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IRJET- Shear Stress Distribution in Beams
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IRJET- Shear Stress Distribution in Beams
1.
International Research Journal
of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 04 | Apr 2019 www.irjet.net p-ISSN: 2395-0072 © 2019, IRJET | Impact Factor value: 7.211 | ISO 9001:2008 Certified Journal | Page 163 Shear Stress Distribution in Beams R. Keerthana1, P. Sathiya Bama2 1M.E Project Student, Dr.Mahalingam College of Engineering and Technology, Pollachi-642003 2Assistant Professor, Dept. of Civil Engineering, Dr.Mahalingam College of Engineering and Technology, Pollachi-642003, Tamil Nadu, India ---------------------------------------------------------------------***---------------------------------------------------------------------- Abstract - All materials fail under a certain loading conditions; it might be in tension, compression, torsion, bending and shear or may be combination of loads. Any type of loads can induce two types of stress. Thestressesarenormal stress and shear stress. In this work, mainly focused on shear stress distribution in rectangular beams by varying depth/breadth ratio. When shear loadisapplied, theimpact of the shearing stress throughout the rectangular cross-section of the beam occurs. It can be resolved by estimating the shearing stress at the particular height from the neutral axis. The distribution of shearing stress on the cross-section of the beam represents a parabolic curve where the maximum shearing stress occurs at the neutral axis of the beam. The analysis of beam is done by using ANSYS software. The analysis of beam is done up to the depth/breadth ratio of 10 and interpretation of results has done. Key Words: shear stress,shearload,depth/breadth ratio 1. INTRODUCTION 1.1 General Any force which tries to shear-off the member is called shear force. Shear force is an unbalanced force, parallel to the cross-section. To resist the shear force, the element will develop the resisting stresses, which is known as shear stress. When a beam is subjected to a transverse loading, a normal and a shearing stresses result in the beam. The influence of shearing stress in the beam does not disturbthe influence of the bending stress. The shearing stress in beam is defined as the stress that occurs due to the internal shearing of the beam that results from shear force subjected to the beam. It is denoted by the symbol and is expressed in the unit N/mm2.The equation of shearing stress is Where, V is shear force b is width of the section d is depth of the section The shear stress distribution will vary based on the sections such as rectangular section, triangular section, circular section, I section, and T section. 1.2 Shear stress distribution of different sections The following shows the shear stress distribution of various sections. 1.2.1 Shear stress distribution for rectangular section Consider a rectangular beam section whose depth of section is d, width of section is b. Fig-1.1 Shear stress distribution of rectangular section Fig-1.2 Rectangular cross-section of beam Shear stress is distributed parabolically across the rectangular section. Shear stress is maximumatneutral axis and will be zero at the extreme ends. The vertical shear stress creates horizontal shear stress. Shear stress is distributed clockwise or anticlockwisethroughoutheight, so shear stress variation will be shown in only on one side. In a rectangular beam, when shear load is applied, the impact of the shearing stress throughout therectangularcross-section of the beam occurs. It can be resolved by estimating the shearing stress on the cross-sectionofthe beamrepresentsa parabolic curve where the maximum shearing stress occurs
2.
International Research Journal
of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 04 | Apr 2019 www.irjet.net p-ISSN: 2395-0072 © 2019, IRJET | Impact Factor value: 7.211 | ISO 9001:2008 Certified Journal | Page 164 at the neutral axis of the beam. The equation for maximum shear stress in rectangular section is Max = 1.5 1.2.2 Shear stress distribution for I section In I section, the web bears the most of the shear stress and according to the bending theory it is said that flange will bear most of the bending stress. Fig-1.3 Shear stress distribution of I section The equation for maximum shear stress in I section is Max (D2 – d2) + 1.2.3 Shear stress distribution for T section T section is not symmetrical over neutral axis, shear stress distribution also will not be symmetrical. The method of finding shear stress distribution in t section is similar to that of I section. Fig 1.4 Shear stress distribution of T section 1.2.4 Shear stress distribution for Triangular section The maximum shear stress is at a distance h/2 fromthe base of the triangle, which is also at a distance of h/6 from the centroidal axis. The maximum shear stress distribution in triangular section is Max = Fig 1.5 Shear stress distribution of Triangle section 1.2.5 Shear stress distribution for Circular section In circular section, the shear stress distribution is parabolic. The maximum shear stress is at the neutral axis and the end of the section has zero stress. The average shear stress in a beam of circular Fig 1.6 Shear stress distribution of I section The maximum shear stress distribution in circular section is Max = 1.3 Factors affecting shear strength of concrete 1.3.1 Size of beam As the depth of the beam increases, theshearstressatfailure decreases. For rectangular section beam, the shear stress distribution is parabolic and maximum shear stress is at neutral axis of the section. The maximumshearstresswill be 1.5times the average shear stress. For circular section beam, the shearstressdistributionhasa parabolic variation. The shear stress is maximum when y=0, at the neutral axis. The maximum shear stress will be 4/3times of average shear stress. For I section, the shear stress distribution is parabolic in the flange and web. From the shear stress diagram for this section, the most of the shear stress is taken by web only. This is very important in the design. For triangular section, the shear stress has a parabolic variation. The maximum shear stress will be 8/3times of average shear stress.
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International Research Journal
of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 04 | Apr 2019 www.irjet.net p-ISSN: 2395-0072 © 2019, IRJET | Impact Factor value: 7.211 | ISO 9001:2008 Certified Journal | Page 165 1.3.2 Shear span to effective depth Its effect is pronounced when span to depth ratio islessthan two and has no effect when it is greater than six. 1.3.3 Tensile strength of concrete The inclined cracking load in shear is a functionofthetensile strength concrete. 1.3.4 Longitudinal reinforcement The shear strength of the RC beams is found to drop significantly if the longitudinal reinforcement ratio decreases below 1.2 to 1.5%. 1.3.5 Axial force Axial tension decreases the inclined cracking load and the shear strength of concrete, whereas axial compression does just the opposite. 1.3.6 Light weight aggregate concrete Light weight aggregate concrete reduces tensile strength than concrete with normal aggregates. 1.3.7 Size of coarse aggregates Increasing the size of coarse aggregates increases the roughness of the crack surfaces, thus allowing the higher shear stresses to be transferred across the cracks. 1.4 Rectangular cross section of beam Considering a rectangular cross section beam where, b is width of the rectangular suction, d is depth of the rectangular section, NA is neutral axis of the beam section, F is shear force , is shear stress, A is area of section CDEF where shear stress to be determined y i distanceoftheC.G of area CDEF from neutral axis of beam, A is area of section CDEF where shear stress to be determined, I is moment of inertia of the given section about the neutral axis Fig 1.7 Rectangular cross section of beam The maximum shear stress will occur at y=0 or at neutral axis and value of shear stress will be zero for the area at the extreme ends. The average shear stressormeanshearstress will be simply calculated by dividing shear force with area. Max = 1.5 Average 2. NUMERICAL INVESTIGATION A simply supported beam was considered withthefollowing parameters. The length of the beam is 1000mm, breadth of the beam is 100mm and depth of beamvaryingfrom100mm to 1000mm.Uniformly distributed load of 100KN/m is applied on the beam. Using the above parameters the beam was modeled with ANSYS software. The shear stress acting in the cross section of the beam is analyzed with varyingd/b ratio. 2.1.1 Dimension of beam 1 The following are the dimension of beam: Length of beam = 1000mm,Width of beam = 100mm,Depth of beam = 100mm Fig-2.1 Dimension of beam 2.1.2 Material property The poi on’ ratio of concrete i 0.18 and young’ modulu is 2E4 is provided. Fig -2.2 Material property for beam 2.1.3 Meshing of beam: The meshing of beam is done. The number of element division provided for meshing is 10.
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International Research Journal
of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 04 | Apr 2019 www.irjet.net p-ISSN: 2395-0072 © 2019, IRJET | Impact Factor value: 7.211 | ISO 9001:2008 Certified Journal | Page 166 Fig-2.3 Meshing of beam 2.1.4 Boundary conditions: The beam is simply supported. The boundary conditionsare one end pinned and other end roller supported. Fig 2.4 Boundary conditions for beam 2.1.5 Load acting on beam: A uniformly distributed load of 100KN/m is applied on the beam. Fig-2.5 Load acting in beam 2.1.6 Solution Preprocessing is done and to solve the beam solution run command is given. The solution is obtained for the given conditions for beam. Fig-2.6 Solution obtained from ANSYS 2.1.7 Path of shear stress acting on beam The shear stress path is plotted along y direction of beam Fig-2.7 Path of shear stress on beam 2.1.8 Graph obtained: The shear stress distribution graph is obtained for d/b= 1 at 250mm. The shear stress distribution is parabolic. Fig-2.8 Graph for shear stress distribution 2.1.9 Shear stress distributioninbeam atL/4,d/bratio= 1 The depth of beam is 100mm, the shear stress at a length of 250mm and maximum shear stress at neutral axis is 375.15N/mm2. Chart-2.1 Shear stress distribution of beam at length of L/4, d/b ratio = 1 2.2 Dimension of beam 2 The following are the dimensions of beam 2:Length of beam = 1000mm,Width of beam = 100mm,Depth of beam = 200mm
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International Research Journal
of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 04 | Apr 2019 www.irjet.net p-ISSN: 2395-0072 © 2019, IRJET | Impact Factor value: 7.211 | ISO 9001:2008 Certified Journal | Page 167 2.2.1 Shear stress distribution in beam at L/4, d/b ratio=2 The depth of beam is 200mm, the shear stress at a length of 250mm and maximum shear stress at neutral axis is 185.12N/mm2. Chart-2.2 Shear stress distribution of beam at length of L/4, d/b ratio = 2 2.3 Dimension of beam 3 The following are the dimensions of beam 3: Length of beam = 1000mm,Width of beam = 100mm,Depth of beam = 300mm 2.3.1 Shear stress distribution in beam at L/4, d/b ratio=3 The depth of beam is 300mm, the shear stress at a length of 250mm and maximum shear stress at neutral axis is 125.65N/mm2. Chart-2.3 Shear stress distribution of beam at length of L/4, d/b ratio = 3 2.4 Dimension of beam 4 The following are the dimensions of beam 4:Length of beam = 1000mm,Width of beam = 100mm, Depth of beam = 400mm 2.4.1 Shear stress distribution in beam at L/4, d/b ratio=4 The depth of beam is 400mm, the shear stress at a length of 250mm and maximum shear stress at neutral axis is 93.75N/mm2. Chart-2.4Shear stress distribution of beam at length of L/4, d/b ratio = 4 2. 5 Dimension of beam 5 The following are the dimensions of beam 5: Length of beam = 1000mm,Width of beam = 100mm,Depth of beam = 500mm 2.5.1 Shear stress distribution in beam at L/4, d/b ratio=5 The depth of beam is 500mm, the shear stress at a length of 250mm and maximum shear stress is 97.35 N/mm2atdepth of 200mm. Chart-2.5 Shear stress distribution of beam at length of L/4, d/b ratio = 5 2.6 Dimension of beam 6 The following are the dimensions of beam 6:Length of beam = 1000mm,Width of beam = 100mm,Depth of beam = 600mm 2.6.1 Shear stress distribution in beam at L/4, d/b ratio=6 The depth of beam is 600mm, the shear stress at a length of 250mm and maximum shear stress is 56.17 N/mm2atdepth of 225mm.
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International Research Journal
of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 04 | Apr 2019 www.irjet.net p-ISSN: 2395-0072 © 2019, IRJET | Impact Factor value: 7.211 | ISO 9001:2008 Certified Journal | Page 168 Chart-2.6 Shear stress distribution of beam at length of L/4, d/b ratio = 6 2.7 Dimension of beam 7 The following are the dimensions of beam 7:Length of beam = 1000mm, Width of beam = 100mm,Depth of beam = 700mm 2.7.1 Shear stress distribution in beam at L/4, d/b ratio=7 The depth of beam is 700mm, the shear stress at a length of 250mm and maximum shear stress is 49.88N/mm2 at depth of 250mm. Chart-2.7 Shear stress distribution of beam at length of L/4, d/b ratio = 7 2.8 Dimension of beam 8 The following are the dimensions of beam 8: Length of beam = 1000mm,Width of beam = 100mm,Depth of beam = 800mm 2.8.1 Shear stress distribution in beam at L/4, d/b ratio=8 The depth of beam is 800mm, the shear stress at a length of 250mm and maximum shear stress is 45.97N/mm2 at depth of 250mm. Chart-2.8 Shear stress distribution of beam at length of L/4, d/b ratio = 8 2.9 Dimension of beam 9 The following are the dimensions of beam 9:Length of beam = 1000mm,Width of beam = 100mm,Depth of beam = 900mm 2.9.1 Shear stress distribution in beam at L/4, d/b ratio=9 The depth of beam is 900mm, the shear stress at a length of 250mm and maximum shear stress is 44.54 N/mm2atdepth of 200mm. Chart-2.9 Shear stress distribution of beam at length of L/4, d/b ratio = 9 2.10 Dimension of beam 10 The following are the dimensions of beam 10. Length of beam = 1000mm, Width of beam = 100mm, Depth of beam = 1000mm 2.10.3 Shear stress distribution in beam at L/4, d/b ratio=10 The depth of beam is 1000mm, the shear stress at a lengthof 250mm and maximum shear stress is 44.25N/mm2 at depth of 200mm.
7.
International Research Journal
of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 04 | Apr 2019 www.irjet.net p-ISSN: 2395-0072 © 2019, IRJET | Impact Factor value: 7.211 | ISO 9001:2008 Certified Journal | Page 169 Chart-2.10 Shear stress distribution of beam at length of L/4, d/b ratio = 10 3. CONCLUSION AND FUTURE WORK This project is specifically concernedaboutinvestigationofa rectangular beam which is modeled and analyzed for different d/b ratios by finite element analysis using ANSYS software. The results obtained from ANSYS, Mechanical APDL 19.0, there is variation in shape of the graphs in Y direction for varying d/b ratio of 1 to 10. The very low values of d/b theories are conservative and for very high d/b ratio, it occurs on unsafe side which is evident from variation in change of conventional shape of graph. The d/b ratio upto 4 (within all practical ranges), the theory holds good. The d/b>4, incase of beam with abnormal size, shear stress graph in Y direction varies and d/b does have some influence. Incase of these beams, due to the high ratio of d/b, beam action itself is modified. For predicting the accuracy in variation of shearstress,thebeamdepthmustbe increased and analysis must be done. REFERENCE 1. Abd El Rahman Megahid , Mohamed M. Rashwan “Effect of hear pan to depth ratio on the tatical behaviour of HSCB ubjectedtotran ver eloading” Journal of Engineering Sciences,2016. 2. M.Ahmadi “Analy i of Rehabilitation effect on the improvement capacity reinforced concrete beams with FEM” the 2nd International Conference on Rehabilitation and Maintenance in Civil Engineering, 2013. 3. Amer M. Ibrahim, WissamD.Salman“Finiteelement analysis of reinforced concrete beams with CFRP in flexural” Diyala Journal of Engineering Science 2009. 4. Biao Hua, Yu-Fei Wub “Effectof hear pan-to-depth ratio on hear trength component of RC beam ” Engineering Structures,2018. 5. Darmansyah Tjitradi, Eliatun Eliatun,Syahril Taufik “3D ANSYS Numerical Modeling of Reinforced Concrete Beam Behavior under Different Collapsed Mechanisms” International Journal of Mechanic and Applications, 2017. 6. Minh Tung Tran “Influence Factors for the Shear Strength of Exterior and Interior Reinforced Concrete Beam-Column Joint ” Procedia Engineering, 2016. 7. Neha S. Badiger “Study of Behaviour of Reinforced Concrete BeamUsingANSYS”3rd WorldCongresson Civil, Structural and Environmental Engineering, 2018. 8. Prof. S. S. Patil, A. N. Shaikh, Prof. Dr. B. R. Niranjan “Experimental and Analytical Study on Reinforced Concrete Deep Beam” International Re earch Journal of Engineering and Technology, 2016. 9. Patel S.K.Dubey K.K.Pathak “Analy i of infilled beams using method of initial functions and compari on with FEM” Engineering Science and Technology, an International Journal, 2014. 10. Tamon UEDA “Behaviour in hear of reinforced concrete beam under fatigue loading” Hokkaido University Collection of Scholarly and Academic Papers, 2016.
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