N. P. Semenyuk

On Design Models in Stability Problems for Corrugated Cylindrical Shells

Three design techniques for stability analysis of longitudinally corrugated cylindrical shells are examined. The first two account for the true geometry of the shell and the third one replaces the corrugated shell with an equivalent orthotropic shell using reduction formulas. The exact formulations employ classical and Timoshenko-type theories. The techniques are analyzed by an example of sinusoidally corrugated shells. It is shown that the exact formulation permits finding practically important relations for corrugation parameters, which raises considerably the specific critical loads.

Timoshenko-Type Theory in the Stability Analysis of Corrugated Cylindrical Shells

A technique is proposed for stability analysis of longitudinally corrugated shells under axial compression. The technique employs the equations of the Timoshenko-type nonlinear theory of shells. The geometrical parameters of shells are specified on discrete set of points and are approximated by segments of Fourier series. Infinite systems of homogeneous algebraic equations are derived from a variational equation written in displacements to determine the critical loads and buckling modes. Specific types of corrugated isotropic metal and fiberglass shells are considered. The calculated results are compared with those obtained within the framework of the classical theory of shells. It is shown that the Timoshenko-type theory extends significantly the possibility of exact allowance for the geometrical parameters and material properties of corrugated shells compared with Kirchhoff–Love theory.

Stability of Axially Compressed Noncircular Cylindrical Shells Consisting of Panels of Constant Curvature

The stability problem is solved for an axially compressed cylindrical shell. Its cross section is formed by circular arcs of radius r with ends supported on a closed circle of radius R. The solution is based on the Flügge equations of the classic theory of deep cylindrical shells. It is shown that the critical axial load for shells of medium length and appropriately chosen cross-sectional profile can be increased by a factor of R/r approximately, compared with the circular shell. The shells length affects considerably the efficiency of noncircular shells of this type. This design model allows us to find out how the local properties of the shell and its stiffness are related

Stability of Orthotropic Corrugated Cylindrical Shells

A technique for stability analysis of cylindrical shells with a corrugated midsurface is proposed. The wave crests are directed along the generatrix. The relations of shell theory include terms of higher order of smallness than those in the Mushtari–Donnell–Vlasov theory. The problem is solved using a variational equation. The Lamé parameter and curvature radius are variable and approximated by a discrete Fourier transform. The critical load and buckling mode are determined in solving an infinite system of equations for the coefficients of expansion of the resolving functions into trigonometric series. The solution accuracy increases owing to the presence of an aggregate of independent subsystems. Singularities in the buckling modes of corrugated shells corresponding to the minimum critical loads are determined. The basic, practically important conclusion is that both isotropic and orthotropic shells with sinusoidal corrugation are efficient only when their length, which depends on the waveformation parameters and the geometric and mechanical characteristics, is small

Stability and efficient design of cylindrical shells of metal composites subject to combination loading

A study is made of the stability of boron-aluminum shells under a combination of axial compression and uniform external pressure. An approximate theoretical model is constructed to describe the deformation of a layer of a fiber composite consisting of elastoplastic components. The model is used to derive the equations of state of multilayered shells reinforced by different schemes. The nonlinear equation describing the subcritical state is solved by the method of discrete orthogonalization with the use of stepped loading. The homogeneous problem is also solved by discrete orthogonalization. It is shown that shells can be efficiently designed for combination loading by plotting the envelope of the boundary curves for specific reinforcement schemes. The envelope is convex for elastic shells and is of variable curvature for elastoplastic shells.

THE STABILITY OF CYLINDRICAL AND CONIC SHELLS MADE OF COMPOSITE MATERIALS WITH AN ELASTOPLASTIC MATRIX

Studies in which problems on the stability of shells made of inelastically deformed composite materials are formulated and solved are generalized. Primary attention is focused on works that employ a structural approach to the description of the deformation of a fibrous composite consisting of elastoplastic components. The load for which the solution of the problem becomes ambiguous (bifurcation) is considered critical. Boroaluminum cylindrical and conic shells subject to external pressure, axial compression, and combined loading by surface and axial forces of different signs are analyzed for stability. The effect of boundary conditions and reinforcement on the critical loads is considered. The effect of temperature on shell stability beyond the elastic limits is investigated by an example of a cylindrical shell.

On the applicability of the relations of the cubic timoshenko-type theory of shells to the study of the postcritical behavior of rods

The question of whether the nonlinear Timoshenko-type theory of shells can be applied to the study of the initial postcritical behavior of a rod under compression is considered. The Koiter asymptotic theory in the Budyanskii form is used. The exact solution of the problem is obtained and a formula for the coefficient of postcritical behavior allowing for the effect of lateral-shear strains is derived. It is shown that the expressions (specified to within cubic terms) for lateral-shear strains and curvature permit us to use the nonlinear theory of shells to analyze the initial supercritical behavior of rods

On accounting for the cubic terms in the equations of the nonlinear Timoshenko theory of plates

Based on the Timoshenko hypotheses, the equations of the nonlinear theory of plates are derived without restriction on the displacements. Then these equations are systematized in the case where the expressions for transverse-shear strains, curvatures, and torsion contain third-order terms. As an example, the equations are used to solve the problem on the initial postcritical behavior of plates. It is demonstrated that cylindrical bending is the case where the theory should necessarily be corrected as proposed

Application of the Timoshenko–Mindlin Theory to the Calculation of Nonlinear Deformation and Stability of Anisotropic Shells

The nonlinear deformation and stability of composite shells are estimated by using the Timoshenko–Mindlin theory of anisotropic shells. The resolving system of equations is presented in a mixed form in displacements, forces, and moments. For its derivation, a modified version of the generalized Hu–Washizu variational principle formulated in rates for a quasi-static problem is used. However, instead of differentiation with respect to time, displacements, stresses, and loads are assumed to depend on a parameter, for which it is advisable to take the length of the arc of equilibrium states, as demonstrated in some studies. On variation of this parameter, the shell-load system can occur either at regular or singular points. A boundary value problem is formulated in the form of a normal system of differential equations in the derivatives of displacements, forces, and moments. In the separation of variables, the Fourier series are used in a complex form. The boundary value problem is solved by the Godunov discrete orthogonalization method in the field of complex numbers. Then, the Cauchy problem is solved by using known methods. Using the methodology developed, an analysis of the influence of composite properties and parameters of the layered structures on the form of the equilibrium curves of cylindrical shells is carried out. The mechanical characteristics of the initial elementary layers of the reinforced material are determined by the micromechanics methods developed by Eshelby, Mori–Tanaka, and Vanin.