# Application of constitutive models in European codes to RC–FRC

Abstract: The recent publication of codes for the design of FRC is a major step towards extending the use of the material. An in depth analysis indicates several differences between the constitutive models proposed in the existing codes. In this study, these models are compared and a numerical simulation is performed to evaluate their differences in terms of the structural behavior predicted and measured in an experimental program of RC–FRC elements. The predictions provided by the models fit satisfactorily the experimental results for elements with steel fibers and with plastic fibers.

## Summary (4 min read)

### 1. Introduction

- Fiber reinforced concrete (FRC) is one of the most relevant innovations in the field of special concretes.
- This makes FRC and the combined solution of traditional reinforced concrete (RC) and FRC (hereinafter RC-FRC) a competitive design alternative both from the technical and the economic point of view [7-8, 10-11].
- A in depth analysis indicate several differences between the constitutive models proposed in codes to design FRC structures.
- Furthermore, a numerical simulation is performed to evaluate their differences in terms of the structural behavior predicted using the model AES [14] and measured in the experimental program presented in [13].

### 2. Constitutive models from the literature

- There are numerous constitutive models proposed in the literature for the design of FRC based on either stress-strain (σ-ε) curves or stress-crack width (σ-w) curves.
- Most of these models are based on an indirect approach, requiring parameters that must be defined each time from experimental data.
- The main advantage of using a σ-w model is that it can be directly compared to the experimental results (e.g. uniaxial tensile tests), thus providing actual physical insight of the mechanisms occurring in the FRC [15].
- Likewise, such approach is more convenient for practical reasons since it is the same used for traditional steel reinforcement.
- Notice that there are several studies in the literature dedicated to the relation between the σ-w diagram and the σ-ε diagram, using characteristic length (lcs) [17-19].

### 2.2 Indirect approach

- Among the several constitutive models following an indirect approach, one of the first proposals regarding a σ-ε curve for FRC was presented in [20].
- Another model proposed in [22] introduced a nonlinear relationship in the pre-cracking stage (depicted in Fig. 1b).
- A study of great relevance was performed by Dupont and Vandewalle [23].
- There are also several proposals concerning models based on a σ-w diagram.
- The contribution of the fibers is considered in the second stage and is defined by two average stresses at certain crack widths.

### 2.3 Direct approach

- The use of a direct approach to simulate the uniaxial behavior of the material requires the definition of parameters defining the constitutive relation [15], which may be determined either from experimental data or from specific material properties.
- The relevance of the study by Li et al. [28] lies in the approach followed to propose a σ-w constitutive model (see Fig. 3a).
- The input parameters in this model are: the characteristic compressive strength, the diameter, the length, the tensile yield strength of the fiber as well as its volume content, the cross section of the structure to be design and the fiber orientation number.
- Concrete properties and fiber geometry are used to obtain the bond-slip response of the fibers.

### 3.1 Identification of the models

- The identification of the most suitable constitutive model to simulate the tensile postcracking behavior represents one of the key steps in the design of FRC structures.
- Table 1 presents the constitutive models proposed by European standards [2-6] grouped according to the type of diagram (namely rectangular, bilinear and trilinear or multilinear).
- This concept takes into account the effect of the height on the bending behavior of the cross section by penalizing the section with larger height.
- Another substantial difference among the design guidelines is the use of the equivalent flexural tensile strength (feq) or the residual flexural tensile strength (fR) to obtain the parameters of the constitutive model.

### 3.2 fib Model Code (2010)

- The deeper knowledge gained on FRC over the past twenty years and the recent publication of design codes and guidelines at a national level led the fib (Fédération Internationale du Béton) to introduce FRC in the updated version of the CEB-FIP Model Code 90, with the aim of providing a tool for the design of FRC structural elements [36].
- The fib Model Code proposes two models for the tensile behaviour of FRC: the rigid-plastic and the linear-elastic behavior (see Fig. 4).
- In order to define the stress-strain constitutive laws (σ-ε) it is necessary to distinguish between softening materials and hardening materials.
- Once introduced the concept of characteristic length, it is important to remark that the the ultimate crack width wu required to estimate fFtu may be calculated as wu=lcs·εFu.
- This is valid for softening or hardening materials.

### 4. Experimental program

- These slabs have a combined reinforcement consisting of a conventional reinforcement and fibers (except in the case of two control elements which are only reinforced with conventional reinforcement).
- In addition to the conventional concrete slabs, eight types of FRC were prepared varying the types and contents of fiber2.
- The fiber content in the elements with mixed reinforcement is 0.25% of the total volume (which corresponds to 20 kg/m3 of steel fibers and 2.28 kg/m3 of polypropylene) and 0.50% of the total volume (40 kg/m3 of steel fibers and 4.55 kg/m3 of polypropylene fibers).
- The deflection at midspan as well as the crack width and spacing were measured in the constant moment zone of 900 mm.
- Anyhow, a visual inspection of the elements did not show any indication of strain or cracking due to shrinkage.

### 5.1. Introduction

- In order to simulate the tests performed in the experimental program, a model capable of carrying out a non-linear sectional analysis and accounting for the cracking, post-cracking and post-failure behavior of the materials is required.
- The model AES (Analysis of Evolutionary Sections) presented in [14] was used.
- Likewise, a numerical subroutine for the structural analysis of the slabs, which includes the AES model, was also developed in this work.
- Such subroutine allows assessing the behavior of the slabs with several combinations of reinforcements under the test setup conditions.
- In this section the main basis and hypothesis implemented in both models are presented aiming at giving a general overview on how these two numerical tools were conceived.

### 5.2. Numerical simulation of the sectional behavior

- The concrete is discretized in layers with constant thickness, whereas steel rebars are simulated as concentrated-area elements.
- On the other hand, the simulation of the postcracking behaviour was performed separately with each model from Table 1.
- The assessment of the crack width (w) depends on the type of reinforcement of the section.
- The following hypotheses have been considered: (1) perfect bond between the materials; (2) sections remain plane before the application of the external forces or after imposing fixed strains and (3) shear strains are negligible and may not be taken into account.

### 5.3 Simulation of the tests

- A subroutine included in AES was implemented in order to assess the P-δ curves considering the test configuration as well as different constitutive equations to simulate the FRC post-cracking behaviour.
- The algorithm implemented in the abovementioned tool to obtain the P-δ laws consists of: 1. Dividing the half span of the slab into intervals of magnitude Δx (see Fig. 9a).
- Obtaining the M-χ (see Fig. 9b) diagram of the cross section considering the mechanical properties of each material.
- Fixing an increment of the midspan displacement Δδ. 4. Fixing tolerances for the values Δδ and ΔP (tolΔδ and tolΔP respectively).
- Calculating the accumulated bending force Mi in each point xi.

### 6.1 Methodology

- The numerical simulation was performed considering only the multilinear and the bilinear models due to their higher accuracy in the SLS (see Table 1).
- The failure mechanisms of the 4-point and 3-point bending test may be schematized as indicated in Fig. 10.
- This correlation can be used to find the equivalence between the experimental results obtained from the test in EN14651:2005 and the results from the test in DIN1048.
- For three of the four experimental cases studied (SF_0.25, PF_0.25 and PF_0.50), the intersection corresponded to a value of strain lower than that for the tensile strength fct.
- The constitutive models (see Table 1) used for the simulation are also presented.

### 6.2 RC slab

- The experimental P-δ curve and the prediction provided by the model AES for the control slab RC are shown in Fig. 12.
- The curves in Fig.12 reveal that the prediction of the response for the control slab RC is satisfactory, particularly at the early stages of the loading and after the yielding of the reinforcement occurs.
- The biggest differences between both curves are detected for values of load over 100 kN and until the yielding of the reinforcement.
- The prediction for 15 mm presented in Table 5 is a 12.2% higher than the load value registered during the test.

### 6.3 RC-SFRC slabs

- In the case of the RILEM model, considerably high values of peak and post-cracking stresses are adopted if compared to other multilinear models.
- This difference can lead to an overestimation of the structural response of the element .
- The model that presents a larger difference with the experimental result is the DBV trilinear that provides a very conservative prediction (the load for 45 mm is nearly 14% lower than those experimentally obtained).

### 6.4 RC-PFRC slabs

- The P-δ curves obtained with each of the multilinear models are shown in figures Fig. 14b and Fig. 14d.
- The lower residual strengths proposed by the CNR-DT 204 in comparison with the EHE results in a slightly lower response in the P-δ curve for large deformations, as shown in Fig. 14f and Fig. 14h.
- The largest overestimation of the experimental data is found if the Model Code is used (8.1 %).
- At this deflection the RILEM, the CNR-DT 204 and the EHE present similar results, underestimating the response of the slabs in 3.5% approximately.

### 7. Conclusions

- The most relevant consitituve models from the literature were analyzed and compared in terms of their capacity to predict the structural response of FRC.
- From the comparative analysis conducted in this study, the following conclusions may be drawed.
- The models included in the DBV present a different approach if compared with the models from other guidelines.
- The estimations performed with the constitutive model from the RILEM differ significatively from the experimental results for small displacements.
- Currently, there is a significant basis upon which to build and promote FRC technology.

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