Cross-Sectional Analysis of the Resistance of RC Members Subjected to Bending with/without Axial Force

This paper deals with the cross-sectional analysis of the resistance of RC members subjected to a bending moment with or without axial forces. To determine section resistance, the nonlinear material law for concrete in compression is assumed according to Eurocode 2, taking into account the effect of concrete softening. It adequately describes the concrete behavior of RC members up to failure. The idealized stress–strain relation for the reinforcing steel is assumed. For the ring cross-section subjected to bending with axial force and for areas weakened by an opening, normalized resistances have been derived by integrating corresponding equilibrium equations. They are presented in the form of interaction curves and compared with the results of testing conducted on RC eccentrically loaded columns. Furthermore, the ultimate normalized bending moment has been derived for the RC rectangle subjected to bending without axial force. It was applied to the cross-sectional analysis of steel and concrete composite beams consisting of the RC rectangular core located inside a reversed TT-welded profile. Comparative analysis indicated good agreements between the proposed section models and experimental data. The objective of the paper is the dimensioning optimization of the considered cross-sections with the fulfillment of structural safety requirements.


Introduction
For structural safety reasons, the load-bearing capacity (resistance) of any designed or existing structures should satisfy the conditions of ultimate limit states. In recent years, intensive developments and advanced applications of RC structural members with the increased resistances are observed in newly designed and erected tower buildings. The resistance of RC (reinforced concrete) members subjected to bending with or without axial force is undertaken both as a structural and a practical task. Such members are commonly encountered in engineering practices, e.g., reinforced concrete columns, tower-like structures, steel and concrete composite columns and beams. In the formulation accepted in this paper, the resistance of RC cross-section is determined by the occurrence of the ultimate strains occurring anywhere in that section, which means that it depends both on the material laws and the geometrical characteristics of the section. With respect to material laws, a simplified approach is the most often used on the basis of the rectangular stress distribution in concrete, represented among others by Knauff [1]. For the design of annular cross-sections, a parabola-rectangle diagram for concrete in compression is commonly assumed, which was introduced by Nieser and Engel [2] in German code DIN 1056 as well as in CICIND Model Code for Concrete Chimneys [3]. The dimensioning diagrams attached to these codes were developed on the assumption of the thin ring's thickness and central layout of reinforcement. The parabolic-trapezoidal stress distribution for concrete in compression was proposed in turn by Hognestad [4] and applied in ACI Standard 307-08 [5]. A review of material laws for concrete and the experimental justification of formulae for the estimation of the complete stress-strain diagram of concrete were presented by Popovics [6,7]. For analysis of the resistance of noncircular cross-sections, deformation models were proposed by Lechman [8] on the basis of the parabola-rectangle stress For both reinforcing and profile steels, the linear elastic-ideal plastic model is applied.

Derivation of Analytical Formulae for the Resistance
• The RC ring cross-section of outer radius R, inner radius r and thickness t = R − r is subjected to axial force N and bending moment M (Figure 1). The section may be unreinforced or reinforced with the reinforcing steel spaced at one or two layers, Materials 2022, 15,1957 3 of 16 which can be replaced by a continuous ring of equivalent area located on the reference circumference of radius r s , r ≤ r s ≤ R. When r s = R or r s = r, this reinforcement is treated as an "external reinforcement". • For the section under combined compression and bending, the relations for strains in concrete ε c (‰) and in reinforcing steel ε s (‰) are given by the following. Taking into account physical and geometrical relationships (1)-(3) in equilibrium Equations (4) and (5), the problem results in a purely mathematical task consisting in searching the indefinite integrals of the following functions of variable ϕ: where the following is the case.
Upon transformation, Equations (6) and (7) result in the following: All angles are measured from the compressive to the tensile zone. The sectional equilibrium equation of the axial forces is described as follows.
The sectional equilibrium equation of the bending moments about the symmetry axis of the section is expressed as follows. Taking into account physical and geometrical relationships (1)-(3) in equilibrium Equations (4) and (5), the problem results in a purely mathematical task consisting in searching the indefinite integrals of the following functions of variable φ: where the following is the case.
Upon transformation, Equations (6) and (7) result in the following: where An indefinite integral of function 1/(cosφ + b) occurring in Equations (9) and (10) has been found as follows.
Having calculated the definite integrals of Equations (9)-(12), normalized ultimate resistances n Rm and m Rm are obtained in the following final form: where the following is the case.
Here, µ is the reinforcement ratio; and α 1 and α 2 are angles determining the depth of the plastified zones of steel in compression and in tension, respectively.
The graphic interpretations of Equations (13)- (15) are the interaction diagrams with normalized resistances n Rm -m Rm valid for the case, when both compressive and tensile strains occur in the analyzed section characterized by the following: the limiting value ε cu = −3.5‰; the concrete grade C20/25 (f cm = 28 MPa); f yk = 500 MPa; E cm = 30 GPa; substitute reinforcement ratio µ f yk /f cm = 0; 0.1; 0 ≤ ε s ≤ 10‰ ( Figure 2). These curves denoted by the solid line (EC2) are compared with those based on the parabola-rectangle diagram (the dashed line, parabola). It is apparent that the coordinates of curves denoted by EC2 are lower than for the parabola due to the effect of concrete softening.

Figure 2.
Interaction curves n Rm -m Rm based on nonlinear relation σ c -ε c (EC2) versus those based on the parabola-rectangle material law (parabola) for the ring section subjected to bending with axial compressive force (ε cu = −3.5‰).
In a similar manner, the corresponding formulae are obtained for the section that is entirely in compression. The relevant relationship for strains is given by the following.
The formulae determining cross-sectional forces n Rm and m Rm are described in the following form: where k 1 = 2ε 1 /(ε 2 − ε 1 ) + 1; and k 2 = −0.5 (ε 2 − ε 1 )/ε c1 shall be substituted. For the section that is entirely in compression, when |ε cu | > |ε c1 |, Equations (17) and (18) describe the ring cross-section at the stage of concrete instability (at failure). This means that Drucker's stability postulates may not be satisfied [38,40]. Therefore, the obtained curves cannot be, in general, regarded as the carrying capacity curves. This is the case presented in [9]. Figure 3 presents interaction curves (EC2) resulting in the relation (1) versus those derived on the basis of the parabola-rectangle (parabola) extended to the ring cross-section that is entirely in compression (ε c1 = −2.0 ‰; ε cu = −3.5‰), with a limitation to the values m Rm ≥ 0. All curves in Figures 2 and 3 satisfy the condition of convexity according to Drucker's postulate. In particular, for ε cu = ε c1 = −2‰, the stability condition by Drucker is satisfied for any values of ε c and ε s . Thus, the obtained curves based on Equations (13)- (15) and Equations (17) and (18) can be regarded as the actual carrying capacity curves accepted in design codes. It is worth highlighting that these interaction diagrams (EC2) are very close to those based on the parabola-rectangle (parabola) (Figure 4).
As the next section model, the ring cross-section weakened by one opening is considered and is subjected to ultimate axial force N u and bending moment M u = N u · e ( Figure 5). The stress distributions in concrete and reinforcing steel are described by design values f cd = f ck /γ c and f yd = f yk /γ s , while the size of opening is denoted by angle α 1 . Following the above outlined algorithm, normalized ultimate resistances n u and m u are derived for this section in the following form: where α a1 and α a2 are the angles determining the depth of the plastified zones of steel in compression and in tension, respectively. The section model is frequently used in the structural design, e.g., flue opening in chimneys, windows in tower walls [8]. In a similar manner, the corresponding interaction diagrams can be constructed for other section shapes.

Experimental Verification with Discussion of the Results
Full-scale tests were carried out on four RC designed columns eccentrically loaded, with annular cross-section, denoted by Typ 2 [15]. The outer radius of all columns was R = 0.3 m, the inner one was r = 0.2 m and the height was h = 2.0 m. The mean compressive strength of the column concrete was determined as fcm = 20 MPa; Ecm = 27 GPa; and εc1 = −1.8‰ by strength test according to Polish Standard PN-EN 12390-3: 2011 [15]. The reinforcement of the columns consisted of longitudinal bars ∅16 mm made of steel B500C (fyk = 500 MPa) and its percentage was μ = 1.024%. The properties of the steel rebars were established in turn by tensile test according to Polish Standard PN-EN ISO 6892-1:2010, [15]. The columns were strengthened in the both support zones by CFRP mats at the length of 0.5 m. The tested specimens and the experimental setup are exhibited in Figure  6. In each load step, the strains were measured in the middle section using strain gauges located along its circumference up to failure. The test results of the examined columns are collected in Table 1. The failure of the columns manifested itself by crushing the concrete and yielding the longitudinal reinforcing steel for all tested members ( Figure 6). The above-described test results have been compared with the cross-section model presented in Section 2.1. with the substitute reinforcement ratio of μ fyk/fcm = 0.2688. Using the derived Equations (13)- (18), the interaction chart nRm-mRm has been plotted. The effect of confinement of the column (stirrups) was not analyzed. The comparisons presented in Figure 7 illustrates a good convergence between the section model and the values of failure loads. The occurring differences between the analytical and the experimental re- The section model is frequently used in the structural design, e.g., flue opening in chimneys, windows in tower walls [8]. In a similar manner, the corresponding interaction diagrams can be constructed for other section shapes.

Experimental Verification with Discussion of the Results
Full-scale tests were carried out on four RC designed columns eccentrically loaded, with annular cross-section, denoted by Typ 2 [15]. The outer radius of all columns was R = 0.3 m, the inner one was r = 0.2 m and the height was h = 2.0 m. The mean compressive strength of the column concrete was determined as f cm = 20 MPa; E cm = 27 GPa; and ε c1 = −1.8‰ by strength test according to Polish Standard PN-EN 12390-3:2011 [15]. The reinforcement of the columns consisted of longitudinal bars ∅16 mm made of steel B500C (f yk = 500 MPa) and its percentage was µ = 1.024%. The properties of the steel rebars were established in turn by tensile test according to Polish Standard PN-EN ISO 6892-1:2010, [15]. The columns were strengthened in the both support zones by CFRP mats at the length of 0.5 m. The tested specimens and the experimental setup are exhibited in Figure 6. In each load step, the strains were measured in the middle section using strain gauges located along its circumference up to failure. The test results of the examined columns are collected in Table 1. The failure of the columns manifested itself by crushing the concrete and yielding the longitudinal reinforcing steel for all tested members ( Figure 6). The above-described test results have been compared with the cross-section model presented in Section 2.1. with the substitute reinforcement ratio of µ f yk /f cm = 0.2688. Using the derived Equations (13)- (18), the interaction chart n Rm -m Rm has been plotted. The effect of confinement of the column (stirrups) was not analyzed. The comparisons presented in Figure 7 illustrates a good convergence between the section model and the values of failure loads. The occurring differences between the analytical and the experimental results are   The mean value εcu = 2.8‰.   sults are caused by measurement uncertainty (strains, eccentricity) as well as by ignorin transverse reinforcement in the section model.  The mean value εcu = 2.8‰.

Derivation of the Ultimate Bending Moment
The RC rectangle of the height t and the width b is subjected to bending moment M without axial force (Figure 8).

The Resistance of Composite Steel and Concrete Beams versus Test Results
The presented section model can be applied for determining the resistance of composite steel and concrete beams, named BH beams, subjected to bending. The considered beams consist of a reinforced (RC) rectangular core placed inside a reversed TT-welded profile, as shown in Figure 9.
Ultimate bending moment MHRm determining the resistance of the considered BH beam is derived in terms of strains upon integrating the equilibrium equation of the bending moments about the horizontal axis of the RC core rectangle [28]. In this derivation, the reversed TT-welded profile is treated as the external reinforcement with respect to the RC rectangular core. To compare the obtained analytical solution with experimental results, four-point bending tests were conducted on three separated BH beams with the length of L = 7.88 m. They were made of concrete with a mean compressive strength of fcm = 68 MPa, which was determined by a strength test according to Polish  Figure 10. The range of the tests included determining failure loads and the relevant strains. In each load step, the strains in concrete εc, in reinforcing steels in compression εs1 and in tension εs2, as well as in the lower flange of profile steel εHf, were measured in the middle section of the BH beam. The relation for strains can be expressed in the following form: where ξ = x'/t and ξ = x/t is the dimensionless coordinate of any point of the rectangle and the dimensionless coordinate describing the location of neutral axis. The equilibrium equation of the bending moments about the horizontal axis of the RC rectangle is described as follows.
As a result of integrating Equation (22), the normalized ultimate bending moment m Rm is obtained in the following final form.

The Resistance of Composite Steel and Concrete Beams Versus Test Results
The presented section model can be applied for determining the resistance of composite steel and concrete beams, named BH beams, subjected to bending. The considered beams consist of a reinforced (RC) rectangular core placed inside a reversed TT-welded profile, as shown in Figure 9.  Ultimate bending moment M HRm determining the resistance of the considered BH beam is derived in terms of strains upon integrating the equilibrium equation of the bending moments about the horizontal axis of the RC core rectangle [28]. In this derivation, the reversed TT-welded profile is treated as the external reinforcement with respect to the RC rectangular core.  Figure 10. The range of the tests included determining failure loads and the relevant strains. In each load step, the strains in concrete ε c , in reinforcing steels in compression ε s1 and in tension ε s2 , as well as in the lower flange of profile steel ε Hf , were measured in the middle section of the BH beam.  The failures of all BH beams manifested themselves by concrete crushing (Figure 11). Table 2  It is worth noting that they are close to failure bending moments Mu (relative differences 4.8-6.5%). This confirms very good agreements between resistances MHRm and the test results in ultimate bending moments Mu. The failures of all BH beams manifested themselves by concrete crushing (Figure 11). Table 2 summarizes the values of strains, failure bending moments M u and resistances M HRm . The compressive strains in concrete reached ultimate values. The value of ε s1 = −2.89‰ indicates that the plastic strains in the rebars in compression may have occurred. The values of M HRm have been calculated in accordance with derived Equations (23)- (26). It is worth noting that they are close to failure bending moments M u (relative differences 4.8-6.5%). This confirms very good agreements between resistances M HRm and the test results in ultimate bending moments M u .

Conclusions
The complete analytical solution has been found for the resistance of RC ring cross-sections subjected to bending with axial force, based on the nonlinear material law

Conclusions
The complete analytical solution has been found for the resistance of RC ring crosssections subjected to bending with axial force, based on the nonlinear material law for concrete and taking into account the effect of concrete softening. It applies both to designed and existing members and structures. In this respect, it can be regarded as a valuable one in the theory of reinforced concrete:

1.
The obtained solutions are presented in the form of interaction diagrams that satisfy the conditions of convexity in accordance with Drucker's postulates. This means that they can be regarded as the actual carrying capacity curves.

2.
The proposed ring section models seem to have a wider application field than the previous ones, due to the assumptions of a noncentral layout of reinforcement and wall-edge strains. As a result, they are suitable for ring cross-sections with both the thin and moderate thicknesses. Furthermore, they can be easily adopted when structure strengthening is required by means of externally bonded CFRP, FRP or GFRP composites as well as for determining the resistance of composite steel and concrete columns. 3.
Using this approach, the similar formulae can be derived for other sections commonly encountered in the engineering practice, for example, a rectangle and the ones weakened by openings.

4.
It was proved that the computational results conform to those obtained by testing on RC eccentrically loaded columns.

5.
The analytical solution was developed for the resistance of RC rectangle subjected to bending without axial forces to determine ultimate bending moment M HRm of the composite steel and concrete beams. 6.
The comparisons made between the computational and test results of BH beams showed good agreements in ultimate bending moments. 7.
The above-developed models enabled the analysis of the behavior of RC members in the postcritical phase. 8.
They have been implemented in Excel to provide a useful tool for the dimensioning optimization of RC-designed members and structures that may result in a reduction in material consumption and a lesser impact on the environment. 9.
Further experimental work is needed concerning the postcritical behavior of RC members.
Funding: This research received no external funding.
Institutional Review Board Statement: Not applicable.

Informed Consent Statement: Not applicable.
Data Availability Statement: Not applicable.

Conflicts of Interest:
The author declares no conflict of interest.