Analysis of reinforced concrete beams with indirect reinforcement according to limit state design

. When analyzing structures within the framework of the theory of limit state design, it is necessary to ensure sufficient reliability and safety at an acceptable level of risk. For these purposes, in engineering calculations, the mean experimental values of material strength are recalculated into design values. This article is devoted to the calculation of reinforced concrete beams with indirect reinforcement of the compressed area with transverse welded meshes for different accepted values of concrete strength (mean value of strength; design value of strength) and comparison of bearing capacity and deformations. For three series of tested beams, deformations were calculated at various stages of loading, and the bearing capacity was determined at the stage before peeling off the protective layer of concrete in the compressed area (operational stage). Calculations according to the method of the deformation model were carried out both on the basis of the mean experimental values of strength of concrete and reinforcement, and on the basis of the design values. It was found that in calculations based on the design values of the material strength, the ultimate deflections in some cases exceed the values obtained on the basis of the experimental mean strength values, which should not be. In this regard, it was indicated the need to calculate the parametric points of the design stress-strain diagram of concrete with indirect reinforcement according to the experimental mean strength values and only after that take into account the necessary reliability coefficients and obtain the final design values.


Introduction
Reinforced concrete is one of the most common materials used in construction, and in this regard, the task of improving calculation methods that allow estimating the stress-strain state for various design solutions as close as possible to real work is extremely important.In addition, when analyzing structures within the framework of the theory of limit state design, it is necessary to ensure sufficient reliability and safety at an acceptable level of risk.[1][2][3].For these purposes, in engineering calculations, the mean experimental values of material strength are recalculated into design values [4].Obtaining design values is an important task that requires mathematical processing of a large sample of experimental data.
One of the constructive solutions that improve properties of reinforced concrete is indirect reinforcement.Such reinforcement, located in the compressed area of concrete perpendicular to the emerging force, inhibits the development of transverse deformations, thereby increasing strength, rigidity and ultimate compressibility.The positive effect of transverse compression is confirmed by a large amount of existing experimental data [5-7, etc.].Currently, various design solutions for indirect reinforcement are used: pipeconcrete elements [8][9][10], spiral reinforcement [11][12][13], welded mesh [14][15][16], etc.When designing bending structure or elements compressed with a large eccentricity, welded meshes are most often used, since they can be placed in that part of the section where compressive stresses arise, and the relatively small size of the mesh cell allows them to work effectively.When determining the strength and deformation characteristics of concrete with indirect reinforcement within the framework of limit state design, a number of problems arise with the adaptation of the calculation formulas originally derived for experimental mean values.This fact was previously noted in articles [17,18].This article is devoted to the calculation of reinforced concrete beams with indirect reinforcement of the compressed area with transverse welded meshes for different accepted values of concrete strength (mean value of strength; design value of strength) and comparison of bearing capacity and deformations.

Materials and Methods
Previously, in the study [14], three series of reinforced concrete beams with indirect reinforcement of the compressed area with welded meshes were tested.Cross-sectional dimensions of the beams are assumed to be 150x200 mm.The series differed from each other in the diameters of reinforcing bars in the tension area.Samples within the series differed in the indirect reinforcement ratio ρ xy .The concrete and reinforcement parameters for the tested samples are presented in Table 1.The beam testing scheme is shown in Fig. 1.For beams, calculations of deformations under various loads were performed, and the bearing capacity was determined at the stage before peeling off the protective layer of concrete in the compressed area (operational stage).The calculations were performed on the basis of the deformation model according to the method presented in [19].For reinforcement and non-reinforced concrete, two-line and three-line diagrams were set, respectively.For concrete in the area of indirect reinforcement, a diagram was set according to the method presented in [20].The main parametric points of the stress-strain diagram were assigned based on the expressions:  where R c is a prismatic concrete compressive strength; ρ xy is an indirect reinforcement ratio; R s,xy is a tensile strength of indirect reinforcement rods; σ c,xy is a lateral compression stress.The calculations were carried out both on the basis of the mean experimental values of strength of concrete and reinforcement, and on the basis of the design values.At the same time, in previous studies [17,18], it was found that the strength and ultimate compressibility calculated by formulas (1) and ( 2), as well as by most other applied dependencies, are overestimated when substituting the design values of the material strength.This is due to the fact that even for unreinforced concrete with lower strength, the ultimate compressibility will be slightly higher.In the case of indirect reinforcement, this difference becomes several orders of magnitude higher (Fig. 2) and, in this regard, the formulas originally derived for the experimental mean strength values are not suitable for substituting the design values of the material strength, which are much less in absolute value than the experimental values.A similar effect (albeit much less influential) is also observed with the calculated values of the strength of concrete with indirect reinforcement.In this regard, it was indicated the need to calculate the parametric points of the stress-strain diagram of concrete with indirect reinforcement according to the experimental mean strength values and only after that take into account the necessary reliability coefficients and obtain the final design values.
In view of the foregoing, in order to confirm the previously obtained conclusions for beams, when plotting the design compression diagrams for concrete with indirect reinforcement, the parametric points were calculated independently in two ways: 1) By substituting into formulas (1) and ( 2) the design values of strength of concrete and reinforcement (R cd , R sd , R xyd ).
2) By substituting in formulas ( 1) and ( 2) the mean values of strength of concrete and reinforcement (R cm , R sm , R xym ), after which the strength of concrete with indirect reinforcement was reduced to the design value.
The process of obtaining design values is described in more detail in study [17].
Fig. 2. Stress-strain relationship for concrete with different compressive strength and equal reinforcement coefficient.

Results and discussion
The results of the calculation of beams at various stages of loading are shown in Fig. 3-4  Table 2 shows the maximum deflections calculated using design material strengths max    As can be seen from the graphs, the form of deformation of the samples is in good agreement with the experimental data.It was recorded in the experiments that the loss of the bearing capacity for samples B-I-1 and B-II-2 began when the tensile reinforcement passed into the yield stage.Similar fracture mechanisms were also obtained from the calculation results by all the considered methods.The loss of bearing capacity for the rest of the samples began with the delamination of the concrete protective layer in the compressed area, which also corresponds to the calculation results.
For the design stress-strain diagrams, the necessary margin for bearing capacity is provided (23.1-29.6% and 25.8-34.1%,respectively).But when considering the obtained deflection values, it can be seen that in calculations based on the design values of the material strength, the ultimate deflections are significantly higher than in the calculation based on mean strength values with subsequent consideration of the reliability coefficients.Moreover, for sample B-I-2, the ultimate strains calculated on the basis of the design strength values exceed those calculated on the basis of experimental mean values, which is contrary to common sense.

Conclusions
The following conclusions can be drawn from the results of the calculations performed:  The method of calculation based on the deformation model is well suited for the calculation of beams with indirect reinforcement of the compressed area, making it possible to evaluate the stress-strain state at various stages of loading;  According to the results of calculation, the character of sample deformation is in good agreement with experimental data.Identical fracture mechanisms were obtained when performing calculations both for the experimental mean strength values and for the design strength values;  When calculating on the basis of experimental mean strength values, the difference in bearing capacity with experimental data was from 2.1% to 11.2%, which is an acceptable accuracy.For most samples, the difference goes to the bearing capacity margin.
 For design compression diagrams, the necessary margin for bearing capacity is provided: 23.1-29.6%and 25.8-34.1%,depending on the chosen method for здщеештп the design stress-strain diagram for concrete with indirect reinforcement.
 When considering the obtained deflection values, it can be seen that in calculations based on the design values of the material strength, the ultimate deflections are significantly higher than in the calculation based on mean strength values with subsequent consideration of the reliability coefficients.Moreover, for sample B-I-2, the ultimate strains calculated on the basis of the design strength values exceed those calculated on the basis of experimental mean values, which is contrary to common sense.This result indicates that the parametric points of the compression diagram for concrete with indirect reinforcement must be obtained from the experimental mean strength values and only after that the transition to the design values should be performed taking into account the specified reliability coefficients.A similar conclusion was previously obtained in the study [17] for compressed prisms.

dF 1 E3S
and compared with deflections obtained using mean values of material strengths max m F .Similar data and comparison for the maximum bending moments obtained before the peeling of the protective layer is given in Table 3. Web of Conferences 410, 02018 (2023) https://doi.org/10.1051/e3sconf/202341002018FORM-2023

Table 1 .
Parameters of tested samples.

Table 2 .
Calculated values of deflections.

Table 3 .
Calculated values of deflections.