The calculation of reinforced concrete sections is a well known and understood problem (Bonet et al. 2006; Daunys and Rimovkis 2006; Choa and Kwonb 2008; Liang and Fragomeni 2009; Ronagh and Baji 2014; Ren et al. 2015). In the particular case of circular sections, the problem is complex. Due to the provision of reinforcement and concrete, the circular sections have substantially the same geometric characteristics in all directions. Therefore, the calculation remains complicated given the position of the armatures which induces several unknown in the expression of the equilibrium equations in the case of a composed bending (Kachi et al. 2014).

Also in the case of fiber reinforced concrete, several studies were conducted in recent years to characterize the behavior of this type of composite, but the problem becomes even more complicated in the case of circular sections subject to combined loads.

Experimental and finite element analysis, in ultimate behavior, on the steel fiber-reinforced concrete beams has been made by Özcan et al. (2009); authors modeled beams using nonlinear properties of materials taken from the experimental study until the ultimate failure cracks. In this study, steel fibers “Dramix-RC-80/0.60-mm” were used as additives in concrete with four different dosages of 30, 40, 50 and 60 kg/m^{3}. The finite element analysis was performed by using the ANSYS program. Results from numerical modeling and experimental analysis are compared to each other, it followed that there is a good agreement between the results of finite element analysis and experimental behavior.

An experimental program was conducted by Ding et al. (2012); the objective was to investigate the influence of steel fibers and the combined effect of fibers and stirrups on the beams deflection and cracking. This study also examined the feasibility of applying the modified compression field theory (MCFT) for the suitable assessment of combined load in fiber and steel rebar reinforced concrete beams. The authors noted that the combined use of stirrups and steel fibers shows great positive composite effect on shear load-carrying capacity, energy absorption capacity and toughness of beams subjected to shear and bending. Additionally, they proposed a new equation of modified compression field theory for fiber reinforced concrete in which cracked fiber concrete is treated as a new material with its own stress–strain characteristics.

Islam and Alam (2013) conducted a study to evaluate the shear strength for steel fiber reinforced concrete beams in a database that is extensive experimental results of 222 specimens with no stirrups. The results of this study showed that the recommended empirical equations were best suited to assess the shear strength of SFRC beams more accurately as compared to those obtained by the previously developed models. The authors concluded that the prediction models of SFRC, suggested by the various studies, were mostly complex and confined to non-linear regression equations.

In another study, six full-scale prestressed concrete I-beams with steel fibers were tested to failure by Tadepalli et al. (2015); the main objective of this study was to determine effects of steel fiber dosage on the shear and flexural modes of beam failure. The beams were subjected to concentrated vertical loads up to their maximum shear or moment capacity. Based on experimental observations, the authors found that the I-beams shear capacity was significantly increased due to the addition of steel fibers in the concrete. The study also showed a complete replacement of conventional shear reinforcement with steel fibers.

On the other hand, Sorensen et al. (2014) presented investigation findings of the fiber content variations in concrete being discharged from a ready-mix truck at the construction site and its effect on the behavior of fiber concrete. The type of steel fiber used in this study consists of fibers with hooked ends that were commonly used in world for decades. Based on the test results and observations, authors concluded that adjustments in batching procedures may be advisable regarding steel fiber mixing routines.

In practice, the calculation of longitudinal reinforcement is often conducted according to the principles of the current regulations, BAEL (Rules: BAEL91 99) and Eurocode2 (Eurocode2, Part 1-1 1999), considering the steel uniformly distributed over the entire section of the concrete and in the case of a normal effort of compression. The normal tensile stress was not considered and the contribution of tense concrete between the cracks is neglected. Then just ignore the tense concrete to remove physical discontinuities corresponding to cracks.

When taking into account the contribution of tense concrete between the cracks, describe the behavior of the element becomes complex considering the stress distribution. A global behavior of the element is then assumed which can be represented by the characterizing each section between two successive fissures by the behavior of an average section, representative of the entire cracking phenomenon.

The objective of this study is the development of numerical methods to simulate the non-linear behavior until rupture taking into account the tense concrete contribution, between two successive fissures. It considers reinforced concrete or fiber concrete in case of circular sections subjected to composed bending. Algorithms are then developed to calculate internal deformations from the forces applied. This has allowed us to establish abacuses giving the reinforcement section based on external forces applied.

For the compression behavior of concrete, the Sargin law is adopted. This allows us to adjust the parameters k_{b} and \( k_{\text{b}}^{{^{\prime } }} \) to the experimental curve (Rules: BAEL91 99). The contribution of tense and cracked concrete is made according to a parabolic law proposed by Grelat (1978).

The behavior of the fiber concrete was taken into account according to the model proposed by Bouafia et al. (2002, 2000). A comparison was made with the behavior of a section considering the conventional diagrams of materials (provided by the BAEL). A second comparative study was performed by using a section armed with fiber.