Keywords: finite element model, non-linear, smeared crack, orthotropic, isoparametric, DIANA, plane stress, principal stress.

This paper describes the formulation of a non-linear finite element (FE) model to simulate the flexural behaviour of simply supported reinforced concrete beams. The beams were reinforced with both conventional steel as well as fibre reinforced polymer (FRP) bars. Concrete was considered isotropic material before cracking. The material model for cracked concrete was based on the modified compression field theory, which was proposed by Vecchio and Collins [1]. A perfect bond was assumed between concrete and reinforcing bars. The secant stiffness moduli were used to evaluate the material stiffness matrix in the direction of the compressive and tensile principal stress axes. The model was implemented in a commercial FE programme DIANA [2]. Only one half of the beam was modelled to take advantage of the symmetry in geometry and loading conditions.

The behaviour of concrete in compression was modelled using the Thorenfeldt [3] constitutive behaviour. The uniaxial stress-strain curve is comprised of an ascending and a descending branch. A linear stress-strain relationship was assumed for the tensile strength of concrete before cracking. The concrete between cracks is considered effective in resisting both compressive and tensile stresses. The cracking and crushing of concrete was simulated with smeared crack model. The effects of tension softening were taken into account. An exponential crack opening law is adopted to model the tension softening and tension stiffening effects in cracked concrete. The softening curve is based on mode I fracture energy through crack bandwidth of the element.

Non-orthogonal crack formation is allowed with the rotating crack model. The formation of secondary cracks was controlled by an additional parameter termed as threshold angle . A value of has been used to allow the formation of non-orthogonal cracks in an integration point. Steel bars were modelled as perfectly plastic after yielding using Von Mises isotropic plasticity. The linear elastic behaviour of FRP bars was described by the Von Mises yield criterion with associated flow and isotropic hardening.

The FE formulation was based on 2D quadrilateral 8 nodded plane stress element. Each node has two degrees of freedom i.e translation in orthogonal directions. The model adopts isoparametric formulation. A 3 x 2 Gaussian integration scheme was used for the elements. The bars were modelled as uniaxial elements, which were embedded in quadrilateral mother FEs. Transverse strength and stiffness characteristics of these bars were not considered. A 2-point Gaussian integration scheme was used for the uniaxial bar elements. The longitudinal strain of bar is compatible with the strain of parent FE, which ensures perfect bond between the rebar and the concrete.

The analysis was carried out with the help of an incremental-iterative linear elastic algorithm. The iteration scheme adopts a regular Newton-Raphson method. A tangential stiffness matrix is formed before each iteration to check the convergence criteria. The loads were applied incrementally in steps. The steps were specified explicitly and were kept small to obtain convergence up to the end. A tolerance of 1E+1 on the displacement norm was defined as the convergence criteria. The modulus of elasticity of concrete was calculated from equation suggested by CP110 [4]. The tensile strength was calculated using a Eurocode 2 equation [5]. Fracture energy was obtained from the relationship proposed by Wittmann [6]. The comparison of analytical results showed a good agreement with the experimental results. The ultimate load and failure modes were accurately predicted. A good correlation for the stiffness of beams at all stages of loading was found.

1

F.J. Vecchio, M.P. Collins, "The Modified Compression-Field Theory for Reinforced Concrete Elements Subjected to Shear", ACI Journal, 83(22), 219-231, 1986.

2

TNO Building and Construction Research, "DIANA Finite Element Analysis", User's Manual release 9, Delft, 2005.

3

E. Thorenfeldt, A. Tomaszewicz, J.J. Jensen, "Mechanical Properties of High-Strength Concrete and Application in Design", in "Proceedings of the Symposium Utilization of High-Strength Concrete", Tapir, Trondheim, 149-159, 1987.

4

British Standards Institution, "Code of Practice for Structure use of Use of Concrete, CP110", London, 1972.

5

Eurocode 2, "Design of Concrete Structures, Part1-1: General Rules and Rules for Buildings", Brussels, 2004.

6

F.H. Wittmann, "Crack formation and fracture energy of normal and high strength concrete", Sadhana, 27(4), 413-423, 2002. doi:10.1007/BF02706991

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