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1.1 Background of the study

With recent developments in structural and material mechanics, assessments of safety margin with respect to non-linear system response and failure, instead of admissible stresses, became possible and even required by several codes . Numerical methods for the accurate simulation of the non-linear behavior of engineering structures have also been developed in last few decades and incorporated into computational tools. This evolution has significantly increased the need for knowledge about the inelastic properties of materials (e.g., plasticity, damage, creep, fracture, etc.) which cannot be assessed, unlike the elastic parameters, by means of non-destructive tests such as those based on ultrasound tests. Furthermore, the assessment of inelastic properties when combined phenomena take place (e.g., plasticity with damage and fracture) is rather difficult, or even impossible, using standardized tests for the evaluation of compressive or tensile strength as a single material property.

Accurate numerical modeling within the non-linear regime is related to the appropriate selection of a constitutive model, capable of accounting for phenomena that are taking place at the material level (e.g., plastic deformation, damage of the material, creep, etc.). Such a constitutive model would offer a framework for the accurate modeling of a structural response in the general context, beyond the one represented by the experiment performed for its calibration. Therefore, the quantification of the parameters that govern the constitutive equations should not be merely reduced to the fitting of a single experimental response.

The importance of appropriate constitutive model selection becomes more evident when a complex material, such as fiber-reinforced concrete (FRC), should be modeled. Owing to the presence of small fibers, the structural response of FRC with respect to conventional reinforced concrete is considerably different. With conventional reinforcement, significant elongation of the steel bar is required, so that it can carry tensile loads, which requires the notable opening of macro cracks within the concrete. In contrast, in FRC the cracks are often barely visible to the naked eye, and are developed in the form of a network, which gives the structural member greater ductility and, at the same time, limits the exposure of fibers to the ambient conditions .

In previous years, considerable research efforts have been devoted to studying the mechanical response of FRC. Significant attention has been devoted to analyzing the influence of fiber orientation on the mechanical response of structures. Since it is recognized that fiber distribution and orientation play an important role in global mechanical properties, several authors have discussed the influence of the casting process on the orientation of fibers, analyzed various methods to measure it, and tried to predict it through flow simulations. There have also been many experimental studies focused on the quantification of the global mechanical properties of structural components made of FRC. However, for the systematic incorporation of this material into structural analysis, a proper constitutive description and related parameter calibration is required.

The mechanical response of the structural components made of FRC depends, to a large extent, on the local distribution and orientation of reinforcing fibers. Such information can be collected through the use of X-ray computer tomography (CT), but its effective incorporation into numerical modeling still needs to be solved. The major difficulty for successful modeling is related to the fact that the existing orthotropic constitutive damage models, which are implemented in commercial finite element codes, are suitable for defining anisotropic material behavior only at the structural level. While this can be an appropriate strategy to model conventional reinforced concrete, it is not appropriate for the FRC, where locally strengthened directions, achieved by reinforcing fibers, vary considerably within different regions of individual structural components.

For reliable numerical simulations, fiber distribution and orientation should be included within the constitutive description, thus requiring multi-scale approaches with the capability of incorporating the inherent variability of the internal structure. For this purpose, discrete models, can be used, with further modifications, to take into account fiber distribution and orientations. These discrete models are capable of addressing material behavior at the micro and meso-scales. For the macro-scale, however, which is of importance for the analysis of large-scale structures, it is desirable to have a continuum phenomenological model. Such models are based on a representative volume element (RVE), treated as a continuum, without the necessity to model smaller constituents (e.g., fibers or grains). The presence of these individual constituents is, instead, taken into account through homogenized, macro-scale mechanical characteristics. These models, necessarily, involve certain assumptions that could limit their applicability. The feasibility of the numerical implementation, however, is significantly improved, since the problem is solved on a single scale. This approach is adopted in the present study.

Considering the nature of the phenomena that take place on the fiber scale, a reasonable approach would be to employ a damage model. The major difficulty related to the employment of existing constitutive damage models within commercial finite element modeling (FEM) codes is that even though they can simulate either isotropic or orthotropic behavior, the orthotropic behavior can only be modeled along the directions defined at the structural level. This can be an appropriate strategy to model conventional reinforced concrete, where reinforcing bars have well established directions with respect to the structure, but is not sufficient for FRC, where locally strengthened directions change from one point to another.

Fibre reinforcement concrete ranks among the category of fibre-concrete composites. Fibre reinforcement concretes may vary by concrete and especially by fibres used, i.e. by shape, length, diameter, and finishing. For building structures such as floors, foundation structures, tunnel linings or underground structures, the use of fibre concrete is often more beneficial that of the ordinary reinforced concrete or plain concrete. Concrete reinforcement by fibres increases the tensile strength and ductility of concrete, which is very small as compared with plain concrete. Other benefits of fibre concrete may include limitation of shrinkage cracks and deformations of concrete, increased ductility and fatigue strength, improved consistency (no falling-off at marginal parts of concrete). For the design of structures from fibre concrete, it is important to know in details its properties, which are most frequently defined in laboratories on testing samples. Fibre concrete itself and its testing are dealt with in the Czech Republic and globally on a long time basis. Research results can be found in many suggestions and national standards, norms, and technical conditions..

Every mechanical system submitted to static, dynamic, or thermal loads presents a response. However, complex analyses must be performed to obtain the mechanical response because the mechanical system is composed of several structural elements that evidence different mechanical behaviors. Some studies have reported the use of different constitutive models for analysis of mechanical concrete systems.

In the context of numerical theories used in computational mechanics, Continuum Damage Mechanics (CDM) deserves a mention, because it shows reliable numerical responses for mechanical systems composed of complex materials like concrete. An important work was developed by La Borderie, who proposed a homogenization rule with a damage model for fiber-reinforced concrete in order to obtain the updated stress in the composite matrix.

On the other hand, Li and Li studied concrete damage models applied to the analysis of tensioned fibers. In their work, the concrete was treated as a medium that shows hardening behavior for increasing strain. The results are satisfactory when compared to experimental ones. The same observation is valid for the work developed by Lee and Liang, who applied CDM to fiber-reinforced cellular concrete.

Moreover, Hameed et al. used damage mechanics to model steel-fiber-reinforced concrete beams, obtaining satisfactory results when compared to experimental tests. Pasa also evaluated the mechanical behavior of steel-fiber-reinforced concrete using the finite element method with smeared crack models.

According to Guello, the nonlinear behavior of the concrete, which takes place even at low stress levels, is influenced by nucleation and propagation of microcracks during the loading process. Thus, the importance of a reliable cracking model can be seen. However, in the context of improved materials for structural application, nowadays, steel-fiber-reinforced concrete is largely used, which reduces the tensile brittle behavior of concrete, leading to a better strain capacity due to the clipping effect of the cracks provided by fibers.

The effect of the addition of steel fibers on the flexural strength of concrete and mortar is more evident with regard to the tensile behavior than the compression strength. This paper intends to present a proposal for modeling the mechanical behavior of fiber-reinforced concrete using the damage model proposed by Pituba and Fernandes, which has already been tested in conventional concrete structures. The one-dimensional version of this proposed modeling is presented and applied to the analysis of fiber-reinforced concrete beams in order to mark out a discussion about the viability and employment restrictions in simplified numerical analyses in the context of structural engineering.

1.2 Problem statement

Experimental and numerical investigations have shown that the inclusion of steel fibres in concrete, when adopted in adequate quantities, can improve the shear resistance of beams by increasing the post cracking strength of the concrete. Fibres embedded within concrete delay the propagation and growth of cracks by improving the effectiveness of the crack-arresting mechanisms present when beams are subjected to high shear stresses.

Many studies have considered the possibility of utilizing SFRC by assigning a proportion of the shear resisting capacity of beams to the fibres. This has been realized by ACI-318 (2008), and more recently by the fib Model Code 2010 (2012) and the Draft Australian Bridge Code: Concrete (2014). Some inconsistencies in some of these approaches, however, have been identified (Foster, 2010; Amin & Foster, 2014).

The role played by fibers is most obvious after matrix cracking has occurred, as fibers offer resistance to crack propagation. As described by Voo and Fos- ter (2003) , for plain concrete, after matrix cracking, the tensile stress immediately decreases. However, after the addition of a cer- tain volume of steel fibers and after matrix cracking, the fibers are able to maintain a certain load bearing capacity, avoiding an abrupt failure of the composite. In addition, the crack widths are less than those of plain concrete ( Deluce, 2011 ). Therefore, the main benefits of the addition of steel fibers in cementitious matrices are directly related to their ability to transfer stresses across cracks.

According to Bentur and Mindess (2007) this process of stress transfer depends on the internal structure of the composite and 

the main factors that influence the composite’s behavior are (i) the structure of the bulk cementitious matrix, (ii) the shape and distri- bution of the fibers and (iii) the fiber-matrix interaction.

Although the application of Steel Fiber Reinforced Concrete (SFRC) has increased in the last years, being very attractive in many structures, such as tunnel linings, bridges, pavements, and pipes, there remains a lack of numerical models for simulating its behavior that consider the contribution of each component (fibers, matrix and fiber-matrix interaction) in a fully independent way.

Several approaches have also been proposed for modeling the behavior of SFRC. Continuum models for SFRC have been devel- oped using results of structural members tested in laboratories, such as 3- and 4-point bending beams and slabs ( S.K. and Ra- maswamy, 2002 ).

In some of these models, stress-strain relations are developed from the inverse analysis of the laboratory test re- sults. These models are very limited because they are only able to reproduce the same conditions applied in the laboratory tests for specific structural members. Moreover, this type of model is highly expensive due to the large number of tests required to calibrate the model.

1.3 Fibre concrete

For the experimental program, fibres Dramix OL13/20 [16] have been selected. General characteristics of these fibres are shown in table 1 and the fibres shape is shown in figure 1. The initial concrete matrix has a formula stated in table 2. Samples were prepared in the laboratory.

Table 1. General characteristics of fibres Dramix® OL13/20 [20].

General characteristicDramix® OL 13/20 (figure 4)
Length [mm]13
Diameter [mm]0.21
Tensile strength [N/mm2]2750
Impact on concrete strength [kg/m3]60
Modulus of elasticity [GPa]200

The concrete mixture used for tested samples may be classified in the category of ordinary concrete of class C25/30.

Figure 1. Dramix® OL13/20 [20].

Table 2. General characteristics of concrete mixture.

SpecificationFormula [kg/m3]
CementCEM II/A-S 42.5
Min. cement content320 kg
Water-cement ration: w/c0.625
Aggregate 0/2 DTK Mankovice525 kg
Aggregate 0/4 DTK Mankovice420 kg
Aggregate 4/8 Tovačov150 kg
Aggregate 8/16 HDK820 kg
Water 200 l
Plasticizer STACHEPLAST3.2 L

1.4 Smeared crack models

Since it has begun to be used in reinforced concrete structures, the Finite Element Method has shown more advantages by representing the cracks through changes of the constitutive equations (smeared models) instead of changes on the finite element mesh (discrete models). The first analyses were based on the idea of fragile failure, that is, to make the material stiffness null in the direction of the maximum tensile stress when it exceeded the tensile strength.

Later on, it was observed that better results in the post-peak phase could be achieved by adopting a gradual reduction in the stress. In order to represent this behavior, the stress-strain diagram started to be defined by experimental values and, thus, several models were proposed (Cedolin e Dei Poli [1], Bazant and Gambarova [2], Gupta and Maestrini [3], and Vecchio [4]).

Though successfully used to represent the behavior of the reinforced concrete structures which presented a crack pattern well distributed, when used to simulate the behavior of structures in which a crack is predominant (plain concrete or deep beams), these sensitivity regarding the mesh, caused by the non-consideration of concepts associated to Fracture Mechanics. So, it was necessary to apply the Fracture Mechanics concepts directly over the concrete structures analysis, creating a series of new models (Bazant and Cedolin [5], Bazant and Oh [6], Feestra and de Borst [7]). Rots and Blaauwendraad [8] presented a comparative study between the discrete and smeared crack models. This study introduced the idea of dividing the smeared crack models into fixed and rotating. In the fixed model, the crack orientation is kept constant during the whole computational process, while in the rotating model the crack orientation may change, following the main directions. There is also an intermediary option which is the concept of multidirectional fixed smeared crack.

There are also the models based on the plasticity theory, which are able to represent well the pre-peak and post-peak phases, consisting of a constitutive model and a failure criterion. In this line it can be pointed out the models by Ottosen [9] and Pramono and Willam [10].

1.5 Application of additional Ultra High Performance Fibre Reinforced Concrete (UHPFRC) layers

A novel technique used to improve the performance of existing structural elements is the application of additional Ultra High Performance Fibre Reinforced Concrete (UHPFRC) layers or jackets in connection to the existing elements. The efficiency of this technique has not been adequately studied, and there are not any published studies on the evaluation of this method with comparisons to other traditional strengthening methods such as the use of Reinforced Concrete (RC) layers and jackets.

The technique of strengthening using additional RC layers and jackets is one of the most commonly used techniques in seismic areas. There are several published experimental and theoretical studies on beams and columns strengthened with conventional concrete. A crucial parameter in this technique, which can considerably affect the durability and the performance of the strengthened structures, is the concrete shrinkage strain of the additional layers/jackets. Additional stresses are induced in strengthened elements, and cracking of the new layer and/or de-bonding may occur. The use of UHPFRC could potentially improve both durability and resistance due to its superior mechanical properties.


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