Two-Dimensional Nonlinear Shell Equations at a Glance1 1Nonlinear shallow shell equations in curvilinear coordinates are not reproduced here, as their justifications rely on asymptotic analyses similar to those of Vol. II, Sects. 4.14 and 5.12; they are nevertheless reviewed in Sect. 11.3.

A great deal of published literature exists which analyzes the free vibrations of turbomachinery blades by means of one-dimensional beam theories. Recently, a more accurate, two-dimensional analysis method has been developed based upon shallow shell theory. The present paper summarizes the two types of theories and makes quantitative comparisons of frequencies obtained by them. Numerical results are presented for cambered and/or twisted blades of uniform thickness. Significant differences between the theories are found to occur, especially for low aspect ratio blades. The causes of these differences are discussed.

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Pressurized Cylindrical Shell With a Rigid Circular Inclusion

Journal of Pressure Vessel Technology ◽

10.1115/1.3454446 ◽

1978 ◽

Vol 100 (2) ◽

pp. 158-163 ◽

Cited By ~ 1

Author(s):

D. H. Bonde ◽

K. P. Rao

Keyword(s):

Differential Equation ◽

Cylindrical Shell ◽

Shallow Shell ◽

Circular Inclusion ◽

Hankel Functions ◽

Curvature Parameter ◽

Shell Equations ◽

Limiting Case ◽

Rigid Circular Inclusion ◽

Good Agreement

The effect of a rigid circular inclusion on stresses in a cylindrical shell subjected to internal pressure has been studied. The two linear shallow shell equations governing the behavior of a cylindrical shell are converted into a single differential equation involving a curvature parameter and a potential function in nondimensionalized form. The solution in terms of Hankel functions is used to find membrane and bending stressses. Boundary conditions at the inclusion shell junction are expressed in a simple form involving the in-plane strains and change of curvature. Good agreement has been obtained for the limiting case of a flat plate. The shell results are plotted in nondimensional form for ready use.

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A 2D Numerical Model for Simulation of Two-Dimensional Circular Dam-Break

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.130-134.2993 ◽

2011 ◽

Vol 130-134 ◽

pp. 2993-2996

Author(s):

Ming Qin Liu ◽

Y.L. Liu

Keyword(s):

Solution Procedure ◽

Dam Break ◽

Curvilinear Coordinates ◽

Two Dimensional ◽

Proposed Model ◽

Curvilinear Coordinate ◽

Orthogonal Curvilinear Coordinate System ◽

Orthogonal Curvilinear Coordinates ◽

Averaged Model ◽

The Mathematical Model

The purpose of this paper is to present a 2D depth-averaged model under orthogonal curvilinear coordinates for simulating two-dimensional circular dam-break flows. The proposed model uses an orthogonal curvilinear coordinate system efficiently and accurately to simulate the flow field with irregular boundaries. As for the numerical solution procedure, The SIMPLEC solution procedure has been used for the transformed governing equations in the transformed domain. Practical application of the model is illustrated by an example, which demonstrates that the mathematical model can capture hydraulic discontinuities accurately such as steep fronts, hydraulic jump and drop, etc.

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