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1.
International INTERNATIONAL Journal
of Mechanical JOURNAL Engineering OF and MECHANICAL Technology (IJMET), ISSN ENGINEERING 0976 – 6340(Print), ISSN 0976 – 6359(Online), Volume 5, Issue 9, September (2014), pp. 374-378 © IAEME AND TECHNOLOGY (IJMET) ISSN 0976 – 6340 (Print) ISSN 0976 – 6359 (Online) Volume 5, Issue 9, September (2014), pp. 374-378 © IAEME: www.iaeme.com/IJMET.asp Journal Impact Factor (2014): 7.5377 (Calculated by GISI) www.jifactor.com 374 IJMET © I A E M E STRESS REDUCTION USING SEMI ELLIPTICAL SLOTS IN AXIALLY LOADED PLATE HAVING CIRCULAR HOLE HARDIK ACHARYA Department of Mechanical Engineering, LDRP Institute of Technology Research, Gandhinagar, India. ABSTRACT When a plate with circular hole is subjected to axial loading, the region in the vicinity of hole is under high stress. To reduce this stress, normally one auxiliary hole is introduced on either side of the main hole. The present work studies how a semi-elliptical slot on each side of the hole effects the maximum stress induced in a isotropic plate with circular hole. Using finite element analysis method, stress calculation is carried out in ANSYS. The effect of variation in the geometric parameters of ellipse on ‘maximum stress’ is presented in graphical form. Stress reduction of 42% is achieved for a particular combination of geometric parameters. A relation for these optimum geometric parameters is obtained with respect to the hole diameter. Keywords: Slot, Stress Reduction, Semi-Elliptical, Optimization. 1. INTRODUCTION As a general practice, in order to reduce the stress concentration around a circular hole, auxiliary holes of smaller dimensions are drilled on either side. But this method has certain limitations. Different approaches for stress reduction are examined and analysed in the literature. H Acharya [1] obtained a relation for the optimum diameter and centre distance of auxiliary hole for 21% stress reduction and showed that thickness and width of the plate does not affect the optimum hole dimensions. N. K. Jain [2] carried out finite element study for reducing SCF by four coaxial holes on either side of the main hole. Using this he revealed that holes helps to smooth flow of the tensile stresses through the main hole. He also studied the effect of length and diameter of two auxiliary holes on each side of the main hole in isotropic and orthotropic plates. V Manur [3] studied the optimum location and diameter of auxiliary holes with respect to different diameter to plate width ratios. Plotting the graph of stress concentration factor (SCF) v/s Central hole diameter-to-plate width ratio, he found that SCF can be relieved by a factor of 19 to 21 percentage. Z Yang [4]
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International Journal of
Mechanical Engineering and Technology (IJME ISSN 0976 – 6359(Online), Volume 5, Issue IJMET), ISSN 0976 T), – 6340(Print), 9, September (2014), pp. 374-378 © IAEME observed the sensitivity of the stress and strain concentration factor with respect to plate thickne thickness as well as the Poisson’s ratio. While the structural dimensions like hole diameter and plate thickness were varied to examine their effect directional stress concentration factor and net deflection of the plate by S. Nagpal [5]. Studying the effect of fiber f orientation of composite materials on stress concentration, B Endigeri [6] found that fibre oriented at 45 concentration. While N.K. Jain [7] analysed the distributions of stresses and deflection in simply supported rectangular isotropic and orthotropic composite plates with central circular hole subjected to transverse static loading using 3- minimise the stress concentration factor, Mansfield [8] determi edge reinforcement around a circular hole in a flat sheet subjected to uniaxial tension. [9] presented a method in which transverse normal pressure created by bolt clamping at open holes can be used to mitigate the hole stress concentration and increase the fatigue life of the plate. D Gunwant [10] studied stress in a plate with elliptical hole for different aspect ratios and compared the results of ANSYS with the analytical method. As per these literatures, introducing circular holes, maximum stress reduction of only 21% can be achieved. Also in order to achieve this reduction, auxiliary holes are to be drilled very near the main hole making it weak. Elliptical holes can be used instead of c of main hole will increase the machining cost. Hence a semi 2. DESCRIPTION OF PROBLEM Fig 1: Model of plate with semi Nomenclature D = Main hole diameter 2a = Length of major axis 2b = Length of minor axis l = Centre distance between ellipse and circle As shown in fig. 1 a plate of width 100 mm and thickness an infinitely long plate, length 2000 mm is taken. While the main hole diameter is D = 40 mm and slot width is kept 5mm. Keeping these dimensions constant, the values of ‘a’, ‘b’ and ‘l’ are varied. A tensile force of 1000N is applied on one end of the plate while keeping the other end f 375 0 and 1350 results in minimum stress -D finite element method. Using numerical analysis, in order to determined optimum distributions of compact gate ese circular ones but very large elliptical holes on both side semi- elliptical slot is studied in this paper. semi-elliptical slots 10 mm is taken. In order to study iber ned P. K. Mallick ircular fixed.
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International Journal of
Mechanical Engineering and Technology (IJME ISSN 0976 – 6359(Online), Volume 5, Issue 3. FINITE ELEMENT ANALYSIS IJMET), ISSN 0976 9, September (2014), pp. 374-378 © IAEME A quadrilateral element is selected with maximum face length of 2mm. The element has eight nodes with each node having two degree of freedom, giving a total of sixteen degree of freedom per element. Proximity and curvature function is used in meshing to give fine meshing near the holes. The study is done for isotropic material. So the material taken is structural steel with Young's modulus 2E+11 Pa and Poisson's ratio 0.3. 4. RESULTS AND DISCUSSION To observe the effect of change in the geometric parameters, a total of 185 design points with different combination of major axis, minor axis and centre distance are calculated for maximum stress in the plate. Using the optimization module of ANSYS, it is found that minimum value ‘Equivalent maximum stress’ is 2.1516 MPa at a= 100mm, b= 20mm, l= 20mm. When the same plate without semi-elliptical slots is used under the same force the ‘Equivalent maximum stress’ is 3.7698 Mpa. Hence by introducing the slots of optimized geometric 42.92% can be achieved. As there are three input and one output parameters, it is not possible to show the relation of all the four parameters in one graph. Hence a graph of variation in maximum stress with respect to ‘a’ is shown in fig 2. In graph (i), the curved lines represent corresponding values of ‘b’ while for l= 20mm in graph (ii) the curved lines represent corresponding ‘l’ values for b= 20mm. From both the graphs, it can be seen that stress goes on de or ‘l’, increasing ‘a’ beyond 100mm does not yield appreciable decrease in stress value hence optimum value of a= 100mm (2.5 D) is taken (i) With respect to ‘b’ The variation in maximum stress with respect to ‘b’ is shown in fig 3. In graph (i), the curved lines represent corresponding values of ‘a’ for l= 20mm while graph (ii) the curved lines represent corresponding ‘l’ values for a= 100mm. From both the graphs, it can be seen that stress is minimum for b= 20 – 21mm thus a relation b = 0.5 D is obtained. 376 hange parameters, a stress reduction of decreasing with increasing ‘a’. But for any value of ‘b’ , (ii) With respect to ‘l’ Fig 2: Stress Vs ‘a’ responding T), – 6340(Print), of creasing
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International Journal of
Mechanical Engineering and Technology (IJME ISSN 0976 – 6359(Online), Volume 5, Issue (i) With respect to ‘a’ T), – 6340(Print), The variation in maximum stress with respect to ‘l’ is shown in fig 4. In graph (i), the curved lines represent corresponding values of ‘a’ for b= 20mm while graph (ii) the curved lines represent corresponding ‘b’ values for a= 100mm. From both the for l= 20 – 22mm thus a relation l = 0.5 D is obtained. (i) With respect to ‘a’ 5. CONCLUSION In this study, by using semi-of 42% can be achieved which is double of what can be achieved using auxiliary (21%). The relation of optimum geometric parameters hole diameter is: a= 2.5D, b= 0.5D and l= 0.5D. IJMET), ISSN 0976 9, September (2014), pp. 374-378 © IAEME 377 (ii) With respect to ‘l’ Fig 3: Stress Vs ‘b’ graphs, it can be seen that stress is minimum (ii) With respect to ‘b’ Fig 4: Stress Vs ‘l’ -elliptical slots it is seen that maximum possible stress reduction of circular holes . for stress minimization imization with respect to main
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International Journal of
Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online), Volume 5, Issue 9, September (2014), pp. 374-378 © IAEME 378 REFERENCES [1] H. Acharya, “Optimization of location and size of auxiliary holes for stress reduction in axially loaded plate with circular hole”. International Journal of Mechanical Engineering Research, Vol. 4, No. 2, 2014, p.135-141. [2] N.K. Jain, “Reduction of stress concentration in a uni-axially loaded infinite width rectangular isotropic/orthotropic plate with central circular hole by coaxial auxiliary holes”. IIUM Engineering Journal Vol. 12, No. 6, 2011, p. 141-150. [3] V Manur, R Shigli, “Finite element method for stress reduction by optimal auxiliary holes in a uniaxially loaded plate”. International Conference on Emerging Trends in Manufacturing Technology, Ernakulam, 2012. [4] Z yang, C Kim, C Cho, H Beom, “The concentration of stress and strain in finite thickness elastic plate containing a circular hole”. International Journal of Solids and Structures, Vol. 45, 2008, p. 713–731. [5] S Nagpal, S Sanyal, N.K. Jain. “Analysis and mitigation of stress concentration factor of a rectangular isotropic and orthotropic plate with central circular hole subjected to in-plane static loading by design optimization”. IJIRSET Vol. 2, Issue 7, 2013, p. 2901-2913. [6] B Endigeri, V Mannur, “Fem for stress reduction by optimal auxiliary holes in a uniaxially loaded composite plate”. Proceedings: International Conference on Research in Electrical, Electronics Mechanical Engineering, Dehradun, 2014. [7] N.K. Jain. “3D analysis of stress concentration factor and deflection in thin isotropic and orthotropic plates with central circular hole subjected to transverse loading”. IJMERD Vol. 1, No. 6, 2011, p. 01-13. [8] E. H. Mansfield, C J Hanson, “Optimum reinforcement around a circular hole in a flat sheet under uniaxial tension”. Reports and Memoranda No. 3723, Structures Dept., R.A.E, Farnborough 1972 [9] P. K. Mallick, “Effects of hole stress concentration and its mitigation on the tensile strength of sheet moulding compound (SMC-R50) composites”. Composites, Vol. 19, No. 4, 1988, p. 283-287 [10] D Gunwant, J. P. Singh, “Stress and displacement analysis of a rectangular plate with central elliptical hole”. International Journal of Engineering and Innovative Technology, Vol. 3, No. 3, 2013, p. 387-392.
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