Finite Element Analysis of the Shear Capacity of Recycled Aggregate Concrete Beams with Corroded Stirrups

. In this paper, a model to predict the residual shear capacity of RAC beams with corroded stirrups was established, and the simulation analysis was conducted on recycled concrete beams with diferent corrosion level using ANSYS. Te reliability of the fnite element analysis model was verifed by comparing the fnite element results with experimental results.


Introduction
Recycled concrete refers to a new type of concrete that is made by using waste concrete, and it can also be called recycled aggregate concrete (RAC).RAC beam is an important structural member, and scholars have carried out a lot of research on its shear behavior.Zhang [1] studied the infuencing factors of the cracking load of the inclined section of recycled concrete beams and ftted the calculation formula of the cracking load of the inclined section of recycled concrete beams.Te authors in [2][3][4][5][6][7][8][9][10][11][12][13][14][15] studied the infuencing factors of shear capacity, and the conclusions are basically consistent: the shear capacity of recycled concrete beams decreases with the increase of shear span ratio and aggregate replacement ratio and increases with the increase of stirrup ratio and concrete strength.Zhang [1] also ftted the shear capacity formula of recycled concrete beams through the shear static test of 8 recycled concrete beams.
Numerical analysis methods have a wide range of applications in engineering structural analysis.Finite element method is a very useful numerical analysis method for the analysis of reinforced concrete structures, which can provide a large amount of structural information, such as structural displacement, stress, strain, bond slip, and failure load [16].
In this paper, the calculation model to predict the residual shear capacity of RAC beams with corroded stirrups is established, and the ANSYS fnite element software was used for structural nonlinear analysis and modeling; a separate model is selected to analyze the reinforced corroded recycled concrete beam, and the reliability of the calculation model is verifed.

Analysis of the Residual Shear Capacity of Corroded RAC Beams
Based on results of the shear behavior experiment of RAC beams with corroded stirrups in laboratory, the relationship between the measured shear capacity V u and the corrosion levels is shown in Figure 1, which illustrates that the measured shear capacity of RAC beams increases frst and then decreases with the increase of corrosion level of stirrups.
Te limit equilibrium method for the residual shear capacity of the oblique section of a beam is a method to solve the problem by establishing the equilibrium of the shear failure limit state of the structure through the study of the shear mechanism.Te theory solves the unknown quantity by establishing the internal force balance equation and deformation equation through the stress analysis of the isolation body.Te total shear ultimate capacity of the corroded stirrup can be expressed by the following equation: where V c is the shear force borne by concrete in shear compression zone; V d is the total pin force of longitudinal reinforcement; V ay is the vertical part of the bite force of the inclined section aggregate; V sv is the shear force borne by stirrup; and Xu and Niu [17] summarized and proposed the calculation method of V cu and V sv , as shown in the following equation: where λ is the shear span ratio; s is the stirrup spacing; f yv is the yield strength of stirrups; A s /bh is the longitudinal reinforcement ratio; A sv ′ is Area of stirrup after corrosion; and In this study, the relationship between axial compressive strength and cubic compressive strength of recycled concrete is f c � 0.76f cu [18].Assuming that the stirrup is uniformly corroded and the average corrosion rate is adopted, the geometric relationship after the stirrup is as follows: where L is the length of stirrup, m L is the weight of stirrup, ρ s is the corrosion level of stirrups, d is the diameter of stirrup after corrosion, ω is the iron density, and D is the diameter of stirrup before corrosion.
where A sv is is area of stirrup, h is the cross-sectional height of the beam, and h 0 is the efective cross-sectional height of the beam.Te relationship between the measured shear capacity V u and the calculated value V u ′ of ( 4) is shown in Figure 2.
Figure 2 illustrates that the measured value is close to the calculated value of (4) but not completely consistent, which is also because the geometric relationship of corroded reinforcement in (3) is obtained under the ideal condition of uniform corrosion.To make the theoretical value of (4) satisfed with measured value, the corroded infuence coefcient c S is introduced, and the ftting relationship between theoretical value and measured value can be expressed as follows: According to (5), the relationship between calculation value of c S and the corrosion levels is shown in Figure 3, and it illustrates that the ft degree is better.Terefore, the   Considering the impact of corrosion on the mechanical properties of beams, this paper simulates the bond slip relationship between steel and concrete using a nonlinear spring element Combin39.Tis unit has two nodes.Tree mutually perpendicular nonlinear springs with diferent stifness and no geometric volume are set at each steel and concrete element node.

Finite Element Analysis of Recycled Concrete Beams with Corroded Stirrups
After establishing diferent elements, the element nodes are merged and then constraints in the X and Y directions are applied, perpendicular to the beam length direction at the support, while freely deforming along the beam length direction.Using displacement loading mode, vertical load is applied at the loading point of the beam.
During the modeling process, the smaller the unit size division, the closer it is to the real component.However, during the simulation process, it is easy to cause stress concentration, which makes the calculation difcult to converge.It is necessary to determine a reasonable grid density by dividing the grid multiple times and comparing it with experimental results.Concrete components often work with cracks during the stress process, and during the cracking process, cracks may close and new crack surfaces may be generated.In the computer simulation process, when internal forces act on the surface of cracks, the program can use the shear transfer coefcient of open cracks to simulate stress loss, or the shear transfer coefcient of closed cracks.Te values of both variables are 0-1, and the analysis is generally based on experience.
Adopting the displacement convergence criterion, the convergence tolerance value has a certain impact on the simulation results.Convergence tolerance has no fxed value range in theory, and by appropriately relaxing, it can accelerate convergence, but by relaxing, it may result in incorrect results.
Based on the abovementioned analysis of infuencing factors, several models were established using a single parameter change method to analyze the impact of each parameter change on the fnite element simulation results (displacement and failure load) and compared with the experimental results to analyze the degree of infuence of each factor.In order to reduce stress concentration, surface loading is used at the loading point.Te reinforcement adopts a dynamic strengthening model, the constitutive relationship of concrete adopts an elastic-plastic MISO type curve, and the failure criterion adopts the William Warner fve parameter failure criterion.Te cracking option is turned on, and the crushing option is turned of.In order to obtain the sensitivity of each parameter to the simulation results, the parameter change amplitude should not be too large.Te diferent model parameter settings are shown in Table 1. 6) for the Shear Capacity.Te ftting formula for shear capacity in Section 2 was obtained when the corrosion rate was less than 13%.To verify the reliability of the ftting formula, a fnite element analysis software was used to increase the corrosion rate of steel bars to establish a model.

Verifcation of the Reliability of Equation (
In the design of fnite element software simulation, the corrosion rates of stirrups are 15%, 19%, 23%, 27%, and 30%.Te strength grade of recycled concrete is designed to be C50, and the other parameters are the same as those in the experiment.Te comparison between the simulated results and equation ( 6) is shown in Figure 5.
From Figure 5, it can be seen that the calculated value of the shear capacity ftting formula is still in good agreement with the fnite element simulation value after increasing the corrosion level of the stirrup, so ftting equation ( 6) is efective.

Model Parameter Analysis.
Te main infuencing factors on the shear capacity of inclined sections are cross-sectional size, shear span ratio, etc. Tis section analyzes the shear capacity of recycled concrete beams by simulating changes in cross-sectional size and shear span ratio.

Infuence of Cross-Sectional Dimensions on Shear
Capacity.To determine the efect of section size on the shear capacity of recycled concrete beams, three sets of models were selected for calculation.Te corrosion level of the stirrups was 0%, and the parameters of the stirrups and concrete were the same as those measured in the experiment in this paper.Te results of simulation calculation using fnite element software are shown in Table 2.
From Table 2, it can be seen that the shear capacity of recycled concrete beams increases with the increase of width and decreases with the increase of height.It can also be seen that the decrease in shear capacity is nonlinear with the Corrosion level of stirrups ρ S (%) Advances in Materials Science and Engineering increase of height.Tis is mainly because when the beam height is large, the tearing cracks are more obvious, the pin bolt efect is greatly reduced, and the width of oblique cracks is also large, weakening the interlocking efect of aggregates, thereby accelerating the speed of reducing shear capacity.

Infuence of Shear Span Ratio on Shear Capacity.
Trough fnite element method, the loading point and support position were changed to simulate beams with shear span ratios of 2 and 3. Te relationship between shear capacity and shear span ratio was obtained as shown in Figure 6.From Figure 6, it can be seen that under the same corrosion level, the ultimate shear capacity decreases with the increase of shear span ratio, which is also because the increase of shear span ratio reduces the number of stirrups per unit volume.

Conclusions
Based on the results of the experiment, the following conclusions are presented: (1) A new calculation model to predict the residual shear capacity of RAC beams with corroded stirrups is established, and the calculated value of the shear capacity ftting calculation model is still in good agreement with the fnite element simulation value after increasing the corrosion level of the stirrup.(2) Several models were established using a single parameter change method to analyze the impact of each parameter change on the fnite element simulation results and compared with the experimental results to analyze the degree of infuence of each factor.
(3) In the fnite element, the shear capacity of recycled concrete beams increases with the increase of width and decreases with the increase of height, and the ultimate shear capacity decreases with the increase of shear span ratio under the same corrosion level.

Figure 1 :
Figure 1: V u versus corrosion levels of RAC beams.

3. 1 .
Establishment of Finite Element Model.Solid65 solid elements are used for the concrete unit.Te Solid65 element has eight nodes, each with three degrees of freedom, which can consider the cracking and crushing of concrete.Te reinforcement unit adopts Link180 rod unit.Tis unit has two nodes, each with three degrees of freedom, does not bear bending moments, and has the function of large deformation.Te concrete and steel elements established are shown in Figure4.

Figure 3 :Figure
Figure 3: c S versus measured corrosion level of RAC beams.

Figure 4 :
Figure 4: Concrete and steel element model.

Figure 5 :
Figure 5: Comparison between the measured and simulated values of equation (6) after increasing the corrosion level.

Figure 6 :
Figure 6: Simulation results of the infuence of shear span ratio on shear capacity.

Table 2 :
Simulation results of the infuence of cross-sectional size on shear capacity.

Table 1 :
Models with diferent parameter settings.