Softened Truss Model Theory for the Analysis of Fibre Reinforced Concrete Deep Beams and Corbels

21  Softened Truss Model Theory for the Analysis of Fibre Reinforced Concrete Deep Beams and Corbels S. A. Al-Ta’an, Professor N. S. H. Al-Husaini, Assistant lecturer Department of Civil Engineering, Dept. of Building and Construction, Mosul University, IRAQ Technical Engineering College, Mosul Corresponding author, S. A. Al-Ta’an, Email: saad.altaan@yahoo.com Abstract Reinforced concrete members may be subjected to axial load, bending moment, shear and torsion. However the behaviour of these members under shear or combined shear and torsion is a complex phenomenon. In this study the softened truss model theory is applied for the analysis of fibre reinforced concrete deep beams and corbels. The theory is more promising than the strut and tie model which satisfies the equilibrium conditions and to some extent materials constitutive relationships. While this theory, considers the equilibrium, compatibility, materials constitutive relationships and the degrading effect of the diagonal tension cracks on the compressive strength of cracked reinforced concrete element when subjected to biaxial compression-tension stresses. The previously developed algorithms for the analysis were modified by incorporating the effect of short discrete steel fibres on the behaviour and strength of concrete subjected to shear. Fibre reinforced concrete deep beams and corbels were analyzed using the adopted algorithm and materials constitutive relationships. The predicted effects of the shear span / depth ratio, volume fraction of steel fibres and the longitudinal steel ratio on the shear strength of fibre reinforced concrete deep beams and corbels showed good agreement with published experimental results.


Introduction
Beams with small span to depth ratio are called deep beams and short cantilever beams with a shear span less than or equal to their effective depth are called corbels or brackets. Their behaviour and strength have been the subject of numerous experimental studies [1][2][3]. Several possible modes of failure have been identified from the published test results, but the predominant one is that of shear failure. The shear action in the beam web leads to compression in a diagonal direction and tension in the perpendicular direction. The observation of web crack before failure leads to a theoretical model that assumes the dependency of shear strength on concrete splitting strength. On the other hand, a simple model that recognizes the arch action in deep beam has been proposed to predict web compression failure. Combining the equilibrium, compatibility and softened stress-strain relationships, a theory was developed which can predict with good accuracy the test results of various types of reinforced concrete structures subjected to shear or torsion [4][5][6]. The theory can predict not only the shear and torsion strengths, but also the deformations of a structure throughout its post-cracking loading history. This theory is called the softened truss model theory to emphasis the importance of the concrete softening phenomenon. It has been successfully used to predict the shear strength of low-rise shear walls, shear strength of framed wall panels, shear transfer strength across an initially uncracked shear plane, torsion strength of beams, and membrane strength of shell elements [7][8][9]. A reliable prediction for the shear strength of shear critical members using the softened truss model theory requires the existence of both longitudinal and transverse reinforcement. Niranjan and Patil [10] introduced an effective transverse compression in the beam web to predict accurately the shear strength of deep beams with vertical web reinforcement. Steel fibres improve the compressive, tensile, shear and impact strength of plain concrete, increase the strain capacity and thus impart ductility to plain concrete [11]. Since shear failure is initiated by diagonal tensile stresses, the presence of steel fibres will retard shear cracks initiation and increase the shear strength in reinforced concrete shear critical members like deep beams, corbels and flat slabs where shear is dominant [1][2][3].
In this study the softened truss model theory was applied to predict the shear strength of fibre reinforced concrete deep beams and corbels. The influence of the shear span/depth ratio, the longitudinal, transverse and steel fibre reinforcement were taken into account in predicting the shear strength.

Materials Constitutive Relationships Fibrous concrete in compression
The stress-strain curve proposed for fibre concrete by Hsu et al. [12] is used in this study: where c f is the stress at strain  , cf f is the compressive strength of fibrous concrete, fo  is the strain at peak stress and taken as proposed by Soroushian and Lee [13]: V f , Ɩ f and d f are the volume fraction, length and diameter of the fibres respectively, and  is a variable that depends on the fibres parameters: where RI is a fibres reinforcing index equals to , where f w is the fibres weight fraction. The strain capacity in compression increases also with the fibres and the Equation proposed by Abdul-Razzaque [14] is used in this investigation: When concrete is subjected to biaxial tensioncompression, softening will occur, i.e.; the compression strength will be reduced due to the presence of tensile stresses in the orthogonal direction. The following softening coefficient proposed by Belarbi and Hsu [15] is used in this investigation: ɛ r = principal tensile strain, this softening coefficient is used for the stress and strain in Eq. (1).

Fibrous concrete in tension
The ascending part of the curve (OA) is similar to that of plain concrete up to first cracking, Fig.(1): E c = initial modulus of plain concrete, ϵ t = tensile strain and f t ' = tensile strength of plain concrete and in this investigation assumed equal to '

5
. 0 c f . A nonlinear region usually follows (AB) due to the increase in microcracks width and propagation up to the peak or the tensile strength of the composite ( tf f ), which is assumed equal to that proposed by Soroushian and Lee [13]: tf  is the corresponding strain.
After the composite reaches its tensile strength f tf , a sudden drop occurs in the tensile stressstrain diagram (BC) due to the formation of a continuous crack and the stress transfer from the matrix to the fibres. The stress at this stage is equal to the post cracking tensile strength and equals to: τ u = bond strength of the steel fibres and in this investigation taken as proposed in Ref. [13]: K f = surface shape factor for the steel fibres = 1.0 for smooth fibres, 1.25 for hooked fibres and 1.5 for crimped fibres [16].
The descending part of the curve (CD) in Fig. (1) is controlled by the following Eq.: The limiting tensile strain (ϵ m ) at which complete separation of the two crack faces occur is taken equal to 0.1 as proposed by Sakai and Nakamura [17].
The stressstrain relationship shown in Fig.(1) is modified to a continuous function as proposed in Ref. [18] as follows The constants a ' , b ' , c ' and d ' can be found from the points A, B and any other point on the descending part CD. The fourth equation is found from the property that the slope of the curve at the peak point equal to zero.

Modulus of Elasticity of Fibrous Concrete
For fibre reinforced concrete, the equation proposed by Hsu et al. [12] is used in this investigation: The Equation proposed by Carraquillo et al. [19] for the modulus of elasticity of plain concrete in the above equation is used in this investigation:

Stress -Strain Relationship of Steel
The reinforcing bars are assumed to have the same properties in tension and compression and follow an elastic (up to the yield point) perfectly plastic behavior.

Solution of Equations of Softened Truss Model Theory
When a reinforced concrete element is subjected to shear stresses and compressive and tensile stresses in two perpendicular directions, cracks will develop at an angle (α) with longitudinal steel direction (Ɩ), Fig.(2). The truss action will be composed by the concrete between the cracks, which resist the compressive stresses, and the reinforcement which resist the tensile stresses. From Fig.(2), three equilibrium equations can be written: Other three compatibility equations can be written for the concrete strains which satisfy Mohr's strain circle:

Analysis of Deep Beams
A deep reinforced concrete beam that is loaded with two symmetrical shear forces is shown in Fig. (3), the shear span should not exceed (2h). The bending moment is resisted by the top and bottom steel while the shear force is resisted by the truss action in the web shown by the dashed rectangle whose depth = d v =d e -d ' . The truss action requires that there are horizontal and vertical reinforcement. When there is no top steel, d ' may be taken as half the depth of the compression stress block (a/2). The small element inside the dashed zone shown in Fig. (3) is subjected to shear and vertical stresses equal to: Since both the shear and vertical stresses are proportional to the shear force V, they can be combined by a factor k: The stress σ r can be found from equations (15 and 17) and the stress-strain relationship of steel: From the stress-strain relationship of steel and equations (15a and 16a) and assuming that (σ Ɩ = 0), the value of (α) can be found: The above equations are solved iteratively to find the values of ( ,

Reinforced Concrete Deep Beams
The above procedure and the materials constitutive relationships are used to analyze the ten fibre reinforced concrete deep beams whose test results are reported in reference [2]. Table (1) show the details of these beams. The longitudinal steel ratio (ρ Ɩ ) for these beams represents the ratio of the tension and horizontal reinforcement. Another twelve fibre reinforced concrete deep beams whose test results are reported in reference by [1] are also analyzed in this investigation. Table (2) show the details of these beams. The longitudinal steel ratio (ρ Ɩ ) for these beams represent the ratio of the tension reinforcement only, since no transverse reinforcement is provided.  Tables (3 and 4 Table (3) show that the transverse reinforcement is more effective in increasing the shear strength than the longitudinal reinforcement for this a/h ratio, while for larger a/h ratios the longitudinal reinforcement is more effective.

Reinforced Concrete corbels
Since the corbel is a short cantilever deep beam, the above procedure and the materials constitutive relationships is used to analyze the twelve fibre reinforced concrete corbels whose test results are reported in reference [3]. Table ( 5) shows the details of these corbels. The longitudinal steel ratio (ρ Ɩ ) for these corbels represents the ratio of the tension and horizontal reinforcement for corbels without steel fibres, while those containing steel fibres (ρ Ɩ ) represent the ratio of the tension reinforcement. Table (6) and Fig. (6) show that the shear strength increases with the longitudinal steel ratio which represent the flexural and shear friction reinforcement. Table (6) and Fig. (7) show that the shear strength increases with the increase of fibres volume and the enhancement is more than that for the deep beams shown in Fig.(5) due to the smaller a/h ratio.