scholarly journals Dynamic Buckling of an Axially Compressed Cylindrical Shell With Discrete Rings and Stringers

1973 ◽  
Vol 40 (3) ◽  
pp. 736-740 ◽  
Author(s):  
C. A. Fisher ◽  
C. W. Bert
Keyword(s):  
Cylindrical Shell ◽  
Circular Cylindrical Shell ◽  
Nonlinear Differential Equations ◽  
Dynamic Buckling ◽  
Algebraic Equations ◽  
Discrete Elements ◽  
Governing Equations ◽  
Aircraft Cabins ◽  
Dirac Delta ◽  
Aircraft Cabin

As an exploratory effort toward improving the crashworthiness of light aircraft cabins, a theoretical analysis was made to predict the dynamic buckling load and buckling time of a stiffened, thin-walled circular cylindrical shell. To provide for the large stiffener spacing in light aircraft, the stiffeners were considered as discrete elements by means of a Dirac delta procedure. The nonlinear governing equations were derived using Hamilton’s principle and the final equations were obtained by means of Galerkin’s method. Solution was carried out by using a Gauss-Jordan technique on the algebraic equations and a Runge-Kutta technique on the nonlinear differential equations. Numerical results are presented for an idealized model of a typical light aircraft cabin.

Free Vibration of a Circular Cylindrical Shell Elastically Restrained by Axially Spaced Springs

1983 ◽  
Vol 50 (3) ◽  
pp. 544-548 ◽  
Author(s):  
T. Irie ◽  
G. Yamada ◽  
Y. Muramoto
Keyword(s):  
Cylindrical Shell ◽  
Free Vibration ◽  
Circular Cylindrical Shell ◽  
Frequency Equation ◽  
Mode Shapes ◽  
Governing Equations ◽  
Structure Matrix ◽  
The Matrix ◽  
Circular Cylindrical Shells ◽  
Elastically Restrained

An analysis is presented for the free vibration of a circular cylindrical shell restrained by axially spaced elastic springs. The governing equations of vibration of a circular cylindrical shell are written as a coupled set of first-order differential equations by using the transfer matrix of the shell. Once the matrix has been determined, the entire structure matrix is obtained by the product of the transfer matrices and the point matrices at the springs, and the frequency equation is derived with terms of the elements of the structure matrix under the boundary conditions. The method is applied to circular cylindrical shells supported by axially equispaced springs of the same stiffness, and the natural frequencies and the mode shapes of vibration are calculated numerically.

Dynamic Buckling of a Damped Externally Pressurized Imperfect Cylindrical Shell

1979 ◽  
Vol 46 (2) ◽  
pp. 372-376 ◽  
Author(s):  
D. F. Lockhart
Keyword(s):  
Cylindrical Shell ◽  
Circular Cylindrical Shell ◽  
Asymptotic Expression ◽  
Simple Relation ◽  
Dynamic Buckling ◽  
Fourier Component ◽  
Buckling Mode ◽  
Buckling Loads ◽  
Imperfect Cylindrical Shell ◽  
Step Loading

The dynamic buckling of a finite damped imperfect circular cylindrical shell which is subjected to step-loading in the form of lateral or hydrostatic pressure is examined by means of a perturbation method. The imperfection is assumed to be small. An asymptotic expression for the dynamic buckling load is obtained in terms of the damping coefficient and the Fourier component of the imperfection in the shape of the classical buckling mode. A simple relation which is independent of the imperfection is then obtained between the static and dynamic buckling loads.

Excitation of an Elastic Cylindrical Shell by a Transient Acoustic Wave

1969 ◽  
Vol 36 (3) ◽  
pp. 459-469 ◽  
Author(s):  
T. L. Geers
Keyword(s):  
Cylindrical Shell ◽  
Acoustic Wave ◽  
Circular Cylindrical Shell ◽  
Elastic Cylindrical Shell ◽  
Sandwich Shell ◽  
Governing Equations ◽  
Fluid Medium ◽  
Method Of Solution ◽  
Plane Step ◽  
Strain Responses

An infinite, elastic, circular cylindrical shell submerged in an infinite fluid medium is engulfed by a transverse, transient acoustic wave. The governing equations for modal shell response are reduced through the application of a new method of solution to two simultaneous equations in time; these equations are particularly amenable to solution by machine computation. Numerical results are presented for the first six modes of a uniform sandwich shell submerged in water and excited by a plane step-wave. These results are then used to evaluate the accuracy of a number of approximations which have been employed previously to treat this and similar problems. The results are also used to compute displacement, velocity, and flexural strain responses at certain points in the sandwich shell.

Free Vibrations of Reinforced Elastic Shells

1969 ◽  
Vol 36 (4) ◽  
pp. 835-844 ◽  
Author(s):  
Hyman Garnet ◽  
Alvin Levy
Keyword(s):  
Cylindrical Shell ◽  
Wide Class ◽  
Circular Cylindrical Shell ◽  
Normal Modes ◽  
Free Vibrations ◽  
Equivalent Model ◽  
Governing Equations ◽  
Interaction Forces ◽  
Elastic Shells ◽  
Loading System

A technique is presented for the analysis of a wide class of reinforced, elastic structures undergoing free vibrations while subject to constraints imposed by the reinforcing elements. The technique consists of replacing the constrained structure by an equivalent model, a structure without reinforcing elements, undergoing free vibrations while subject to a loading system which consists of the structure-reinforcing element interaction forces. These forces are introduced as displacement-dependent loads, whose magnitudes reflect the elastic and inertial properties of the reinforcing elements. The displacements of the constrained body are expanded in terms of the normal modes of the unconstrained body. This approach leads to a set of manageable governing equations describing the behavior of the reinforced body exactly. A solution to these equations may then be obtained to any desired degree of accuracy. The technique is illustrated by computations performed for the case of a ring reinforced, circular cylindrical shell.

Vibration of a Cracked Cylindrical Shell of Rectangular Planform

1985 ◽  
Vol 52 (4) ◽  
pp. 927-932
Author(s):  
R. Solecki ◽  
F. Forouhar
Keyword(s):  
Cylindrical Shell ◽  
Fourier Transformation ◽  
Circular Cylindrical Shell ◽  
Homogeneous System ◽  
Algebraic Equations ◽  
Discontinuous Functions ◽  
Harmonic Vibrations ◽  
Linear Algebraic Equations ◽  
Special Cases ◽  
Gauss Theorem

Harmonic vibrations of a circular, cylindrical shell of rectangular planform and with an arbitrarily located crack, are investigated. The problem is described by Donnell’s equations and solved using triple finite Fourier transformation of discontinuous functions. The unknowns of the problem are the discontinuities of the slope and of three displacement components across the crack. These last quantities are replaced, using constitutive equations, by curvatures and strain in order to improve convergence and to represent explicitly the singularities at the tips. The formulas for differentiation of discontinuous functions are derived using Green-Gauss theorem. Application of the boundary conditions at the crack leads to a homogeneous system of linear algebraic equations. The frequencies are obtained from the characteristic equation resulting from this system. Numerical results for special cases are provided.

Stress Concentration in a Stretched Cylindrical Shell With Two Equal Circular Holes

1975 ◽  
Vol 42 (1) ◽  
pp. 105-109 ◽  
Author(s):  
P. Seide ◽  
A. S. Hafiz
Keyword(s):  
Cylindrical Shell ◽  
Stress Concentration ◽  
Stress Distribution ◽  
Circular Cylindrical Shell ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Wide Range ◽  
Circular Holes ◽  
Limiting Case ◽  
Infinite Set

In this investigation, the stress distribution due to uniaxial tension of an infinitely long, thin, circular cylindrical shell with two equal small circular holes located along a generator is obtained. The problem is solved by the superposition of solutions previously obtained for a cylinder with a single circular hole. The satisfaction of boundary conditions on the free surfaces of the holes, together with uniqueness and overall equilibrium conditions, yields an infinite set of linear algebraic equations involving Hankel and Bessel functions of complex argument. The stress distribution along the boundaries of the holes and the interior of the shell is investigated. In particular, the value of the maximum stress is calculated for a wide range of parameters, including the limiting case in which the holes almost touch and the limiting case in which the radius of the cylinder becomes very large. As is the case for a flat plate, the stress-concentration factor is reduced by the presence of another hole.

The limit state of concrete and reinforced concrete rods at complex and longitudinal-transverse bending

2020 ◽  
pp. 60-73
Author(s):  
Yu V Nemirovskii ◽  
S V Tikhonov
Keyword(s):  
General Solution ◽  
Least Squares ◽  
Nonlinear Differential Equations ◽  
Limit State ◽  
Algebraic Equations ◽  
Method Of Least Squares ◽  
Elasticity Limit ◽  
Limit Values ◽  
Symbolic Calculations ◽  
Deformation Diagrams

The work considers rods with a constant cross-section. The deformation law of each layer of the rod is adopted as an approximation by a polynomial of the second order. The method of determining the coefficients of the indicated polynomial and the limit deformations under compression and tension of the material of each layer is described with the presence of three traditional characteristics: modulus of elasticity, limit stresses at compression and tension. On the basis of deformation diagrams of the concrete grades B10, B30, B50 under tension and compression, these coefficients are determined by the method of least squares. The deformation diagrams of these concrete grades are compared on the basis of the approximations obtained by the limit values and the method of least squares, and it is found that these diagrams approximate quite well the real deformation diagrams at deformations close to the limit. The main problem in this work is to determine if the rod is able withstand the applied loads, before intensive cracking processes in concrete. So as a criterion of the conditional limit state this work adopts the maximum permissible deformation value under tension or compression corresponding to the points of transition to a falling branch on the deformation diagram level in one or more layers of the rod. The Kirchhoff-Lyav classical kinematic hypotheses are assumed to be valid for the rod deformation. The cases of statically determinable and statically indeterminable problems of bend of the rod are considered. It is shown that in the case of statically determinable loadings, the general solution of the problem comes to solving a system of three nonlinear algebraic equations which roots can be obtained with the necessary accuracy using the well-developed methods of computational mathematics. The general solution of the problem for statically indeterminable problems is reduced to obtaining a solution to a system of three nonlinear differential equations for three functions - deformation and curvatures. The Bubnov-Galerkin method is used to approximate the solution of this equation on the segment along the length of the rod, and specific examples of its application to the Maple system of symbolic calculations are considered.

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