Renovation, restoration, remodeling, refurbishment, and the retrofitting of buildings often imply applying forces  to beams that before were subjected to distributed loads only. In the case of reinforced concrete structures, the new condition causes a beam to bear a concentrated load with the crack pattern that resulted from the distributed loads which had acted before. If the concentrated load is applied at or near the beam’s midspan, the new shear demand reaches the maximum where cracks are vertical or quasi-vertical, and where inclined bars are not common according to any standards. So, the actual shear capacity can be substantially lower than new shear demand due to the concentrated load. This paper focuses on reinforced concrete beams whose load distribution has to be changed from distributed to concentrated and presents a design method to bring the beam’s shear capacity up to the new demand. The method consists of applying fiber composites  with fibers at an angle of 45° bonded to the beam’s web. This kind of external reinforcement arrangement has to comply with some practical measures, which are presented as well. The paper also provides the analytical model that predicts the concentrated load-carrying capacity of a beam in the strengthened state. The model accounts for the crack’s verticality, which nullifies the contributions of steel stirrups, aggregate interlock, and dowel action, and for the effective bond length of each fiber, which depends on the distance between the ends of the fiber and the crack it crosses. There is a time in their service life at which constructions no longer meet modern architectural demand. At that time, the construction may be subjected to renovation, restoration, remodeling, refurbishment, or retrofitting in order to meet modern demand .Those activities may include placing new columns, walls, beams, or trusses into the building. Such new structural elements may be made supported by existing beams that previously carried floor slabs, which implies changing the load condition of a beam from distributed to concentrated. Moreover, the new concentrated load may be applied at or near the midspan of the existing beam.If the beam is made of reinforced concrete (RC), the new load condition may be demanding even if the new concentrated load is lower than the resultant force of the previous distributed load. The reason lies in the crack pattern around the midspan, which is composed of stabilized vertical or quasi-vertical cracks, given that they were generated by the distributed loads that acted on the beam. That crack pattern implies a weakness, which often entails strengthening the beam.This paper is devoted to those RC beams and presents a design method to increase the concentrated load-carrying capacity, including the relevant analytical model that predict the concentrated load-carrying capacity of RC beams both in the strengthened and unstrengthened condition.This section is devoted to framing the topic this paper deals with and to presenting the study’s statement of purpose, to which all elements of the paper relate logically.A lot of research work has been accomplished since the end of the nineteenth century on the shear behavior of RC structures, and outstanding papers are available in the literature.An essential advance in knowledge was the well-known analogy between the shear strength of a web-reinforced concrete beam and a flat truss, postulated by Ritter 1899) [899 [1], Another significant advance in knowledge was the strut-and-tie modeling, which divides a structure in B-regions (“B” standing for either beam or Bernoulli) and D-regions (“D” standing for either discontinuity or disturbed). The latter ones are in the vicinity of loads or geometric discontinuities, whereas the former ones are between the latter. Only the analysis and design of B-regions can proceed on a sectional basis for which plane sections remain plane, whereas the analysis and design of D-regions must proceed on a regional basis. Some of the many references are ,25,26,27Then, a lot of research activity has been devoted to studying beam action and arch action. Some of the many references are ,42,43,44The principal theoretical shear resisting system of an RC structure is an ideal parallel chord truss made of concrete and steel. The truss is composed of a top compression chord , a bottom tension chord , vertical tension ties , and 45° inclined compression struts (the concrete between the shear cracks).The joints of the members that compose the truss are assumed to be hinges. Accordingly, that system is called ‘pin-joined truss’. The pin-joined truss is statically determinate and stable. Consequently, the axial forces in the truss’s components can be calculated using the equilibrium only.The truss mechanism exists only after the formation of the shear cracks, which cause the diagonal tension stresses to disappear.This point merely reviews some well-known generalities about the cracking behavior of a simply supported RC beam with web-reinforcement, under uniform loading ,37,40,43A uniform load induces a two-dimensional state of stress in the beam, whose maximum principal stress causes the concrete to crack if it exceeds the concrete tensile strength.At midspan, the principal stresses are dictated by the bending moment only (which is true for the doubly fixed RC beam as well). Therefore, the principal stresses act on vertical planes and reach their maximum at the bottom of the cross-section. As a consequence, a crack at midspan initiates at the bottom face of the beam and propagates upwards vertically, up to the new compression zone in the upper part of the cross-section .From the midspan to the supports, the interaction between bending moment and shear force increases. However, cracks are still initiated by bending moment, unless the beam is particularly deep. Each crack hence starts from the bottom face of the beam and propagates upwards. Nonetheless, the shear force is not nil, so propagation is driven by both bending and shear. That combined action causes cracks to deviate from verticality . Deviation increases from around the midspan, where cracks are quasi-vertical, to around the supports, where cracks are approximately at 45°.At a support of a simply supported RC beam, the principal stresses are dictated by the shear force. Thus, the principal stresses act on planes at 45° and reach their maximum at the centroid. A crack due to shear force hence initiates at the centroid and propagates inwards at 45°, reaching the bottom edge of the beam (shear crack). Ultimately, the crack pattern of a simply supported RC beam exhibits vertical cracks at midspan, quasi-vertical cracks around the midspan, and progressively more splayed cracks toward the supports.At and near the supports of a doubly fixed RC beam, there is some interaction between shear and bending. As a result, cracks can be either vertical  or at 45° (shear cracks). There is, however, a much lower chance of cracks that initiate at the centroid of the cross-section (shear cracks) appearing around the supports, whatever the restraints.The intended structural use of the vast majority of beams is to bear uniformly distributed loads, which requires that the shear force around midspan is small or nil. Accordingly, web reinforcement around the midspan of the vast majority of RC beams consists of relatively widely spaced steel stirrups, while neither bent-up longitudinal bars nor inclined tension ties are present around the midspan (no standards prescribe them), unless the design loads include concentrated loads.The existing RC beams that this paper deals with were designed to bear uniform loads. So, the web reinforcement around the midspan has the above-described arrangement.If a concrete strut is intersected by a substantial crack, it does not transmit any significant compression force, since the concrete strut cannot close the crack that it intersects. This behavior implies that the ideal pin-joined truss is unstable.Ergo, the classical analogy, whose concrete struts are at 45°, is valid only around the supports, but it does not hold true at the beam’s midspan. Given that a strut cannot be intersected by a substantial crack, each strut has to coincide with the concrete between two consecutive cracks—namely, the inclination of struts cannot be chosen arbitrarily but is dictated by the crack pattern. Accordingly, away from a beam’s support, the classical pin-joined truss analogy (whose concrete struts are at 45°) must be replaced by a pin-joined truss whose concrete struts are progressively less splayed from a support towards the midspan. Namely, the inclination of the struts must increase from 45° to 90° from a support to the midspan. Such a pin-joined truss extends the validity of the analogy to a greater fraction of the beam’s span. However, at and around the midspan, even that pin-joined truss loses its validity. Around the midspan, indeed, this concrete strut is vertical, so it has the same inclination (and the same position too) as a stirrup . Ergo, around the midspan, even this pin-joined truss is unstable. Ultimately, neither the classical pin-joined truss analogy nor the modified pin-joined truss analogy justifies any shear capacity of the beam’s segment where cracks are vertical. Nevertheless, the pin-joined truss analogy is valid at and around the supports, where the shear capacity plays a substantial role in the structural capacity, and it loses validity only where the shear demand of distributed loads is small. So, that deficiency plays a substantial role only in the specific case that this paper is devoted to.Neither the fact that the pin-joined truss analogy does not hold true, nor the absence of web reinforcement automatically entails that an RC beam fails by shear ,33,39,44In the case of RC beams without web reinforcement, if cracking is initiated by shear force, the crack propagates inclined along the entire height, up to severing the beam. Ergo, without web reinforcement a shear crack causes the RC beam to fail—namely, the beam cannot bear the load that has cracked the concrete. Conversely, if cracking is initiated by the bending moment, the crack propagates either vertically or inclined along the section, up to stabilization (as long as the amount of tension reinforcement is adequate). Ergo, even without web reinforcement, flexural cracks and flexural-shear cracks allow a stabilized crack pattern to be reached—namely, the beam can bear the load that has cracked the concrete.The shear resistant mechanisms of an RC beam with flexural and flexural-shear cracks without web reinforcement are arch action and beam action ,42,43,44Arch action consists of an ideal concrete arch, given by the internal principal compression stresses. The shear capacity derives from the inclination of the internal compression, which is substantial at the supports but is negligible around the midspan . So, arch action provides the segment around the midspan with marginal shear capacity.Beam action triggers when the cracks divide the tension zone into blocks, each of which consists of the concrete between two consecutive cracks. A block is usually referred to as ‘concrete cantilever’, since it acts as a cantilever with the base at the compression zone of the concrete and the free end in the concrete cover, and with the bond force variation applied at the free end.Beam action mechanism can transfer the transverse shear force by means of the uncracked part of a cracked cross-section and can transfer the bond force variation between two consecutive cracks by means of the concrete cantilever.In more detail, a flexural or flexural-shear cracked section of an RC beam without web reinforcement resists the transverse shear force by means of tangential stresses acting on the uncracked concrete, boosted by dowel action and aggregate interlock. Thus, the shear force is equilibrated by shear stresses on the transverse sections of the uncracked concrete, on the faces of the crack, and in the longitudinal tension reinforcement that crosses the crack. However, the tangential stresses acting on the uncracked concrete meet their limit when the principal tension stress reaches concrete tensile strength, which entails that this failure mode is brittle. At failure, consequently, the transverse relative displacements between the crack faces are small. Therefore, dowel action and aggregate interlock boost no more than moderately the shear transfer across the uncracked concrete zone of a cracked cross-section.An RC beam without web reinforcement resists the bond force variation between two consecutive flexural or flexural-shear cracks by means of the bending, axial, and shear strength of the concrete cantilever, boosted by dowel action and aggregate interlock. Thus, the bond force variation is equilibrated by normal and shear stresses on the transverse sections of the concrete cantilever, and by shear stresses on the crack faces of the two adjacent cracks and in the length of the longitudinal tension reinforcement that cross those cracks.The concrete cantilever meets its limit when the principal tension stress reaches concrete flexural strength, which requires that this failure mode is relatively ductile . However, the greater the angle of the cantilever to the beam’s axis, the lower the transverse relative displacements between the faces of the crack and, in turn, the lower the contributions of aggregate interlock and dowel action.At failure, consequently, shear stresses transferred across flexural cracks due to aggregate interlock are no more than marginal, and dowel forces transferred across the cracks due to flexural stiffness of longitudinal reinforcement are negligible, whereas across flexural-shear cracks, those stresses are large, and those forces are significant.In an RC beam with web reinforcement (with steel stirrups), beam action still takes place and acts in the same way as in an RC beam without web reinforcement, but more intensely, as stirrups make crack widths smaller, which boosts aggregate interlock, and make the concrete cover stronger, which boosts dowel action. In particular, beam action not only takes place for shear cracks too , but also, aggregate interlock and dowel action provide the cantilever with even greater contributions than for flexural-shear cracks due to the lower crack’s inclination angle to the axis.However, for flexural cracks of RC beams with stirrups the contributions of aggregate interlock and dowel action are negligible, as in the case of beams without stirrups.Ultimately, around the midspan, where cracks are vertical , the transverse relative displacements of the faces of a crack are negligible, and consequently, the contributions of aggregate interlock and dowel action are negligible as well, independently of the stirrups.The conclusion that can be drawn relevant to the issue being addressed in this paper is that shear strength around the midspan of an RC beam derives only from the uncracked part of the cracked cross-sections, and from the concrete cantilevers . As a result, shear strength around the midspan of an RC beam is small. However, a concentrated load that is applied around the midspan induces the maximum shear force in that zone.According to the conclusion of Increasing the shear strength of a beam’s segment with vertical cracks requires referring to a mechanism different than the pin-joined truss. In fact, neither the classical analogy nor the truss model with variable strut inclination is able to predict shear strength where cracks are vertical. Moreover, not even aggregate interlock and dowel action mechanisms can be referred to, as they provide no more than marginal contributions where cracks are vertical.Unfortunately, shear strengthening techniques that can be borrowed from the literature are based on the pin-joined truss and/or aggregate interlock and dowel action. Such strengthening techniques and the relevant models do not suit shear strengthening around the midspan, where cracks are vertical or quasi-vertical. That is the gap in knowledge that this paper aims to fill.The focus of the present research activity was to define a method for increasing the concentrated load-carrying capacity of RC beams, including the analytical model that predicts the capacity in the strengthened state.The RC beams considered here are existing. So, they are cracked, and cracking is stabilized. Moreover, they do not have any reinforcement purposely designed for resisting concentrated loads around the midspan . The content and results of this paper, including the analytical model, apply to every RC beam, except for the beams embedded in the floor slab . The reference beam, which is introduced in The main traditional shear strengthening techniques for RC beams are steel plate bonding , additional steel bars , and section enlargement. However, those techniques are complex to install and expensive . Moreover, they may be heavy and aesthetically unpleasant ,47,48,49An innovation of the nineties of the past century was externally bonded fiber composite systems ,67,68,69FRP and FRCM systems for strengthening RC structures are an excellent alternative to the above-mentioned traditional strengthening techniques, as the former offer substantial advantages over the latter: FRP and FRCM systems are easy to install, lightweight, very thin, noncorroding, and relatively economical.In the case of FRP systems, supplemental externally bonded reinforcement for shear strengthening consists of either continuous textile fabric sheets or discontinuous fiber strips, whereas in the case of FRCM systems, it consists of continuous textile fabric sheets ,71,72.The configuration that reproduces the stirrups would consist of strips or sheets totally wrapped to the concrete section (fibers that envelop the cross-section). However, the cross-section (T-beam) or the floor slab (rectangular beam) implies that wrapping the beam with closed external reinforcement require breaking the flange or the floor slab, which is particularly invasive and expensive. Actually, wrapping the entire cross-section with the FRP or FRCM jacket is much less convenient than inserting new bars into the concrete, totally embedded from the top face to the bottom face of the beam, with the inclination that is wanted.It follows that the typical shear strengthening configuration is the U-shaped reinforcement with the ends at the bottom of the beam’s flange or of the floor slab. Accordingly, shear strengthening of RC beams with external reinforcement is typically obtained by bonding either a continuous sheet with vertical fibers or spaced vertical strips onto the web  of the beam.Unfortunately, U-shaped reinforcement cannot act as the web tension member of the pin-joined truss, since the external reinforcement does not reach the compression zone (top chord) or is not sufficiently anchored to the top chord. Namely, U-shaped reinforcement does not give rise to the vertical tension ties.The topic of beams whose shear capacity is not enough or without web reinforcement needs in-depth analysis in order to frame the issues of beam’s shear strengthening around the supports (this section) and of midspan’s shear strengthening (next section). Although the first issue is well-known, the latter one is part of the gap that this paper aims to fill.It is barely the case to observe that, if shear capacity is not enough, the beam with web reinforcement behaves as the beam without web reinforcement, since the stirrups have already provided the maximum force.V, (2) and the bond force variation between two consecutive cracks , which is denoted by ΔsT.The variation of bending moment along an RC beam implies two internal forces, namely: (1) the transverse shear force across a cracked cross-section (across the concrete compression zone and crack), which is denoted by According to V that a cracked cross-section can bear is dictated by concrete tensile strength. When the maximum principal stress in the compression zone  reaches concrete tensile strength, the cross-section reaches the maximum V that it can bear. A greater V initiates an oblique crack, which propagates inwards up to reaching the top of the beam in a direction, and the apex of the crack in the other direction and results in the severing of the beam into two parts.(1) The maximum (ultimate) sT that a concrete cantilever can bear is dictated by the bending strength, since the shear strength of the cantilever is almost always greater . When the maximum principal stress at the built-in section of the concrete cantilever reaches concrete flexural strength, the cantilever reaches the maximum ΔsT that it can bear. A greater ΔsT breaks the built-in transverse section of the concrete cantilever, which causes the cantilever to fail and the beam to collapse by horizontal sliding of the lower part with respect to the upper part.(2) The maximum (ultimate) ΔV or the maximum (ultimate) ΔsT that the beam can bear is the ultimate load.The lowest value of the load that makes the beam reach either the maximum V and/or the maximum ΔsT that the RC beam can bear.Generally speaking, hence, increasing the shear capacity of an RC beam requires increasing the maximum V that a cracked cross-section can bear is obtainable only by wrapping the external reinforcement around the entire cross-section, which is an inconvenient technique, as previously observed. Actually, no U-shaped reinforcement allows the maximum tolerable V to be increased, since the uncracked concrete (compression zone) lies entirely, or almost entirely, in the flange, whereas the external reinforcement is bonded only on the web .Using external reinforcement, an increase in the maximum V that a cracked cross-section can bear to be increased. Namely, external reinforcement does not work at the support, let alone around the midspan.Ultimately, no external reinforcement allows the maximum transverse shear force V is greater than the load that makes the beam reach the maximum ΔsT that can be carried by the RC beam.Nevertheless, in the vast majority of RC beams, the load that makes the beam reach the maximum sT and not by the transverse shear V.Ultimately, shear strength is usually dictated by the bond force variation ΔsT that a cantilever can bear, since the maximum ΔsT occurs at the support and the fibers cross the cracks at the support with an angle different than zero.U-shaped external reinforcement can increase the maximum bond force variation ΔsT that a cantilever can bear to be increased, which, in turn, allows the shear strength of an RC beam to be increased.In conclusion, typical shear strengthening techniques with externally bonded reinforcement—either sheets with vertical fibers or vertical strips bonded on the web (U-jacket configuration)—allow the maximum bond force variation ΔThe conclusion of U-shaped reinforcement is vertical, so around the midspan, that angle is zero, since cracks are vertical too. Thus, U-shaped reinforcement does not provide any increase in shear strength around the midspan, just as a steel stirrup does not, and for the same reason a.For the sake of completeness, U-shaped external reinforcement (or steel stirrups) would act as longitudinal reinforcement bonded onto (or embedded into) the concrete cantilever. Nevertheless, that behavior strengthens the built-in end section of the cantilever no more than marginally, since the cantilever’s cracking moment is greater than the cantilever’s plastic moment . Thus, that contribution can be ignored.sT tolerable by a vertical cantilever to be increased. Ultimately, no typical U-shaped reinforcements—neither sheets with vertical fibers nor vertical strips—allow the maximum bond force variation ΔsT that a vertical cantilever can bear, the reinforcement has to provide the concrete cantilever with an oblique force. Ergo, the fibers of the web reinforcement have to be inclined  at a lateral face and an erroneous angle of +45° (or −45°) at the opposite lateral face . In doinIn other words, the angle of the fibers on a lateral side of a web must be opposite to the angle of the fibers on the opposite lateral sides of the web. Namely, fibers on the two sides are parallel. So, fibers cannot be continuous from one side to the opposite side of a web.Ultimately, two different fabric sheets must be bonded onto the two lateral faces of the part of the beam on the left of the concentrated load, and other two different fabric sheets onto the part of the beam on the right, with fibers angle whose sign is opposite the sign of the opposite fibers.k), the sheets overlap each other for the length λ > 2·0.707·effL. The anchorage length (bond length) of an FRP sheet at the bottom face of the cross-section is λ’ > 2·0.707·effL. At the bottom face, a sheet overlaps the sheet attached onto the other lateral face of the concrete section (unless the web is particularly wide). The figure also shows the anchorage length fL of a generic fiber.At the cross-section where the concentrated load is applied , a sheet has to be turned up onto the opposite lateral side .The method proposed in this paper to increase the concentrated load-carrying capacity of RC beams is that described above. It is represented in On one hand, the content of this paper is general. On the other hand, a reference structure saves the potential readers to invest too much effort in understanding the paper without losing generality.L, the distance between the concrete extreme tension fiber (edge) and the center of the longitudinal bars  by t, and the effective height by d.The basic reference structure is the beam diagrammed in The cross-section of the reference beam is constant along the span (prismatic beam), and the restraints are symmetric. The reference beam has either tee cross-section (T-beam) or rectangular section with depth greater than the depth of the floor slab it supports. Hence, the model directly includes not only the simply supported beam of P (a force) applied at a point K around the midspan . The position of point K is defined by β, i.e., the abscissa of K from the left support is β·L. The position of K is fixed, i.e., the new loading demand defines it (mobile load are not included).The reference beam was subjected to a distributed load .The total thickness of the external reinforcement bonded onto one lateral face is denoted by udP from the equation of vertical translational equilibrium ). So, modeling is directed at predicting the resisting shear udV, where the first subscript indicates that it is the ultimate (maximum) transverse shear force which a cross-section can bear without triggering the shear failure mode and the second subscript indicates that it is the design shear force of the RC beam . Shear failure mode occurs by the bending failure of the shadowed concrete cantilever of The model derives the concentrated load-carrying capacity Mmax = maximum bending moment in the beam due to udP and q (whereas the role played by q in producing Mmax is negligible compared to that of udP), which occurs at the distance β·L from the support that provides the beam with the maximum vertical reaction  does not imply losing generality, since, in the marginal cases where the distributed load Classical research about crack patterns of RC beams ,74,75,76Combining assumptions (3) and (4) gives:The last assumption that is made is about crack faces. Each face of a crack is assumed to be plane and to remain plane when loads make crack open, even when the two faces of a crack are tied  by externally bonded reinforcement.Ft of each layer is very small.According to the above assumption, the crack profile is always triangular. This is a realistic assumption as well, since axial stiffness of fiber composite reinforcement is small, given that the thickness This section describes the mathematical development of the model, i.e., shows the equations and their derivation, providing explanation.F, has the direction of the fibers, i.e., ±45°.The side-bonded fiber composite reinforcement ties (sustain) the crack. The opening of a crack thus makes the fibers stretch . Fiber eSince strains in the fibers are induced by crack opening, the strain profile of the external reinforcement at a crack is equal to the crack opening profile. By virtue of the last mechanical assumption , the strεFmax .F is produced by the triangular area of the above-described strain profile. In order to calculate F, two provisional assumptions are made, which will be removed in the following. The first is that there is no bond-slip between externally bonded reinforcement and concrete . As is well-known, ‘bond-slip’ is an oxymoronic term that means the relative movement of the reinforcement with respect to the surface it is attached onto or embedded into.The force The second provisional assumption is that the external reinforcement coats the entire crack depth.F turns out to be:Under those provisional assumptions, F provided by Equation (6) is the upper bound of the force, not the actual value. The reason why the actual value of F is lower is that there is some slip between the side-bonded reinforcement and the concrete lateral face it is attached onto  up to the loss of adhesion between side-bonded reinforcement and concrete .Failure can initiate at one of the ends of the side-bonded reinforcement or at the crack. According to , the foreffL .The strain that causes end debonding to occur depends on the bond length of the fiber. The greater the bond length, the greater the strain at which failure by end debonding occurs. However, end debonding strain can increase only to a certain limit. Any external reinforcement is characterized by an ‘optimal bond length’, which is the length, if exceeded, having no increase in the force transferred between external reinforcement and concrete. The optimal bond length is here denoted by Fd. The maximum strain that a given external reinforcement can reach when bonded to a given surface is herein called ‘full end-debonding strain’ and is denoted by εeffL and εFd can be borrowed from the literature, in particular by codes [ctdf is the design value of the concrete cylinder tensile strength, and cdf is the design value of the concrete cylinder compressive strength.Both by codes ,71,72. TFE and F-tott individually, but their product. On one hand, FE and F-tott (as well as Ft) are hard to define and difficult to measure individually. Actually, they are ideal quantities borrowed from continuum mechanics, whereas fibers and textile fabrics are discontinuous. So, those quantities are fictitious. On the other hand, however, their product is the axial stiffness of the reinforcement, which is a quantity clearly defined and easy to be measured.Equations (7) and (8) do not incorporate Equations (7) and (8) are written in a form that allows them to be used for collapse analysis, since the material strength values plugged into those equations (can) include partial factors for material properties to be used in the ultimate limit state verifications, as indicated by the subscript ‘d’.effL, the lower the maximum strain that the external reinforcement can reach with respect to εFd.Due to bond-slip, the shorter the actual bond length with respect to Each fiber of the external reinforcement spans diagonally across the lateral side of the beam’s web  and contUltimately, every fiber ranges from the bottom face of the flange to the bottom side of the beam .Each fiber of the FRP reinforcement crosses a crack (infrequently two cracks). A fiber is hence anchored at the two sides of a crack, at a certain distance from the crack.FL, is binding for the end-debonding. Consequently, FL dictates the force that the fiber can provide the crack with. Namely, if and how much the force is lower than the maximum value that can be transferred between external reinforcement and concrete depends on FL compared to effL.As explained at the beginning of this subsection, the force that a fiber provides a crack with also depends on bond-slip, which, in turn, depends on the length of anchorage of the fiber (bond length). The bond length of a fiber is equal to the distances between the end of the fiber and the intersection between fiber and crack. However, a fiber has one anchorage at each side of the crack, so it has two anchorages and then two bond lengths. In general, the two bond lengths are different from one another. The shortest of those two bond lengths, which is denoted by FL ≥ effL, where effL is provided by Equation (7), fiber debonding strain can reach the maximum value that it can, i.e., εFd, for a given external reinforcement and bonding surface—namely, that provided by Equation (8)—whereas for FL < effL, fiber debonding for strains is lower than εFd, and the extent of this depends on effL − FL.Ultimately, for FL ≥ effL, Equation (6) would provide the force in the side-bonded reinforcement at a crack plugging εFd for FL greater than effL, the fibers can reach the full end-debonding strain, and F can be obtained using that resisting strain as maximum strain.If every fiber guaranteed FL ≥ effL is more important at the bottom edge, where the strain is at maximum .Obviously, the requirement FL ≥ effL is obtained by bonding each fabric sheet onto the bottom face of the concrete section, so that the two side-bonded reinforcements overlap each other, as detailed in effL, each fabric sheet has to be turned up onto the opposite lateral side.For the bottom end of the external reinforcement, the requirement FL ≥ effL cannot be guaranteed, since the flange (or the floor slab) often prevents the fabric sheet from being prolonged beyond the crack apex for a length equal to the anchorage (and sometimes even from reaching the apex). Thus, the model has to account for that behavior.For the top end of the external reinforcement, the requirement effL.The effective bond length of the external reinforcement is different from the effective bond length of the fiber, because the latter is along the fiber, whereas the former is along the height of the web. Namely, due to the 45° angle of the fibers, the former is lower than the latter. More specifically, the effective bond length of the external reinforcement is equal to 0.707·The side-bonded reinforcement coats the entire vertical crack only if:ded onto .effL < ξ’, at and near the crack apex, the fabric sheet has a bond length that does not allow the fibers to reach the full end-debonding strain, but only a fraction of it. In that case, the crack is not completely tied (stitched) by the side-bonded reinforcement. The greater the difference between the right term minus the left term of (9), the lower that fraction.If μ’ − 0.707·F is not produced by a triangular stress profile but by the area of the trapezoid inscribed into that triangle. The value of F is hence lower than that provided by Equation (6).In that case, FL.The value of μ’ depends on the depth of the flange (or of the floor slab). The effective depth μ of the side-bonded external reinforcement is therefore  μ = μ’ −The analytical model refers directly to μ = ξ’ and account for the actual value of μ indirectly, using a reduction factor η. This research work obtained η analytically. However, the analytical expression is cumbersome. All the more, η has a secondary weight on the results. With all that considered, a parametric analysis was carried out, so as to obtain approximate values of η. Those values are shown in The strengthening design should guarantee μ/ξ’ ≥ 0.20. Ratios lower than 0.20 should be avoided, because they would imply that only a minor part of the external reinforcement is well-bonded. Equation (6) can hence be replaced by:F.Bonding the external reinforcement onto the cracked concrete surface poses another bond problem, which needs solution—namely, intermediate debonding caused by flexural crack. A crack entails a discontinuity in the strain field of the concrete, which causes the reinforcement around a crack to detach. The point is whether and how much that detachment reduces dL denote the length of side bonded reinforcement that has detached at each side of a crack  transforms that equation into :(11)F = dL < 80 mm [dL = t, Equation (11) turns into:Observation has proven that 20 <  < 80 mm . On the F from the neutral axis is 2·ξ/3 and the distance of F from the steel reinforcement is ξ/3.The distance of the force FdF denote the maximum value that the force F can reach. The force FdF is dictated by end debonding:Let Fd and FdF occurs at the crack that transmits the maximum bending moment.The side-bonded reinforcement elongates in proportion to the crack width. So, εsT induces a bending moment ΔsT·ξ at the built-in end of the vertical concrete cantilever . The longitudinal bond force variation Δntilever . The glosT and the shear action V in the cracked section:Once the internal lever arm is fixed, the sole equilibrium furnishes the relation between ΔF in the side-bonded reinforcement varies along the span, since it is in proportion to the width of the crack that it ties .The force  it ties . In turnF and sT reach their maximum at midspan, and that those forces decrease from the midspan to the supports. It follows that the variation of F at the two sides of the concrete cantilever induces a bending moment in the built-in end of the concrete cantilever that has the same sign of the bending moment induced by the variation of sT. This fact would seem to invalidate the proposed strengthening technique.The above sequence of implications leads to the conclusion that sT at the two sides of the concrete cantilever (one at a crack and the other at the consecutive crack) are coaxial. Namely, the lever arm between those forces is zero. On the contrary, the two forces F at the two sides of the cantilever are inclined, so the lever arm between those forces is not zero.Nevertheless, the two forces F applied to a concrete cantilever, one force at the right side and the other force at the left side .FdF at the lateral side of the cantilever closer to the midspan and F’ at the lateral side more distant from the midspan (with F’ < FdF). Tforce F’ will be defined in the following. The inclination of forces FdF and F is +45° , and bending moment in the concrete cantilever. The section of the concrete cantilever that dictates failure is the built-in end (the section of the cantilever that is fixed to the compression zone of the beam).The forces eN, the transverse shear force eT, and bending moment eM given by the following expressions Ne=eT of Equation (15) and eM of Equation (16) are resisting internal actions—namely, they make tension stresses due to ΔsT lower. So, eT and eM belong to the resisting system. On the contrary, the tension axial force eN of Equation (14) belongs to the force system (and not to the resisting system), since this induces tension stresses in the concrete cantilever, as does ΔsT. In other words, with all the other parameters being the same, the greater eT and eM, the greater the increase in concentrated load-carrying capacity provided by the externally bonded reinforcement. On the contrary, the greater eN, the lower the increase in concentrated load-carrying capacity provided by the externally bonded reinforcement.This is to highlight that sT and the axial force, shear force, and bending moment due to the external reinforcement. Eventually, the internal actions of Equations (14)–(16) allow the RC beam to carry greater concentrated loads around the midspan.Ultimately, the internal actions in the vertical concrete cantilever are the shear force and bending moment due to the longitudinal bond force variation ΔsT is constant along the span, since V, α, and d are constant. The effects induced by ΔsT are thus the same in each vertical concrete cantilever. It follows that failure is dictated by the concrete cantilever where the effects induced by Equations (14)–(16) are minimum.Equation (13) shows that the modulus of ΔF acting on a cantilever are minimum, and vice versa. Since F is in proportion to crack width, and crack width is in proportion to the bending moment, that minimum occurs at the vertical  concrete cantilever that is farthest from point K , and vice versa.The bending moment of Equation (16) reaches its minimum where the two forces F acting on a cantilever are maximum, and vice versa. Again, F is in proportion to crack width, and crack width is in proportion to the bending moment, so that maximum occurs at the vertical  concrete cantilever that is nearest point K , and vice versa.The tension axial force of Equation (14) reaches its maximum where the two forces eN is substantial. As a result, the combination between eN and the bending moment eM induces the lowest effect in the vertical cantilever where eN is at maximum, although in that cantilever, eM is at maximum too.The tension axial force sT occurs in the vertical cantilever at point K or nearest point K . Ergo, failure is dictated by the concrete cantilever at point K (or that is nearest point K). If the concentrated load is at midspan, as in the reference beam, failure is dictated by the concrete cantilever at midspan.To sum up the above analysis, the minimum compression stress that reduces the maximum tension stress induced by ΔDebonding is a very brittle failure mode. The classical U-shaped external reinforcement used to strengthen in shear RC beams around the ends  debonds before dowel action and aggregate interlock have given substantial contributions. As a result, the actual increase in shear strength of the beam provided by the classical external reinforcement is moderate.According to The possible failure modes of an RC beam with side-bonded reinforcement whose configuration is that described in FdF and F’, which results in a reduction in the resisting internal actions of Equations (14)–(16) to zero, which, in turn, causes the cantilever to fail. In fact, without those resisting internal actions, the concrete cantilever cannot resist the bending moment that is induced by the steel reinforcement at its built-in end. Once the cantilever fails, the whole beam fails.Failure triggers because the side-bonded reinforcement stops providing the concrete cantilever with the forces Fd. 3. To fail is the vertical concrete cantilever whose lateral side at (near) the midspan is a face of the crack where the external reinforcement exhibits Fd. 4. Failure occurs at the built-in  section of that cantilever. 5. Debonding causes that cantilever to lose the forces exchanged with the external reinforcement, and without those resisting forces, the cantilever cannot equilibrate the bending moment induced by the variation of longitudinal force in the steel bars . 6. When the cantilever fails, the beam collapses.To sum up: 1. the capacity of carrying a concentrated load applied around the midspan is dictated by the debonding of the external reinforcement, since the concrete cantilever, although not slender, fails by bending moment, and moreover, it displays some plasticity. 2. Debonding occurs when the maximum strain in the external reinforcement, sT is resisted by two bending moments acting on the built-in section—namely, the resisting bending moments provided (a) by the forces in the side-bonded reinforcement, and (b) by the stresses acting on the built-in section of the cantilever.At debonding, the bending moment produced by ΔFdF and F’ and their lever arm, Ω. The former force is already known, i.e., it is given by Equation (12), whereas the latter force is calculated in the following. As previously detailed (FdF and F’.(a) The resisting bending moment provided by the forces in the side-bonded reinforcement derives from detailed , the res(b) The resisting bending moment provided by the stresses acting onto the built-in horizontal section of the cantilever requires knowing the flexural behavior of the concrete cantilever at debonding of external reinforcement.Theoretical analysis carried out within this research activity demonstrates that at debonding, the maximum tension strain in the concrete cantilever is beyond the elastic limit, although always lower than the cracking tensile strain. Accordingly, the stress distribution on the concrete section is elasto-plastic.ctdf. That assumption is simultaneously realistic and conservative, because the bending moment given by the stresses that are ignored is small and would increase the strength.Modeling assumes that stresses acting onto the built-in section at debonding have a bi-triangular profile, with maximum equal to the design value of the concrete cylinder tensile strength cm denote the maximum tension stress in the concrete cantilever. The stress σcm occurs at one edge of the built-in end of the cantilever (the edge more distant from the midspan), as shown by Let σcm is equal to:W is the section modulus of the built-in end of the concrete cantilever.The bi-triangular stress profile, which implies a linear-elastic behavior of the concrete cantilever, requires that σPlugging Equation (13) into Equation (17) leads to:On substituting Equation (4) for α and ξ, Equation (18) becomes:Rearranging terms, Equation (19) can be expressed in the following form:K or nearest point K . Accordingly, the vertical face of the concrete cantilever at point K (at midspan) is subjected to the maximum bending moment Mmax, which causes the side-bonded reinforcement at that face to reach the full end debonding strain εFd.According to K (from the midspan) is subjected to a bending moment, which is denoted by M’. That bending moment derives from the variation of M between the two consecutive cracks that define the boundaries of the concrete cantilever. The moment M’ causes the external reinforcement to reach a maximum tension strain, which is denoted by The lateral face of that concrete cantilever more distant from point M’ = Mmax − α·udV.The first modeling assumption allows the following relationship to be established: Mmax, M’, and εFd are known, the strain Given that Modeling assumptions allow Equation (21) to be written in the following form:Again, modeling assumptions allow Equation (22) to be rewritten:F’ in the external reinforcement at the side of the concrete cantilever more distant from the midspan derives from the triangular area of the strain profile whose maximum is The force cm in the concrete cantilever at debonding of the external reinforcement is equal to the concrete tensile strength ctdf. Plugging ctdf into Equation (20):According to udV. In so doing, the ultimate shear force is found:Equation (25) may be solved for Equation (26) may be expressed in the following form:W to be expressed as a function of the geometric parameters. On substituting α into the formula of the section modulus W (rectangular section): W = 0.074·b·d2. On substituting the expression of W into Equation (27):Modeling assumptions represented by Equations (3)–(5) allow the section modulus Equation (27) can be rewritten as:The sum and the difference of the terms into the three round brackets can be obtained plugging Equations (12) and (24) into each round bracket. In so doing, the round bracket with the minus sign turns out to be:Equation (30) can be simplified into:In the same fashion, the round bracket with the plus sign turns out to be:Equation (32) can be simplified into:Plugging Equations (31) and (33) into (29):ud,V to be finally worked out:Equation (34) allows the ultimate design resisting shear, ud,P is:udV is given by Equation (35).Equation (35) permits the maximum concentrated load that the strengthened RC beam can carry to be derived. The final formula that predicts the concentrated load-carrying capacity of a strengthened RC beam, Ultimately, Equations (35) and (36) constitute the model that allows the RC beams strengthened in the fashion described in The model is based on some assumptions, which, on one hand, are realistic, but on the other hand, suggest checking the actual predictive capacity of the model. Checking was accomplished by comparing the theoretical predictions to experimental results obtained from laboratory tests on four RC beams . The four beams were identical, i.e., they had the same geometry and reinforcement and were made of the same concrete. The beams were rectangular. The height of the cross-section was 200 mm, and the width was 600 mm. Each specimen was simply supported over a span of 3900 mm .The steel reinforcement was designed in order to tie the cracks the less that was possible. Namely, the longitudinal steel bars that were placed allowed vertical cracks to attain large widths.As explained below, the cracks were obtained applying to each beam two equal forces significantly greater than the theoretical forces that implied cracking of the concrete. Given that the greater the tension strain in the longitudinal steel reinforcement induced by a load, the greater the widths of the cracks that are produced by the load, the flexural steel reinforcement was reduced to an essential level. The longitudinal steel reinforcement consisted of bars with the minimum number and diameter necessary to make the cage and to cast the concrete, and the flexural reinforcement necessary to carry the loads was obtained with FRP external reinforcement.On the contrary, the stirrups were made of steel bars with a particularly large diameter, in order to confirm that they play virtually no role in providing shear strength where cracks are vertical.In geometrical terms, the steel reinforcement consisted of four longitudinal bars with diameters of 5 mm (two at the top and two at the bottom) and stirrups with diameter of 16 mm at the spacing of 130 mm .The reinforcement that provided the beam with bending strength was obtained by epoxy-bonding longitudinal FRP strips onto the entire lower face of the beam (three layers).As mentioned above, the beams underwent a preliminary procedure devoted to producing vertical cracks around the midspan. Every beam was subjected to two concentrated vertical forces, each one applied 0.975 m apart the midspan . The magnitude of each force was 31.50 kN. It is of note that the theoretical value of the two forces that caused first cracking was 17.29 kN, and the collapse value was 36.50 kN.Then, those two concentrated forces were taken away . The result was the obtainment of four RC beams with quasi-vertical cracks for about 1950 mm around the midspan.Then, two beams were strengthened in the fashion described in Lastly, a concentrated load was applied at midspan of each beam, and it was increased up to failure. So, two beams were subjected to a collapse test without side-bonded reinforcement, and the other two beams were subjected to a collapse test with the side-bonded reinforcement described in N = 6.The side-bonded reinforcement consisted of three layers of textile fabric per lateral face of the beam. Namely, The side-bonded reinforcement used for the tests consisted of an FRCM material. FRCM reinforcement was preferred to FRP reinforcement since debonding of FRCM consists of fibers that slip with respect to the matrix they are embedded into, whereas debonding of FRP consists of the rupture of a layer of concrete cover. For that reason, debonding of an FRCM reinforcement is more controllable in a laboratory test than debonding of an FRP reinforcement. The strain of the FRCM reinforcement at debonding was accurately measured, whereas the debonding strain of an FRP reinforcement could not have been measured so accurately. Moreover, the debonding strain of an FRCM reinforcement can be better predicted that debonding strain of an FRP reinforcement.Fd that had been predicted before executing the tests was in the range 2.75–2.85‰, while the average debonding strain that was measured in the two tests of the strengthened beams was 2.801‰. Moreover, the two experimental debonding strains  differed marginally from one another. Those evaluations were confirmed by the experimental campaign. The FRCM debonding strain εFt = 0.177 mm and FE = 244,000 N/mm2. The characteristics of the external reinforcement as standalone component were: FL ≥ effL both at the top and the bottom of each sheet. That condition is represented by η = 1 of The side-bonded external reinforcement was anchored onto the top and bottom faces of the beam. Accordingly, the side-bonded reinforcement guaranteed 2 and 4.2 N/mm2, respectively.The concrete compressive strength and tensile strength measured by cylinder tests were 41.6 N/mmUltimately, the one-point load experiments tested up to collapse four beams made of concrete with the ultimate stresses in tension and compression reported above, and with vertical cracks around the midspan. Two beams were tested without side-bonded reinforcement and two beams with the side-bonded reinforcement described above.The initial two-point load procedure and the final one-point load test used high-tensile steel cables and actuators. The two-point load procedure applied the two forces with four prestressing tendons and four pistons (one high tensile steel cable and actuator per side and per point) to develop two constant vertical forces that caused the beam to crack by pure bending moment and the cracks to open widely. The one-point load test applied the force with two prestressing tendons and two pistons (one high tensile steel cable and actuator per side and per point) to develop an increasing vertical force up to collapse . Failure of the two beams without the side-bonded reinforcement was triggered by the concrete cantilever at midspan. Tension stresses on the built-in section reached concrete cracking stress.Failure of the two beams with side-bonded reinforcement was triggered by debonding of the side-bonded external reinforcement at midspan crack. Tension strains in the fibers at the bottom of the lateral side reached the debonding strain.The collapse loads of the tests of the two beams without side-bonded external reinforcement were 158.04 kN and 163.70 kN, respectively .udV = 74.592 kN. Since β = 0.5, Equation (36) gives udP = 149.184 kN (Setting to zero the terms of Equation (35) that reproduce the contribution given by the side-bonded reinforcement, the equation provides 9.184 kN . The maximum difference between the two experimental failure loads and the theoretical failure load is 8.9%, which is marginal. A theoretical prediction lower than the experimental value is in agreement with the term that reproduces the contribution of the concrete cantilever. In fact, that term uses concrete tensile strength and not concrete flexural strength (the latter is 1.1–1.2 times greater than the former).Not only do the tests confirm the predictive capacity of the model, but also that stirrups are ineffective when cracks are vertical. It is of note that the diameter of the stirrups was large. Despite this, the stirrups provided shear strength around the midspan with no contribution.The collapse loads of the tests of the two beams with side-bonded external reinforcement were 240.40 kN and 235.18 kN, respectively .Fd = 2.8‰ into Equation (35), the equation provides udV = 108.377 N. Since β = 0.5, Equation (36) yields udP = 216.754 kN ; for each of them, the ultimate value udP of P is calculated using the model. The geometric and mechanical characteristics of the case studies are shown in The analytical model was applied to two virtual cases. Each case study consisted of an RC beam with vertical cracks around the midspan, strengthened in the fashion described in udP of P provided by the model for the two case studies allow considerations to be drawn . Moreover, each RC beam was strengthened with an externally bonded FRP reinforcement attached onto both the side surfaces of the beam’s web, for the entire span . The sidOn assumptions (3) and (4), cracks exhibit the following pattern:The concrete cover is 40 mm . So, craF-tott = 3·0.177 = 0.531 mm. The effective bond length turns out to be:The total thickness of the FRP reinforcement attached onto each side surface is FL ≥ effL, each sheet has to be anchored onto the whole width of the beam’s bottom side (b = 150 mm) and must then be bent onto the other side of the concrete section for no less than 8 mm.To obtain effL ≥ 313 mm, in which effL of Equation (38) has been used. The fabric sheet height μ’ should therefore satisfy the following relationship: μ’ ≥ 313 + 0.707·158 → μ’ ≥ 425 mm.The most efficient arrangement of the side-bonded FRP reinforcement needs μ ≥ ξ’. Namely, μ’ − 0.707·Unfortunately, the flange bounds the height of the sheet to: μ’ = 450 − 200 = 250 mm < 425 mm :The ultimate shear force is derived from Equation (35):udP = 92765.8 N ≡ 92.8 kN.The concentrated load-carrying capacity is derived from Equation (36), plugging the result of Equation (41) and β = 0.5: The real role played by the side-bonded FRP reinforcement in strengthening this RC beams is represented by the ratio between the fraction of strength provided by the concrete cantilever and that provided by the FRP reinforcement:Ratio (42) confirms that the FRP reinforcement plays the major role in providing the beam with concentrated load-carrying capacity.udP would have been only 24.9 kN. It is of note that the above value of udP (that without side-bonded FRP reinforcement) was derived using a concrete flexural strength to concrete tensile strength ratio equal to 1.20 (which is appropriate for this case).Without side-bonded FRP reinforcement, udP with side-bonded FRP reinforcement is 3.7 times greater than without. In other words, the side-bonded FRP reinforcement has provided an increase in udP of approximately 450%. Ultimately, the concentrated load-carrying capacity Considering that both the concrete section and steel reinforcement of this beam are adequate for carrying uniform loads, this case study confirms that the strengthening technique is a viable method for the cases it is devoted to.Assumptions (3) and (4) allow the following condition to be establish:According to Equation (6) gives:effL of Equation (44) in the relevant expression, the effective depth of the side-bonded FRP reinforcement is: μ = 460 − 0.707·195 = 322 mm. The flange thickness  imposes The adopted side-bonded FRP reinforcement configuration implies that the crack is uncoated for 160 mm, i.e., μ/ξ’ = 0.67. That condition is accounted for by the coefficient η of Equation (7) yields:udV is derived from Equation (35):The ultimate shear force Equation (46) becomes:udP = 187919.7 N ≡ 187.9 kNThe concentrated load-carrying capacity is derived from Equation (36), plugging the result of Equation (47) and β = 0.5 into it: The actual role played by the side-bonded FRP reinforcement in strengthening this RC beams is represented by the ratio between the fraction of strength provided by the concrete cantilever and the fraction provided by the FRP reinforcement:Ratio (48) confirms that the FRP reinforcement plays a major role in providing the beam with concentrated load-carrying capacity.udP would have been only 46.6 kN. Again, udP without side-bonded FRP reinforcement was derived using a ratio between flexural strength and tensile strength of concrete equal to 1.20.Without side-bonded FRP reinforcement, udP with side-bonded FRP reinforcement is 4.0 times greater than without. In other words, the side-bonded FRP reinforcement has provided an increase in udP of approximately 400%. Ultimately, the concentrated load-carrying capacity Considering that both the concrete section and steel reinforcement of this beam are adequate to carry uniform loads, even this case study confirms that the technique is a viable method for the cases it is devoted to strengthening.The case studies of the preceding section are emblematic for beams of building, road, rail, or waterway structures. As such, those case studies allow interpretation and general conclusions to be drawn. Concrete sections and steel reinforcement arrangements that are adequate for carrying uniform loads conversely are inadequate to carry concentrated loads applied around the midspan. So, if the applied load has to be changed from distributed along the span to concentrated around the midspan, the beam must be strengthened.Side-bonded composite reinforcement with fibers at ±45° , dowel action and aggregate interlock boost shear strength no more than marginally. Moreover, the cantilever reaches the concrete tensile strength. So, the resisting bending moment of the concrete cantilever is only slightly lower than the bending moment it could reach due to the plasticity it could display .On one hand, the high ratio between the concentrated load-carrying capacity of the strengthened beam and that of the unstrengthened beam obtained for the case studies is due to the fact that the denominator is small. In fact, the shear-carrying capacity around the midspan is small, and so is the concentrated load-carrying capacity.On the other hand, however, the externally bonded reinforcement makes concentrated load-carrying capacity increase significantly, not only relative to the value it had in the unstrengthened condition, but also in absolute terms. As a matter of fact, the concentrated load-carrying capacity that can be reached allows an RC beam to carry not only the same resultant force it carried before though now in a concentrated fashion rather than in the preceding distributed fashion, but also substantially higher concentrated forces.The above evaluation underlies a general reflection. Externally bonded fiber composite materials can increase the distributed load-carrying capacity, bending-carrying capacity, shear-carrying capacity at the beam’s ends, axial-carrying capacity of a column, and dissipation capacity no more than moderately. On the contrary, they can increase the concentrated load-carrying capacity substantially.The sensitivity analysis has shown that increasing the full end debonding strain of the side-bonded FRP reinforcement would allow the concentrated load-carrying capacity to increase in proportion. That relationship between debonding strain and capacity is, however, typical for any FRP reinforcement.Full end debonding strain can be increased by anchoring the external reinforcement with stud shear connectors ,57,58,72udV and d is not far from linear. That result may drive the design of the reinforcement.The model shows that the relationship between The conventional shear capacity models of RC beams have been used for many decades with satisfactory results, which proves their reliability . Those models have, however, mainly been used for new structures, and their use for existing structures started only few decades ago. Moreover, in the case of existing RC beams, the shear capacity models have been mainly applied to predict the increase in ultimate load factor that a certain strengthening technique allows the beam to obtain.On the contrary, shear capacity models of RC beams have not been tested on beams whose load distribution is to be changed (or has been changed) from distributed to concentrated. This paper has proven that, in those cases, the conventional shear capacity models of RC beams fail. Not only are those models unsuccessful, but also conventional strengthening techniques fail in those cases, including the most common one—namely, externally bonded fiber composite reinforcement.Aiming at filling those gaps, the paper has presented a strengthening technique together with the relevant predictive model. The strengthening technique consists of side-bonded reinforcement with fibers at 45°. The model consists of an analytical formulation that predicts the concentrated load-carrying capacity of the RC beams in the strengthened state.d/1.5 and α = ξ) proves that the error would be moderate even if ξ and α were substantially far from the assumed values. The assumptions that the model is based on relate to crack spacing and crack depth. Those assumptions are in accordance with real observation, so they are realistic. Nonetheless, the expansion of the shear capacity provided by the model in a Taylor series about the assumed conditions (ξ = The last significant modeling assumption is that, at debonding failure, stresses acting on the built-in end of the concrete cantilever have a bi-triangular distribution, and that the maximum stress of the distribution is equal to concrete tensile strength. Research activity has proven that, at debonding failure of external reinforcement, the stress distribution is elasto-plastic, which means that the assumed stress profile is both realistic and conservative.No further substantial assumptions were made. The proposed model hence provides reliable predictions. That statement has also been proven by the experimental campaign that was performed in order to check the model, which confirmed that the theoretical predictions are in agreement with the experimental results. The case studies have proven that the proposed externally bonded reinforcement can increase the concentrated load-carrying capacity of an RC beam substantially, which is a result that externally bonded reinforcement cannot guarantee when used to increase other types of capacities of RC structures.The strengthening technique is also applicable to structures for which the application point of the concentrated load is not fixed but can vary its position. In that case, nevertheless, the fabric sheets have to be bi-directional—namely, fibers both at +45° and −45° on each side of the web.


Petromyzon marinus), one of the oldest living vertebrates, have a complex metamorphic life cycle. Following hatching, SL transition into a microphagous, sediment burrowing larval stage, and after 2–10+ years, the larvae undergo a dramatic metamorphosis, transforming into parasitic juveniles that feed on blood and bodily fluids of fishes; adult lamprey cease feeding, spawn, and die. Since gut microbiota are critical for the overall health of all animals, we examined the microbiota associated with SLs in each life history stage. We show that there were significant differences in the gut bacterial communities associated with the larval, parasitic juvenile, and adult life stages. The transition from larval to the parasitic juvenile stage was marked with a significant shift in bacterial community structure and reduction in alpha diversity. The most abundant SL-associated phyla were Proteobacteria, Fusobacteria, Bacteroidetes, Verrucomicrobia, Actinobacteria, and Firmicutes, with their relative abundances varying among the stages. Moreover, while larval SL were enriched with unclassified Fusobacteriaceae, unclassified Verrucomicrobiales and Cetobacterium, members of the genera with fastidious nutritional requirements, such as Streptococcus, Haemophilus, Cutibacterium, Veillonella, and Massilia, were three to four orders of magnitude greater in juveniles than in larvae. In contrast, adult SLs were enriched with Aeromonas, Iodobacter, Shewanella, and Flavobacterium. Collectively, our findings show that bacterial communities in the SL gut are dramatically different among its life stages. Understanding how these communities change over time within and among SL life stages may shed more light on the role that these gut microbes play in host growth and fitness.Sea lamprey (SL;  Petromyzon marinus, often generically referred to as the sea lamprey (SL), have a complex metamorphic life cycle. Following hatching, SL transition into a microphagous (filter-feeding-like) larval stage that burrow in soft sediment in or near streams. After several years (ranging 2–17) (Lampreys (Petromyzontiformes) are one of the oldest living groups of vertebrates . Along wng 2–17) , the larng 2–17) , transfong 2–17) . Parasitng 2–17) .Sea lamprey are endangered in parts of its native range in Europe , but areGut microbiota are critical for the overall health of nearly all animals species studied to date . Each spThe objective of the current study was to identify and characterize the bacterial communities associated with landlocked SL during different life stages using a 16S rRNA gene-based sequencing approach. We hypothesized that bacterial communities would differ among life stages and become less complex and more specialized in parasitic juvenile SL, because they feed exclusively on blood and bodily fluids. A thorough understanding of bacterial communities associated with each SL life stage may help inform restoration of valued populations, as well as control invasive populations.Sea lamprey representing each life stage were obtained from the Lake Huron watershed. Equal numbers of SL from each life stage could not be obtained due to difficulty procuring juvenile and adult life stages. Samples of the surrounding water were not collected nor were samples of the skin of SL hosts, although these could be avenues of future research to understand how external sources of microbiota influence the gut microbiota of SLs.via backpack electrofishing    in BoltoSalvelinus namaycush) or Chinook salmon (Oncorhynchus tshawytscha) that were angled from Lake Huron near Cheboygan, Michigan . Juveniles were collected from June 30 to August 12, 2019 and were frozen the day of capture. SL were transported to the lab at the HBBS, thawed, the intestines were removed and dissected into nine 5 mm by 5 mm pieces. Three pieces were placed in a 2.0 ml microcentrifuge tube such that three triplicate samples were collected per individual. Tubes were filled with 1.7 ml RNA-later and frozen at −20°C. Scalpel blades were changed between lampreys, and tweezers and cutting boards were sterilized using a 10% solution of bleach . Intestines were sampled from 7 juvenile SL , having a mean length of 263 mm (range: 185–370 mm), and mean weight of 53 g (range: 16–136 g). The sex of the juveniles was not determined.Juvenile SL were removed from lake trout , a tributary to Lake Huron. Adults were captured between April 24 and May 24, 2019. When a lamprey was captured, fecal samples were collected at U.S. Geological Survey, HBBS by applying pressure to the abdomen. A sterile swab was used to collect fecal material and the swab was placed directly in a microcentrifuge tube containing 1.5 ml of RNA-later . Samples were stored at −20°C until processed. To minimize risk of DNA contamination, samples were collected by personnel wearing latex gloves. The urogenital region of each SL was washed with deionized water prior to swabbing. Fecal samples were obtained from 35 adult SL having a mean length of 477 mm (range: 320–560 mm) and a mean weight of 208 g (range: 69–340 g). Sixty percent (21 of 35) of the captured SL were females.After shipping samples on ice to the University of Minnesota , the dissected tissue samples were transferred into PowerBead tubes containing Solution 1  and were ground for 2 min using a sterile micro pestle. DNA was extracted from the suspension using the DNeasy PowerSoil DNA Isolation Kit (Qiagen) and starting with the bead beating step, according to the manufacturer’s instructions. DNA concentrations were measured using a Qubit 2.0 Fluorometer  and all samples were stored at −20 °C until further analyzed.n = 109) were sequenced at the University of Minnesota Genomics Center  by using universal primers: 515f (5′-GTGCCAGCMGCCGCGGTAA-3′) and 806r (5′-GGACTACHVGGGTWTCTAAT-3′) targeting the V4 region of the 16S rRNA gene as described elsewhere . Alpha diversity measures were calculated using the Chao1, Shannon H, observed species, and Simpson E indices. Bray–Curtis dissimilarity matrices were used for principal coordinate analysis (PCoA). These matrices were also used to assess differences in beta diversity by analysis of similarity (ANOSIM) . CanonicR = 0.881; p-value = 0.001). Notably, the transition from larval to the parasitic juvenile stage was marked with a significant shift in bacterial community structure  among the larval, parasitic, and adult lamprey stages; only taxa with linear discriminant analysis (LDA) effect sizes ≥3.75 are shown in fe stage .A total of 116, 2, and 6 OTUs were identified in ≥90% of the larval, parasitic, and adult SLs, respectively, which were defined as members of their core microbiota . None ofMicrobial functional groups in larval, parasitic, and adult SLs were predicted by comparing taxonomic information of OTUs against the FAPROTAX database. This analysis revealed that parasitic SLs showed significantly higher relative abundances of animal parasite- and human-associated taxa, as well as those involved in ureolysis, compared to larval and adult SLs . RelativSea lampreys have a complex metamorphic life cycle consisting of three distinct stages each with a unique feeding behavior: (a) larvae typically feed on detritus and algae; (b) juveniles feed on fish blood; and (c) adults essentially cease feeding, spawn, and then die. We hypothesized that distinct intestinal microbiota in SLs likely correspond with their three different life stages. In this study we show that the lamprey gut microbiota change in concert with its dietary needs. For clarity and ease of discussion, the relationship between lamprey life stages and gut bacterial community composition is presented sequentially.Larval SL are filter feeders burrowing in sediments and slowly draw water across their pharynx, resulting in the trapping and concentration of suspended particles in mucus . SuspendPrionace glauca; Cetorhinus maximus; via FAPROTAX database revealed an enrichment of animal parasite- and human-associated bacterial taxa in parasitic SLs compared to the larval and adult SLs  in complex media supplemented with blood products. Bacterial genera, such as Streptococcus, and Hemophilus, which are capable of hemolytic activity  can be found below: Sea lamprey were captured and maintained in conditions as recommended under the American Fisheries Society Guidelines for the Use of Fishes in Research (2014).PM, MB, NJ, and MS conceived of the studies, wrote, and edited the manuscript. NJ obtained the samples. MB and PM prepared the samples. PM analyzed the data. All authors contributed to the article and approved the submitted version.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.


Implementation of antimicrobial stewardship programs (ASPs) within long-term acute care facilities (LTACs) is challenging due to limited resources and missing patient data from transferring facilities. In October 2018, an ASP was established within a 43-hopital system consisting of LTACs and rehabilitation hospitals. Despite the presence of a restricted antimicrobial policy, increased utilization was observed for five restricted antimicrobials. The system ASP committee implemented a multipronged approach to optimize utilization of these five agents. Investigators sought to assess the impact of an antimicrobial intake process on antimicrobial consumption.This was a retrospective analysis within a 43-hospital system of LTACs and rehabilitation hospitals, comparing use of five restricted antibiotics before (Jul19-Jun20) and after (Jul20-Apr21) implementation of a data-collection and system review process. An antibiotic intake form and process for review for five restricted antibiotics  was approved at the system ASP committee. The intake form consisted of a restricted antibiotic form, cultures and susceptibilities, physician notes, and other pertinent data. Any orders for the five antibiotics required completion of an intake form and submission to system ASP members for review and recommendations. Antibiotic consumption was measured in cost per acute patient day (cost/pd) using a 2-sided t-test.± 2.29) vs. (&8.48 ± 1.45); p=0.0003]. The cost/PD of the five restricted antibiotics decreased 42.52% .Post-implementation, the five restricted antibiotics comprised 29.1% of the total antibiotic expenditure for the healthcare system compared to 35.6% pre-implementation. Ten months after program implementation, the total antibiotic cost/PD decreased 29.45% [(&12.02 Implementation of an antimicrobial intake process within a post-acute medical system resulted in a significant reduction in antibiotic consumption for five targeted antibiotics as well as overall antibiotic expenditure.All Authors: No reported disclosures


Recycled aggregate concrete-filled steel tubular (RACFST) columns are widely recognized as efficient structural members that can reduce the environmental impact of the building industry and improve the mechanical behavior of recycled aggregate concrete (RAC). The objective of this study is to investigate the behavior of recycled aggregate concrete-filled circular steel tubular (RACFCST) stub columns subjected to the axial loading. Three-dimensional finite element (FE) models were established using a triaxial plastic-damage constitutive model of RAC considering the replacement ratio of recycled aggregates. The FE analytical results revealed that the decreased ultimate bearing capacity of RACFCST stub columns compared with conventional concrete infilled steel tubular (CFST) columns was mainly due to the weakened confinement effect and efficiency. This trend will become more apparent with the larger replacement ratio of recycled aggregates. A practical design formula of the ultimate bearing capacity of RACFCST stub columns subjected to axial load was proposed on the basis of the reasonably simplified cross-sectional stress nephogram at the ultimate state. The derivation process incorporated the equilibrium condition and the superposition theory. The proposed equation was evaluated by comparing its accuracy and accessibility to some well-known design formulae proposed by other researchers and some widely used design codes. The past several decades have witnessed the rapid progress of engineering construction technology and worldwide expedite urbanization, which leads to a continuous updating of design codes to meet the changing requirements of society. Therefore, the Construction & Demolition (C&D) waste is becoming an inevitable problem due to the upgrading of existing engineering structures. As the most commonly used engineering materials by far, concrete has the merits of low price and high compressive strength. According to available survey studies, around 20 billion metric tons concrete were consumed in 2009 , and thiHowever, compared with structural concrete utilizing natural aggregates, RAC has some drawbacks verified by experimental studies such as the low strength and elastic modulus, high shrinkage and creep, inferior durability, and large scatter of its tested mechanical properties ,5,6,7,8.One of the most promising structural members in its form that can provide sufficient confinements on concrete and reduce their shrinkage and creep is the concrete-filled steel tubular (CFST) columns . CorrespTo sum up, past studies partially addressed the understanding of mechanical behavior of RACFST columns under axial loading and verified its prospect in engineering application. However, more in-depth investigations are required to clearly explain the structural behavior such as the confinement effect and efficiency and their relations with the design parameters. In the meantime, the promotion of RAC usage in engineering practice requires more practical and simple yet accurate design tools. To date, the specific calculation method of the ultimate bearing capacity of RACFST stub columns is still rare, and the current design formulae of CFST stub columns were reported as not appropriate. The differences between the confinement effect of the steel tube and infilled RAC and that of normal concrete counterparts should be noticed, including the discrepancy of stress variation of infilled RAC and normal infilled concrete and the correspond steel tube stress variation.To this end, on the basis of past research of CFST stub columns under axial loading , this stABAQUS is widely regarded as the most powerful finite element software for the analysis of complex solid mechanical structural mechanical systems, especially for the control of very large and complex problems and the simulation of highly non-linear problems.Non-linear FE models of axially compressed RACFCST stub columns were established making use of the commercially available software ABAQUS version 6.14 . The outD/25 to D/5, where D means the outer diameter of RACFCST stub columns. D/10 mesh size and those with smaller mesh size predicted similar axial-bearing capacity with the difference being less than 1%. The results predicted by the larger mesh size are sensitive to the change of element size. Therefore, the mesh size is selected as D/10 in the modeling. The mesh generation and boundary conditions of the established FE model are shown in The mesh size will affect the accuracy of the FE model and computation time. Choosing a proper mesh size is important to reasonably replicate the test results. In order to identify the optimal mesh size, a convergence study was performed. The scope of the mesh size is from 12 MPa and the Poisson’s ratio was set as 1.0 × 10−7.The interaction between the steel tube and the infilled RAC was conducted in a surface-to-surface contact model. The main surface is the inner surface of steel pipe, and the secondary surface is the outer surface of concrete. The sliding formulation was chosen as “finite sliding”, and the discretization method was chosen as “surface-to-surface”. The contact properties between the steel pipe and infilled RAC were simulated by normal and tangential behavior. In the normal behavior setting, the “Pressure-Overclosure” was set to “hard” contact mode, and the separation after contact was allowed. The penalty function was set as the tangential behavior of the contact attribute, in which the friction coefficient was 0.5 . A tie cr, Ar)111ccuc are the parameters to determine the uniaxial behaviors of the RAC; their detailed definition can be found in the reference [σ and ε are the stress and strain of RAC, respectively. In this study, the uniaxial full-range stress–strain relation of RAC proposed by Ding et al.  was appleference . The σ aσ-ε relation of RAC under uniaxial compression and the strength criterion parameters of concrete under multiaxial stress [Combining the l stress , the tril stress ,29.The diagrams of confined and unconfined plastic-damage model of recycled aggregate concrete are shown in iσ is the equivalent stress; fu is the ultimate strength of steel, which is equal to 1.5 times the yield strength fs; the Young’s modulus Es is 2.06 × 105 MPa; iε stands for the equivalent strain of steel; yε means the yield strain; stε means the hardening strain and is equal to 12 times yε; εu is the ultimate strain and is equal to 120 times yε; and ζ is a parameter chosen as 1/216. This steel model has been verified in previous studies of concrete-filled steel tubular columns subjected to axial loading [The adopted stress–strain model of an outer steel tube is an elasto-plastic model. Meanwhile, this model considers the Von Mises yield criteria, Prandtl–Reuss flow rule, and isotropic strain hardening. The expression is as follows :(3)σi={E loading ,32,33,34 loading , torsion loading , and cyc loading . The strfcu [fcu,100 to 150 mm fcu,150 [fcur150 in The validity of the FE models was verified by comparing the existing RACFCST axial compression stub column test data ,44,45,46ngth fcu , cube wi fcu,150 ) are shoNu,exp/Nu,FE) are presented in Nu,exp/Nu,FE) was 0.978, and the variable coefficient is 0.048, which indicates that the ultimate bearing capacity predicted by the established FE model is in good agreement with the experimental results. Meanwhile, To verify the applicability and accuracy of the established FE model, the calculated ultimate bearing capacity of collected specimens was compared with the test results. The ratios of the test results to FE results ). The composite action can be assessed by the occurrence of the intersection point of the axial stress–strain curve and the transverse stress–strain curve of a steel tube. In general, the greater the radial concrete confinement coefficient is, the better the utilization efficiency of the steel tube on the infilled concrete. The composite action can be reflected by the axial and transverse stress–strain curves of the steel, and the intersection of axial and lateral stress–strain curves indicate the best composite action point of the composite structures. The two stress–strain curves intersecting early means the composite action and the confinement efficiency were better [The confinement effect is the most valued characteristic of CFST columns caused by the interaction between the steel tube and infilled concrete. The confinement effect can be evaluated by the radial stress e better . Hence, fcu) is shown in σr,c and the radial concrete confinement coefficient ξr,c of fcu,0 = 70 MPa are slightly lower in the early stage but are greater in the late stage. Meanwhile, the longitudinal and the lateral stress–strain curve of fcu,0 = 70 MPa intersected earlier. It can be concluded that the effect and efficiency of confinement increase with the increase of concrete strength.The influence on the confinement effect and efficiency of the RACFCST stub columns caused by different concrete strength (fs) is shown in ξr,c of fs = 235 MPa is higher than that of fs = 345 MPa in fs = 235 MPa intersect before those of fs = 345 MPa in The influence on the confinement effect and efficiency of the RACFCST stub columns caused by diverse steel strength (ρ) is shown in ξr,c and postpones the cross of the longitudinal and the lateral stress–strain curves, which means the decrease of confinement efficiency. In conclusion, a thicker steel tube causes a weaker confinement efficiency on the infilled concrete.The influence on the confinement effect and efficiency of the RACFCST stub columns caused by various steel ratios (r) on the confinement effect correlation curves. The comparison states follow the mechanical behavior of RACFCST stub columns:(1)r results in the radial concrete confinement coefficient r,cξ of RACFCST stub columns being smaller at the initial phase of the loading process. However, at the end of loading, r,cξ with different r tends to be the same.The increase of replacement ratio of recycled aggregate (2)The axial stress decreased after reaching the peak point, the transverse stress gradually increased under continuous axial loading, and the axial stress–strain curve intersects with the transverse stress curve in the end. However, the intersection point of the longitudinal and the lateral stress–strain curve is slightly postponed with the increase of the recycled aggregate replacement ratio.r has less of an effect on the confinement effect and efficiency contrasted with material strengths and steel ratio investigated in this paper.The above observations indicate that the replacement ratio of recycled aggregate D = 500 mm, L = 1500 mm, t = 6 mm, fs = 345 MPa, fcu,0 = 50 MPa) and changing the replacement ratio of recycled aggregate r, the concrete stress nephogram values at the middle height cross-section of CFCST and RACFCST stub columns were obtained by ABAQUS calculation, as shown in r slightly weakens the confinement effect of the steel tube, but the reduction is not obvious.After fixing parameters σr,c=Lσ,s and θσ,s represents the longitudinal stress and lateral stress of the steel tube, respectively. Lσ,c and r,σc represents the axial compressive stress and radial stress of core concrete, respectively.For the RACFCST stub columns, the following expressions were derived. fsc = Nu/Asc, Asc = Ac + As), the average ratio of Lσ,s and θσ,s of the steel tube could be obtained as:As shown in The longitudinal stress of confined concrete could be given as:k is the coefficient of the lateral pressure and k = 3.4 was suggested herein, according to Ding et al. [g et al. .Nu of RACFCST stub columns under axial compression could be put as:Based on the static equilibrium method, the ultimate bearing capacity Nu could be obtained as:K is the proposed confinement coefficient of RACFCST stub columns subjected to the axial load. Based on the aforementioned FE model analysis, the K of the RACFCST stub columns obtained from Equation (13) is 1.61. It worth mentioning that Ding et al. [Nu = fcAc + 1.7 fsAs) of the CFCST stub column according to the limit equilibrium method, which was validated by 115 CFCST stub columns test results. Meanwhile, Ding et al. [K (K = 1.62) of CFCST stub columns as well. This discrepancy between the two methods was considered due to the constitutive relation of the steel tube in the FE analysis, which could not perfectly reflect the stress–strain curves of the steel tube in the test, especially in the hardening stage. According to the existing experimental tests, local buckling could be occasionally observed on the outer steel tube, while the specimen reached the ultimate loading capacity [K of the RACFCST stub columns obtained in this study is close to the confinement coefficient K (K = 1.62) of CFCST stub columns [Substituting Equations (7)–(12) into Equation (13), g et al.  establisg et al.  combinedcapacity , and the columns  obtainedNu), test results , and FE results  for RACFCST stub columns under axial loading. For comparison, the average stress of the cross-sectional area of the column specimens (N/(As + Ac)) is used as the horizontal axis. Nu,exp and Nu,FE. The mean value of the ratios of Nu,exp/Nu,FE is 0.978 with the corresponding dispersion coefficient of 0.048. The average ratio of Nu,exp to Nu is 1.024 with the corresponding dispersion coefficient of 0.042. The average ratio of Nu,FE to Nu is 1.049 with the corresponding dispersion coefficient of 0.041. As a result, the ultimate axial bearing capacity of RACFCST stub columns calculated by Equation (13) is accurate.Nu,exp to Nu,ACI-318-11, Nu,T/CECS 625-2019, Nu,EC4, Nu,AIJ, and Nu,AISC 360-16 shown in In order to visually illustrate the advantages of the proposed equation, it is compared with some current international design codes and well-known design formulae proposed by other scholars. Firstly, as shown in Nu,exp to Nu,Chen, Z.P., Nu,Chen, M.C., and Nu,Xu, W. are 1.035, 1.268, and 1.025 with the corresponding dispersion coefficients of 0.072, 0.064, and 0.079, respectively. As the result, only the formulae proposed by Xu et al. [Then, the performance of the proposed design equation was further evaluated by comparing the accuracy of the predicted results using design formulae proposed by other scholars. The details of those formulae are show in u et al.  show a cThe FE models of RACFCST stub columns under axial compression can be modeled utilizing the triaxial RAC plastic-damage and steel constitutive relations. The good agreement between the experimental results and the FE calculation results indicates the validity of the FE model.On the basis of verifying the FE modeling method, parameter analysis was conducted on the full-scale RACFCST stub columns. The analytical results indicate that the recycled aggregate slightly weakened the confinement effect between the steel tube and infilled RAC.ξr,c–axial strain curves and steel stress–axial strain curves indicate that the materials’ strength  and specimen dimensions (ρ) impact on the confinement efficiency in the axial loading process. However, the effect of replacement rate of recycled aggregate (r) on the confinement effect and efficiency of RACFCST in the limit state can be ignored.The conclusions can be drawn from parametric analysis, the K) of RACFCST axial compression stub columns is 1.7. The proposed equation is more precise and compendious than current international design codes and other scholars’ literature.An equation for calculating the ultimate bearing capacity of RACFCST stub columns was proposed by the superposition method. The confinement coefficient (In this paper, numerical study was conducted using the FE models validated by existing experimental results to investigate the confinement effect and confinement efficiency of RACFCST stub columns under axial loading. A simple and practical design equation is proposed to predict the ultimate bearing capacity of RACFCST stub columns. The following conclusions could be stated:There are still some limitations in this paper. The influence of different kinds of recycled aggregates on the bearing capacity of the RACFCST columns and the application of recycled concrete in practical engineering will be carried out in the future.