Analyse the Stress Concentration Effect of a Perforated Plate under Uniaxial Loading Using Ansys

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1 Analyse the Stress Concentration Effect of a Perforated Plate under Uniaxial Loading Using Ansys Manju Saroha Assistant Professor (Mechanical Engg.), Department of CSE/IT, BPSMV, Khanpur Kalan ABSTRACT It is a well-known fact that the presence of a hole introduces stress concentration around the hole boundary. The knowledge of stress concentration is very important to designers in solving many practical problems. The goal of this research was to create some comparative results with varying shape of hole and other dimensions like length and breadth using finite-element method with experimental results. For this we choose HT-9/2239 as material and finite element technique to get accurate results. Finite Element Method (FEM) is one of the numerical technique and very suitable in solving the continuum problem. A series of ANSYS elastic-plastic large deflection finite element analysis are carried out with varying the cutout size as well as plate dimensions. Than compare the results of all dimensions with varying conditions. At last the analysis of results gives us some graphical data and some graphics of stress concentration of figure under load. Keywords: Stress concentration, Finite-Element method, ANSYS, Mesh, uni-axial loading. 1. INTRODUCTION It is often necessary to provide openings in thin plated structures such as cold-formed steel members, aero plane fuselages, plate and box grinders and ship structures for access and services. The presence of holes in such structures results in a redistribution of the membrane stresses accompanied by a change in the strength characteristics. These perforations could reduce the ultimate strength and yield strength of the plates. Stress concentration factor may be defined as the ratio of maximum stress developed around the hold boundary to the animal uniform stress in the member. The stress concentration factor may be defined either based upon the gross area or net area of the member at the location of the cutout. The knowledge of stress concentration is very important to designers in solving many practical problems. Two-dimensional problems of the theory of elasticity may be formulated either as plane stress or plane strain problems. When the plate thickness is very small and is loaded by force applied at the boundary parallel to the thickness then it may be assumed that a state of plane stress exists in the plate. When a hole of circular profile is introduced in the plate then the uniform distribution of stresses in the plate is disturbed. The redistribution of stresses depends upon the dimensions of hole and the plate. In solving this type of problem, numerical techniques appear to be a very effective way. Most of the numerical techniques are based on the principle that it is possible to derive some equations and relationships that actually describe the behaviour of a small part of the body. Technique and Tool Used By discreatizing the entire continuum into smaller elements and link those elements by using appropriate compatibility and equilibrium relationships. It is possible to obtain a reasonably accurate prediction of the values of variables such as stress and strain. Finite Element Method (FEM) is one of the numerical technique and very suitable in solving the continuum problem. In FEM, the entire continuum will be divided into finite numbers of elements and hence this method is called finite element method (FEM). Over each element, the behaviour is described by the displacements of the element and the material law. Like others numerical techniques, al the elements in FEM is also assembled together with continuity and equilibrium relationships. The FEM is very suitable for practical engineering problems of complex geometries. To obtain good accuracy in regions of rapidly changing variables, a large number of elements must be used. FEM is very useful tool in analyzing the stress distribution and displacements in a continuum problem. The material, which is used in this thesis, is HT-9/

2 Alloy HT-9 is a Cr containing martensitic stainless steel, also known as 12Cr-1MoVW. It contains 12 weight percent of Cr. Cr provides significant resistance to atmospheric corrosion, while Mo enhances the resistance of this alloy to seawater corrosion. A small Ni content in this alloy offset the ferritizing effect of low carbon content. Alloy HT-9 has a body centered cubic (BCC) lattice structure. 2. LITERATURE REVIEW The presence of hole could reduce the ultimate strength and yield strength in a perforated plate. That s why the knowledge of stress concentration is very important to designers in solving many practical problems. A lot of work has been done in this field. The problem or determination of stress concentration around a circular cut out in an infinite plate under tension was studied first of all by kirsch (1). He found that the stress concentration is maximum at a point on the hole boundary where the tangent to the hole boundary is parallel to the applied stress where as at a point on the hole boundary, where tangent is perpendicular to the applied stress the stress concentration is minimum. The problem to determine the stress concentration in a semi infinite plate with a circular hole under simple tension was investigated by Jeffary(2). Whereas Howland (3) determine the stress concentration around a circular hole in a plate of finite width under dimple tension. When an increasing tensile load is applied to the ends of the plate and the hole is positioned symmetrically in the plate width, Sokalov (4) determined the plastic fond around the circular hole in the case of biaxial tension. Feverbery (5) devised a technique to determine the plastic zone around a circular hole in the case of uniaxial tensile forces in a plate of finite dimensions. Thxsris and Merketas (6) experimentally determined the elastic plastic stress and strain distribution occurring in a thin strip of a strain-hardening alloy with a central hole under uniaxial tension. Stresses were evaluated by introducing the assumption of plane stress in a material conforming to the misses yield criteria and the incremental flow stress strain relation. Maroal (7) used elastic finite element methods to determine the size of the plastic zones developing around the notches. Griffith (8) also estimated the elastic plastic deformation in edge notched tensile specimen under plane stress condition by the finite element method. Ibrahim and Mc challian (9) elastic plastic deformation around a circular hole in a plate under cyclic loading. Cookerhers and Baton (10) have devised a method to estimate the size of the plastic zone developing around the central hole in a finite width plate subjected to plane stress condition and to be made from elastic, linearly strain hardening material. Many other research workers for example Byre Gowda and Topper (11), W.A.Box(12), Reigh (13), Mc Clintok and Rohed (14), Budhinsty (15), Tuba(16), Sherbourne and Haydl (17), Ishikawa (18), Ibrahim Ital(19) have analysed to determine plastic stress concentration around the hole boundary under unaxial and bioxial loading. Narayanan and Chow (20) developed design charts based on ultimate capacity of uniaxially compressed perforated plates with square and circular openings. Yettram and Brow (21) studied the stability behaviour of flat square plates with central square perforations. Xianzqiao Yan (22) gives a study about the analysis a crack emanating from a corner of a square hole in an infinite plate using the hybrid displacement discontinuity method. An analytical investigation is used to study the stress analysis of plates with different central cutout. Particular emphasis is placed on Flat Square plates subjected to a uni-axial tension load. The results based on analytical solution are compared with the results obtained using finite element methods. The main objective of this study is to demonstrate the accuracy and simplicity of presented analytical solution for stress analysis of composite plates with central cutout. The effect of cutout geometry (circular, square, or special cutouts), material properties (isotropic and orthotropic), fiber angles, and cutout curvature are considered. 3. PROBLEM FORMULATION & GEOMETRY GENERATION For ships and aeroplane fuselages structures, cutouts are typically made in plates of ballast water tanks. In a continuous steel stiffened plate structure, a plate is surrounded by support members, which are typically designed so that they should not fail prior to the plate. We are using the plate for study with cutout at center of plate. It is assumed that the cutout is located at the center of plate only. The effects of cutout types other than circular type are also considered here. The diameter of the circular hole is varied as d/b = 0.4, 0.3, 0.2, and 0.1. The area of rectangular hole is same as the area of the circular hole. The details of the parametric study are given in Table 3.1. The plate dimensions are 2400*800*15 mm. The value of young s modulus is 160GPa, the material yield strength is MPa & the Poisson ratio is The diameter of the cutout is 320 mm. It is assumed that the plate material follows the elastic-perfectly plastic scheme. A series of ANSYS elastic-plastic large deflection finite element analysis are carried out with varying the cutout size as well as plate dimensions. A convergence study with varying the number of finite elements was carried out to determine a relevant fine mesh model for ANSYS nonlinear FEA of 9

3 the plate. Four node eighty-four elements (solid) available in the ANSYS element library are used for discretization of perforated plate. The element has three degrees of freedom per each node i.e. (Ux, Uy, and Uz). This element is well suitable for analyzing the linear strain and nonlinear applications. The finite element model of the plate with circular and rectangular opening is shown in Figure 1 and 2. Both geometric and material nonlinearities are considered in the analysis. Small displacement static analysis with NEWTON RAPHSON criteria is used for the solution of non-linear analysis. Table1: Details of Parametric Study S.No. Specimen Length Width Thickness d/b Area of Shape (mm) (mm) (mm) cutout (mm 2 ) cutout (mm 2 ) circular 1 P1 2 P2 3 P3 4 P4 5 P square circular square circular square circular square circular square circular square circular square circular square of One must choose the type of elements that are to be used, which depends on the type of analysis. One type of element differs from another insofar as the variables that it deals with and how each variable varies spatially in the element. For example, for doing a two-dimensional elasticity analysis, in any particular finite element program their will only is one or two element types that are appropriate. The element has three degree of freedom for each node. In case of FEA of perforated plate we use solid element, four node eighty-four nodes and plane stress with thickness condition. Here we have to mention the value of thickness as 15mm, which is fixed for all cases in our work. The perforated plates are subjected to uniform compression in the x direction. The four edges of the perforated plates are either simply supported or clamped. The two axes of symmetry (i.e., x and y axes) of the plate have no in-plane motions. The lower and the upper horizontal edges of the plate are free to move in the loading direction. The two unloaded edges are either constrained from the transverse in-plane motion, or unconstrained from the transverse inplane motion. Table 2: Properties of Materials for Elastic Linear Material Model Property Alloy HT-9 Thermal Conductivity, W/m*K 28 Modulus of Elasticity, GPa (10 6 psi) 160 Poisson s Ratio 0.33 Coefficient of Thermal Expansion per C *

4 Table 3: Values for Inelastic/Nonlinear Material Model STRESS STRAIN Geometric Modelling Two numbers of FE models were analyzed. Both of them were two-dimensional models. The first model was the two dimensional plate with circular hole of the solid specimen. The second model was the two dimensional plate with square hole of the solid specimen. This model was generated to confirm the properties assign for steel plate can give a consistent respond in term of stress and strain compared to the experimental values. The model was generated to study the behaviour of specimen with steel plate of 15mm thickness in both cases for circular and square hole. First of all make the key points with proper distance on the model with a reference key point (0, 0, 0). And then add all key points of the geometry. Then create area from key points. Now make perforated plate with subtract command from menu bar. The body must be broken up into elements and nodes. The mesh will consist of a set of node points distributed around the body. Lines will connect the node points and the lines form the boundaries of the elements. In our study we choose the mapped meshing with quadrilateral element. And size of the element is (40, 12).Usually this involves describing the external influences across the boundary of the body (If gravity is important, this force acts at all points of the body, not just the boundary). External influences acting across the boundary are usually represented by specifying the displacement or the force over each node point on the boundary. Sometimes a distributed force can be defined over a part of the boundary. FEA has a default assumption on the boundary conditions. While finite element programs allow you to enter distributed loads (force per length or per area), it is sometimes useful to be able to convert a distributed load to forces at the nodes. The net force from the distribution, which acts over that portion of the boundary, is applied to the node. When doing FEA, this solves the equations of elasticity approximately. When doing FEA, one can view the results in a various ways. We can plot the results in different ways. For example in stress, strain etc. Figure 1 Displacements 11

5 Fig. 2: Meshing for a plate with a centrally located circular hole Figure 3: Meshing for a plate with a centrally located square hole 4. FE ANALYSIS OF PERFORATED STEEL PLATE Validation of the developed FE model is done with the published results of Frank Thilo Trautwein. For this purpose, a perforated plate under uniaxial tension is considered for the study. The size of the plate is 8 x 40 mm. The diameter of the hole is 10 mm. The Young s modulus of elasticity (E) as 205.8GPa and Poisson s ratio (υ) of 0.3. Table 4: Comparison between published result & ANSYS (present study) Sr. No Stress/Strain(%) Published result Ansys (present Study) % Error 1 σu x εu x Boundary conditions specify the strength of the perforated plate. It is important that they are specified appropriately. The boundary conditions in ANSYS are classified as follows:- Selection of element type Properties of the material: linear properties (value of E, value of υ) and non-linear properties. Type of Loading at end points: axial (X-direction-direction), biaxial, shear. Mesh independency 12

6 Stress (MPa) International Journal of All Research Education and Scientific Methods (IJARESM) 5. RESULTS AND DISCUSSION Comparison of published results with developed results experimental data Ansys data Strain (%) Figure 4: Stress-Strain Curve of perforated plate under tensile loading Fig.5: A sample of the FE mesh for a plate with a centrally located circular hole Perforated plate has been analysis by many researchers. During these researches, load has been applied like axial, biaxial compression/tension and shear. But no work so far has been done on perforated plate under uniaxial tensile loading with change in perforation area, change in length of plate and change the breadth and change of shape of hole. The present research work is wholly concentrated on effect of perforation area, length of plate and change in the breadth and change of shape of hole. Effect of Change in Length on the Yield Strength of the Plate The increase in length results in increased yield strength of the plate. It is due to decrease in stress concentration at the edge of the perforation. In present work, the above effect has been analyzed in case of circular and square perforation area on mm, mm and plate for uniaxial tension. 13

7 Table 5: Values of σ x for different value of length of plate for circular hole under uniaxial tension S. No. Length of plate (mm) σ x (MPa) Table 6: Values of σ x for different value of length of plate for square hole geometry under uniaxial tension S. No. Length of plate (mm) σ x (MPa) Effect of Change in Breadth on the Yield Strength of the Plate The increase in breadth results in increased yield strength of the plate. It is due to decrease in stress concentration at the edge of the perforation. Table 7: Values of σ x for different value of breadth of plate for circular hole under uniaxial tension S. No. Breadth of plate (mm) σ x (MPa) Table 8: Values of σ x for different value of breadth of plate for square hole geometry under uniaxial tension S. No. Length of plate (mm) σ x (MPa) Fig 6: Contour of perforated steel plate for L=800mm for square hole 14

8 Fig 7: Contour of perforated steel plate for B=800mm for circular hole CONCLUSION AND FUTURE SCOPE In the present research work, results have been obtained for perforated plate with different length of plate, different breadth of plate, and different shape of perforation area. Size of the length of the plate affects the value of the yield stress of the plate. As the length of the plate increases, the value of yield stress increase. Size of the breadth of the plate affects the value of the yield stress of the plate. As the breadth of the plate increases, the value of yield stress increase. As it is concluded that plate with square geometry have yield stress approximately 50% of the yield stress of the plate having circular geometry. Size of the length of the plate affects the value of the yield stress of the plate in case of square hole geometry. A lot of research works are going on perforated plate under different loading, including shape of perforation area, which may be square, rectangular, circular and ellipse. This work including shape of perforation area, lengthwise, breadth wise, under uniaxial loading only. But we can further proceed with biaxial loading, loading ratio, other shapes of perforation etc. It can be of great importance in future. REFERANCES [1]. G. Kirsch, Plane stress solution for a thin plate of infinite width with a hole under tension ZVDI Vol. 42, 1898, pp [2]. G. B. Jeffery, Plane stress and plane strain in bi-polar co-ordinates Phil. Trans. Of the royal sec. Series A.221, London (1921), pp [3]. R. C. J. Howland, One the stresses in the neighborhood of a circular hole in a strip under tension. Phil. Trans. A 229, 1929, pp [4]. A. A. Sokolov, The elastic plastic state of a plate, Doki. Akad. Mask SSR, Vol.x, No. 1, [5]. I. I. Feverberg, Tension applied to a plate with a circular hole with stresses exceeding the limit of elasticity, Trudy, Ixst, No. 615 (1947). [6]. N. Merketos, Elastic plastic analysis of perforated thin strip of a strain hardening material, Int. J. Mech. Phys. Solids, vol. 12, 1964, pp [7]. P. V. Maroal and J.P. King, Elastic plastic analysis of two dimensional stress system by the finite element method, Int. J. Mech. Vol. 9, 1967, pp.143. [8]. S. B. Griffith, Experimental investigation of the effect of plastic flow in a tension panel with a circular hole, NACA, TB 1705, [9]. S. N. Ibrahim and Mc.Callion, Elastic plastic deformation around a circular hole in a plate under cyclic loading,appl. Mech. [10]. Convention, Int. J. Mech. Eng. London, Proc , Vol. 180, Part 31, pp [11]. G. Cookerhers and D. H. Baton, An estimation of plastic zones in a plate with central hole, Strain, Vol. 12, No. 4, Oct. 1976, pp [12]. C. V. Byre Gowda and T. N. Topper, On the relation between stress and strain concentration factor in a notched member in a plane stress, J. Appl. Mech., Vol. 37, No. 1, March 1970, pp [13]. W. A. Box, The effect of the plastics strains on the stress concentrations, Proc. SESA. Vol. 8, No. 2, 1950, pp

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