Studying Decreased Methods of Stress Concentration around Holes and Openings of Plate and Shell Structures Body

International Research Journal of Applied and Basic Sciences 01 Available online at www.irjabs.com ISSN 51-88X / Vol, 5 (8): 1008-1015 Science Explorer Publications Studying Decreased Methods of Stress Concentration around Holes and Openings of Plate and Shell Structures Body Mahdi Mirzagoltabarroshan 1 1. Sama technical and vocational training college, Islamic Azad University, Sari Branch, Sari, Iran Corresponding author s email: m.goltabar@yahoo.com ABSTRACT: Phenomenon of stress concentration and creating it in shells is one of the scientifically and practically critical issues in structural and mechanical engineering. Submarines, vessels, wings and fuselages of airplanes, missiles and rockets, steam boilers and pressure tanks, pipes fittings in pipelines of fluids like gas, steam, water and wastewater are all examples of this phenomenon application in industry. Generally, creating an opening in a cylindrical shell or junction of two shells or even in plates, results in turbulence in stress field and creates stress concentration. The stress may be increased up to many times more than the stress created in body of shell structure (sometimes more than a hundred times) which is important from the standpoint of analysis and design. In this research, while reviewing previous studies and classifying the problems, we present the ruling equation in mixed form by partial derivative in fourth order; then, we present numerical solving approach with finite difference model as well as numerical solving approach with finite element model (with Ansys software). Then, we investigate trends of coefficient changes of stress concentration in plates and shells for different openings diameters and coefficient of curvatures (a dimensionless coefficient containing shell size including radius, thickness, and material as well as opening size and stress concentration depends on it) in single-hole or more mode and different types of openings such as circular, oval, square and rectangular influenced by different loadings to identify the effects of hole and opening on stress concentration phenomenon and its coefficient changes in plates and the shells. In addition, decrease of stress concentration is an important problem in theoretical, industrial, and applicable dimensions. Then, we offer some economical and practical solutions based on analysis and experimental tests. In these structures, one can decrease stress concentration using provisions like change of material, thickness (uniform and step form), using shell of different thickness (according to changes of stress diagram), using an amplification ring or loop around the opening and also inside of it. In this research, we investigate the effects of these parameters and show the results in diagrams and tables. Considering different methods for decrease of stress concentration (regarding limitations of laboratory and economic parameters), we introduce the optimum method for decreasing stress concentration of each shell with every type of openings or holes. Key words: stress concentration, decrease methods, shell structure, opening, theoretical and numerical analysis. INTRODUCTION Due tosudden increase of stress in a limited area, phenomenon of creating stress concentration in shells is one of the important issues in structures design and analysis of structure and mechanical engineering. Generally, an opening in a shell created for joints or exploitation, forming turbulence in stress fields and can increase the stress many times (sometimes more than one hundred times) in the body of shell structure and causes in damages or fractures in the mentioned location. Some industrial applications are aerospace structures (missiles and airplanes bodies), submarines bodies, steam boilers, and pressure tanks, pipes and. History & Related Literature In most materials, material strength depends significantly on a fracture of opening, which shows the effect of the openings with coefficient. Dimensionless coefficient of curvature is one of the most important

Intl. Res. J. Appl. Basic. Sci. Vol., 5 (8), 1008-1015, 01 1 1v a Rh coefficients for expressing magnitude of hole and cylinder in relation to each other and has a direct relationship with curvature parameter. This coefficient defines as: (1) where a, ν, R and h are radius of circular hole, Poisson s ratio, radius and thickness of the shell, respectively. The shells of different dimensions and sizes but of the same curvatures, opening rate and a certain loading have similar features. For a shell of two openings, stress changes depend on curvature rate as well as distance x coefficient that defines as follows: () where x and a are distance between two holes and radius of a circle, respectively. Stress Concentration Factor (S.C.F.) defines to investigate the stress in the shell with hole and opening and is shown with K as well as α. This factor obtained from dividing the stress in the shell with opening by the stress in the same shell without opening with the same loading; depending on the type of loading and O geometrical conditions, value of this factor is different. The main stress in axial tension O loading defines as h where N 0 and h are internal tensile force and thickness of the cylinder, respectively. Maximum Stress Concentration Factor expresses as follows: () N O h O k tot ~ Max t ~ 0 ~ N t M t 1h 1h t N t M t h k tot k membrane k bending N N where σ t, Ñ and M are total stress, membrane force, and bending moment, respectively. Several researchers studied holed shells. Lure [1] used a complex method to obtain stresses surrounding the holes with β less than one. In his solution, he simplified his problem as a plate with a curvature which accuracy is as maximum percentage error of 0.5β. This method is suitable for very small holes in which a is very smaller than square root of multiplied shell radius in its gradient; it means that when α<<. Using Fourier Series and a complex technique, Withum [] developed a solution for shells with β greater than and under torsion loading. Savin [] developed general formula for calculating the stress near medium-sized circular and oval holes of shell structures. He investigated results of empirical studies on holes of cylindrical and spherical shells. He concluded that one can study the shell with holes of positive or zero Gaussian Curvature, with a hole limited to a soft environment (with no angle like a circle) which bending is enough soft and flexible, as a shallow shell. In addition, there is a direct relationship between SCF and curvature parameter and according to shallow shells theory, the results depend on small ratio of a/r. To fulfill the terms of this theory, Van Dyke [] suggested that proportion of radius comparing to cylinder thickness has to be large enough. In these cases, experimental solution based on shallow shells theory is well coordinated with laboratory results. It should be mentioned that regarding the empirical limitation and inapplicability of analytical and even laboratory methods, numerical methods are preferred in solving these problems. Regarding unique properties and applicability of FEM (Finite Elements Method) for any desired shapes of different properties of materials and sections and/or any types of loading arrangements, is suitable for this case. Among the advantages of FEM to other methods are analyzing models with complicated geometry and boundary conditions, low cost as well as spending less time. Modeling and Networking In shells modeling, structure s materials given as having linear and elastic properties and values of its Poisson coefficient and modulus of elasticity are ν = 0. and E =.l e 10 (Kg/m ), respectively. In addition, we used FEM method along with ANSYS software to analyze shells. In thin-shell structures, membrane reactions are of more importance, but in thick-shell structures and modes of entering centered loads and releasing edges, bending reactions are important, too. Therefore, in modeling one is to apply an element to include the both modes in solving shell problems. This is why we used structural elements for modeling in software and two dimensional eight-node plane elements in modeling -dimension shells. This element is of solid group and quad 8-node type. We chose shell element in three dimensional shell stresses to solve shell problems in space. This element is of shell group and elastic 8-node type. Regarding the symmetry of the most shells, one may divide the structure into four or eight parts along with symmetry axis and network and analyze only one part of the structure to decrease volume of calculations, accelerate, save time and costs. It should be mentioned that problem modeling has to be done considering needed arrangements on cut lines in structure s symmetry axis, meeting change of location conditions; loading, factual boundary conditions to have no problem in modeling and the boundary conditions are different in several modes of loading. Given the stress concentration, its serious changes near the hole and the fact that points away from the hole are less influenced, dimensions of O N O 1009

Intl. Res. J. Appl. Basic. Sci. Vol., 5 (8), 1008-1015, 01 elements are considered smaller near the opening (at least once as hole diameter); more away from the opening along the cylinder axis, decreasing its due effects and unifying the stresses, the larger elements we have. Moreover, for more accurate evaluation of the results, we modeled and analyzed networking as two modes of Auto-mash and manual that is shown in figure 1. Figures & show shallow shell and cylinder shell which mashing are investigated considering the symmetry in the model. At last, figure shows a geometry modeling of the structure and its hole, meeting boundary conditions, restrains and loadings. In this study, we attempted to provide such shell s dimensions, anchor conditions, and loading to be interpreted and compared easily. Generally, in this research we analyzed more than 100 modes of shell structures of different dimensions of structure s geometry, holes diameters, and holes arrangements. Figure 1. Auto-mash and mandatory networking of a shell with opening Figure. Geometrical shape of a shallow shell Figure. Networking of a cylinder shell with opening and mashing around the hole getting smaller Figure. Meeting boundary and loading conditions on the shell Models Analysis Considering the mentioned theories, problem conditions and shell load, we studied several structures based on several geometrical proportions. Having compared stresses with the primary stresses, we calculated SCF for holed plates, and determined critical paths for the distance between the edge of the hole to the edge of the shell and stress changes of it. In addition, considering several modes of holes arrangements, we analyzed critical arrangement of two holes to study stress concentration in cylinder shells of two holes. The results show that the most critical arrangement of two holes to each other is when connecting line of the two holes is perpendicular to the producer of cylinder shell. In holed plates, analysis show that limited length and width influence SCF. Increasing the ratio of the hole s radius to the plate width (a/w), results in increasing SCF. This increasing trend accelerates in higher ratios of (a/w) which reach up to 9 times more than the initial stress. In lower ratios of the plate s length to its width (L/w), SCF is higher and all tending-to-zero lines (a/w) tend to, value of SCF in unlimited isotropic plates, for isotropic materials. In addition, the graphs show that bending stresses comparing to membrane stresses are negligible for βs less than one, but by increase of this factor, the importance of bending stresses as well as stress concentration will be increased. Studies show that stress concentration range surrounding the hole is limited to an area of equals to the hole s radius. In addition, increase of curvature coefficient results in increase of SCF. Moreover other results show that change trend of stress in cylindrical shells is more than single hole mode for different coefficient of curvature and distance. The maximum increase of stress is related to increase of stress in internal edge of the hole. SCF in external edge of the shell with two openings is independent from λ and the same as single-hole shell. Also in double-hole opening shells, both β and λ parameters influence stress. Increase of the distance between two holes will decrease the stress; this trend continue such that the shell moves toward single hole mode and if the distance of internal edges of two holes is suitable to each other, interactions of two holes to each other disappear. In result, in shell structures having several openings, one can decrease stress 1010

Intl. Res. J. Appl. Basic. Sci. Vol., 5 (8), 1008-1015, 01 concentration and has no interactions of holes on each other like single hole shell, by calculating and observing the distance of holes interactions to each other, if there is no geometrical and application limitations. However, if observing the distance of the holes to each other is not possible, one can decrease SCF by reinforcing the shells. Followings are some examples of shells analysis results and distribution of stress around the shells with one or more openings (figure 5). Table 1 shows SCF of holed shells for different curvature coefficients and tensile loading. Figure 5. Stress distribution around a shell of one, two, and four openings and different arrangements of openings. Figure. Graph of stress concentration changes for different β (a/r=0.1) Figure 7. Graph of stress concentration changes for different β (a/r=0.) Figure 8. Graph of stress concentration changes for different β (a/r=0.5) Figure 9. Graph of stress concentration changes in critical path from the edge of the hole outwardly Figure 10. Graph of stress concentration changes for changes of hole diameter to plate width 1011

Intl. Res. J. Appl. Basic. Sci. Vol., 5 (8), 1008-1015, 01 Decrease Methods of Stress Concentration Around the Holes In shell structures under internal stress loadings and uniform axis tensile, if there is no hole, there will be no bending stress; only a fixed membrane stress develops in everywhere, one type of material and thickness is applicable for all points of the structure. Therefore, a hole makes the stress concentrated around it and changes stress diagram seriously such that stress near the hole is several times more than stress in far points. This issue makes designing difficult; because if we are to design a shell of fixed thickness, we have to design maximum stress in the edge of the hole. While stress away from the hole decreases quickly and considering this decrease of stress the section is no more economic and suitable for design. The most ideal and economic design is that we have change of thickness only in the areas near the hole due to serious changes of stress near the holes and decrease the stress of related areas and have a fixed stress in all points designing changing thickness around the hole. Ideally, this fixed stress equals to membrane stress, which will be created in the shell if there is no hole. Decrease of Stress around Small Holes Using Reinforcement Area We used a cylindrical model with radius of 100 cm, thickness of 0.077 cm and curvature coefficient of 0. and reinforcement area near the edge of the hole to study the shell under axis tensile. We chose a model of a circular small hole with radius of 10 cm which reinforcement area contains a row of elements with width of 0.571 cm around the edge of the hole. Figures 11 & 1 show comparison of membrane and tangential bending SCF for times, 5 times, 7 times, and 10 times more than cylinder thickness in reinforcement and nonreinforcement modes. In the figures, we can see that membrane SCF will decrease with increase of reinforcement percentage. However, in some cases bending SCF will increase by reinforcing edge of the hole especially in angle of 0. We used again S model including reinforcing area of two-row elements with width of 1.8 cm around the hole edge to study effect of width of reinforcing area on the related results. In these models, volume of added materials in reinforcing area equals to the similar volume in series one and behavior similarity of the models is the same as models of series one. The behavior similarity of these models with models of series one is apparent, but according to figure 1, for tangential bending stress we see that behavior of models of series one is betters than series two behavior. Figure 11. Comparison of tangential membrane SCF in reinforcement single row mode Figure 1. Comparison of tangential bending SCF in reinforcement single row mode Table 1. Table of stress concentration changes for shell with opening for different curvature coefficients and tensile loading major axis parallel to generator 0 1. 9 M. - 5 - - B. 0.9 1. 0.9 1.5 0.7 1.1 M. 80.1 - - B..1 0..0 1.1 1.5 M. 90.5 10. - B. 1.9 1.9 0.9 1. 8 M. 90.5 0. B. 1.8 81.8 0. 1. 0 7. 50.5 0. 5-0.8 0.5 9-1.8 0.9 0 -. 1.1 1. 0. 1.8-0.5 0.8 0. - 0.1-0.0 75. 5.1 1.0 5. 1.5. - 1.7.7 5-1.7 90.9-1. 7. 1 -.0 9.1 -. 10. 9 -. 101

Intl. Res. J. Appl. Basic. Sci. Vol., 5 (8), 1008-1015, 01 Figure 1. Comparison of tangential membrane SCF for cylinders of series 1 & Decrease of Stress Concentration around Large Holes Using Reinforcement Area We carried on a similar research on large holes. We chose S5 model with hole radius of 50 cm and cylinder thickness of 0.57 cm, but the rest of geometrical dimensions and materials are the same as the cylinders with small holes. We chose widths of reinforcement areas according to table. Table. Chosen dimensions for cylinders with reinforcement area near large hole with radius of 50 cm Width of Thickness of Percentage of Model's Reinforcement Area Reinforcement Area to Reinforcement Name (cm) Cylinder Thickness 5.5 8.1 17.5 91.08 1. 9.99.1 8.7 1. 9.5.9. 11.5 7.9 15.58 77.8 10. 17. 8. 10. 17. 8..17 5. 10. 17. 8..17 5. 10. 17. 8. 1 S5 S5 S5 S55 S5 S5 S55 S51 S5 S5 S5 S55 S51 S5 S5 S5 S55 Having compared membrane and tangential bending SCF of different models, we concluded that increase of reinforcement percentage decrease SCF. However, ratio of decrease for models of double reinforcement, S5, S55 and S5, is less than 50% and therefore, reinforcement of areas near the hole is not effective for the above-mentioned models. Considering high percentage of reinforcement, percentage of decrease is significant in fourfold reinforcement models of S55, S5, S5, S5, and S51 but is less than in small holes. Figure 1 shows comparison of tangential membrane SCF for these models and figure 15 shows this comparison more clearly for angle of 90. Figure 1 shows results for sextuple reinforcement models of S55, S5, S5, S5, and S51. These results are similar to the results showed in figure 1 for fourfold reinforcement models only with a difference in quantity. In summary, there is an inverse relationship between decrease of stress concentration and percentage of reinforcement in cylinders; however, in small holes, its rate is more quickly than in large holes. In bending cases, initial low percentage of reinforcement increases bending concentration factor, but by more reinforcement, concentration factor moves toward a constant value. In addition, we observed two following phenomena in the cylinders with large holes: A- The area influenced by concentration is larger than cylinder mode of smaller hole. B- Stress quantity is more than in cylinder with small hole. Figure 1. Comparison of membrane SCF for the models 101

Intl. Res. J. Appl. Basic. Sci. Vol., 5 (8), 1008-1015, 01 Figure 15. Reinforcement effect on SCF for models of reinforcement area of four times than cylinder thickness Figure 1. Comparison of tangential membrane SCF for the models Using Shell of Changing (Stepwise) Thickness to Decrease Stress Considering that the most economic reinforcement method (theoretically) is reinforcement according to stress changing trend surrounding the hole, therefore, one may more reinforce areas around the hole and decrease the reinforcement while moving away from the hole. For this purpose, we used S5 models with cylinder thickness of 0.57 cm. Table shows two modes of thickness stepwise changing. The results of these two models are the same as the results in the models with reinforcement area of sextuple. Therefore, these two models do not cause in more economic design. Percentage of Reinforcement 18.87 89.78 Table. Thickness stepwise changing Width of Thickness of Reinforcement Reinforcement Area to Area Cylinder Thickness.17..90 7..17..90 7. 11 5 Row No. 1 1 5 Model's Name S5V S55V Using Reinforcement Ring to Decrease Stress We may use rings around the openings to reinforce plates and shells. The results show that using rings to reinforce the area around the holes, significantly influence stress concentration decrease. We used model S5 with cylinder thickness of 0.57 cm and the same model but with reinforcement rings with diameters of 0.5, 1, and cm. Figure 17 shows comparison of tangential membrane SCF for these four models. In this figure, decrease of membrane SCF is less than the mode of reinforcement area in figure 11. Figure 17. Comparison of tangential membrane SCF using reinforcement ring RESULTS And DISCUSSIONS Our results of finite elements method for single hole mode was compatible with other researchers results[5,,7,8,9 and 10]. Meanwhile, change of each parameters of thickness, diameter of hole, radius of shell 101

Intl. Res. J. Appl. Basic. Sci. Vol., 5 (8), 1008-1015, 01 curvature, which interferes with β curvature factor, influences significantly the change of the results. General review of the results show that SCF adjacent to small holes and cylinder shells with β 1 resemble to plate structures results. In all modes of shells, increase of β will increase stresses and in all studied modes, bending stresses are less important than membrane stress for β less than one; however, increase of β increase importance of bending stresses. The most membrane stresses in all βs happen in angle of 90 and general shape of stress changing based on the angles are the same in all ratios. Studying the results shown that decrease of distance factor of λ will increase significantly stress concentration in internal edge of small hole in shells of two openings and for a certain β. For βs less than or equals 1, small hole has less influence on large hole; but, increase of β will increase the influence of small hole on the large hole. On the other hand, stresses of external edge of the hole show a behavior dependent from λ and its stress approximately equals to shell mode of single hole cylinder. Summarizing the results of studying some examples of shells with opening, one may argue that one can study a single hole shell in shells of two holes with center distance of three times more than hole radius. USA regulations for pressure boilers and steam boilers suggest several formulas for different modes to calculate approximately thickness of reinforced area. According to beam theory, theoretical fundamentals of these formulas are based on elastic context. Therefore, it is expecting that all turbulent stresses surrounding the hole will be removed. This issue is not true about case of large hole or a cylinder with R/h ratio of more than 100. Therefore, a method with high capability of predicting stresses and lacking limitations in experimental and analyzing methods no further exist and it is possible to consider loads and boundary conditions more real and accurate. In this paper, we created a suitable network and provided an analyzing method of acceptable accuracy. In small hole modes, comparing method of replacing level provided in American Mechanic Society Regulations (reinforcing percentage of approximately 100%) with this paper results, we concluded that using this method is desired for an reinforced area with width of less than or equals to one twentieth of hole radius. In large hole modes, referring the results of this paper it is concluded that one obtains better results with more reinforcement of the area near the hole. This issue is in compatible with the suggestions of American Mechanic Society Regulations. On the other hand, using shells with changing (stepwise) thickness or shells with reinforcement rings to decrease the stress do not cause in more economic designs. CONCLUSION Considering that an opening in plates or shells can make turbulence in stress fields and can increase the stress many times in the body of this structures and causes in damages or fractures in the mentioned location, In this paper we tried to investigate: Coefficient changes of stress concentration (for different openings diameters and different types of openings) in plates and shells and compare them with other researchers results. The effect of existence two or more holes with several modes of holes arrangements on stress concentration phenomenon. Coefficient changes of S.C.F. by different loadings on plate and shell structures with hole. The method of reinforcement of holed plates and shells with different material, thickness and shape (uniform and step form) to decrease stress concentration factor. 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