The Stresses Around Square Holes with Rounded Corners

1958 ◽  
Vol 2 (03) ◽  
pp. 37-41
Author(s):  
Joseph S. Brock
Keyword(s):  
Stress Distribution ◽  
Analytical Solution ◽  
Infinite Plate ◽  
Mapping Function ◽  
Mapping Technique ◽  
Radius Of Curvature ◽  
Stress Concentrations ◽  
Method Of Solution ◽  
Christoffel Transformation ◽  
Square Hole

This paper presents an analytical solution for the stress distribution around a square hole with rounded corners in an infinite plate subjected to pure tension. The method of solution is a combination of a conformal mapping technique and the complexvariable method of Muskhelishvili. The form of the mapping function is obtained from the Schwarz-Christoffel transformation. The mapping function is general and gives a good approximation to square holes with rounded corners of arbitrary radius of curvature. The ratios of the corner radius to the width of the opening considered cover the range from 0.03 to 0.5. This is considered 1o be the interesting range for ship structures. The results are given in terms of stress concentrations around the boundary of the opening.

scholarly journals A Non-Singular Analytical Technique for Reinforced Non-Circular Holes in Orthotropic Laminate

2020 ◽  
Vol 25 (1) ◽  
pp. 92-105
Author(s):  
Pradeep Mohan ◽  
R. Ramesh Kumar
Keyword(s):  
Stress Distribution ◽  
Analytical Solution ◽  
Power Series ◽  
Orthotropic Shell ◽  
Infinite Plate ◽  
Stress Singularity ◽  
Power Series Solution ◽  
Element Analysis ◽  
Full Field ◽  
Analytical Technique

AbstractThe intricacy in Lekhnitskii’s available single power series solution for stress distribution around hole edge for both circular and noncircular holes represented by a hole shape parameter ε is decoupled by introducing a new technique. Unknown coefficients in the power series in ε are solved by an iterative technique. Full field stress distribution is obtained by following an available method on Fourier solution. The present analytical solution for reinforced square hole in an orthotropic infinite plate is derived by completely eliminating stress singularity that depends on the concept of stress ratio. The region of validity of the present analytical solution on reinforcement area is arrived at based on a comparison with the finite element analysis. The present study will also be useful for deriving analytical solution for orthotropic shell with reinforced noncircular holes.

scholarly journals Analytical solution and numerical verification for the pressure-relief method of a circular tunnel

2018 ◽  
Vol 38 (3) ◽  
pp. 338-351
Author(s):  
Shunchuan Wu ◽  
Miaofei Xu ◽  
Yongtao Gao ◽  
Shihuai Zhang ◽  
Fan Chen
Keyword(s):  
Stress Distribution ◽  
Analytical Solution ◽  
High Stress ◽  
Mapping Function ◽  
Mapping Method ◽  
Numerical Verification ◽  
Fast Method ◽  
Circular Tunnel ◽  
Variable Theory ◽  
Releasing Effect

This paper presents an elastic analytical solution to a circular tunnel with releasing slots at high stress areas near the hole by using a conformal mapping method and the complex variable theory. Compared to the original stress distribution around the circular hole, the releasing effect on elastic stresses is evaluated. After grooving slots, low stress area is generated where the high stress concentration is located. This is agreeable with what was predicted by the finite difference FLAC2D. Besides, displacements are obtained along the periphery of the released hole and are in accordance with those of FLAC2D. In addition to the intersection of the mapping contour, the influences of the sampling points distribution, series number in mapping function, and slot shape are discussed. It is inevitable that the mapping accuracies for the slot and the circle cannot be satisfied at the same time The mapping effect on the circle has to be considered primarily since the stress distribution around the circle is much more significant than the tunnel stability. The analytical solution can be available and fast method of estimating the releasing effect of the application on the tunnel without rock parameters.

Quasi-Square Hole With Optimum Shape in an Infinite Plate Subjected to In-Plane Loading

1981 ◽  
Vol 103 (4) ◽  
pp. 866-870 ◽  
Author(s):  
A. J. Durelli ◽  
K. Rajaiah
Keyword(s):  
Stress Concentration ◽  
Circular Hole ◽  
Biaxial Loading ◽  
Optimization Technique ◽  
Infinite Plate ◽  
Efficiency Factor ◽  
Optimum Shape ◽  
Stress Concentrations ◽  
Large Plate ◽  
Square Hole

This paper deals with the optimization of the shape of the corners and sides of a square hole, located in a large plate and subjected to in-plane loads, with the object of minimizing stress concentrations. Appreciable disagreement has been found between the results obtained previously by other investigators. In this paper new tests have been conducted and discrepancies have been corrected. Using an optimization technique, the authors have developed a quasi square shape which introduces a stress concentration of only 2.54 in a uniaxial field, the comparable value for the circular hole being 3. The efficiency factor of the proposed optimum shape is 0.90 whereas the efficiency factor of the best shape developed previously was 0.71. The shape also is developed that minimizes the stress concentration in the case of biaxial loading when the ratio of biaxiality is 1:-1.

BENDING OF A SUBTLE-FINISHED PLATE ON ELASTIC BASE WITH SPECIAL CONDITIONS OF ITS WORK

2019 ◽  
pp. 424-430
Author(s):  
A. T. Marufiy ◽  
A. S. Kalykov
Keyword(s):  
Analytical Solution ◽  
Working Conditions ◽  
Fourier Transforms ◽  
Infinite Plate ◽  
Elastic Base ◽  
Generalized Solutions ◽  
Middle Plane ◽  
Actual Working

In this article, an analytical solution is obtained for the problem of bending a semi-infinite plate on an elastic Winkler base, taking into account incomplete contact with the base and the influence of longitudinal forces applied in the middle plane of the plate. The analytical solution is obtained by the method of generalized solutions using integral Fourier transforms. Any analytical solution is the result, approaching the actual working conditions of the designed structures.

scholarly journals Application of the Schwarz-Christoffel Transformation to the Solution of Harmonic Dirichlet Problems in Electrostatics

2020 ◽  
Vol 3 (2) ◽  
pp. 168-178
Author(s):  
ST Swem ◽  
P Ogwola ◽  
E Otene
Keyword(s):  
Electric Field ◽  
Mapping Function ◽  
Mapping Method ◽  
Infinite Strip ◽  
Dirichlet Problems ◽  
Suitable Alternative ◽  
Conformal Mapping Method ◽  
Straight Line ◽  
Christoffel Transformation ◽  
Electrostatic Problems

In this paper, a purely conformal mapping method for efficiently solving harmonic Dirichlet problems of electrostatic in domains free of charge and with charge whose boundaries have inconvenient geometries consisting of straight-line segments is presented. The method which uses the inverse of an appropriately determined Schwarz-Christoffel transformation as the mapping function, was applied to harmonic Dirichlet problems in an infinite strip and infinite sector and the solution or electrostatic potential for the problem obtained for each case. Furthermore, the equipotential lines of the electric field were also obtained in order to show the features of the solution and the field analysed accordingly. The electric field intensity was also analysed to show its variation in the field. This method could therefore be a suitable alternative method for solving Laplace's equation in two dimensional electrostatic problems.

An Anisotropic Generalization of the Bridgman Analysis of Tensile Necking

1983 ◽  
Vol 105 (4) ◽  
pp. 264-267 ◽  
Author(s):  
M. A. Eisenberg ◽  
C. F. Yen
Keyword(s):  
Stress Distribution ◽  
Analytical Solution ◽  
Plastic Flow ◽  
Approximate Analytical Solution ◽  
Parameter Family ◽  
Isotropic Case ◽  
Anisotropic Plastic

Tensile necking in anisotropic bars is analyzed in the spirit of P. W. Bridgman’s treatment of the isotropic case. Anisotropic plastic flow causes an initially axisymmetric bar to develop an elliptical neck. Using physical approximations analogous to Bridgman’s, an approximate analytical solution for the stress distribution is obtained. The solution is shown to be asymptotically correct in two important limiting cases: (a) the fully developed anisotropic neck, and (b) the isotropic limit. In the latter case it is shown that the solution is a member of a one-parameter family of solutions, which includes the Bridgman and the Davidenkov and Spiridonova solutions.

The Stresses in a Thick Cylinder Having a Square Hole Under Concentrated Loading

1958 ◽  
Vol 25 (4) ◽  
pp. 571-574
Author(s):  
Masaichiro Seika
Keyword(s):  
Stress Distribution ◽  
Complex Variable ◽  
Complex Variable Method ◽  
Variable Method ◽  
Numerical Examples ◽  
Concentrated Loading ◽  
Thick Cylinder ◽  
Square Hole

Abstract This paper contains a solution for the stress distribution in a thick cylinder having a square hole with rounded corners under the condition of concentrated loading. The problem is investigated by the complex-variable method, associated with the name of N. I. Muskhelishvili. The unknown coefficients included in the solution are determined by the method of perturbation. Numerical examples of the solution are worked out and compared with the results available.

A Sequel to Technical Note 13: The Curved Tetrahedronal and Triangular Elements TEC and TRIC for the Matrix Displacement Method

1969 ◽  
Vol 73 (697) ◽  
pp. 55-65 ◽  
Author(s):  
J. H. Argyris ◽  
D. W. Scharpf
Keyword(s):  
Computational Analysis ◽  
Three Dimensional ◽  
Technical Note ◽  
Radius Of Curvature ◽  
Two Dimensional ◽  
Displacement Method ◽  
Stress Concentrations ◽  
The Past ◽  
The Matrix ◽  
Triangular Elements

It is by now well established that the computational analysis of significant problems in structural and continuum mechanics by the matrix displacement method often requires elements of higher sophistication than used in the past. This refers, in particular, to regions of steep stress gradients, which are frequently associated with marked changes in geometry, involving rapid variations of the radius of curvature. The philosophy underlying the idealisation of such configurations into finite elements was discussed in broad terms in ref. 1. It was emphasised that the so successful, constant strain, two-dimensional TRIM 3 and three-dimensional TET 4 elements do not, in general, prove the best choice. For this reason elements with a linear variation of strain like TRIM 6 and TET 10 were originally evolved and followed up with the quadratic strain elements TRIM 15, TRIA 4 (two-dimensional) and TET 20, TEA 8 (three-dimensional) of ref. 2. However, all these elements are characterised by straight edges and necessitate a polygonisation or polyhedrisation in the idealisation process. This may not be critical in many problems, but is sometimes of doubtful validity in the immediate neighbourhood of a curved boundary, where stress concentrations are most pronounced. To overcome this difficulty with a significant (local) increase of elements does not always yield the most economical and technically satisfactory solution. Moreover, there arises another inevitable shortcoming when dealing with TRIM and TET elements with a linear or quadratic variation of strain. Indeed, while TRIM 3 and TET 4 elements permit a very elegant extension into the realm of large displacements, this is not possible for the higher order TRIM and TET elements. This is simply due to the fact that TRIM 3 and TET 4 elements, by virtue of their specification, always remain straight under any magnitude of strain, but this is not so for the triangular and tetrahedron elements of higher sophistication.

Interfacial Stresses of a Coated Square Hole Induced by a Remote Uniform Heat Flow

2020 ◽  
Vol 12 (06) ◽  
pp. 2050063
Author(s):  
S. C. Tseng ◽  
C. K. Chao ◽  
F. M. Chen
Keyword(s):  
Heat Flow ◽  
Shape Factor ◽  
Coating Layer ◽  
Infinite Plate ◽  
Interfacial Stress ◽  
Graphical Form ◽  
Shear Stresses ◽  
Interfacial Stresses ◽  
Uniform Heat ◽  
Square Hole

This paper presents an analytical solution of a coated square hole embedded in an isotropic infinite plate under a remote uniform heat flow. Based on conformal mapping, analytic continuation theorem and the alternation technique, temperature and stress functions are derived in a compact series form. Results of temperature contours and interfacial stresses are validated using the finite element method. The comparison indicates the high accuracy of the proposed method. Numerical results of both the interfacial normal and shear stresses for different properties and geometric parameters of a coated layer are provided in a graphical form. The results indicate that the interfacial stresses are highly dependent on the thermal expansion coefficient, thickness of the coating layer and shape factor of the coated square hole. In conclusion, the interfacial shear stresses exhibit a significant increase at the corners with abrupt geometrical changes, which would cause the delamination of the coating layer system. Furthermore, increasing the thickness of the coating layer and the shape factor results in a higher interfacial stress.

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