Parametric study of the strength of reinforced concrete polygonal sections submitted to oblique composite flexion

abstract: This work aims to verify the influence of characteristic compressive cylinder strength ( f c k), section geometry and eccentric axial load on the strength of square, cross, “T” and “L” reinforced concrete sections, under oblique composite flexion. A computational algorithm was created to calculate sections interaction diagram of bending strength, taking into account NBR 6118 idealized parabola-rectangle stress-strain relationships for 20 to 90 MPa f c k concretes. The results show that f c k influence is stronger for higher values of axial load and that the failure surface shape in interaction diagrams depends directly on the f c k and on the rebars distribution in the section. Furthermore, under lower compressive axial loads, higher oblique composite flexion strengths are reached when there is more reinforcement area in tension regions but, as the compression increases, the reinforcement presence and larger concrete areas in compression zones provide higher bending moment strengths.


INTRODUCTION
Sections subjected to combined bending and axial load are recurrent in reinforced concrete structures. Generally, situations such as columns located in building corners, or even supporting two-way slabs, lead to such types of internal forces [1]. Usually, the representation of reinforced concrete sections strength of combined biaxial bending and axial load is made by axial force ( N ) -bending moment ( x M and y M ) interaction diagrams. These diagrams calculation involves an internal section equilibrium iterative process, due to the physical nonlinearity problem because steel and concrete constitutive relationships. There are, in the literature, a variety of methods proposed to reach such strengths. Dall'Asta and Dezi [2] for example, develop an iterative method in which the main unknown of the problem is a minimum required reinforcement steel area, given a section geometry (arbitrary polygon), the number and location of the rebars, and the mechanical behavior materials. Fafitis [3] generates an algorithm based on the Green's theorem analytical integration of concrete compressive stresses, transforming the double integral integrals into integral line along the compressed polygons. Vaz Rodrigues [4], in turn, implements an algorithm based on the section subdivision into trapezoidal elements and uses the Gauss-Legendre stress integration over these elements.
In addition to the ways to reach the sections strengths, there is a need to evaluate the material properties (reinforced concrete) and section geometry effects on the results of these calculation algorithms. In 2014, NBR 6118 [5] began to contemplate concrete characteristic compressive cylinder strength ( ck f ) from 50 to 90 MPa and proposed a parabolarectangle constitutive relationships to these materials. The equations proposed in this standard idealize the concrete stress-strain behavior as a parabola-rectangle curve and establish the possible deformation domains for reinforced concrete sections under combined bending and axial load by strains c2 ε (concrete compressive strain at the end of parabolic region) and cu ε (concrete ultimate compressive strain), calculated according to ck f , combined with steel yield and ultimate strains, according to Figure 1. In this context, there are analytical and experimental works that help the understanding of the new formulations. Torrico [6], for example, studies the behavior of high strength concrete slender columns, analyzing, among other factors, the concrete ultimate limit states of rupture both experimentally and numerically.
With the investigation of section geometry effects there are also the possibility of optimizing reinforced concrete structural elements under combined bending and axial load, according to applied forces. Campione et al. [7] perform a study of the parameters such as concrete strength, section shape, confining level, reinforcement ratio and rebars distribution influence. The authors conclude, among other things, that increasing concrete strength does not modify the interaction diagram shape of the section with maximum curvature, although it provides an increase in section deformation capacity by reducing the dimensionless axial force ν , calculated according to Equation 1. Based on what already exists in the literature and continuing the study of Souza [8], the present work objectives are: a) the verification of the influence of concrete ck f variation on the oblique composite flexion of square, cross-format, "T" and "L" reinforced concrete sections, applying the parabola-rectangle of NBR 6118 ck f [5]; b) a comparison of the oblique composite flexion strength among different strength shapes with constant steel and concrete areas, considering the variation of concrete and axial force.

METHOD
The analysis of the work is based on the results produced by a computational routine developed in Python 3 language, which reads the input data of the problem, generates the moment-axial force-curvature interaction diagrams and finally calculates the moment-axial force-curvature interaction diagrams for reinforced concrete sections, as shown in Figure 2.

Input data
The main input data of the problem are the sections geometry, materials constitutive relationships, neutral axis inclination angles and axial forces, for which the moment-axial force-curvature interaction diagrams were calculated.
In view of the definition of the problem geometry, we chose to analyze the square, cross-format, "T" and "L" sections of Figure 3 which have the same reinforcement area equal to 12.57 cm 2 and concrete area equal to 625 cm 2 (convenient values). The definition of x H , y H , x b e y b dimensions and of all sections departed from the square section of dimensions 25 x 25 cm. Thus, it was arbitrated that, in cross-format, "T" and "L" the section dimensions  The definition of the rebars distribution and respective area followed a similar script, in which the first decision is the fixation of the square section reinforcement detailing, with 4 rebars of diameter l φ equal to 2 cm, symmetrically distributed, with d' distances between section edges and rebars centroid equal to the sum of the concrete nominal cover nom c , equal to 2.5 cm, with the transverse reinforcement diameter t φ , equal to 0.5 cm, and with longitudinal rebars radius (1 cm), resulting in d' equal to 4 cm.
By fixing the reinforcement area s A at 12.57 cm 2 and using the same value to nom c and t φ from all sections, a longitudinal rebars area for the sections with other shapes is calculated, and the criterion to define the quantity of rebars was the minimum quantity required to enable transverse fixation reinforcement.
Concrete characteristic compressive cylinder strength ( ck f ) range from 20 to 90 MPa by 10 MPa, i.e. the range of values encompassed by NBR 6118 [5].
The equations for the mathematical representation of the concrete and steel constitutive relationships are also defined from NBR [5]. Concrete tensile strength is neglected and its compressive behavior is represented by the idealized parabola-rectangle diagram, described by Equations 2 to 5 and generalized for application in concretes up to 90 MPa. In turn, the CA-50 longitudinal reinforcement steel has its behavior described by NBR 6118 [5] bilinear elastic perfectly plastic stress-strain relationship, represented by Equation 6, for both tensile and compressive strains.
. , where c σ = concrete compressive stress; cd f = concrete compressive strength design value; c ε = concrete compressive strain; c2 ε = concrete compressive strain at the end of parabolic region; n = parabolic region exponent; cu ε = concrete ultimate compressive strain; ck f = concrete characteristic compressive cylinder strength; s σ = rebar stress; s ε = rebar strain; s E = rebar steel elastic modulus design value; yd f = rebar steel design yield strength. Axial force values applied to sections, N , vary depending on the type of analysis desired. In the analysis of the influence of ck f , values shift from 500 to 1200 kN by 100 kN for square and cross-format sections. For "T" sections, N values shift from 500 to 1000 kN by 100 kN and, for "L" sections, from 500 to 1100 kN by 100 kN. Different limit superior values (1000, 1100 and 1200 kN) are used because maximum section axial strength is different for each section shape.
Flexural strength comparisons among different section shapes are done for sections subjected to the same dimensionless axial force ν , considering that sections compared have the same concrete area and ck f .

Moment-axial force-curvature interaction diagrams calculation
Moment-axial force-curvature interaction diagrams determine the correspondence between the section curvature and the respective flexural strength, based on the axial force N , neutral axis inclination angle and section geometry. Figure 4 iterative process, based on the bisection method, is used to find the section maximum allowed curvature before concrete crushing or reinforcement steel elongation limit. The interval containing the solution is repeatedly bisected until the relative variation between two consecutive iterations is less than 1%.
Rupture verification, inside the section maximum allowed curvature calculation, is done by another iterative algorithm, that aims to calculate neutral axis equilibrium position LN y α . This procedure is applied to find the section strain field that balances the section curvature ( ) k 1 r and the applied axial force N . As in the algorithm that achieves a section maximum allowed curvature, the bisection method is used until the relative variation between the section axial strength and the applied axial force is less than 1%. Concrete compressive stress integration is solved by using numerical integration to determine the compressed area, the c2 y ε α coordinate is calculated according to Equation 7.
where c2 y ε α = y α coordinate that corresponds to c2 ε strain; The compressed areas vertex coordinates are determined according to Figure 5, using the Python's Shapely package (geometry.polygon.intersection), from the python shapely package, developed by Gillies [9]. The area pol intersec results from the intersection between the infinite region pol 2 above neutral axis and section region pol 1 , using ( x α , y α ) coordinates.
Then, pol intersec area is meshed with triangles, using Python's MeshPy package (meshpy.triangle),developed by Klöckner [10]. Mesh generation is controlled by triangles maximum area set as 10 cm 2 . This value represents a balance between computational processing time and results expected precision. The mesh generation output data is a set of t n matrices [ ]      ) ( )   and a neutral axis inclination angle α , subjected to an applied axial force N .

Bending moment-axial force interaction diagrams calculation
Bending moment-axial force interaction diagrams curves are sets of points represented by combined loads (bending and axial) that lead to section failure. These points are determined based on moment-axial force-curvature data processing. Each ( N , α ) pair determine a maximum flexural strength rd M , its respective θ inclination angle from the x axis and its respective neutral axis equilibrium position, as illustrated in Figure 5b. Difference between α and θ angles is due to the asymmetric distribution of concrete compressive stresses and of the rebars stresses in relation to the neutral axis perpendicular line.
The interaction diagrams points coordinates are the applied axial force   Figure 7 present the results for "L" sections compressed by N = 500 kN (minimum applied axial load), N = 800 kN and N = 1100 kN (maximum applied axial load), respectively. Table 1 summarizes the interaction diagrams results presented in Figures 6 and 7, for θ angles varying from 0 to 7 4 π rad, shifted by 4 π rad, in order to facilitate the numeric reading of figures results.
The use of axial force absolute values instead of dimensionless axial force ν was chosen because ck f is variable and, consequently, for the same section subjected to a constant ν , N absolute value would also vary with the ck f , something that could distort comparisons.

Shape influence on section flexural strength
Shape influence on section flexural strength is evaluated by comparing side by side the flexural strength interaction diagrams of square, cross-format, "T" and "L" sections of Figure 3, that are composed of concretes of the same ck f , same reinforcement steel area and subjected to the same dimensionless axial force ν . It is noteworthy that, in this case, the analysis by the rates of dimensionless axial force ν is possible because the graphs deal with the same ck f and the same concrete area.    The visualization of the neutral axis equilibrium position at the flexural strength interaction diagrams limits allows to see the stress and strain states in each case. It appears in Figure 10 plot for N = 500 kN that neutral axis intercepts the section in all interaction diagrams for all θ angles (NBR 6118 [5] domains 2, 3 and 4) at sections equilibrium limit states and that high strength concretes require smaller compressed areas. It is also interesting to point out that the concrete flexural strength contribution is higher with the increase of the ck f . When θ=0°, the respective concrete and reinforcement contributions on flexural strength are 47% and 53% for 20 MPa concrete sections and 65% and 35% for 90 MPa concrete sections.   The relative proximity of the square sections interaction diagrams with different strength concretes in Figure 6 with N = 500 kN can then be associated with the predominance of bending effects, where low concrete flexural strength contributions can be offset by the increase in the reinforcement steel contribution, since the applied axial force is relatively low.
On the other hand, the Figure 6 square sections interaction diagrams with N = 800 kN now indicate some differences in relation to N = 500 kN diagrams and greater distance between the interaction diagrams is perceived, being that the order of the flexural strengths agrees the ascending order of the ck f . The only exception is the comparison of the interaction diagrams of the square sections with 50 and 60 MPa ck f , which almost coincide outside the principal axis of inertia and move away only when are on them.
It is attested that for square sections with N = 800 kN, there is an increase in concrete flexural strength with θ = 0°, which is 56% for a section with ck f = 20 MPa and increases up to 68% for those with ck f = 90 MPa. In addition, Figure  10 plot for N = 800 kN shows that neutral axis equilibrium positions continue to intercept the sections even if the compressed area predominates in the case of concretes with ck f from 20 to 70 MPa. The increase of concrete contribution on section flexural strength is, in this case, the main factor that interferes in the greatest distance between the Figure 6 square sections interaction diagrams with N = 800 kN.
The differences between the curves are greater in Figure 6 square section diagrams with N = 1200 kN, due to the increase of the compressed areas and, consequently, the concrete contribution on section flexural strength. However, when analyzing square sections equilibrium configuration when the angle α is zero in Figure 10 plot for N = 1200 kN, it is seen that only the square section with ck f = 20 MPa has neutral axis totally external to the section. All other sections with ck f values above 20 MPa indicate partial compression, with the neutral axis intercepting the section, even with N =1200 kN.
The same phenomenon is observed with the variation of the N for sections with other shapes, showing that ck f the influence grows as the compressive axial force becomes predominant in relation to the bending moment. Another interesting perception in Figure 6 refers to the change in the interaction diagrams shape, described in point (b). It is observed that the ck f ≤ 50 MPa square sections interaction diagrams come out of a format next to a diamond ( N =500 kN) to a more rounded format ( N =800 kN). Meanwhile, curves of sections with 60 to 90 MPa ck f change inversely, having shapes with rounded corners under compression of 500 kN and tending to diamonds when the axial force increases to 800 kN, with sharp corners in the principal axis of inertia. These tendencies become more explicit in Figure 6 diagrams in which the square sections are subjected to N = 1200 kN.  In the case of square sections, the change of interaction diagram shapes is reversed when changing the formulation used to calculate the concrete compressive strength. As shown in Equations 2 to 5, the stress calculation parameters are different for concretes with ck f below and above 50 MPa, which is a limit that coincides with the different behaviors identified in the curves.
Interpreting Observing the square section equilibrium configuration with ck f = 20 MPa under N = 500 kN, with an θ angle equal to 45° (outside the principal axis of inertia), it is perceived that flexural strength is about 18% lower than that for θ = 0°. Although neutral axis stabilizes in similar positions in relation to the section centroid when θ=0° and θ=45° and the maximum concrete and reinforcement strains present close values, θ=0° strength is higher due to the better reinforcement and compressed concrete areas distribution, making the concrete and the axial reinforcement strengths lever arm grater in relation to the centroid section.
The increase of ck f tends to reduce the relative differences of flexural strength between θ=0° and θ = 45° to N = 500 kN, which softens the contours of the interaction diagrams. In the most extreme case, which is that of section with ck f = 90 MPa, the concrete contribution on section flexural strength prevails, representing more than 65% of section flexural strength for θ=0° and for θ=45°. As the plastic stretch does not exist for ck f = 90 MPa and the respective stress-strain curve is almost linear, the magnitude and the position of section axial strength change little and so the interaction diagrams become more rounded.
For high compression, as in Figure 6  in the various θ directions, i.e., tend to a constant value and the interaction diagrams look rounded.
In contrast, square sections with ck f > 50 MPa have strains between 0 and cu ε when subjected to high compression, in addition to the fact that neutral axis equilibrium position is closer to centroid. Thus, c σ also tends to vary in a way closer to the linear and θ angle variation significantly influences the value and lever arm of rd N . Therefore, such concretes under N =1200 kN have interaction diagrams with straight lines and live corners in the principal axis of inertia.
Thus, a direct correlation between concrete constitutive relationship and the reinforced concrete sections behavior under combined bending and axial load is perceived. The sections internal equilibrium is sensitive to c σ distribution over concrete areas and cu ε values of which, in turn, interfere in the reinforcement deformation resulting in different interaction diagrams as the concrete constitutive relationship is modified.
Extending the analyses to the cross-format sections under low compression (Figure 6, N = 500 kN), ck f ≤ 50 MPa concretes provide interaction diagrams with format approaching a square with rounded corners, in which θ = (2 n + 1) π /4 directions, where n = 0, 1, 2..., present the maximum flexural strengths. At the same time, ck f > 50 MPa concretes lead to nearly round interaction diagrams shapes.
Shifting N to 1200 kN, the shapes of ck f ≤ 50 MPa diagrams of cross-format sections become almost round, and this becomes sharper the smaller the ck f . Meanwhile, the shapes of ck f > 50 MPa concretes become almost a smooth square. Comparing such observations with those made for square sections, it is perceived that the same factors guide the process, i.e., the concrete stress-strain relationship and the neutral axis position, associated with compressed areas and reinforcement distribution.
For "T" and "L" sections, the increase of N also lead to a change in interaction diagrams shape and behaviors continue to be distinct for the two ranges of the Equations 2 to 5. However, both sections have only one axis of symmetry, unlike square and cross-format sections. This implies, among other things, greater reinforcement concentration in some regions (see Figure 3), something that should be taken into account in the analyses. In the case of "T" sections the reinforcement ratio is higher in the meeting of the flange with the web and, in the "L" sections, in the meeting between the two legs.