Calculation of the christoffel-schwartz integral in the solution of two-dimensional problems of the theory of elasticity with the use of conformal mappings

1990 ◽  
Vol 22 (12) ◽  
pp. 1784-1786
Author(s):  
I. A. Bilyk ◽  
A. L. Shestopal
Keyword(s):  
Theory Of Elasticity ◽  
Conformal Mappings ◽  
Two Dimensional

A Study on Optimum Topology Using Conformal Mapping

2000 ◽  
Author(s):  
Satoshi Kitayama ◽  
Hiroshi Yamakawa
Keyword(s):  
Conformal Mapping ◽  
Fluid Mechanics ◽  
Laplace Equation ◽  
Coordinate Transformation ◽  
Theory Of Elasticity ◽  
Conformal Mappings ◽  
Two Dimensional ◽  
Numerical Examples ◽  
Planar Structures ◽  
Optimum Topology

Abstract This paper presents a method to determine optimum topologies of two dimensional elastic planar structures by using conformal mappings. We use the conformal mappings which is known to be effective in two dimensional fluid mechanics, electromagnetics and elasticity by complex coordinate transformation. We show that two invariants of stress can satisfy the Laplace equation, and then we clarify that corresponding relationships between fluid mechanics and electromagnetics can also be valid in the theory of elasticity. Then, presented a method to obtain optimum topologies is easier than by the conventional methods. We treated several numerical examples by the presented method. Through numerical examples, we can examine the effectiveness of the proposed method.

Equivalent Circuits of the Elastic Field

1944 ◽  
Vol 11 (3) ◽  
pp. A149-A161
Author(s):  
Gabriel Kron
Keyword(s):  
Steady State ◽  
Reference Frames ◽  
Three Dimensional ◽  
Elastic Field ◽  
Companion Paper ◽  
Theory Of Elasticity ◽  
Equivalent Circuits ◽  
Two Dimensional ◽  
Arbitrary Boundary ◽  
Nonlinear Stress

Abstract This paper presents equivalent circuits representing the partial differential equations of the theory of elasticity for bodies of arbitrary shapes. Transient, steady-state, or sinusoidally oscillating elastic-field phenomena may now be studied, within any desired degree of accuracy, either by a “network analyzer,” or by numerical- and analytical-circuit methods. Such problems are the propagation of elastic waves, determination of the natural frequencies of vibration of elastic bodies, or of stresses and strains in steady-stressed states. The elastic body may be non-homogeneous, may have arbitrary shape and arbitrary boundary conditions, it may rotate at a uniform angular velocity and may, for representation, be divided into blocks of uneven length in different directions. The circuits are developed to handle both two- and three-dimensional phenomena. They are expressed in all types of orthogonal curvilinear reference frames in order to simplify the boundary relations and to allow the solution of three-dimensional problems with axial and other symmetry by the use of only a two-dimensional network. Detailed circuits are given for the important cases of axial symmetry, cylindrical co-ordinates (two-dimensional) and rectangular co-ordinates (two- and three-dimensional). Nonlinear stress-strain relations in the plastic range may be handled by a step-by-step variation of the circuit constants. Nonisotropic bodies and nonorthogonal reference frames, however, require an extension of the circuits given. The circuits for steady-state stress and small oscillation phenomena require only inductances and capacitors, while the circuits for transients require also standard (not ideal) transformers. A companion paper deals in detail with numerical and experimental methods to solve the equivalent circuits.

Separation of the 3D stress state of a loaded plate into two-dimensional tasks: bending and symmetric compression of the plate

2021 ◽  
Vol 103 (3) ◽  
pp. 53-62
Author(s):  
Victor Revenko ◽  
Andrian Revenko
Keyword(s):  
Boundary Conditions ◽  
Harmonic Functions ◽  
Three Dimensional ◽  
Partial Solution ◽  
Theory Of Elasticity ◽  
Two Dimensional ◽  
Normal Stresses ◽  
Equilibrium Equations ◽  
Dimensional Boundary ◽  
The Right

The three-dimensional stress-strain state of an isotropic plate loaded on all its surfaces is considered in the article. The initial problem is divided into two ones: symmetrical bending of the plate and a symmetrical compression of the plate, by specified loads. It is shown that the plane problem of the theory of elasticity is a special case of the second task. To solve the second task, the symmetry of normal stresses is used. Boundary conditions on plane surfaces are satisfied and harmonic conditions are obtained for some functions. Expressions of effort were found after integrating three-dimensional stresses that satisfy three equilibrium equations. For a thin plate, a closed system of equations was obtained to determine the harmonic functions. Displacements and stresses in the plate were expressed in two two-dimensional harmonic functions and a partial solution of the Laplace equation with the right-hand side, which is determined by the end loads. Three-dimensional boundary conditions were reduced to two-dimensional ones. The formula was found for experimental determination of the sum of normal stresses via the displacements of the surface of the plate.

Stress Analysis and the Characteristics of T-Shape Bolted Joints Under Tensile Loadings

2005 ◽  
Author(s):  
Toshiyuki Sawa ◽  
Seiichi Hamamoto
Keyword(s):  
Bending Moment ◽  
Theory Of Elasticity ◽  
Bolted Joints ◽  
Load Factor ◽  
Two Dimensional ◽  
Bolted Joint ◽  
Dimensional Theory ◽  
Matching Method ◽  
Point Matching ◽  
Bolt Stress

In designing a bolted joint, it is important to examine the interface stress distribution (clamping effect) and to estimate the load factor, that is the ratio of an additional axial bolt force to a load. In order to improve the clamping effect raised faces of the interface have been used. But these interfaces in bolted joints have been designed empirically and the theoretical grounds are not made clear. In the present paper, in the case of T-shaped flanges with raised faces the clamping effect is analyzed by a two-dimensional theory of elasticity and the point matching method. Then, the load factor is analyzed. Moreover, with the application of the load a bending moment is occurred in bolts and the stress is added due to this bending moment. The bending moment in the bolt is also analyzed. In order to verify these analyses experiments to measure the load factor and the maximum bolt stress were carried out. The values of the load factor and the load when interface start to separate are compared with those of the joints with flat-faces. The analytical results are in fairly good agreements with the experimental ones.

Exact Stress Distribution in Standard Gear Teeth and Geometry Factors

1973 ◽  
Vol 95 (4) ◽  
pp. 1159-1163 ◽  
Author(s):  
C. N. Baronet ◽  
G. V. Tordion
Keyword(s):  
Stress Distribution ◽  
Gear Tooth ◽  
Theory Of Elasticity ◽  
Tooth Profile ◽  
Two Dimensional ◽  
Dimensional Theory ◽  
Concentrated Load ◽  
Gear Teeth ◽  
Pressure Angle ◽  
Surface Stresses

Using the two-dimensional theory of elasticity and an appropriate transform function, the stress distribution in a gear tooth acted on by a concentrated load has been obtained. Computations were carried out for the 20 and 25-deg pressure angle, standard full-depth system, for numbers of teeth ranging from 20 to 150. The intensities of the maximum static surface stresses along the root fillets are given for different loading positions on the tooth profile. Some of the results are compared with others found in the literature.

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