Gravitational Stresses in a Circular Ring Resting on Concentrated Support

1955 ◽  
Vol 22 (1) ◽  
pp. 103-106
Author(s):  
Yi-Yuan Yu
Keyword(s):  
Complex Variable ◽  
Analytic Functions ◽  
Present Author ◽  
Circular Ring ◽  
Complex Variable Method ◽  
Variable Method ◽  
Two Dimensional ◽  
Numerical Example ◽  
Gravitational Stress

Abstract Muschelišvili’s complex variable method in two-dimensional elasticity has been extended by the present author to solve a few gravitational stress problems. A further problem of a heavy circular ring resting on a concentrated support is solved by the same method in the present paper. The two analytic functions which constitute the solution of the problem are determined. A numerical example is given for a ring the radii of which are in the ratio of 1 to 2.

scholarly journals Complex potentials in two-dimensional elasticity

1945 ◽  
Vol 184 (997) ◽  
pp. 129-179 ◽  
Keyword(s):  
Complex Variable ◽  
Body Force ◽  
Displacement Function ◽  
The Body ◽  
Complex Variable Method ◽  
Complex Potentials ◽  
Variable Method ◽  
Two Dimensional ◽  
Special Cases ◽  
Airy’S Stress Function

This paper gives an approach to two-dimensional isotropic elastic theory (plane strain and generalized plane stress) by means of the complex variable resulting in a very marked economy of effort in the investigation of such problems as contrasted with the usual method by means of Airy’s stress function and the allied displacement function. This is effected (i) by considering especially the transformation of two-dimensional stress; it emerges that the combinations xx + yy , xx — yy + 2 ixy are all-important in the treatment in terms of complex variables; (ii) by the introduction of two complex potentials Ω( z ), ω( z ) each a function of a single complex variable in terms of . which the displacements and stresses can be very simply expressed. Transformation of the cartesian combinations u + iv , xx + yy , xx — yy + 2 ixy to the orthogonal curvilinear combinations u ξ + iu n , ξξ + ηη, ξξ - ηη + 2iξη is simple and speedy. The nature of "the complex potentials is discussed, and the conditions that the solution for the displacements shall be physically admissible, i.e. single-valued or at most of the possible dislocational types, is found to relate the cyclic functions of the complex potentials. Formulae are found for the force and couple resultants at the origin z = 0 equivalent to the stresses round a closed circuit in the elastic material, and these also are found to relate the cyclic functions of the complex potentials. The body force has bhen supposed derivable from a particular body force potential which includes as special cases (i) the usual gravitational body force, (ii) the reversed mass accelerations or so-called ‘centrifugal’ body forces of steady rotation. The power of the complex variable method is exhibited by finding the appropriate complex potentials for a very wide variety of problems, and whilst the main object of the present paper has been to extend the wellknown usefulness of the complex variable method in non-viscous hydrodynamical theory to two-dimensional elasticity, solutions have been given to a number of new problems and corrections made to certain other previous solutions.

scholarly journals Elastic Analysis for Subaqueous Tunnel Surrounding Rock via the Complex Variable Method

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Li-yuan Yu ◽  
Hong-wen Jing ◽  
Ying-chao Wang
Keyword(s):  
Complex Variable ◽  
Classical Problem ◽  
Image Plane ◽  
Circular Ring ◽  
Surrounding Rock ◽  
Complex Variable Method ◽  
Variable Method ◽  
Surface Boundary ◽  
Displacement Fields ◽  
Underwater Tunnel

Generally speaking, the subaqueous tunnels can be regarded as the shallow-buried ones. Consequently, the classical problem of an elastic half plane with a round cavity, loaded arbitrarily along the surface boundary, can be used to obtain the stress and displacement fields of the surrounding rock for this type of tunnels. The solution uses the complex variable method, with a conformal mapping onto a circular ring in the image plane. Because of the convergence of the complex potentials throughout the annular region, the coefficients in the Laurent series expansion form for complex functions can be determined by a system of liner recurrent equations, obtained from both the horizontal and the cavity boundary conditions. The stresses and deformations of the surrounding rock can then be calculated via some relevant equations. The whole calculation program should be coded by Fortran language. As an example, the case of a specific underwater tunnel is considered in some detail eventually.

A Complex-Variable Method for Two-Dimensional Internal Stress Problems and Its Applications to Crack Growth in Nonelastic Materials: Part I—Theory

1987 ◽  
Vol 54 (1) ◽  
pp. 59-64 ◽  
Author(s):  
K. C. Wu ◽  
C. Y. Hui
Keyword(s):  
Internal Stress ◽  
Crack Growth ◽  
Complex Variable ◽  
Complex Variable Method ◽  
Variable Method ◽  
Two Dimensional ◽  
Systematic Method

In Part I of this work, a systematic method based on a nonanalytic complex calculus is developed for two-dimensional internal stress problems. The method is employed to derive the internal stress due to a field of nonelastic strain, which can be nonzero at infinity, in an infinite uncracked or cracked body.

Complex Variable Method for Eigensolution Sensitivity Analysis

AIAA Journal ◽  
2006 ◽  
Vol 44 (12) ◽  
pp. 2958-2961 ◽  
Author(s):  
B. P. Wang ◽  
A. P. Apte
Keyword(s):  
Sensitivity Analysis ◽  
Complex Variable ◽  
Complex Variable Method ◽  
Variable Method

On Bending of a Flat Slab Supported by Square-Shaped Columns and Clamped

1954 ◽  
Vol 21 (3) ◽  
pp. 263-270
Author(s):  
S. Woinowsky-Krieger
Keyword(s):  
Stress Distribution ◽  
Conformal Mapping ◽  
Complex Variable ◽  
Complex Variable Method ◽  
Variable Method ◽  
Flat Slab ◽  
Flat Slabs ◽  
At Will

Abstract A solution is given in this paper for the problem of bending of an infinite flat slab loaded uniformly and rigidly clamped in square-shaped columns arranged to form the square panels of the slab. The complex variable method in connection with conformal mapping is used for this aim. Although not perfectly rigorous, the solution obtained is sufficiently accurate for practical purposes and, besides, it can be improved at will. Stress diagrams traced in a particular case of column dimensions do not wholly confirm the stress distribution, generally accepted in design of flat slabs.

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.

scholarly journals Complex Variable Solutions for Forces and Displacements of Circular Lined Tunnels

2018 ◽  
Vol 2018 ◽  
pp. 1-12 ◽  
Author(s):  
Xingbo Han ◽  
Yongxu Xia ◽  
Xing Wang ◽  
Lunlei Chai
Keyword(s):  
Rock Mass ◽  
Complex Variable ◽  
Earth Pressure ◽  
Complex Variable Method ◽  
Complex Potentials ◽  
Variable Method ◽  
Lateral Earth Pressure ◽  
Series Expression ◽  
Stress Boundary Conditions ◽  
Stress Boundary

A complex variable method for solving the forces and displacements of circular lined tunnels is presented. Complex potentials for the stresses and displacements are expressed in the term of series expression. The undetermined coefficients of the complex potentials are obtained according to the stress boundary conditions along the lining inner surface and the displacement and surface traction boundary condition along the lining and rock-mass interface. Solutions for the stresses and displacements of the tunnel lining and rock-mass are then established by applying Muskhekishvili’s complex variable method. In addition, forces solutions for linings are presented based on the tangential stress at the two boundaries. Examples are finally established to reveal the applicability and accuracy of the proposed method. The effects of the degrees from the tunnel crown to the invert, coefficient of the lateral earth pressure, and distance from the rock-mass to the interface on the regulations of the lining forces and rock-mass stresses are also thoroughly investigated.

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