % and 25 %, respectively. Based on these results, columns in the DS3 range likely

3 experienced an actual axial load ratio closer to 0.05, potentially enhancing their shear performance

4 compared with predictions. Based on the section model in Fig. 3, Campione et al. [5] proposed the

5 following model to predict the degraded axial load-bearing capacity:

𝑃𝑚𝑎𝑥,𝑐𝑜𝑟𝑟 = 𝜓𝑝 𝑓𝑐′ 𝐴𝑐𝑜𝑣𝑒𝑟 + 𝜓𝑝 𝑓𝑐𝑐 𝐴𝑐𝑟𝑎𝑐𝑘𝑒𝑑 𝑐𝑜𝑟𝑒 + 𝑓𝑐𝑐 𝐴𝑐𝑜𝑟𝑒 + 𝐴𝑙 𝜎𝑠 (31)

6 Where 𝐴𝑙 𝜎𝑠 accounts for the reduction of longitudinal buckling resistance. 𝐴𝑐𝑜𝑣𝑒𝑟 , 𝐴𝑐𝑟𝑎𝑐𝑘𝑒𝑑𝑐𝑜𝑟𝑒 , and

7 𝐴𝑐𝑜𝑟𝑒 are derived from Fig. 3. 𝜓𝑝 is the ratio of the degraded compressive strength for each zone to

8 the uncorroded compressive strength, proposed by Vecchio and Collins [60]. The reduction factor,

9 𝜓𝑝 , derived through the modified compression field theory, is functionally equivalent to the softening

10 coefficient, 𝜁.

11 The magnitude of the axial component (Eq. 3) is determined by the horizontal component of the

12 diagonal compression strut, which carries the shear stress from the application of axial force to the

13 base of the column, illustrated in Fig. 4c [16]. The axial component of the shear resistance mechanism

14 (Eq. 3 and Fig. 4c) is not directly analogous to the axial capacity (Eq. 31). However, assuming that

15 as the axial capacity declines, the shear resistance provided by the axial force is indirectly affected

16 by the smaller dependable section. Thus, an equivalent relationship is proposed based on the

17 dependable strength (𝐴𝑔 𝑓𝑐′ ) of the deteriorated section.

𝜁𝑢𝑐 (𝐷 − 𝑑) + 𝜁𝑐𝑐 (𝑑 − 𝑑 ′ ) + 𝑑 ′ , 𝑖𝑓 𝐷𝑆2

𝐷′ = { (32)
𝜁𝑐𝑐 (𝑑 − 𝑑′ ) + 𝑑′ , 𝑖𝑓 𝐷𝑆3

18 Displacement Ductility

19 Four key points characterize the curve presented in Fig. 1a: 1) the initial shear strength, 𝑉𝑖 (𝛾𝑖 ) before

20 significant plasticity is reached; 2) the dependable displacement ductility limit, 𝜇𝑖 ; 3) the flexural

21 displacement ductility capacity, 𝜇𝑓 ; and 4) the residual shear capacity, 𝑉𝑓 (𝛾𝑓 ), assumed to be reached

22 once the displacement ductility 𝜇∆ ≥ 𝜇𝑓 . After exceeding 𝜇𝑖 (2.0 for an uncorroded column under

17
 1 uniaxial bending), the concrete shear resistance is assumed to degrade linearly due to excessive crack

2 dilation. More participation is demanded from the yielding transverse reinforcement to compensate

3 for the degradation in concrete strength, which can only be achieved by lowering the diagonal strut

4 inclination of the analogous truss [13]. Hence, the extent of the degradation depends heavily upon the

5 degree of core confinement. The greater the supplied transverse reinforcement and confinement, the

6 later the onset of transverse yielding, leading to less degradation and larger displacement ductility

7 [18]. The following sections propose estimations for deriving a member’s design-specific shear-

8 ductility envelope.

9 Shear Strength Before Degradation, 𝑽𝒊,𝒄

10 In an uncorroded state, 𝑉𝑖 = 𝑉𝐴 (Eq. 1), is related to the displacement ductility factor, 𝛾𝑖 , through

11 Eq. 33 [14]. Therefore, the reduced tensile capacity of the concrete in a corroded state is related to

12 the softened compressive strength by:

𝑓𝑐𝑡,𝑐𝑜𝑟𝑟 = 0.29√𝜁𝑢𝑐 𝑓𝑐′ (33)

𝛾𝑖 = 0.29√𝜁𝑢𝑐 (34)

13 Where 𝜁𝑢𝑐 governs both DS2 and DS3 zones since 𝜁𝑢𝑐 ≤ 𝜁𝑐𝑐 .

14 Residual Shear Strength, 𝑽𝒇,𝒄

15 The residual shear strength describes the level of underlying basic shear stress that can be sustained

16 by sufficient confined strength [13, 14, 23]. 𝑉𝑓,𝑐 is obtained through the additive method in Eq. 1,

17 accounting for the complete degradation of the concrete shear resistance:

𝑉𝑓,𝑐 = 𝑉𝑠𝑓,𝑐 + 𝑉𝑝𝑓,𝑐 + 𝑉𝑐𝑓,𝑐 (35)

18 𝑉𝑠𝑓 is characterized by a change in strut angle to compensate for the degradation in concrete

19 resistance. The lowered angle can be calculated using Eq. 28 and 29. From Ang [13], the value of the

20 web effectiveness factor, 𝜈, for a column failing in ductile flexural bending was calibrated to be 0.193.

18
 1 The residual axial component, 𝑉𝑝𝑓 , is assumed to keep the same form as Eq. 3. The residual concrete

2 shear resistance is taken as:

𝑉𝑐𝑓 = 𝑣𝑐𝑓,𝑐 𝐴𝑒

𝑉𝑐𝑓,𝑐 = 𝑣𝑐𝑓,𝑐 ∙ 0.8(√𝜁𝑢𝑐 𝐴𝑐𝑜𝑣𝑒𝑟 + √𝜁𝑐𝑐 ∙ 𝛼𝐾 𝐴𝑐𝑟𝑎𝑐𝑘𝑒𝑑𝑐𝑜𝑟𝑒 + √𝛼𝐾 𝐴𝑐𝑜𝑟𝑒 ) (36)

3 Where the concrete shear stress, 𝑣𝑐𝑓 , is taken as half of the basic shear stress, 𝑣𝑏 = 0.37√𝑓𝑐′ (for

4 𝜌𝑙 ≥ 1.3 %) if the section is sufficiently confined [18]. It can be empirically assumed that:

0.185√𝜁𝑢𝑐 𝑓𝑐′ , 𝑖𝑓 𝜌𝑠𝑝,𝑟𝑒𝑠 ≥ 1 %

𝑣𝑐𝑓,𝑐 = { (37)
18.5𝜌𝑠𝑝,𝑟𝑒𝑠 √𝜁𝑢𝑐 𝑓𝑐′ , 𝑖𝑓 𝜌𝑠𝑝,𝑟𝑒𝑠 < 1 %

5 Assuming that the confined concrete core can still provide a fragment of aggregate interlock at high

6 displacement ductility (Eq. 36), the [16] base limit of 𝛾𝑓 = 0.05 can be maintained as a reasonable

7 lower-bound estimate.

8 Dependable Displacement Ductility, 𝝁𝒊

9 The dependable displacement ductility, 𝜇𝑖 , defined by the onset of degrading concrete shear resistance

10 due to accumulative plasticity and loss of aggregate interlock, is taken as 2.0 for a column in an

11 uncorroded state [13]. Due to increased bond damage and bond-slip in corroded members, less energy

12 is dissipated for the same lateral deflection measured in an equivalent uncorroded member.

13 Furthermore, since the elastoplastic yield displacement is defined by the secant stiffness of the force-

14 displacement response at 0.75𝑃𝑝𝑒𝑎𝑘 [61], some curvature is experienced (initial yielding of the outer

15 bars) before complete yield is achieved (characteristic of circular cross-sections). Therefore, because

16 a larger proportion of the lateral deflection is attributed to bond-slip in corroded situations, less energy

17 is being dissipated at the same lateral deflection, and the actual displacement ductility due to yielding

18 of the longitudinal bars is reduced – ultimately causing a reduction in the dependable displacement

19 ductility.

20 When shear deformations begin to dominate the performance, the curvature profiles along the

21 member height tend to become more irregular [13]. Thus, 𝜇𝑖 = 2 describes the final ductility at which

19
 1 curvature regularity is maintained, which characterizes the transition in failure from shear with

2 limited ductility (2 < 𝜇 ≤ 4) to brittle shear (1< 𝜇 ≤ 2) (Fig. 6). Therefore, the degradation of the

3 dependable displacement ductility limit can be qualitatively estimated through a curvature assessment

4 of columns under corroded conditions. It is assumed that all members failing in brittle modes (DS3)

5 do not achieve the dependable ductility limit before reaching 𝜇∆ , serving as a conservative lower-

6 bound estimate and forming the scope of this analysis to [𝜇𝑐 ≤ 𝜇 < 2.0].

7 Fig. 6 presents example curvature profiles for DS0, DS2, and DS3 specimens. For DS2 members

8 failing with limited ductility, the minimum measured ductility at which an irregular curvature

9 distribution was recorded was found to be 𝜇 = 1.7. In DS3, where all members failed in brittle modes,

10 the maximum ductility that maintained a regular curvature profile was determined to be 1.7, with an

11 average of 1.5. Therefore, a conservative lower-bound estimate of, 𝜇𝑖 = 1.5, is adopted for members

12 within DS3.

1.70 , 𝑖𝑓 𝐷𝑆2
𝜇𝑖,𝑐 = { (38)
1.50 , 𝑖𝑓 𝐷𝑆3

13 The initial assumption that all ductilities over 𝜇 = 2.0 initiate degradation of the concrete shear

14 resistance is also qualitatively verified by the uncorroded curvature profiles in Fig. 6.

15 Flexural Ductility Capacity, 𝝁𝒇

16 The flexural displacement ductility capacity, 𝜇𝑓 , describes the ductility required to ensure a pure

17 bending failure with no shear influence (Fig. 1a). The UCSD model considers 𝜇𝑓 = 8.0. To

18 characterize an equivalent degraded flexural displacement ductility factor, 𝜇𝑓,𝑐 , a database of 44

19 additional circular columns tested under quasi-static lateral cyclic loading, was compiled from six

20 research programs [6, 8, 30 – 32, 62]. Ductility capacities, aspect ratios, and longitudinal mass losses

21 ranged between [1.31 ≤ 𝜇 ≤ 11.9], [2.0 ≤ 𝐿/𝐷 ≤ 6.0], and [0 ≤ 𝜂𝑚𝑙 ≤ 30.71 %], respectively.

22 A degradation relationship was derived between the normalized displacement ductility capacity

23 (𝜇∆,𝑐 /𝜇∆,0) and 𝜂𝑚𝑙 for DS2 and DS3 zones, shown in Fig. 7. All tests failing in brittle modes without

20
 1 developing notable ductility were removed from the regression analysis. The linear regressions for

2 DS2 and DS3 are given by Eq. 39 (Fig. 7).

0.75 ≤ (1.95 − 0.2419𝜂𝑚𝑙 )𝜇∆,0 ≤ 1.0 , 𝑖𝑓 𝐷𝑆2

𝜇∆,𝑐 = { (39)
(0.99 − 0.0216𝜂𝑚𝑙 )𝜇∆,0 ≤ 0.75 , 𝑖𝑓 𝐷𝑆3

3 Thus, the degraded flexural ductility capacity, 𝜇𝑓,𝑐 , can be determined by substituting 𝜇𝑓,0 = 8.0 into

4 Eq. 39. The imposed limits, 0.75 < 𝐷𝑆2 ≤ 1.0 and 𝐷𝑆3 ≤ 0.75, ensure continuity between damage

5 states. The Eq. 39 regression produced predictions with an R2 of 0.850 and 0.768 for DS2 and DS3,

6 respectively, with calculated MAPEs of 13.9 % and 21.4 %.

7 Bond Failure Penalization

8 Brittle shear-bond failures are characterized by severe bond cracking through the concrete cover,

9 originating from longitudinal corrosion cracks, which influences the initial loss of concrete strength

10 at the point of spalling (often at the peak load). Shear failure is then distinguished by rupturing of the

11 spiral reinforcement, resulting in an acute drop in load-bearing strength as the steel contribution is

12 abruptly lost. In a member failing in shear-bond, diagonal struts cannot adequately develop due to

13 significant vertical cracking (Fig. 4b), redistributing imposed stresses immediately after loading

14 begins – i.e., the concrete has no opportunity to establish the desirable inclined stress pathways. Thus,

15 the maximum shear resistance of the transverse steel cannot be fully developed. Table 2 shows that

16 those members failing with bond characteristics typically demonstrated greater losses in peak shear

17 capacity relative to the uncorroded benchmarks. The maximum crack width (𝑤𝑐𝑟,𝑚𝑎𝑥 ) provides the

18 strongest correlation to the onset of bond-oriented failure modes, as shown in Fig. 8. Excluding

19 specimen WD-WC-M1, a threshold maximum crack width of 1.0 mm within the DS3 range signifies

20 the onset of bond cracks dominating the concrete performance. Hence, a penalization factor, 𝜓𝐵 , is

21 proposed for those members whose concrete resistance is lost prematurely through bond before the

22 complete shear resistance can be developed.

21
 1.0 , 𝑖𝑓 𝐷𝑆2 (40)
𝜓𝐵 = { 1.0 , 𝑖𝑓 𝐷𝑆3 𝑎𝑛𝑑 𝑤𝑐𝑟,𝑚𝑎𝑥 < 1.0𝑚𝑚
0.85, 𝑖𝑓 𝐷𝑆3 𝑎𝑛𝑑 𝑤𝑐𝑟,𝑚𝑎𝑥 ≥ 1.0𝑚𝑚

1 Crack Correction Factor

2 At low corrosion, concrete resistance is overly penalized by softening coefficients of 𝜁 < 0.75. Given

3 the same initial concrete compressive strength, a 20 % reduction in 𝑉𝑐 is measured between BM-WC

4 and WD-WC-L1 (𝜂𝑚𝑙 = 4.38 %). The increased crack area is more likely to supply a small

5 compensatory effect, whereby the early redistribution of stress provides a greater opportunity for

6 aggregate interlock, dowel action, and shear transfer before significant deflection is reached. This

7 effect would be an amplification of the concrete shear resistance (before exceeding 𝜇𝑖 ), based on the

8 total crack strain and remaining confining steel volume.

9 Another important consideration is the documented effect of corrosion rate on morphology and crack

10 growth [63, 64]. Larger current densities are known to increase the likelihood of uniformly distributed

11 corrosion patterns while decreasing the frequency of localized pitting, which is characteristic of

12 naturally occurring corrosion [65]. A corrosion rate correction is therefore proposed to account for

13 the relationship between crack growth and the rate of material loss per year (𝑅𝑖𝑐𝑜𝑟𝑟 ). An empirical

14 correction factor, 𝜈𝑐 , is proposed in the following form:

𝜀𝑐𝑟
1.0 ≤ 𝜈𝑐 = 1.05 + − (𝑅𝑖𝑐𝑜𝑟𝑟 )2 ≤ 1.3 (41)
𝐿⁄ ∙ 𝜌
𝐷 𝑠𝑝,𝑟𝑒𝑠
𝐴
𝑅(𝑖) = (42)
𝑍𝐹𝛾𝐹𝑒

15 Current density is measured relative to actual impressed time for those subjected to wet-dry cycling.

16 𝐴 is the molar mass of iron (55.847 g), 𝑍 is the ionic valency of iron (2), 𝐹 is Faraday’s constant

17 (96,487 A.s), and 𝛾𝐹𝑒 is the density of iron (7.86 g/cm3). Hence, the material loss per year for a given

18 current density is 𝑅(𝑖) = 1161.1 cm/year.

22
 1 MODELING RESULTS

2 Table 3 presents the results of models M-0 through M-3. A summary of model M-3 is presented in

3 Appendix C. Model M-0 accurately estimates the peak shear capacity of columns within the DS2

4 range while under-predicting the uncorroded benchmark members and drastically over-predicting

5 severely deteriorated members. An important observation is the substantial increase in test variability

6 at higher levels of corrosion (DS3) compared to DS2. Given similar corroded properties, there is

7 considerable variation in the experimental test performances within the DS3 range. For example, UC-

8 M1 and UC-S1 had measured peak capacities of 588 kN and 495 kN (16 % variance), respectively,

9 with average transverse respective mass losses of 𝜂𝑚𝑠𝑝 = 37.5 % and 43.0 %. Observations taken

10 during testing show that corrosion damage location becomes a critical parameter at larger

11 deterioration when considered relative to the load direction. The likely reason UC-M1 performed so

12 well was that most of the corrosion damage was located outside of the shear planes and was more

13 heavily concentrated toward the push-pull extreme fibers. The location and distribution of the

14 corrosion damage relative to the load direction are not adequately captured by an average mass loss

15 measurement, which is the prevailing measurement of experimental corrosion. Therefore, to account

16 for test variance and improve model predictions, a distribution parameter considering the global

17 distribution with respect to the location of the shear planes would be required. Global corrosion

18 distribution was not quantitively measured in the program presented in this work, so a location

19 parameter must be considered in future research.

20 Fig. 9 compares the predictive performance of each model against the measured experimental

21 capacity. The final M-3 model efficiently estimates the peak shear capacity, particularly within the

22 DS3 range. All but two test specimens (WD-WC-S1 and WD-WC-S2) were conservatively under-

23 predicted, with a mean prediction ratio of 8.60 %. DS2 predictions demonstrate a strong model

24 consistency with a variance of 5.53 %. The predictions are still conservative, however, with a mean

25 ratio of 13.13 %. This is likely the result of an over-penalization of the steel truss component with

26 transverse mass losses ranging between 12.7 % and 28.8 %. Despite the relatively high transverse
23
 1 deterioration, DS2 columns still performed exceptionally well compared to their uncorroded

2 counterparts. Models M-1 and M-2 show high scatter and poor predictive accuracy. Therefore,

3 models M-1 and M-2 are unsuitable predictive tools for the degraded cyclic shear capacity of corroded

4 shear-critical RC circular columns.

5 From Fig. 9e, the effect of the crack enhancement on the prediction ratio can be seen to have a mild

6 negative trend. Justifying the need to impose limits on Eq. 39, which may be iteratively improved as

7 more data becomes available. Fig. 10 illustrates the shear-ductility envelopes of a selection of test

8 specimens from each damage state, highlighting the change in slope as a function of design and

9 corrosion damage. Additional verification should be sought for the proposed modifications to the

10 displacement ductility curve, implementing force-displacement responses and numerical analyses

11 over a greater range of ductility capacities and failure modes.

12 Finally, to ensure that all predictions provide a conservative prediction of the actual shear capacity, a

13 strength reduction design factor is proposed based on Table 3 in the form of:

𝑉𝐷,𝑐 = ϕ𝑉𝐴,𝑐 (43)

ϕ = 0.80

14 CONCLUSIONS AND LIMITATIONS

15 An adaptation of an existing analytical model for calculating the cyclic shear capacity of circular RC

16 columns considering the effects of chloride-induced corrosion is presented in this research. The

17 modifications account for material and mechanical degradation effects within the corroded RC

18 column. The proposed model categorizes RC columns by the severity of the corrosion damage,

19 represented as different damage states. Damage states 0 (DS0) and 1 (DS1) represent effective

20 uncorroded conditions, such that the model maintains its original structure. DS2 and DS3 describe

21 moderately to severely corroded states, defined by the average longitudinal mass loss ranges

22 [1.5 % < 𝜂𝑚,𝑙 ≤ 7.5 %] and [𝜂𝑚,𝑙 > 7.5 %]. Where 7.5 % longitudinal mass loss corresponds to

23 approximately 27.5 % average transverse mass loss. New empirical decay law coefficients are

24
 1 proposed to explain the reduction in material properties with increasing corrosion based on a database

2 of over 1,380 corroded tensile tests.

3 The changes in diagonal strut inclination, degradation of axial compression capacity, neutral axis

4 depth, the relationship between concrete shear resistance and displacement ductility, bond

5 degradation, and shear stress transfer through concrete cracking are considered in the model. An

6 additional database of 44 corroded circular columns is compiled to derive new shear-ductility

7 relationships for each corroded damage state.

8 The final proposed model was shown to improve the predictive performance and variance compared

9 to simplified degradation models developed for comparison, particularly in more severely corroded

10 ranges (DS3). Due to the limited experimental data available on corroded RC circular columns failing

11 in shear-dominated modes, a degree of scatter remains in the final model. It was observed that the

12 prevailing location of the corrosion damage relative to the load direction (and, therefore, the

13 formation of shear planes) heavily dictated the final performance. Thus, a corrosion distribution

14 parameter relative to the shear planes should be explored to account for test variance and improve the

15 final model predictions. Global corrosion distribution was not quantitively measured in the

16 experimental program presented in this work, so a location parameter must be considered in future

17 research. Nevertheless, with a mean predictive ratio of ± 8.60 % compared to the experimental data,

18 the model provides a strong predictive tool for the assessment of aging and compromised RC columns

19 subjected to chloride-induced deterioration. Final DS2 predictions demonstrated a conservative mean

20 predictive accuracy of +13.13 %. Through a comprehensive comparison between the original model,

21 two simplified degradation models, and the final proposed model, the original (uncorroded) model

22 proposed by Kowalsky and Priestley [16] provides the strongest predictive accuracy for columns

23 within the DS2 range, with a mean accuracy of 0.05 % and standard deviation of 4.53 %.

24 Although many of the proposed modifications involve well-founded concepts from the literature, the

25 final proposed model has been built upon a limited pool of data, only considering a small range of

25
 1 aspect ratios, confinement degrees, failure modes, and only one axial load ratio. More testing must

2 be conducted across key experimental variables to improve predictive ability.

3 ACKNOWLEDGEMENTS

4 The authors would like to acknowledge the Civil and Natural Resources Engineering Department and

5 University of Canterbury Structural Engineering Laboratory technical staff for their contributions and

6 financial support, without whom this work would not be possible.

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32
 1 LIST OF TABLES

2 Table 1 - Column specimen geometric, reinforcement, and corrosion design parameters.

3 Table 2 - Structural results from cyclic testing.

4 Table 3 - Model M-0 to M-3 results and comparison with experimental values, ordered by
5 ascending longitudinal mass loss.

7 LIST OF FIGURES

8 Fig. 1 – ASCE/ACI 426 [17] shear-ductility model [18] (a); UCSD γ model for the degradation of
9 concrete shear resistance with increasing displacement ductility [16] (b).

10 Fig. 2 – Standard under-confined column design and well-confined design (a); Cyclic shear test
11 setup (b).

12 Fig. 3 – Corroded section model for material modifications.

13 Fig. 4 – Original inclined strut-compression zone relationship (a); change in strut formation with
14 the inclusion of vertical corrosion cracks (b); contribution of the axial compression load to the shear
15 capacity of an RC column (c) [16].

16 Fig. 5 – Average measured shear strut angle for DS2 and DS3 test members.

17 Fig. 6 – Curvature profiles for uncorroded benchmark specimens (BM-UC and BM-WC); typical
18 curvature profiles of DS2 members (UC-L1 and CS-L2); and DS3 columns (UC-M2 and UC-S1).

19 Fig. 7 – Displacement ductility degradation with increasing corrosion damage using data collected
20 from [6, 8, 30 - 32, 62].

21 Fig. 8 – Maximum crack width and transition in failure mode. Note: SLD = shear limited ductility;
22 BS = brittle shear; BS-B = brittle shear-bond.

23 Fig. 9 – Model M-0 prediction results compared with experimental results (a); M-1 (b); M-2 (c); M-
24 3 (d); influence of crack correction factor on model M-3 predictions (e). SLD = shear limited
25 ductility, BS = Brittle shear, BS-B = brittle shear-bond.

26 Fig. 10 - Example shear capacity-ductility envelopes for several DS0, DS2, and DS3 members
27 based on design and corrosion properties.

28

29

30

31

38
1 Table 1. Column specimen geometric, reinforcement, and corrosion design parameters.
𝒇′𝒄 𝝆𝒔𝒑 𝒔 𝒅𝒔𝒑 𝑳 𝒊𝒄𝒐𝒓𝒓 𝒕 𝜼𝒎,𝒍 𝜼𝒎,𝒔𝒑
Column ID
(MPa) (%) (mm) (mm) (mm) (μA/cm2) (days) (%) (%)
BM-UC 37.4 0.57 125 10 1075 0 0 0 0
UC-L1 47.1 0.57 125 10 1075 200 20 4.87 13.20
UC-L2-2.95 47.1 0.57 125 10 1475 300 13 4.56 12.32
UC-M1 46.7 0.57 125 10 1075 200 69 7.88 37.49
UC-M2 47.1 0.57 180 12 1075 200 69 8.97 27.16
UC-S1 46.7 057 125 10 1075 200 123 10.51 42.96
UC-S2-2.95 47.1 0.57 125 10 1475 300 80 9.45 30.44
BM-WC 35.9 1.10 65 10 1075 0 0 0 0
CS-L1 56.7 1.10 65 10 1075 200 20 4.96 16.30
CS-L2 46.7 1.10 65 10 1075 300 13 5.10 15.22
CS-S1 56.7 1.10 65 10 1075 200 123 13.21 58.51
CS-S2 56.7 1.10 65 10 1075 300 82 12.83 52.43
WD-WC-L1 56.7 1.10 65 10 1075 200 20 4.38 12.16
WD-WC-L2 38.6 1.10 65 10 1075 300 13 4.40 15.48
WD-WC-M1 56.7 1.10 65 10 1075 200 69 5.81 28.68
WD-WC-M2 47.1 1.10 95 12 1075 200 69 7.68 27.70
WD-WC-S1 56.7 1.10 65 10 1075 200 123 8.87 37.69
WD-WC-S2 46.7 1.10 65 10 1075 300 82 7.61 35.70
2 Note: CS = constant saturation; WD=wet-dry phasing; BM = benchmark uncorroded; UC =
3 under-confined; WC = well-confined; L = low target corrosion; M = moderate corrosion; S =
4 severe target corrosion; and the prefix 2.95 refers to the change in aspect ratio.

5 Table 2. Structural results from cyclic testing.

𝑷 𝜽 𝜟𝒚 𝑽𝒚 𝑽+
𝒎𝒂𝒙 𝑽−
𝒎𝒂𝒙 𝜟𝒖
Specimen ID 𝝁∆ Failure Mechanism
(kN) (deg) (mm) (kN) (kN) (kN) (mm)
BM-UC 147 27.8 7.57 433 500 444 32.1 4.24 Shear limited ductility
UC-L1 185 24.0 8.13 422 483 513 19.2 2.36 Shear limited ductility
UC-L2-2.95 185 25.8 24.81 465 498 424 35.6 1.44 Brittle shear
UC-M1 183 24.0 11.53 509 588 535 15.2 1.32 Brittle shear
UC-M2 185 24.4 7.66 422 478 408 12.1 1.59 Brittle shear-bond
UC-S1 183 23.7 8.63 414 474 495 15.2 1.77 Brittle shear
UC-S2-2.95 185 22.1 16.01 339 391 345 23.3 1.45 Brittle shear-bond
BM-WC 141 38.1 13.00 639 689 640 36.2 2.78 Shear limited ductility
CS-L1 223 25.0 14.81 621 671 638 23.2 1.57 Brittle shear
CS-L2 183 26.4 13.37 610 670 680 27.8 2.08 Shear limited ductility
CS-S1 223 0.0 9.28 424 518 491 15.4 1.65 Brittle shear-bond
CS-S2 223 0.0 13.00 476 507 501 19.7 1.52 Brittle shear-bond
WD-WC-L1 223 30.9 14.40 652 725 764 30.8 2.14 Shear limited ductility
WD-WC-L2 152 28.6 13.51 537 637 592 28.2 2.09 Shear limited ductility
WD-WC-M1 223 25.9 11.35 577 671 632 19.2 1.69 Brittle shear
WD-WC-M2 185 25.2 15.87 612 706 645 27.1 1.71 Brittle shear
WD-WC-S1 223 20.3 7.16 370 420 430 10.5 1.47 Brittle shear-bond
WD-WC-S2 183 30.1 9.97 427 493 444 16.0 1.61 Brittle shear-bond
+
6 Note: 𝑉𝑚𝑎𝑥 represents the maximum shear force experienced in the push direction, and likewise,
−
7 𝑉𝑚𝑎𝑥 in the pull direction.

8
9

39
 Table 3. Model M-0 to M-3 results and comparison with experimental values, ordered by ascending
longitudinal mass loss.
Damage 𝜼𝒎𝒍 𝑽𝒎𝒂𝒙,𝒆𝒙𝒑 𝑽𝒎𝒂𝒙,𝒆𝒙𝒑 𝑽𝒎𝒂𝒙,𝒆𝒙𝒑 𝑽𝒎𝒂𝒙,𝒆𝒙𝒑
Specimen ID 𝑽𝒎𝒂𝒙,𝒆𝒙𝒑 𝑽𝑨,𝑴−𝟎 𝑽𝑨,𝑴−𝟏 𝑽𝑨,𝑴−𝟐 𝑽𝑨,𝑴−𝟑
State (%) 𝑽𝑨,𝑴−𝟎 𝑽𝑨,𝑴−𝟏 𝑽𝑨,𝑴−𝟐 𝑽𝑨,𝑴−𝟑
BM-UC 0.00 499.8 470.0 - - - 1.06 - - -
BM-WC 0.00 689.0 618.9 - - - 1.11 - - -
DS0 Mean 1.09
Mean (%) 8.83
Std. (%) 3.54
WD-WC-L1 4.38 764.3 705.3 659.3 579.8 665.3 1.08 1.16 1.32 1.15
WD-WC-L2 4.40 636.8 632.0 590.3 477.8 591.0 1.01 1.08 1.33 1.08
UC-L2-2.95 4.56 498.2 503.9 467.7 438.1 451.0 0.99 1.07 1.14 1.10
UC-L1 4.87 512.7 511.6 468.8 442.0 472.4 1.00 1.09 1.16 1.09
CS-WC-L1 4.96 671.0 705.3 643.9 544.1 613.5 0.95 1.04 1.23 1.09
DS2
CS-WC-L2 5.10 680.0 667.5 606.0 515.5 569.3 1.02 1.12 1.32 1.19
WD-WC-M1 5.81 670.8 705.3 621.3 477.2 552.4 0.95 1.08 1.41 1.21
Mean 1.00 1.09 1.27 1.13
Mean (%) 0.05 9.15 27.23 13.13
Std. (%) 4.53 3.87 9.85 5.53
WD-WC-S2 7.61 492.7 667.5 542.9 413.3 522.8 0.74 0.91 1.19 0.94
WD-WC-M2 7.68 706.1 663.5 537.8 444.8 597.0 1.06 1.31 1.59 1.18
UC-M1 7.88 587.8 509.9 409.4 375.3 444.4 1.15 1.44 1.57 1.32
WD-WC-S1 8.87 430.4 705.3 540.0 445.3 539.4 0.61 0.80 0.97 0.80
UC-M2 8.97 477.8 511.1 389.3 398.1 432.6 0.93 1.23 1.20 1.10
UC-S2-2.95 9.45 390.9 503.9 374.7 383.1 390.8 0.78 1.04 1.02 1.00
DS3
UC-S1 10.5 494.9 509.9 358.8 367.5 435.4 0.97 1.38 1.35 1.14
CS-WC-S2 12.8 507.0 705.3 434.7 409.0 452.7 0.72 1.17 1.24 1.12
CS-WC-S1 13.2 518.0 705.3 424.5 398.9 443.8 0.73 1.22 1.30 1.17
1 Mean 0.86 1.17 1.27 1.09
Mean (%) -14.45 16.55 26.86 8.60
2 Std. (%) 18.20 21.34 21.26 15.29
Note: A negative mean prediction represents a mean prediction ratio less than 1.0.
3

10

11

12

13

14

15

16

40
 a) b)
Figure 1. ASCE/ACI 426 [17] shear-ductility model [18] (a); UCSD γ model for the degradation of
concrete shear resistance with increasing displacement ductility [16] (b).
1

a)

b)

b)
Figure 2. Standard under-confined column design and well-confined design (a); Cyclic shear
test setup (b).

41
1

2
Figure 3. Corroded section model for material modifications.

Figure 4. Original inclined strut-compression zone relationship (a); change in strut formation
with the inclusion of vertical corrosion cracks (b); contribution of the axial compression load
to the shear capacity of an RC column (c) [16].
5

42
 Figure 5. Average measured shear strut angle for DS2 and DS3 test members.

Figure 6. Curvature profiles for uncorroded benchmark specimens (BM-UC and BM-WC);
typical curvature profiles of DS2 members (UC-L1 and CS-L2); and DS3 columns (UC-M2
and UC-S1).
4

43
1 Figure 7. Displacement ductility degradation with increasing corrosion damage using data
2 collected from [6, 8, 30 - 32, 62].

Figure 8. Maximum crack width and transition in failure mode. Note: SLD = shear limited
ductility; BS = brittle shear; BS-B = brittle shear-bond.

44
 1

a) b) c)

d) e)

Figure 9. Model M-0 prediction results compared with experimental results (a); M-1 (b); M-2
(c); M-3 (d); influence of crack correction factor on model M-3 predictions (e). SLD = shear
limited ductility, BS = Brittle shear, BS-B = brittle shear-bond.
2

10

45
 Figure 10. Example shear capacity-ductility envelopes for several DS0, DS2, and DS3
members based on design and corrosion properties.
1

46