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carried out using finite element method to study the behavior of deep beam by considering flexural stress, flexural strain,

and shear stress variations at different sections for various effective lengths to depth ratio and compared with EulerBernoulli Theory. The effective span to depth ratios of the beams considered were 1.25, 1.375 and 1.5

Keywords: Finite Element Method (FEM), Reinforced Concrete (R.C) deep beam, Shear strength.

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- 1. International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol. 2, Issue. 6, Nov.-Dec. 2012 pp-4664-4667 ISSN: 2249-6645 Analysis of R.C Deep Beam by Finite Element Method B.R. Niranjan, 1 S.S.Patil2 1 Professor, Civil Engineering Department, U.V.C.E. Bangalore University Bangalore. India.) 2 Associate Professor, Civil Engineering Department, Walchand Institute of Technology. Solapur IndiaABSTRACT : The analytical study of reinforced concrete simply supported deep beams subjected to two point loads wascarried out using finite element method to study the behavior of deep beam by considering flexural stress, flexural strain,and shear stress variations at different sections for various effective lengths to depth ratio and compared with Euler-Bernoulli Theory. The effective span to depth ratios of the beams considered were 1.25, 1.375 and 1.5Keywords: Finite Element Method (FEM), Reinforced Concrete (R.C) deep beam, Shear strength. I. INTRODUCTION The use of Reinforced deep beam has become more prevalent in recent years. In IS-456 (2000) clause 29, a simplysupported beam is classified as deep when the ratio of its effective span L to overall depth D is less than 2. Continuous beamare consider as deep when the ratio L/D is less than 2.5.The effective span is defined as the center to center distance betweenthe supports or 1.15 times the clear span whichever is less. Deep beam often appear in form of transfer girders in high-rise building as well as pile caps, foundation walls,water tanks, bins, folded plate roof structures, floor diaphragms, shear walls & brackets or corbels. They are characterized asbeing relatively short & deep, having thickness that is small relative to their span or depth and being primarily loaded in theplane of the member. They are “two dimensional” members in a state of plane stress in which shear is dominant feature. Theinternal stresses cannot be determined by ordinary beam theory & ordinary design procedures for determining strength donot apply. The behavior of deep beams is significantly different from that of beams of more normal proportions, requiringspecial consideration in analysis, design and detailing of reinforcement. A deep beam is in fact a vertical plate subjected toloading in its own plane. The strain or stress distribution across the depth is no longer a straight line, and the variation ismainly dependent on the aspect ratio of the beam. Stresses in deep beams can be studied using the methods of two dimensional elasticity or Finite Element analysis.Plane sections before bending remaining plane after bending does not hold good for deep beams. Significantly warping ofthe cross-sections occurs because of high shear stresses consequently flexural stresses are not linearly distributed, even in theelastic range, and the usual methods for calculating section properties and stresses cannot be applied. Shear strength of deepbeams may be as much as 2 to 3 times greater than that predicated using conventional equations developed for members ofnormal proportions. The stresses in isotropic homogeneous deep beams before cracking can be determined using finite element analysisor photo elastic model studies. It is found that the smaller the span/depth ratio (i.e., less than 2.5), the more pronounced thedeviation of the stress pattern from that of Bernoulli and Navier. II. OBJECTIVES OF STUDIEa) To analyze the deep beam by using Finite Element Method. The finite element model consists of subdividing the Deep beam into small 2D rectangular elements. Each element has four nodded plane stress Element to study the behavior of deep beam through flexural stresses, flexural strains, and shear stresses variation with various L/D Ratio.b) To compare the flexural steel requirement as per codal provisions with that calculated using the finite element method. III. ANALYTICAL STUDY OF DEEP BEAMSA deep beam having following data is analyzed for various L/D ratios. Beam is simply supported subjected to two pointsloading as shown in Figure 1.Depth = 400 mm and Width of beam = 150 mm (constant for all cases);L/D ratios is varied from 1.25, 1.375, 1.5 i.e. (Span = 500, 550, 600) mm L Fig. 1 Deep beam with two point loads www.ijmer.com 4664 | Page
- 2. International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol. 2, Issue. 6, Nov.-Dec. 2012 pp-4664-4667 ISSN: 2249-6645Deep beam is an example of plane stress conditions. It is assumed that no stress component varies across the thickness. The x , y xystate of stress is then specified by and only. Y V1 V2 ε V1 U2 U1 V1 r V3 b 3 U3 a XFig.2a-Deep beam subdivided into rectangular element Fig.2b-Typical rectangular finite Element Fig 2a shows Deep Beam which has been subdivided into small rectangular finite element for analysis purpose. Theselection of mesh size for element depend on accuracy desired, the finer the mesh, the better the result but at the same timethe larger the computation required. Rectangular elements are found to be very convenient due to simplicity in formulation, The accuracy of FEM depends on the determination of stiffness matrix of Rectangular element. The twodimensional four noded Rectangular element is as shown in fig.2b. It has four nodes located at the corners and has twotranslational degrees of freedom at each node, displacement component in X and Y directions with constant σx, σy and τxy. The analysis of Deep Beam is carried out with the Finite Element Method by using the SAP (Structural Analysisprogram me) software. The variation of flexural stresses, strains at mid span and shear stresses near support is valuated forall these cases. (a) Variation of flexural strain Fig.3a to Fig. 3c show that the flexural strain at mid span of simply supported deep beam for different span –to-depth ratios. The beams have disturbed region in flexural strain distribution. Deep beams behave differently from shallowbeams. In these members, the distribution of strain across the depth of the cross section is nonlinear and a significant amountof load is carried to the supports by a compression strut joining the load and the reaction. It is found that the smaller thespan/depth ratio (i.e., less than2.0), the more pronounced the deviation of the strain pattern from that of Euler Bernoullitheory. FLEXURAL STRAINS DISTRIBUTION FLEXURAL STRAIN DISTRIBUTION 8 8 FEM 7 6 DEPT 4 EBT 5 EBM 4 3 H 2 FEM 0 1 0 -4.00E-05 -2.00E-05 0.00E+00.00E-05 .00E-05 2 4 -1 -4 -4.01E-0 -29E-07 .14E-05 -1.96E-0 2 -3 -4 -5 -6 Strain L/D-8 1.25 = Strain L/D = 1.375 Fig. 3a Flexural Strain Distribution Fig. 3bFlexural Strain Distribution FLEXURAL STRAIN DISTRIBUTION 6 DEPTH 5 4 3 2 EBT 1 FEM 0 -1 -3E-05 -2 -5E-05 -1E-051E-053E-055E-05 -3 -4 -5 -6 strain L/D = 1.5 Fig.3c Flexural Strain Distribution www.ijmer.com 4665 | Page
- 3. International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol. 2, Issue. 6, Nov.-Dec. 2012 pp-4664-4667 ISSN: 2249-6645 (b)Variation of flexural stress Fig. 4a to Fig. 4c shows the flexural stress at mid span of simply supported deep beam for two different shear span–to-depth ratios. The compressive stresses increase rapidly at the top and neutral axis moves towards soffit of the beam. Fromthe variation of flexural stress graphs the definition of simply supported deep beam as per IS 456:2000 i.e. when L/D ratio isless than or equal to 2.0 is reasonably accurate. FLEXURAL STRESS DISTRIBUTION FLEXURAL STRESS DISTRIBUTION 3 10 DEPTH 2 EBM 5 FEM DEPTH 1 EBM FEM 0 0 -4000 -2000-1 0 2000 4000 -5000 -5 0 5000 -2 -10 Stress in N/mm2 L/D = 1.25 Stress in N/mm2 L/D 1.375 Fig.4a Flexural Stress Distribution Fig.4b Flexural Stress Distribution FLEXURAL STRESS DISTRIBUTION 10 DEPTH 5 FEM EBM 0 -6000 -4000 -2000 0 2000 4000 -5 -10 Stress in N/mm2 L/D =1.5 Fig.4c Flexural Stress Distribution(c) Variation of shear stress Fig.5a to Fig.5c shows the shear stress near support of simply supported deep beam for different shear span to depthratios .The beams have drastic change in shear stress distribution. Deep beams behave differently from shallow beams. Theshear stress patterns have also changed in case of deep beam. It is found that the smaller the span/depth ratio (i.e., lessthan2.0), the more pronounced the deviation of the shear stress distribution from that of Euler Bernoulli theory 20 EBT Shear Stress in N/mm2 8 EBM 20 FSM 15 FEM 15 10 FSM 4 10 EBT 5 5 0 0 0 -5 0 0 200 400 -5 0 200 400 600 500 -10 -4 -10 -15 -15 -8 -20 -20 Shear stress in N/mm L/D=1.25 Shear Stress inn Shear Stress in N/mm2 N/mmL/D=1.375 Fig.5a Shear Stress Distribution Fig.5b Shear Stress Distribution Fig. 5c Shear Stress Distribution From the variation of shear stress graph it is clear that shear effect is predominant in beams having L/D ratio lessthan or equal to 2.0 which may lead to warping of the section.The reinforcement required as per Finite Element Method based on flexural stress distribution graphs for various L/D ratioswas presented in table1. Table1. Reinforcement required as per FEM Sr.No. Depth from Center of Reinforcement required mm2 Beam ( mm) L / D Ratio 1.50 1.375 1.25 200 39.23 46.56 56.73 www.ijmer.com 4666 | Page
- 4. International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol. 2, Issue. 6, Nov.-Dec. 2012 pp-4664-4667 ISSN: 2249-6645 175 23.57 25.79 26.70 150 15.59 17.24 18.97 125 11.65 12.56 12.05 100 6.39 8.692 7.36 75 3.923 5.493 4.2 50 1.987 2.77 1.205 25 0.123 0.398 1.5 Total 102.45 119.503 128.75 TABLE 2: AREA OF FLEXURAL STEEL REQUIRED Case No. Case 1a Case 1b Case 2a Case 2b Case 3a Case 3b Design Method I.S.456 ACI 318 I.S.456 ACI318 I.S.456 ACI318 Span L (mm) 500 500 550 550 600 600 Span to depth (L/D) ratio 1.25 1. 25 1.375 1.375 1.50 1.50 Z (Lever Arm) 280 333 270 333 260 333 Flexure steel Required in mm2` 149.37 179.42 154.89 179.42 160.85 179.42 Flexural steel required in mm2 102.45 102.45 119.5 119.5 128.72 128.72 IV. CONCLUSION Flexural stress and strain variation graphs indicate that the definition of simply supported deep beam as per IS 456:2000 i.e. when L/D ratio is less than or equal to 2.0 is reasonably accurate. With Finite element analysis, Flexural strain and stress of deep beams is not linear is confirmed. Shear effect is predominant in beams having L/D ratio less than or equal to 2.0 which may lead to warping of the section. The flexural steel requirements inversely proportionally to the Effective span to Depth Ratio of deep beam. The flexural steel required by finite element method is approximately 21 % less than I.S.456 and A.C.I.318. REFERENCES Journal Papers[1] Mohammed Reza Salamy. Hiroshi Kobayashi and Shigeki Unjoh, Experimental and Analytical Study on Deep Beams.[2] R.S. Londhe(2010-11), Experimental studiesin RC Transfer(Deep) Beam for High rise building‟,The structural Engg.Vol.37No.5,pp.305-312[3] Mr. Varghese and Mr. Krishnamoorthy,(1966),Strength and Behaviour of Deep Reinforced Concrete Beams, Indian Concrete Journal, 104-108. Books[4] N. Krishna Raju. Advanced reinforced concrete design, CBS Publications.[5] C. S. Krishnamoorthy(1995), „ The finite Element Analysis‟ Tata McGraw-Hill Education, 1995.[6] F. K. Kong. „Reinforced Concrete Deep Beam‟, Van Nostrand Reinhold, New York. Standard Codes[7] „Indian Standard Code of Practice Plain and Reinforced Concrete‟ IS 456:2000-. Bureau of Indian Standards, Manak Bhavan, New Delhi, India.[8] ACI 318–05. „Building Code Requirements for Structural Concrete and Commentary‟ American Concrete Institute, Detroit, USA. www.ijmer.com 4667 | Page

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