Keywords: concrete, beams, FRP rebars, bond modelling, material nonlinearity.

The objective of this study is the effect of bond models in the analysis of concrete beams reinforced with fiber-reinforced plastic (FRP) under monotonic loading. Modeling the behavior of such beams requires constitutive models for concrete, FRP, and their interaction. Since, material properties of concrete and FRP are different, the analysis is based on separate material models for reinforcing FRP rebars and concrete, which are then combined along with models of the interaction between the two constituents to describe the behavior of the composite material.

A piecewise linear elastic model with variable moduli simulates the nonlinear response of concrete. A modified version of the Balakrishnan and Murray [1] model is proposed in view of the fact that the response of typical reinforced concrete structures is much more affected by the tensile than by the compressive behavior of the concrete. A number of damage regions in which the material properties are altered to match the state of stress is used. Within each region the modulus of elasticity and Poisson's ratio are considered to be constant. The behavior depends on the location of the present stress' state in the principal stress space. In the biaxial compression region the model remains linear elastic for stress combinations inside the failure surface described by Kupfer [2]. When the biaxial stresses exceed Kupfer's failure envelope, concrete failure occurs by crushing of the concrete. The shear modulus for non-cracked concrete is assumed to be proportional to the elastic modulus in the elastic region with a constant Poisson's ratio.

Cracking is assumed to be smeared and occur on the plane of maximum principal stress, with the axes of orthotropy are parallel and orthogonal to the crack. Poisson's ratio is set to be zero when cracking is initiated. The problems encountered with the usage of the fixed crack model are overcome by introducing a cracked shear modulus. A variable cracked shear modulus to represent the change in shear stiffness, as the principal stresses in the concrete vary from tension to compression is implemented. The tension stiffening effect is included by assigning a descending linear branch in the tension portion of the concrete stress-strain.

The properties of FRP rebars, unlike concrete, are uniaxial. Thus, a single stress-strain relation is sufficient to define the material properties needed in the analysis.

Complete compatibility of strains between concrete and FRP rebars could be assumed, which implies perfect bond. In reality there is no strain compatibility between FRP and surrounding concrete near cracks. This incompatibility gives rise to relative displacements between FRP and concrete, which are known as bond-slip. Tests have shown that the bond is different from that of steel reinforcing bars and varies with the type of reinforcing bar as well [3,4]. Because of the empirical nature of bond models, many of these models proposed for the bond of steel bars could be applied to the bond of FRP rebars. Based on experimental study conducted by many researchers propose a few bond models for FRP are proposed [3,4]. One of the better models for the bond of FRP bars was proposed by Malvar [3], which is adopted in the present study.

In the present study, the beams analyzed are considered to be two-dimensional plane stress elements with layered cross sections to take into accounted progressive cracking and the changing material properties through the depth. Element stresses are assumed to be constant through the thickness. Both concrete and FRP elements are represented by four-node rectangular elements. Since, the overall structural behavior is of primary interest link elements, which connects a node of a concrete element with a node of an adjacent FRP element to represent the relative slip is implemented [5]. The load is applied in sufficiently small load increments to obtain the associated displacement increments. The tangent stiffness method, which requires the smallest number of iterations, is used to improve the displacement increments by satisfying a specified convergence criterion.

The proposed models are used to simulate the experimental tests performed by other researchers to predict the behavior of concrete beams reinforced with FRP rebars. Four examples are considered and a correlation study between numerical and experimental results and the parametric study associated with them is presented. Since bond-slip increases with loading, while tension stiffening does not, consistent results are obtained when both effects are included in the model and a reduced tension stiffening is introduced. The exclusion of the bond-slip effect can lead to significant overestimation of the stiffness of the member. Introducing bond models for conventional steel reinforcement to simulate the bond slip of FRP rebars may lead to an acceptable prediction. Implementation of the Malvar model leads to good prediction for most of the tested examples.

1

S. Balakrishnan and D. Murray, "Concrete Constitutive Model for NLFE Analysis of Structures", Journal of Structural Engineering, ASCE, Vol.114, No.7, 1449-1466, 1988. doi:10.1061/(ASCE)0733-9445(1988)114:7(1449)

2

W.F. Chen, "Plasticity in Reinforced Concrete", McGraw-Hill, New York, 1982.

3

L.J. Malvar, "Tensile and bond properties of GFRP reinforcing bars", ACI Material Journal, V. 92 No. 3, 276-285, 1995.

4

L.J. Malvar, J.V. Cox and K.B. Cochran, "Bond between Carbon Fiber Reinforced Polymer Bars and Concrete. I: Experimental Study", Journal of Composites for Construction, 154-163, 2003. doi:10.1061/(ASCE)1090-0268(2003)7:2(154)Malvar

5

D. Ngo. and A.C. Scordelis, "Finite Element Analysis of Reinforced Concrete Beams", Journal of ACI, Vol. 64, No. 3, 152-163, 1967.

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