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In this paper, stress distribution around a triangular hole in finite isotropic plate under in-plane loading is studied. With the assumption of plane stress conditions, the method employed is based on the analytical solution of Muskhelishvili’s complex variable method and conformal mapping. The finite plate (the ratio of the length of the biggest side of the hole to side of the plate is greater than 0.2) can be considered as isotropic and linearly elastic. For solving the problem, the finite area with a triangular hole in z plan is mapped onto finite area outside a unite circle in ζ plan using the conformal mapping function. The stress function in finite plate with triangular hole is presented by superposition of the stress function for an infinite plate with a triangular hole and ones for a finite plate without a hole. The unknown coefficients in stress function are obtained by using the least square boundary collocation method and applying the appropriate boundary conditions. The effect of hole curvature, hole orientation, plate’s aspect ratio, hole size, type of loading as the effective parameters is investigated. The results based on analytical solution are in a good agreement with the results obtained from the finite element method . The results show that the analysis of the stress distribution in perforated plates that the ratio of the length of the biggest side of the hole to the smallest side of the plate is greater than 0.2, by using the infinite plate theory has a great error.

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Shape sensitivity analysis of finite element models is useful for structural optimization and design modifications. Within numerical design optimization, semi-analytical method for sensitivity analysis is frequently applied to estimate the derivative of an objective function with respect to the design variables. Generally numerical sensitivity analysis widely suffers from severe error due to the perturbation size and find a method which is not sensitive to the perturbation size is topics under study. Complex variable methods for sensitivity analysis have some potential advantages over other methods. For first order sensitivities using the complex variable method, the implementation is straightforward, only requiring a perturbation of the finite element mesh along the imaginary axis. This paper uses a complex variable and combine it with discrete sensitivity analysis, thus present new method to obtain derivatives for linear structure. The advantage of this method are quickly, accuracy and its simple implementation. The methodologies are demonstrated using two dimensional finite element models of linear elasticity problems with known analytical solutions. Obtained sensitivity derivatives are compared to the exact solution and also finite difference solutions and show that the proposed method is effective and can predict the stable and accurate sensitivity results.

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This paper aims at optimizing the parameters involved in stress analysis of finite isotropic plates, in order to achieve the least amount of stress around a quadrilateral cutout located in a finite isotropic plate under in- plane loading using a novel Swarm Intelligence optimization technique called Ant lion optimizer. In analysis of finite isotropic plate, the effective parameters on stress distribution around quadrilateral cutouts are cutout bluntness, cutout orientation, plate’s aspect ratio, cutout size and type of loading. In this study, with the assumption of plane stress conditions, analytical solution of Muskhelishvili’s complex variable method and conformal mapping is utilized. The plate is considered to be finite (proportion of cutout side to the longest plate side should be more than 0.2), isotropic and linearly elastic. To calculate the stress function of a finite plate with a quadrilateral cutout, the stress functions in finite plate are determined by superposition of the stress function in infinite plate containing a quadrilateral cutout with stress function in finite plate without any cutout. Using least square boundary collocation method and applying appropriate boundary conditions, unknown coefficients of stress function are obtained. Moreover, the finite element method has been used to check the accuracy of results. The obtained results show that the mentioned parameters have a significant effect on stress distribution around a quadrilateral cutout and that the structure’s load- bearing capacity can be increased by proper selection of these parameters.

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One of the concerns of designers of engineering structures is structural failure due to stress concentration caused by geometric discontinuities in the structures. Therefore, by considering that perforated composite plates are used in most engineering structures, their study is very important. The purpose of this paper is to present a new model based on the regression method for estimating stress concentration factor of a circular hole in orthotropic plates. One of the important applications of providing stress distribution around holes in terms of mechanical properties is the use of these relationships in the stress analysis of perforated viscoelastic plate using the effective modulus method or Boltzmann's superposition principle. First, using different values of the mechanical properties of the composites plates, and employing an analytical solution based on the complex variable method, the stress concentration factor of circular hole is calculated for a number of these materials. Then, using multiple linear regression, an explicit expression for the stress concentration factor is given in terms of mechanical properties. The results show that the multiple regression model is able to predict the circumferential stress with a maximum error of less than 1%.

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The semi-analytical method (SAM) is an approach that computationally efficient and easy to implement. That's why this method often used for the sensitivity analysis of finite element models. However, SAM is not without defect especially in problems that rigid body motions are relatively large reveals severe inaccuracy. Such errors outcome from the pseudo load vector calculated by differentiation using the finite difference method. In the present paper, a new semi-analytical approach based on complex variables is proposed to compute the sensitivity of nonlinear finite element models. This method combines the complex variable method with the discrete sensitivity analysis to obtain the response sensitivity accurately and efficiently. The current approach maintains the computational efficiency of the semi-analytical method with higher accuracy. In addition, the current approach is insensitive to the choice of step size, a feature that simplifies its use in practical problems. The method can be used to nonlinear finite elements only requires minor modifications to existing finite element codes. In this paper, the authors demonstrate that the discrete sensitivity analysis and the complex variable method are equivalent and solve the same equation. Finally, the accuracy of the method is investigated through the various numerical examples by comparing by other methods and will show that this method is reliable and independent of step size.

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