Determination of the bearing capacity of biaxially bended beams based on the design strength of reinforced concrete

Based on the design assumptions according to Eurocode 2, the concept of determining the value of the design strength of reinforced concrete has been developed for calculating the bearing capacity of reinforced concrete members subjected to biaxial bending. The theorem on the parallelism of the planes of internal and external forces action is applied. The considered design cases are for the most expected forms of the compressed concrete areas, which are characteristic for biaxial bending of beam members. The value of the coefficient of influence of biaxial bending on the characteristics of the member of rectangular section with triangular and trapezoidal forms of the compressed concrete area is obtained. A methodology for calculating the bearing capacity of biaxially bended members based on the application of the design strength of reinforced concrete is brought to the engineering level of application. An example of a calculation is presented, which demonstrates the convenience and effectiveness of the developed method. The proposed method of calculating the bearing capacity of beam member under biaxial bending is approved by comparing the results of calculations with experimental data for 12 samples of biaxially bended beams.


Introduction
Complex deformation of reinforced concrete members is widespread in the practice of operation of many building structures.Wherein, it may be caused by both force and random factors of a constructive, technological or operational nature.As a rule, the main attention of scientists is devoted to the study of reinforced concrete members subjected to axial load and biaxial bending.Proposals were developed for the construction of universal diagrams for determination the bearing capacity of columns under biaxial bending [1][2][3][4][5], while not always acceptable simplifications were used.
Meanwhile, many bending members that are operated in the absence of axial force are subjected to complex deformation.Biaxial bending is exposed by individual members, for example, reinforced concrete girders, crane beams, horizontal elements of frameworks, elements of shells, bridges, staircases and underground structures, crossbars of transport galleries and overpasses, foundation and framing beams, wall panels, etc. and entire spatial systems of buildings and structures.
The results of studies of the strength of biaxially bended reinforced concrete members are given in publications [17,18].The developed strength calculation algorithms and their results are compared with experimental data.
The concept of the design strength of reinforced concrete and its application are given in the work [19] for calculating the bearing capacity of members subjected to plane bending.The essence of the considered approach is that reinforced concrete is considered as a composite that has a certain strength characteristic -the design strength of reinforced concrete.At the same time, the strength problems for reinforced concrete members are solved on the basis of the classical theory of resistance of materials.The implemented characteristic is an integral value that takes into account the strength characteristics of concrete and reinforcement, the amount of reinforcement in the section and its location.
It is obvious that it can be used for any type of deformation of a reinforced concrete member, in particular for biaxial bending.The problem of this implementation is complicated by an additional unknown parameter, namely the angle of inclination of the neutral axis, as well as the presence of different forms of the compressed concrete area in the cross-section of the member.To date, no practical proposals have been developed regarding the application of the theory of design reinforced concrete strength for biaxially bended members in this aspect.

Results
Obtaining the design strength of reinforced concrete in biaxial bending is considered for members of rectangular section with single reinforcement.At the same time, the assumptions for design according to Eurocode 2 [20] are used.In particular, a rectangular stress distribution in the concrete of the compressed area is accepted, the stress-strain diagram of the reinforcing steel with a physical yield point is assumed to be bilinear with a horizontal section of unlimited length.The deformation criterion is used as a failure criterion.
In general, it is proposed to solve strength problems using the design strength of reinforced concrete on the basis of the dependence where f zM is the design strength of a reinforced concrete member of rectangular section with single reinforcement in plane bending; W = bd 2 /6 -moment of resistance of a rectangular section; γ is a reducing coefficient for the design strength of a reinforced concrete at biaxial bending.
On the basis of the work [19], the design strength of reinforced concrete for a rectangular section, taking into account the accepted assumptions, is obtained in the form where ρ l is the longitudinal reinforcement ratio.Using dependence (2), it is possible to tabulate the design strength of reinforced concrete for different classes of concrete and reinforcement at different of reinforcement ratios.For example, table 1 shows the value of f zM for reinforcement of A500C class.It should be noted that the values in table 1 are determined under the condition that at the moment of failure, the stresses in the tensile reinforcement reach the yield point.
To determine the coefficient γ, which takes into account the decreasing of the design strength of reinforced concrete of biaxially bended members, it is first considered the case of biaxial bending of a reinforced concrete beam, in which the compressed concrete area has the form of a trapezoid.According to the accepted prerequisites, the design scheme of the section has the form presented in figure 1. Taking into account the assumptions outlined above and figure 1, the design equations of equilibrium in the plane of the vertical coordinate axis Y are written in the form: where N s , N c -resultant forces, in the reinforcement and in the compressed concrete area, respectively; d h is the distance from the most compressed concrete fiber of the section to the point of application of the resultant N s ; y c is the coordinate of application of the resultant N c ; M Rd,Y , M Ed,Y -values of bending moments from the action of internal and external forces, respectively, in the plane of the coordinate axis Y at the moment of exhaustion of the strength of the reinforced concrete member in the normal section.
To obtain the necessary design formulas, the expressions of the resultant N c , as well as the distance y c from the most compressed concrete fiber to the point of application of N c , were first realized.The desired expressions for the trapezoidal form of the compressed concrete area (figure 1) are obtained in the following form: where θ is the angle of inclination of the neutral axis; X is the neutral axis depth.Since the problem is solved for the case when the stresses in the reinforcement of the tensile area reach the values σ s = f yd , then the resultant force in the tensile reinforcement is determined by the dependence After substituting ( 6) -( 7) in ( 3) and ( 5) for the trapezoidal form of the compressed area, the formulas for determining the neutral axis depth X and the ultimate value of the moment M Rd,Y in the plane of the coordinate axis Y are obtained: Substituting the value of X from equation ( 9) into equation (10) and making the substitution By equating the right-hand side of inequality (1) and equation ( 9) taking into account (2), it is possible to obtain the coefficient of reduction of the design strength of reinforced concrete at biaxial bending for a rectangular cross-section with a trapezoidal form of the compressed concrete area To obtain the dependence θ = f (β), which can be used to calculate the angle θ of inclination of the neutral axis, the condition of parallelism of the planes of the internal M Rd,β and external M Ed,β bending moments action is applied.Those planes are inclined at an angle β to the vertical Y axis of symmetry of the section.According to this condition in the XOY coordinate system (figure 1) the ratio is valid where d b , d h are the effective heights, respectively, in the direction of the X and Y axes of the cross-section of the biaxially bended member; x c , y c are the coordinates of the point of application of the resultant in the compressed concrete area in XOY coordinate system.The coordinate y c is determined by dependence (7), and the coordinate x c is determined by the following dependence After substituting ( 9) into ( 7) and ( 14), it is obtained that the formulas for calculating the coordinates of the application of resultant force in the compressed area take the form Substituting ( 15) and ( 16) into the original formula (13) makes it possible to obtain an equation whose solution with respect to tan θ will be a formula that allows calculating the angle θ depending on the angle β of inclination of the external load plane where Having singled out the ratio b/d h as one that can be specified, and performing the substitution ρ l , the formula for determining the angle θ is written in the form where k = d b /b.Thus, using dependencies ( 12) and ( 18), it is possible to tabulate the coefficient of decreasing of the strength of reinforced concrete at biaxial bending when the angle β of the external load plane inclination to the vertical axis of symmetry of the section is changed for a certain value of the b/d h ratio.
In a similar way, the dependencies for the triangular form of the compressed concrete area are found.In this case the design scheme is shown in figure 2.
The general equilibrium equations for the considered design diagram (figure 2) will have the form (3) -( 4).
The sought expressions of the resultant N c in (3), as well as the coordinates y c of its application in (4) for the triangular form of the compressed concrete area (figure 2) take the following form: Since the problem is solved for the case when the stresses in the reinforcing steel of the tensile area reach the values σ s = f yd , the resultant force in the tensile reinforcement is determined by dependence (8).
After substituting ( 19) -( 20) in ( 3) and ( 5) for the triangular form of the compressed area, the formulas for determining the neutral axis depth X and the ultimate value of the moment M Rd,Y in the plane of the coordinate axis Y are obtained: By substituting the value of X from equation (21) into equation ( 22) and performing the substitution ρ l , it is obtained By equating the right-hand side of inequality (1) and equation ( 23) taking into account (2), an expression for calculating the coefficient γ is obtained, which takes into account the influence of biaxial bending on the values of the member characteristics in (1) for a rectangular cross-section with a triangular form of a compressed concrete area To obtain the dependence θ = f (β), condition ( 13) is applied, in which the coordinate y c is determined by dependence (20), and the coordinate x c is determined by the following dependence After substituting ( 21) into ( 20) and ( 25), it is obtained that the formulas for calculating the coordinates of the application of resultant force N c in the compressed area take the form: Substituting ( 26) and ( 27) into the original formula ( 13) makes it possible to find an equation whose solution with respect to tan θ will be a formula that allows calculating the angle θ depending on the angle β of inclination of the external load plane tanθ = where c = 3d b − 3d h tan β.
Having singled out the ratio b/d h as one that can be specified, the formula for determining the angle θ is written in the form where Full calculation of biaxially bended members according to the described method is possible only if the form of the compressed concrete area is known.For its definition, a prerequisite is adopted, on the transformation of the trapezoidal form of the compressed area into a triangular one on the border of the limiting state of its existence.The considered state is characterized by one case of the position of the neutral axis, namely, when the neutral axis crosses the less compressed edge.This position of the neutral axis, as can be seen from the comparison of the design diagrams shown in figure 1 and figure 2, is the boundary between these schemes.
According to the design diagram (figure 2), at the limit position of the neutral axis, the equation ( 3) using ( 19) and (8) will have the form From the analysis of equation (30), it is obvious that the condition for delimiting the forms of the compressed area will be inequality If condition (31) is fulfilled, the compressed concrete area has the form of a triangle, if it is not fulfilled, the compressed area has the form of a trapezoid.To use this condition in practice, it is necessary to know all the parameters included in inequality (31).When solving the bearing capacity check problem, two parameters X and θ are unknown.
To determine the unknowns, the condition of the parallelism of the planes of action of internal and external forces is applied, which, based on the design diagram, is written in the form (13).
Substituting ( 25) and ( 20) into ( 13) under the condition that for the limiting case of the position of the neutral axis λX = b sin θ it is obtained After substituting (32) into inequality (31) and performing mathematical transformations, the condition for delimiting the forms of the compressed area is reduced to a simple form Therefore, if inequality (33) is fulfilled, the form of the compressed area of concrete is triangular, if it is not fulfilled, it is trapezoidal.
Using the obtained dependencies ( 12), ( 18), ( 24), ( 29) and (33), it is possible to determine the coefficient γ of taking into account the influence of biaxial bending on the values of the member characteristics in (1) for a rectangular section depending on the angle of inclination of the external load plane.At the same time, the geometric characteristics of the cross-section in the form of ratios b/d h and k = d b /b (figures 1, 2) and the longitudinal reinforcement ratio ρ l of the cross-section are also taken into account.The values of the coefficient γ for reinforcement of class A500C are given in table 2 for the case of the tensile reinforcement reaching the yield point at the time of member failure.
Using the developed method, the bearing capacity of the biaxially bended beams studied in [18] was determined.The calculation results and experimental data are shown in table 3.
To illustrate the developed calculation method, the following example is considered.
Given: a reinforced concrete beam of a rectangular profile with cross-sectional dimensions b = 200 mm, h = 450 mm, the beam is made of concrete class C25/30 (f cd = 17 MPa); in the beam, tensile reinforcement (4Ø16) with a cross-sectional area of A s = 804 mm 2 of class A500C (f yd = 435 MPa) is located symmetrically relative to the vertical axis at a distance of its centre of gravity from the lower face of the cross-section a = 50 mm, the angle between the vertical axis of symmetry of the cross-section and the plane of action of the external load β = 5 • .It is necessary to determine the maximum value of the bending moment that the beam may perceive.Using formula (1), the bending moment that might be perceived by the beam in the vertical plane is calculated M Rd,Y = f zM W γ = 22.76 × 5333333 × 0.962 = 116.77× 10 6 Nmm = =116.77kNm.
The moment in the external load plane action passing at an angle β to the vertical axis of symmetry of the section is M Rd,β = M Rd,Y cos β = 116.77× cos 5

Conclusions
The concept of determining the value of the design strength of reinforced concrete for calculating the bearing capacity of reinforced concrete members subjected to biaxial bending has been developed.The value of the coefficient of influence of biaxial bending on the characteristics of the member of rectangular section with triangular and trapezoidal forms of the compressed concrete area is obtained.The methodology is brought to the engineering level of application.The proposed method of calculating the bearing capacity of beam members under biaxial bending ensures a satisfactory convergence of the calculation and experimental data.

Figure 1 .
Figure 1.Design diagram of the cross-section of a biaxially bended member with a trapezoidal form of a compressed concrete area.

Figure 2 .
Figure 2. Design diagram of the cross-section of a biaxially bended member with a triangular form of a compressed concrete area.

Table 1 .
Design values of reinforced concrete strength for bending members of rectangular section with single A500C reinforcement (f yd = 435 MPa).

Table 3 .
Results of comparison of experimental and theoretical values of the destructive bending moment of biaxially bended beams (b/d h = 0.8; k = 0.62).