A computer program for nonlinear analysis of pressure vessels

This paper describes the nonlinear analysis of pressure vessels necessary for taking into account the large deformations that take place at the junctions of shells of different geometries. Specifically, a computer program has been developed, based on both the linear and nonlinear theories of shells, which obtains numerical solutions for the most commonly used types of pressure vessels, namely those with spherical, ellipsoidal or conical heads and also flat-end pressure vessels. A multisegment integration technique has been used to obtain the solutions of the governing equations. The computed solutions are found to be highly accurate when compared with the known results of simple shells, as no nonlinear analysis is reported in the literature on the shell junctions in pressure vessels.     

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.     

Nonlinear thermal stability of eccentrically stiffened FGM double curved shallow shells

This article presents analytical solutions for the nonlinear static and dynamic stability of imperfect eccentrically stiffened functionally graded material (FGM) higher order shear deformable double curved shallow shell on elastic foundations in thermal environments. It is assumed that the shell’s properties depend on temperature and change according to the power functions of the shell thickness. The shell is reinforced by the eccentrically longitudinal and transversal stiffeners made of full metal. Equilibrium, motion, and compatibility equations are derived using Reddy’s higher order shear deformation shell theory and taking into account the effects of initial geometric imperfection and the thermal stress in both the shells and stiffeners. The Galerkin method is applied to determine load–deflection and deflection–time curves. For the dynamical response, motion equations are numerically solved using Runge–Kutta method. The nonlinear dynamic critical buckling loads are found according to the criterion suggested by Budiansky–Roth. The influences of inhomogeneous parameters, dimensional parameters, stiffeners, elastic foundations, initial imperfection, and temperature increment on the nonlinear static and dynamic stability of thick FGM double curved shallow shells are discussed in detail. Results for various problems are included to verify the accuracy and e?ciency of the approach.     

FREE VIBRATIONS OF A CONICAL SHELL WITH TEMPERATURE-DEPENDENT MATERIAL PROPERTIES

This article presents an analysis of the free vibrations of a truncated conical thin shell subjected to thermal gradients. The governing equations of the shell are based on the Donnell-Mushtari theory of thin shells. Simply supported and clamped boundary conditions are considered at both ends of truncated conical shell. Temperature loading due to supersonic flow is assumed to vary along the meridian and across the thickness of the shell Hamilton's principle is used to derive the appropriate governing equations of a conical shell with temperature-dependent material properties. The shell material has a kind of inhomogeneity due to the varying temperature load and temperature dependency of material properties. The resulting differential equations are solved numerically using the collocation method. The results are compared with certain earlier results. The influence of temperature load on the vibration characteristics is examined for the conical shells with various geometrical properties.     

THERMOELASTIC BUCKLING OF THIN CYLINDRICAL SHELLS BASED ON IMPROVED STABILITY EQUATIONS

The nonlinear strain-displacement relations in general cylindrical coordinates are simplified by Sander's assumptions for the cylindrical shells and substituted into the total potential energy function for thermoelastic loading. The Euler equations are then applied to the functional of energy, and the general thermoelastic equations of nonlinear shell theory are obtained and compared with the Donnel equations. An improvement is observed in the resulting equations as no length limitations are imposed on a thin cylindrical shell. The stability equations are then derived through the second variation of potential energy, and the same improvements are extended to the resulting thermoelastic stability equations. Based on the improved equilibrium and stability equations, the magnitude of thennoelastic buckling of thin cylindrical shells under different thermal loadings is obtained. The results are extended to short and long thin cylindrical shells.     

Stresses and stability for the cone-cylinder shells with toroidal transition

The stresses in the cone-cylinder shells with a toroidal segment as a transition, subjected to external hydrostatic pressure, have been calculated by both analytical and finite element methods, and compared with those in cone-cylinder shells without tratisition. The stability of the shells with transition have been also analyzed by the Rayleigh-Ritz's method. The numerical results obtained show that the smooth shell formed by inserting a toroidal segment between the cone and cylinder, has much lower stresses and slightly higher buckling-loads than the corresponding cone-cylinder shell without transition.     

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.     

Stress concentration factors for nozzles in ellipsoidal pressure vessel heads due to thrust loads

A theoretical elastic analysis of the stresses due to a thrust applied to a radial nozzle in an ellipsoidal pressure vessel head has been carried out. The nozzle can be either flush or protruding. The results, in the form of stress concentration factors and shear stress concentration factors in the ellipsoidal shells, are presented as graphs of non-dimensional shell parameters.     

Stability analysis of a toroidal pipe-reducer under uniform external pressure

The stability of a toroidal pipe-reducer system is determined here from the solution of non-linear governing equations of axisymmetric deformations of shells of revolution. Numerical solutions are obtained by a modified version of the computer program developed by Uddin for solving the governing equations of axisymmetric shells by the multisegment method of integration. The interpretation of instability of the toroidal reducers is based on Thompson's theorems I and II. Critical pressures for the toroidal reduers are calculated over useful ranges of the curvature ratio, the thickness ratio, and the diameter ratio. It has been found that the critical pressure of these reducers varies almost linearly with the diameter ratio and that the long toroidal reducers are prone to local instability near the larger end. But this critical zone occurs near either one of the two ends as the reducer becomes shorter. The results of stability and stress analysis of toroidal pipe-reducers are compared here with those of conical reducers obtained by Ali and parabolic reducers obtained by Rahman. Comparison shows that toroidal reducers develop uniform stresses of lower magnitude compared to the other two. Further, toroidal reducers are found to sustain higher critical pressure than parabolic reducers except at higher diameter ratio.     

Thermal stress analysis of laminated structures by a variable kinematic MITC9 shell element

A linear static thermal stress analysis of composite shell structures is carried out by means of a shell finite element with variable through-the-thickness kinematic. The refined models used are both Equivalent Single Layer (ESL) and Layer Wise (LW) and they are grouped in the Unified Formulation by Carrera (CUF). These models permit the distribution of displacements, stresses and temperature along the thickness of the multilayered shell to be accurately described. The Principle of Virtual Displacement (PVD) is employed to derive the governing equations. The Mixed Interpolation of Tensorial Components (MITC) method is used to contrast the membrane and shear locking phenomenon for a nine-node shell element. Cross-ply plate, cylindrical and spherical shells with simply supported edges and subjected to bi-sinusoidal thermal load are analyzed and various thickness ratios are considered. The results, obtained with different theories contained in the CUF, are compared with both the elasticity solutions given in the literature and the analytical solutions obtained using higher-order models and Navier's method. From the analysis, one can conclude that the shell element based on the CUF is very efficient, and its use leads to reach higher accuracy than classical models in the study of layered structures.     

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.     

Geometrically nonlinear dynamic analysis of doubly curved isotropic shells resting on elastic foundation by a combination of harmonic differential quadrature-finite difference methods

The nonlinear dynamic response of doubly curved shallow shells resting on Winkler–Pasternak elastic foundation has been studied for step and sinusoidal loadings. Dynamic analogues of Von Karman–Donnel type shell equations are used. Clamped immovable and simply supported immovable boundary conditions are considered. The governing nonlinear partial differential equations of the shell are discretized in space and time domains using the harmonic differential quadrature (HDQ) and finite differences (FD) methods, respectively. The accuracy of the proposed HDQ-FD coupled methodology is demonstrated by numerical examples. The shear parameter G of the Pasternak foundation and the stiffness parameter K of the Winkler foundation have been found to have a significant influence on the dynamic response of the shell. It is concluded from the present study that the HDQ-FD methodolgy is a simple, efficient, and accurate method for the nonlinear analysis of doubly curved shallow shells resting on two-parameter elastic foundation.     

Elasto/visco-plastic dynamic response of thin shells of revolution subjected to mechanical or thermal loads or both

An analytical method for the elasto/visco-plastic dynamic problems of axisymmetrical thin shells subjected to mechanical or thermal loads or both is developed. The equations of motion and the relations between the strains and displacements are derived by extending Sanders' elastic-shell theory. For the constitutive relations, Perzyna's elasto/visco-plastic equations, including the temperature effect, are employed. The derived fundamental equations are numerically solved by the finite-difference method. As numerical examples, simply supported cylindrical shells made of mild steel are treated, and the following two cases are analyzed: a non-uniformly heated cylindrical shell subjected to impulsive internal pressure, and an internally pressurized cylindrical shell subjected to impulsive thermal load. In both cases, the variations of displacements and internal forces with time are discussed.     

Transient stress analysis of sandwich plate and shell panels with functionally graded material core under thermal shock

A finite element formulation for stress analysis of functionally graded material (FGM) sandwich plates and shell panels under thermal shock is presented in this work. A higher-order layerwise theory in conjunction with Sanders’ approximation for shells is used to develop the finite element formulation for transient stress analysis of FGM sandwich panels. The top and the bottom surfaces of FGM sandwich panels are made of pure ceramic and metal, respectively, and core of the sandwich is assumed to be made of FGM. The temperature profile in the thickness direction of the panels is considered to be varying as per the Fourier’s law of heat conduction equation for unsteady state. The heat conduction equations are solved using the central difference method in conjunction with the Crank–Nicolson approach. Transient thermal displacements of the sandwich panels are obtained using Newmark average acceleration method and the transient thermal stresses are obtained using stress–strain relations, subsequently. Results obtained from the present layerwise finite element formulations are first validated with available solutions in literature. Parametric studies are taken up to study the effects of volume fraction index, temperature dependency of material properties, core thickness, panel configuration, geometric and thermal boundary conditions on transient thermal stresses of FGM sandwich plates and shells.     

THERMOELASTICITY OF A REGULARLY NONHOMOGENEOUS THIN CURVED LAYER WITH RAPIDLY VARYING THICKNESS

A regularly nonhomogeneous (composite), anisotropic, thin curved layer with rapidly oscillating material parameters and thickness is considered for the case when mean thickness and period scale have small magnitudes of the same order. A three-dimensional thermoelasticity problem for this layer is reduced to a homogenized shell model by means of an asymptotic homogenization method for periodic structures. The effective thermoelastic and thermal material parameters of this shell are expressed in terms of solutions for auxiliary local problems in the cell of periodicity. Using the solution of the boundary-value problem for the homogenized shell and the solutions of the local problems, one can obtain a three-dimensional microstructure of the stresses, displacements and temperature with a high accuracy

This general model is applied to the derivation of thermoelastic and thermal constitutive equations for network periodic shells. The relations obtained lay the foundation for a new continuous model of thermoelasticity and heat conductivity for network periodic shells and plates.     

Thermal Post-Buckling and Vibration Analysis of Composite Conical Shell Structures Using Layerwise Theory

The thermal post-buckling and vibration characteristics of composite conical shells are investigated using a finite element method. Based on the layerwise theory and the von Karman displacement strain relationships, the nonlinear finite element equations of motion are derived for the thermoelastic response of the composite conical shell structure. The cylindrical arc-length method is used to account for the snapping phenomenon. The influence of the structural parameters, such as the semi-cone angle, thickness ratio, and shallowness angle (curvature), on the structural stability of the composite conical shell subjected to the thermal load is also observed.     

A lower bound analysis for the calculation of limit pressure for a thick spherical vessel with a radial cylindrical nozzle

Using a three-dimensional stress formulation a lower bound analysis is presented for evaluating the limit internal pressure for a pressure vessel consisting of a thick spherical shell and a thick radial cylindrical nozzle. The analysis—which deals directly with stresses and the material yield criterion—can, in general, be applied to vessels of any thickness. The stresses are expressed in terms of an independent set of variables and the Von Mises yield criterion is used. The limit pressure is optimised using a non-linear programming method. Computed results are presented for a limited range of geometric parameters and a comparison is made with thin shell results.     

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.     

The contact stress analysis of plate and shell structures in pressure vessels

This study introduces the concept of contact stresses into an analysis of the local stresses in calculations relating to plate and shell structures in pressure vessels. A new computational method for local stresses is proposed in the light of the contact between pad and shell and the compatibility at the pad edges due to the fillet weld. Using the developed program of the contact stress analysis of plates and shells, which is based on the flexibility mixed FEM, the authors have studied the contact features of a pad structure under a concentrated load. Contact stress analyses of a saddle support model and a suspended tower structure have been made, which could not have been achieved with previous methods. The theoretical calculations showed reasonable agreement with the experimental results.     

Thermoelastic Response of Cross-Ply Laminated Shells Based on a Rigorous Shell Theory

Thermal deformations in cross-ply laminated shells are investigated. The state space approach is used to generate exact solutions for the thermoelastic response of cross-ply spherical, cylindrical and doubly curved shells for various boundary conditions and subjected to general temperature field. The shells possess two parallel edges simply supported and the remaining ones having any possible combination of boundary conditions: free, clamped or simply supported. A rigorous thick deep first order shear deformation shell theory is used in the analysis. Deflections are computed for shells with various boundary conditions undergoing uniform and linearly varying temperature through the thickness. The exact solutions for thermal deflections can be used as benchmarks for approximate solutions such as Rayleigh-Ritz and finite element methods.