Stress analysis of the multiple circular holes with the rhombic array using alternating method

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Abstract

This paper presents the alternating method to study the stress distributions of the multiple circular holes with the rhombic pattern in the infinite domain. For the purposes of the efficiency and simplicity, one must use an analytical solution for a single circular hole in an infinite domain, subjected to the arbitrary traction on the circle boundary. Then, the analytical solution correlates with a successive iterative superposition process capable of satisfying the prescribed boundary conditions in the plane problems with the multiple holes. Comparisons between the solutions of the present procedure and the conventional boundary element technique demonstrate the accuracy and the advantages. Effects of the number of holes and the spaces on the stress concentration factors also are evaluated in detail herein.

Introduction

In the realistic mechanical components in fields of chemical and nuclear engineering, such as pressure vessels, heat exchangers, steam generator, etc., the stress concentration problems with thousands of circular holes play an important and even a primary roles. The reliable evaluation for reducing the accompanying stress concentration factor around these holes is necessary to assure the structural integrity.

Several analytical methods have been devoted to evaluate the stress concentration factors for the multiple circular holes in an infinite domain. However, these solutions were restricted to the multiple equal holes with specific patterns [1], [2], [3], [4], [5], [6]. The effective Young's modulus and Poisson ratio were used to simplify the discontinuity of the perforated plate [7]. The accuracy of this approximate method is another questionable problem. The finite element method (FEM) can extensively study the arbitrary arrangement of holes with various sizes. Numerical experiments indicate that the accuracy of the stress distribution around the discontinuities strongly depends on the mesh size taken. Thus, the discretization of the meshes among the adjacent holes makes these problems complicated and always consumes more computing time. Boundary element method (BEM) also provides another numerical method. The drawbacks of the strong singularity near the boundary make the solutions of the neighboring holes incorrect. Recent research revealed the potential of the alternating method solving the arbitrary discontinuities of the fracture problems [8], [9], [10], [11]. Nevertheless, this efficient procedure has paid a little attention to the issue of the multiple holes[12]. Thus the extension of the previous works to analyze the interactions among the general rhombic array of the multiple circular holes is of interest, and is the main objective of this work.

For the purpose of simplicity and efficiency, one must use an analytical solution of the single circular hole in an infinite domain subjected to the arbitrary surface tractions over the circumference of the hole. The complete elastic solutions corresponding to the arbitrary tractions expressed by the traditional trigonometric Fourier series are drawn. In the process of the alternating method [13], these analytical solutions are then used for the successive iterative superposition procedures to satisfy the prescribed boundary conditions.

This paper deals with an infinite solid containing the rhombic arrays of multiple circular holes. Two typical square and the equilateral triangular arrays of holes are considered. Numerical calculations are carried out for various hole spaces subjected to the remote loads in the arbitrary directions to discuss the strongly interacting effects. Owing to the inadequacy of the referenced solutions, the results of the proposed alternating method compare with the boundary element solutions. The method put forth in this work demonstrates to be quite efficient and rigorous for the evaluation of accurate stress concentration factor in the realistic design.

Section snippets

Analytical solution for single hole with arbitrary surface tractions in an infinite domain

Consider a circular hole with radius R in an infinite domain and let the origin of coordinates be at the center of the hole, as shown in Fig. 1. This boundary-value problem is most effectively formulated in terms of the complex stress potentials Φ and Ψ of Muskhelishvili [14]. With respect to polar coordinates r, θ, the normal stresses σr and σθ, and the shear stress τrθ, and the displacements ur and uθ, are given byσr+σθ=2[Φ(z)+Φ(z)]σθ−σr+2iτrθ=2[zΦ′(z)+Ψ(z)]e2iθ2μ(ur+uθ)=e−iθ[κϕ(z)−zϕ′(z)−ψ(z)

Alternating method solving multiple holes in an infinite domain

The two-dimensional infinite solid containing k circular holes is subjected remote stresses, as shown in Fig. 2(a). This problem can be expressed as a superposition of two cases, as illustrated in Fig. 2(b) and (c). Fig. 2(b) represents the case when external forces are acting on the infinite plane without any holes and Fig. 2(c) the case with multiple circular holes under the fictitious tractions along the hole boundaries. Hence, based on the analytical solution of a single circular hole under

Results and discussion

This paper deals with the interaction effects of the multiple holes with the rhombic pattern in an infinite domain under remote arbitrary loads as shown in Fig. 4. Let the radius of the holes be a, and the spaces between adjacent holes in horizontal and vertical rows be 2b and 2c, respectively. The origin of global axes X and Y is at the center of the main hole. Two parameters λ=a/b μ=b/c describe the spaces among holes. Numerical calculations are carried out for two typical rhombic arrays of

Conclusions

In this paper, the alternating method demonstrates the validity in the calculations of the stress concentration factor of the multiple holes with the rhombic array in the infinite domain subjected the remote arbitrary loads. The interaction effects among holes can be accurately evaluated through very simple iteration procedure even when the holes are very close. The computer cost of the present computation is much less than the traditional boundary and finite element methods. This work can

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