Post-Cracking Shear Stiffness Model of Reinforced Concrete Beams

: Macro diagonal cracks can signiﬁcantly reduce the stiffness of slender reinforced concrete (RC) beams, which results in excessive deﬂection compared with limitations from design speciﬁ-cations. To evaluate the post-cracking stiffness of slender RC beams with diagonal cracks, a shear degradation model that considers shear deformation is proposed. Based on the variable angle truss model, this study deduced the strut angle formula based on the minimum energy principle. Then, the relationship between the stirrup yielding shear stiffness and elastic shear stiffness was modeled. Finally, the calculation procedure was developed by quantifying the stiffness degradation tendency. The comparison between the experimental results of deﬂection and the proposed analytical method showed good agreement. Additionally, the proposed method can capture the full-range features of shear strain curves.


Introduction
Diagonal cracks are commonly seen in cast-in-place and precast reinforced concrete (RC) box girder bridges, which results in additional shear deformation besides flexural deflection.Therefore, accurately computing the total deformation is a perquisite for the design of RC box girder bridges.Typically, the total deformation can be computed by adding the deformations caused by shear and flexural deformations.In order to decouple the contribution of diagonal cracking to deflection from other time-varying factors (such as concrete shrinkage and creep), it is necessary to evaluate the shear stiffness of diagonal cracked bridges.However, due to the complex influencing mechanism of shear cracks on shear stiffness degradation, quantitative and practical evaluation methods that are suitable for engineering applications are particularly needed.
From a practical perspective, current deflection models for RC beams, whether cast in situ or prefabricated, typically ignore the contribution of shear deformations [1,2].However, some prior shear tests on slender beams revealed that the measured deflections are generally much larger than the prediction-by-code method [3][4][5].One possible reason for these underestimations may be from the neglect of the contribution of shear deformations, especially the contribution of shear deformation after shear cracking [6][7][8][9][10][11].
In the face of frequently occurring excessive deflections and shear failure in large-span concrete and precast concrete bridges and buildings, it is crucial to build rational shear models.The shear stiffness of a diagonally cracked RC beam is approximately 10% to 30% of the shear stiffness of the uncracked one, depending on the web reinforcement provided [12].This means that diagonal cracking has a much larger effect on shear stiffness than flexural cracking on bending stiffness.However, unlike the calculation of bending stiffness, which is based on the Branson's effective moment of inertia model and is accurate enough for design application [13], there is still no widely accepted shear stiffness model for cracked RC beams.Therefore, a simple, rational, and practical shear stiffness model for RC beams with shear cracks is needed.
Recently, the lower-bound theory of plasticity [9] and minimum strain energy principle [7] have been used for calculating the strut angle.Based on the minimum strain energy principle, truss models with variable strut angles were used to investigate the effective shear stiffness of RC beams with diagonal cracking [8, 24,25].
For the shear stiffness degradation criteria, scholars have proposed various rules, such as linear tangent degradation [7], linear secant stiffness degradation [26], hybrid stiffness degradation [9], exponential degradation [8], adopted Branson's degradation [3], etc.However, due to the complex influencing mechanism of shear cracks on shear behavior [27][28][29][30], there is a need for shear stiffness theories that can properly reflect the gradual degradation of shear stiffness with the development of shear cracks.
This paper aims to propose an energy-based strut angle calculation method and a practical degradation rule of the effective shear stiffness of diagonally cracked RC beams.With the adoption of the minimum strain energy principle and the additional consideration of the contribution of moment, an implicit analytical calculation formula for strut angle is established.Furthermore, through a parameter analysis and comparative study with implicit expressions, a simplified practical strut angle calculation formula is obtained.Experimental validation shows that the proposed method yields good and consistent predictions of post-cracking shear stiffness and shear deformation.

Intial Elastic Shear Stiffness K e
Beam shear can be divided into two stages: pre-cracking stage and post-cracking stage.At the first stage, the elastic shear stiffness K e can be calculated using elasticity theory [12], as shown in Equation (1).
where K e is the elastic shear stiffness before cracking, G c is the shear modulus of concrete, E c is the modulus of elasticity of concrete, µ is the Poisson's ratio of concrete, A v is the effective shear area (A v = b w d v ), b w is the web width of the beam, and d v is the effective shear depth.
In the post-cracking stage, as diagonal cracking violates the continuity of concrete, an alternative approach must be adopted for the post-cracking shear stiffness calculation.

Stirrup Yielding Shear Stiffness K y Based on Truss Model
A truss model gives good depiction of slender beams in shear, which provides a possibility that the post-cracking shear stiffness can be depicted by the truss model.For slender beams, when the stirrups are yielded and the diagonally shear cracks are well developed, the truss model shown in Figure 1 can be used for analyses.As the observed shear cracks are generally parallel to each other in principle, the inclination of the strut angle θ y can be assumed to be the same.The shear deformation of the truss is mainly induced by the deformation of web members δ web , including the elongation of stirrups δ s and the vertical deformation δ c caused by the shortening of the inclined strut.The deformation of stirrups and inclined concrete struts are listed in Table 1, in which A s , A g , and A cm are the areas of the longitudinal reinforcement, cross-section, and compression zone, respectively.

Stirrup Yielding Shear Stiffness Ky Based on Truss Model
A truss model gives good depiction of slender beams in shear, which provides a possibility that the post-cracking shear stiffness can be depicted by the truss model.For slender beams, when the stirrups are yielded and the diagonally shear cracks are well developed, the truss model shown in Figure 1 can be used for analyses.As the observed shear cracks are generally parallel to each other in principle, the inclination of the strut angle θy can be assumed to be the same.The shear deformation of the truss is mainly induced by the deformation of web members δweb, including the elongation of stirrups δs and the vertical deformation δc caused by the shortening of the inclined strut.The deformation of stirrups and inclined concrete struts are listed in Table 1, in which As, Ag, and Acm are the areas of the longitudinal reinforcement, cross-section, and compression zone, respectively.Equal to the shear distortion (shear deformation per unit length) induced by a unit shear force, the shear stiffness of the equivalent truss can be expressed as ( ) where n is the ratio of the elastic modulus of the reinforcement and the concrete (n = Es/Ec), and ρv is the stirrup ratio.Equation (2) shows that the shear stiffness is mainly affected by the material type (n, Es, Ec), geometric properties (Av), stirrup ratio (ρv), and strut angle (θy).
Table 1.Member deformation of the variable-angle truss model.In order to evaluate the degradation of the shear stiffness of the beam after the yielding of stirrups, it is necessary to establish the relationship between the stirrup yielding

Shear force V Stirrups
Equal to the shear distortion (shear deformation per unit length) induced by a unit shear force, the shear stiffness of the equivalent truss can be expressed as where n is the ratio of the elastic modulus of the reinforcement and the concrete (n = E s /E c ), and ρ v is the stirrup ratio.Equation (2) shows that the shear stiffness is mainly affected by the material type (n, E s , E c ), geometric properties (A v ), stirrup ratio (ρ v ), and strut angle (θ y ).
In order to evaluate the degradation of the shear stiffness of the beam after the yielding of stirrups, it is necessary to establish the relationship between the stirrup yielding shear stiffness K y and the initial shear stiffness K e .Combine Equations (1) and (2), the ratio of K y to K e can be expressed as where the ratio factor ξ y is defined as the degradation coefficient of shear stiffness.Apparently, the parameters that have an effect on the shear stiffness degradation are the stirrup ratio ρ v and the strut angle θ y .For a specified specimen, n, µ, and ρ v are all known; if θ y is determined, the degradation coefficient can be easily calculated from Equation (3).Similarly, the yielding shear stiffness can be obtained from Equation (2).
Considering that the stirrup ratio ρ v of practically used RC beams is generally between 1.0% and 2.0%, n = E s /E c ≈ 6 and the Poisson's ratio of the concrete µ ≈ 0.2.For θ y between 25 • and 45 • , the shear stiffness degradation coefficient is between 0.12 and 0.29.That is to say, for concrete beams with severe web shear cracks, the yielding shear stiffness can be reduced by more than 70% compared with the elastic shear stiffness.

Shear Stiffness Degradation Rules for Partially Shear-Cracked RC Beams
Because the transition mechanism from an elastic shear stiffness to a stirrup yielding shear stiffness is affected by many factors, such as reinforcement ratio, section size, concrete strength, etc., it is unrealistic to give an accurate and quantitative expression to describe this degradation process.
Scholars have developed some empirical or simplified degradation rules to depict the degraded shear stiffness of partially shear-cracked RC beams, such as linear secant stiffness degradation [30], linear tangent stiffness degradation [7], mixed degradation [9], exponential degradation [8], regression fitting degradation [10], adopted Branson's degradation [3], etc. Below are four representative effective shear stiffness models, where K eff,1 , K eff,2 , K eff,3 , and K eff,4 are the effective shear stiffness considering linear secant degradation, linear tangent degradation, mix degradation, and adopted Branson's degradation, respectively; V cr and V y are the diagonal cracking shear force and stirrup yielding shear force, which can be analytically calculated using Equations ( 8) and ( 9).
V cr = 0.17 After determining the key parameters of the yield shear stiffness and the degradation rule of the shear stiffness, the effective shear stiffness of a partially shear-cracked RC beam can be calculated according to the following steps: a.
Before shear cracking, the elastic shear stiffness K e can be calculated using Equation (1) and used for the calculation of elastic shear deformation; b.The cracking shear force V cr and stirrup yielding force V y of the critical section are calculated using Equations ( 8) and ( 9); c.
After shear cracking, the yielding shear stiffness K y can be obtained from Formula (2), and which the shear stiffness degradation coefficient ξ y can be calculated using Equation (3); d.The shear increment V-V cr under the shear load V at different loading levels is calculated and substituted into the selected formula from Equations ( 4)-( 7) to calculate the effective shear stiffness K eff of the RC beam with shear cracks; e.
The average shear strain or shear deformation of the corresponding beam segment is calculated according to the effective shear stiffness K eff under a specific shear load.

Determination of Strut Angle, θ y 3.1. Strut Angle Based on Minimum Energy Principle
It is believed that the angle θ y will occur at an inclination that requires the minimum potential energy [16].The strain energy stored in the truss members can be determined using the virtual work principle.Assume that a unit shear force is applied to the truss, the internal force and deformation of each member (including web members and chord members) of the truss can be obtained (See Figure 1 and Table 1).The total vertical deformation of the truss is the sum of the members' deformations listed in Table 1, that is: The strain energy stored in the truss is equal to the external shear work (EWD) applied to the truss, which is the distortion angle of the truss induced by a unit load.The distortion angle can be obtained by dividing the vertical deformation of the truss by its horizontal length: Based on the minimum strain energy principle, the rational strut angle leads to a minimum energy.By differentiating Equation (11) with respect to θ y and equating it to zero: Through simplification, Equation ( 12) becomes where 1 + nρ s , and M Vd v = λ MV , in which ρ s (=A s /A g ) is the longitudinal reinforcement ratio.The analytical solution of Equation ( 13) is too complex for expression.So, a trial-and-error procedure is used for the θ y calculation.
The factors R 1 , R 2 in Equation ( 13) remain constant for a specified section.For the rectangular section, the effective shear depth d v is the maximum of {0.72h, 0.9d} [14]; therefore, A v /A g can be assumed to be 0.72 for simplicity; the depth of the compression zone c can be simply assumed to be 0.35d [31]; therefore, A cm /A g is equal to 0.35.For the I-section, A v /A g is equal to A web /A g , and A cm /A g is still simply assumed to be 0.35.
The factor λ MV in Equation ( 13), which can be regard as the shear span-to-depth ratio, has a larger influence on shear deformation than the stirrup ratio ρ v during the pre-cracking stage but less influence during the post-cracking stage [3,6].This interesting phenomenon is further discussed in the next subsection and in the experimental validation.

Simplification Formula for Strut Angle
As is mentioned above, the shear-to-depth ratio λ MV has little effect on the postcracking shear deformation.To validate this supposition, first assume that the influence of moment is neglected, that is, λ MV = 0.Then, Equation ( 13) can be rearranged and worked out: The strut angles calculated using Equations ( 13) and ( 14) are compared to investigate the influence of λ MV (see Figure 2).

Shear Deformation Test in the Literature
To evaluate the proposed shear stiffness model, three series of shear test data were used for validation, in which the shear deformation was directly measured, and its corresponding equivalent shear stiffness was deduced.A total of 23 lattices/zones of 10 beams are used for comparison.It should be noted that many RC beam test data in which only the total deformation was measured cannot be used for validation and, therefore, were excluded.
The authors conducted shear deformation tests on six large thin-webbed RC beams.Self-tailored strain measuring lattices were installed on one side of the beams, and each measuring lattice was composed of five mechanical dial gauges.The mean shear strain of each lattice was obtained through a strain analysis.The detailed procedures are illustrated in reference [6].
Debernardi and Taliano [5] reported an experiment comprising six RC I-section beams with a thin web.Square lattices made up of transducers were arranged to measure the mean curvature and the mean shear strain, from which the shear deformations can be decoupled from flexural deformation.Fifteen lattices in four specimens (TR1, TR2, TR3, and TR6) with detailed shear strain curves were selected for validation.The specimens were simply supported, subjected to two symmetrical loads (TR1 and TR2), a symmetrical load (TR3), or a non-symmetrical load (TR6).
Hansapinyo et al. [4] tested four RC beams with web reinforcement to investigate the shear deformation after diagonal cracking.Electronic transducers were also used to measure the shear strains of each panel based on the rosette concept.All of the four specimens were selected for validation.The main parameters of all the specimens are listed in Tables 2 and 3.As is shown in Figure 2, the factor λ MV has an effect on the strut angle θ y , but slightly so, especially for a beam with a higher longitudinal reinforcement ratio ρ s .For example, when the longitudinal reinforcement ratio ρ s > 2.0%, the calculated strut angle of Equation ( 13) (taking into account the influence of λ MV ) is approximately 1.0 to 1.1 times of the calculated angle obtained using Equation ( 14) (assuming λ MV = 0).The relative difference is very small and can be ignored.
Particularly, when we set A v = 0.72 A g and A cm = 0.35 A g , the value of R 1 is approximately equal to 1 for cross-sections with ordinary longitudinal reinforcement arrangements.Therefore, Equation ( 14) can be simplified to:

Shear Deformation Test in the Literature
To evaluate the proposed shear stiffness model, three series of shear test data were used for validation, in which the shear deformation was directly measured, and its corresponding equivalent shear stiffness was deduced.A total of 23 lattices/zones of 10 beams are used for comparison.It should be noted that many RC beam test data in which only the total deformation was measured cannot be used for validation and, therefore, were excluded.
The authors conducted shear deformation tests on six large thin-webbed RC beams.Self-tailored strain measuring lattices were installed on one side of the beams, and each measuring lattice was composed of five mechanical dial gauges.The mean shear strain of each lattice was obtained through a strain analysis.The detailed procedures are illustrated in reference [6].
Debernardi and Taliano [5] reported an experiment comprising six RC I-section beams with a thin web.Square lattices made up of transducers were arranged to measure the mean curvature and the mean shear strain, from which the shear deformations can be decoupled from flexural deformation.Fifteen lattices in four specimens (TR1, TR2, TR3, and TR6) with detailed shear strain curves were selected for validation.The specimens were simply supported, subjected to two symmetrical loads (TR1 and TR2), a symmetrical load (TR3), or a non-symmetrical load (TR6).
Hansapinyo et al. [4] tested four RC beams with web reinforcement to investigate the shear deformation after diagonal cracking.Electronic transducers were also used to measure the shear strains of each panel based on the rosette concept.All of the four specimens were selected for validation.The main parameters of all the specimens are listed in Tables 2 and 3. "*"-The yielding shear force V y was calculated using Equation ( 9), as the specimen failed in flexure before the stirrups yielding in shear span.

Comparison of Strut Angle and Degradation Coefficient
The stirrup yielding strut angles θ y , θ P , θ H were calculated using the proposed method, Pan et al. [7], and He et al. [9], respectively.The stirrup yielding strut angles θ y , θ P , θ H and corresponding calculated degradation coefficient ξ y , ξ P , ξ H are listed in Table 4. Overall, the average ratio of the predicted values to the observed degradation coefficient ξ exp and its coefficient of variation (CV) are 0.95 and 0.19, 0.76 and 0.25, and 0.95 and 0.20 for the proposed method, Pan's method, and He's method, respectively.
As is shown in Table 4, Pan's method obtain a higher prediction of strut angle and lower prediction of degradation coefficient than the other two methods, and it has the largest value of CV.The other two methods obtain a relatively good prediction of shear stiffness degradation.However, He's prediction method ignores the influence of the longitudinal reinforcement ratio and may cause larger deviations for specimens with a low longitudinal reinforcement ratio.

Comparison of Effective Shear Stiffness and Shear Strain
The measured and calculated effective shear stiffness reduction factor ξ and shear strain of 23 zones of 10 beams are shown in Figures 3-7.All the shear force-shear strain curves have a distinct turn point, which means the shear stiffness decreases abruptly after the first diagonal cracking.Behind the turn point, the curve stays nearly linear before the stirrup yielding.Based on the experimental results, here, the linear tangent degradation model, which depicts the main features of shear strain curves of slender RC beams, was adopted to evaluate the post-cracking shear stiffness and was used for comparison.

Comparison of Effective Shear Stiffness and Shear Strain
The measured and calculated effective shear stiffness reduction factor ξ and shear strain of 23 zones of 10 beams are shown in Figures 3-7.All the shear force-shear strain curves have a distinct turn point, which means the shear stiffness decreases abruptly after the first diagonal cracking.Behind the turn point, the curve stays nearly linear before the stirrup yielding.Based on the experimental results, here, the linear tangent degradation model, which depicts the main features of shear strain curves of slender RC beams, was adopted to evaluate the post-cracking shear stiffness and was used for comparison.As is known, the shear span-to-depth ratio λMV has a significant effect on shear strain before cracking.However, as is observed from the tests, the influence becomes very small for RC beams at the post-cracking stage (See A to H zones with different λMV values of As is known, the shear span-to-depth ratio λ MV has a significant effect on shear strain before cracking.However, as is observed from the tests, the influence becomes very small for RC beams at the post-cracking stage (See A to H zones with different λ MV values of specimens TR1 to TR6 in Figures 4 and 5).
Three methods for determining the strut angle are used for comparison: the proposed method, method by Pan et al. [7], and method by He et al. [9].As is shown in Figures 3-7, all of the three methods give a rational evaluation of effective shear stiffness, while Pan's method gives a more conservative result than the two other methods.Compared to He's method, the proposed method accounts for the influence of ρ s and gives a better prediction for beams with a lower ρ s (Specimen TR2 in Figure 5).Based on the assumption that the longitudinal reinforcement will not yield before stirrup yielding, He's method tends to give a similar prediction of effective shear stiffness with the proposed method for RC beams, with (ρ s f ys )/(ρ v f yv ) > 3.
From Figures 3-7, it can be seen that the proposed shear stiffness degradation model simulates the degradation process of shear stiffness very well and provides better stiffness prediction results in most cases.Although the stiffness prediction results of specimen S2 are slightly larger (about 20%), the shear stiffness degradation law still conforms well.It should be noted that for specimen S2, the deviation between the predicted value calculated using the analytical method in the original literature [4] and the measured shear strain is also the largest, which may be due to the shear stiffness anomaly caused by the specimen's own defects.
For thin-webbed beams, a linear tangent stiffness degradation tends to give safe predictions.Therefore, the authors suggest that linear tangent degradation criteria are used for the concrete box girder bridge's shear stiffness evaluation in engineering practice.
In addition, although there is a significant difference in the shear span-to-depth ratio between different beam segments of the same specimen in the experiment (such as TR1~TR6), there is no clear connection between the measured shear stiffness degradation coefficient and the shear span-to-depth ratio.The stiffness degradation curves of each beam segment are also relatively close, indicating that the shear span ratio has little effect on the post-cracking shear stiffness degradation, and its impact on the shear stiffness and strain after cracking can be ignored.
Finally, the proposed shear stiffness model for RC beams is only validated using 23 beam segments.More shear deformation data are needed for further validation.The digital image correlation (DIC) technique based on shear deformation measurements may be widely used in a future study [32][33][34][35].

Conclusions
To evaluate the effective shear stiffness of RC beams with diagonal cracks, this paper proposed a post-cracking shear stiffness degradation model.Based on the variable truss model, and using the minimum strain energy principle, the strut angle was analytically deduced.Then, the linear tangent stiffness degradation rules were found to be more suitable for thin-webbed concrete beams.With experimental validation, the following conclusions can be drawn: 1.
Based on the variable-angle truss model, the relationship between the stirrup yielding shear stiffness and elastic shear stiffness was established.Then, a practical linear tangent shear stiffness degradation model, which depicts the main features of shear strain curves of slender RC beams at the post-cracking stage, was adopted to evaluate the effective shear stiffness.

2.
The strut angle θ y , which was found as a function of the stirrup ratio, longitudinal reinforcement ratio, and elasticity modulus ratio, was determined using the minimum energy principle.Compared with other two methods in the literature, the proposed angle equation tended to give a moderate prediction of strut angles and degradation coefficients for varying beam parameters.

3.
A turning point occurs in the shear strain curves corresponding to the first diagonal crack.Behind the turn point, the tangent slope of the curve remains nearly constant before the stirrup yielding, especially for thin-webbed beams.Additionally, the shear span-to-depth ratio λ MV has little effect on the shear deformation of slender RC beams at the post-cracking stage.4.
The analytical prediction was compared with the shear strain data of 23 zones.The results showed that the proposed method gives a good and consistent prediction of the effective shear stiffness and shear strain.The proposed degradation model can be used for the post-cracking shear stiffness evaluation of shear-cracked RC beams. 5.
In practice, the procedure introduced in Section 2.3 can be used for a quick shear stiffness evaluation of concrete box girder bridges in service, on which performing experimental tests may be impractical and costly.

Figure 1 .
Figure 1.Truss model for shear response of diagonally cracked RC beams.

Figure 1 .
Figure 1.Truss model for shear response of diagonally cracked RC beams.

Figure 2 .
Figure 2. The influence of λ MV on angle θ y .

Table 1 .
Member deformation of the variable-angle truss model.

Table 2 .
Details of specimens.

Table 3 .
Shear-to-span depth ratio of each measuring zone.

Table 4 .
Strut inclination θ y and degradation coefficient ξ y at stirrup yielding status.

Table 4 .
Strut inclination θy and degradation coefficient ξy at stirrup yielding status.
Author Contributions: Conceptualization, K.Z.; Data curation, S.N. and J.G.; Funding acquisition, K.Z.; Investigation, K.Z. and S.N.; Methodology, K.Z.; Project administration, Y.Z. and Y.W.; Resources, Y.W.; Software, J.G. and M.G.; Supervision, Y.Z. and Y.W.; Validation, S.N., J.G. and M.G.; Writingoriginal draft, K.Z.; Writing-review and editing, Y.Z. and Y.W.All authors have read and agreed to the published version of the manuscript.This research was funded by the National Natural Science Foundation of China (52108150), China Postdoctoral Science Foundation (2021M691606), State Key Laboratory of Mechanical Behavior and System Safety of Traffic Engineering Structures (KF2020-17), S&T Program of Hebei (215676145H), and Nanjing Forestry University Undergraduate Innovation Training Program (202310298024Z).
Funding:Data Availability Statement: Data are contained within the article.