Prediction of the monotonic load–displacement curve of slender RC beams with corroded transverse and longitudinal reinforcements

The present paper proposes a mechanical model for the prediction of the response of slender reinforced concrete beams with corroded transverse and longitudinal reinforcements. The proposed model is based on the Modified Compression Field Theory and is able to predict the load–displacement curve of members subjected to monotonic displacement‐controlled tests up to their peak load. The corrosion‐damage effects are considered by means of an effective width of the beam and by means of reduced values of both cross‐sectional area and strain capacity of steel bars. The accuracy of the model is evaluated by comparison with results of tests reported in the literature. Moreover, a simple relation is suggested for the estimate of the shear strength of reinforced concrete beams with low values of the effective geometric percentage of (corroded) stirrups.


| INTRODUCTION
2][3] In such conditions, degradation of physical and mechanical properties of RC members is often due to corrosion of longitudinal and transverse steel reinforcements.Corrosion causes a reduction of the cross-sectional area and deformation capacity of the steel bars and loss of bond between concrete and steel bars and cracking, or even spalling, of the concrete cover. 4any research studies have proved that corrosion effects on structures largely depend on the level and morphology (uniform or localized) of corrosion.In particular, several laboratory tests on statically determinate beams endowed with corroded longitudinal rebars and designed to fail in flexure, for example, 5 have highlighted a considerable reduction of the loading and deformation capacities of the member.In addition, laboratory tests on statically indeterminate beams 6 have revealed that corrosion of the longitudinal reinforcement may also prevent such structures from developing significant moment redistribution.
5][16] In turn, the degradation of the mechanical properties of the corroded transverse steel bars causes a reduction of the confinement effects on concrete and favors the growth of wide shear cracks.All these effects unfavorably characterize the response of corroded RC members designed to fail in shear, in serviceability and ultimate loading situations as well.
Due to the reduction of the cross-sectional area of the transverse reinforcement, corrosion can also be responsible for a change in the failure mode of members designed to fail in flexure, 17,18 from a ductile flexural mode to a brittle shear mode.Further, corrosion of the longitudinal reinforcement and deterioration of bond between concrete and steel bars can reduce the loading capacity and require direct transfer of loads to the supports by means of arch action.In any case, the significant reduction of the deformation capacity of corroded rebars may limit the loading capacity of members because of anticipated rupture of corroded rebars.In this latter regard, it is worth noting that most of the shear models proposed in the literature are able to predict the sole load-bearing capacity of corroded RC members, without any check of the deformation demand of the steel bars being carried out.These models are usually not based on mechanics, and are merely calibrated by comparison between laboratory tests results and numerical predictions, for example. 19he present study aims to extend a theoretical procedure, previously proposed by one of the authors for RC beams strengthened with FRP and based on the Modified Compression Field Theory (MCFT), 20 to predict the loaddeformation curve of slender RC beams failing in shear and endowed with corroded stirrups.The procedure predicts the response of beams subjected to monotonic tests up to their peak load.
The proposed model considers an RC cracked element with longitudinal and transverse smeared steel bars and computes strain and stress fields by means of equilibrium and compatibility equations.The constitutive laws of concrete and steel are defined by means of nonlinear relations.Corrosion effects are incorporated by means of an effective width of the concrete cross-section and by means of reduced values of both cross-sectional area and strain capacity of steel bars.The accuracy of the numerical results is assessed by comparison with the results of experimental tests collected from the literature.The values obtained by some significant response parameters at the achievement of the loading capacity in the numerical model are also compared with those resulting from the Compression Chord Capacity Model (CCCM) recently formulated by Cladera et al. 21Finally, a parametric analysis is carried out by means of the proposed numerical model to derive more general observations regarding the effects of the level of corrosion of stirrups on some key response parameters.In particular, these latter results are used to obtain a simple relation for the prediction of the shear strength of slender RC beams with low values of the effective geometric percentage of (corroded) stirrups.

| MODELED STRUCTURAL CORROSION EFFECTS
The reduction of the cross-sectional area of corroded steel bars is often evaluated by means of the cross-sectional area loss ratio, η a : where A v0 and ϕ v0 are the cross-sectional area and diameter of a steel bar before corrosion, respectively; A v and ϕ v are the area and diameter of a specific cross-section of a steel bar after corrosion.As proved by several research studies, the maximum cross-sectional area loss ratio η am along the entire steel bar significantly influences the axial force capacity of the rebar.For this reason, the area of this cross-section is calculated here based on η am .Corrosion also affects the deformation capacity of steel bars, mainly because of localization of the inelastic response.To be on the safe side, the effects of corrosion on the strain capacity of the steel bars are considered in this paper as proposed by Imperatore et al. 7 for the case of pitting corrosion.In particular, the strain capacity is assumed equal to ε vu,0 e b 1 η w , where ε vu,0 is the strain capacity of the uncorroded transverse reinforcement (assumed here equal to 60 mm/m), b 1 is equal to À0.0546993, and η w is the weight loss ratio.
To obtain the weight loss ratio η w from the maximum cross-sectional area loss ratio η am (and vice versa when necessary), the following equation has been used in the present paper, which was calibrated by Lu et al. 19 based on RC specimens subjected to accelerated corrosion and slightly modified by Cladera et al. 21: It should be noted that higher values of the ratio η am /η w have sometimes been suggested by other researchers 22 and are mostly representative of localized corrosion.The use of Equation ( 2) is unfortunately necessary because data concerning the corrosion-damage obtained by different campaign of tests are collected in an inhomogeneous manner.A shared data collection protocol would be desirable.
Regarding the effects of steel corrosion on cracking or even spalling of the concrete cover, Higgins et al. 23 have observed that these effects primarily depend on stirrup spacing (s v ) and concrete cover (c v ). 24Assuming that the angle of discrete spalls is 20 , Higgins et al. 23 proposed two relations, based on empirical data and theoretical computations, to estimate the effective width of the beam (b w,eff ), that is, the width of the beam that is effective in bearing shear stresses.Recently, these relations have been slightly modified by Cladera et al. 21as follows: These latter relations are considered in the proposed model.As suggested by Higgins et al., 23 the beam width (b w ) is reduced only if the average cross-sectional area loss ratio of stirrups is equal or higher than 10%.

| ANALYTICAL MODEL OF CORRODED RC BEAMS
The proposed model predicts the monotonic shear forcedisplacement response of slender RC beams with corroded reinforcements, up to their load bearing capacity.As assumed by other researchers, 25 the flexural and shear responses are separately evaluated and the flexural response is obtained by means of a preliminary sectional analysis.

| Flexural model
Once the top axial strain (ε ct ) has been assigned, the flexural strain field is calculated by means of an iterative process. 26The application of the above analysis leads to the evaluation of the axial strain (ε xf ) from flexure at the centroid level of the cross-section and hence to the calculation of curvature (χ) and bending moment (M) (Figure 1).

| Shear model
As assumed in the MCFT, the equilibrium conditions require that: where σ cv is the concrete stress in the vertical direction; σ c1 and σ c2 are the concrete principal tensile and compressive stresses, respectively; τ is the shear stress and θ is the angle of inclination of the diagonal compressive stress field of concrete.The equilibrium along the vertical direction provides the equation of clamping stresses (σ v ): where is the geometric percentage of the transverse reinforcement, A v is the total cross-sectional area of stirrups, s v is the spacing of stirrups, and σ sv is the axial stress of stirrups.
In slender beams, σ v can be neglected if referring to a part of the member away from the loading point. 27In such conditions, substituting Equation ( 6) (σ cv = Àρ v σ sv ) into Equation (5) gives the following relation of the shear stress: Rearranging Equation ( 5) gives an alternative relation of the shear stress: Substituting Equation (7) into Equation ( 8) provides the angle θ: In addition, the compatibility equations of the MCFT require that: where ε x and ε v are the axial and vertical strains, respectively; and ε 1 and ε 2 are the principal tensile and compressive strains, respectively.Equations ( 10) and ( 12) lead to the same values of the angle θ.Therefore, substituting Equation (10) into Equation (12) gives an expression for ε v , which is independent of angle θ.In this regard, two cases are distinguished for the state of stress of the transverse reinforcement.
If stirrups are elastic, the stirrups stress is σ sv = E s ε v and a second-order equation is obtained from Equations (10) and (12) to calculate ε v .In such conditions, the close-solution for ε v is: where coefficients A e , B e , and C e are: If stirrups have yielded, the axial stiffness of the stirrups is null (A e = 0) and ε v is obtained as the solution of the following linear equation: where The stress-strain relation of the compressive concrete in a diagonally cracked web is a function of the principal strain.Neglecting the confinement of the transverse reinforcement on concrete, the following equations are used for concrete in compression 28 : where f ce is the compressive strength of the diagonally cracked concrete and ε c0 is the strain at peak stress of concrete.
A linear-elastic response is assumed for concrete in tension until occurrence of first crack (σ c1 = E c ε 1 ).After first crack, concrete is still assumed to be able to carry tensile forces and the tensile stress is equal to the maximum corresponding to the mechanisms of tension softening (σ c1,a ) and tension stiffening (σ c1,b ).The stress due to tension softening can be predominant in beams without or with very low geometric percentages of stirrups and is evaluated by a decreasing exponential function: where f ct is the tensile strength of concrete and w is the average crack width (in mm).The stress due to tension stiffening derives from the interaction between rebars and surrounding concrete and is evaluated as: where c t = 2.2m t and m t is given by the following relation: where ρ lb is the geometric percentage of the longitudinal tension reinforcement.
The contribution of σ c1 to the shear strength is limited by the capacity of the element to bridge forces across crack sides.The maximum value of the principal tensile stress in concrete is given by the minimum of the expressions derived from equilibrium of forces across the crack in the longitudinal and transverse directions: where f yl and f yv are the yield strength of longitudinal and transverse rebars, respectively, and σ slb is the stress in the longitudinal tension reinforcement.The local shear stress τ i on the crack surface is given by the relation: where d g is the maximum coarse aggregate size (in mm).The average crack width is calculated as w = ε 1 S mθ where the average diagonal crack spacing S mθ is obtained as 1/(sin θ/S ml + cos θ/S mv ) and depends on the average crack spacing along the two orthogonal directions S ml and S mv .Parameters S ml and S mv are assumed equal to s v and d, respectively.In Equations ( 24) and ( 25), the residual stresses of reinforcements at the crack, that is, (f yl À σ slb ) and (f yv À σ sv ), are taken equal to zero if the yield strength has been reached.

| The analytical procedure
The steps of the flexural analysis are: 1. Assign a value to ε ct .2. Solve the flexural problem and calculate ε xf , χ, and M.
3. Calculate the shear stress at the critical cross-section.
In simply supported beams subjected to concentrated transverse forces (like those considered in the present research study) the critical cross-section is at the end of the loading plate, away a v from the support as indicated by the yellow dashed line in Figure 1.In addition, in such beams the shear stress is constant along the shear span (a) and is equal to where d is the effective depth of the beam.
In the flexural numerical model, failure is achieved because of excessive deformation of either longitudinal tension rebars or concrete.In the shear model, failure is achieved because of excessive crack width, stirrup rupture (ε v > ε vu ), or concrete crushing.
The assumption that the critical cross-section is close to the loading point, where shear and flexure are maximum, implicitly considers that the maximum corrosion level is constant along the shear span.It is an assumption that can lead to conservative results because, in general, corrosion levels are variable along the longitudinal axis of the member and the critical cross-section could be in a different position.1).The specimens considered in this paper are the same as those in the database recently collected by Cladera et al. 21This latter database has been carefully revised by the authors and amended in some cases, as reported in reference 30.
0][11][12][13] The examined beams are over-reinforced in bending and characterized by ρ lb in the range from 1.77% to 3.02%.Conversely, ρ v is fairly low and in the range from 0.10% to 0.52%.The compressive strength f cm is in the range from 20.8 to 44.4 MPa, and f yv is in the range from 300 to 626 MPa.Corrosion is present in the transverse reinforcement, and sometimes also in the longitudinal reinforcement.All the selected beams have been corroded by means of accelerated corrosion, but the details of the corrosion procedure are not reported in all cases.The current density applied to reinforcements varies from the minimum value of 100 μA/cm 2 assumed by Rodriguez et al. 5 to 1000 μA/cm 2 used by Xue et al. 10 A current density of 100 μA/cm 2 , which corresponds to up to 10 times the maximum corrosion measured in concrete structures, 5 provides corrosion products not different from those in natural environments.By contrast, the use of high values of the current density reduces the time necessary to obtain the desired corrosion level, but determines artificial structural effects, such as first cracking of concrete cover, crack width evolution, and bond deterioration, which can be very different from those in natural environments. 31Note that the beams tested by Rodriguez et al. 5 suffered spalling of the top concrete cover because of effects due to corrosion products and transverse loads.To highlight the effects of the top concrete cover on the shear strength, the numerical analyses of these beams have been performed with or without top concrete cover, as also reported in Cladera et al. 21eferring only to the six beams tested by El-Sayed et al., 13 where the level of corrosion of the steel bars was calculated by means of the weight loss ratio η w , the authors note that the maximum cross-sectional area loss ratio η am of the rebars has been obtained by means of Equation (2).
To assess the effectiveness of the proposed model against other mechanical models, the CCCM, recently modified by Cladera et al. 21to predict the shear strength of corrosion-damaged RC beams, has been applied to all the specimens of the database.The shear strength resulting from the CCCM is calculated as the sum of concrete (V cu ) and transverse reinforcement (V su ) contributions and is limited to be not higher than the shear strength corresponding to concrete crushing (V max ).Details regarding the formulae of the CCCM are summarized in Table 2.

| Validation of the response at peak load and comparison with the CCCM
As shown in Figure 2, the shear strength of the examined beams is predicted by the proposed model with sufficient accuracy.Similar results are also obtained from the application of the CCCM.To evaluate the accuracy in numbers, the following parameter is calculated with regard to all the experimental tests: where V num is the shear strength predicted by the numerical model and V exp is the maximum shear force recorded during the laboratory test.The single values of R v are reported in Table 3 along with the failure mode predicted by the proposed model (labeled as: ST for shear tension, SF for shear-flexure, and SR for stirrups rupture).The authors of the test campaigns considered in the database describe the failure mode of groups of beams and provide general information (i.e., significative reduction of stirrup section due to pitting, reduction of the effective width of beam, etc.), but without declaring the type of shear failure (ST, SF, or SR) for each specimen, thus this information is not included in Table 3, but the reader is referred to the referenced papers.In particular, referring to the specimens tested by Rodriguez et al., 5 this table reports the results referring to the cases with (referred to as the standard case) and without (in brackets) top concrete cover.As expected, the shear strength prediction provided by the model without top concrete cover is safer than that of the model with top concrete cover.Following figures and comments only refer to the results of the model with top concrete cover.Specifically, the ratios R v resulting from the application of the proposed model range from 0.58 to 1.68, with a mean value equal to 1.07 and a coefficient of variation (CoV) equal to 0.30.Similarly, the results obtained from the CCCM show minimum and maximum values equal to 0.53 and 1.59, a mean value equal to 1.00, and a CoV equal to 0.26.To offer a more comprehensive view of the response at the achievement of the shear strength, Figure 3 plots-for each specimen and numerical model-the ratio R v , the concrete and stirrup contributions to the shear stress (τ c / τ and τ s /τ) and the cotangent of the angle θ.In addition, but only with regard to the proposed model, the same figure (Figure 3e) reports the vertical to yield strain ratio ε v /ε yv .
As evident from the visual analysis of Figure 2, the values of R v are virtually the same for the proposed model and the CCCM (see Figure 3a).Neglecting the top concrete cover in the specimens tested by Rodriguez et al. 5 leads the proposed model to a safer prediction of the shear strength.A similar trend is also generally shown by the concrete and stirrup contributions predicted by the two models to the shear stress (see Figure 3b).However, the stirrups contributions provided by the proposed model are in most cases higher than those obtained by the CCCM.This result is in line with the assumptions of the considered mechanical models.In fact, the CCCM assumes that the external load is mainly resisted by shear stresses in the compression chord, whereas the proposed model assumes that the load bearing capacity is limited by the shear stress along the crack, T A B L E 2 Summary of the CCCM equations for slender beams.
which is resisted by means of the tensile strength and aggregate interlock of cracked concrete in the web.The angle θ (see Figure 3c) varies in the CCCM in a narrower range of values than in the proposed model.Specifically, the angle θ obtained from the CCCM ranges from 33 to 38.4 (cot θ = 1.54 Ä 1.26) whereas that obtained from the proposed model ranges from 25.1 to 59.5 (cot θ = 2.13 Ä 0.59).
The vertical strain demand to yield strain ratio ε v /ε yv resulting from the proposed model at the achievement of the shear strength is finally plotted in Figure 3d.As previously noted, this parameter is reported for the sole proposed model because it is not estimated by the CCCM.To identify any possible influence of the reduced strain capacity of stirrups on the peak responses, at any single step of the numerical analysis the ratio ε v /ε yv resulting from the proposed procedure is compared with the strain capacity ratio ε vu /ε yv calculated as suggested by Imperatore et al. 7 for the case of pitting corrosion and the analysis is stopped if ε v /ε yv is equal or higher than ε vu /ε yv .The strain capacity ratios of the transverse reinforcement of all the specimens are plotted in Figure 3d by black dots.However, note that the black dots referring to numerous specimens are not visible because the corresponding values are out of scale (i.e., ε vu /ε yv ≥ 5).In every case the maximum ratio ε v /ε yv is lower than the corresponding strain capacity ratio.In this regard, the authors note that the value considered for the strain capacity ratio is particularly low because estimated on the basis of a localized corrosion assumption.The analogous strain capacity ratio corresponding to a uniform corrosion 7 would be even higher than the considered value.
The values of the ratios ε v /ε yv obtained at the achievement of the shear strength are also plotted in Figure 4 as a function of η am , cot θ, and ρ v .The figure shows that ε v /ε yv generally decreases with the increase of η am and, more evidently, increases with ρ v .Only in the presence of values of ρ v > 0.2% and low levels of corrosion (≤10%-20%), most of the specimens develop a vertical strain higher than two times the yield strain.In the presence of ρ v < 0.1%, the ratio ε v /ε yv is lower than unity and cot θ is lower than about 1.5.In this regard, also note that Chen et al. 14 experimentally observed that the deformation capacity of corroded steel bars drastically reduces in the presence of cross-sectional area loss ratios greater than a critical value (η am,crit ).This latter value was estimated by Chen et al. 14 as 100 (1 À f yv /f uv ) where f yv and f uv are the yield strength and the ultimate strength of the transverse reinforcement, respectively.Assuming f uv /f yv = 1.15typical of common steel reinforcements-the η am,crit is about 13.3%.

| Comparison with laboratory tests
In this section, the proposed model is applied to replicate the experimental shear force-displacement curves obtained for some beams tested in the past by other researchers.In particular, the procedure is first applied to some specimens tested by Xue et al. 10 that failed in shear.Then, it is applied to simulate the response of some specimens tested by Rodriguez et al. 32 and failed in shear or in flexure.
In any case, at the generic load step, the mid-span deflection is estimated as the sum of the flexural and shear contributions, that is, δ = δ f + δ s .To calculate the flexural contribution to the deflection δ f , the first moment of area of the area under the curvature diagram of the cross-sections between the support and the considered cross-section (i.e., the midspan cross-section) is needed as reported in the following relation 33 :

Reference
In the present work, δ f is evaluated in a closed form to avoid the numerical integration of the curvature diagram.
It is assumed that it is the sum of elastic and plastic components, that is, δ fe and δ fp , respectively (Figure 5).The elastic component is calculated by taking advantage of the linearity of curvature between the support and the crosssection at which the bottom longitudinal reinforcement has not yielded yet as , where a m is half of the length of the central part of the beam subjected to pure moment (= 0 for three-points load scheme) and, therefore, at constant curvature.If the bottom longitudinal reinforcement is in its inelastic range of behavior, also the plastic component of deflection needs to be estimated.For a three points test, it is assumed that the plastic hinge has a length equal to the cross-section depth (i.e., L p = d), while for a four points test the plasticized portion of the beam is between the two load points.Then, the plastic contribution of deflection is δ fp = (χ À χ y ) (a À L p /4)L p /2 for a three points test and δ fp = (χ À χ y )(a + a m /2)a m for a four points test, respectively.In the previous formulas χ y is the curvature at yield and L p is the plastic hinge length.The shear deflection is obtained from the shear analysis as δ s = 2a Â (ε x À ε 2 )cot θ.
The specimens tested by Xue et al. 10 have a rectangular cross-section (120 Â 240 mm) and are strongly reinforced in bending (two D19 rebars) to fail in shear.The specimens are classified into two series (B-39 and B-52) and are characterized by different values of ρ v , i.e., 0.39% and 0.52%, respectively.The mechanical properties of the materials and the corrosion levels of the transverse reinforcement of the considered specimens are reported in Table 1.The experimental and numerical shear forcedeflection curves of the specimens 10 are plotted in Figure 6.The peak load of the specimens is predicted with sufficient accuracy by the proposed model and a failure mode for shear tension is achieved in all the cases (Table 3), while the elongation of stirrups is always lower than the limit value calculated as proposed by Imperatore et al. 7 However, the curves show that the slope of the curves of the numerical models is higher than that of the laboratory specimens.To evaluate the accuracy of the results obtained by means of the proposed model, an additional numerical analysis for uncorroded specimens (B-39-0 and B-52-0) has been performed by means of the program Response2000 (r2k) and plotted in Figure 6 by El-Sayed et al  means of black square dots.The numerical results of the two models are comparable, in terms of both loads and deflections.Note that also Xue et al. 10 tried to reproduce their laboratory experiments by means a finite element analysis and that also the results of their analyses exhibited an overestimate of the slope of the shear forcedeflection curves.The specimens tested by Rodriguez et al. 32 in series 31 have a rectangular cross-section (150 Â 200 mm); the beams considered in this analysis are named 313, 314, 315, and 316.The longitudinal reinforcement is characterized by ρ lb = 1.77%, whereas the transverse reinforcement is characterized by a value of ρ v equal to 0.22% or 0.44%.The mechanical properties of the materials and the corrosion levels of the transverse reinforcement of the considered specimens are reported in Table 1.Note that only specimen 315 failed in shear.The others specimens failed in flexure because of concrete crushing.The experimental and numerical responses of the above specimens tested by Rodriguez et al 32 are plotted in Figure 7.For each specimen, the experimental curve is reported by solid line whereas the numerical curve is reported by solid line and circle symbols.In addition, a yellow solid triangle is plotted in Figure 7, which identifies the point of the curve corresponding to a stirrup strain equal to the ultimate strain according to the proposal by Imperatore et al. 7 for pitting corrosion.
The response of specimen 315 is reproduced with sufficient accuracy, in terms of stiffness, deflection, and load corresponding to failure.The numerical responses obtained for the other tests are characterized by a lower slope of the curve and by a lower ultimate load (with the only exception of specimen 313).These results can be explained by analyzing some of the assumption at the basis of the model.In fact, due to medium-high levels of the weight loss ratio (higher than 10%), the effective width of the beam is lower than the nominal width and this leads to a reduction of the stiffness of the element.Further, the proposed model assumes that the maximum cross-section loss is equal in all the steel bars of either the longitudinal or transverse reinforcement.Owing to this, no distinction is made between external and internal longitudinal rebars or stirrups positioned close or far from the point load.A conservative estimate of stiffness and ultimate load of the beams is thus expected.
It is also to note that in the application of the proposed model to the above tests carried out by Rodriguez et al., 32 the cross-section area of the bars is prudently calculated removing from the nominal cross-section the circular crown with thickness equal to the maximum pit depth.However, in order to calculate the residual crosssection area of corroded steel bar different and less conservative approaches than that of the herein assumed circular crown are present in the literature to calculate the residual cross-section area of corroded steel bars and they start from the assumption of a different shape of pit.To evaluate the effects of these conservative approaches, Val's model 34 is considered in the proposed procedure and the corresponding load-displacement curves are plotted in Figure 6 by solid lines with square symbols.The use of the model proposed by Val 34 for the estimation of the residual cross-section area of bars leads to larger residual cross-section areas and thus to higher load bearing capacities.Further, referring to specimens 313, 314, and 316 (which failed in flexure), the proposed procedure with Val's model reproduces the slope of the experimental curves with better accuracy.However, in specimen 315, the use of the proposed procedure with Val's model would lead to a flexural crisis, which is in contrast with the experimental evidence.This is because, at this stage of the research, the proposed procedure is not able to capture the limitation of the load bearing capacity because of deterioration of bond between tensile longitudinal bars and concrete.In particular, when significant loss of bond occurs, an arch mechanism can develop which resists the external forces by means of the compressive strength of concrete and the tensile strength of the longitudinal bars at the supports.
To prove that the shear capacity of the selected beams may, in the case of significant deterioration of bond between steel bars and concrete, be limited to values that are close to those of the laboratory tests, the strut and tie model recently proposed by Coronelli et al. 35 has been applied to the considered tests.In particular, the model proposed by Coronelli et al. 35 is formulated for the estimation of the shear strength of RC beams with corroded reinforcement and takes into account the detrimental effects of bond deterioration.
The results of the approach proposed by Coronelli et al. 35 are shown in Figure 7 in terms of shear strength by means of a horizontal blue solid line and in terms of failure mode (reported in words close to the blue line).The failure mode predicted by Coronelli et al. 35 is governed by flexure in specimens 313 and 314 and is in good accord with the results of the proposed model.Coronelli et al. 35 predicted a shear failure mode due to arch mechanism for specimens 315 and 316.This mechanism cannot be reproduced by the present proposed model; however, the shear strength value calculated with the proposed model with the more conservative description of the residual cross-section area is comparable with the experimental shear strength.A further development of the proposed model could include the corrosion-induced bond deterioration effects, as recently assumed in more complex finite element model. 36,37

| PARAMETRIC ANALYSIS
To investigate more in depth the role of η am on ε v /ε yv and cot θ, a parametric analysis has been carried out on a RC beam with rectangular cross-section.The cross-section is 300 Â 500 mm in size and the ratio a/d is equal to 3. The geometric percentages of the longitudinal and transverse reinforcements (ρ lb and ρ lt ) are equal to 1.94% and 0.48%, respectively.Stirrups spacing s v is obtained assuming ρ v in the range from 0.10% to 0.30% (typical of existing buildings and/or structures weak for shear).The maximum cross-sectional area loss ratio η am ranges from 0% to 90%.The values of f cm and f y are 35 and 450 MPa, respectively.
The τ-δ curves of all the cases are reported in Figure 8.The slope of the ascending branch of the curves referring to the cases with η am > 10% is higher than the slope in the case with uncorroded stirrups, mainly because in the cases with moderate-to-high corrosion a reduced width of the beam cross-section has been considered in accordance with Higgins et al., 23 see Equations ( 3) and ( 4).As expected, the shear stress τ increases with the geometric percentage of (uncorroded) stirrups ρ v0 and decreases with the increase of η am .The displacement obtained at peak load increases with the geometric percentage of (uncorroded) stirrups ρ v0 .
To investigate further the effects of stirrups corrosion on the shear strength of the RC beams, the response parameters that appear in Equation ( 9) at peak load, that is, σ sv /f yv , cot θ, and σ c1 /f ct , have been plotted in Figure 9 as a function of the mechanical percentage of (corroded) stirrups (ω v = ρ v f yv /f cm ) for different values of ρ v0 (0.10%Ä0.30%).Corrosion influences ω v because ρ v is calculated considering both A v [Equation ( 1  As is evident from the figure, the ratio σ sv /f yv increases with ω v .For values of ρ v0 < 0.15% the stirrups remain elastic, in the occurrence of all the considered values of η am (Figure 9a).With the increase of ρ v0 above 0.15%, the shape of the τ-δ curve changes and stirrups start yielding at some values of ω v .These latter values increase with ρ v0 .In particular, stirrups yield for ω v > 0.010 if ρ v0 = 0.175% and for ω v > 0.018 if ρ v0 = 0.30%.
The parameter cot θ generally increases with ω v .In the cases where the transverse reinforcement remains elastic (due to low geometric percentages of stirrups and/or to medium-high levels of corrosion) the angle θ is higher than about 33.2 (cot θ ≤ 1.53) (Figure 9b) and thus negligible re-orientation of the diagonal compression stress of concrete takes place.The relationship between cot θ and ω v follows a nonlinear trend.In particular, the cot θ first increases with ω v , until a peak value has been reached.Then, slowly decreases with the increase of ω v .While this final trend of cot θ is in accord with the results reported by Walraven et al. 38 for beams with higher values of ρ v , the first trend of the relationship is not reported by other researchers and is useful for a more in-depth comprehension of the response of slender beams with low values of ρ v .
A numerical regression analysis of the data leads to the simple relations (see Figure 9): Equations ( 29) and ( 30) suggest a simple formulation for the prediction of the shear strength of RC beams with F I G U R E 1 1 Comparison between experimental shear strength (V exp ) and numerical shear strength obtained by means of the simplified proposed formulation (V prop ) for the Reineck et al. 39 database.
low values of ρ v (≤0.30%).Indeed, introducing these two equations into Equation ( 9) and considering that stirrups have yielded at peak load-as also assumed in many codes-the shear strength can be easily calculated as follows: To validate the simplified proposed relation, a comparison with the previously considered database of experimental test results is plotted in Figure 10.The ratios R v resulting from the application of the simplified proposed relation range from 0.80 to 1.86 (see The effects of medium-high corrosion levels of stirrups can mainly be interpreted as a reduction of the geometric percentage of stirrups.Owing to this, the proposed equation for the calculation of the shear strength has also been validated with reference to the beams present in the database collected by Reineck et al. 39 and characterized by a/d ≥ 2.5 and ρ v ≤ 0.30%.Based on these criteria, a set of 74 specimens has been obtained and used for validation.The experimental versus numerical shear strength values are reported in Figure 11.The proposed simplified relationship leads to a slightly conservative prediction with a mean value of 1.13 and a CoV of 0.23 of the experimentalto-numerical shear strength ratio R v .These values are similar to those obtained for the selected corroded beams and give more validity to the proposed simple formulation.

| CONCLUSIONS
This research study presents an analytical procedure for the evaluation of the monotonic response of slender RC beams failing in shear and endowed with corroded stirrups.The proposed model considers the reduction of both the cross-sectional area and the strain capacity of the transverse reinforcement because of steel corrosion, along with the reduction of the beam width because of spalling of the concrete cover.The numerical model is validated by comparison with the results of 62 tests on slender RC beams with corroded stirrups and with the results deriving from the application of the CCCM to the same set of beams.Finally, a simple relation for the prediction of the shear strength of RC beams with low values of the effective geometric percentage of (corroded) stirrups has been derived.
The research findings are summarized as follows: • The proposed model is able to consider the effects of the strain capacity of corroded steel bars, recognizing the failure modes governed by either stirrup yielding or rupture.• The proposed model provides sufficient predictions of the shear strength of RC beams with corroded stirrups; in particular, the mean value of the numerical to experimental shear strength ratio is equal to 1.07 while the CoV is equal to 0.30.The response at peak load is in good agreement with that resulting from the application of the Capacity Compression Chord Model.• In all the considered beams, the maximum strain demand of the stirrups is lower than the strain capacity evaluated for localized corrosion.

F I G U R E 1
Model of the RC beam: (a) lateral view, (b) crosssection, (c) flexural model, and (d) shear model.RC, reinforced concrete.
Comparison between experimental and numerical shear strength: (a) proposed model and (b) CCCM.CCCM, Compression Chord Capacity Model.T A B L E 3 Numerical results.

F I G U R E 3
Peak response of the beams in the considered database: (a) experimental to numerical shear strength ratio, (b) concrete and stirrups contributions to the shear stress, (c) cotangent of the angle of inclination of the diagonal compressive stress field of concrete, and (d) vertical strain to yield strain ratio.F I G U R E 4 Vertical strain to yield strain ratios obtained at peak load by means of the proposed model versus (a) cross-sectional area loss ratio, (b) cotangent of the angle of inclination of the diagonal compressive stress field of concrete, and (c) geometric percentage of the (corroded) transverse reinforcement.

F
I G U R E 5 Calculation scheme of elastic and plastic contributions of deflection for (a) three-points test and (b) four-points test.
)] and b w,eff [Equations (3) and (4)].Owing to this, in Figure8the dots with the same color identify cases characterized by the same values of ρ v0 but different η am .

F I G U R E 9
Results of the peak response of the parametric analysis: (a) stirrups stress to yield strength ratio, (b) cotangent of the angle of inclination of the diagonal compressive stress field of concrete, and (c) principal tensile stress to tensile strength ratio versus the mechanical percentage of corroded stirrups.F I G U R E 1 0 Comparison between experimental shear strength (V exp ) and numerical shear strength obtained by means of the simplified proposed formulation (V prop ).
The proposed model is validated by comparison with results of tests available in the literature (Table

Table 2 )
, with a mean value equal to 1.16 and a CoV equal to 0.19.The mean value indicates a slightly conservative estimate of the shear strength.The range between minimum and maximum values is similar to those obtained by means of both the proposed numerical model and the CCCM.The CoV obtained by the proposed relation, however, is significantly lower than those obtained by means of the above other models.

•
In the beams with ρ v < 0.10%, stirrups remain elastic and cot θ < 1.5.In such cases, a negligible reorientation of the diagonal compression stress of concrete occurs.•The proposed model does not consider steel-concrete bond degradation due to corrosion and is not able to reproduce the strut mechanism that can be triggered by failure of the steel-concrete bond.Further developments of the proposed model could involve accurate modeling of the corrosion-induced degradation of the steel-concrete bond response.• Based on the results of a parametric analysis performed with the proposed numerical model, a simple relation is suggested for the prediction of the shear strength of RC beams with ρ v ≤ 0.30%, typical of existing building.The application of this relation to the above laboratory specimens proves that it is able to provide accurate estimates of the load-bearing capacity of such RC beams.The mean value of the V exp /V prop ratios resulting from the application to the above database is equal to 1.16, whereas the CoV is equal to 0.19.To validate the proposed relation further, this relation has also been applied to a database of 74 uncorroded RC slender beams with a low percentage of stirrups (ρ v ≤ 0.30%), achieving satisfactory results both in terms of mean (=1.13) and CoV (=0.23) of the V exp /V prop ratio.