Stability of a toroidal fluid-containing shell

This paper presents a solution to the stability problem of a sectorial toroidal shell subject to fluid loading. The solution is based on the Budiansky shell stability theory specialized for toroidal coordinates, and is developed using the two-dimensional differential quadrature method. Numerical results are presented and these results are compared with results from the Donnell shell theory, and with ones from a finite element method solution. The study establishes the suitability of the differential quadrature method for problems of shell stability.     

Ram bending of a cylindrical pipe in the elastic range

The problem of ram bending of a straight cylindrical pipe is considered. Separate shell theory and finite element method (FEM) solutions are presented. The loading is idealized as a set of pads of uniform radial pressure, and results are given for the elastic range. Particular attention is paid to the FEM solution characteristics and the pipe springback behavior. The present study is a necessary preliminary step to the full elastic-plastic solution of the problem.     

On a Thermodynamic Theory of Porous Cosserat Elastic Shells

This paper presents a theory for porous thermoelastic shells using the model of Cosserat surfaces and the Nunziato–Cowin theory for materials with voids. To describe the porosity of the thin body, we introduce two scalar fields: one field accounts for the changes in volume fraction along the middle surface of the shell, and the other field characterizes the porosity variations along the shell's thickness. First, we postulate the principles of thermodynamics for these two-dimensional continua and we obtain the equations of the nonlinear theory. Then, we consider the linearized theory and prove the uniqueness of solution to the boundary initial value problem with no definiteness assumption on the constitutive coefficients. Finally, we consider the deformation of isotropic and homogeneous shells and determine the constitutive coefficients for Cosserat surfaces, by comparison with the results obtained from the three-dimensional approach to shell theory.     

THERMOELASTIC BUCKLING OF THIN SPHERICAL SHELLS

Thermoelastic stability of thin perfect spherical shells based on deep and shallow shell theories is presented. To derive the equilibrium and stability equations according to deep shell theory, Sanders's nonlinear kinematic relations are substituted into the total potential energy function of the shell and the results are extremized by the Euler equations in the calculus of variation. The same equations are also derived based on quasi-shallow shell theory. An improvement is obtained for equilibrium and stability equations related to the deep shell theory in comparison with the same equations related to shallow shell theory. Approximate one-term solutions that satisfy the boundary conditions are assumed for the displacement components. The Galerkin-Bubnov method is used to minimize the errors due to this approximation. The eigenvalue solution of the stability equations is obtained using computer programs. For several thermal loads it is found that the deep shell theory results are slightly more stable as compared to the shallow shell theory results under the same thermal loads. The results are compared with the Algor finite element program and other known data in the literature.     

Dynamic response of curved pipes

The problem of the linear elastic response of a thin curved pipe subjected to a local impulsive loading is considered. A series solution based on the Sanders shell theory is developed in the toroidal coordinate system. This solution covers the case of a pipe with arbitrary circular curvature of the torus center line. The loading is represented as a double series in the geometric variables, and direct time integration is carried out using the Newmark method. Sample results are presented for loadings concentrated at the intrados and extrados for both short and long pipes.     

Stress analysis of a storage vessel in the form of a complete triaxial ellipsoid: hydrostatic effects

Closed-form expressions for membrane stress resultants in a liquid-filled triaxial ellipsoidal storage vessel are derived. Unlike the ellipsoid of revolution, the triaxial ellipsoid has three different semi-axes, and hence does not possess axi-symmetry, necessitating a somewhat different analysis approach to that normally adopted for shells of revolution in general. However, instead of treating the vessel as a shell of completely arbitrary shape, and hence pursuing the method of stress functions usually employed for such shells, advantage is taken of the affine relationship between the geometries of the general ellipsoid and the sphere, by deriving the stress resultants in the ellipsoid directly from those for an equivalently-loaded spherical shell, the solution to the latter problem being readily obtainable on the basis of the theory of nonsymmetrically loaded shells of revolution (for which the general method of stress functions need not be used, as a more straightforward procedure is applicable). While the technique of using an affine transformation (to deduce the stresses in a complex shell from a solution for a related but simpler shell) is itself well known, the closed-form results for hydrostatic pressure that are presented herein are new, being readily applicable to the design of storage vessels and water tanks. In storage-vessel design, where both hydrostatic pressure and uniform internal pressure may be equally important, the present results complement those for the latter (and simpler) loading which, for the vessel geometry in question, are already available in the literature.     

Buckling analysis of an orthotropic thin shell of revolution using differential quadrature

A method is developed to predict the buckling characteristics of an orthotropic shell of revolution of arbitrary meridian subjected to a normal pressure. The solution is given within the context of the linearized Sanders–Budiansky shell buckling theory and makes use of the differential quadrature method. Numerical results for buckling pressures and mode shapes are given for complete toroidal shells. Both completely free shells and shells with circumferential line restraints are covered. The loadings considered consist either of uniform pressure or circumferential bands of constant pressure. It is demonstrated that the differential quadrature method is numerically stable and converges. For isotropic toroidal shells, good agreement is observed with previously published analytical and finite element results. New results for buckling pressures and mode numbers are given for orthotropic shells and for band loaded shells.     

Thermoelastic analysis of laminated composite and sandwich shells considering the effects of transverse shear and normal deformations

Abstract

In the present study, thermoelastic analysis of laminated composite and sandwich shells (cylindrical/spherical) is presented using fifth-order shear and normal deformation theory. The significant characteristic of the present theory is that it includes the effects of both transverse shear and normal deformations. The mathematical formulation uses the principle of virtual work to derive the variationally consistent governing equations and traction free boundary conditions. To obtain the static solution, these governing equations are solved by employing Navier’s solution technique. The shell is subjected to a mechanical/thermal load sinusoidally distributed over the top surface of the shell. The thermal load linearly varies across the thickness of the shell. The present results are compared with other higher-order models and 3D elasticity solution wherever possible. Thermal stresses presented in this study will act as a benchmark for the future work.     

Deformation and perforation of cylindrical shells struck normally by blunt projectiles

An approximate theory is proposed in this paper to predict the deformation and perforation of metallic cylindrical shells struck normally by blunt projectiles at velocities up to 229 m s⁻¹. On the basis of the experimental observations of quasi-static load–displacement characteristics, the problem of a cylindrical shell impacted normally by a blunt projectile can be tackled through the solution of an equivalent clamped circular plate struck transversely by the same projectile. It is shown that the approximate theoretical predictions are in good agreement with the experimental results for cylindrical shells in terms of the maximum permanent transverse displacements and the critical impact velocities (ballistic limits) when material strain rate sensitivity is taken into account.     

Free vibration analysis of rotating cylindrical shells using discrete singular convolution technique

Free vibration analysis of rotating cylindrical shells is presented. Discrete singular convolution (DSC) method has been proposed for numerical solution of vibration problem. The formulations are based on Love's first approximation shell theory, and include the effects of initial hoop tension and centrifugal and Coriolis accelerations due to rotation. Frequencies are obtained for different types of boundary conditions and geometric parameters. In general, close agreement between the obtained results and those of other researchers has been found.     

Transient Thermal Stresses of a Cross-Ply Laminated Cylindrical Shell Using a Higher-Order Shear Deformation Theory

This paper is concerned with the theoretical treatment of transient thermoelastic problem of a cross-ply laminated cylindrical shell due to axisymmetrical heat supply. We analyze the transient thermoelastic problem of a cross-ply laminated cylindrical shell under a simply supported condition using a higher-order shear deformation theory. Some numerical results for the temperature change, the displacements and the stresses are shown in figures. Furthermore, numerical results from this formulation are compared with those obtained using classical and first-order shear deformation theories.     

A method for solving the buckling problem of a thin-walled shell

A weighted solution for the critical load of a cylindrical shell is presented. To determine the weights, some special known results are applied. The method can be used to solve generally complicated buckling problems by making use of the solutions of special simple problems. Lastly, some numerical solutions for the same problem are obtained by finite elements. Comparison between the solution method in this paper, the finite element solution and cited results in the literature, shows that the weighted solution has good precision.     

Stress concentration in cone–cylinder intersection

The finite element method and shell theory were employed to investigate cone–cylinder shell intersections. The developed special-purpose computer program Sais (stress analysis in intersecting shells) was used for elastic stress analysis of branch connections. A comparison of calculated results with experimental data is presented. A parametric study of non-radial models of the cone–cylinder shell intersection subjected to internal pressure loading was performed. The intersections of thin and middle thickness shells were analysed. The results are presented in graphical form. Non-dimensional geometric and angular parameters are considered to analyse the effects of changing these parameters on stress ratios in the shell intersection.     

Three-dimensional analysis of a cross-ply cylindrical shell subjected to a localized circumferential shear force

The objective of this work is to provide a three-dimensional elasticity solution with tabulated numerical results for a laminated composite cylindrical shell subjected to a localized circumferential shear force. The three-dimensional solution is valuable because it enables the validity of simple two-dimensional laminate theories such as the classical lamination theory to be judged for such analysis. It is shown here that the classical theory yields unacceptably erroneous predictions of the stresses in the vicinity of the local load even if the shell is thin.     

Thermal Bending Analysis of Doubly Curved Composite Laminated Shell Panels with General Boundary Conditions and Laminations

Thermal bending analysis of doubly curved laminated shell panels with general boundary conditions and laminations is presented. The equations of equilibrium are derived in the form of two coupled sets of ordinary differential equations based on a general shell theory and solved through the state-space approach in a repeated manner. It is depicted that the results of the present method are in great agreement with analytical solutions. Cylindrical shell panels with general boundary conditions and laminations, where no analytical solution is available, are solved. It is found that the present method exhibits a high convergence rate as well as presenting accurate results in all cases.     

Thermal Instability of Functionally Graded Shallow Spherical Shell

In this paper, thermal instability of shallow spherical shells made of functionally graded material (FGM) is considered. The governing equations for a thin spherical shell based on the Donnell–Mushtari–Vlasov theory are obtained. The equations are derived using the Sanders simplified kinematic relations and variational method. It is assumed that the mechanical properties vary linearly through the shell thickness. The constituent material of the functionally graded shell is assumed to be a mixture of ceramic and metal. Analytical solutions are obtained for three types of thermal loading including Uniform Temperature Rise (UTR), Linear Radial Temperature (LRT), and Nonlinear Radial Temperature (NRT). The results are validated with the known data in the literature.

     

Investigation Methods for Thermal Shock Crack Problems of Functionally Graded Materials–Part I: Analytical Method

In most published papers, in order to obtain the analytical solution of the crack problems in functionally graded materials (FGMs), the thermomechanical properties of FGMs are usually assumed to be very particular functions and, hence, may not be physically realizable for many actual material combinations. Very few analytical methods can be used to solve the thermal shock crack problem of an FGM cylindrical shell or plate with general thermomechanical properties. In this article, a set of analytical methods is proposed for the thermal shock crack problem of an FGM plate or cylindrical shell with general thermomechanical properties. The crack problem of a cylindrical shell is modeled by a plate on an elastic foundation. Greatly different from previous studies, a set of analytical methods using both the perturbation method and a piecewise model are developed to obtain the transient temperature field and thermal stress intensity factor (TSIF). The perturbation method is applied to deal with the general thermal properties and the piecewise model is used to deal with the general mechanical properties. In the analytical procedure, integral transform, the residue theorem, and the theory of singular integral equation are used. Several representative examples are considered to check the capability of the present method. The transient thermal shock behavior of a ZrO₂/Ti-6Al-4 V FGM plate with a surface crack and a Rene 41-Zirconia FGM cylindrical shell with a circumferential crack are analyzed.     

Thermal Buckling of Imperfect Functionally Graded Cylindrical Shells Based on the Wan–Donnell Model

ABSTRACT

A thermal buckling analysis of an imperfect functionally graded cylindrical shell is considered using the Wan–Donnell model for initial geometrical imperfections. Derivation of the equations is based on the first-order classical shell theory using the Sanders nonlinear kinematic relations. Results for the buckling loads are obtained in the closed form. The effects of shell geometry and volume fraction exponent of functionally graded material on the buckling load are investigated. The results are validated with known data in the literature.     

Fractional phase-lag Green–Naghdi thermoelasticity theories

The constitutive equations are given with a fractional Maxwell–Cattaneo heat conduction law using the Caputo fractional derivative and the fractional order heat transport equation is given. The uniqueness theorem is proved, the reciprocity relation is deduced and the variational characterization of solution is given. Seven thermoelasticity theories result from the given problem as special cases (Coupled, Lord–Shulman, phase-lag Green–Naghdi theories, Green–Naghdi theory without energy dissipation and the Green–Naghdi theory of type III).     

ASSESSMENT OF SHELL THEORIES FOR HYBRID PIEZOELECTRIC CYLINDRICAL SHELL UNDER THERMOELECTRIC LOAD

This study presents an assessment of classical lamination shell theory and first-order shear deformation theory for a simply supported finite circular cylindrical hybrid shell with a cross-ply laminate as an elastic substrate under thermoelectric static load. Navier-type solutions for these shell theories are obtained and used in three-dimensional (3D)equilibrium equations and transverse strain-displacement relations to obtain transverse stress components and an improved value of deflection. These solutions are assessed by comparison with the 3D solution.