Equations For Cylindrical Shells Research Articles

Deformation and Strength of a Horizontal Reservoir Having an Oval Cross Section

A reservoir supported horizontally and partially by saddles yields a very large deformation. Tolerable limits of the deformation may vary according to the working conditions of a reservoir. In the present paper, the authors tried to find the shape of a reservoir which does not yield too much deformation under partial support condition. Reservoirs having an oval cross section which is formed by three circular arcs are investigated. Stresses in a reservoir and its deformations are found as the solutions of the differential equations for cylindrical shells introduced by one of the present authors. Experimental study is made along with the theoretical analysis. The results show that there exists an optimum oval form which holds the deformation of a reservoir under desired level.

Accurate Fourth Order Equation for Circular Cylindrical Shells

A pair of complex conjugate fourth-order partial differential equations which govern the deformation of circular cylindrical shells is presented. This pair of equations has at least the same accuracy as the Flugge equation and is as accurate as an equation can be within the scope of the Kirchhoff assumptions. Because they are complex conjugates, only one of the equations need be considered. The advantage in using the present pair of equations, besides their accuracy, is that the solutions of these equations can be readily found in simple closed forms as shown in the paper. This simplifies the calculations considerably in solving problems of cylindrical shells. Unlike the accurate Flugge equation and other equally accurate equations for cylindrical shells, their solutions can only be found approximately by trial and error.

Dynamic Analysis of Axially Non-Uniform Thin Cylindrical Shells

Part 1 of this paper presents a new theory for the dynamic and static analysis of axially non-uniform, thin cylindrical shells. It is a hybrid of finite element and classical shell theories: the shell is subdivided into cylindrical finite elements, and the displacements within each (expressed in terms of nodal displacements), i.e., the displacement functions, are obtained using Sanders' equations for thin cylindrical shells in full. Sanders' theory gives zero strains for small rigid-body motions, so that displacement functions based on it satisfy the convergence criteria of the finite-element method.Expressions for the mass, stiffness and stress-resultant matrices are obtained, and the method for constructing the equivalent global matrices is given. This paper is supported by Part 2, where the eigenvalues of a number of shells are calculated and compared with other theories and experiments.In Part 2, the free flexural vibration characteristics of thin cylindrical shells are studied by this new, hybrid finite-element theory, where the displacement functions are determined by solution of the cylindrical thin-shell equations. Uniform shells with simply-supported, clamped and free ends are studied, as well as ring-stiffened shells and shells with thickness discontinuities. The frequencies of vibration are compared with those obtained by other theories and with the experiments of others. Agreement with other theories is good and, in the majority of cases, is even better with the experiments.

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.

Strength of a Horizontal Reservoir Supported Partially by Equidistant Saddles

In a horizontal reservoir, partially supported by saddles, great stresses arise near the supports, and the magnitude of these stresses is closely related with the form of the saddle supports. In this paper, we discussed the strength of such a reservoir and its relationship to the form of the saddle supports, utilizing the fundamental differential equation of cylindrical shells, and we pointed out the importance of the form of the saddle supports. To decrease the magnitude of the stresses arising near the supports, we further made researches on the optimum form of the saddle supports. From this study, we knew that the deformation of such a reservoir was very large, and so on.

Cylindrical Shell Subject to Longitudinal Line Load

The utilization of a single Fourier series provides a solution to the problem of a simply-supported, closed circular cylindrical shell subjected to a varying longitudinal line load. Few of the previous studies of stresses and deformations of such a shell lead to practical results. One useful method for solving the problem under consideration is to first obtain an exact closed form solution for a sinusoidal line load by applying singularity functions. Next, a single Fourier series is formed to furnish a solution for a line load with an arbitrary intensity. The Novozhilov's equation of cylindrical shells is the most suitable for these procedures. Improvement of the convergence of the series for a segmental line load shorten the process of evaluating the critical stresses. Included in the article are mathematical proof of the convergence of the improved series, an estimate of error, and some numerical results.

Stability equations of cylindrical shells with finite precritical deflections

Stability equations of cylindrical shells with finite precritical deflections

Dynamic Thermoelastic Response of Cylindrical Shells

The dynamic, thermoelastic response of cylindrical shells to suddenly applied and rotating thermal inputs is investigated. Fully coupled, dynamic, thermoelastic cylindrical shell equations are derived using Galerkin’s method. Identical results were obtained independently using a variational theorem. Analytical solutions to these equations are formulated for finite-length and infinite-length cylinders. Numerical results for the response of infinite-length cylindrical shells to suddenly applied and rotating longitudinal lines of heat flux are presented. It is shown that for many thermoelastic problems involving moving thermal inputs that the maximum ratio of dynamic to quasi-static deflection can be much greater than two, that dynamic effects can be important for all thicknesses within the realm of thin shell theory, and that semicoupled theory gives incorrect results in some cases for which a fully coupled theory is required.

Thermoelasticity of heterogeneous orthotropic cylindrical shells

The axisymmetric linear quasi-static thermoelastic equations for composite orthotropic cylindrical shells are solved in closed form for fixed-end boundary conditions.The case of semi-infinite shell is explicitly evaluated and various combinations of laminated shells are examined. It is shown that 1.(1) shell heterogeneity, or the layer reversal effect for two-layer shells, may significantly affect the stress field obtained and its level;2.(2) the combined action of the composed layers transcends the sum of the individual properties and provides, in many cases, new performance unattainable by the constituents when acting as homogeneous shells;3.(3) a cross-thermoelastic effect is exhibited by the stress distribution for the considered composite shells.

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].

An approximate solution to the problem of a circular cylindrical shell with a circular hole subjected to an arbitrary self-equilibrated edge loading

In this investigation, we shall use Donnell's equation of shallow circular cylindrical shell to obtain an approximate solution for stresses in an infinitely long circular cylindrical shell with a circular hole about which there acts a self-equilibrated load. To satisfy the edge conditions, we first shall expand the appropriate solutions of the differential equation into series containing powers, products of powers and logarithm, of the characteristic parameter β ρ0 defined in (1). We then shall represent the unknown coefficients of the solutions in similar series including new constants which depend only on the material property and the type of loading. We determine the new unknown constants through simple algebraic equations. In conclusion, we shall present two examples for symmetric and anti-symmetric loadings, including expressions which illustrate their accuracy, and expressions for stress concentration.

Accurate buckling equations for arbitrary and cylindrical elastic shells

Accurate equations for the neutral equilibrium of arbitrary elastic shells are derived. These equations involve no displacement components and thus complement alternate sets of neutral equilibrium equations proposed by Koiter and Budiansky. The equations are specialized to a uniformly loaded circular cylindrical shell and reduced to a single partial differential equation. For a simply supported cylindrical shell, expressions for the buckling loads are obtained which are as simple as those predicted by the Donnell equations, yet as accurate as those predicted by the more elaborate Flügge equations.

Buckling of cylindrical shells with eccentric spiral-type stiffeners.

Buckling equations for orthotropic cylindrical shells with eccentric spiral stiffeners were derived through the variation method of the total potential. Loadings analyzed are axial compression, hydrostatic pressure, bending, and torsion. For each loading type, the conventional ring-stiffened, stringer-stiffened, and ring-and-stringer-stiffened cylinders are compared, through a model cylinder, with spiral types on an equal-weight basis and an equalstrength basis. The limited numerical study indicated a possibility that the spiral-type stiffening might be stronger than the corresponding conventional-type stiffening of equal weight and that, in addition, the strength ratio of the two types of cylinder might increase when the amount of stiffening is increased. The merit of placing spiral stiffeners internal or external to the cylinder is also evaluated. Correlation between the present analysis and existing test data for spiral-stiffened waffle cylinders shows good agreement.

Bending of a simply supported circular cylindrical shell subjected to uniform line load along a generator

The complex differential equation of cylindrical shells, given by Novozhilov [1], is used to obtain the state of stress for a simply supported closed thin circular cylindrical shell acted upon by a uniform inward radial line load along a generator. The problem is solved by obtaining a closed form particular integral of the differential equation, and satisfying the edge conditions with the aid of complementary solutions in the form of a single Fourier series which converges very rapidly for the region near the middle of the shell. For comparatively long shells an approximate expression, in closed form, is derived for the region far away from the edges. A mathematical proof of the convergence of the series is given, and numerical results for several ratios of length to radius are presented.

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.

A set of simple, accurate equations for circular cylindrical elastic shells

The Sanders' equations for a circular cylindrical elastic shell of constant thickness are reduced to a single, simple, fourth order partial differential equation for the complex-valued function W+i√(A/D)F, where W is the midsurface normal deflection, F is a stress function, and A/D is an elastic constant. Auxiliary equations expressing the tangential midsurface displacements, stress resultants, stress couples, and Kirchhoff edge forces in terms of W, F, and surface load integrals are also derived. Approximations are introduced only in the stressstrain relations, but the resulting errors are shown to be negligible by Koiter's arguments. Work of previous writers is reviewed and compared with results of the present paper,

Postbuckling equilibrium of axially compressed circular cylindrical shells.

Earlier numerical solutions of the von Karman-Donnell large-displacement equations for thin circular cylindrical shells have been extended by considering larger numbers of terms in the double Fourier series representing the radial displacements after buckling. In the most comprehensive one of the calculations whose results are presented here a total potential energy expression consisting of about 1100 terms was minimized with respect to 16 unknowns. The results of the computations as well as theoretical considerations indicate that the solution for long shells of the von Karman-Donnell equations with the aid of the von Karman-TsienLeggett procedure leads in the limit to the trivial solution in which the amplitude of the displacements tends to zero, the number of waves around the circumference tends to infinity, and the average axial compressive stress capable of maintaining equilibrium in the postbuckling state tends to zero.

Free Vibrations of Conical Shells

Linear shell theory was used to develop an analysis for predicting the frequency spectrum of isotropic simply supported conical shells. An experimental investigation was performed in conjunction with this analysis. The experimental frequencies were compared with the developed theory. The results were in good agreement with the theoretical analysis. The experimental data were also compared with equivalent cylindrical shell theory. This approximation gave good results for the minimum frequencies and frequencies at low circumferential wave numbers. A secondary experimental investigation was conducted for cantilevered conical shells.

Deformation of and Stress in a Short Pipe Grooved Axially under Torsion

When a thin cylindrical pipe of finite length with a groove cut out through the length axially is twisted with both ends fixed on grips as shown in Fig. 1, the influence of fixed ends on the angle of twist and the stress is discussed, using the fundamental differential equation for cylindrical shell introduced by one of the present authors. The angle of twist per unit length of the short pipe with a groove is very small compared with that of the pipe of infinite length and this difference is so extremely great that the application of the equations for the infinitely long pipe to a short pipe may be inadequate. Besides this remarkable difference, there exist great axial thrusts at both fixed ends. These theoretical results are ascertained by experiments.

Thin Plates and Shallow Cylindrical Shells Subjected to Hot Spots

Using Lagrange’s equation of motion for thin plates, the equations of plane stress, and the equations for shallow cylindrical shells, the deformations of thin plates and cylindrical shells subjected to bending and plane hot spots are studied. The interrelation between different singular functions associated with these problems has been indicated also.