Simplified Evaluation of Shear Stiffness Degradation of Diagonally Cracked Reinforced Concrete Beams

Shear cracking in concrete box-girder bridges, which could cause excessive deflection during the serviceability limit state, cannot be effectively avoided by code-guided design. While elastic shear deformation only accounts for a small proportion of total deformation for un-cracked reinforced concrete (RC) beams, the magnitude of after-cracking shear deformation becomes comparable to flexural deformation for RC beams. However, there is still a lack of practical models to predict the after-cracking shear deformation of RC beams. First, six thin-webbed I beams were tested to investigate the shear stiffness degradation mechanism and the decrease ratio. Then, a very simple truss strut angle formula, which is the crucial parameter for shear stiffness, was established. Furthermore, a stiffness degradation rule for partially cracked beams was proposed considering the influence of concrete tension stiffening, which is essential for predicting the development process of after-cracking shear deformation. Finally, directly measured shear strains were used to validate the proposed shear stiffness model. The results showed that the shear stiffness drops to about 30~40% of the original stiffness after the first diagonal crack, and the remaining shear stiffness is only about 10% of the original one when the stirrup yields. Increasing the stirrup ratio is a more effective method to control shear stiffness degradation for diagonally cracked RC beams. Also, the proposed shear stiffness model well captures the main features of the shear stiffness degradation, and it provides a relatively accurate prediction of the equivalent shear stiffness at the post-cracking stage.


Introduction
As is well known, the deformation of concrete beams mainly consists of two parts: bending deformation and shear deformation [1]. Generally, the deformation of beams is mainly bending deformation, and the magnitude of shear deformation is small, which can be ignored [2,3]. However, for long-span concrete box-girder bridges commonly used in bridge engineering, this assumption may cause deviations [4]. Especially for the thinwebbed box girder with diagonal cracking, its shear deformation may be equivalent to the bending deformation [5].
According to a finite element analysis of a thin-webbed box-girder bridge with large web height [4], the magnitude of elastic shear deformation under load is considerably large and may reach a level that cannot be ignored. Scholars' experiments on thin-webbed concrete beams [1,6] have shown that: after the appearance of diagonal cracks, the shear deformation of the beam will significantly increase, with a shear deformation ratio of over 30% of the total deformation. Further, according a survey on an existing concrete box-girder

Test Object and Design Concept
To study the impact of diagonal cracks on the degradation of shear stiffness and development of shear deformation of concrete thin web beams, a direct shear measurement test was conducted on six concrete thin-webbed constrained beams. The main experimental objectives include: (1) Achieve continuous direct measurement of shear deformation before and after diagonal cracks in the concrete web; (2) Analyze the amplitude of changes in shear deformation values before and after shear cracking and study the degree of influence of diagonal cracks on shear deformation; (3) Study the degradation law of shear stiffness after the development of diagonal cracks. In order to achieve the above experimental objectives, the main experimental ideas used include: (1) Using large-scale thin-webbed I-shaped cross-section specimens to better simulate the stress behavior of thin web bridges, while facilitating the testing of web strain and the observation of diagonal cracks. (2) Adopting a reinforcement design with "strong bending and weak shear" concept, ensuring the priority occurrence and full development of diagonal cracks, with a focus on observing the impact of diagonal cracks on shear deformation and shear stiffness. (3) Constrained beams are used to investigate the shear performance of concrete beams under different combinations of bending and shear internal forces. (4) The effects of inclined bottom chord on diagonal crack and shear strength were investigated by using two types of specimens, namely, equal-height beam and variableheight beam. (5) Propose a strain-based shear deformation calculation method for arbitrary quadrilateral lattices, achieving direct peeling measurement of bending deformation and shear deformation.

Specimen Parameters and Setup
A total of 6 I-shaped cross-section specimens were made and divided into two groups: Group BC consisted of 2 beams of equal height; Group BV consisted of 4 variable-height beams, with the upper flange of the cross-section horizontal and the lower flange height varying in a parabolic manner. All the specimens were constructed in two batches on site ( Figure 1). The width of the beam web is 100 mm, and the beam length is 5400 mm.
In order to achieve the above experimental objectives, the main experimental ideas used include: (1) Using large-scale thin-webbed I-shaped cross-section specimens to better simulate the stress behavior of thin web bridges, while facilitating the testing of web strain and the observation of diagonal cracks. (2) Adopting a reinforcement design with "strong bending and weak shear" concept, ensuring the priority occurrence and full development of diagonal cracks, with a focus on observing the impact of diagonal cracks on shear deformation and shear stiffness. (3) Constrained beams are used to investigate the shear performance of concrete beams under different combinations of bending and shear internal forces. (4) The effects of inclined bottom chord on diagonal crack and shear strength were investigated by using two types of specimens, namely, equal-height beam and variableheight beam. (5) Propose a strain-based shear deformation calculation method for arbitrary quadrilateral lattices, achieving direct peeling measurement of bending deformation and shear deformation.

Specimen Parameters and Setup
A total of 6 I-shaped cross-section specimens were made and divided into two groups: Group BC consisted of 2 beams of equal height; Group BV consisted of 4 variableheight beams, with the upper flange of the cross-section horizontal and the lower flange height varying in a parabolic manner. All the specimens were constructed in two batches on site ( Figure 1). The width of the beam web is 100 mm, and the beam length is 5400 mm.
All the specimens are simply supported with a cantilever, which were loaded with two concentrated loads at the cantilever end and within the simply supported span, respectively. The ratio of cantilever load to span load is 1:2 for BC specimens, and 1:1 for BV specimens. The detailed dimensions and elevation layout of the component are shown in Figures 2 and 3, The detailed arrangement of the measuring lattice and corresponding calculation method could refer to a pre-publication focusing on the measuring technology [26].    All the specimens are simply supported with a cantilever, which were loaded with two concentrated loads at the cantilever end and within the simply supported span, respectively. The ratio of cantilever load to span load is 1:2 for BC specimens, and 1:1 for BV specimens. The detailed dimensions and elevation layout of the component are shown in Figures 2 and 3, The detailed arrangement of the measuring lattice and corresponding calculation method could refer to a pre-publication focusing on the measuring technology [26].    1200  2400  500  500  600  600  600  800   G1  G3  G5  G4  G2   200  400  1200  2400  1200  500  500  600  600  600  800   G1  G3  G5  G4  G2   560  800   400   100   2P  P   75 250 75  325  75  BC1, BC2 BV1, BV2, BV3, BV4     1200  2400  500  500  600  600  600  800   G1  G3  G5  G4  G2   200  400  1200  2400  1200  500  500  600  600  600  800   G1  G3  G5  G4  G2   560  800   400   100   2P  P   75 250 75  325  75  BC1,  For the specimens, D25 bars and D12 bars with a total cross section area of 3624 mm 2 and average yield strength of 497 MPa were used as longitudinal bars, which were arranged symmetrically in top and bottom flanges. D8 round bars (two legs) with average For the specimens, D25 bars and D12 bars with a total cross section area of 3624 mm 2 and average yield strength of 497 MPa were used as longitudinal bars, which were arranged symmetrically in top and bottom flanges. D8 round bars (two legs) with average yield strength of 326 MPa were arranged as stirrups at a spacing of 200 mm (ρ v = 0.5%) or 250 mm (ρ v = 0.4%) for deferent beams along the whole span. The average concrete cylinder strengths were 39.0 MPa and 36.0 MPa for batch I (BC1, BV1, BV2) and batch II (BC1, BV1, BV2), respectively. It must be mentioned that excessive longitudinal reinforcement is arranged to prevent early flexural failure. The specimen number and main design parameters are shown in Table 1, with the changing parameters being concrete strength and reinforcement ratio.

Resources
Specimen NO.

Specimen Failure Modes
The final failure mode of all 6 specimens is shear failure, manifested as the yielding of web stirrups and concrete crushing at the bending reverse point, as shown in Figure 4.
The test reveals that the development of bending cracks was slow or almost nonexistent during the loading process, and the strain increment of longitudinal reinforcement was small and did not reached yield [10]. With the appearance of diagonal cracks, their development is relatively rapid, quickly developing from the middle of the web to the upper and lower flanges, and gradually penetrating the entire web. With the formation of the main diagonal crack, the stirrups at the diagonal crack also yielded quickly. Afterwards, the number, width and range of diagonal cracks further expanded, and the stirrups of different shear span also yielded one after another. Finally, the concrete of the beam web at the bending reverse point collapsed, and the deformation increased sharply, declaring the failure of the specimen.
Though the first diagonal crack occurs at the cantilever span with maximum shear forces, all specimens failed in shear uniformly at the bending reverse point The positive and negative bending moment on both sides of the bending reverse point intensifies the shear deformation of the concrete web, ultimately leading to the tearing of the concrete web and the diagonal compression of the concrete web. The phenomenon reflects that the bending reverse point section is the weakest position for shear failure of continuous beams (constrained beams).

Specimen Failure Modes
The final failure mode of all 6 specimens is shear failure, manifested as the yielding of web stirrups and concrete crushing at the bending reverse point, as shown in Figure 4. The test reveals that the development of bending cracks was slow or almost nonexistent during the loading process, and the strain increment of longitudinal reinforcement was small and did not reached yield [10]. With the appearance of diagonal cracks,

Observed Shear Stiffness Degradation
Besides failure modes, the sequence of shear crack, strain development and yielding of stirrup, and failure load of each shear span can refer to an earlier publication [26]. Here, we only focus on the degradation of shear stiffness. Shear stiffness is the most important measure of structural shear deformation. In the elastic stress stage, the shear stiffness K e of the section can be expressed as GA v , and considering the Poisson's ratio, it becomes EA v /[2(1 + µ)]. After cracking, due to the destruction of structural continuity, the shear stiffness is no longer equal to elastic shear stiffness. Generally, the nominal shear stress of the test beam after cracking can be taken as τ, which is the ratio of shear force P to the shear cross-sectional area A v . Based on the stress-strain relationship, the equivalent shear stiffness K eq of the shear cracked specimen is defined by the following equation, To study the variation of shear stiffness after the development of diagonal cracks, a shear stiffness degradation factor is defined as λ, namely, the ratio of equivalent shear stiffness K eq to elastic shear stiffness K e , Considering that the tested shear strain in the elastic stage is very small, if the calculated λ > 1, take λ = 1. The shear stiffness reduction factor λ of each observation lattice can be plotted as a function of nominal shear stress τ, as shown in Figure 5. It can be seen that except for specimen BV1, the shear stress levels of all specimen frame G3 and G4 lattices with significantly reduced shear stiffness are around 2MPa, which is in good agreement with the initial shear crack load. This indicates that the shear stiffness reduction factor λ can properly reflect the impact of diagonal web cracks on shear stiffness. web and the diagonal compression of the concrete web. The phenomenon reflects that the bending reverse point section is the weakest position for shear failure of continuous beams (constrained beams).

Observed Shear Stiffness Degradation
Besides failure modes, the sequence of shear crack, strain development and yielding of stirrup, and failure load of each shear span can refer to an earlier publication [26]. Here, we only focus on the degradation of shear stiffness. Shear stiffness is the most important measure of structural shear deformation. In the elastic stress stage, the shear stiffness Ke of the section can be expressed as GAv, and considering the Poisson's ratio, it becomes EAv/[2(1 + μ)]. After cracking, due to the destruction of structural continuity, the shear stiffness is no longer equal to elastic shear stiffness. Generally, the nominal shear stress of the test beam after cracking can be taken as τ, which is the ratio of shear force P to the shear cross-sectional area Av. Based on the stress-strain relationship, the equivalent shear stiffness Keq of the shear cracked specimen is defined by the following equation, To study the variation of shear stiffness after the development of diagonal cracks, a shear stiffness degradation factor is defined as λ, namely, the ratio of equivalent shear stiffness Keq to elastic shear stiffness Ke, eq e / KK  = (2) Considering that the tested shear strain in the elastic stage is very small, if the calculated λ > 1, take λ = 1. The shear stiffness reduction factor λ of each observation lattice can be plotted as a function of nominal shear stress τ, as shown in Figure 5. It can be seen that except for specimen BV1, the shear stress levels of all specimen frame G3 and G4 lattices with significantly reduced shear stiffness are around 2MPa, which is in good agreement with the initial shear crack load. This indicates that the shear stiffness reduction factor λ can properly reflect the impact of diagonal web cracks on shear stiffness. Comparing the degradation curves of shear stiffness of specimens with different concrete strength grades (BV1 and BV3, BV2 and BV4) and different reinforcement ratios (BV1 and BV2, BV3 and BV4), it was found that increasing the concrete strength grade (mainly the elastic modulus) and increasing the reinforcement ratio can both improve the shear stiffness after cracking. The contribution of increasing the reinforcement ratio to the re- Comparing the degradation curves of shear stiffness of specimens with different concrete strength grades (BV1 and BV3, BV2 and BV4) and different reinforcement ratios (BV1 and BV2, BV3 and BV4), it was found that increasing the concrete strength grade (mainly the elastic modulus) and increasing the reinforcement ratio can both improve the shear stiffness after cracking. The contribution of increasing the reinforcement ratio to the remaining shear stiffness is more significant after shear cracking, and the suppression of shear deformation is more effective. Generally, after the diagonal cracking of the specimen, the shear stiffness is about 30~40% of the original, but when the stirrup yields, the remaining shear stiffness is only about 10%.

Fully Diagonally Cracked Shear Stiffness
To evaluate the cracked shear stiffness, the truss model is recommended by scholars [33][34][35][36]. For slender beams, the inclined cracks are roughly parallel to each other. Therefore, a standard VATM can be used for analysis. As the truss model ignores the tensile stresses between cracked concrete, it is only suitable for calculating the shear stiffness of fully diagonally cracked RC beams. If the strut angle θ u is determined, the corresponding shear stiffness K u can be expressed as, where K u is the fully diagonally cracked shear stiffness, n is the ratio of E s to E c , E s is the modulus of elasticity of reinforcing steel, ρ v is the stirrup ratio, and θ u is the strut angle.

Ultimate Shear Stiffness Degradation Factor
Similarly to Equation (2), we can define ultimate shear stiffness degradation factor λ u , which is equal to the ratio of K u and K e , The factor λ u is defined as the shear stiffness degradation factor, which reflects the decreasing magnitude of shear stiffness at the fully cracked stage. As shown in Equation (4), the main parameters that influence λ u are the stirrup ratio ρ v and the strut angle θ u . If the only unknown parameter θ u is determined, λ u can be easily calculated by Equation (4).

Determination of Strut Angle θ u
Scholars have already proposed various solving methods for the strut angle [33][34][35], most of which employed the minimum energy principle or plasticity theory. Their findings imply that the strut angle closely relates to the stirrup ratio, longitudinal reinforcement ratio, or concrete strength.
For calculating the shear stiffness calculation of fully diagonally cracked slender RC beams, Pan et al. [8] suggested that the strut angle can be calculated by Equation (5). It accounts for the influence of the web and longitudinal reinforcement.
In addition, He et al. [9] derived the strut angle of slender beams based on the lowerbound theorem of plasticity (Equation (6)). It accounts for the influence of the stirrup ratio and the concrete strength while assuming that the longitudinal reinforcement will not Materials 2023, 16, 4752 9 of 16 yield before shear failure. The key parameter for the equation ω is the mechanical web reinforcement ratio, and ω = ρ v f yv /f' c , f' c is the compressive strength of concrete.
For simplification, compression field theory [27] is adopted for beam shear analysis to determine the strut angle (see Figure 6). The formulation process assumes that concrete beams are subject to service load, under which the steel bars and the inclined concrete struts behave linearly elastically. In addition, He et al. [9] derived the strut angle of slender beams based on the lo bound theorem of plasticity (Equation (6)). It accounts for the influence of the stirrup and the concrete strength while assuming that the longitudinal reinforcement wi yield before shear failure. The key parameter for the equation ω is the mechanica reinforcement ratio, and ω = ρvfyv/f'c, f'c is the compressive strength of concrete.
For simplification, compression field theory [27] is adopted for beam shear ana to determine the strut angle (see Figure 6). The formulation process assumes that con beams are subject to service load, under which the steel bars and the inclined con struts behave linearly elastically. Figure 6. CFT adopted for shear analysis of RC beam.
The equilibrium of average stress and compatibility of average strain in a beam tion are summarized in Equations (7)- (13), in which the longitudinal strain εx at the dle height is assumed to be 0.5 times the longitudinal strain εs at the center of longitu steel bars [3]. The equilibrium of average stress and compatibility of average strain in a beam section are summarized in Equations (7)- (13), in which the longitudinal strain ε x at the middle height is assumed to be 0.5 times the longitudinal strain ε s at the center of longitudinal steel bars [3].
ρ s f ls = 0.5v cot θ u (8) where v, f v and f ls are the shear stress, stirrup stress and longitudinal reinforcement stress, respectively, and ε x , ε s , ε z and ε 2 are the longitudinal strain, longitudinal strain at the center of longitudinal steel bars, vertical strain and main compression strain, respectively. From Simultaneous Equations (7)-(13), we can obtain, Equation (14) is mainly influenced by the web and longitudinal reinforcement ratio.

Comparison of θ u with Other Methods
To investigate the validity of angle prediction for slender beams, parameter analysis and comparison are performed according to Pan et al. [8], He et al. [9] and the proposed Equation (14). Parameter values of the reference specimen are ρ v = 0.5%, ρ s = 2%, f' c = 50MPa and f yv = 400MPa. Figure 6 shows how the strut angle changes as a function of stirrup ratio ρ v , longitudinal reinforcement ratio ρ s and concrete strength f' c , respectively.
As is shown in Figure 7, the proposed equation gives an intermediate prediction of strut angle θ u , while θ Pan predicts the highest value, and θ He predicts the lowest. All three angles are affected by stirrup ratio ρ v , along with which the predicted angle grows (Figure 7a). Meanwhile, both θ u and θ Pan reflect the inverse relationship between longitudinal reinforcement ratio ρ s and the strut angle (Figure 7b). θ He ignores the influence of longitudinal reinforcement ratio ρ s but emphasizes the importance of concrete strength f' c (Figure 7c), and it may cause larger deviations for specimens with low longitudinal reinforcement ratio or high concrete strength.

Proposed Degradation Rules
In spite of the fact that we have obtained the elastic shear stiffness and the fully cracked shear stiffness, it is still difficult to evaluate the effective shear stiffness Keff of a partially diagonal cracked RC beam. As the transition from elastic stiffness to post-cracking stiffness is very complicated and is controlled by many parameters, establishing an exact and quantified expression for Keff is almost impossible.
As is shown in Figure 8, the simplest and most ideal shear stiffness degradation model is the secant stiffness linear degradation model, which assumes that the post-cracking secant shear stiffness will degrade linearly with the shear force (Figure 8a), and it tends to give an unsafe prediction of shear deformation under service state. However, another shear stiffness model, namely, the constant tangent stiffness degradation model, which assumes that the post-cracking tangent shear stiffness will keep constant before stirrup yielding (Figure 8b), is only suitable for thin-webbed beams and tends to give a larger shear deformation prediction.

Proposed Degradation Rules
In spite of the fact that we have obtained the elastic shear stiffness and the fully cracked shear stiffness, it is still difficult to evaluate the effective shear stiffness K eff of a partially diagonal cracked RC beam. As the transition from elastic stiffness to post-cracking stiffness is very complicated and is controlled by many parameters, establishing an exact and quantified expression for K eff is almost impossible.
As is shown in Figure 8, the simplest and most ideal shear stiffness degradation model is the secant stiffness linear degradation model, which assumes that the post-cracking secant shear stiffness will degrade linearly with the shear force (Figure 8a), and it tends to give an unsafe prediction of shear deformation under service state. However, another shear stiffness model, namely, the constant tangent stiffness degradation model, which assumes that the post-cracking tangent shear stiffness will keep constant before stirrup yielding (Figure 8b), is only suitable for thin-webbed beams and tends to give a larger shear deformation prediction.
For RC beams, tension stiffening arises from tension carried by the concrete between the cracks (whether flexural or shear cracks). This contribution decreases with an increasing load after the member has cracked. To simulate the tension stiffening effect, the effective moment of inertia I e approach introduced by Branson [37] facilitates a gradual transition from un-cracked to a fully cracked section as the ratio of service load moment M a to cracking moment M cr increases. model is the secant stiffness linear degradation model, which assumes that the post-cracking secant shear stiffness will degrade linearly with the shear force (Figure 8a), and it tends to give an unsafe prediction of shear deformation under service state. However, another shear stiffness model, namely, the constant tangent stiffness degradation model, which assumes that the post-cracking tangent shear stiffness will keep constant before stirrup yielding (Figure 8b), is only suitable for thin-webbed beams and tends to give a larger shear deformation prediction.
(a) Shear deformation curve (b) Shear stiffness degradation mode For RC beams, tension stiffening arises from tension carried by the concrete between the cracks (whether flexural or shear cracks). This contribution decreases with an increasing load after the member has cracked. To simulate the tension stiffening effect, the effective moment of inertia Ie approach introduced by Branson [37] facilitates a gradual transition from un-cracked to a fully cracked section as the ratio of service load moment Ma to cracking moment Mcr increases.
The ACI 318 standard [2] adopts Branson's degradation method when calculating the bending stiffness after flexural cracking. To make the deformation calculation equation uniform and simple, we recommend using a degradation criterion similar to Branson to reflect the effect of tensile hardening. The recommended shear stiffness degradation formula is as follows, The ACI 318 standard [2] adopts Branson's degradation method when calculating the bending stiffness after flexural cracking. To make the deformation calculation equation uniform and simple, we recommend using a degradation criterion similar to Branson to reflect the effect of tensile hardening. The recommended shear stiffness degradation formula is as follows, where the diagonal cracking load V cr and stirrup yielding load V u can be calculated by Equations (16) and (17), respectively.

Experiment Introduction
The shear deformation of 12 beam lattices in 6 thin web-restrained beams was directly measured and analyzed with self-designed strain-measuring lattices. Further, the shear deformation test of six beams conducted by Hansapinyo [24] was used to verify the proposed degradation model. Data for a total of 16 shear deformation measurement lattices are used for experimental verification. The main parameters of the specimens are listed in Table 1. Hansapinyo's test gives detailed shear strain test results, in which electronic transducers were also used to measure the normal and shear strains of each lattice based on the rosette concept.
As the stress distribution is disturbed by local point load in D regions (discontinuity regions, such as the lattice regions of G1, G2 and G5 shown in Figure 2), the direct strut component joins in the force transfer mechanisms in addition to the flexural and shear components. Consequently, the total deformation consists of not only flexural deformation and shear deformation but also the deformation of direct strut compression. Mean shear strain in these D regions becomes insignificant. Therefore, shear strain analyses are only performed on the lattice where the local point load disturbances are negligible (Lattice G3 and G4).

Comparing Results and Discussion
The measured and calculated shear strain of 16 zones of 10 beams are shown in Figures 9-11, respectively. Compared with the measured shear force-strain, the theoretical prediction results are in good agreement with the measured values. It can be concluded that the proposed shear stiffness degradation model simulates the degradation process of shear stiffness very well, and it tends to give a conservative prediction for the shear strain after diagonal cracking.  (c) S3 (d) S4 Figure 11. The shear strain of each measuring lattice in specimens S1, S2, S3 and S4.
For the specimen CV series ( Figure 9) and S series ( Figure 11) with constant depth, the proposed degradation model is a little conservative and gives very acceptable accuracy. Meanwhile, the degradation rule well captures the main characteristics of the shear strain curves, such as the turning point for the first diagonal crack and the gradual evolution from elastic stiffness to fully diagonally cracked stiffness.
For the specimen BV series with variable depth (Figure 10), it should be noted that the inclined lower chord bears part of the shear force, which is not completely consistent with the assumption for constant depth beam, for which the concrete web bears most of the shear force based on elastic beam theory. Therefore, it can be foreseen that the shear deformation prediction of the BV series specimens will be slightly larger than experimental results in the early loading stage, but the degradation law of shear stiffness is still in good agreement among each specimen.
In addition, although the shear span ratios of different observed beam lattices have large changes (such as lattices G3 and G4 of BV series in Figure 9, and lattices S1 and S2 in Figure 10), there is no obvious relationship between measured shear strain and shear depth-to-span ratio. This shows that the shear depth-to-span ratio has little effect on the degradation of the shear stiffness, and its effect on the after-cracking shear deformation is less.
For the specimen CV series ( Figure 9) and S series ( Figure 11) with constant depth, the proposed degradation model is a little conservative and gives very acceptable accuracy. Meanwhile, the degradation rule well captures the main characteristics of the shear strain curves, such as the turning point for the first diagonal crack and the gradual evolution from elastic stiffness to fully diagonally cracked stiffness.
For the specimen BV series with variable depth (Figure 10), it should be noted that the inclined lower chord bears part of the shear force, which is not completely consistent with the assumption for constant depth beam, for which the concrete web bears most of the shear force based on elastic beam theory. Therefore, it can be foreseen that the shear deformation prediction of the BV series specimens will be slightly larger than experimental results in the early loading stage, but the degradation law of shear stiffness is still in good agreement among each specimen.
In addition, although the shear span ratios of different observed beam lattices have large changes (such as lattices G3 and G4 of BV series in Figure 9, and lattices S1 and S2 in Figure 10), there is no obvious relationship between measured shear strain and shear depth-to-span ratio. This shows that the shear depth-to-span ratio has little effect on the degradation of the shear stiffness, and its effect on the after-cracking shear deformation is less.
As the beams crack randomly during the loading stage, they cannot get into a fully diagonally cracked status that matches the theoretical assumptions. The real shear stiffness of the beams might be larger than theoretical values, while the real shear deformations are just the opposite. In total, the proposed model will give a relatively conservative shear stiffness prediction for partially diagonal cracked beams.
Here, specimen S3 (ρ s = 2.13%) is taken as an example to show the validation process. The calculated V cr = 45kN, and the calculated V y for the proposed method, Pan method and He method are 179.4 kN, 148.7 kN and 144.3 kN, respectively, while the values of λ u are 0.169, 0.127 and 0.178, respectively. The predicted ultimate shear stiffness degradation reduction factor λ u and shear strain calculated by three methods are compared to the measured data (Figure 12), which shows that the proposed method is a little better than others for the after-cracking shear stiffness evaluation.
As the beams crack randomly during the loading stage, they cannot get into a fully diagonally cracked status that matches the theoretical assumptions. The real shear stiffness of the beams might be larger than theoretical values, while the real shear deformations are just the opposite. In total, the proposed model will give a relatively conservative shear stiffness prediction for partially diagonal cracked beams.
Here, specimen S3 (ρs = 2.13%) is taken as an example to show the validation process. The calculated Vcr = 45kN, and the calculated Vy for the proposed method, Pan method and He method are 179.4 kN, 148.7 kN and 144.3 kN, respectively, while the values of λu are 0.169, 0.127 and 0.178, respectively. The predicted ultimate shear stiffness degradation reduction factor λu and shear strain calculated by three methods are compared to the measured data (Figure 12), which shows that the proposed method is a little better than others for the after-cracking shear stiffness evaluation.  In summary, the shear stiffness degradation model proposed in this paper can better evaluate the shear stiffness degradation of each beam lattice, and it can give a reasonable prediction of shear deformation, which can be used for the evaluation of the shear stiffness of diagonally cracked RC beams.

Conclusions
This study aims to obtain the shear stiffness degradation law and to propose a simplified and practical prediction method of thin-webbed beams, which can be further applied to existing concrete box-girder bridges with diagonal web cracks. Six large-scale thin-webbed concrete beams were tested in this paper to investigate the shear stiffness degradation of RC beams before and after shear cracking. Considering the effect of concrete tension stiffening, a practical shear stiffness degradation model was proposed and validated. The main conclusions can be drawn as follows: 1. The shear deformation test showed that the shear stiffness drops to about 30~40% of the original stiffness following the occurrence of the first main diagonal crack, and it further drops to only about 10% of the original stiffness when the stirrup yields. 2. The strut angle θu was deduced by combining CFT and elastic beam theory. Compared with two other methods from the literature, the proposed angle tends to give a moderate prediction of strut angles and shear deformation with higher accuracy. 3. Considering the tensioning stiffness effect, a simplified shear stiffness degradation rule was suggested for a diagonally cracked RC beam. A cubic form degradation equation consistent with the degradation form of flexural stiffness was established and validated.  Comparison of three methods in prediction of shear stiffness and shear strain [8,9].
In summary, the shear stiffness degradation model proposed in this paper can better evaluate the shear stiffness degradation of each beam lattice, and it can give a reasonable prediction of shear deformation, which can be used for the evaluation of the shear stiffness of diagonally cracked RC beams.

Conclusions
This study aims to obtain the shear stiffness degradation law and to propose a simplified and practical prediction method of thin-webbed beams, which can be further applied to existing concrete box-girder bridges with diagonal web cracks. Six large-scale thin-webbed concrete beams were tested in this paper to investigate the shear stiffness degradation of RC beams before and after shear cracking. Considering the effect of concrete tension stiffening, a practical shear stiffness degradation model was proposed and validated. The main conclusions can be drawn as follows: 1.
The shear deformation test showed that the shear stiffness drops to about 30~40% of the original stiffness following the occurrence of the first main diagonal crack, and it further drops to only about 10% of the original stiffness when the stirrup yields.

2.
The strut angle θ u was deduced by combining CFT and elastic beam theory. Compared with two other methods from the literature, the proposed angle tends to give a moderate prediction of strut angles and shear deformation with higher accuracy.

3.
Considering the tensioning stiffness effect, a simplified shear stiffness degradation rule was suggested for a diagonally cracked RC beam. A cubic form degradation equation consistent with the degradation form of flexural stiffness was established and validated. 4.
Data for a total of 16 zones of lattice shear deformation from 10 beams were measured or collected for verification. The results showed that a turning point occurs in the shear deformation curve corresponding to the first diagonal crack. And, rather than the pre-cracking stage, the shear span-to-depth ratio has little effect on the shear deformation of RC beams in the post-cracking stage.

5.
The results showed that the proposed method gives a good and consistent prediction of the effective shear stiffness and shear strain development. The proposed model could capture the development characteristics of shear deformation curves. However, for the BV series, the bottom flanges bear part of the shear force, which will cause a larger predicted shear strain. 6.
In general, the proposed simplified shear degradation model tends to give a conservative prediction of shear stiffness, and it is very practical for the early evaluation of diagonally cracked box-girder bridges in service.
Moreover, the contribution of the inclined flanges of variable depth specimens on shear capacity and shear deformation could be an interesting topic for further study. Also, the digital image correlation (DIC) technique maybe a good choice for shear deformation tests in future studies [38][39][40][41].