Analysis of Plane Stress in Polar Co-ordinates and With Varying Thickness

1959 ◽  
Vol 26 (3) ◽  
pp. 437-439
Author(s):  
H. D. Conway
Keyword(s):  
General Solution ◽  
Plane Stress ◽  
Stress Function ◽  
Constant Thickness ◽  
Numerical Example ◽  
Arbitrary Power ◽  
Varying Thickness

Abstract The biharmonic stress-function equation for plane stress in polar co-ordinates is generalized by assuming that the thickness varies with the radius. In order to make this equation tractable, the thickness is assumed to be proportional to an arbitrary power of the radius. Michell’s general solution for the constant-thickness case is then generalized for this variation of thickness. As a numerical example, the solution is given for the problem of a segmental plate bent by end couples, the constant-thickness version of this well-known problem being due to Golovin.

Axially Symmetrical Plates With Linearly Varying Thickness

1951 ◽  
Vol 18 (2) ◽  
pp. 140-142
Author(s):  
H. D. Conway
Keyword(s):  
General Solution ◽  
Closed Form ◽  
Variable Thickness ◽  
Practical Problem ◽  
Constant Thickness ◽  
Circular Plates ◽  
Varying Thickness ◽  
Small Deflection ◽  
Special Case ◽  
Plates Of Variable Thickness

Abstract The most practical problem in the bending of symmetrically loaded circular plates of variable thickness is probably that in which the thickness decreases linearly with the distance from the center of the plate. A general solution of the small-deflection problem of such plates is given here in closed form for the special case when Poisson’s ratio is 1/3. Numerical results are given for two particular examples, and these are compared with the results for corresponding plates of constant thickness.

Exact solution for free vibration analysis of linearly varying thickness FGM plate using Galerkin-Vlasov’s method

2020 ◽  
pp. 146442072098049
Author(s):  
V Kumar ◽  
SJ Singh ◽  
VH Saran ◽  
SP Harsha
Keyword(s):  
Free Vibration ◽  
Variable Thickness ◽  
Vibration Analysis ◽  
Shear Deformation Theory ◽  
Free Vibration Analysis ◽  
Functionally Graded ◽  
Constant Thickness ◽  
Engineering Structures ◽  
Property Variation ◽  
Varying Thickness

The present paper investigates the free vibration analysis for functionally graded material plates of linearly varying thickness. A non-polynomial higher order shear deformation theory is used, which is based on inverse hyperbolic shape function for the tapered FGM plate. Three different types of material gradation laws, specifically: a power (P-FGM), exponential (E-FGM), and sigmoid law (S-FGM) are used to calculate the property variation in the thickness direction of FGM plate. The variational principle has been applied to derive the governing differential equation for the plates. Non-dimensional frequencies have been evaluated by considering the semi-analytical approach viz. Galerkin-Vlasov’s method. The accuracy of the preceding formulation has been validated through numerical examples consisting of constant thickness and tapered (variable thickness) plates. The findings obtained by this method are found to be in close agreement with the published results. Parametric studies are then explored for different geometric parameters like taper ratio and boundary conditions. It is deduced that the frequency parameter is maximum for S-FGM tapered plate as compared to E- and P-FGM tapered plate. Consequently, it is concluded that the S-FGM tapered plate is suitable for those engineering structures that are subjected to huge excitations to avoid resonance conditions. In addition, it is found that the taper ratio is significantly affected by the type of constraints on the edges of the tapered FGM plate. Some novel results for FGM plate with variable thickness are also computed that can be used as benchmark results for future reference.

Generalized plane stress in a semi-infinite strip under arbitrary end-load

1964 ◽  
Vol 281 (1385) ◽  
pp. 184-206 ◽  
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Boundary Conditions ◽  
Fourth Order ◽  
Plane Stress ◽  
Stress Function ◽  
Infinite Strip ◽  
Order Ordinary Differential Equation ◽  
Orthogonal Functions

The problem involves the determination of a biharmonic generalized plane-stress function satisfying certain boundary conditions. We expand the stress function in a series of non-orthogonal eigenfunctions. Each of these is expanded in a series of orthogonal functions which satisfy a certain fourth-order ordinary differential equation and the boundary conditions implied by the fact that the sides are stress-free. By this method the coefficients involved in the biharmonic stress function corresponding to any arbitrary combination of stress on the end can be obtained directly from two numerical matrices published here The method is illustrated by four examples which cast light on the application of St Venant’s principle to the strip. In a further paper by one of the authors, the method will be applied to the problem of the finite rectangle.

A System of Matrix Equations over the Quaternion Algebra with Applications

2017 ◽  
Vol 24 (02) ◽  
pp. 233-253 ◽  
Author(s):  
Xiangrong Nie ◽  
Qingwen Wang ◽  
Yang Zhang
Keyword(s):  
General Solution ◽  
Quaternion Algebra ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Matrix Equations ◽  
Electronic Networks ◽  
Consistency Conditions ◽  
Numerical Example ◽  
System Of Matrix Equations ◽  
Necessary And Sufficient

We in this paper give necessary and sufficient conditions for the existence of the general solution to the system of matrix equations [Formula: see text] and [Formula: see text] over the quaternion algebra ℍ, and present an expression of the general solution to this system when it is solvable. Using the results, we give some necessary and sufficient conditions for the system of matrix equations [Formula: see text] over ℍ to have a reducible solution as well as the representation of such solution to the system when the consistency conditions are met. A numerical example is also given to illustrate our results. As another application, we give the necessary and sufficient conditions for two associated electronic networks to have the same branch current and branch voltage and give the expressions of the same branch current and branch voltage when the conditions are satisfied.

Spherical Symmetry in the Theory of Elasticity

1958 ◽  
Vol 25 (1) ◽  
pp. 136-140
Author(s):  
F. Goded
Keyword(s):  
General Solution ◽  
Stress Tensor ◽  
Axial Symmetry ◽  
Spherical Symmetry ◽  
Stress Function ◽  
Theory Of Elasticity ◽  
Plane Symmetry ◽  
The Subject ◽  
The Common

Abstract Part 1 of the paper presents a study of the common characteristics of both plane symmetry and axial symmetry in the theory of elasticity; viz., the form of the stress tensor and the existence of further analogous symmetries. In Part 2, the subject deals with the possibility of the existence of further analogous symmetries which are found to be possible only in some specific cases. In particular, spherical symmetry is treated. The method of obtaining the stress function of this new symmetry and the equation which this function must satisfy also are discussed, together with the stresses expressed by means of this stress function. The paper ends with a brief review of a general solution of the stress function and an example of the application of this stress function to a given problem.

Influence Coefficients for Hemispherical Shells With Small Openings at the Vertex

1955 ◽  
Vol 22 (1) ◽  
pp. 20-24
Author(s):  
G. D. Galletly
Keyword(s):  
Constant Thickness ◽  
Influence Coefficient ◽  
Circular Opening ◽  
Central Angle ◽  
Calculation Time ◽  
Numerical Example ◽  
Hemispherical Shell ◽  
Influence Coefficients ◽  
Hemispherical Shells

Abstract Three methods of obtaining the influence coefficients for a thin, constant-thickness, hemispherical shell with a circular opening at the vertex were investigated and utilized in a numerical example. Bearing in mind both accuracy and calculation time, it was concluded that when the total central angle subtended by the opening is less than approximately 30 deg, good results for the influence coefficient calculation will be obtained by using Method II in the text of the paper.

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