Microsoft Word - 228-1053 vetted IIUM Engineering Journal, Vol. 12, No. 6, 2011: Special Issue in Science and Ethics Jain 141 THE REDUCTION OF STRESS CONCENTRATION IN A UNI-AXIALLY LOADED INFINITE WIDTH RECTANGULAR ISOTROPIC/ORTHOTROPIC PLATE WITH CENTRAL CIRCULAR HOLE BY COAXIAL AUXILIARY HOLES JAIN N.K. Department of Applied Mechanics, National Institute of Technology, Raipur (C.G.)-492010, India. nkjmanit@rediffmail.com, nkjnitrr@rediffmail.com ABSTRACT: A comprehensive plane stress finite element study is made for reduction of stress concentration factor (SCF) in a uni-axially loaded infinite width rectangular isotropic/orthotropic plate with central circular hole. The finite element formulation is carried out by the ANSYS package. With the help of present work, stress concentration can be reduced up to 24.4 % in an isotropic and 31 % in an orthotropic plate by introducing four coaxial auxiliary holes on either side of main hole. The study reveals that the introduction of these holes helps to smooth flow of the tensile stresses passes through the main hole and results, a reduction in stress concentration factor. With such reduction in maximum stress levels, the improvement in fatigue life of a component can be significant. ABSTRAK: Satu kajian komprehensif tentang unsur terhingga telah dibuat keatas factor pemusatan tegangan bagi plat isotropic/ortotropik segi empat tepat lebar infinit yang dimuat secara uni-aksial berlubang pusat bulat. Rumusan unsur finit telah dibuat menggunakan pakej ANSYS. Dengan kajian masa kini, pemusatan tegangan dapat dikurangkan sehingga 24.4 % dalam plat isotropik and 31 % dalam plat ortotropik dengan membuat empat lubang auksiliari sepaksi di kedua belah lubang utama. Kajian ini menunjukkan penggunaan empat lubang ini telah melicinkan aliran tegasan tegangan melalui lubang utama. Ini mengakibatkan pengurangan factor pemusatan tegangan. Dengan pengurang tahap tegangan maksimum ini peningkatan hayat kelesuan komponen boleh menjadi signifikan. KEYWORDS: stress concentration factor; finite element method; plate; composite 1. INTRODUCTION An infinite width rectangular isotropic/orthotropic plate with central circular hole have found widespread applications in various fields of engineering such as aerospace, marine, automobile and mechanical. Stress concentration arises from any abrupt change in geometry of plate under loading. As a result, stress distribution is not uniform throughout the cross section. Failures such as fatigue cracking and plastic deformation frequently occur at points of stress concentration. Hence, for the design of a plate with central circular hole, stress concentration factor plays an important role and accurate knowledge of stresses and stress concentration factor at the edges of hole under in plane or transverse loading are required. Analytical solutions are available in the literature for prediction of IIUM Engineering Journal, Vol. 12, No. 6, 2011: Special Issue in Science and Ethics Jain 142 SCF in different types of abrupt changes in shape, but there are limited methods for reduction in SCF. Meguid [1] presented a technique for reducing stress concentration factor by introducing defence hole system in a uni-axially loaded plate with two coaxial holes by finite element method, where reduction in SCF ranging from 7.5 to 11 % maximum have been achieved. But to achieve the maximum possible reduction, these auxiliary holes have to be introducing very near to the original holes, which is not practically possible in all cases. Giare et al. [2] presented a method for the reduction of stress concentration around the main hole in an isotropic plate by using a ring of composite material around the main hole, method is not suitable for all designing conditions and also no techniques were presented for reduction in SCF for orthotropic plate. Providakis and Sotiropoulos [3] have developed a boundary element approach to the reduction in stress concentration factor in visco-plastic plates by multiple holes. Tenchev et al. [4] presented a design procedure for reducing stress concentration around holes in laminated composite by increasing the thickness in this area. Younis [5] investigated by reflected photoelasticity method that the assembly stress contributes to reducing stresses around the holes in a plate. Mahiou and Bekaou [6] studied for local stress concentration and for the prediction of tensile failure in unidirectional composites. Toubal et al. [7] studied experimentally for stress concentration in a circular hole in composite plate. Wu and Mu [8] analysed stress concentrations for isotropic/orthotropic plates and cylinders with a circular hole by finite element method. Peterson [20] has developed good theory and charts on the basis of mathematical analysis and presented excellent mythology in graphical form for evaluation of stress concentration factors, but no techniques were presented for reduction of SCF. This paper is concerned with the reduction of SCF in isotropic and orthotropic infinite width rectangular plates with central circular hole under in-plane static loading. In general stress concentration factor can be reduced up to 12-18 % by two coaxial auxiliary holes on either side of the main hole, to achieve the maximum possible reduction; these auxiliary holes are to be introducing very near to the original hole, which is not practically possible. It has been seen that the reduction in SCF at the edges of main hole become directly proportional to diameter of auxiliary holes, increasing the diameter of auxiliary holes results in reduction of normal stress (σx) at the edges of main hole, but this is also accompanied by an increase of σx at the edges of auxiliary holes. At a certain diameter of auxiliary holes, σx at the edges of auxiliary holes attained more than σx at the edges of main hole i.e. maximum value of σx is shifted from edges of main hole to edges of auxiliary holes. This problem can avoid by introducing two more coaxial holes on either side of first two coaxial holes and the diameter of first two auxiliary holes can increase more. This results more reduction in stress concentration. The work has been done firstly for isotropic plates, and then study is further expanded for orthotropic plates. This work is most appreciated in aerospace and marine industry where the introduction of holes will not only reduce the stress concentration around main hole, but also reduce the weight of the components. This study will also provide guidelines technique to designer for reduction of stress concentration up to maximum level. 2. DESCRIPTION OF PROBLEM IIUM Engineering Journal, Vol. 12, No. 6, 2011: Special Issue in Science and Ethics Jain 143 To study the influence of auxiliary holes upon the stress concentration around the main hole, three models A, B and C are described in Fig. 1, where model A is basic model (a plate with central circular hole of diameter D), model B is modification of model A (plate with two coaxial auxiliary holes of diameter D1, numbered as auxiliary hole 1) and model C is modification of model B (plate with four coaxial auxiliary holes) i.e. two more coaxial auxiliary holes of diameter D2 (numbered as auxiliary hole 2) are created in model B. The centre distance between main hole and auxiliary hole 1 is taken as X and the centre distance between auxiliary hole1 and auxiliary hole 2 is taken as 0.8 X. All the details of dimensions are described in Fig 1. A plate under in-plane static uniformly distributed load of P, is analyzed by finite element method for D/A ratio of 0.1, where A is plate width. The dimensions are selected such a way that it can be considered as infinite width plate. (a) Model A (Basic plate with central circular hole of diameter D ). (b) Model B (c) Model C Fig. 1: Details of three models analysed in this study. 3. FINITE ELEMENT ANALYSIS IIUM Engineering Journal, Vol. 12, No. 6, 2011: Special Issue in Science and Ethics Jain 144 An eight nodded quadrilateral element with element length of 1 mm was selected and used through out the study. Each node has two degrees of freedom, making a total 16 degrees of freedom per element. In order to construct the graphical image of the geometries of the different models of plate examined using the ANSYS (Advanced Engineering Simulation), it was necessary to input the basic geometric elements such as points, lines and arcs. Due to the symmetric nature of different models investigated, it was only necessary to discretize and analyse the one quarter of the problem of each model for finite element analysis. Main task in finite element analysis is selection of suitable element type. Number of checks and convergence test are made for selection of suitable element type from different available element. Results were then displayed by using post processor of ANSYS programme. 4. RESULTS AND DISCUSSION Numerical results are presented for isotropic and orthotropic plates with central circular hole under in-plane static loading. The SCF is calculated on nominal area in all cases. The reduction in SCF is achieved firstly for isotropic plates and then work is extended for orthotropic plates. 4.1 Isotropic Plate In the basic model (model A) maximum stress concentration is always occurred at the edges of main hole. At the edges of main hole; normal stress in X- direction (σx) is placed 3.003 times of P and SCF (Calculate at nominal area) is appeared as 2.70, which is very close to theoretical value of 2.72. Stress concentration at the edge A is reduced by two coaxial auxiliary holes of diameter D1 in model B. The analysis for reduction in stress concentration at the edges of main hole has been done for different centre distances of main hole and auxiliary hole 1. It was very necessary to analyse, the influence of auxiliary holes on stress concentration around main hole as well as around of auxiliary holes. In model B, edges of main hole (edge A) and edges of auxiliary hole 1 (edge B) are zone of stress concentration. Table 1 shows the effect of D1/D on σx/P at the edges A and B for X/D=1.2, 1.5, 1.8 and 2.0 respectively in model B. For all cases of X/D, results illustrate clearly that increasing D1/D results in continuously reduction of stress concentration at the edges of main hole, but this is also accompanied by an increase in stress concentration at the edges of auxiliary holes. The following observations can be made from Table 1. For X/D=1.2; increasing D1/D from 0.4 to 0.85 results in reduction of σx/P at edge A from 2.895 to 2.451, this is also accompanied by an increase of σx/P at edge B from 2.058 to 2.435. But when D1/D increases from 0.85 to 0.86, σx/P is increased from 2.435 to 2.441 at edge B and attained more than σx/P at edge A i.e. maximum value of σx is shifted from edge A to edge B. The critical value of D1/D is 0.85 for this case. Table 1: Variation of σx/P at the edges of main hole and auxiliary hole 1 with D1/D for different values of X/D in model B. IIUM Engineering Journal, Vol. 12, No. 6, 2011: Special Issue in Science and Ethics Jain 145 X/D=1.2 X/D=1.5 X/D=1.8 X/D=2.0 D1/D Edge A Edge B D1/D Edge A Edge B D1/D Edge A Edge B D1/D Edge A Edge B 0.40 2.895 2.058 0.40 2.892 2.320 0.40 2.904 2.503 0.40 2.912 2.598 0.60 2.747 2.215 0.60 2.751 2.439 0.60 2.779 2.581 0.60 2.800 2.645 0.80 2.521 2.407 0.80 2.551 2.507 0.70 2.698 2.589 0.70 2.728 2.660 0.84 2.466 2.436 0.82 2.530 2.523 0.74 2.663 2.606 0.76 2.681 2.666 0.85 2.451 2.435 0.84 2.505 2.537 0.78 2.626 2.600 0.78 2.664 2.661 0.86 2.437 2.441 ----- ----- ----- 0.80 2.607 2.609 0.80 2.646 2.661 If D1/D increases beyond critical value, maximum stress concentration occurs at edge B and increases with increase of D1/D. For X/D=1.5; increasing D1/D from 0.4 to 0.82 results in reduction of σx/P at edge A from 2.892 to 2.530, this is also accompanied by an increase of σx/P at edge B from 2.320 to 2.523. But when D1/D increases from 0.82 to 0.84, σx/P is increased from 2.523 to 2.537 at edge B and attained more than σx/P at edge A i.e. maximum value of σx is shifted from edge A to edge B. The critical value of D1/D is 0.82 for this case. For X/D=1.8; increasing D1/D from 0.4 to 0.78 results in reduction of σx/P at edge A from 2.904 to 2.626, this is also accompanied by an increase of σx/P at edge B from 2.503 to 2.600. But when D1/D increases from 0.78 to 0.80, σx/P is increased from 2.600 to 2.609 at edge B and attained more than σx/P at edge A i.e. maximum value of σx is shifted from edge A to edge B. The critical value of D1/D is 0.78 for this case. For X/D=2.0; increasing D1/D from 0.4 to 0.78 results in reduction of σx/P at edge A from 2.912 to 2.664, this is also accompanied by an increase of σx/P at edge B from 2.598 to 2.661. But when D1/D increases from 0.78 to 0.80, σx/P at edge B is attained more than at edge A i.e. maximum value of σx is shifted from edge A to edge B. The critical value of D1/D is 0.78 for this case. Table 2 shows the effect of D1/D and D2/D on σx/P at the edges A, B and C for X/D=1.2, 1.5, 1.8 and 2.0 respectively in model C. For all cases of X/D, Model C is prepared by creating two more coaxial auxiliary holes in model B (with critical value of D1/D). The following observations can be made from Table 2. For X/D=1.2; increasing D2/D from 0.4 to 0.74, while maintaining the D1/D unchanged as 0.85 (critical value in model B), results in reduction of σx/P at edge A from 2.434 to 2.359, this is also accompanied by a decrease of σx/P at edge B from 2.362 to 2.142 and an increase of σx/P at edge C from 1.943 to 2.343. But when D2/D increases beyond 0.74, σx/P at edge C is attained more than σx/P at edge A i.e. maximum value of σx is shifted from edge A to edge C. Again, increasing D1/D from 0.85 to 0.92, while maintaining the D2/D unchanged as 0.74, results in reduction of σx/P at edge A from 2.359 to 2.271, this is also accompanied by an increase of σx/P at edge B from 2.142 to 2.241 and a decrease of σx/P at edge C from 2.343 to 2.255. For X/D=1.5; increasing D2/D from 0.4 to 0.7, while maintaining the D1/D unchanged as 0.82 (critical value in model B), results in reduction of σx/P at edge A from 2.507 to 2.454, this is also accompanied by a decrease of σx/P at edge B from 2.440 to 2.226 and an increase of σx/P at edge C from 2.226 to 2.448. But when D2/D increases beyond 0.7, σx/P IIUM Engineering Journal, Vol. 12, No. 6, 2011: Special Issue in Science and Ethics Jain 146 at edge C is attained more than σx/P at edge A i.e. maximum value of σx is shifted from edge A to edge C. Again, increasing D1/D from 0.82 to 0.90, while maintaining the D2/D unchanged as 0.7, results in reduction of σx/P at edge A from 2.454 to 2.369, this is also accompanied by an increase of σx/P at edge B from 2.226 to 2.330 and a decrease of σx/P at edge C from 2.448 to 2.365. For X/D=1.8; increasing D2/D from 0.4 to 0.6, while maintaining the D1/D unchanged as 0.78 (critical value in model B), results in reduction of σx/P at edge A from 2.603 to 2.574, this is also accompanied by a decrease of σx/P at edge B from 2.520 to 2.421 and an increase of σx/P at edge C from 2.461 to 2.563. But when D2/D increases beyond 0.6, σx/P at edge C is attained more than σx/P at edge A i.e. maximum value of σx is shifted from edge A to edge C. Again, increasing D1/D from 0.78 to 0.88, while maintaining the D2/D unchanged as 0.6, results in reduction of σx/P at edge A from 2.574 to 2.482, this is also accompanied by an increase of σx/P at edge B from 2.421 to 2.474 and a decrease of σx/P at edge C from 2.563 to 2.457. Table 2: Variation of σx/P at the edges of main hole, auxiliary hole 1 and auxiliary hole 2 with D1/D and D2/D for different values of X/D in model C. X/D D1/D D2/D Edge A Edge B Edge C 1.2 0.85 0.40 2.434 2.362 1.943 0.85 0.60 2.400 2.257 2.181 0.85 0.70 2.371 2.181 2.281 0.85 0.74 2.359 2.142 2.343 0.88 0.74 2.324 2.195 2.297 0.90 0.74 2.298 2.205 2.282 0.92 0.74 2.271 2.241 2.255 1.5 0.82 0.40 2.507 2.440 2.226 0.82 0.60 2.476 2.333 2.400 0.82 0.70 2.454 2.226 2.448 0.84 0.70 2.433 2.279 2.415 0.86 0.70 2.413 2.308 2.411 0.90 0.70 2.369 2.330 2.365 1.8 0.78 0.40 2.603 2.520 2.461 0.78 0.60 2.574 2.421 2.563 0.80 0.60 2.557 2.431 2.537 0.82 0.60 2.539 2.440 2.518 0.86 0.60 2.502 2.467 2.474 0.88 0.60 2.482 2.474 2.457 2.0 0.78 0.40 2.642 2.587 2.585 0.78 0.50 2.631 2.539 2.583 0.78 0.56 2.623 2.514 2.616 IIUM Engineering Journal, Vol. 12, No. 6, 2011: Special Issue in Science and Ethics Jain 147 0.80 0.56 2.606 2.521 2.599 0.82 0.56 2.590 2.532 2.579 0.84 0.56 2.574 2.539 2.542 0.86 0.56 2.556 2.546 2.536 For X/D=2.0; increasing D2/D from 0.4 to 0.56, while maintaining the D1/D unchanged as 0.78 (critical value in model B), results in reduction of σx/P at edge A from 2.642 to 2.623, this is also accompanied by a decrease of σx/P at edge B from 2.587 to 2.514 and an increase of σx/P at edge C from 2.585 to 2.616. But when D2/D increases beyond 0.56, σx/P at edge C is attained more than σx/P at edge A i.e. maximum value of σx is shifted from edge A to edge C. Again, increasing D1/D from 0.78 to 0.86, while maintaining the D2/D unchanged as 0.56, results in reduction of σx/P at edge A from 2.623 to 2.556, this is also accompanied by an increase of σx/P at edge B from 2.514 to 2.546 and a decrease of σx/P at edge C from 2.616 to 2.536. Figure 2 illustrates the variation of SCF at the edges of main hole (edge A) in model B and C with D1/D for different values of X/D. These are maximum SCF in plate, attaining at the edges of main hole. Fig. 2: Effects of D1/D on SCF at the edges of main hole (edge A) in Model B and C. In model B, SCF is reduced up to 2.21, 2.28, 2.36 and 2.4 for X/D = 1.2, 1.5, 1.8 and 2.0 respectively. Where, in model C, SCF is reduced up to 2.04, 2.13, 2.23 and 2.3 for X/D = 1.2, 1.5, 1.8 and 2.0 respectively. Figure 3 shows the % reduction in SCF at the edges of main hole in model B and C with D1/D for different values of X/D. In model B; 18.39, 15.75, 12.57 and 11.31 % reduction in SCF is achieved for X/D = 1.2, 1.5, 1.8 and 2.0 respectively. IIUM Engineering Journal, Vol. 12, No. 6, 2011: Special Issue in Science and Ethics Jain 148 Fig. 3: Effects of D1/D on % reduction in SCF at the edges of main hole (edge A) in Model B and C. Where, in model C; 24.39, 21.13, 17.36 and 14.90 % reduction in SCF is achieved for X/D = 1.2, 1.5, 1.8 and 2.0 respectively. Results indicate clearly that maximum possible % reduction in SCF in model C is almost 6.0 % greater than model B for all values of X/D. It has been observed that maximum possible % reduction in SCF decreases with increase of X/D in model B and C both. 4.2 Orthotropic Plate Thirteen different composite materials are selected for analysis of influence of auxiliary holes in model B and C, upon reduction of stress concentration around main hole in an orthotropic plate. SCF in model A and % reduction in SCF in model B and C for X/D=1.2 and 1.5, are listed in Table 3. The elastic properties of different composite materials used in study are also listed in Table 3. Table 3: SCF in model A and achieved % reduction in SCF in model B and C for orthotropic plates. Material Type E1 E2 G12 µ SCF % Reduction in SCF X/D=1.2 X/D=1.5 Model B Model C Model B Model C E-glass/epoxy 39 8.6 3.8 0.28 3.816 21.13 29.60 19.36 27.15 S-glass/epoxy 43 8.9 4.5 0.27 3.789 21.20 29.58 19.40 27.10 Woven glass/epoxy 29.7 29.7 5.3 0.17 3.169 19.83 27.11 17.61 24.23 Kelvar/epoxy 87 5.5 2.2 0.34 5.017 18.70 29.42 17.73 27.95 Carbon/epoxy 142 10.3 7.2 0.27 4.488 20.81 30.55 19.52 28.60 Carbon/PEEK 131 8.7 5 0.28 4.696 20.11 30.27 18.96 28.51 Carbon/epoxy 177 10.8 7.6 0.27 4.637 20.46 30.48 19.27 28.66 Carbon/polyimide 216 5 4.5 0.25 5.384 17.97 29.15 17.18 27.95 IIUM Engineering Journal, Vol. 12, No. 6, 2011: Special Issue in Science and Ethics Jain 149 Graphite/epoxy 294 6.4 4.9 0.23 5.585 16.89 28.28 16.13 27.26 Boron/epoxy 201 21.7 5.4 0.17 4.889 18.80 29.36 17.77 27.83 Boron/aluminum 235 137 47 0.3 3.169 20.08 27.20 17.82 24.28 SiC/Al 204 118 41 0.27 3.175 20.10 27.23 17.84 24.31 SiC/ceramic 121 112 44 0.2 2.783 18.73 24.95 16.17 21.74 The reduction in SCF is calculated with critical value of D1/D and D2/D of isotropic plate. Following observations can be made from these results. For X/D=1.2; in case of model B, reduction in SCF is between the range of 17 to 21 % for all materials, maximum in case of S-glass/epoxy with 21.20 % and minimum in case of Graphite/epoxy with 16.89 %. In case of model C, reduction in SCF is between the range of 25 to 31 % for all materials, maximum in case of Carbon/epoxy (AS4/3501-6) with 30.55 % and minimum in case of Sic/ceramic with 24.95 %. For X/D=1.5; in case of model B, reduction in SCF is between the range of 16 to 20 % for all materials, maximum in case of Carbon/epoxy (AS4/3501-6) with 19.52 % and minimum in case of Graphite/epoxy with 16.13 %. In case of model C, reduction in SCF is between the range of 22 to 29 % for all materials, maximum in case of Carbon/epoxy (IM6SC1081) with 28.66 % and minimum in case of Sic/ceramic with 21.74 %. 5. CONCLUSION The reduction in SCF depends on diameter and position of auxiliary holes in isotropic or orthotropic plates, two parameters X/D and D1/D in model B and three parameters X/D, D1/D and D2/D in model C influence the reduction in SCF. For an isotropic plate; in case of model C, maximum reduction in SCF is achieved for X/D=1.2 as 24.39 %, where, in case of model B, maximum reduction in SCF is achieved for X/D=1.2 as 18.39 % only. For a convenient design, for X/D=1.5, up to 21.13 % reduction in SCF can be achieve in model C, but in case of model B, it can be only 15.75 %. In case of orthotropic plate; maximum reduction in SCF is between the range of 25.0 to 31.0 % for different materials in model C, where in model B, maximum reduction in SCF is between the range of 17.0 to 21.0 % for different materials. It has been observed that maximum possible % reduction in SCF decreases with increase of X/D in model B and C both, for isotropic and orthotropic plates. 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