Three-Dimensional Stress Concentration Factor in Finite Width Plates with a Circular Hole

Abstract

The three-dimensional stress concentration factor (SCF) at the edge of elliptical and circular holes in infinite plates under remote tension has been extensively investigated considering the variations of plate thickness, hole dimensions and material properties, such as the Poisson’s coefficient. This study employs three dimensional finite element modeling to numerically investigate the effect of plate width on the behavior of the SCF across the thickness of linear elastic isotropic plates with a through-the-thickness circular hole under remote tension. The problem is governed by two geometric non-dimensional parameters, i.e., the plate half-width to hole radius (W/r) and the plate thickness to hole radius (B/r) ratios. It is shown that for thin plates the value of the SCF is nearly constant throughout the thickness for any plate width. As the plate thickness increases, the point of maximum SCF shifts from the plate middle plane and approaches the free surface. When the ratio of plate half-width to hole radius (W/r) is greater than four, the maximum SCF was observed to approximate the theoretical value determined for infinite plates. When the plate width is reduced, the maximum SCF values significantly increase. A polynomial curve fitting was employed on the numerical results to generate empirical formulas for the maximum and surface SCFs as a function of W/r and B/r. These equations can be applied, with reasonable accuracy, to practical problems of structural strength and fatigue, for instance.

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M. Vaz, J. Cyrino and G. Silva, "Three-Dimensional Stress Concentration Factor in Finite Width Plates with a Circular Hole," World Journal of Mechanics, Vol. 3 No. 3, 2013, pp. 153-159. doi: 10.4236/wjm.2013.33013.

1. Introduction

Many applications in engineering employ components with a circular hole. In the specific case of perforated plates under cyclical load, the effect of stress concentration can propagate cracks and compromise their structural integrity. The stress concentration near a geometric discontinuity in a plate is frequently described by the stress concentration factor (K), defined as the ratio of the actual stress acting on that region to the stress applied to the plate extremity.

Howland [1] studied the stress around the central hole on finite width plates, using bipolar coordinates and biharmonic functions. The solution is iterative, so accuracy may be successively improved and the results were compared with photo-elastic experiments. Timoshenko and Goodier [2] and Muskhelishvili [3] presented classical solutions for bi-dimensional analysis of stress concentration along the hole edges on infinite width plates.

Based on the Theory of Linear Elasticity for plane strain or stress problems, Koiter [4] demonstrated that when the diameter of a centered, through-the-thickness hole approaches the plate width, the ratio between the maximum stress at the hole edge and the average stress in the reduced section (σmaxav) is in the limit equal to 2. Parks and Mendoza [5] employed experimental analysis with strain gages to study the behavior of plates when the ratio between the hole diameter and the plate width is equal to 0.98 and 0.99, and showed that σmaxav approaches 2. Wahl [6] used a simple non-linear formulation, applied to small relations between the hole wall minimum thickness and the plate width and also showed that σmaxav tends to 2 and it approaches 1 as the load is increased. Cook [7] employed a geometrically non-linear finite element model to analyze elastic plates with Young’s Modulus and Poisson’s coefficient respectively equal to 100 GPa and zero, and varied the remotely applied stress from zero to 1000 MPa and the ratio between the hole wall minimum thickness and the plate width from 102 to 10−6. The results indicated that σmaxav decreases from 1.94 to 1 as the load intensity is increased. Pradhan [8] employed a plane stress state finite element model for isotropic and composite plates and showed that the maximum stress at the edge of a centered, through-the-thickness hole strongly depends on the material property. The compilation of the stress concentration factors published by Pilkey [9] is usually considered an important reference for design.

Fatigue analysis methods for cracks and other forms of stress concentration are generally developed using bidimensional (2D) models. However, when these models are applied to certain types of problems where the geometry or material properties may lead to accentuated tri-dimensional (3D) stress concentrations the results can be inaccurate, as demonstrated by Bellett et al. [10] and Bellett and Taylor [11]. A variation of the SCF through the wall thickness of infinite width plate with elliptic and circular holes in isotropic material under remote tensile stress has been systematically investigated using finite element methods by Altenhof and Zamani [12], She and Guo [13,14], Yu et al. [15] and Yang et al. [16]. These analyses show that the maximum value of stress concentration occurs near but not on the plate surface as the thickness increases and this effect is more significant for elliptic holes. Kubair and Bhanu-Chandar [17] employed a FEM to investigate the effect of inhomogeneous material properties on the SCF for plates with a central circular hole subject to a remote stress. The material is functionally graded, that is, its properties vary spatially. A parametric study indicated that the SCF reduces when the Modulus of Young is progressively increased towards the center of the hole and that the angular position of maximum stress on the plate surface is unaffected by the material inhomogeneity. More recently, Chao et al. [18] presented an analytical solution for the stress field at an infinity plate with reinforced elliptical hole subjected to an inclined uniform remote tensile stress. The material properties for the reinforcement material may differ from the plate properties, and the reinforcement layer is limited by two confocal ellipses. Rezaeepazhand and Jafari [19] carried out an analytical investigation for the stress distribution in isotropic plates with centered holes with different shapes, subjected to remote uniaxial tensile stress. In the parametric study circular, triangular, square and pentagon holes were considered, furthermore, the cut-out shape, bluntness and orientation were also taken into account. The analytical results were compared with FEM simulations and showed that the SCF for square holes are smaller than for a similar plate with circular holes and that triangular and pentagon holes yield higher SCFs.

The objective of this study is to evaluate the variation of the stress concentration factor through the thickness for isotropic plates, with through-the-thickness circular holes, subject to remote tensile stress and to investigate the effect of plate width on the results. A finite element model was elaborated with various widths and thicknesses to allow a comprehensive parametric evaluation of this variation.

2. Definition of Geometric Parameters

Figure 1 shows a plate with a through-the-thickness circular hole, submitted to uniaxial stress, where the geometric parameters used in this study can also be seen. The plate width, thickness and length are respectively equal to, and. The circular hole, located at the center of the plate, has a radius equal to. The fixed geometric parameters are r = 5 mm and H = 100 mm, therefore the ratio is equal to 20, which ensures that the stress is applied far enough from the hole. The dimensionless variable ratios employed in the analyses are: = 1.2, 1.4, 1.6, 1.8, 2.0, 2.2, 2.5, 3, 4, 5, 6, 10 and 20 and = 0.2, 1, 2, 3, 4, 5, 6, 10, 20 and 30, totaling 130 simulation cases. The results in the analyses are related to these dimensionless parameters in such a way that an ample range of plate widths and thicknesses is investigated. Nevertheless, whenever convenient the results will be only presented for a smaller set of data.

Conflicts of Interest

The authors declare no conflicts of interest.

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