A numerical procedure simulating RC structures reinforced with FRP using the serial/parallel mixing theory
Introduction
The first known report in which carbon fiber reinforced polymers (CFRP) were used to retrofit a damaged structure is from 1991, when they were used to strengthen the Ibach Bridge in Lucerne, Switzerland [1]. Since then, this technology has become far more widely used and is now one of the main applications of composite materials in civil engineering.
Most current knowledge about the structural reinforcement and/or retrofitting of reinforced concrete (RC) structures with fiber reinforced polymers (FRP) is based on experimental simulations, that are used to analyze different reinforcement applications such as bending reinforcements [2], shear reinforcements [3], [4], column wrapping [5] or anchorage of the reinforcement to the existing structure [6]. These studies use experimental techniques that are supported and complemented by analytical calculations. If the problem is treated numerically, material nonlinearities are usually linearized and the FRP composite is considered as a single material (i.e. [7]).
On the other hand, composite materials have been, and still are, one of the principal areas of research in computational mechanics throughout the last few decades. Main efforts are focused on the study of composite plates and shells [8], [9], as these are the structural elements commonly used in aeronautical, nautical and automotive structures, which are the engineering fields in which composites are most widely applied.
Traditionally, numerical simulations of composites have been performed using orthotropic materials with average properties from their constituents. With this approach, no model has been found that is able to function beyond the elastic limit state of its constituents. As a result, numerical simulations are limited to elastic cases. Different theories have been proposed to solve this problem which take into account the internal configuration of the composite to predict its behavior. The two most commonly used are described below.
Homogenization theory: This method deals with the global composite structure problem in a two-scale context. On the macroscopic scale the composite materials determine the global response of the structure. Composites are considered to be homogeneous materials in this scale. The microscopic scale represents an elemental characteristic volume in which the microscopic fields inside the composite are obtained. This scale deals with the component materials of the composite, each one with its own constitutive equation. Homogenization theory relates these two scales by assuming a periodic configuration of the composite material [10], [11].
Mixing theory: The first formulation of the mixing theory was developed by Truesdell and Toupin [12] and it is based on two main hypotheses: 1. All composite constituents are subject to same strains. 2. Each constituent contributes to the composite behavior according to its volumetric participation. The main drawback of the mixing theory is the iso-strain condition which enforces a parallel distribution of the constituents in the composite. Some improvements to the original formulation can be found in [13], [14].
Despite all the existing studies on both subjects, experimental tests of FRP reinforcements and numerical characterization of composite materials, little research has used a numerical approach to analyze the structural reinforcement of RC structures with FRP. Therefore, the main goal of this paper is to combine both fields, developing a numerical procedure for computing RC structures reinforced with FRP. The developed formulation is based on the finite element method and enables determining the structural performance of existing structures when they are reinforced and/or retrofitted with FRP. This performance is calculated taking into account material nonlinearities. The developed formulation also identifies the performance of each constituent material in the structure (for example, it is possible knowing the stress state of the fiber in the composite reinforcement when the structure collapses). The code can be used to study the same structure with different FRP configurations, to determine the most suitable option for the case considered. It can also apply the reinforcement to structures that are already damaged, reproducing with more accuracy the conditions found in real applications.
The numerical formulations proposed in this paper use the serial/parallel rule of mixtures, developed by Rastellini [15], to analyze composite materials. The code also includes a construction-stages algorithm that is used to consider the case of structural retrofitting. Section 2 contains a detailed description of both of these features. In Section 3 the experimental data reported in [2] for a RC beam reinforced with CFRP is used to validate the proposed code. Two different numerical simulations are then described to illustrate the potential of the formulation developed: The first case, in Section 4, shows the results obtained when the RC beam used to validate the code is retrofitted. The second case, in Section 5, uses the code to simulate a concrete frame structure in which different FRP reinforcements are applied to the beam, column and beam–column connecting joint. This simulation illustrates how the developed code can be used to determine which FRP reinforcement configuration achieves better results. Finally, in Section 6, are presented the conclusions about the numerical tool developed and the conclusions drawn from the calculations performed.
Section snippets
Serial/parallel rule of mixtures
The serial/parallel rule of mixtures is an improvement of the classical mixing theory, in which the iso-strain hypothesis is replaced by an iso-strain condition in the fiber direction and an iso-stress condition in the transversal directions. This theory was developed by Rastellini and is explained in detail in [15].
Validation of the numerical procedure
To validate the proposed formulation, a numerical model of a RC beam reinforced with FRP was developed. The beam considered is the same as the one defined by Spadea et al. in [2]. The numerical results obtained with PLCd are compared with the experimental results given in [2].
CFRP retrofitting of RC structures
Two different numerical models have been developed to study the effect of retrofitting a structure, depending on the existing level of damage in the beam when the CFRP reinforcement is applied. The beam retrofitted is the same that has been used in previous section. The models developed are:
Sp3D-Rt2: The CFRP reinforcement is applied when the damage appears in the concrete material.
Sp3D-Rt3: CFRP reinforcement is applied when the steel starts to yield.
Results obtained with these two models are
Concrete frame structure simulation
The main aim of this simulation is to apply the formulation developed to verify the ability of CFRP reinforcements to increase the strength of concrete frame structures. The connecting joints between the beams and columns can be often subject to greater stress than other zones of concrete frame structures and in most cases these joints are the cause of structural weakness. The frame joint is reinforced in the models developed for this study with two different CFRP configurations to analyze the
Conclusions
The results of the numerical procedures and simulations developed in this paper show that the numerical tool developed to simulate FRP reinforcements of RC structures performs well. The results are in good agreement with existing experimental results. The code is prepared to compute real structures that are reinforced or retrofitted with CFRP. Last simulation presented has shown that different numerical simulations with different FRP reinforcements can be performed with PLCd in order to obtain
Acknowledgements
This work has been supported by CEE-FP6 (LESSLOSS Project, Ref. FP6-50544 (GOCE)), by the Spanish Ministry of Science and Technology (RECOMP Project, Ref. BIA2005-06952 and DECOMAR Project, Ref. MAT2003-08700-C03-02) and by the Spanish Ministry of Public Works (project “Retrofitting and reinforcement of reinforced concrete structures with composite materials. Numerical and experimental developments applied to joint of bars and composites anchorage proposal”). X. Martinez was awarded a
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