# Constitutive Behavior and Finite Element Analysis of FRP Composite and Concrete Members

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Constitutive Model for FRP Composite and Concrete Circular Section

#### 2.1. Laminate Composite for FRP Composite

**T**indicates the transformation matrix; the σ and ε are for the stress and strain vectors in the local coordinate (x, y, and z), respectively; while the σ and ε

_{1}are for the stress and strain vectors in the structure coordinate (1, 2, and 3), respectively. Simultaneously, the coefficients of the constitutive matrix

**Q**can be rendered as follows:

_{1}is for the modulus of the lamina in the fiber direction; E

_{2}for the modulus normal to the fiber direction; ${\nu}_{ij}$ for the Poisson’s ratio for the deformation in the $j$ direction in loading in the $i$ direction; and G

_{12}for the in-plane shear modulus of the lamina.

**Figure 1.**(

**a**) fiber reinforced polymer (FRP) laminated composite; and (

**b**) FRP jacketed concrete cylinder in compression.

_{L}and E

_{H}are for the elastic modulus of the longitudinal and hoop direction, respectively; and ν

_{LH}for the Poisson’s ratio of the FRP laminate composite.

_{L}and E

_{H}are the elastic moduli in the longitudinal and hoop directions, respectively; while ν

_{HL}and ν

_{LH}for the Poisson’s ratios, respectively; and $t$ for the thickness of the laminate.

**Figure 2.**(

**a**) In-plane stresses in FRP jacket; (

**b**) triaxial stresses in concrete; and (

**c**) equilibrium in cross-section.

#### 2.2. Stresses and Strains of FRP Composite and Concrete Circular Section

_{l}and E

_{r}are the concrete tangent moduli in the axial and radial directions, respectively; and μ

_{ij}is derived from the Poisson’s ratios ν

_{ij}as

_{iu}is the concrete equivalent uniaxial strain. The material in the strain in direction $i$ would exhibit if subjected to a uniaxial stress σ

_{i}with other stresses equal to zero, and derived as follows

_{r}; the increment of longitudinal stress in the FRP composite, dσ

_{L}; and the increment of radial strain in concrete, dε

_{r};, can be derived, respectively

#### 2.3. Constitutive Model of Concrete in Triaxial Stress State

_{r}and σ

_{r}= σ

_{h}. In the test of confined concrete, it is known that the radial and hoop stresses of concrete were mostly identical and thus the assumption has been commonly adopted in the majority of previous studies [2,3,10].

**Figure 3.**(

**a**) Five-parameter failure surface of concrete; and (

**b**) confined and unconfined stress-strain curve of concrete in compression.

_{ci}in the equivalent uniaxial stress and strain curve of concrete as shown in Figure 3b. Concrete becomes more ductile when the level of confinement is increased. In the present research, the strain surface of concrete wrapped by FRP composite is defined as the equation of strength enhancement factor [1,15]. To model the equivalent uniaxial stress-strain relation of the unconfined and confined concrete is shown in Figure 3b, Saenz’s curve is adopted to describe the equivalent uniaxial compressive stress-strain curve of concrete [16]. On the other hand, the Poisson’s ratio of concrete was adopted as a cubic function that predicted the volume expansion observed during experimental studies of concrete and this model was well matched with the experimental results [12]. Hence, after reaching the failure strength of the FRP composite in a section, it is assumed that the member does not resist additional loads and finally fails. In the proposed model, the Tsai-Wu failure criterion was applied to predict the failure of the FRP composite [17]. Tsai and Wu [17] postulated a general form of a failure surface in six-dimensional stress space. For the case of the in-plane composite behavior as an FRP jacket, the criterion could be minimized to have three parameters of a composite material; two axial strengths both in 1- and 2-directions and one shear strength.

## 3. Mixed Beam Finite Element with Layered FRP Concrete Composite Section

#### 3.1. Finite Element Formulation

**D**is the section force vector consisting of section moments;

**d**

^{i}for the section deformation vector consisted of section curvature; y

^{i}for the displacement fields at the ith Newton-Raphson iteration; and

**f**for the section flexibility matrix, respectively. By the principle of virtual work, the equilibrium can be formulated in relation of the section deformation vector,

**d**= ∂y

**d**is the virtual section deformation vector caused by the virtual displacement δy;

**Q**and q for the nodal force and nodal displacement vectors, respectively; while p for the applied load vector per unit length. The Hellinger-Reissner mixed formulation can be derived by a combination of the compatibility and equilibrium equations as:

**D**are interpolated in terms of the force degrees of freedom

**Q**. Also the displacement fields y are interpolated in terms of the nodal displacements q.

**K**

^{i}

^{−1}is the element stiffness matrix at the end of the last iteration defined as follows:

_{e}and q

_{r}

^{i}

^{−1}for the equivalent nodal load and nodal residual vectors, respectively;

**Q**

_{r}

^{i}

^{−1}is the element resisting forces defined as

#### 3.2. Layered Section Discretization

**Figure 4.**(

**a**) Layered isoparametric beam finite element; and (

**b**) four-point bending test of FRP and concrete circular beam.

## 4. Analysis of FRP and Concrete Composite Cylinder in Compression

**Figure 5.**(

**a**) Prediction of compression test for CFRP thickness of 2.29 mm; and (

**b**) prediction of compression test for CFRP thickness of 4.57 mm.

**Figure 6.**(

**a**) Prediction of compression test for CFRP thickness of 0.676 mm; and (

**b**) prediction of compression test for CFRP thickness of 0.338 mm.

## 5. Finite Element Prediction of FRP and Concrete Composite Beams

**Figure 8.**(

**a**) Axial strains and bending moment relationship at center span; and (

**b**) hoop strains and bending moment relationship at center span.

## 6. Conclusions

## Acknowledgments

## References

- Cho, C.G.; Kwon, M.; Spacone, E. Analytical model of concrete-filled fiber-reinforced polymer tubes based on multiaxial constitutive laws. J. Struct. Eng.
**2005**, 131, 1426–1433. [Google Scholar] [CrossRef] - Burgueno, R. System characterization and design of modular fiber reinforced polymer (FRP) short- and medium-span bridges. Ph.D. Thesis, University of California, San Diego, CA, USA, 1999. [Google Scholar]
- Davol, A.; Burgueno, RJ.M.; Seible, F. Flexural behavior of circular concrete filled FRP shells. J. Struct. Eng.
**2001**, 127, 810–817. [Google Scholar] [CrossRef] - El-Tawil, S.; Ogunc, C.; Okeil, A.; Shahawy, M. Static and fatigue analysis of RC beams strengthened with CFRP laminates. J. Compos. Constr.
**2001**, 5, 258–267. [Google Scholar] [CrossRef] - Fam, A.Z.; Rizkalla, S.H. Confinement model for axially loaded concrete confined by circular fiber-reinforced polymer tubes. ACI Struct. J.
**2001**, 98, 451–461. [Google Scholar] - Hosotani, M.; Kawashima, K.; Hoshikuma, J. A stress–strain model for concrete cylinders confined by carbon fiber sheets. J. Mater. Concr. Struct. Pavement JSCE
**1998**, 39, 37–52. [Google Scholar] - Kawashima, K.; Hosotani, M.; Hoshikuma, J. A Model for Confinement Effect for Concrete Cylinders Confined by Carbon Fiber Sheets; Technical Report NCEER-97-0005; State University of New York: Buffalo, NY, USA, 1997; pp. 405–430. [Google Scholar]
- Mimiran, A.; Shahawy, M. Behavior of concrete columns confined by fiber composites. J. Struct. Eng.
**1997**, 123, 583–590. [Google Scholar] [CrossRef] - Spoelstra, M.R.; Monti, G. FRP-confined concrete model. J. Compos. Constr.
**1999**, 3, 143–150. [Google Scholar] [CrossRef] - Mander, J.B.; Priestley, M.J.N.; Park, R. Theoretical stress–strain model for confined concrete. J. Struct. Eng.
**1988**, 114, 1804–1826. [Google Scholar] [CrossRef] - Pantazopoulou, S.J.; Mills, R.H. Microstructural aspects of the mechanical response of plaine concrete. ACI Mater. J.
**1995**, 92, 605–616. [Google Scholar] - Kupfer, H.B.; Hildorf, H.K.; Rusch, H. Behavior of concrete under biaxial stresses. ACI J.
**1969**, 66, 656–666. [Google Scholar] - Darwin, D.; Pecknold, D.A. Nonlinear biaxial law for concrete. J. Eng. Mech. Div.
**1977**, 103, 229–241. [Google Scholar] - Willam, K.J.; Warnke, E.P. Constitutive model for the triaxial behavior of concrete. In Proceedings of International Association for Bridge and Structural Engineering, Bergamo, Italy, 17–19 May, 1974; IABSE: Zurich, Switzerland, 1975; Volume 19, pp. 1–30. [Google Scholar]
- Cho, C.G.; Park, M.H. Finite element prediction of the influence of confinement on RC beam-columns under single or double curvature bending. Eng. Struct.
**2002**, 13, 313–328. [Google Scholar] - Saenz, L.P. Discussion of equation for the stress–strain curve of concrete by Desayi and Krishman. ACI J.
**1994**, 61, 1229–1235. [Google Scholar] - Tsai, S.W.; Wu, E.M. A general theory of strength for anisotropic materials. J. Compos. Mater.
**1971**, 5, 58–80. [Google Scholar] [CrossRef] - Spacone, E.; Filippou, F.C.; Taucer, F.F. Fibre beam-column model for nonlinear analysis of R/C frames: Part I. Formulation. Earthq. Eng. Struct. Dyn.
**1996**, 25, 711–725. [Google Scholar] [CrossRef]

© 2013 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Ann, K.Y.; Cho, C.-G.
Constitutive Behavior and Finite Element Analysis of FRP Composite and Concrete Members. *Materials* **2013**, *6*, 3978-3988.
https://doi.org/10.3390/ma6093978

**AMA Style**

Ann KY, Cho C-G.
Constitutive Behavior and Finite Element Analysis of FRP Composite and Concrete Members. *Materials*. 2013; 6(9):3978-3988.
https://doi.org/10.3390/ma6093978

**Chicago/Turabian Style**

Ann, Ki Yong, and Chang-Geun Cho.
2013. "Constitutive Behavior and Finite Element Analysis of FRP Composite and Concrete Members" *Materials* 6, no. 9: 3978-3988.
https://doi.org/10.3390/ma6093978