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- 1. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME 37 A CLOSED-FORM SOLUTION FOR STRESS CONCENTRATION AROUND A CIRCULAR HOLE IN A LINEARLY VARYING STRESS FIELD Raghavendra Nilugal1 , Dr. M. S. Hebbal 2 1 Lecturer, Sri Venkateshwara College of Engineering, Bengaluru, Karnataka, INDIA. 2 Professor, Dept. of Mechanical Engineering, Basaveshwar Engineering College, Bagalkot, Karnataka, INDIA. ABSTRACT This paper presents a closed-form solution, based on ‘Theory of Elasticity’ to determine the stress concentration around a circular hole in an infinite isotropic plate subjected to linearly varying stress. Numerical solutions such as Finite Element Method, Finite Difference Method and Boundary Element Method can be employed to solve this problem. But these requires are approximate methods, and requires high skill, computers and computer programs to obtain the solution. Whereas the method present here does not need computers and computer programs. The equation developed in the present work can be used to determine the stress field around the circular hole. The results obtained are compared with FEA results and are in close agreement. Keywords: Closed-form solution, linearly varying stress field, stress concentration. 1. INTRODUCTION The study of distribution of stress field in elastic medium has been going on since 1910 in different part of the world undertaken by different research teams. Kolosov G. V. is the oldest name in this field who initiated the study of stress concentration around holes of arbitrary shape and size in elastic medium, which is subjected to different stresses. Cracks in structures often initiate and propagate from the locations of stress or strain concentration. The stress and strain concentration locations are the critical structural details to determine the crack initiation, growth and life of engineering structures. Despite of careful detail- design, practically many structures contain stress and strain concentration due to discontinuities such as holes. Publication by Pilkey (1997), Young (Young and Budynas, 2002) and Pilkey (2005) are widely known for data on stress concentration factor for a wide variety of possible specimen configurations. INTERNATIONAL JOURNAL OF MECHANICAL ENGINEERING AND TECHNOLOGY (IJMET) ISSN 0976 – 6340 (Print) ISSN 0976 – 6359 (Online) Volume 4, Issue 5, September - October (2013), pp. 37-48 © IAEME: www.iaeme.com/ijmet.asp Journal Impact Factor (2013): 5.7731 (Calculated by GISI) www.jifactor.com IJMET © I A E M E
- 2. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME 38 Shivakumar (1992) studied three dimensional results for plates with straight shank holes subjected to various types of loadings using a FROTRAN program. They found that the maximum stress concentration lies at the mid-thickness of the isotropic plate and drops near the free surface. The formulation of the compatibility conditions on the boundary of an elastic continuum was presented by Patnaik (2001). Kotousov (2002) presented analytical solutions for (3-D) stress distribution around typical stress concentrators in an isotropic plate with arbitrary thickness. They showed that for wide range of thickness to radius ratios stress concentration factor for a circular hole is slightly perturbed from the plane strain solution. Yang (2008) showed that the location of maximum stress concentration factor occurs on the mid plane only in thin plates and it moves away from the mid plane of plate by increasing the plate thickness. Singh (2007) and Xu (2008) published the Kirsch’s problem for stress-concentration due to a circular hole in a stressed plate. Solution strategies to elasticity problems to get Beltrami-Michell compatibility equations and Navier’s equations are published by Sadd (2009) and Saada (2009). Batista (2011) investigated the stress concentration around a hole in an infinite plate using modified Muskhelishvili complex variable method. They showed that concave corners of polygonal holes are stress free while convex corners gave the circumferential stress component and also the stress concentration depends on the radius of curvature of a convex corner. Mohammadi (2011) presented a general form for the stress concentration factor in case of biaxial tension and Frobenius series solution for pure shear case to calculate the stress concentration factor around a hole in an infinite plate made up of functionally graded material subjected to uniform biaxial tension and pure shear. Sharma (2012) investigated and reported that by controlling the corner radius of the polygonal hole, the stress concentration can be kept within some permissible limit. As the corner radii approaches zero, stress concentration factor approaches infinity. However, these literatures do not include information on stress concentration around a circular hole in a plate subjected to linearly varying stress field. Hence, this paper presents a closed- form solution to the stress concentration around a circular hole in an infinite isotropic plate subjected to linearly varying stress field. Numerical methods like Finite Element Method, Finite Difference Method and Boundary Element Method can be employed to compute the stress concentration around holes but the advantage of closed-form (analytical) solution is that it provides one step best possible solution without including the infinite series, integrals or limits. And also it does not require the computers and computer programs to obtain the solution. Nomenclature tK Stress Concentration Factor maxσ Maximum Stress in N/mm2 nomσ Nominal Stress in N/mm2 a Hole Radius in mm r Radius at any Point in the Plate in mm θ Angle in degree h Height of the Plate in mm tσ Stress at the Top Edge of the Plate in N/mm2 bσ Stress at the Bottom Edge of the Plate in N/mm2 ,x yσ σ Normal Stress in Cartesian Co-ordinates in N/mm2 xyτ Shear Stress in Cartesian Co-ordinates in N/mm2 φ Stress Function
- 3. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME 39 ,r θσ σ Normal Stress in Polar Co-ordinates in N/mm2 rθτ Shear Stress in Polar Co-ordinates in N/mm2 b Radius very much larger than Hole Radius in mm 1 16toC C Constants 0 , uσ σ Uniform Stress in N/mm2 2. STRESS CONCENTRATION AROUND A CIRCULAR HOLE The localization of maximum stress due to irregularities or abrupt change of cross section is known as the stress concentration. Stress concentration is measured by stress concentration factor, which is theoretically defined as the ratio between the maximum stresses in the neighborhood of a geometric discontinuity to nominal applied stress. Mathematically it is representing as, max t nom K σ σ = (1) 2.1. Closed-form Solution Redistribution of stress will take place if a small hole is made inside the elastic body subjected to the stress distribution. However, such a phenomenon of stress concentration is of a localized character, i.e., the effect of the hole is negligible at distances which are large in comparison with the dimension of the hole. Consider a problem of stress concentration in a rectangular plate with a small circular hole of radius ܽ and subjected to variable tensile stress of intensity ߪ௧ at one end to ߪ at another end varying linearly in the x direction as shown in Fig. 1. A small hole made in the middle of the plate will cause stress redistribution in the plate. Since the hole is small compared with the plate width, the change in the stress distribution will be localized in the neighborhood of the hole. Hence, the stresses acting far away from the hole, i.e., at a distance several times the hole diameter, will remain unchanged as in the plate with no hole. In other words, according to Saint-Venant’s principle, the change in stresses is negligible at large distances of radius b compared with the radius of the hole a. Fig. 1: Plate with a circular hole subjected to linearly variable stress field
- 4. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME 40 Considering plane stress with no body forces, the stresses with no hole are: ( ) ( ) ( ) 1 1 1 2 0 t b x b y xy h y h σ σ σ σ σ τ − = + + = = (2) We have 2 1 2 2 t b x b h y y h σ σφ σ σ −∂ = = + + ∂ (3) Integrating Equation (3) twice with respect to y. We get, 3 2 1 6 4 t b t b y y h σ σ σ σ φ − + = + (4) We have, siny r θ= in polar co-ordinates. Hence Equation (4) becomes, ( ) ( )3 2 1 3sin sin3 1 cos2 24 8 t b t b r r h σ σ σ σ φ θ θ θ − + = − + − (5) Hence the stresses in the problem of plate with no hole subjected to a uniaxial varying tensile stress field is ( ) 2 1 1 2 21 1 1 r r r r φ φ σ θ ∂ ∂ = + ∂ ∂ ( ) ( )1 sin3 sin cos2 4 4 4 t b t b t b r r h σ σ σ σ σ σ σ θ θ θ − + + ∴ = + + + (6a) ( ) 2 1 21 r θ φ σ ∂ = ∂ ( ) ( )1 3sin sin 3 cos2 4 4 4 t b t b t b r h θ σ σ σ σ σ σ σ θ θ θ − + + ∴ = − − + (6b) ( ) 1 1 1 r r r θ φ τ θ ∂∂ = − ∂ ∂ ( ) ( )1 cos cos3 sin 2 4 4 t b t b r r h θ σ σ σ σ τ θ θ θ − + ∴ = − − + (6c) Now suppose that a hole of radius a in the centre at the origin ‘o’ is drilled through the plate. A new system of stresses rσ , θσ and rθτ will be produced inside the plate. These stresses must satisfy the following boundary conditions: When r a= , 0r rθσ τ= = (7a) When r b= = ∞ , ( )1r rσ σ= , ( )1θ θσ σ= (Ragab (1998)) and ( )1r rθ θτ τ= (7b)
- 5. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME 41 These stresses can be derived from the biharmonic equation. 2 2 2 2 4 2 2 2 2 2 2 1 1 1 1 0 r r r r r r r r φ φ φ φ θ θ ∂ ∂ ∂ ∂ ∂ ∂ ∇ = + + + + = ∂ ∂ ∂ ∂ ∂ ∂ (8) Assume this unknown stress function φ to be of the same form as the stress function given by Equation (5), thus ( ) ( ) ( ) ( ) ( )1 2 3 4, sin sin3 cos2r f r f r f r f rφ θ θ θ θ= + + + (9) Substituting Equation (9) in Equation (8), we get, 2 22 2 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 22 2 3 3 4 4 4 2 2 2 2 2 2 1 91 1 1 1 9 1 sin sin3 41 1 1 4 1 cos2 d f df f d f df fd d d d dr r dr r dr r dr r dr r dr r dr r dr r d f df d f df fd d d d dr r dr dr r dr dr r dr r dr r dr r θ θ θ + − + − + + − + − + + + + + − + − = 0 Or 22 1 1 1 2 2 2 2 22 2 2 2 2 2 2 2 22 3 3 2 2 22 4 4 4 2 2 2 2 1 1 1 0 91 9 1 0 1 1 0 41 4 1 0 d f df fd d dr r dr r dr r dr r d f df fd d dr r dr r dr r dr r d f dfd d dr r dr dr r dr d f df fd d dr r dr r dr r dr r + − + − = + − + − = + + = + − + − = (10) The general solution of the above equations are, ( ) ( ) ( ) ( ) 33 1 1 2 4 3 55 7 2 6 83 2 2 3 9 10 11 12 2 4 15 4 13 14 162 ln ln ln C f r C r C r r C r r C C f r C r C r r r f r C C r C r C r r C f r C r C r C r = + + + = + + + = + + + = + + + (11) Hence the function ( ),rφ θ becomes. ( ) 3 3 53 5 7 1 2 4 6 83 2 2 2 4 15 9 10 11 12 13 14 162 ln sin sin3 + ln ln cos2 C C C C r C r r C r C r C r r r r C C C r C r C r r C r C r C r φ θ θ θ = + + + + + + + + + + + + + + (12)
- 6. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME 42 Using Equation (12) and the stress-Airy Stress Function relationships 2 2 2 1 1 r r r r φ φ σ θ ∂ ∂ = + ∂ ∂ (13a) 2 2 r θ φ σ ∂ = ∂ (13b) 1 r r r θ φ τ θ ∂ ∂ = − ∂ ∂ (13c) We get, ( ) 33 5 72 4 6 83 3 5 10 15 16 11 12 132 4 2 2 10 12 2 sin 6 4 sin3 6 4 + 2 1 2ln 2 cos2 r C C CC C r C r C r r r r r C C C C C r C r r r σ θ θ θ = − + − + + + + + + − + + (14a) ( ) 33 5 72 4 6 83 3 5 210 15 11 12 13 142 4 2 2 12 6 sin 6 20 sin3 6 + 2 3 2ln 2 12 cos2 C C CC C r C r C r r r r r C C C C r C C r r r θσ θ θ θ = + + + + + + − + + + + + + (14b) 33 5 72 4 6 83 3 5 2 15 16 13 14 4 2 2 6 12 2 cos 6 12 cos3 6 2 + 2 6 sin 2 r C C CC C r C r C r r r r r C C C C r r r θτ θ θ θ = − − + − − + − + + − − (14c) Where C’s are constants. Since for r → ∞ , rσ , θσ and rθτ must be finite. 8 12 14C 0C C∴ = = = (15) The remaining constants are determined from boundary conditions given by Equations (7). Here single terms are took separately from the Equations (14) and solved for the constants for ease of evaluation. Consider the terms having sinθ and cosθ as coefficient from Equations (14) and apply boundary conditions from Equations (7). We get, 32 42 4 2 2 0 CC C a a − + = (16a) 3 32 2 4 42 4 2 4 2 2 2 0 2 0 C CC C C C a a a a − − + = ⇒ − + = (16b)
- 7. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME 43 32 42 4 2 2 sin CC C r b b θ − + sin 4 t b r h σ σ θ − = 32 42 4 2 2 4 t bCC C b b h σ σ− ⇒ − + = (16c) 32 42 4 2 6 sin CC C r b b θ + + 3 sin 4 t b r h σ σ θ − = 32 42 4 2 6 3 4 t bCC C b b h σ σ− ⇒ + + = (16d) 32 42 4 2 2 cos CC C r b b θ − − + cos 4 t b r h σ σ θ − = − 32 42 4 2 2 4 t bCC C b b h σ σ− ⇒ − + = (16e) By taking 0 a b = and solving Equations (16) we get, 2 0C = , 4 3 8 t b C a h σ σ− = , 4 8 t b C h σ σ− = (17) Similarly by considering the terms sin3θ and cos3θ , 1 and cos2θ and sin 2θ as coefficients from Equations (14) and applying boundary conditions from Equations (7) we get, 4 5 8 t b C a h σ σ− = , 6 24 t b C h σ σ− = − , 6 7 12 t b C a h σ σ− = − (18) 2 10 4 t b C a σ σ+ = − , 11 8 t b C σ σ+ = (19) 13 8 t b C σ σ+ = − , 4 15 8 t b C a σ σ+ = − , 2 16 4 t b C a σ σ+ = (20) Respectively. Substitute Equations (17), (18), (19) and (20) in Equations (14) and by solving we get the stress distribution around circular hole subjected to the linearly variable stress field in polar co-ordinate system. 4 6 4 4 6 4 2 4 2 2 4 2 4 5 = 1 sin 1 sin3 4 4 3 4 1 1 cos2 4 4 t b t b r t b t b a a a r r h r h r r a a a r r r σ σ σ σ σ θ θ σ σ σ σ θ − − − + + − ∴ + + + − + + − (21a) 4 6 4 4 6 4 2 4 2 4 4 = 3 sin 1 sin3 4 4 3 1 1 cos2 4 4 t b t b t b t b a a a r r h r h r r a a r r θ σ σ σ σ σ θ θ σ σ σ σ θ − − + − + − + + + + − + (21b)
- 8. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME 44 4 6 4 4 6 4 4 2 4 2 4 3 = 1 cos 1 cos3 4 4 3 2 1 sin 2 4 t b t b r t b a a a r r h r h r r a a r r θ σ σ σ σ τ θ θ σ σ θ − − − − + − + + − − + (21c) 2.2. Localized Stress Concentration Factor From the present work, nominal stresses cannot be obtained that are used to find the stress concentration factor given by the Equation (1). So a new term can be used to calculate the stress concentration factor and it is called as localized stress concentration factor. Localized SCF is defined as the ratio of stress at a point with hole to the stress at the same point without hole. Mathematically it can be expressed as: Stress With Hole at a Point Localized Stress Concentration Factor Stress Without Hole at a Point = (22) 3. VERIFICATION OF THE PROPOSED CLOSED-FORM SOLUTION Conditions of the stress concentration around circular hole subjected to uniform stress are applied to the new Equations (21) for verification of the present work. Uniform stress case is the well-known method. Nominal condition for uniform stress is t bσ σ= . Stress concentration equations around a circular hole subjected to uniform stress given by Kirsch’s problem are 2 4 2 0 0 2 4 2 3 4 1 1 cos 2 2 2 r a a a r r r σ σ σ θ = − + + − (23a) 2 4 0 0 2 4 3 1 1 cos 2 2 2 a a r r θ σ σ σ θ = + − + (23b) 4 2 0 4 2 3 2 1 sin 2 2 r a a r r θ σ τ θ = − − + (23c) Apply the condition t b uσ σ σ= = in the Equations (24) we get, 2 4 2 2 4 2 3 4 1 1 cos 2 2 2 u u r a a a r r r σ σ σ θ = − + + − (24a) 2 4 2 4 3 1 1 cos 2 2 2 u ua a r r θ σ σ σ θ = + − + (24b) 4 2 4 2 3 2 1 sin 2 2 u r a a r r θ σ τ θ = − − + (24c)
- 9. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME 45 By comparing Equations (23) and (24), it can be said that the present method gives well agreement with the Kirsch’s problem. Hence present method can be used to determine the stress concentration around a circular hole subjected to linearly varying stress. 4. NUMERICAL EXAMPLE Consider an example to solve the Equations (21) to observe the behavior of stresses around a hole subjected to the linearly varying stress field and to compare the results obtained by the equations with ANSYS by using following parameters: Plate size = 300݉݉ ൈ 300݉݉ Height, h = 300݉݉ Hole radius, ܽ ൌ 25݉݉ Normal Stress, ߪ௧ ൌ 100 ܰ ݉݉ଶ⁄ Normal Stress, ߪ ൌ 30 ܰ ݉݉ଶ⁄ Young’s Modulus, ܧ ൌ 200 ൈ 10ଷ ܰ ݉݉ଶ⁄ Poisson’s Ratio, ߥ ൌ 0.3 Same values are taken for the model generation in ANSYS. PLANE82 element is used for meshing. The parameters prescribed above are substituted in the Equations (21) for the analysis. Half- symmetry of the model shown in Fig. 1 is considered for the evaluation. At the inner side of the hole there is no applied stress is acting, hence ߪ is zero at ݎ ൌ ܽ (Equation (7a)) and it is minimum along ߠ ൌ 90° and ߠ ൌ െ90° as there is no circumferential stress is acting on the plate. Radial stress approaches to the applied stress at the edge of the plate as ݎ ܽ⁄ increases as shown in Fig. 2. Fig. 2: Radial Stress (ߪ) vs. ݎ ܽ⁄ at different Angles (ߠ). 0 10 20 30 40 50 60 70 0 1 2 3 4 5 6 7 RadialStress,σr(N/mm2) r/a θ=-90 θ=-60 θ=-30 θ=0 θ=30 θ=60 θ=90
- 10. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME 46 Fig. 3: Circumferential Stress (ߪఏ) vs. ݎ ܽ⁄ at different Angles (ߠ) Circumferential stress (ߪఏ) for ߠ ൌ 90° and ߠ ൌ െ90° at the edge of the hole is maximum because force flow lines tend to modify their path around hole and it concentrate around edge of the hole perpendicular to the load direction, as the radius increases stress decreases and at the end of the plate it reaches the applied stress as shown in the Fig. 3. Shear stress (߬ఏ) is the resultant of the radial stress (ߪ) and the circumferential stress (ߪఏ), hence it has the nature of both the normal stresses as shown in the Fig. 4. Fig. 4: Shear Stress (߬ఏ) vs. ݎ ܽ⁄ at different Angles (ߠ) -100 -50 0 50 100 150 200 250 0 1 2 3 4 5 6 7 CircumferentialStress,σθ(N/mm2) r/a θ=-90 θ=-60 θ=-30 θ=0 θ=30 θ=60 θ=90 -50 -40 -30 -20 -10 0 10 20 30 40 50 0 1 2 3 4 5 6 7 ShearStress,τrθ(N/mm2) r/a θ=-90 θ=-60 θ=-30 θ=0 θ=30 θ=60 θ=90
- 11. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME 47 Fig. 5: Localized SCF vs. ݎ ܽ⁄ at different Angles (ߠ) Stress concentration factor is maximum at the hole edge (ݎ ܽ⁄ ൌ 1) because the stress flow lines tend to modify their paths around the discontinuities (hole), this increases the stress around hole and stress concentration factor. As the radius increases, stress reduces; this decreases the stress concentration factor. At the plate edge stress is nearly equal to the applied stress hence the stress concentration factor is around one as shown in Fig. 5. Fig. 6 shows the comparison of present work with the ANSYS of normal stress (ߪ௫) versus angle at radiusݎ ൌ 25. Comparison shows that the results are well agreed. Here only one comparison is presented, but the comparison of remaining stresses for different radii is also give the close results. Fig. 6: Comparison of Normal Stress (ߪ௫) vs. Angle (ߠ) with ANSYS -0.5 0 0.5 1 1.5 2 2.5 3 3.5 0 1 2 3 4 5 6 7 LocalizedStress ConcentrationFactor r/a θ=-90° θ=-60° θ=-30° θ=30° θ=60° θ=90° -50 0 50 100 150 200 250 -180 -120 -60 0 60 120 180 NormalStress,σx(N/mm2) Angle, θ (Degrees) Present Work for r=25 FEA for r=25
- 12. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME 48 5. CONCLUSIONS New equations for stress distribution around circular hole in an isotropic infinite plate in a linearly varying stress field are formulated using closed-form solution and it is extended from the Kirsch’s problem for ‘Stress-concentration due to a circular hole in a stressed plate’. Newly found equations (21) show that it satisfies the boundary conditions from Fig. 2 to Fig. 4. The maximum stress concentration is found at the edge of the hole at an angle of 90° from the load direction. Localized stress concentration factor is calculated and it is observed that the localized SCF is maximum at the hole and it decreases as radius increases. At the end of the plate localized SCF is equal to 1 that means stresses are converged. The results obtained from new equations are compared with the ANSYS and it is in closed agreement. REFERENCES [1] Pilkey W. D., Peterson’s stress concentration factors Second edition, John Wily and Sons, United States, America, 1997. [2] Young W. C., Budynas R. G., Roark’s formulas for stress and strain Seventh edition, McGraw-Hill, United States, America, 2002. [3] Pilkey W. D., Formulas for stress, strain, and structural matrices Second edition, John Wily and Sons, Hoboken, New Jersey, 2005. [4] Shivakumar K. N., Newman J. C., Jr., Stress concentrations for straight-shank and countersunk holes in plates subjected to tension, bending, and pin loading, NASA TP-3192, 1992. [5] Patnaik S. N., Hopkins D. A., Stress formulation in three-dimensional elasticity, NASA TP- 210515, 2001. [6] Kotousov, Wang C. H., Three-dimensional stress constraint in an elastic plate with a notch, International Journal of Solids and Structures, 39, 4311-4326, 2002. [7] Yang Z., Kim C. B., Cho C., Beom H. G., The concentration of stress and strain in finite thickness elastic plate containing a circular hole, International Journal of Solids and Structures, 45, 713-731, 2008. [8] Singh S., Theory of elasticity Fourth edition, Khanna Publishers, Delhi, India, 2007. [9] Xu Z., Applied Elasticity New Age International Publishers, New Delhi, India, 2008. [10] Sadd M. H., Elasticity theory, applications, and numerics Elsevier, United States, America, 2009. [11] Saada A. S., Elasticity theory and applications Second edition, J. Ross Publishers, U. S. A., 2009. [12] Batista M., On the stress concentration around a hole in an infinite plate subject to a uniform load at infinity, International Journal of Mechanical Sciences, 53, 254-261, 2011. [13] Mohammadi M., John R. Dryden, Liying Jiang, Stress concentration around a hole in a radially inhomogeneous plate, International Journal of Solids and Structures, 48, 483-491, 2011. [14] Sharma D. S., Stress distribution around polygonal holes, International Journal of Mechanical Sciences, 65, 115-124, 2012. [15] Ragab A. R., Bayoumi S. E., Engineering solid mechanics-fundamentals and applications First edition, CRC Publishers, 1998. [16] Akash.D.A, Anand.A, G.V.Gnanendra Reddy and Sudev.L.J, “Determination of Stress Intensity Factor for a Crack Emanating from a Hole in a Pressurized Cylinder using Displacement Extrapolation Method”, International Journal of Mechanical Engineering & Technology (IJMET), Volume 4, Issue 2, 2013, pp. 373 - 382, ISSN Print: 0976 – 6340, ISSN Online: 0976 – 6359.

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