Equations For Circular Cylindrical Shells Research Articles

Free vibration analysis of porous laminated rotating circular cylindrical shells

In this research, investigations are presented of the free vibration of porous laminated rotating circular cylindrical shells based on Love’s shell theory with simply supported boundary conditions. The equilibrium equations for circular cylindrical shells are obtained using Hamilton’s principle. Also, Navier’s solution is used to solve the equations of the cylindrical shell due to the simply supported boundary conditions. The results are compared with previous results of other researchers. The numerical result of this study indicates that with increase of the porosity coefficient the nondimensional backward and forward frequency decreased. Then the results of the free vibration of rotating cylindrical shells are presented in terms of the effects of porous coefficients, porous type, length to radius ratio, rotating speed, and axial and circumferential wave numbers.

Simplified non-linear dynamical equations of circular cylindrical shells

Simplified non-linear dynamical equations of circular cylindrical shells are obtained on the basis of asymptotic analysis. The original partial differential equations (PDEs), which have eighth order in the longitudinal coordinate, are replaced by two sets of equations that have fourth order on this coordinate. The first set contains non-linear and dynamic PDEs. They describe the oscillations in the inner region of the shell. For small non-linearities, they coincide with the well-known linear semi-membrane dynamical equations. Ordinary differential equations of the edge effect allow one to satisfy all the boundary conditions. They are linear and quasistatic. An important question about the boundary conditions for the simplified equations is considered. We also discuss the solution of the obtained boundary value problems.

Stress analysis of axisymmetric shear deformable cross-ply laminated circular cylindrical shells

A generalized mixed theory for bending analysis of axisymmetric shear deformable laminated circular cylindrical shells is presented. The classical, first-order and higher-order shell theories have been used in the analysis. The Maupertuis–Lagrange (M–L) mixed variational formula is utilized to formulate the governing equations of circular cylindrical shells laminated by orthotropic layers. Analytical solutions are presented for symmetric and antisymmetric laminated circular cylindrical shells under sinusoidal loads and subjected to arbitrary boundary conditions. Numerical results of the higher-order theory for deflections and stresses of cross-ply laminated circular cylindrical shells are compared with those obtained by means of the classical and first-order shell theories. The effects, due to shear deformation, lamination schemes, loadings ratio, boundary conditions and orthotropy ratio on the deflections and stresses are investigated.

ON THE NATURAL FREQUENCIES OF THIN SIMPLY SUPPORTED CYLINDRICAL SHELLS

Within the context of the Donnell - Mushtary - Vlasov equations for thin circular cylindrical shells, the fundamental frequency of simply supported circular cylindrical shells is determined. In this work, it is shown that with regard to the frequency equation for circular cylindrical shells, there exists a quantity Ω = (ωmna)2 which must remain constant and invariant once we fix the slenderness and the thickness ratios l/a and a/h. The quadruple logarithmic coupling among the parameters ω, l/a, a/h and m is fully displayed by means of a set of parallel straight lines. Furthermore, the nature and influence of the longitudinal wave number m on the natural frequencies ωmn is fully discussed.

On A Formulation of the Elastic Stability Equations of Circular Cylindrical Shells Under Variable Internal Pressure

The mathematical model of the stability analysis of circular cylindrical shells under arbitrary internal pressure is presented. The paper consists of a direct analysis of the equilibrium modes in the neighbourhood of the unperturbed principal equilibrium path. The final stability condition results in a completely symmetric differential operator which is then compared with current theories found in the literature.

Theory of Orthotropic and Composite Cylindrical Shells, Accurate and Simple Fourth-Order Governing Equations

A pair of complex conjugate fourth-order differential equations that govern the deformation of orthotropic circular cylindrical shells is presented. As shown in the paper, this pair of equations is as accurate as equations can be within the scope of the Kirchhoff assumptions. Also presented for the first time are several pairs of accurate and simple fourth-order equations which can be systematically and explicitly deduced from the previously mentioned pair of equations. Because of their accuracy and simplicity, these simple equations are of practical importance. The advantage in applying those fourth-order equations presented herein is that their solutions can be easily found in simple closed forms. This considerably simplifies calculations for solving problems of orthotropic and composite cylindrical shells as well as isotropic shells as a special case. Unlike other equations known in the literature, their general solutions remain unknown because of the algebraic complexities involved. The present method of deducing simple fourth-order equations improves upon the one used for isotropic shells in the first author’s previous paper entitled “Accurate Fourth-Order Equations for Circular Cylindrical Shells,” Journal of the Engineering Mechanics Division, ASCE, 1972.

Influence of loading behaviour on the stability of cylindrical shells

In order to investigate the influence of motion-dependency of loading on the form of stability equations of circular cylindrical shells, certain types of deformation-dependent forces are characterized. The stability equations of rings and columns acted upon by motion-dependent (and possibly non-conservative) forces are obtained as special cases. Some new aspects of ring instability such as tension-induced modes of instability and rigid body modes are also studied.

Isotropic Cylindrical Shells under Line Load

Morley's equations for circular cylindrical shells are solved for uniform line load along a generator. The solution is compared with those obtained from the equations by Flugge, Novozhilov, and Donnell. Extensive experimental work on cylindrical shells is reported and detailed data are provided. The new solution produces good results for most practical shell geometries. The results indicate that root comparisons of the characteristic equations are not reliable in predicting the accuracy of a particular set of equations. Based on the Flugge equations the applicable range of parameters for which the solutions are valid is given and the computational effort of the four solutions is assessed.

On Asymptotic Expansions and Error Bounds in the Derivation of Two‐Dimensional Shell Theory

Known results on asymptotic two‐dimensional equations for circular cylindrical shells, including the effects of transverse shear and normal stress deformation, are supplemented by upper‐ and lower‐bound determinations of influence coefficients, using minimum‐potential and complementary energy principles in conjunction with asymptotic‐expansion results. The new bound analysis shows that the consequences of the asymptotic two‐dimensional theory are in exact agreement, except for terms which are small of higher order, with the corresponding consequences of three‐dimensional theory, for some classes of edge conditions. The analysis also shows the nature of the differences between results of two‐ and three‐dimensional theory, as a function of geometrical and elastic parameters, where this difference is of importance because of the effect of a St. Venant boundary layer.

Bending of thin circular rings

The basic equations for the bending of circular rings are deduced from a set of accurate equations for circular cylindrical shells. The advantages in using these differential equations as compared with the customary energy method are shown through examples. It turns out that solutions of these equations can be as easily obtained as solutions of the well-known differential equation for straight beams. It is also shown that the center line of the ring is essentially inextensible, which is assumed ab initio in the classical ring theory.

On an Accurate Theory for Circular Cylindrical Shells

The accuracy of the characteristic roots obtained from solving a pair of complex conjugate fourth-order differential equations which govern the deformation of circular cylindrical shells is studied. Roots of the characteristic equations of the pair of fourth-order equations can be readily obtained in simple closed forms, unlike the Flu¨gge and other unreduced equations for circular cylindrical shells for which roots of the characteristic equations can only be found approximately by complicated numerical processes. In the present paper, roots from the pair of equations are computed for a range of significant parameters and comparisons are made with solutions based on other known equations. These results show that the pair of complex conjugate fourth-order equations has at least the same accuracy as the Flu¨gge equation and is as accurate as an equation can be within the scope of the Kirchhoff assumptions. The pair of complex conjugate governing equations for the homogeneous solutions of circular cylindrical shells is as follows: Lw=0andLw¯=0, in which L=(∇2+1)∇2+i1k∂2∂α2+k(1−ν)∂2∂β2−∂2∂α2∇2+∂2∂β2,L¯=thelinearcomplexconjugateoperatorofL,k=h2a3(1−ν2),h=thickness,a=radiusofcylinder,ν=Poisson’sratio,w=radialdisplacement. The particular solutions can be found from nonhomogeneous equation LLw¯=a4D∇4Z. It may be shown that the simplified equations, such as Donnell, Morley, Novozhilov, and other equations can be readily obtained from the present equation.

On reductions of the differential equations for circular cylindrical shells

The paper begins with a statement of the differential equations of equilibrium and compatibility for circular cylindrical shells in the form first given in a pioneering work by Schaefer, and with constitutive equations incorporating transverse shear deformations and moment components normal to the shell surface as stated earlier by the present author. It is then shown that this system of equations may be reduced, for a rather general class of orthotropic shells, to four simultaneous third-order equations for two stress functions and two displacement variables. Subsequent to this it is shown that for the case that transverse shear deformations and shell surface normal moment components are absent the four simultaneous equations degenerate in such a manner that a further reduction to two simultaneous fourth order equations, for one stress function and one displacement variable, becomes possible. For the case of the isotropic shell these are equivalent to results by Morley, Simmonds, and Wan which were obtained by somewhat different reduction procedures.

On the Equations of the Linear Theory of Elastic Conical Shells

In a recent article [19], it was shown that the equations of the linear theory of elastic shells of revolution under arbitrary loads, after a harmonic analysis in the polar angle of the base plane of the shell, can be reduced to two simultaneous fourth order ordinary differential equations for a stress function and a displacement function. The exact reduction procedure of [19] was applied to a circular cylindrical shell. Also mentioned was the possibility of a simplified procedure based on the fact that there is an inherent error in shell theory of the order h/R, where h is the shell thickness and R is a representative magnitude of the two principal radii of curvature. In the present paper, the exact and the simplified procedure of [19] will be applied to a conical shell frustrum. The simplified procedure leads to two differential equations as well as to auxiliary equations for the calculation of stress resultants and couples which differ from those of the exact procedure only in terms which are O(h/R). The differential equations obtained here are very similar in form to those of shallow shell theory [6,8] as well as to those of the Donnell type theory [15]. They do in fact reduce to the shallow shell equations when the slope angle /!? between a meridian and the base plane is very small. On the other hand, for /3 = z/2, they reduce to the “correct” equations for circular cylindrical shells of [16,19] while the results of [15] reduce only to the Donnell equation. Aside from their usefulness in obtaining the solution of specific problems, our results also delimit the range of applicability of the Donnell type equations for conical shells previously obtained in [5,15].

Asymptotic Solution for Infinite Cylinder Reinforced by Radially Loaded Ring

An asymptotic expansion procedure proposed by E. Reissner is applied to the simple Donnell equations for circular cylindrical shells, and a simple explicit solution is obtained for the interaction problem of an infinite cylinder reinforced by a central ring subjected to a nonuniform radial load. It is shown that the sign of the dominant stresses depends entirely upon whether the shell is reinforced by inside or outside rings. In addition, even when only the lower harmonic contributions are retained, good agreement with an exact solution for the case of concentrated loads is achieved.