# Effect of wire rupture on the mechanical behaviour of an external prestressing tendon

Jonas Aparicio, Gwendal Cumunel, Tien Hoang, Gilles Foret

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Jonas Aparicio, Gwendal Cumunel, Tien Hoang, Gilles Foret. Effect of wire rupture on the mechanical behaviour of an external prestressing tendon. Construction and Building Materials, 2025, 504, pp.144628. 10.1016/j.conbuildmat.2025.144628. hal-05385951

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# Effect of wire rupture on the mechanical behaviour of an external prestressing tendon

Jonas Aparicio<sup>1,*</sup>, Gwendal Cumunel<sup>1,†</sup>, Tien Hoang<sup>1,‡</sup>, Gilles Foret<sup>1,§</sup>

Navier, Ecole des Ponts, Univ Gustave Eiffel, CNRS, 6 et 8 avenue Blaise Pascal, 77455, Marne-la-Vallée, France

## ARTICLE INFO

**Keywords:**

Posttensionned bridge

Prestress loss

Seven-wire strand

Wire rupture

Interwire friction

## ABSTRACT

External prestressing is a construction technique widely employed since the 1980s for large civil engineering structures to reduce the cross-sectional area of structures and thus their permanent load, and more recently to reinforce structures. External prestressing tendons are composed of strands, each comprising 7 wires with a very high elastic limit, laid in a HDPE sheath, anchored in reinforced concrete blocks and generally tensioned to 80% (excluding creep and shrinkage) of their ultimate tensile strength. To safeguard the cables, grout was injected into the sheaths. However, this protection system has not always prevented the corrosion of the strands. When wires and strands re-anchor through friction in the grout after fracturing due to corrosion, the stress on the remaining tendon increases until the breaking load of the entire tendon is reached. The sudden release of energy from such a break can jeopardize the structure and endanger any personnel present on-site.

Therefore, understanding the state of the cable at a given level of degradation and the permissible degradation before failure is crucial for structure managers.

Several 46 m long cables between supports were tested and brought to failure by the progressive cutting of the wires composing the strands. Initially, three tendons, each comprising a single strand of seven wires, were tested. Subsequently, three tendons, each comprising five strands, were examined.

The experiments conducted and the analysis of the results have enhanced the understanding of the evolution of tension in the tendon as it becomes damaged. To this end, a model was developed to estimate the length affected by wire ruptures. This work also presents findings regarding the transfer of stresses within the tendon and strands resulting from these ruptures.

Finally, a model that estimates tension loss due to the sliding of wires after they break is presented.

## 1. Introduction

Prestressed concrete is widely used in long-span bridges due to its efficiency in reducing structural mass and enhancing integrity. This paper focuses on prestressing tendons composed of steel wire strands encased in a plastic sheath and subsequently grouted. Each strand consists of a straight core wire surrounded by six helical wires. These strands are typically tensioned to approximately 70% of their fracture strength, which is the usual residual tension in prestressing tendons after accounting for creep and relaxation. The strands are protected from external aggressions by a plastic sheath, commonly made of high-density polyethylene (HDPE). Historically, the sheath was injected with grout to protect the steel from corrosion. However, since the 2000s, wax or grease has been used instead. Issues during construction or operation can compromise this protection, leading to steel corrosion. The primary durability concern for prestressing tendons is the loss of wire due to corrosion [1]. Wire breaks result in stress redistribution due

to the friction between the broken wire, the grout, and the adjacent wires. When a wire is cut, it regains its original tension at a certain distance from the break. This distance is known as the re-anchoring or transfer length. The redistribution of stress can lead to catastrophic tendon failure as individual wires reach their fracture strength, triggering a cascade of breaks. In-service tendon failures typically occur after approximately 30% section loss. This is the critical point at which the remaining steel cross-section is insufficient to withstand the residual tension, leading to tendon rupture. Damaged tendons often exhibit wire breaks distributed across multiple strands.

The simplest approach to modelling damage in a tendon is to consider it on a strand-by-strand basis. In this scenario, damage is treated as the complete failure of a strand, which then re-anchors in the grout similarly to how strands anchor in internal prestressing. However, this approach is not accurate for external post-tensioning systems. In these systems, multiple strands are bonded together with

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: jonas.aparicio@enpc.fr (J. Aparicio).</footnote>
relatively little grout, and wire breaks are often distributed across multiple strands. Most existing literature focuses on damage occurring at the strand level, with some studies examining damage at the wire level. However, no studies have been found that address the specific configuration of damage observed in external prestressing systems.

The mechanism of strand re-anchoring is as follows: when a strand breaks, if there is sufficient bond between the strand and the surrounding concrete and the tendon is sufficiently long, the strand will regain its original stress at a certain distance from the break. Alternatively, a point will be reached where the stress state in the strand ceases to evolve with distance from the damage.

This distance is referred to by various terms in the literature, for strand re-anchoring: transmission [2], re-anchoring [3], recovery [4], and length of stress-decreasing region [5]; and for wire re-anchoring: affected length [6], effective stress transfer length [7], and anchorage length [8].

The Eurocode (EN 1992-1-1:2004, Section 8.10) distinguishes between transmission length and anchorage length. The former refers to the length required to transfer stress to the concrete, while the latter refers to the length at which the strand reaches its state of equilibrium. In this paper, this length will be referred to as the re-anchoring length.

Re-anchoring of a strand occurs through friction, which is enhanced by two phenomena: the Poisson effect and the Hoyer effect. The Poisson effect causes the ruptured end of the broken strand to form a wedge, thereby increasing friction with the grout. Additionally, the loss of tension in the wires causes them to unfold and increase contact with the grout, a phenomenon known as the Hoyer effect [9]. Together, these effects create friction between the wires and between the external wires and the grout, facilitating the transfer of stress back into the cut strand.

The article [3] has shown that the transfer length for a grouted strand is typically on the order of one metre, based on a review of various studies on this phenomenon. The mechanical characteristics of steel and grout are identified as the primary parameters influencing the re-anchoring length. A sensitivity analysis of the factors affecting the re-anchoring length can be found in the article [5], which follows an experimental campaign of cutting strands in different configurations. The article [2] demonstrates that in case of inefficient grouting, the transfer length increases significantly, from approximately 1 m with good grouting to 1.7 m with incomplete grouting. Several other analytical and numerical models address the re-anchoring of strands [10,11], utilizing axisymmetry and modelling damage to the grout.

The re-anchoring of an entire strand, as observed in internal prestressing, is addressed in multiple construction codes, including the American Concrete Institute (2011), the Eurocode EN 1992-1-1 (2004) (§8.10.2), and the British code BS 855 (§8.3.1). According to these codes, the transfer length of a cut strand primarily depends on the grout strength, the strand diameter, and the surface condition of the strand. This length is typically on the order of one metre.

However, detected damage in tendons reveals that deterioration occurs on a wire-by-wire basis, with damaged strands often breaking only after multiple wires have already ruptured. The codes rely on assumptions about grout cover that are rarely applicable to external prestressing.

Deriving analytical formulations becomes more challenging when investigating the effect of a wire break within a strand, as the symmetry of the strand is disrupted. Consequently, models used for intact strands, such as those presented in [12,13], are no longer valid. These models rely on the symmetrical arrangement of helical wires around the central wire to express equilibrium in the cross-section. As a result, there are fewer studies addressing the breaking of individual wires. One of the most notable contributions in this area is the work of MacDougall and Bartlett, although they have not studied grouted strands. In [14], they investigated the re-anchoring of one or two wires within a strand. Their experiments involved cutting individual wires in a strand with concrete deviations, leading to transfer lengths exceeding 10 m. The transfer of tension was found to be partial, as the tension in the strand decreased by 2%–4% with each cut. In a 7-wire strand, each wire accounts for

<figure><img src="image_2.png" /><figcaption>(a) Strain distribution in a strand with a broken wire extracted from [14]</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Fig. 21. Multiple wire breaks.</figcaption></figure>

(b) Numerical model of a strand with wires break extracted from [4]

Fig. 1. State of the art of wire re-anchoring modelization.

approximately 14% of the total tension. The study demonstrated that stress redistribution in the wires is symmetric relative to the cut wire but not homogeneous. The tension in each wire after the cut depends on its relative position to the cut wire (Fig. 1(a)). In subsequent works, MacDougall and Bartlett analytically and numerically established how stress evolves in the wires of a strand with one wire broken [6,15].

Beltran et al. [16,17] employ analytical and Finite Element (FE) computations to derive the static response of a strand with a broken wire, assuming the strand is unbonded. Both articles posit that damage to a broken wire induces a helical displacement of the strand to maintain equilibrium. A study involving the confinement of the strand was conducted by Abdullah et al. [4] (Fig. 1(b)), where unstressed strands surround the stressed strand containing cut wires. The tension loss in the tendon was found to be only 10% less with confinement compared to without. There was minimal stress transfer from the cut wire to the others, and the re-anchoring phenomenon was not reproduced due to the inefficiency of the confinement, as the transfer length was measured to be 30 m between the cut and the end of the bench. This experiment was also reproduced numerically [18], where the wires are confined but without bonding between the wire and the confinement.

The study that most closely replicates re-anchoring behaviour in grouted tendons is the experimental campaign conducted by Lee and Kang [19], in which four grouted tendons (10 m long, 6-strand configuration) were subjected to controlled corrosion-induced damage. Tension measurements were obtained using load cells, and the experiments successfully demonstrated effective stress transfer between individual wires, although tension losses in the laboratory setting remained higher than those typically observed in field conditions. Since the primary focus of this study was the evolution of frequency response
<figure><img src="image_4.png" /><figcaption>Fig. 2. Sketch of the experimental set-up for a tendon with one strand.</figcaption></figure>

characteristics rather than re-anchoring mechanisms, the analysis of damaged tendon behaviour was limited to its impact on dynamic response properties.

The literature review has shown that while the behaviour of the healthy strand is well known, the behaviour of a damaged strand is only partially understood. To the best of the authors' knowledge, and there is no study on the behaviour of damaged strands that manage to reproduce the near complete stress transfer between wire that is found in grouted external prestressing tendon.

Understanding this behaviour will help to better the assessment of the remaining capability of a detected damaged tendon. Furthermore, the improved knowledge on a damaged tendon could be used to improve control and detection of damage in the tendon.

In this paper, an experiment is presented in which realistically sized tendons were built then broken by cutting their wires one at a time until the tendon failure. The aim of this experiment is to better the understanding of stress transfer inside the damage tendon and to have an estimation of the length affected by the damage. The experiments have been repeated using variable number of strand, brittleness of sheath and cutting schedule in order to perform a sensitivity analysis on these different parameters. Mechanical results have been obtained from a load cell, photographs and films.

## 2. Methods and experiments

The goal of the experiment was to understand the impact of the bond with the grout, the presence of multiple strands, and the distribution of wire break locations on the evolution of tension and residual durability of external prestressing tendons.

To achieve this, a series of tests was financed by ASFA (Association des Sociétés Françaises d'Autoroutes). These tests were carried out as part of a collaboration between the group SANEF, École nationale des ponts et chaussées (ENPC), and Gustave Eiffel University. The construction of the cables and the wire-cutting phase were performed by Freyssinet, which also financed a test with a similar protocol but using a geopolymer-injected cable. The results of this test are also presented. In total, six tendons were tested: three with one strand (1T15) and three with five strands (5T15). For the first two 1-strand tests, the bond between the wire and the grout was lost over a 5 m length during the experiment. Additionally, different cutting orders were used for the tendons.

The tests were conducted on the 200 m test bench in the MAST-SMC laboratory at Gustave Eiffel University in Nantes [20]. These tests involved producing instrumented prestressing cables approximately 50 m long, which were then placed under tension. 10 cm-wide trap doors were created in the cables to allow the strand wires to be cut by removing the sheath and grout locally in the centre of the cable. These trap doors enabled the wires to be cut one by one until the cable failed.

The cables were instrumented with a load cell (Freyssicel500 [capacity 500 kN] for 1-strand cables and Freyssicel1700 [capacity 1700 kN] for 5-strand cables, the combined error is 0.3% of the capacity), accelerometers, one or more mechanical excitation systems,

and strain gauges glued to the steel wires of the strands and to the sheath. The instrumentation is summarized in Fig. 2. Additionally, Freyssinet installed a high-speed camera, the SMC laboratory installed acoustic listening systems on the cables, and IXO France installed an optical fibre on one of the cables. The procedure for conducting the tests evolved throughout the experimental campaign to incorporate feedback. A summary of the changes is provided at the end of this section.

This article focuses on the results obtained using a high-frequency camera, the load cell, and photographs of the central trap. Other sensors were used for monitoring or damage detection and will not be presented here.

The cable was constructed by threading the strand through the passive anchorage and then through HDPE sheath segments with a diameter of 63 mm and a thickness of 5.8 mm, which were then thermally welded. For the last 1-strand test and subsequent tests, electro-weldable sleeves were used to connect the ducts. For the first two tests, the HDPE sheath was replaced with a transparent PVC sheath of the same diameter in the central part of the duct to observe the cracking of the grout. The connection with the transparent PVC sheath of the first 1-strand was made with screw fittings.

A vent was also fitted to the sheath to improve grouting quality. The cable anchoring systems were then installed at both headers, with a load cell on the passive header side (see Fig. 3). A jack was used to tension the cable at the active anchorage system to a tension equal to $0.72F_m$ ($F_m$ being the theoretical maximum force supported by the cable), i.e. a jack tension of 200.9 kN for 1-strand cables (1T15) and 1004.4 kN for 5T15 cables (composed of 5 strands, see Table 1). The tension read on the load cell on the day of the test, before cutting, was between 186 kN and 204 kN for the 1-strand cables and between 988 kN and 1024 kN for the 5T15 cables. Finally, the cables were injected with Holcim Superstresem cement grout, the only grout currently authorized for construction in France, except for the last cable (5T15-3), which was injected with a geopolymer cement grout. To facilitate the identification of the wires to be cut, each perimeter wire in the 1T15 cables (except the second) or each strand in the 5T15 cables was painted with a different colour in the breaking and anchoring zones.

Great precautions were taken to protect the operators, particularly the operator responsible for cutting the wires, by safeguarding against the whiplash effect of the tendon during its rupture.

Data acquisition from the measuring instruments (accelerometers, load cell, and strain gauges) was carried out using NI acquisition cards connected to a computer running a program based on LabVIEW 2020 for displaying and recording the data. The strain gauge wires were soldered to a terminal block made in the laboratory and then connected to NI 9235 cards installed on a NI cDAQ 9174 acquisition box, which was linked via Ethernet to the measurement computer. The accelerometers were connected via coaxial cables to an NI 9234 card installed on a NI cDAQ 9174, which also housed an NI 9219 card connected to the load cell and an NI 9263 board controlling the fall of a ball used to excite the cable. The excitation and frequency response acquisition and results for the 1-strand cables can be found in the
<figure><img src="image_5.png" /><figcaption>Fig. 3. Photos of the experimental bench at various step of the experiment set up of the 1T15-1.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 4. Photos of cutting.</figcaption></figure>

following proceedings [21]. This acquisition box was positioned close to the accelerometers and connected to the measurement computer by a 30 m long USB cable.

The wires were always cut at the same point, in the central part of the cable (Fig. 4), 24 m from the end of the bench where the passive anchorage system had been installed, one week after injection (6 or 7 days, except for the 5T15-3 cable). To cut the wires, a trap door was created by cutting the sheath over a length of 20 cm for the 1T15-1 cable, then a length 10 cm for the other tendons, and the strand was freed by removing the grout using a chisel. The wires were then cut one by one using a Dremel rotary saw (Fig. 4). Care was exercised to ensure that only one wire was damaged during each cutting operation, with

wire rupture occurring before the rotary tool had completely severed the cross-section. As cable damage accumulated, the time to rupture progressively decreased, reflecting the increased average tension carried by each remaining intact wire. For the 5T15 cables, a larger sheath with a diameter of 75 mm and a thickness of 5.6 mm was installed in the central part of the cable over a length of 36 m to accommodate the installation of gauges on the strand wires.

Six cables were tested: three 1T15 cables (1T15-1, 1T15-2, and 1T15-3) and three 5T15 cables (5T15-1, 5T15-2, and 5T15-3), each subjected to different wire cutting sequences. The tests on the three 1T15 cables and the first two 5T15 cables were conducted as part of the ASFA-funded program, while the third 5T15 cable test was
<table><caption>**Table 1**
List of experiments.</caption><thead><tr><th>Strand</th><th>Sheath</th><th>Initial tension (kN)</th><th>Theoretical/Experimental wire loss needed for tendon failure</th><th>Others</th></tr></thead><tbody><tr><td>1T15-1</td><td>PVC</td><td>186</td><td>3/6 (43%/86%)</td><td>Brittle sheath</td></tr><tr><td>1T15-2</td><td>PVC</td><td>204</td><td>3/7 (43%/100%)</td><td>Brittle sheath</td></tr><tr><td>1T15-3</td><td>HDPE</td><td>204</td><td>3/4 (43%/57%)</td><td></td></tr><tr><td>5T15-1</td><td>HDPE</td><td>1018</td><td>10/16 (29%/46%)</td><td>Sequential cutting</td></tr><tr><td>5T15-2</td><td>HDPE</td><td>988</td><td>12/15 (34%/43%)</td><td>Alternate cutting on 3 strands</td></tr><tr><td>5T15-3<sup>a</sup></td><td>HDPE</td><td>1024</td><td>10/16 (29%/46%)<sup>b</sup></td><td>Alternate cutting on 4 strands</td></tr></tbody></table>

<figure><img src="image_7.png" /><figcaption>**Fig. 6.** Cutting and rupture schedule of 5T15-1 (Sequential Wire cuts), secondary ruptures causes by the cut 13 are denoted (13), count restart at 15.</figcaption></figure>

performed in collaboration with Freyssinet. For this final test (5T15-3), the standard Superstresemes grout was replaced with geopolymer cement grout to evaluate its performance in damaged cable conditions. Test specimens were fabricated on the day of injection and tested until a satisfactory compressive strength was achieved, ranging between 50 and 60 MPa, thereby ensuring comparable mechanical properties of the grout or geopolymer across all tests. Different cutting strategies were employed across both cable types to investigate their influence on re-anchoring behaviour. For the 1T15 tendon, the different cutting schedules arose from the necessity to maintain unobstructed access to individual wires, ensuring that only one wire was damaged during each cutting operation (Fig. 5). For the 5T15 cable tests, the cutting strategies were deliberately varied to assess whether distributed or localized damage patterns are more detrimental to tendon durability. During the first test (5T15-1), wires were cut sequentially within a single strand until complete strand failure occurred, after which cutting proceeded to the next strand. This approach is termed the sequential cutting method (Fig. 6). In contrast, the subsequent two tests employed distributed cutting patterns across multiple strands: three strands for the second test (5T15-2) and four strands for the third test (5T15-3) (Fig. 7 and Fig. 8, respectively). Due to the effects of tensioning and grouting processes, the precise relative positioning of individual wires between different strands could not be determined. At one point during the final experiment (5T15-3), four of the five strands ruptured. This point is considered as the tendon failure, as the cable had lost its ability to support its operating tension within the structure, even though complete tendon rupture required five additional wire cuts.

**Evolution of the cable construction protocol.** The cable construction protocol evolved during the tests:

* The first two single-strand cables tested were fitted with a 6 m long transparent PVC sheath in the central part of the cable to observe the cracking front in the grout. These sheaths broke suddenly during the second wire break due to the brittle nature of the rigid PVC used. Cracks were initiated at the first rupture, they propagated at each rupture due to the chock waves. Thus breaking the confinement and allowing the grout to be in the traction domain, which cause the grout to crack. An HDPE sheath was chosen for the entire length of the cable starting from 1T15-3.

* Injection quality also improved. The first cable (1T15-1) had injection defects, including a gully about 1 cm deep at the top of the cable. The second cable was better injected but had a strand centring problem that was detrimental to the re-anchoring phenomenon.

* To address the centring problem, starting with the 1T15-3 cable, the cables were supported almost entirely during the grouting period to ensure that the strands were properly centred in the sheath. These supports were made using straps installed on the bench and the creation of temporary supports. The supports were removed before cutting to avoid altering the cable's vibratory response. The strands of the 5T15 cables were positioned in the HDPE sheath using supports made via 3D printing at the Navier Laboratory and placed on either side of the supposed re-anchoring

<footnote>a Experiment funded by Fressynet using a geopolymer grout.</footnote>

<footnote>b One strand survives the failure of the tendon.</footnote>
<figure><img src="image_8.png" /><figcaption>**Fig. 7.** Cutting schedule of 5T15-2 (Distributed Wire Cuts)</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>**Fig. 8.** Cutting and breaking schedule of 5T15-3 (Distributed Wire Cuts), secondary ruptures caused by the cut 16 are denoted (16), breaking count restart at 29.</figcaption></figure>

zone, i.e. after the gauges furthest from the cutting zone. This was
done to improve the centring of the strands and ensure that the
strands had a known position.

* The electric wires of the gauges glued to the wires of the strands were gradually changed to provide stronger wires. The gauges

<figure><img src="image_10.png" /><figcaption>**Fig. 9.** Tension in the 1-strand according to the number of wire cut.</figcaption></figure>

were also put farther to the cut zone that lead to more exploitable
gauges result for the 5T15-3.

* Lastly, the sheath was replaced by a larger diameter sheath (75 mm x 5.6 mm) in the central part for the 5T15 experiments to accommodate the gauges.

3. Results

The main results of the experiments are the number of cut needed
for the breaking of strands and tendons depending on their position, the
evolution of the tension of the tendon and the evolution of the sliding
of the wire after cutting.

3.1. *Evolution of tension in the tendon*

The tension in the tendon decreases over the course of the exper-
iment, indicating that stress transfer from the cut wire to the tendon
is not complete, as some tension is lost. Due to this loss of tension,
the number of wires that need to be cut until the tendon ruptures will
be greater than the number required for the tendon to reach its tensile
limit, assuming that the tension in intact wires is homogeneous and the
tension in broken wires is null due to equilibrium (Table 1). The tensile
limit of the steel is 1.96 GPa, according to the certificate provided by the
strand manufacturer.

Most of the stress in a wire is transferred when that wire is cut,
as the tension loss is only a few kilonewtons, while each wire initially
holds a tension of the order of 25 to 30 kN. However, when the grout
is severely damaged (see Fig. 4(a)) or when an entire strand is broken
(see the square in Fig. 10a 10 ), the tension loss can be much higher,
exceeding 30 kN.

3.2. *Evolution of the sliding of the wires*

Sliding is defined as the distance between two cut ends of a wire
after its break and is denoted $g_i$.

Fig. 12 illustrates different degrees of damage observed both on-site and during the experiment. It demonstrates that the sliding of the wires is of the same magnitude in both the experiment and on-site. The extent of sliding depends on the state of the tendon, with greater sliding occurring when entire strands are cut. The sliding of each wire continues to increase with each subsequent cut, as the shock and new damage to the grout lead to another redistribution of stress.
<figure><img src="image_11.png" /><figcaption>**Fig. 10.** Tension in the 5T15 according to the number of wires broken.</figcaption></figure>

<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /><figcaption>**Fig. 12.** On the left photos of damage on opened tendon on a bridge ©IFST-TAR for Sanef, on the right photos of broken wire during the experiments.</figcaption></figure>

Sliding on the wires was estimated after the experiments using videos. The length between two matching points on a broken wire was measured in pixels and then converted to centimetres using the length of the trapdoor or graduations on the tendon as a reference. When using the camera, the pixel size of the reference length was measured before any cuts occurred.

This method of measurement is subject to uncertainty for several reasons. Sliding is a three-dimensional phenomenon, yet the photographs are two-dimensional, leading to projection errors. Additionally, there are inaccuracies due to shifts in the size of the trapdoor used as a reference.

When the sliding of a wire cannot be measured because one end is not visible in the photograph, the average sliding measured on the strand is used as the value. There are no results for the tendons 5T15-2 and 5T15-3 because no photographs and film taken from one direction capture all the damaged wire due to the cutting schedule causing damage on multiple side of the tendon.

All these inaccuracies on the value of sliding led to important uncertainty. They are represented on **Fig. 19** by an arbitrary error bar, taken as the maximum between ±10% of the sliding or ±1 cm for each wire on the measured value. While doing two independent measurement of the sliding using two different sources when available, the standard deviation was between 1 cm and 2 cm, the repeatability being better on 1T15-3 and 5T15-1 due to the broken wire not unwinding.

### 3.3. Result of the strain gauges

Kyowa KFGS-1N-120-C1-11 has been used on the sheath and Kyowa KFP-2-120-C1-65 for the wire. The strain gauges glued to the sheath showed significant strain caused by the wire cutting, reaching almost 1 % when the tendon was close to breaking. The furthest gauges, placed
<figure><img src="image_14.png" /><figcaption>Fig. 13. Strain on the sheath for the tendon 1T15-3.</figcaption></figure>

more than 5 m from the cutting zone in the 5T15-3 experiment, registered changes in strain after about a dozen wires were cut. The strain in the sheath is complex to model, involving a superposition of twist, bending, compression, and local helical strain due to re-anchoring. The authors did not manage to disambiguate all those mechanicals phenomenon so gauges data are provided only as indirect measurement of transfer length.

On the other hand, the strain gauges on the wires were too few to survive the grouting and the first wire cut, making them unsuitable for deriving an experimental model. The strain in the wire is influenced by the distance to the cut, the relative angular position of the wire to the cut wire (by extending MacDougall and Bartlett's equilibrium in the strand to every strand), the position of the strand in the tendon, and the position of the wire in the strand, which periodically shifts due to stranding. The authors lack a model to disambiguate between the impact of bending and twist in the tendon to the strand, the impact of localized grout damage and the impact of re-anchoring on the strain. Only the data for the three first rupture of 5T15-3 are provided for an indirect observation of transfer length as this are the only useable data which can be interpreted easily.

Strain gauges were positioned on the upper surface of the sheath (circles) or on the lower surface of the sheath (only for 1T15-3, stars) in the longitudinal direction at locations whose distance from the cutting zone corresponded to integer multiples of the lay pitch (one pitch equals 24.4 cm) in order to minimize the influence of strand geometry on the measurements.

The gauges on the 1T15-3 sheath (Fig. 13) show that the strain increases with the number of cut wires, that the strain is greater at the upper surface of the sheath — which may be attributed to the fact that wires were cut at the top of the tendon for accessibility reasons — and that the strains measured at the upper and lower surfaces of the sheath converge.

The results for the 5T15-1 sheath are more complex (Fig. 14). Wire ruptures induce significant strain variations; however, depending on the location of the cut wire, the strain measured by the gauges may either increase or decrease. Due to the complexity of the phenomena involved, strain does not necessarily vary monotonically with distance from the cutting zone. The measured strains are lower than for the

<figure><img src="image_15.png" /><figcaption>Fig. 14. Strain on the sheath for the tendon 5T15-1.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Fig. 15. Arrangement of strain gauges on wires of cable 5T15-3.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>Fig. 16. Strain profiles in the strands of cable 5T15-3 from the 1st to the 3rd cut.</figcaption></figure>

previous configuration, as the additional strand increases the overall
tendon stiffness.

Strain gauges were bonded on one wire of strand 1 (yellow) and
one wire of strand 5 (black) of cable 5T15-3 according to the arrange-
ment presented in Fig. 15. All gauge measurements were shifted to
correspond to the theoretical initial strain (0.067%) at the start of the
experiment.

During the first three cuts, the strain values obtained on the two
instrumented wires (out of thirty-five) are consistent with the hypoth-
esis of a homogeneous redistribution of forces from the cut wires to
the intact wires (Fig. 16). Indeed, by performing a linear regression on
the strain values obtained by the gauges, we obtain for the first cut a tensile strain variation in the intact wires ranging from $1.8 \times 10^{-4}$ to $2 \times 10^{-4}$ at the cutting point, for the second cut from $3.7 \times 10^{-4}$ to $4.5 \times 10^{-4}$, and for the third cut from $5.3 \times 10^{-4}$ to $6.3 \times 10^{-4}$. A strain value of $2 \times 10^{-4}$ in the 34 intact wires yields a force transfer of 34 kN (using the formula for the tension variation due to cuts $\Delta T = ES\Delta\epsilon$, where E is the Young's modulus of the strand, S is the cross-sectional area of the steel wires, and $\epsilon$ is the strain). This value is of the same order of magnitude as the initial tension in each wire (29 kN). The strain variation then decreases progressively until reaching zero. By taking the point where the linear regression intersects the abscissa axis, we obtain a re-anchorage length for these three failures of 11, 14, and 15 lay pitches respectively, corresponding to values between 2.2 and 3.2 m. All Pearson correlation coefficient of the linear regression (R-value) range from -0.95 to -0.86.

# 4. Models and discussion

The models presented at the end of this section rely on strong simplifying assumptions that may lead to some inaccuracies but are necessary to develop tractable models that require minimal measurements of material characteristics to provide information on the tendon state.

## 4.1. Repeatability

Repeatability represents a major challenge in this experimental campaign, as the experimental protocol was modified for each tendon. Furthermore, exact replication of the same 46 m tendon is nearly impossible. The strands would not follow identical trajectories from one tendon to another, despite our best efforts to ensure trajectory consistency. Most of the variation between tendons of the same category (Table 1) derives from geometric variations in the tendon and damage configuration, with additional variation arising from the different filler materials used in the sheath of 5T15-3, although care was taken to mitigate this variation as well. Despite these variations, common properties emerge that are shared within tendon categories, such as very similar evolutions of tension and wire sliding. Notably, tendon failure occurring at 16, 15, and 16 wire losses for the 5T15 specimens, comparable tension evolutions, and sliding of similar magnitude depending on the damage pattern are strong indicators of common behaviour despite different cutting schedules and sheath fillers. However, this paper does not provide sufficient data to quantify the evolution of mechanical parameters with associated uncertainties. Its objective is rather to elucidate the underlying mechanisms and describe different phases and categories of damage along with their mechanical implications.

## 4.2. Sensitivity analysis on the quality of stress transfer according to the confinement and cutting schedule

When a wire is cut, some of the tension it was carrying is lost, and the remainder is transferred to other wires in the tendon. In these experiments, the transfer of stress to the sheath and grout can be ignored because they were removed to perform the cutting of the wires. As the continuity of the grout and sheath was ended to free access to the strand, the tension from the cut wire cannot be transferred to the grout and sheath, thus the stress is redistributed to the other wires. This simplification is unlikely to introduce significant bias because the tensile strength of the grout and the stiffness of the sheath are very low compared to those of the steel. The efficiency of stress transfer appears to be directly correlated with the bond between the wires and the grout. As the confinement of the steel decreases due to damage to the grout, the stress transfer decreases until it becomes almost non-existent (Fig. 9). Sheath failure, which occurred with the brittle PVC sheath in the first two experiments, causes the grout to rupture and the strand and grout to split over several metres (Fig. 4(a)). Generally, grout degradation due to multiple ruptures makes stress transfer less and less efficient as damage accumulates in the tendon (Fig. 9 and Fig. 10(a)).

**1-Strand.** The experiments revealed two distinct behaviours for 1-strand cables, depending on the type of sheath used (Fig. 5). When a PVC sheath was used for the first two strands, the wire breaks caused the sheath to rupture, followed by large cracks in the grout. This resulted in a loss of bond between the grout and the tendon over an increasing length with each cut. When the bond between the strand and the grout was broken over multiple metres, which occurred at the 4th cut for 1T15-1 and at the 2nd cut for 1T15-2, each subsequent cut caused the tension in the tendon to drop by a value only slightly less than the initial tension of the cut wire. This indicates that the tension transfer during wire cutting was minimal, with most of the tension lost (Fig. 9).

Conversely, when an HDPE sheath was used, only small cracks were observed in the grout, indicating that the bond between the grout and the strand remained intact. The tension loss was only a small fraction of the tension in the cut wire, with most of the tension being redistributed to the other wires. The relationship between bond strength and confinement is consistent with the slip-pipe test conducted by [22], which demonstrates that the materials used for the sheath (steel, PVC, aluminium, heat shrink) significantly impact the bond strength of cable bolts. Pull-out test results should be interpreted with caution, as the Hoyer and Poisson effects can cause adherence to decrease in pull-out tests and increase in re-anchoring scenarios. The smaller tension loss associated with damage causes the well-confined 1-strand tendon to require fewer cuts to rupture than the weakly confined 1-strand tendon, as the more gradual decrease in tension allows the tendon to reach the remaining resistance of the strand sooner—approximately equal to the sum of the resistances of its remaining intact wires.

**Tendons with multiple strands.** Three experiments were conducted on tendons with 5 strands each, using three different cutting schedules.

In the first experiment, wires were cut sequentially in the same strand until the strand broke before moving on to another strand (Fig. 6). The second strand broke on the sixth cut on this strand (the thirteenth cut overall), and the entire tendon broke on the second cut in the third strand. In the second experiment, wires are cut on three different strands. The original plan was to cut three wires on each strand, then two, and finally one. However, accessibility constraints necessitated an adjustment to this plan (shown on Fig. 7). In the last 5T15 experiment (Fig. 8), three wires were cut on each of the four outer strands, followed by three cuts on one of the damaged strands, causing the four damaged strands to break. The central strand survived the tendon failure, an event unlikely to occur on-site, as it is highly improbable for all but one strand to be extensively damaged.

There was minimal tension loss at each cut until the very end, although the tension loss from individual cuts tended to increase as the damage to the tendon accumulated and when a strand finally broke. There is strong evidence suggesting that the stress from the cut wire is redistributed not only within the strand to which it belongs but also among the other strands. Sequential cutting resulted in strand failure after six and seven wire cuts, failure would have occurred earlier if the stress had been redistributed only within the strand being cut. However, the redistribution of tension was not homogeneous among the strands. High-speed filming of the rupture of tendons subjected to alternating cutting patterns showed that the outer four strands failed before the central one, even when only three of the outer strands contained cut wires in specimen 5T15-2. This can be explained by the significant bending and torsion generated by the requirement for static equilibrium to accommodate the loss of tension in the broken wires, which induces greater stress in the outer strands than in the central strand.

In the three 5T15 experiments, the evolution of tension loss was very similar (Fig. 11) even through the cutting schedule was different with a good redistribution of tension for the first ten ruptures with a tension loss by wire of $5.4 \pm 3.7$ kN. As the rupture caused damage in the grout, the bond with the strand decrease and the tension loss start
<table><caption>**Table 2**
Impact of grout bond and geometry of damage.</caption><thead><tr><td>Tendon</td><td>Grout bond</td><td>Tension loss</td><td>Strand breaking</td><td>Experiments</td></tr></thead><tbody><tr><td>1-strand</td><td>Weak</td><td>Important</td><td>6-7 wires cut</td><td>1T15-1, 1T15-2</td></tr><tr><td>1-strand</td><td>Strong</td><td>Small</td><td>4 wires cut</td><td>1T15-3</td></tr><tr><td>5 strands (seq)</td><td>Strong</td><td>Small</td><td>6-7 wires cut</td><td>5T16-1</td></tr><tr><td>5 strands (alt)</td><td>Strong</td><td>Small</td><td>Not until the tendon failure</td><td>5T15-2, 5T15-3</td></tr></tbody></table>

increasing monotonously — only one outlier when the second strand breaks for 5T15-1- at each new rupture (Fig. 11) but remain less than the actual average tension supported by every remaining healthy wire until the rupture of the tendon. Doing a linear regression from the 10th wire broken onward yield a slope of 5.4±1.0 kN with an R-Value of 0.87. This statistical analysis shows that the tension evolution is similar for the three 5T15 tendon. It should be noted that the amplitude of tension loss and the number of rupture needed for a regime of degrading bond will probably change for tendon with a different number of strands. The influence of the trap door on tension loss behaviour is also unknown. Its absence would likely result in reduced sliding and tension loss.

This experiment has led to the creation of the following four categories to describe the behaviour of prestressing tendons under damage (summarized in Table 2):

1. **1-Strand tendons with Weak Confinement (1T15-1 and 1T15-2):** The weak confinement leads to the failure of the bond. Most of the stress in the broken wires is lost rather than transferred. Approximately six to seven ruptures are needed to break the strand, as the tension decreases with each rupture by an amount close to the tension in the broken wire.

2. **1-Strand tendons with Strong Confinement (1T15-3):** The maintenance of the bond leads to good stress transfer with minimal but still significant tension loss. The number of ruptures needed to break this tendon was four instead of three if no stress were lost.

3. **Multiple Strands tendons with Sequential Wire Cuts (5T15-1):** There is little loss of tension. A strand requires the rupture of six to seven wires to break, indicating that a significant share of the stress is transferred to the other strands.

4. **Multiple Strands cables with Distributed Wire Cuts (5T15-2 and 5T15-3):** There is little loss of tension. The strand can hold with only two wires remaining, showing that the tension of the broken wires is transferred to the other strands. The kinematics of the break, with the outer wires breaking before the central wire, shows that the tension distribution is not homogeneous among the strands. The outer strands, even if intact, are more impacted.

The authors argue that depending on the bond quality, the strand will range from perfect stress transfer to no stress transfer. The tension of the broken wire is lost if the strand is totally unbonded. The bond depends on grouting quality and sheath quality, as the breaking of the sheath will cause the grout to break regardless of its quality due to the poor traction resistance of the grout. No tendon with multiple strands and poor bond was tested, but it would probably lead to almost no stress redistribution, and each individual strand would follow the behaviour of the first category of 1-Strand tendon with Weak Confinement.

This framework can be applied to the existing studies that have tried to reproduce re-anchoring in grouted tendon. The experiment by Abdullah et al. [4] present a strand bonded with 6 surrounding strands by the use of a tie wire. The tension loss at each wire rupture is 5 to 10% less than the tension loss for a unbonded strand. This behaviour is closely related to the first category that were defined above of 1-Strand tendon with Weak Confinement. The experiment by Lee and Kang [19], present four 6T15 grouted tendon. About a third of the tension of each ruptured wire re-anchor into the tendon which is less

than the above experiments where more than two third of the tension is consistently re-anchored at each rupture except the last few one for the 5-strands. This difference in bond efficiency between their and our experiment is probably not due to the different number of strand but to the difference of the size of the windows where the strands were unbonded in order to be able to damage them. Lee and Kang had realized 30 cm-long trap doors in order to damage the tendon whereas our trap doors are 10 cm long. 30 cm is longer than the pitch of the strand, usually around 25 cm, which may has a significant impact on re-anchoring. Furthermore, their tendons are 10 m-long whereas ours are 50 m-long. Two more parameters that could impact the bond is the difference of length of the tendon, due to the greater length our strands are tangled together before tension and the fact that Lee and Kang use forced corrosion to cause rupture while we use cut. These two reasons imply that tension loss on in-service tendon will probably be even lower than on these experiments due to the absence of the window and the tangling of the strands which may explain why the tested tendon required a bit more rupture to break than what is observed during post-mortem of in-service tendon.

The experiments have shown that there are discrepancies on the exact tension loss at each wire cut, which translates into uncertainty about the number of wire cuts that will cause the tendon to break. This uncertainty arises because the transfer depends on strong dynamic effects, the cracking of the grout, the exact position of the wire, the relative position of every other wire, and, in a tendon where the strands are threaded one by one, the dispersion of tension between the strands. This will lead to a small discrepancy in results, even though the experiments show that the first two 1-strand tendons and the three tendons with 5T15 have similar behaviour, despite the 5T15 tendons being subjected to three different cutting schedules (Figs. 9 and 10(a)) and the 5T15-3 tendon having a different grout.

### 4.3. Discussion on the impact of corrosion

The experimental campaign does not account for the impact of corrosion. In the experiments, wires exist in only two states: intact or broken. In damaged in-service tendons, wires can exhibit a wide array of corrosion-induced damage states. The coupling between corrosion and high stress in the strand causes corrosion to develop faster than in steel under lower loading conditions and to have a significant impact on the remaining durability.

The primary impact of corrosion on mechanical properties is a dramatic reduction of the ductile phase [23], meaning that ultimate strain decreases depending on the level of corrosion, until the failure mode transitions from ductile to brittle fracture at 17% corrosion loss [24].

In the presence of high corrosion levels in the tendon, the number of ruptures preceding the final cascade of ruptures leading to tendon failure will be lower. In our experiments, intact wires enter the ductile phase starting from approximately the 10th rupture and fail at 15 to 16 ruptures. Surrounding corroded wires would have ruptured earlier than the intact wires, and thus the final cascade of ruptures would have occurred with fewer wires broken.

Conversely, the presence of corroded wires would likely have only a minor impact on tendon behaviour when the damage level is low, as the wires remain firmly within their elastic regime.

It should also be noted that corrosion may additionally impact prestressing loss in the tendon, as Xiao et al. found that internally
<figure><img src="image_18.png" /><figcaption>(a) Linear evolution of strain and minimal distance</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>(b) Exponential evolution of strain and minimal distance</figcaption></figure>

<caption>**Fig. 17.** Strain in the broken wire according to the distance to the rupture location.</caption>

prestressed concrete blocks exhibited prestress loss when subjected to corrosion [25], which would be compounded by the prestressing loss from wire ruptures.

## 4.4. Estimation of transfer length using sliding of the wire

The transfer length is defined as the distance from the rupture location where the impact of the rupture has vanished. This section introduces a model that estimates the transfer length based on the measurement of the sliding of the wire that occurs after the cut but can be more generally applied to wire rupture due to other causes like corrosion, assuming the corrosion has not dissolved the wire to a point were the sliding can no longer be measured. This model employs a purely kinematic approach and assumes that strain variation is both regular and monotonic. The transfer length is derived from an assumed strain expression using compatibility conditions with the observed sliding behaviour. This model assumes that rupture and subsequent equilibrium do not induce plastic strain in the wires, that the strain returns to or approaches its initial value far from the damage zone, and that out-of-plane displacement of the outer wires in the new static equilibrium has no impact on strain. All of these assumptions are substantial and likely to introduce inaccuracies in the model results. Despite these limitations, this model represents, to the authors' knowledge, the first attempt to account for how different levels of damage influence the transfer length.

Let $g_i$ be the sliding of a broken wire $i$, i.e., the distance between the two ends of the wire created by the rupture, and $x_c$ be the distance to the rupture location. The strain is assumed to be symmetric with respect to the rupture location which is a reasonable assumption if the rupture is located far enough from the two anchorage of the tendon. The sliding on each side is then $g_i/2$. Different expressions can be selected for the strain distribution. The strain in each helical wire is $\cos^2 \alpha \epsilon$ [26], where $\epsilon$ is the strain along the axis of the strand, initially at $\epsilon_0 = \Delta L/L$, with $L$ being the length of the strand and $\Delta L$ the jacking distance. A length $l$ on the strand translates into a length $l/\cos^2 \alpha$ on the helical wire due to the wire curvature, where $\alpha$ is the lay angle of the helix (7.8° for the experiment). This model does not consider strand rotation. Assuming sliding symmetry with respect to the rupture location and letting $L_h$ be the length from the cut to the anchorage in a chosen direction, the kinematic compatibility can be written as

$$ \frac{g_i}{2} = \int_{0}^{L_h} \delta\epsilon(x_c)dx_c = \int_{0}^{L_h} \epsilon_0 - \epsilon_i(x_c)dx_c \quad (1) $$

where $\delta\epsilon(x_c)$ is the variation of the strain, projected on the axis of the strand, due to the rupture of wire $i$. On the $x_c$ axis, 0 represents the position of the rupture location and $L_h$ is the axial distance between

the rupture location and the position of the anchorage in the chosen direction. Let $\epsilon_0$ be the generalized strain in the strand far from the rupture location where the strain has converged. This is also the original strain of the strand computed as the ratio of the strand displacement to the strand length. It is assumed that $\epsilon_0$ is constant, which is valid only when tension loss is very small compared to the total tendon tension and when wire tension is homogeneous far from the rupture location. Both assumptions become problematic as the number of broken wires increases. The smallest transfer length that accounts for the sliding is (dashed curve in Fig. 17),

$$ L_0 = \frac{g}{2\epsilon_0} \quad (2) $$

assuming that strain is monotonic, non-negative, and of value $\epsilon_0$ far from the rupture location. This is achieved with

$$ \epsilon(x_c) = \begin{cases} 0 & \text{if } x_c < L_0 \\ \epsilon_0 & \text{if } x_c \ge L_0 \end{cases} \quad (3) $$

Assuming a linear evolution of the strain from 0 to $\epsilon_0$, the transfer length is $2L_0$ (left solid curve in Fig. 17)

$$ \epsilon(x_c) = \begin{cases} \frac{x_c}{2L_0} & \text{if } x_c < 2L_0 \\ \epsilon_0 & \text{if } x_c \ge 2L_0 \end{cases} \quad (4) $$

The assumption of an exponential evolution of the strain would result from a simple model assuming exponential friction between the broken wire and the central wire. This is achieved by introducing a Coulomb friction law in the model, where the friction force between the wires depends on the contact force in the wires, which is dominated by the tension. The evolution of the strain with respect to the distance from the rupture location is (right solid curve in Fig. 17):

$$ \epsilon(x_c) = \epsilon_0(1 - \exp(-\frac{x_c}{L_0})) \quad (5) $$

With exponential evolution of the strain, the wire regains 90% of its strain at a distance to the rupture location of $2.3L_0$ and 99% at $4.6L_0$.

If an anchorage of the tendon is closer to the rupture location than the estimated transfer length then the model is no longer valid and tension loss of the tendon may be more important due to partial re-anchoring of the broken wire.

Due to the failure of the strain gauges on the wires, the experimental campaign did not yield direct measurements of transfer length that could be used for validation of this model.

Indirect measurements are available in the form of the length at which the sheath strain gauges cease to detect changes and the strain measurements on one wire of two intact strands in the only experiment where wire-mounted gauges yielded useable results (5T15-3).

These two indirect measurements are expected to underestimate transfer length due to detection limits. The very small strain changes in wires at the end of the transfer length would not be detectable on the sheath, and the strain on one of the more than thirty remaining intact wires would necessarily converge to the noise level faster than the strain of a ruptured wire on another strand.

Nevertheless, the presented transfer length model does not fit the indirect measurements well, as the evolution of the indirectly measured transfer length does not match the evolution of sliding.

For specimen 1T15-3, the maximum sliding on the wire is 1.6 cm, 3.3 cm, and 7.6 cm, respectively (precision on sliding is $\pm 1$ cm), which would correspond to a minimum transfer length $L_0$ of 1.2 m, 2.5 m, and 5.7 m, while the positions of the strain gauges where the stress has converged yield indirect transfer length measurements of 1.2 m, 1.5 m, and 2.2 m (precision is the distance between two gauges, 25 cm). A maximum growth of 375% in maximum sliding does not correspond to the 80% growth in the indirect transfer length measurement.

The same deviation is observed for specimen 5T15-1, where the maximum sliding at the first rupture is 0.9 cm (associated with a minimum transfer length of 0.7 m) while the last affected gauge position
is at 1.7 m. The fifth rupture exhibits a maximum sliding of 4.4 cm (corresponding $L_0$ of 3.3 m), but the last affected gauge is located at 2.2 m.

Here again, the evolution of sliding is not matched by the evolution of the indirect transfer length measurement.

Finally, using the wire strain gauges on specimen 5T15-3 for the first three cuts, the maximum sliding values are 3.8 cm, 5.4 cm, and 5.4 cm (corresponding to minimum transfer lengths of 2.8 m, 4.1 m, and 4.1 m), while the strain has converged at the gauge positions of 2.2 m, 3.0 m, and 3.2 m.

None of the indirect measurements of transfer length match the evolution of sliding except for small damage increments, and nearly all indirect length measurements yield smaller values than predicted by the transfer length model (even when the measurement approaches $L_0$, the model predicts a transfer length of $2L_0$ as more probable than $L_0$).

Direct transfer length measurements are needed to validate or refute the model assumptions. In the meantime, due to the poor agreement with the available data, this transfer length model should be considered as a proposition rather than a fully validated model.

### 4.5. Computation of an upper bound of the remaining prestress strength due to sliding

In this section, mechanical models linking sliding to tension loss are introduced.

Three models are derived to estimate the loss of tension caused by sliding. They differ in their assumptions about the impact of torque and twist as we have observed that the most significant variation of tension is correlated with rotation of the strand. In the first model termed *Only sliding*, only the change of strain in the broken wire is considered. As twist is not permitted during tension, a significant torque is created in the strand [27].

In the second model, the loss of the contribution of the ruptured wires to the torque is considered, as the tension and the torque of the strand are coupled, this loss of torque in the strand translate into a loss of tension. This second model is termed *Sliding + Torque* and include both the impact of sliding and torque relaxation leading to tension loss.

The third model, termed *free rotation* considers the free rotation of the strand to account for a phenomenon that occurred in the two tendons 1T15-1 and 1T15-2 (Fig. 18), where the grout is gone over a long span and the remaining wires could no longer prevent the rotation of the two halves of the strand, allowing them to rotate freely to relax the important torque in the strand created by the tensioning with blocked rotation. Due to the coupling of the tension of the strand and the torque, this major decrease of the torque will translate into a major decrease of tension of the strand. This model keep the impact of sliding on tension loss but assumes that every damaged strand has fully relaxed its torque.

In all three models, the tension in the grout and sheath is neglected. While the stiffness of the sheath is negligible compared to that of the steel, this is not the case for the grout when intact; however, its damage state is unknown, the bias introduced by this assumption will be discussed when presenting the results. Furthermore, the influence of bending effects is neglected, and the wires are assumed to remain in their elastic domain. Both of these assumptions are inaccurate and should be relaxed in a more sophisticated model.

In order to account for the coupling between the twist and the tension, we use a stiffness matrix relating the displacement and angle of twist of a healthy strand to its tension and torque.

We will use the stiffness model from McConnel and Zemek [28], which is the less complex model still in good agreement with numerical simulations for small lay angles such as those of the strands found in prestressing tendons [27]. It states that:

$$ \begin{pmatrix} T \\ C \end{pmatrix} = \begin{pmatrix} K \\ \frac{u'}{\theta'} \end{pmatrix} \qquad (6) $$

<figure><img src="image_20.png" /><figcaption>Fig. 18. 1T15-1 after the 6th cut.</figcaption></figure>

with

$$ \begin{align} K &= \begin{pmatrix} k_{rr} & k_{r\theta} \\ k_{\theta r} & k_{\theta\theta} \end{pmatrix} \tag{7} \\ \equiv & \begin{pmatrix} k_{rr} & k_{r\theta} \\ k_{\theta r} & k_{\theta\theta} \end{pmatrix} \nonumber \end{align} $$

and

$$ k_{rr} = ES_0 + 6ES_h \cos^3 \alpha \qquad (8) $$

$$ k_{r\theta} = k_{\theta r} = 6ES_h R_h \cos^2 \alpha \sin^2 \alpha \qquad (9) $$

$$ k_{\theta\theta} = GJ_0 + 6GJ_h + 6ES_h R_h^2 \sin^2 \alpha \cos \alpha \qquad (10) $$

where $S$ and $S$ are the cross-sectional area of a wire, with indexes 0 and $h$ denoting respectively the area of the central and helical wires, $J$ is the polar moment of Inertia, $E$ is the Young modulus of steel and $G$ is its shear modulus. $R_h$ is the distance between the centre of the central wire and the centre of the helical wires, $\alpha$ is the helix angle of the strand, $T$ and $C$ are respectively the tension and the torque, $u$ and $\theta$ are the displacement of the strand and its angle of twist, respectively, with the apostrophe denoting the derivative with respect to the direction of the strand. All of this value can be derived, using McConnel and Zemek [28], from the material parameters of the steel ($E = 200$ GPa and $\nu = 0.3$ the Poisson coefficient) and the geometrical parameters of the strand. For the strand used the radius of the central and helical wires respectively 2.7 mm and 2.6 mm and the helix angle $\alpha = 7.8^\circ$.

For the first model *only sliding*, a wire denoted $i$ has slid a length $g_i$, if the wire is not broken $g_i = 0$,

$$ g_i = \int_0^L \delta\epsilon_i(x)dx \qquad (11) $$

Along the tendon, assuming the strain is still in the elastic domain and there is no plastic strain, the tension is equal to:

$$ T = \sum_i ES_i \epsilon_i(x) \qquad (12) $$

thus, the variational form

$$ \Delta T = \sum_i ES_i \delta\epsilon_i(x) \qquad (13) $$

where $E$ is the Young modulus of steel, $S_i$ the surface area of the central wire if $i$ is a central wire or $\cos \alpha^3 S_h$ with $S_h$ the surface of a helical wire if $i$ is a helical wire. The factor $\cos \alpha^3 \approx 0.97$ is neglected for the sake of clarity.

By integrating along the tendon:

$$ L\Delta T = \sum_i ES_i g_i \qquad (14) $$

$$ \Delta T = \sum_i ES_i \frac{g_i}{L} \qquad (15) $$

Which give a unitary tension loss by wire according to sliding of

$$ \delta T_{\text{Sliding}} = ES_i \frac{g_i}{L} \qquad (16) $$

Participation of the grout can be incorporated to the model with the same reasoning, adding a $\delta T_{\text{grout}} = \frac{E_g S_g g_g}{L}$ term to the variation of tension with $E_g$, $S_g$ and $g_g$ represent the Young modulus, area and sliding of the grout. The grout is too damaged to contribute in the 1T15-1 and 1T15-2 experiments. On experiments 5T15-1 sliding of the grout reached 1 cm at the 14th which gives, assuming intact grout and with a Young Modulus of 20 GPa, and grout cross-sectional area of
<table><caption>**Table 3**
Root mean square error of the tension loss models compared with the force sensor.</caption><thead><tr><th>Tendon</th><th>Only sliding (kN)</th><th>Sliding + Torque (kN)</th><th>Sliding + Free rotation (kN)</th><th>Mixed (kN)</th></tr></thead><tbody><tr><td>1T15-1</td><td>26</td><td>22</td><td>34</td><td>7</td></tr><tr><td>1T15-2</td><td>42</td><td>34</td><td>24</td><td>12</td></tr><tr><td>1T15-3</td><td>7</td><td>3</td><td>37</td><td>–</td></tr><tr><td>5T15-1</td><td>37</td><td>28</td><td>56</td><td>–</td></tr><tr><td>5T15-1 (twelve first cut)</td><td>11</td><td>13</td><td>60</td><td>–</td></tr></tbody></table>

2500 mm² would yield 10 kN to compare with a tension loss of 166 kN. The grout elastic modulus is highly unlikely to be homogeneous due to damage near the rupture zone. Incorporating the grout into the models would require an assumption on grout damage evolution and strain distribution as the simplification used for (14) can no longer be used. Accounting for grout participation would have improved the fit of the tension loss prediction as Fig. 19 shows the models underestimating the tension loss. There is a strong coupling between torque and tension in the strand; the prediction of tension loss can therefore be improved by introducing assumptions regarding the torque. The two following models employ torque assumptions to yield an additional corrective term for tension loss. These two models are mutually exclusive.

For the second model sliding + torque, the variation of tension due to the application of torque variation caused by the breaking of a wire is considered. The contribution of the torque of an outer wire, using the simplified expression given by [12], is:

$$ \delta C = R_h \frac{T_i}{\tau} \qquad (17) $$

with $\tau = \sin \alpha \cos \alpha \frac{1}{R_h}$. For simplicity $T_i$ is taken as $T_0/N_{wire}$ with $N_{wire}$ the number of wire, respectively 7 or 35. The $T_i$ used for the treatment of the experimental data is given in Table 1. The change of tension for each broken outer wire due to torque relaxation in the strand is:

$$ \delta T_{\text{Torque}} = \frac{k_{r\theta}}{k_{\theta\theta}} \delta C \qquad (18) $$

For the Free rotation model, it is assumed that the strand can freely rotate such that the initial torque is entirely relaxed. The initial torque induced by the elongation is $k_{r\theta}\epsilon_0$. Relaxation of this torque yields an angle of twist of $\theta_{\text{Free rotation}} = \frac{k_{r\theta}\epsilon_0}{k_{\theta\theta}}$ which translates into a change in tension for each strand once free rotation is permitted:

$$ \delta T_{\text{Rotation}} = \frac{k_{r\theta}^2}{k_{\theta\theta}} \epsilon_0 \qquad (19) $$

At each cut, a new wire slides, and the sliding of the already cut wires increases. This phenomenon accounts for the quasi-quadratic evolution of the total tension loss ($\Delta T_n$) for the first few cuts (Fig. 19). Fig. 19 shows the comparison between measured tension loss $\Delta T_n$ and the estimated tension loss due to sliding according to the number of cuts $n$ from the models. The sliding of the grout was not used for the estimation, as the grout broke and fell into the bench during the experiment, leading to an underestimation of the tension loss.

Fig. 19 indicates that the model considering only sliding performs well only for the initial cuts (the first cut for the 1T15 cables, the first six cuts for the 5T15-1 cable). 5T15-2 and 5T15-3 are not used to validate those models as a lot of wires sliding occurs out of the camera field due to the cutting schedule. The model incorporating the impact of torque performs well in cases most likely to occur in in-service prestressing tendons (1T15-3 and 5T15-1 until the second strand breaks). The fact that the model is still accurate even with one strand completely broken shows that there is considerable torque transfer between the strand and the grout while the bond with the grout is still holding. The free rotation model becomes applicable only in very specific cases where the bond with the grout is compromised along a significant length (5 m for 1T15-1 and 1T15-2 due the failure of the PVC sheath) and after the rupture of the second strand for the 5T15-1, when the bond damage is sufficient to permit rotation.

The rotation of the two end of the wires relative to each other is indicative of the free rotation of the strand when its angle of twist

become close to $\theta_{\text{Free rotation}}$, which start from the 4th cut for 1T15-1 and 3rd cut for 1T15-2. The free rotation of the strand is never achieved in the 1T15-3 and 5T15-1 experiments. The large discrepancy between the model and measurements for specimen 5T15-1 after the 2nd strand rupture (Fig. 19(d)) is attributed to wires in the rupture zone exceeding their elastic domain and entering the plastic strain regime. The displacement of 1 cm over a 10 cm zone is incompatible with the elastic domain of the wires, which extends to 0.82% strain.

The validity of the model is assessed using the root mean square error (Table 3) between the force sensor measurements and the tension loss predicted by the model. Accounting for torque consistently yields improved accuracy, as predicted. The experimental tension loss prediction is also more accurate when accounting for the transition from the Sliding + Torque model to the Free Rotation model, which occurs at the fourth cut for specimen 1T15-1 and the third cut for specimen 1T15-2. This corresponds to the mixed column in Table 3.

From the second strand rupture onward, there is a considerable loss of accuracy in the 5T15-1 tension loss prediction due to plasticity. The model performs very well when limited to ruptures on the first two strands.

When the appropriate model is selected — Sliding + Torque or Sliding + Free Rotation for strands exhibiting considerable rotation, represented by the mixed column on Table 3 — the proposed tension loss model demonstrates excellent performance.

## 4.6. Usage

These models can be applied to on-site tendons. We present three use cases involving low and severe damage scenarios for a 50 m long tendon with 19 15.7 mm-diameter strands (19T15). Table 5 shows the unitary tension loss computed using the geometric parameters of the strand and tendon, the material parameters of the steel, and the initial load. This value differs slightly from those used in the experimental validation due to variations in length and initial tension. The strands are assumed to be identical to those used in the previous section.

The models require an estimation of the residual tension in the tendon under undamaged conditions. For both scenarios, we assume that theoretical computation of prestressing loss indicates a remaining tension of 3500 kN in the healthy tendon.

The associated strain in the wires is 0.61% ($\frac{T}{ES}$, where T is the tendon tension, E is the steel elastic modulus of 200 GPa, and S is the steel cross-sectional area), neglecting the strain difference between central and helical wires.

Table 4 shows the relationship between sliding and different predictions of transfer length for varying damage levels under different assumptions regarding the evolution of strain in the broken wire along the tendon, with a generalized strain of 0.61 mm/m. For the exponential strain evolution, the lengths given represent the distances from the rupture zone where the strain in the broken wire has recovered 90% and 99% of the generalized strain, respectively. These predictions are not yet fully validated and should be used with appropriate caution.

To estimate the tension loss, we need to choose between the Sliding + Torque model or the Sliding + Free rotation model. The only Sliding model is a simpler but less accurate alternative. The selection can be done by at the angle of twist between the ruptured wires of a strand. The angle of twist caused by the free rotation of the strand ($\frac{k_{r\theta}}{k_{\theta\theta}}\epsilon_0$) would range from one quarter of a turn to one half for usual strand
<figure><img src="image_21.png" /><figcaption>Fig. 19. Comparison tension loss measured by a force sensor and estimated using different models based on the sliding of the wires.</figcaption></figure>

<table><caption>Table 4
Sliding and transfer length for $\epsilon_0$ = 6.1 mm/m.</caption><thead><tr><th rowspan="2">Damage</th><th rowspan="2">Typical sliding<br>(cm)</th><th colspan="4">Transfer length in one direction (m)</th></tr><tr><th>Minimal<br>$g_i/(2\epsilon_0)$</th><th>Linear<br>$g_i/\epsilon_0$</th><th>Exp. (90% strain)<br>$2.3g_i/(2\epsilon_0)$</th><th>Exp. (99% strain)<br>$4.6g_i/(2\epsilon_0)$</th></tr></thead><tbody><tr><td>1–2 wire cut</td><td>1</td><td>0.8</td><td>1.6</td><td>1.9</td><td>3.8</td></tr><tr><td>4–5 wires cut</td><td>5</td><td>4.1</td><td>8.2</td><td>9.4</td><td>18.9</td></tr><tr><td>1 strand cut</td><td>10</td><td>8.2</td><td>16.4</td><td>18.8</td><td>37.7</td></tr></tbody></table>

<table><caption>Table 5
Tension loss in a 50 m 19T15 tendon with a residual tension of 3500 kN.</caption><thead><tr><th>Cause</th><th>Symbol</th><th>Value</th></tr></thead><tbody><tr><td>Sliding<br>(by cm of wire sliding)</td><td>$\delta T_{\text{Sliding}}/g_i$</td><td>0.8 kN cm<sup>-1</sup></td></tr><tr><td>Torque relaxation (by outer wire)</td><td>$\delta T_{\text{Torque}}$</td><td>1.8 kN</td></tr><tr><td>Free rotation (by strand)</td><td>$\delta T_{\text{Rotation}}$</td><td>38 kN</td></tr></tbody></table>

and strain. In our case that would be 120°, not accounting for rotation induced by unwinding of the ruptured wire.

In the first scenario, five broken outer wires were identified with 1 cm of sliding on each wire. The transfer length would then be in the range of 1.6 m ($\frac{1 \text{ cm}}{0.0061}$) in each direction from the rupture zone. As the damage level is likely insufficient for the grout to be sufficiently

cracked to permit full rotation of damaged strands, the Sliding + Torque model provides more accurate tension estimates. The tension loss can then be estimated as 13 kN using Table 5, with a sliding contribution of 4 kN (5 m total sliding of the wires) and a torque relaxation contribution of 9 kN. This demonstrates that the tension loss is negligible compared to the estimated 3500 kN of tension.

In the second scenario, three complete strands have ruptured, and another strand has lost five outer wires, for a total of twenty-six broken wires or 18% of the total wire count. The broken strands have slid 10 cm, while the other broken wires have slid 5 cm. The transfer length is computed using the highest sliding value and is therefore in the range of 16 m ($\frac{10 \text{ cm}}{0.0061}$) in each direction. If an anchorage is closer, then the tension loss model is no longer valid and the tension loss is likely higher. None of the five strands have rotated, so the tension loss can be estimated as 229 kN, with a sliding contribution of 188 kN (235 cm of sliding) and a torque relaxation contribution of 44 kN (23 outer wires broken).
The third scenario is identical to the second except that the five damaged strands have rotated. The tension loss can then be estimated as 378 kN, with a sliding contribution of 188 kN and a free rotation contribution of 190 kN (5 strands affected). The free rotation model supersedes the torque relaxation component for the outer wires.

In both the second and third scenarios, the tension loss remains within the operational margin for a typical bridge. However, in the second case, the lower tension loss makes tendon durability more critical, as there is greater tension in the remaining intact wires than in the third scenario, making repair or replacement more urgent. Of course, the tension estimate should be validated through direct measurement whenever possible.

## 5. Conclusion

An experimental campaign was conducted on multiple prestressing tendons to deepen the understanding of damage mechanisms in external prestressing tendons. The results of this campaign explored the impact of grouting quality, strand multiplicity, and wire breaking on the behaviour of damaged tendons. The main finding is the critical importance of the bond between a strand and the tendon. Tension loss due to damage is shown to be very minor when the bond between the strand and the grout remains intact. However, when this bond fails, most of the tension supported by a wire is lost upon breakage. Bonded 5-strand tendons have been shown to fail after the rupture of approximately 40% of the wires, with exceptions occurring when bond failure caused significant tension loss. The location of the rupture has only a minor impact on tension loss and tendon durability, even though strand rupture causes slightly more tension loss than the rupture of an equivalent number of wires distributed across multiple strands. When a wire breaks while the bond is maintained, its stress is transferred not only to the wires of its strand but also to the other strands of the tendon. This has major implications for tendon durability, but care must be taken as the stress transfer is not homogeneous in the intact wires. While some insights are provided about how the stress is distributed in the remaining wire no clear model have emerged from our date due to a multiplicity of concurrent mechanical phenomenon including friction, helical bending and torsion. The bond depends on grouting quality. When the grout has lost its capacity to constrain the strands over a length of several metres due to cracking, every strand acts as if it were unbonded. A model was developed based on the kinematic compatibility of the wire to estimate the transfer length, depending on assumptions about the strain and displacement of the cut wires in the damaged tendon. A closed-form formula linking the sliding of the wire, assumptions on the form of the strain, and the transfer length was derived. This model may prove useful for assessing the extent of damage in in-service tendons, but it requires further validation. Additional models were presented to estimate tension loss using the sliding of the wire after a break. The first model, considering only the sliding of the broken wires, provides a lower bound for the tension loss. The second model, which also accounts for the loss of tension caused by the loss of the torque carried by the broken wires, gives accurate results when the bond between the grout and the strand prevents strand rotation. This model is most likely to provide the most accurate description of tension loss due to wire damage in the 12 or 19T15 tendons commonly found in transportation structures. The third model, which assumes complete relaxation of the torque in the strand, accurately describes unbonded 1T15 cables or multiple-strand tendons with broken strands that are unbonded close to the rupture due to grout damage. The first model provides a lower bound for tension loss and the third model provides an upper bound, so these models can be used to compute upper and lower bounds of the remaining tension in the tendon from a picture of the damaged area, providing another tool to assess the residual durability of damaged tendons. These models offer significant practical advantages due to their ease of implementation, as they require only the sliding measurements of the wires, which can be readily obtained

once damage is detected in the tendon. However, the presented models require further experimental validation, particularly the transfer length model. Additionally, their accuracy can be enhanced by incorporating more complex behavioural phenomena, most importantly plasticity.

### CRediT authorship contribution statement

**Jonas Aparicio:** Writing – original draft, Visualization, Methodology, Investigation, Conceptualization. **Gwendal Cumunel:** Writing – review & editing. **Tien Hoang:** Writing – review & editing, Software, Resources, Investigation. **Gilles Foret:** Writing – review & editing, Validation, Supervision, Resources, Project administration, Methodology, Investigation, Funding acquisition.

### Declaration of Generative AI and AI-assisted technologies in the writing process

During the preparation of this work the authors used large language model in order to improve grammar and sentence flow. After using this tool/service, the authors reviewed and edited the content as needed and take full responsibility for the content of the published article.

### Declaration of competing interest

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

### Acknowledgement

This experiment has been funded by the ASFA for the first five tendons and Freyssinet for the last one.

### Data availability

Data will be made available on request.

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