Critical load of a bilayered orthotropic elastic–plastic conical shell with the change of the shell basic surface location

The paper presents the derivation of stability equations and the comparison of critical loads for an orthotropic elastic–plastic conical shell with proper location of the shell basic surface. Prandtl–Reuss plastic flow theory is accepted in the analysis. To derive the stability equations the variational method was accepted and Ritz method was used to solve the equations. Numerical results were obtained by the use of a special iterative algorithm of the elastic–plastic analysis implemented on the PC.     

Free vibration analysis of functionally graded cylindrical shells including thermal effects

Free vibration analysis of simply supported FG cylindrical shells for four sets of in-plane boundary conditions is performed. The material properties are assumed to be temperature-dependant and gradually changed in the thickness direction of the shell. The effects of temperature rise are investigated by specifying arbitrary high temperature on the outer surface and the ambient temperature on the inner surface of the cylinder. Distribution of temperature across the shell thickness is found from steady state heat conduction only in the thickness direction. The equations of motion are based on Love's shell theory and the von Karman–Donnell-type of kinematic nonlinearity. The static analysis is first performed to determine the prestressed state induced by the thermal loadings, using the exact solution of the governing equations and then the equations of motion are solved by Galerkin's method. The results are obtained to indicate the effects of power law index on the natural frequencies and corresponding mode shapes in the thermal environment.     

Natural frequencies and mode shapes of an orthotropic thin shell of revolution

A method is developed to determine the free vibration characteristics of an orthotropic thin shell of revolution of arbitrary meridian. A solution is given within the context of the Sanders–Budiansky shell theory and using the differential quadrature method (DQM). Numerical examples for frequencies and mode shapes are given for a complete toroidal shell. Both completely free shells, and shells with circumferential line supports are considered. Close agreement is observed in comparisons with previously published results and with results obtained using the finite element method. The paper ends with a set of appropriate conclusions.     

Stability of elastic–plastic conical shells under shear loading

A method of determination of critical loads for thin-walled conical shells loaded by shear forces developed by moment of twist is presented. The three governing equations of neutral equilibrium with respect to basic displacement vector components u, v, and w are used. It is assumed that effective stress in the prebuckling state of stress in the shell can exceed the yield limit of the shell material. The use is made both of physical relations of Nadai–Hencky small elastic–plastic deformation theory of plasticity, and Prandtl–Reuss J₂ incremental plastic flow theory. Also, a bilinear stress–strain material model, material compressibility and Shanley approach will be accepted in the analysis. Galerkin method is applied to solve the problem equations and iterative techniques are accepted in numerical algorithm to determine critical loads for elastic–plastic shells.     

An elastic–plastic analysis of large deflection of thin shell structure using a delta-sequence function

The analysis of thin shell under large deflection has complex problem associated with the geometrical and material nonlinearities, in which the solutions for stress and deformation are desired to obtain the same level accuracy. One of ordinary and powerful method for this subject is a finite element method that has generally inconvenience to be necessarily extensive calculation due to large number of freedom.This paper is concerned with the elastic–plastic analysis of thin shell structures by the hybrid method using a functional for the principle of modified complementary energy. For elastic–plastic materials, the numerical calculation could be well executed by the way that the stress distribution across the panel thickness is expressed as continuous function using the delta-sequence function. This approach introduces a considerable way for the reduction of computing volume. The proposed method was applied to discuss the resisting mechanism of thin shell structure under large deflection.     

Extracting the bulk effective parameters of a metamaterial via the scattering from a single planar array of particles

In this paper we present a method for retrieving the effective parameters of a metamaterial composed of a regular rectangular orthorhombic lattice of linear biaxially anisotropic particles suspended in free space. By assuming the point–dipole interaction approximation, equations are derived which extract the electric and magnetic polarizabilities of the individual particles given the measured or simulated scattering parameters of a single planar array of particles. These results are in turn substituted into the Clausius–Mossotti equations to find the bulk effective permittivity and effective permeability. To demonstrate our approach, the extraction method is applied to a metamaterial consisting of a cubic arrangement of magnetodielectric spheres using the scattering parameters obtained by simulating the structure with Ansoft HFSS. Our results show good agreement with a known analytical solution at frequencies in which the Clausius–Mossotti approximation is valid.     

Viscoplastic axisymmetrical buckling of spherical shell impulse subjected to radial pressure

The solution of viscoplastic axisymmetrical buckling of a complete thin spherical shell subjected to radial pressure impulse is presented. Analytically, the problem is formulated as a superposition of small perturbations on the basic unperturbed motion. The amplitudes of perturbed motion are restricted to be so small that the homogeneous compressive deformation predominates over the local bending. This condition allows the constitutive equations to be linearized by the expansion into Taylor's series in the vicinity of unperturbed motion. As a result, a set of two linear partial differential equations of the fifth order is obtained for describing the perturbed motion of the shell. The functional coefficients of these equations are determined by the solution for the unperturbed motion. The governing set of equations is solved using the series of Legendre's functions. As a result, for the time-dependent amplitudes in the series, the set of two ordinary equations with variable coefficients is arrived at. It turns out that certain harmonics grow very rapidly and cause the shell to exhibit a characteristic wrinkled shape which is characterized by a critical mode number. This property of the amplitudes is used to determine the threshold impulse that a shell can tolerate without excessive deformation.The influence of the meridional displacement on the magnitude of radial displacement, buckling mode and critical impulse is investigated. Also, the influence of the viscosity and the initial imperfections of the geometry and loading is shown. The numerical results for a steel shell are presented diagrammatically.     

Elastic tubes—Assumptions, equations, edge conditions

This paper is concerned with further development of tube-analysis methods which are as simple as necessary for design and as accurate as possible in the shell theory. The basic assumptions are stated in Section 2. For ‘long’ tubes their accuracy is estimated in the course of derivation and simplification of nonlinear equations of axisymmetric flexure (Section 3). These equations and the estimates of the assumptions are extended to two-dimensional problems in Section 4. Equations of the semi-momentless theory encompassing shear deformation and appropriate edge conditions are discussed in Sections 5 and 6. Their solution, including a simple approximation, is in Sections 7–9 preceded by known calculations and experiment.     

板锥网壳结构的拟三层壳分析法

板锥网壳结构是一种受力性能合理 ,技术经济效益良好的新型空间结构形式。本文对板锥网壳结构提出了一种连续化的分析方法 ,根据其受力特点按照刚度等效的原理 ,采用弹性力学的基本方法推导出板锥网壳结构的等代壳体刚度 ,将其连续化为能共同作用的特殊形式的三层薄壳 ,按薄壳理论建立其位移法和混合法的基本微分方程。通过对微分方程的求解 ,计算其整体位移及结构内力。该法具有一定的精度 ,可宏观地了解结构的力学性能 ,并可用于板锥网壳结构的初步设计。     

Buckling of elastic cylindrical shells considering the effect of localized axisymmetric imperfections

The effect of localized axisymmetric initial imperfections on the critical load of elastic cylindrical shells subjected to axial compression is studied through analytical modeling. Some classical results regarding sensitivity of shell buckling strength with respect to distributed defects having axisymmetric or asymmetric forms are recalled. Special emphasis is placed after that on the more severe case of localized defects satisfying axial symmetry by displaying an analytical solution to the Von Kármán–Donnell shell equations under specific boundary conditions. The obtained results show that the critical load varies very much with the geometrical parameters of the localized defect. These variations are not monotonic in general. They indicate, however, a clear reduction of the shell critical load for some defects recognized as the most hazardous isolated ones. Reduction of the critical load is found to reach a level which is up to two times lower than that predicted by general distributed defects.     

Elastic–brittle–plastic analysis of circular openings in Hoek–Brown media

A closed-form solution is presented in this paper for the prediction of displacements around circular openings in a brittle rock mass subject to a hydrostatic stress field. The rock mass is assumed to be governed by Hoek–Brown yield criterion and a non-associated flow rule is used. For the elastic–brittle–plastic analysis of circular openings in an infinite Hoek–Brown medium, the existing analytical solutions were found to be incorrect. The present closed-form solution is based on a theoretically consistent method and the solution does not require the use of any numerical method.The present closed-form solution was validated by using the finite element method. In the finite element analysis, the infinite boundary was simulated “exactly” by using the newly developed elastic support method. Several cases were analyzed and the present closed-form solutions for stresses and displacements were found to be in an excellent agreement with those obtained by using the finite element method.     

Free vibration of non-uniformly ring stiffened cylindrical shells using analytical, experimental and numerical methods

In this research, the free vibration analysis of cylindrical shells with circumferential stiffeners, i.e. rings with non-uniform stiffeners eccentricity and unequal stiffeners spacing is investigated using analytical, experimental and finite elements (FE) methods. Ritz method is applied in analytical solution while stiffeners treated as discrete elements. The polynomial functions are used for Ritz functions and natural frequency results for simply supported stiffened cylindrical shell with equal rings spacing and constant eccentricity is compared with other's analytical and experimental results, which showed good agreement. Also, a stiffened shell with unequal rings spacing and non-uniform eccentricity with free–free boundary condition is considered using analytical, experimental and FE methods. In experimental method, modal testing is performed to obtain modal parameters, including natural frequencies, mode shapes and damping in each mode. In FE method, two types of modeling, including shell and beam elements and solid element are used, applying ANSYS software. The analytical and the FE results are compared with the experimental one, showing good agreements. Because of insufficient experimental modal data for non-uniformly stiffeners distribution, the results of modal testing obtained in this study could be as useful reference for validating the accuracy of other analytical and numerical methods for free vibration analysis.     

Postbuckling analysis of cross-ply laminated cylindrical shell panels under parabolic mechanical edge loading

Postbuckling equilibrium paths of simply supported cross-ply laminated cylindrical shell panels subjected to non-uniform (parabolic) inplane loads are traced in this paper. Love's shell theory with higher order shear deformation theory and von Kármán nonlinear strain–displacement relations are used in the mathematical formulation of the problem. In the first step, the plate membrane problem is solved to evaluate the stress distribution within the prebuckling range as the applied inplane edge load is non-uniform. The governing shell panel postbuckling equations are derived from the principle of minimum total potential energy using the above stress distributions. Adopting multi-term Galerkin's approximation, the governing equations are reduced into a set of non-linear algebraic equations. Newton–Raphson method in conjunction with Riks approach is employed to plot the postbuckling paths through limit points. Numerical results are presented for symmetric (0/90/0) crossply laminated cylindrical shell panels under parabolic inplane load, lateral distributed load and initial imperfections. Limit loads and snap-through behavior of shell panels are studied.     

Auto-parametric resonance in thin cylindrical shells using the slow fluctuation method

Internal auto-parametric instabilities in the non-linear vibrations of an undamped and unloaded cylindrical shell are studied. The focus is on the coupling between two modes that can combine to break the in–out symmetry and give an energetically favourable pattern of deformation. When the ratio of the natural frequencies is close to 2, internal resonance triggers significant energy transfer between the modes. This energy exchange has both regular and chaotic features, which were previously investigated by other methods of analysis in [J. Sound Vibrat. 227 (1999) 65; J. Sound Vibrat. 248 (2001) 395]. A Rayleigh-Ritz discretization of the von Kármán–Donnell equations is employed to derive the Hamiltonian equations in R⁴. A perturbation analysis using action-angle coordinates gives slow fluctuation equations that can be solved by quadrature in Jacobian elliptic functions: and, a simple analytical criterion for the parametric instability is presented. This analytical solution gives a good approximation to the regular motions, and provides a framework in which to examine the chaotic motions that are studied in [J. Sound Vibrat. 248 (2001) 395].     

Exact solutions for thin-walled open-section composite beams with arbitrary lamination subjected to torsional moment

The exact solutions for torsional analysis of thin-walled open-section composite beams with arbitrary lamination subjected to torsional moment are presented for the first time. For this, a general thin-walled composite beam theory with arbitrary lamination is developed by introducing Vlasov's assumption and the equilibrium equations and the force–deformation relations are derived from the energy principle. Applying the displacement state vector consisting of 14 displacement parameters and the nodal displacements at both ends of the beam, the displacement functions are derived exactly. Then, the exact stiffness matrix for torsional analysis is determined using the force–deformation relations. As a special case, the closed-form solutions for symmetrically laminated composite beams with various boundary conditions are derived. Finally, the finite element procedure based on Hermitian interpolation polynomial is developed. To demonstrate the validity and the accuracy of this study, the numerical solutions are presented and compared with the closed-form solutions and the finite element results using the Hermitian beam elements and ABAQUS's shell elements.     

Stability of orthotropic elastic–plastic open conical shells

The paper presents a derivation of the stability equation and the solution method of the problem for an orthotropic elastic–plastic open conical shell. The use is made of the constitutive relations of incremental plastic flow theory, elastic compressibility of the material, and Shanley concept of increasing load are taken into account in the consideration. A variation, strain energy method is used to derive the stability equation for bilayered open conical shell with nonuniform pre-critical stress distribution. The shell is free supported at the edges and the load acting the shell, in the form of longitudinal force and lateral pressure, is active one, i.e. unloading is not considered.     

Ultimate strength of submarine pressure hulls with failure governed by inelastic buckling

The analysis of submarine pressure hull assumes great importance among structural engineers due to the complexity involved in the collapse mechanism of stiffened shell structures. In most of the cases, the failure of stiffened shell structures occurs due to elastic buckling. But for some combinations of shell-stiffener geometry and material characteristics, the structure can fail by inelastic buckling, for which the methods of analysis are meagre. In this paper, the analysis of submarine pressure hull structure in which the failure gets governed by inelastic buckling is demonstrated. Three different approaches have been employed to investigate the ultimate strength of the ring stiffened submarine pressure hull structure with inelastic buckling modes of failure. The methods used are ‘Johnson–Ostenfeld inelastic correction’, ‘imperfection method’ and ‘finite element approach’. A typical submarine shell structure has been analysed for the inelastic buckling failure using these three approaches and the results are discussed.     

A systematic approach towards the structural behaviour of a lightweight deck–side shell system

Lightweight structures are increasingly used for high-speed ships. This paper presents a systematic approach to analyse the structural behaviour of a lightweight deck–side shell system using high strength steel. An analytical model of the deck–side shell system was first given, which includes the effects of stiffeners for the deck and side shell, the support conditions of the centreline girder (CL-girder), the influence of transverse beams, and the interaction between the side shell and the lightweight deck as parts of problems to the solution. By changing several geometric parameters, the sensitivity of both overall and local stress and deflection for the deck–side shell system was investigated. The different geometric parameters analysed comprise the influence for variation in the thickness of the web for transverse beams, longitudinal stiffeners and the CL-girder, the thickness of lower flange for the transverse beam and, the thickness for the panel. Furthermore, the influence of the lightweight deck and loads from the deck above on the side shell, the effects of the side shell and loads from top deck on the deck, the support conditions for the CL-girder, and the influence of deck loads on the eigenmodes were also analysed. By evaluating the results obtained from FE simulation, the support conditions of the CL-girder, the thickness of the panels and the lower flange of the transverse beams were found to be the most relevant parameters affecting both the stress and the deflection distribution of the structure. The dynamic characteristics of the structure were also analysed. The FE analysis concerning buckling of the structure was present. The results enable naval architects and structural engineers to design new extreme lightweight deck structure more reliable and economical. And some suggestions for future research are also given.     

Design charts for the general instability of ring-stiffened conical shells under external hydrostatic pressure

The paper describes experimental tests carried out on three ring-stiffened circular conical shells that suffered plastic general instability under uniform external hydrostatic pressure. In this mode of failure, the entire ring–shell combination buckles bodily in its flank. The cones were carefully machined from EN1A mild steel to a very high degree of precision.Using the results obtained from these three vessels, together with the results obtained from elsewhere, the paper also provides two-design charts, which are much easier to use than older design charts. The design charts allow the possibility of obtaining a plastic knockdown factor, so that the theoretical elastic buckling pressures for perfect vessels, can be divided by the plastic knockdown factor, to give the predicted buckling pressures. Although similar design charts have been produced in the past, the design charts presented here are based on using the simpler ring-stiffened circular cylinder, which has been made equivalent to the much more complex ring-stiffened circular conical shell. The advantage of using this method is that it is simpler and the design time is reduced by a factor of about 10 with little loss of precision. This method can also be used for the design of full-scale vessels.     

Coupled axial–torsional vibration of thin-walled Z-section beam induced by boundary conditions

The coupled axial–torsional vibration of thin-walled Z-section beam induced by the boundary conditions is investigated. The value of the warping function is not zero at centroid for Z-section beam. If the axial displacement of the pin end is restrained at the centroid of the Z-section for thin-walled Z-section beam, the axial vibration and torsional vibration may be coupled. The governing equations for linear axial and torsional vibration of a thin-walled Z-section beam are derived by the d’Alembert principle and the virtual work principle. The bending vibration is uncoupled from axial and torsional vibrations and is not dealt with in this paper. For harmonic vibration, the general solution of these equations with undetermined constant coefficients may be obtained. Substituting the general solution into the displacement and force boundary conditions, a set of homogeneous equations can be obtained. The natural frequencies and the coefficients of the general solution may be obtained by solving the homogeneous equations using the bisection method.Numerical examples are studied to verify the accuracy of the proposed method and to investigate the effect of boundary conditions and the value of warping function at centroid on the coupled axial and torsional natural frequency of Z-section beam.