Forensic assessments of the influence of reinforcement detailing in reinforced concrete half-joints

This paper presents the results of forensic assessments on reinforced concrete half-joint beams tested within the literature through the application of nonlinear finite element analyses. The software package ATENA Science was utilised, in which critical evaluations were conducted into the accuracy of the software in predicting the response of reinforced concrete half-joints with inadequate reinforcement detailing. It was shown that the load capacity of the half-joints could be predicted accurately, with an average experimental-over-predicted ratio of 1.04 and a coefficient of variation of 7.3%. The predicted crack patterns and modes of failure consistently coincided with those obtained experimentally, exhibiting localised failures at the re-entrant corner of the nib, along with failures located within the full depth of section due to an inadequate amount of shear reinforcement. Moreover, the combination of bar reductions to replicate severely corroded reinforcement at critical zones highlighted the influence deterioration mechanisms have on the structural capacity, with reductions of more than 50% observed in comparison to newly constructed half-joints with no deterioration.


Introduction
Reinforced concrete half-joint beams (also known as dapped-end beams [1,2]) adopt a geometric setup consisting of a full depth section and a reduced section at the joint.The half-joint is achieved when the beam element is reduced at the corner near the joint (referred to as the dap) and the region of concrete above and/or below known as the nib (see Fig. 1).By achieving this detail, the bearing location can therefore be moved higher into the cross-section, thus providing greater lateral stability of the structural element at the supports than a conventional beam [3].Historically, this structural layout was favoured in bridge construction due to the ease of installation of pre-cast elements.Furthermore, the overall reduced depth of section, which is obtained by recessing the depth into the supporting beam, offers a construction method which is both economical and efficient [4].
Over the years however, catastrophic failures have encouraged investigations into the safety of these structural layouts.The collapse of the De la Concorde overpass in Quebec, after nearly 40 years of service, is perhaps one of the best-known cases of a structural failure involving half-joints, tragically killing five people, and injuring six others [5].Failure of the overpass was sudden and was preceded by the formation of a major crack which was spotted by road users before the incident [6].Unfortunately, the overpass failed shortly after in a brittle manner, literally under its own self-weight.From a report of commission of inquiry that followed, it was noted that the failure was attributed to a combination of various aspects including improper detailing and placement of hanger reinforcement (and the omittance of shear reinforcement in the full depth of section) and deterioration of the steel and concrete due to inadequate drainage of water containing chloride from de-icing salt used on roads and freeze-thaw cycles [5].Another example of a structural collapse is the failure of the Annone overpass in the North of Italy (Lecco) in 2016, which resulted in one fatality and several injuries [7].The failure of the overpass was again attributed to a combination of the accelerated deterioration of steel and concrete due to chloride induced corrosion and design flaws with insufficient reinforcement in the half-joint [7].
Considering the catastrophic nature of recent half-joint failures, a thorough analysis was undertaken on the inspection data of half-joint structures across the strategic road network throughout England [8].It was found that there were 428 structures with half-joint elements across the Highways England Road Network, in which 252 structures displayed half-joint-related defects.These structures were then interrogated further to categorise the defects observed, with 87% of the 252 structures displaying structural and deterioration mechanisms.It was found between 33 and 38% displayed corrosion, spalling, and deterioration mechanisms, and 61% of structures displayed varying degrees of cracking.Subsequently, based on recent experimental findings, six individual zones were proposed which could aid in identifying potential distress in half-joints [8].The occurrence of cracking at a certain orientation within a given zone could therefore indicate a higher probability of being associated with specific issues.
Cracking under service loads can occur in other types of structures adopting half-joint detailing.Fig. 2 displays an example of cracking in a multi-storey reinforced concrete frame structure, with extensive diagonal cracking in the vicinity of the re-entrant corner along with several other diagonal cracks further in-span within the full depth portion of the beam.Such crack formations are not only aesthetically unpleasing but may also indicate that the beam is under significant distress.Given that failure of the half-joint could be potentially catastrophic, prompt structural assessment is required to evaluate the remaining strength of the joint and hence the overall safety of the structure.
When analysing reinforced concrete half-joints, laboratory experiments have generally been adopted with replica specimens being cast and tested in a controlled laboratory environment [1,2,5,[9][10][11][12][13][14][15][16][17][18][19], although field testing had also been undertaken in a few circumstances [7,20,21].Mattock and Chan [1] tested eight half-joint beams and demonstrated that nib failures along with shear, due to cracking originating from the bottom corner of the full depth of section, could occur.They emphasised the importance of not only including the correct amount of hanger reinforcement but also anchorage of longitudinal  reinforcement in the nib.Clark [10] conducted experimental testing including and excluding diagonal reinforcing bars and concluded that diagonal bars strongly influenced the serviceability response of reinforced concrete half-joints.Similarly, Moreno-Martínez and Meli [16] reported that the omittance of diagonal bars could not only contribute to cracking at lower loading, but also displayed larger initial crack widths.Moreover, along with diagonal bars, the distance of the first stirrup from the nib was observed to aid in controlling the propagation of diagonal cracks.Wang et al. [14] proposed that the distance from the nib to the first stirrup should be less than 40 mm.More recently, fifteen half-joint specimens tested [19] illustrated an increase in reinforcement ratios from 50 to 100% yielded an increase in strength of 15-60%, respectively.
Whilst experimental testing is an ideal way to replicate real world problems, it is expensive and time-consuming and often suffers from restrictions on the size and number of specimens that can be tested.The availability of a powerful finite element analysis software and the exponential growth of computing power, however, can allow for the analysis of complex engineering problems to a very high degree of accuracy with significantly less human effort.Work conducted by [20] utilised experimental data to calibrate a nonlinear finite element model of a typical reinforced concrete half-joint.The results obtained were deemed satisfactory, but emphasis was placed on the suitability of the model, with applications being limited to specimens of similar geometric and loading conditions.Moreover, half-joint specimens exhibiting extensive cracking in critical regions and/or improper detailing were recommended to implement caution due to the localised nature of the response.Aswin et al. [21] observed that the implementation of nonlinear finite element analyses, herein referred to as NLFEA, on four half-joint beams provided a more accurate prediction for the failure load of the specimens in comparison to other analysis methods, with the experimentally obtained load-displacement responses coinciding with the analyses.Mitchell and co-workers [5] implemented NLFEA to simulate the progressive concrete degradation observed on the De la Concorde Overpass, with the results obtained providing an explanation as to how the structure catastrophically failed under minimal live loading.More recently, Santarsiero et al. [22] simulated chloride ingress induced concrete degradation and the subsequent corrosion of steel reinforcement of reinforced concrete half-joints located on the Musmeci bridge in Italy.The analyses illustrated further that corrosion can not only influence the ultimate load bearing capacity, but also the collapse mechanism of reinforced concrete half-joints.Moreover, work conducted by [23] and [24] investigated the influence of corrosion induced bond deterioration through both experimental testing and the application of NLFEA.The results obtained illustrated that the presence of deterioration of concrete and steel near the re-entrant corner can significantly alter the response of a half-joint, thereby making accurate predictions of the capacity and remaining life difficult.
Thus, there is evidence to suggest that the application of NLFEA is a powerful alternative to the traditional methods of laboratory testing.In this paper, existing experimental data reported recently by the University of Cambridge [17] is analysed, to further cement the value of implementing NLFEA for forensic assessments of reinforced concrete half-joints.To do so, internal mechanisms under loading and the implications of reinforcement layout due to, for example, inconsistencies between as-built and as-designed reinforcement detailing and/or deterioration of concrete and steel, on the load and deflection capacities of reinforced concrete half-joints are assessed to provide additional insights that complement the original experiment and allow for additional factors to be investigated.

Overview of cambridge half-joint specimens
In 2016, Desnerck and co-workers tested four half-joints with the aim to investigate the influence of reinforcement arrangements on the load capacity and failure modes of half-joints [17].The geometry and the reinforcing details of all half-joint specimens are presented in Fig. 3.
All specimens had a span of 3.14 m and an overall width of 0.4 m.The depth over the main span was 0.7 m and near the support, this was reduced to 0.325 m to form the nib.All specimens were reinforced with five 20 mm top longitudinal bars and five 25 mm bottom longitudinal bars to provide the required flexural strength.In the refence specimen (NS-REF), three U-shaped 12 mm bars were provided in the nib region and extended ~ 1.2 m into the full depth of section to ensure sufficient anchorage.Four hanger reinforcing bars were provided near the end of the full depth section, each in the form of 2 legs of 10 mm closed-loop shear links, provided at a spacing of ~ 115 mm.In the nib region, four 12 mm diameter diagonal bars were also provided; with their ends bent and aligned parallel to the main longitudinal reinforcement to provide sufficient anchorage.Three-legged 10 mm closed-loop shear links were provided over the remaining part of full depth section at a spacing of 200 mm.As illustrated in Fig. 3, three alternative half-joint detailing were studied to highlight the role of reinforcement near the re-entrant corner: (i) half-joint without the inclusion of diagonal bars (NS-ND); (ii) half-joint without the inclusion of U-bars (NS-NU); and (iii) half-joint with reduced shear links (NS-RS).
Apart from the omittance of a specific bar as listed above, the remaining detailing is identical to the reference specimen (NS-REF).In this paper, the omittance of specific bars was adopted as a conservative approach, in which severely corroded bars were simply removed as opposed to modelling bond deterioration and localised bar reductions, along with bars which have been unbonded due to cracking/spalling of the surrounding concrete.
Table 1 lists the properties of steel reinforcement used in the halfjoint tests [17], illustrating the yield and ultimate stresses for each respective bar.

Preliminaries
The nonlinear responses of the half-joint beams were investigated by nonlinear finite element analyses using ATENA-Science, developed by Červenka Consulting [25,26].Part of the ATENA-Science software includes two software packages, namely GiD and ATENA which are fully integrated.GiD is a user-friendly pre-processor which can be used for defining the geometric, material, and loading properties along with the generation of the finite element mesh.When a finite element model is run in GiD, the model and its respective data are loaded automatically into either ATENA Studio or ATENA Console, which is used for the nonlinear finite element analysis.In this work, ATENA Studio was used as it also has the post-processing capability with an integrated real-time display of results during an analysis, providing observations which are not readily available in many other analysis packages.
In ATENA, the fracture-plastic model is used to represent the nonlinear behaviour of concrete [25,26].The formulation within the constitutive models implements the small strain approach with the strain being decomposed into elastic ε e ij , fracture ε f ij , and plastic ε p ij components [26], as given by The stress increment is described by the rate equations considering progressive damage of concrete due to cracking and plastic accumulation of concrete due to concrete crushing [25]: where D ijkl is the elastic modulus of concrete; εkl is the total strain W. Don et al.
rate; εf kl and εp kl are the fracture and plastic strain rates, respectively [27].The rates of fracture εf ij and plastic strains εp ij are determined using the fracture and plasticity models, respectively, using the flow rule: where λf and λp are the inelastic fracturing and plastic multipliers; m f ij and m p ij specify the directions of inelastic fracturing and plastic flow rule [28]; g f and g p are the potential function defining the direction of inelastic fracturing and plastic strains, respectively.For further details about how the plasticity and fracture models are implemented and combined in ATENA, the reader is referred to [25,26].In the next section, the main constitutive models incorporated in the fracture and plasticity models are briefly described.

Crack representation and constitutive models
The formulation within the constitutive models adopted in ATENA implements the smeared crack approach.Three crack models are available within the software's library: a fixed crack, a rotated crack, and a combination of the two [26].In the fixed crack formulation, the direction of cracking is fixed based on the direction of the principal stresses at the onset of cracking.During loading, the direction of the principal stresses may rotate and no longer coincides with the crack direction, leading to the development of shear stress and crack slip on the cracked plane.For the rotated crack model, the direction of cracking is set to follow the direction of principal stress and hence no consideration of shear stress is required.In this work, the fixed crack model is
adopted, due to providing a more general representation of crack formations in reinforced concrete half-joints which are subjected to monotonic and cyclic loading conditions.Although the scope of the work presented in this paper has its focus primarily on the behaviour under monotonic loading, the adoption of the fixed crack model would allow further extension to non-stationary cyclic loading i.e., to replicate traffic loading on a highway bridge adopting half-joint detailing.The stresses and strains in the constitutive models described below are evaluated along the crack directions.
In the fracture model (see Fig. 4(a)), the Rankine failure criterion is adopted to define crack initiation and the crack band approach [29] is then followed to describe the post-cracking response of concrete [25,26].This is formulated using a softening law which considers the crack opening displacement w and fracture energy G f based on the experimentally derived empirical expressions proposed by Hordijk [30]: where σ t is the tensile stress (MPa); f t is the concrete tensile strength (MPa); w is the crack opening displacement (mm); w c is the crack opening at the complete release of stress (i.e., zero stress) (mm); L t is the characteristic length obtained from the finite element mesh size projected into the normal crack direction (mm), with the mesh orientation bias minimized using an orientation correction factor [31]; G f is the fracture energy required to create a unit area of stress-free crack (N/m); G f0 = 30 N/m is the base value of fracture energy based on the maximum size of aggregate of 16 mm [32] (in this study, however, this value is reduced to 26.3 N/m to account for the 10 mm max aggregate size used in the experiment [17]); c 1 and c 2 are empirical constants, and the values of c 1 = 3 and c 2 = 6.93 were proposed based on the analysis of different sets of experimental data [30].
In this work, the concrete compressive strength was taken as f ' c = 36.8MPa, following the mean strength measured from the cylinder specimens [17].The tensile strength was estimated from the compressive strength using equation (5b) and taken as f t = 2.96 MPa.This value is notably lower than the reported flexural tensile strength of 3.83 MPa [17] and was considered to account for residual tensions developing in the beam prior to testing due to restrained shrinkage [33].The fracture energy was taken as G f = 65.3N/m, determined using equation (5e).
As illustrated in Fig. 4(b), the Menétry-William failure plasticity model [34] incorporating a non-associated flow rule is adopted to represent the behaviour of concrete in compression under multiaxial stress conditions [25].In this model, the position of failure surfaces is not fixed and can move depending on the value of the strain parameter [26].In addition, the failure surface is controlled by hardening and softening models, which are defined based on the response of concrete under uniaxial compression.In the hardening part, the ratio of compressive stress σ c (MPa) to the cylinder compressive strength f ' c (MPa) is related to the compressive stress beyond the elastic limit f co (MPa), the equivalent concrete plastic strain ε eqp (mm/mm), and the plastic strain at the peak stress ε p c (mm/mm) in the following manner [26]: In the above formulations, the compressive stress beyond the elastic limit, f co , is taken as twice the tensile strength of the concrete (=5.92 MPa) [26]; f cu and f ' c are the cube and cylinder compressive strengths of the concrete, respectively; and E c is the concrete elastic modulus, taken as 35.1GPa, based on the measured concrete strength.
Whilst the hardening part of the compression model is computed based on the strain, the softening part is computed based on the displacement and the crush band approach is adopted to ensure mesh objectivity [35].In this model, it is assumed that the post-peak compressive stress decreases linearly from the peak stress f ' c to zero stress at a prescribed displacement w d , in which a value of w d = 2.5 mm is adopted [35].The crush band size L c is determined based on the finite element size projected into the direction of the minimum compressive stress, with the mesh orientation bias minimized using an orientation correction factor [35].The minimum crush band size is taken as the minimum beam dimension (=400 mm) to reduce the dependency of results on mesh size [35].
The reduction of concrete strength due to crack formations in the transverse direction is considered through a simple reduction factor: In this work, it is assumed that there is no reduction in compressive strength when the tensile strain normal to the crack ε 1 is less than 0.1%.
Between 0.1% and 0.5%, the value of r c is assumed to decrease linearly to 0.6; the value thereafter is taken as 0.6 [36].
To represent the reduction in the shear modulus of concrete after cracking, a shear retention factor following the expression proposed by Kolmar [26,37,38] is adopted (see Fig. 4(c)).Here, the shear modulus is related to the strain normal to the crack ε 1 which is indicative of the crack opening.The governing equations used to describe the shear retention factor are ) c 2 (8b) where G is the shear modulus after cracking (MPa); r g is the shear retention factor; G c is the initial shear modulus (MPa); ρ is the transformed ratio of steel reinforcement to the crack plane; c 1 and c 2 are parameters which are dependent on the steel bar(s) crossing the direction of the crack; and c 3 is a user scaling factor (by default c 3 = 1).In this study, ρ is taken as zero as the restraining effects from the reinforcing steel could be considered automatically in the analysis.It is worth noting that the above model is not related to mesh size and its use would be limited to the range of mesh sizes used in this work [38].Another way to compute the shear retention factor is by relating the post-cracking shear stiffness to the stiffness along the crack opening direction, using a constant scaling factor [39].The former approach is, however, adopted in this paper.In addition to the reduction in shear stiffness, the maximum shear stress τ max that can be transmitted across a crack is related to the crack opening displacement w and the maximum aggregate size a g in the following manner [26,38,40]: where f ' c is the concrete compressive strength (=36.8MPa).In this work, a g was taken as 10 mm, following the maximum aggregate size used in the experiment [17].
In ATENA, steel reinforcement can be modelled in two distinct forms: discrete or smeared reinforcement.In the discrete representation, the reinforcement is treated as a one-dimensional truss element, possessing axial stiffness only.The compatibility between the truss element and the solid element representing the concrete is achieved by implementing a kinematic constraint between the displacement at each end of the truss element and the surrounding nodal points in the solid element.In the smeared representation, the reinforcement is smeared over the volume of an element and hence, the ratio between the area of the reinforcing bar to the area of the considered mesh needs to be inputted.In this work, the discrete approach is adopted.
Fig. 4(d) indicates a schematic diagram of the stress-strain relationship for steel reinforcement which is adopted.For each reinforcing bar, the yield and ultimate strengths are inputted from the values reported in the relevant experiments.Moreover, the strain capacity of the steel reinforcement is taken as 0.1, which is a typical value in a nonseismic region.

Finite element mesh and boundary conditions
All beam specimens were modelled utilising an eight-node hexahedral element, as illustrated in Fig. 5, with a mesh size of 25 mm.In all specimens, the bond between the reinforcing bars and concrete was modelled by adopting the bond stress-slip recommendation provided in CEB-FIP MC1990 [32], following the procedures developed in [41].The slip at bond failure was taken as 0.6 mm [32], whereas the peak bond stress for bottom bars was taken as 12.1 MPa, and for top bars conservatively as 6.1 MPa accounting for the top cast effects [42,43].
The full beams were modelled with the reinforcement detailing in the right-hand side of the specimen following the NS-REF detailing, while the left-hand side adopting different reinforcement layouts (i.e., with omittance of either diagonal or U-bars, or the reduction of shear links) to allow for a direct comparison in crack patterns to a properly designed half-joint.The steel plates at midspan loading and at the supports were modelled utilising a tetrahedral element, with linear elastic properties assigned to prevent localised yielding.
The load was applied to the centre of the top plate as an incrementally imposed deflection of 0.3 mm per step until failure.Several monitoring points were setup; one at the underside of the beam at midspan to monitor the beam deflection, one at support on the underside of the nib to monitor the reaction load, and several monitoring points on the bars at the re-entrant corner to monitor maximum bar stresses.

Response of NS-REF half-joint
In this section, the response of specimen NS-REF is critically evaluated to explore the accuracy of the nonlinear finite element modelling implemented in ATENA in predicting the response of a half-joint beam.This specimen was chosen as it was the only specimen which was designed in accordance with the relevant codes and design principles.The predicted load-deflection response for specimen NS-REF is presented in Fig. 6(a), together with the predicted principal tensile strain and crack patterns at various increments of loads in Fig. 6(b).
Comparison between the predicted and observed responses presented in Fig. 6(a) indicates a good agreement.The initial loading closely follows the measured linear elastic response and there is a significant loss of stiffness thereafter, with both the predicted and observed load-deflection responses following an approximately straight line at a slope of only 15-20% of the initial stiffness.This near identical trend continues up to approximately 250kN, and the analysis then starts to produce a consistently higher load than that obtained experimentally, and this discrepancy continues at approximately the same magnitude up to the peak load.Regarding the series of results presented in Fig. 6(b), it is evident that crack formations after first cracking (i.e., between 80 and 150kN) are only limited to the re-entrant corner and at the bottom tensile region of the beam at midspan due to flexure.As the load increases to 400kN, flexural cracking within the full-depth of section progresses upward past the mid-region of the beam.Moreover, the formation of inclined cracking and near parallel cracking at the end of the full depth section are observed which are consistent with the experimental evidence [17].At loads greater than 400kN, the magnitude of the principal tensile strain at the re-entrant corner of the nib increases significantly, due to yielding of the reinforcement at the reentrant corner.Accordingly, there is only a small increase in load under further increasing deflection.The analysis predicts a maximum load of 419.2kN at a midspan deflection of 10.9 mm which is in general agreement with the measured peak load of 402.3kN at a midspan deflection of 10.2 mm [17].The failure mode is also correctly predicted, due to the diagonal crack that originates from the re-entrant corner and further extends upwards towards the top surface of the beam [17].The good correlation between the predicted and experimentally observed load-deflection response and crack patterns, along with the mode of failure, highlights the accuracy of the FE models in providing an accurate assessment of a half-joint beam.

Comparison of all half-joints
Fig. 7(a) presents the predicted and observed load-deflection responses for all specimens tested by [17], together with the predicted failure crack patterns and principal tensile strain in Fig. 7(b) and the crack patterns observed in the experiment in Fig. 8 [17].Overall, there is a good agreement between the solid lines representing the predicted Fig. 5. Typical finite element mesh used in the analysis displaying the detailing adopted for half-joint beam with NS-NU (left) and NS-REF (right) detailing.

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response and the dashed line representing the measured response.As summarised in Table 2, the average value of the experimental-topredicted load capacity is 1.04 with a coefficient of variation of 7.3%.The analysis is capable of reproducing the post-cracking stiffness reasonably well, while the accuracy in predicting the peak load varies depending on the reinforcement layout.
Regarding the load capacity and overall ductility of Specimen NS-ND, it is evident from Fig. 7(a) that the omittance of diagonal bars yields the lowest load carrying capacity and ductility amongst all four test specimens, which agrees with the experimental evidence.The analysis underestimates the measured load by ~ 15%, but correctly predicts diagonal tension cracking at the re-entrant corner as the primary mode of failure (see Fig. 7(b)).It is worth noting that the analysis can capture the subsequent reduction in the number of cracks developing when compared to the properly detailed half-joint (NS-REF), clearly indicating that the inadequate detailing in the nib hinders the overall progression of cracking in the full depth of the beam.Consequently, crack formations are predicted to occur primarily at the regions close to the re-entrant corner and at the bottom of the beam at midspan, which is consistent with the observed crack pattern.
With reference to the response of Specimen NS-NU, it is interesting to note from Fig. 7  carry the vertical force applied to the nib, which would, in turn, increase the nib capacity.The increased load-carrying capacity of NS-NU (compared to NS-ND) would indicate that the net effect of the latter is higher and would also highlight the importance of diagonal bars in the behaviour of half-joints.
Similar to the NS-ND specimen, the poorly detailed end-region in Specimen NS-NU is predicted to result in the formation of localised cracking at the re-entrant corner and limited cracking in the full-depth of section (see Fig. 7(b)).While the presence of the re-entrant corner cracking still governs the overall response of the beam throughout the loading process, the analysis predicts that the failure occurs due to the formation of a near vertical crack at the junction between the nib and full-depth section, which is in excellent agreement with the failure crack pattern obtained from the experiment (see Fig. 8).This is due to the reduced level of horizontal restraint at the re-entrant corner (discussed above).It is worth noting that this analysis case produces the most accurate representation of the observed load-deflection response.
Regarding the predicted response of NS-RS presented in Fig. 7(a), a good agreement is again obtained although the analysis exhibits a slightly larger post-cracking load than that obtained experimentally.Both the analysis and experiment show that there is a significant increase in load capacity than that of specimens NS-ND and NS-NU, which can be attributed to the proper detailing of reinforcement in the nib.However, due to the excessively large spacing between the shear links, the analysis shows that this leads to the formation of diagonal shear cracking at the bottom corner of the full depth section (see Fig. 7(b)) which is consistent with the experimental evidence.

Bar stresses and force distribution at the inner nib
Fig. 9 provides a visual representation of the magnitude and distribution of maximum bar stresses throughout the four half-joints.Predicted values of maximum bar stresses lie in the range 550-570 MPa, which is in good agreement with the 565-595 MPa range measured experimentally [17].The location of a high concentration of stresses displays a strong correlation with the location of cracking near the reentrant corner (compare Fig. 7(b), 8 and 9).In the NS-REF specimen, for example, the primary concentration of tensile stresses can be notably observed in the U-bars, diagonal bars, and the first two shear links from the re-entrant corner.A high concentration of stresses is also evident near the re-entrant corner of all other three specimens, with both NS-ND and NS-NU specimens exhibiting a more localised stress development, in particular NS-NU which displays the concentration of stresses on the diagonal bars and the first shear link only.In contrast, the concentration of stresses in NS-RS specimen extends up to the third link (i.e., the second shear link from the nib in this case), due to its intersection with the diagonal shear crack originating from the bottom corner of the full depth section.
The maximum bar stresses discussed above were then used to evaluate the contribution of each bar type to the overall load carrying capacity.This was done by multiplying the tensile stresses in the bar with the corresponding cross-sectional area to determine the bar forces.The total vertical forces were then computed based on the total contribution of the vertical component of the force in the diagonal bars (when present) and the first two shear links from the inner nib.Fig. 10(a)-(d) presents the evaluated total vertical forces against the applied shear load    for all four half-joint specimens.It is predicted that a large percentage of the applied load in the NS-REF specimen is carried by the diagonal bars, followed by the first and second shear links (see Fig. 10(a)).The contribution of the shear links is shown to increase when the diagonal bars have reached their elastic limit, from the applied shear of ~ 350kN.When the diagonal bars are omitted (see Fig. 10(b)), the first shear link from the inner nib is shown to carry the largest internal stresses, accompanied closely by the second shear link, along with a significant reduction in load carrying capacity.Furthermore, with the omittance of U-bars as in the case of NS-NU specimen, the diagonal bars are predicted to transfer the bulk of the applied load, and the contribution from the first shear link is shown to increase in magnitude until the peak loading is reached (see Fig. 10(c)).Almost all the applied shear is being transferred through these bars at peak loading.Finally, the reduction of shear links in the NS-RS specimen displays a similar distribution of forces amongst the diagonal bars and the first shear link (when compared to that of NS-REF displayed in Fig. 10(a)).The contribution of the third link becomes more apparent from the applied shear of ~ 340kN (see Fig. 10  (d)), after yielding of reinforcement at the re-entrant corner and formation of the diagonal shear crack that originates from the bottom corner of the full depth section.
To gain a deeper understanding of the internal mechanisms that are created to resist the applied load, it would be of significant value to evaluate the minimum principal stress distribution in the concrete, in addition to the internal tie actions created by the reinforcing bars.Fig. 11 presents the minimum principal stresses in the concrete at peak loading for all half-joint specimens.Inspection of the properly detailed NS-REF specimen highlights two major formations of compressive stresses in the concrete: (i) compressive stress path originating from the point of load application at midspan, running near horizontal along the top face of the beam and continuing downwards diagonally towards the support at the nib; and (ii) compressive stress bulb projecting from near the point of load application at midspan diagonally down towards near the bottom left-hand corner of the beam.
The NS-RS specimen displays a similar distribution of principal compressive stresses, but with a less distinguished diagonal compressive stress formation close to the nib (i.e., compressive stresses developing along path (i) are lower in magnitude).In contrast, specimen NS-ND only displays the distinguishable pattern of the top compressive stress from the midspan point of loading down towards the support and no diagonal compressive stress formation over the full depth of section is observed.It is worth noting in this case that the concrete in the nib and along the top region of the full depth of section are utilised as the load carrying mechanism in this specimen.Moreover, the NS-NU specimen displays a similar distribution of stresses, but with a weak formation of diagonal compressive stresses over the full depth of section, signifying the localised failure at the re-entrant corner of the nib as noted previously.The magnitude and location of the minimum principal stress distributions in the concrete and the maximum stresses in the steel provide the ability to place compressive struts and tension ties more accurately, thus fine-tuning strut-and-tie models which are adopted as standard practice in industry.Work is now continuing to develop a suitable physical model for half-joints which considers the effect of concrete cracking and deterioration mechanisms.

Influence of further reinforcement reductions
To further investigate the influence of reinforcement reductions in half-joints, combinations in bar reductions closest to the external faces of the beams were implemented.Reasoning for which is due to corrosion typically occurring on bars that are the closest to the surface.With respect to the diagonal bars, of which there are four in total, the two outer bars were removed.This was to replicate a scenario in which the extensive deterioration of the outer bars due to corrosion essentially deems their contribution to be negligible, and as such the ratio of diagonal bars was reduced by 50%.Regarding U-bars, of which there are three in total, the two outer bars were removed, similarly to the diagonal bars to replicate extensive deterioration, but with a reduction in reinforcement of ~ 67%.Moreover, with respect to the shear links, the extent in which the deterioration from the nib occurs is determined by the deterioration zone 3 proposed by [8], which extends into the full Fig.11.Minimum principal stress distribution in the concrete for all half-joints.

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depth of section beyond the first, second, and third shear links, respectively.Thus, the following bar reductions were implemented; HD and HU referring to half the number of diagonal and U-bars present; NS1 to 3 referring to the omittance of the first, second, and third shear links from the face of the inner nib, and various combinations of each.
Table 3 lists the specimens analysed with the corresponding peak loading and midspan deflection, along with a direct comparison to the reference (NS-REF) specimen.Moreover, Fig. 12(a) and (b) present the load-deflection response for all additional simulations.Inspection of Fig. 12(a) reveals a reduction in both peak shear and midspan deflection of 23% and 10% with half the number of diagonal bars.When reducing the number of U-bars from three to one, a reduction in peak shear and midspan deflection of 20% and 10%, respectively, is observed.With respect to the omittance of the shear links in chronological order from the re-entrant corner, the peak shear reduces as the number of omitted links is increased (particularly the first two), which highlights the importance of vertical reinforcement close to the re-entrant corner.It is worth noting that despite the total omittance of the first shear link from the face of the inner nib, a better response was obtained compared to the reduction of the number of diagonal or U-bars.
Regarding Fig. 12(b), a combination of local reductions and omittance of bars is shown to result in further significant reductions in both shear capacity and midspan deflection.When halving the number of both diagonal and U-bars, the shear capacity and midspan deflection of the half-joint is essentially halved, with a reduction of 45% and 24%, respectively (see Table 3).The combined reduction of diagonal bars and the omittance of the first two shear links, resulted in the peak shear and midspan deflection being 44% and 40% lower than the reference specimen, further signifying the importance of diagonal bars in the overall capacity of half-joints.Moreover, adopting a critical scenario with a reduction in diagonal and U-bars along with the omittance of the first and second shear links resulted in a significant reduction of shear and midspan deflection of 59% and 43%, respectively.

Concluding remarks
The work presented highlights the application of nonlinear finite element analyses to provide a critical investigation into the influence of reinforcing bar reductions on the response of reinforced concrete halfjoints.From the results presented, the following conclusions can be drawn: (1) Nonlinear finite element analysis adopting the smeared crackand crush-band approaches, along with incorporating fractureplastic models of concrete, provides a powerful platform to perform forensic analyses of reinforced concrete half-joints.Accurate predictions of load capacities were obtained throughout the simulation of four half-joints, with an observed-to-predicted ratio of 1.04 and a coefficient of variation of 7.3%.Moreover, comprehensive post-processing visualisation was found to be particularly useful to locate areas of distress at various stages of loading.(2) A reduction in the amount of reinforcement near the re-entrant corner has a significant influence on the response of reinforced concrete half-joints.The largest reduction was observed in a halfjoint with the omittance of the diagonal reinforcing bars, exhibiting a reduction in both load carrying and deflection capacities of approximately 50% of the reference specimen with no bar reductions.(3) All four half-joints displayed initial diagonal cracking at the reentrant corner, which is consistent with the experimental evidence.The angle and extent of this crack was found to vary depending on the reinforcement layout (i.e., extent of bar reductions near the re-entrant corner).This aspect could, therefore, be potentially used as an indicator for the structural assessment of deteriorated bridges comprising half-joint detailing.(4) An accurate assessment of failure modes was observed throughout the four half-joints.In all specimens, failures were preceded by limited yielding of reinforcement at the re-entrant corner of the nib, thus providing little warning of imminent failure.Furthermore, a reduction in the number of shear links resulted in a shear failure due to diagonal cracking originating from the bottom corner of the full depth of section.

Table 3
Comparison between a combination of bar reductions and reference specimen.(5) Limited cracking was observed in the full depth section of halfjoints with reductions of reinforcement near the re-entrant corner, due to premature failure of the nib.(6) In the four half-joints investigated, a large proportion of the applied shear at the re-entrant corner is carried by the diagonal bars, followed by the first two shear links from the nib.The contribution of the shear links is shown to increase when the diagonal bars (and shear links closer to the nib) have reached their elastic limit.(7) Half-joints comprising properly detailed reinforcement exhibit two major formations of compressive stress paths in the concrete, with one displaying a direct transfer of compressive stresses to the support at the nib, and the other projecting from the point of the load application towards the bottom left-hand corner of the beam.The omittance and/or reduction of reinforcing bars can alter the distribution of stresses in a reinforced concrete half-joint and hence the removal of one of the stress paths.The presence of the stress paths would also be influenced by both the distance to the point of loading and the depth of the member.(8) A combination of local reductions and the total omittance of bars is shown to result in further reductions of up to 60% and 40% in load carrying and deflection capacities, respectively when compared to the properly detailed reference specimen.
Reinforced concrete half-joints are susceptible to the ingress of water containing dissolved chlorides, yielding favourable conditions for corrosion [44][45][46].This can, eventually, lead to the reduction in crosssectional area of the steel reinforcement, cracking or even spalling of concrete, and loss of bond.In this work, the problem was simply considered as the reduction and/or omittance of some of the reinforcing bars embedded in the concrete.Further work is directed toward investigating the influence of synergistic deterioration effects of concrete and steel, and bond deterioration on the structural response of half-joints.

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.

Fig. 2 .
Fig. 2. Example of reinforced concrete frame structure adopting half-joints: (a) cracking in the end regions and in the nib forming the bottom supporting seat; and (b) schematic of frame structure and cantilevered span adopting half-joint.

Fig. 6 .
Fig. 6.(a) Comparison of predicted and observed load-deflection response for NS-REF half-joint; and (b) predicted principal tensile strain and crack patterns at different stages of loading to failure (only cracks larger than 0.1 mm are displayed).
(a) that the omittance of U-bars results in a higher predicted load-capacity than the NS-ND (~30% higher) but retains the same level of overall ductility (approximately only half of the NS-REF).Despite the omittance of U-bars, the increased capacity of NS-NU (compared to NS-ND) can be attributed to the presence of the diagonal bars which provide restraint against the diagonal tension cracking that propagates from the re-entrant corner.Referring to the bar diagram presented earlier in Fig. 3, the equivalent area of 4H12 diagonal bars along the horizontal direction at the re-entrant corner can be calculated as 4 × 0.25 × π × 12 2 × 0.68 = 307.6 mm 2 , with 0.68 being the ratio of the projected horizontal length of the diagonal bars from the bottom lefthand corner of the beam (115 + 120 + 115 = 350 mm) to the actual length of the diagonal bars from the re-entrant corner to the base (~515 mm).This equivalent area equals to ~ 90% of the total area of the missing 3H12 U-bars in Specimen NS-NU (3 × 0.25 × π × 12 2 = 339.3mm 2 ) and would indicate that almost the same horizontal internal tensile forces can develop at the re-entrant corner of NS-NU and NS-ND specimens i.e., the U-bars only provide an additional ~ 10% of horizontal resisting force at the re-entrant corner.The presence of diagonal bars in Specimen NS-ND, however, comes at an advantage as the vertical component of the force developing in these bars can also contribute to

Fig. 7 .
Fig. 7. (a) Comparison of predicted and observed load-deflection response for half-joint specimens with different bar configurations.(b) Predicted failure crack patterns (only cracks larger than 0.1 mm are displayed).

W
. Don et al.

Fig. 12 .
Fig. 12. Load-deflection response for a combination of bar reductions.

Table 2
Comparison of calculated and experimental results for all half-joints.