Postbuckling of Shear Deformable Laminated Cylindrical Shells

A postbuckling analysis is presented for a shear deformable laminated cylindrical shell of finite length subjected to compressive axial loads. The governing equations are based on Reddy’s higher-order shear deformation shell theory with a von Kármán–Donnell type of kinematic nonlinearity. The nonlinear prebuckling deformations and initial geometric imperfections of the shell are both taken into account. A boundary layer theory of shell buckling, which includes the effects of nonlinear prebuckling deformations, large deflections in the postbuckling range, and initial geometric imperfections of the shell, is extended to the case of shear deformable laminated cylindrical shells under axial compression. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling response of perfect and imperfect, unstiffened or stiffened, moderately thick, cross-ply laminated cylindrical shells. The effects of transverse shear deformation, shell geometric parameters, total number of plies, fiber orientation, and initial geometric imperfections are studied.     

Imperfection Sensitivity of Laminated Cylindrical Shells According to Different Shell Theories

The imperfection sensitivity of laminated cylindrical shells is considered—via the initial postbuckling analysis—on the basis of three different shell theories: Donnell in 1933; Sanders in 1963; and Timoshenko in 1961. The procedure involves nonlinear partial differential equations, which are converted into a sequence of three linear sets. The equations are solved with the variables expanded in Fourier series in the circumferential direction and in finite differences in the axial directions. A general code is developed and used in studying the effect of higher exactness of the shell theory on the sensitivity behavior, and in a parametric study of the sensitivity of anisotropic angle-ply cylindrical shells.     

Buckling of Vertical Cylindrical Shells Under Combined End Pressure and Body Force

This paper is concerned with the elastic buckling of vertical cylindrical shells under combined end pressure and body force. Such buckling problems are encountered when cylindrical shells are used in a high-g environment such as the launching of rockets and missiles under high-propulsive power. The vertical shells may have any combination of free, simply supported, and clamped ends. Based on the Goldenveizer-Novozhilov thin shell theory, the total potential energy functional is presented and the buckling problem is solved using the Ritz method. Highlight in the formulation is the importance of the correct potential energy functional which includes the shell shortening due to the circumferential displacement. The omission of this contributing term leads to erroneous buckling solutions when the cylindrical shell is not of moderate length (length-to-radius ratio smaller than 0.7 or larger than 3). New solutions for body-force buckling parameters are presented for stubby cylindrical shells to long tube-like shells that approach the behavior of columns. The effects of the shell thickness and length on buckling parameter are also investigated.     

Vibration of Axially Loaded Rotating Cross-Ply Laminated Cylindrical Shells via Ritz Method

This paper presents the free vibration analysis of axially loaded rotating cross-ply laminated cylindrical shells with the consideration of the effects of centrifugal and Coriolis forces as well as the initial hoop tension due to the rotation. The Ritz method is employed for the solution of this problem. Adopting the trigonometric series as the admissible displacement functions, a set of frequency characteristics equations is derived. The frequency characteristic analysis for shells of simply supported boundary conditions is examined and the frequency characteristics of various lamination schemes are investigated. The results from the present analysis are compared with the available solutions to validate its accuracy.     

Thermomechanical Postbuckling Analysis of Cross-Ply Laminated Cylindrical Shell Panels

The postbuckling analysis of symmetric and antisymmetric cross-ply laminated cylindrical shell panels subjected to thermomechanical loading is examined in this paper. The formulation is based on an extension of Reissner’s shallow shell simplifications and accounts for parabolic distribution of transverse shear strains. Adopting a multiterm Galerkin’s method, the governing nonlinear partial differential equations are reduced into a set of nonlinear algebraic equations. The nonlinear equilibrium paths through limit points are traced using the Newton–Raphson method in conjunction with Riks approach. Numerical results are presented for symmetric [?start0/90/0end?] and antisymmetric [?start0/90end?] cross-ply laminated cylindrical shell panels, that illustrate the influence of mechanical edge loads, lateral distributed load, initial imperfection, and temperature field on the limit loads and snap-through behavior.     

Shear Failure of Pultruded Fiber-Reinforced Polymer Composites under Axial Compression

When structural elements are subjected to compressive loads, the shear forces and stresses induced by second-order effects may lead to shear failure prior to compressive failure. This is particularly likely to occur in the case of pultruded glass fiber-reinforced polymer profiles, which normally exhibit low shear strength in relation to compressive strength. This paper analyzes the effects of initial imperfection, slenderness, shear-to-compressive strength ratio, shear coefficient, and type of shear failure criterion on ultimate load and failure mode (shear or compressive failure). A formulation for predicting ultimate load based on shear failure and second-order deformation is proposed. The results obtained compare well with similar results obtained using other methods and experimental data available in literature. The proposed method is based strictly on mechanics and thus requires no fitting to experimental data.     

Buckling of Variable Section Columns under Axial Loading

In this paper, the static stability of the variable cross section columns, subjected to distributed axial force, is considered. The presented solution is based on the singular perturbation method of Wentzel-Kramers-Brillouin and the column is modeled using Euler-Bernoulli beam theory. Closed-form solutions are obtained for calculation of buckling loads and the corresponding mode shapes. The obtained results are compared with the results in the literature to verify the present approach. Using numerous examples, it is shown that the represented solution has a very good convergence and accuracy for determination of the instability condition.     

Buckling of Homogeneous Isotropic Cylindrical Shells under Axial Stress

A 2D higher-order shell theory that can take into account the complete effects of higher-order deformations is applied to the buckling problems of a thick circular cylindrical shell subjected to axial compression. The effects of higher-order deformations such as shear deformations and thickness changes on buckling stresses of homogeneous isotropic circular cylindrical shells are studied. Based on the power series expansion of displacement components, a set of fundamental equations of a 2D higher-order shell theory is derived through the principle of virtual displacements. Several sets of truncated approximate theories are applied to solve the buckling problems of a simply supported thick circular cylindrical shell. To assure the accuracy of the present theory, the convergence of the buckling stresses is examined in detail, and the results are compared with those obtained in existing theories.     

Vibration of Open Cylindrical Shells with Stepped Thickness Variations

This paper presents the first-known exact solutions for vibration of open circular cylindrical shells with multiple stepwise thickness variations based on the Flügge thin shell theory. An open cylindrical shell is assumed to be simply supported along the two straight edges and the remaining two opposite curved edges may have any combination of edge support conditions. The shell is subdivided into segments at the locations of thickness variations. The state-space technique is adopted to derive the homogenous differential equations for a shell segment and the domain decomposition method is employed to impose the equilibrium and compatibility requirements along the interfaces of the shell segments. The correctness of the proposed method is checked against existing results in the open literature and results generated from finite element package ANSYS and excellent agreement is achieved. Several open shells with various combinations of end boundary conditions are studied by the proposed method.     

Thermomechanical Postbuckling of Cross-Ply Laminated Rectangular Plates

Postbuckling analysis is presented for shear deformable cross-ply laminated composite rectangular plates subjected to the combination of in-plane edge compressive mechanical loading and thermal loads due to a linearly varying temperature across the thickness. The formulation is based on the first-order shear deformation theory and von-Karman-type nonlinearity. The analysis uses a quadratic extrapolation technique for linearization and Chebyshev polynomials for spatial discretization. An incremental iterative approach is employed to estimate the critical load. The boundary conditions consisting of clamped, simply supported, free edge, and their combinations are considered. The effects of the thinness ratio, aspect ratio, lamination scheme, the number of layers, and the modulus ratio on the critical load/limit load and postbuckling behavior are studied.     

Nonlinear Response of Laminated Cylindrical Shell Panels Subjected to Thermomechanical Loads

The nonlinear response of multi-layered composite cylindrical shell panels subjected to thermomechanical loads are studied in this article. The structural model is based on the first order shear deformation theory incorporating geometric nonlinearities. The nonlinear equilibrium paths are traced using the arc-length control algorithm within the framework of finite element method. Hashin’s failure criterion has been adopted to predict the first-ply failure of cylindrical laminates. Both temperature independent and temperature dependent elastic properties are considered in the analysis. Specific numerical results are reported to show the effect of radius-to-span ratio, thickness-to-span ratio, laminate stacking sequence, and boundary condition on stability characteristics of laminated cylindrical shell panels subjected to combined thermal and mechanical transverse loads.     

Construction Method for Cylindrical Latticed Shells Based on Expandable Mechanisms

This note proposes a cost-effective construction method for large span cylindrical latticed shell structures. The method converts a latticed shell into a six-bar linkage, allowing the major part of the assembling work to be carried out at ground level. The structure is then lifted into position by activation of the predetermined mechanism mode and the structure is completed by adding additional members. In the note, we present the detailed construction procedures, together with some technical challenges that were encountered in practice. A discussion of the possible solutions to these challenges is also given, together with some suggestions for future similar applications of the method. From this study, researchers may borrow ideas to explore new construction methods, and practitioners of this technology can apply it more safely and properly.     

Free Vibration of Open Circular Cylindrical Composite Shells with Point Supports

A variational full-field method is presented in this paper for the free vibration analysis of open circular cylindrical laminated shells supported at discrete points. A differential equation in matrix form is developed using the first-order shear deformable theory of shells, and rotary inertia is included. The displacement fields are defined by using very high-order interpolating polynomials and a large number of preselected nodal points on the reference surface of the shell. Each nodal point has 5 degrees of freedom, three displacement components, and two components of the rotation of the normal to the reference surface. The stiffness and mass matrices are obtained using the strain and kinetic energy functions. The numerical results are calculated for shallow and deep circular cylindrical panels with four-, six-, and eight-point supports along the two parallel straight edges. The values of the natural frequency obtained from the present method show good agreement with published data in the literature.     

Buckling of Multiwalled Carbon Nanotubes Using Timoshenko Beam Theory

A Timoshenko beam model is presented in this paper for the buckling of axially loaded multiwalled carbon nanotubes surrounded by an elastic medium. Unlike the Euler beam model, the Timoshenko beam model allows for the effect of transverse shear deformation which becomes significant for carbon nanotubes with small length-to-diameter ratios. These stocky tubes are normally encountered in applications such as nanoprobes or nanotweezers. The proposed model treats each of the nested and concentric nanotubes as individual Timoshenko beams interacting with adjacent nanotubes in the presence of van der Waals forces. In particular, the buckling of double-walled carbon nanotubes modeled as a pair of double Timoshenko beams is studied closely and an explicit expression for the critical axial stress is derived. The study clearly demonstrates a significant reduction in the buckling loads of the tubes with small length-to-diameter ratios when shear deformation is taken into consideration.     

Free Vibration Analysis of Cylindrical Shells Supported on Parts of the Edges

Numerical studies of the free vibration analysis of open skewed circular cylindrical shells supported only on selected segments of the straight edges are presented in this paper. The uniform thickness shell geometry is defined by the radius, subtended angle and the length, all with reference to the middle surface. The open skewed circular cylindrical shell is modeled by dividing the reference surface into few patches and introducing upon them displacement nodal points and also five degrees of freedom in accordance with the first order shear deformable shell theory are assigned to each of these nodal points. The free vibration analysis of the shell structure is performed using two types of interpolating polynomials, viz. simple high order algebraic and Bezier, respectively. The number of nodal points per patch determines the order of the displacement polynomials. As a consequence considerably high-order polynomials are used in computations for the accurately converged results. Convergence studies are carried out to validate the method for cases in which the skewed cylindrical shell is supported only on the third of each of the two straight edges. Additionally, the performance of the present method is assessed and discussed by comparing frequency results with those from standard finite element methods using linear and parabolic quadrilateral elements.     

Closed-Form Solutions for Buckling of Long Anisotropic Plates with Various Boundary Conditions under Axial Compression

Closed-form solutions for buckling of long plates with flexural/twist anisotropy with the short edges simply supported and with the longitudinal edges simply supported, clamped, or elastically restrained in rotation under axial compression are presented. An energy method (Rayleigh–Ritz) is employed to obtain the critical buckling loads. The critical buckling loads are expressed in terms of minimum nondimensional buckling coefficients and stiffness parameters. The new closed-form solutions show an excellent agreement when compared to existing solutions and finite-element analysis. Due to their simplicity and accuracy, the new closed-form solutions can be confidently used as an alternative to computationally expensive structural analysis to assess buckling in the preliminary design phase of composite structures.     

Postbuckling of Axially Loaded Functionally Graded Cylindrical Panels with Piezoelectric Actuators in Thermal Environments

A compressive postbuckling analysis is presented for a functionally graded cylindrical panel with piezoelectric actuators subjected to the combined action of mechanical, electrical, and thermal loads. The temperature field considered is assumed to be of uniform distribution over the panel surface and through the panel thickness and the electric field considers only the transverse component EZ. The material properties of the presently considered functionally graded materials (FGMs) are assumed to be temperature-dependent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents, whereas the material properties of the piezoelectric layers are assumed to be independent of the temperature and the electric field. The governing equations are based on a higher-order shear deformation theory with a von Kármán-Donnell-type of kinematic nonlinearity. A boundary layer theory for shell buckling is extended to the case of hybrid FGM cylindrical panels of finite length. The nonlinear prebuckling deformations and initial geometric imperfections of the panel are both taken into account. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the compressive postbuckling behavior of perfect and imperfect FGM cylindrical panels with fully covered piezoelectric actuators, under different sets of thermal and electrical loading conditions. The effects due to temperature rise, volume fraction distribution, applied voltages, panel geometric parameters, in-plane boundary conditions, as well as initial geometric imperfections are studied.     

Buckling of Circular Mindlin Plates with an Internal Ring Support and Elastically Restrained Edge

This paper is concerned with the elastic buckling problem of circular Mindlin plates with a concentric internal ring support and elastically restrained edge. In solving this problem analytically, the circular plate is first divided into an annular segment and a core circular segment at the location of the internal ring support. Based on the Mindlin plate theory, the governing differential equations for the annular and circular segments are then solved exactly and the solutions brought together by using the interfacial conditions. New exact critical buckling loads of circular Mindlin plate with an internal ring support and elastically restrained edge are presented for the first time. The optimal radius of the internal ring support for maximum buckling load is also found. An approximate relationship between the buckling loads of such circular plates based on the classical thin plate theory and the Mindlin plate theory is also explored.     

Integrated Optimization of Control System for Smart Cylindrical Shells Using Modified GA

In this paper, modeling of the vibration of cylindrical shell components of space structures incorporating piezoelectric sensor/actuators (S/As) for optimal vibration control is proposed and formulated. The parameters of the control system, which include the placement and sizing of the piezoelectric S/As and the feedback control gains, were considered as design variables and optimized simultaneously. The effect of the amount of piezoelectric patches was investigated as well. The criterion based on the maximization of energy dissipation was employed for the optimization of the control system. A modified real-encoded genetic algorithm (GA) dealing with various constraints has been developed and applied to search for the optimal placement and size of the piezoelectric patches as well as the optimal feedback control gains. The results of three numerical examples, which include a simply supported plate, a simply supported cylindrical shell, and a clamped-simply supported plate, demonstrated significant vibration suppression based on the optimal design of the control system. It was also found that for specific controlled vibration modes, the optimal distribution of the piezoelectric S/As should be located at the areas separated by the nodal lines to achieve the optimal control effect. This finding would be useful for the practical design of smart structures.     

Nonlinear Buckling of Composite Shells of Revolution

By adopting the energy method, a method of calculating the stability of the rotational composite shell is presented that takes into account the influence of nonlinear prebuckling deformations and stresses on the buckling of the shell. The relationships between the prebuckling deformations and strains are calculated by nonlinear Karman equations. The numerical method is used to calculate the energy of the whole system. The nonlinear equation is solved by combining the gradient method and the amended Newton iterative method. A computer program is also developed. Examples are given to demonstrate the accuracy of the method presented in this paper.