ANONLINEAR ANALYSIS OF FIBROUS REINFORCED CONCRETE SLABS BY ASSUMED STRAIN AND HETEROSIS ELEMENTS

In the present work, the finite element method has been used to investigate the behavior of fibre reinforced concrete slabs in the pre and post-cracking levels up to the ultimate load. Assumed transverse shear strain is used in the formulation to overcome the shear locking, and Heterosis elements are employed in the analysis. Both an elastic-perfectly plastic and strain hardening plasticity approach have been employed to model the compressive behavior of the fibre concrete. The yield condition is formulated in terms of the first two stress invariants. Concrete crushing is a strain-controlled phenomenon, which is monitored by a fracture surface similar to the yield surface. A layered approach is adopted to discretize the concrete through the thickness. A tension stiffening model has been suggested by making a regression analysis of the experimental results, with index of determination (90.61%). The steel is considered either as an elastic perfectly plastic material or as an elastic-plastic material with linear strain hardening. Steel reinforcement is assumed to have similar tensile and compressive stress-strain relationship.


Introduction
The purpose of the present paper is to study the behavior of fiber reinforced concrete slabs by using heterosis elements [1] and assumed strain elements [2]. Heterosis elements are a 9-node quadrilateral, which employs serendipity shape function for the transverse displacement, and Lagrange shape function for the rotations.
To enhance the weak properties of concrete such as tensile strength and ductility, steel, glass, polymer and other types of fibres have been added as reinforcement, thus producing a composite material. The mechanical behavior of the resulting composite material is substantially different from that of the matrix material and possessing higher tensile strength, ductility and fracture toughness.
Huang [2] used an artificial method for the elimination of shear locking by interpolating new shear strain fields from the strain values at the sampling points which are appropriately located in individual elements.

The Behavior of Fibrous Concrete in Uniaxial Compression
The additional of fibre to the concrete increased the compressive strength and the peak strain as defined below [3]: , c cf f f is the compression strength of standard cylinder for plain and fibre reinforced is fibre length and diameter.
The modulus of elasticity of fibre reinforced concrete calculated using the following equation [4]: E s , E c is modulus of elasticity of steel fibre and concrete respectively .
The ultimate strain in compression can be calculated using the following equation [5]:

The Behavior of Fibrous Concrete in Uniaxial Tension a) The ascending part:
A simple expression that has been used for defining uniaxial tensile stress-strain curve up to the peak stress value of plain concrete [6].
Where: N f is the number of fibres crossing a unit area b) The descending part: By making regression analysis of the experimental results of references [9 and 10], the following model is adopted to express the relationship between post-peak stress and strain [7], the comparison between experimental and numerical model shown in After crack happened the elastic modulus and Poisson's ratio are reduced to zero in the direction perpendicular to the crack and a reduced shear modulus is employed to simulate aggregate interlock. Two different approaches are used to reduce the shear modulus: (i) Approach named G1: This approach was proposed initially by Al-Mahaidi [11]. and modified by many investigations [5,12 and 13] by substituting e tf for e t . The shear modulus of cracked concrete G can be calculated as follows: (ii) Approach named G2: This approach was proposed initially by Abdul-Razzak [5]. According to this approach, for concrete cracked in direction 1: Where: 3

Fig.(2) The descending part of tension model
Numerical Application

Example 1.
A simply supported square slab was tested by Ali [14]. The test specimen has (1800*1800)mm with (125)mm thickness and an effective depth of (100)mm to steel reinforcement. The material properties of tested slab are summarized in Table(1). The slab was modeled by four elements. Due to the symmetry of the slab, only are quarter of the slab is analyzed. Four steel layers are used to represent the reinforcement and eight concrete layers are found to be enough for the analysis. All dimension and detail of the slab shown in Fig.(3).

Slab
No. * Crimped steel fibre 0.5*50 mm Fig. (4) shows a comparison between two models of cracked shear modulus G1 and G2. These two models give good agreement compared with experimental results. However, G1 model give ultimate load less than G2 model for both the assumed and heterosis elements. Both of G models give ductile results compared to experimental results. Fig.(5) shows a comparison between the assumed strain element and 9-node Lagrangian degenerate element. The used of 9-node Lagrangian degenerate element shows stiff results and larger failure loads, this may be due to the shear locking that happened in 9node Lagrangian degenerate element. Fig.(6) shows a comparison between the assumed strain element and heterosis element. Both elements show a good agreement compared with experimental results.

Fig.(3) Dimension and detail of slab No.(S-2) [14]
Al-Rafidain Engineering Vol.18 No.5 October 2010 19 Fig.(7) shows the cross section of the slab (S-2), it is seen the cracks that happened in the slab at the ultimate load, and the yield of concrete under the load. Fig.(8) shows crack patterns at 84kN, 156kN, 192kN and at ultimate load using G2 model, it is seen that the cracks increased when the load increased.

Example 2.
A simply supporting circular slab was tested by Walraven J. et al [15]. The diameter of slab was (1750) mm and the total depth was (140) mm, and the effective depth of (110) mm to steel reinforcement. The material properties of the tested slab are summarized in Table  (2). The slab is modeled by three elements. Due to the symmetry of the slab, only are quarter of the slab is analyzed. Two steel layers are used to represent the reinforcement and eight concrete layers are found to be enough for the analysis. All dimensions and details of the slab shown in Fig.(9), and finite element mesh shown in Fig.(10).   11) shows a comparison between two models of cracked shear modulus G1 and G2. These two models give small softening when compared with experimental results. However, G2 gives results of ultimate load more than G1 model. Fig.(12) shows a comparison between the used of assume strain elements and heterosis elements. The used of two elements show a good agreement with experimental results.

Conclusions
1. The developed models for fiber concrete concerning the strain hardening plasticity and tension stiffening models proved to give satisfactory results for the analysis of reinforced concrete slabs subjected to incremental loading up to failure. 2. Two shear moduli of cracked fibre concrete is used and it is concluded that the shear crack approach G2 gives better results than G1 approach compared to experimental results. 3. The assumed strain and heterosis elements proved to be efficient for nonlinear analysis of fibrous reinforced concrete slabs, and no locking was detected in these types of elements.  8. Shah, S.P., and Gopalatnam, V.S., " Micromechanical and model for the tensile fracture of stell fibre reinforced concrete ", in RILEM Symp. on Developments in fiber