Buckling and Postbuckling of Antisymmetrically Laminated Cross-ply Shear-Deformable Cylindrical Shells under Axial Compression

The generalized Donnell-type equations governing large deflection of antisymmetrically laminated cross-ply cylindrical shells counting for transverse shear deformations are derived and presented. An asymptotic series solution is constructed by regular perturbation technique for postbuckling behaviors of the cylindrical shells with simply supported edges subjected to axial compression. Boundary layer influence at both ends of the shells on overall buckling and postbuckling are considered, and for consistency of the boundary valued problem, the boundary layer solutions are also designed to match the out-of-plane edge conditions by singular perturbation approach. Effects of transverse shear deformation, Batdorf’s parameter, elastic moduli ratio, and initial geometric imperfection on buckling and postbuckling performance of the shells are examined. Some numerical examples are taken for comparison of the present results of buckling loads and load–deflection curves of the shells with corresponding theoretical predictions to show effectiveness and accuracy of the present asymptotic perturbation solution.     

Three-Dimensional Elasticity Solutions of Laminated Annular Spherical Shells

An asymptotic theory is presented for the analysis of laminated annular spherical shells by the perturbation method. By means of proper nondimensionalization, asymptotic expansion, and then successive integration of the basic equations through the thickness direction, we obtain the recursive sets of governing equations for the bending analysis of annular spherical shells. The method of differential quadrature is adopted for solving the problems of various orders. Illustrative examples are given to demonstrate the performance of the present asymptotic theory.     

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.     

Nonlinear Zigzag Theory for Buckling of Hybrid Piezoelectric Rectangular Beams under Electrothermomechanical Loads

A new efficient electromechanically coupled geometrically nonlinear (of von Karman type) zigzag theory is developed for buckling analysis of hybrid piezoelectric beams, under electrothermomechanical loads. The thermal and potential fields are approximated as piecewise linear in sublayers. The deflection is approximated as piecewise quadratic to explicitly account for the transverse normal strain due to thermal and electric fields. The longitudinal displacement is approximated as a combination of third order global variation and a layerwise linear variation. The shear continuity conditions at the layer interfaces and the shear traction-free conditions at the top and bottom are used to formulate the theory in terms of three primary displacement variables. The governing coupled nonlinear field equations and boundary conditions are derived using a variational principle. Analytical solutions for buckling of symmetrically laminated simply supported beams under electrothermal loads are obtained for comparing the results with the available exact two-dimensional (2D) piezothermoelasticity solution. The comparison establishes that the present results are in excellent agreement with the 2D solution which neglects the prebuckling transverse strain effect.     

Buckling and Postbuckling Behavior of Cross-Ply Composite Plate Subjected to Nonuniform In-Plane Loads

This paper presents a study of buckling and postbuckling behaviour of simply supported composite plates subjected to nonuniform in-plane loading. The mathematical model is based on higher order shear deformation theory incorporating von Kármán nonlinear strain displacement relations. Because the applied in-plane edge load is nonuniform, in the first step the plane elasticity problem is solved to evaluate the stress distribution within the prebuckling range. Using these stress distributions, the governing equations for postbuckling analysis of composite plates are obtained through the theorem of minimum potential energy. Adopting Galerkin’s approximation, the governing nonlinear partial differential equations are reduced into a set of nonlinear algebraic equations in the case of postbuckling analysis, and homogeneous linear algebraic equations in the case of buckling analysis. The critical buckling load is obtained from the solution of associated linear eigenvalue problem. Postbuckling equilibrium paths are obtained by solving nonlinear algebraic equations employing the Newton-Raphson iterative scheme. Explicit expressions for the plate in-plane stress distributions within the prebuckling range are reported for isotropic and composite plates subjected to parabolic in-plane edge loading. Buckling loads are determined for three plate aspect ratios (a/b = 0.5, 1, 1.5) and three different types of in-plane load distributions. The effect of shear deformation on the buckling loads of composite plate is reported. The present buckling results are compared with previously published results wherever possible.     

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.     

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.     

Bounds on Flexural Properties and Buckling Response for Symmetrically Laminated Composite Plates

Nondimensional parameters and equations governing the buckling behavior of rectangular symmetrically laminated plates are presented that can be used to represent the buckling resistance, for plates made of all known structural materials, in a very general, insightful, and encompassing manner. In addition, these parameters can be used to assess the degree of plate orthotropy, to assess the importance of anisotropy that couples bending and twisting deformations, and to characterize quasi-isotropic laminates quantitatively. Bounds for these nondimensional parameters are also presented that are based on thermodynamics and practical laminate construction considerations. These bounds provide insight into potential gains in buckling resistance through laminate tailoring and composite-material development. As an illustration of this point, upper bounds on the buckling resistance of long rectangular orthotropic plates with simply supported or clamped edges and subjected to uniform axial compression, uniform shear, or pure in-plane bending loads are presented. The results indicate that the maximum gain in buckling resistance for tailored orthotropic laminates, with respect to the corresponding isotropic plate, is in the range of 26–36% for plates with simply supported edges, irrespective of the loading conditions. For the plates with clamped edges, the corresponding gains in buckling resistance are in the range of 9–12% for plates subjected to compression or pure in-plane bending loads and potentially up to 30% for plates subjected to shear loads.     

Boundary Condition Effects in Buckling of “Soft” Core Sandwich Panels

The effects of boundary conditions on the critical load level and the corresponding deflection mode shape of sandwich panels with a “soft” core due to in-plane loads are presented. The study is conducted using a closed-form high-order linearized buckling analysis that includes the influence of the transverse flexibility of the core as well as of the localized effects on the overall sandwich panel behavior, and allows the use of different boundary conditions for the upper and lower skin at the same section. The panel construction is general and consists of two skins (not necessarily identical), metallic or composite-laminated symmetric, and a soft core made of foam or a low-strength honeycomb. The closed-form high-order analysis yields the general buckling behavior of the structure, which means that the solutions obtained allow for interaction between the skins and the core. The solutions are general and are not based on separation of the buckling response on several types of uncoupled buckling modes, such as overall buckling, skins wrinkling, etc., as commonly used in the literature. The numerical scheme consists of finite differences to approximate the governing equations of the closed-form high-order formulation and to transform the set of linearized governing differential equations into an eigenvalue problem that is solved using the deflated iterative Arnoldi procedure. The influence of a general type of boundary conditions, including different conditions throughout the height of the same section and nonidentical conditions at the upper and lower skin, as well as of the core properties, on the buckling behavior of the sandwich panels is considered. The discrepancy between the Timoshenko-Reissner model and the present formulation is discussed. In particular, a partial fixity phenomenon due to the existence of the pinned boundary conditions, i.e., simply supported conditions, at the upper and lower skins at the edge is demonstrated. It is shown that the core properties affect the buckling loads and the corresponding modes of the panel in such a way that the structures with identical boundary conditions but with different cores may undergo different types of buckling such as overall and local as well as interactive loss of stability. The effect of an edge concentrated moment, induced by a couple and exerted on the skins only is also studied.     

Hierarchical High-Fidelity Analysis Methodology for Buckling Critical Structures

A hierarchical high-fidelity analysis methodology for predicting the critical buckling load of compression-loaded thin-walled isotropic shells is described. This hierarchical procedure includes three levels of fidelity for the analysis. Level 1 assumes that the buckling load can be predicted by the classical shell solution with simply supported boundary condition, and with a linear membrane prebuckling solution. Level 2 includes the effects of a nonlinear prebuckling solution and the effects of traditional clamped or simply supported boundary conditions. Level 3 includes the nonlinear interaction between nearly simultaneous buckling modes and the effects of boundary imperfections and general boundary conditions. Various deterministic and probabilistic approaches are used to account for the degrading effects of unavoidable shell-wall geometric imperfections. The results from the three solution levels are compared with experimental results, and the effects of the assumptions and approximations used for the three solution levels are discussed. This hierarchical analysis approach can be used in the design process to converge rapidly to an accurate prediction of the expected buckling load of a thin-shell design problem.     

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.     

Postbuckling Analysis of Geometrically Imperfect Conical Shells

A suitable postbuckling analysis, based on geometrically nonlinear behavior, is developed for arbitrary imperfect conical shells. The conical shell was chosen as a representative case exhibiting the entire range of sensitivity to imperfection. A general symbolic code (using the MAPLE compiler) was programmed to create the differential operators of the nonlinear partial differential equations, based on Donnell’s type shell theory. The code then uses the Galerkin procedure, the Newton-Raphson and arc-length procedures, and a finite-differences scheme for automatic development of an efficient FORTRAN code. The code is used for parametric study of the nonlinear behavior and yields the sensitivity characteristic for a wide range of cone semivertex angles. A typical nonlinear behavior of a conical shell is investigated. Comparison with a simpler procedure, based on the initial postbuckling analysis (Koiter’s theory), confirms the need for the present more accurate one, especially for shells with prebuckling nonlinear behavior. The present investigation summarizes the sensitivity behavior with respect to imperfection shapes and amplitudes for the entire range of cone semivertex angles.     

Inelastic Plate Buckling

Current models to determine the local buckling stress of inelastic plates under in-plane loading are based on plastic deformation theory and semirational or empirical relationships. A successful J2 flow theory describing inelastic local buckling of initially perfect plates needs to avoid two well-known pitfalls known as the “inelastic column buckling paradox” and the “plastic buckling paradox.” While the former problem, which found its origin in 1895 in Engesser’s double modulus approach, was resolved by Shanley in the late 1940s, a convincing solution of the plastic buckling paradox has not yet been presented. This paper proposes a modification to the J2 flow theory which hinges on the determination of the shear stiffness from second-order considerations. A differential equation is derived which describes the incremental plate deformations at the inelastic local buckling load. The differential equation is studied for two cases of boundary conditions: a plate simply supported along four edges and a plate simply supported along three edges with one longitudinal edge free.     

Influence of Shear Deformation on Buckling of Pultruded Fiber Reinforced Plastic Profiles

Theoretical studies of the influence of shear deformation on the flexural, torsional, and lateral buckling of pultruded fiber reinforced plastic (FRP)-I-profiles are presented. Theoretical developments are based on the governing energy equations and full section member properties. The solution for flexural buckling is consistent with the established solution based on the governing differential equation. The new solutions for torsional and lateral buckling incorporate a reduction factor similar to that for flexural buckling. The solution for lateral buckling also incorporates the influence of prebuckling displacements. Closed form solutions for a series of simply supported, pultruded FRP I-profiles, based on experimentally determined full section flexural and torsional properties, indicate the following conclusions. For members subjected to axial compression, shear deformation can reduce the elastic flexural and torsional buckling loads by up to approximately 15% and 10%, respectively. For members subjected to bending, prebuckling displacements can increase the buckling moments by over 20% while shear deformation decreases the buckling moments by less than 5%.     

Buckling of Laminated Composite Rectangular Plates

The present study estimates the critical/buckling loads of laminated composite rectangular plates under in-plane uniaxial and biaxial loadings. The formulation is based on the first-order shear deformation theory and von-Karman-type nonlinearity. Chebyshev series is used for spatial discretisation and quadratic extrapolation is used for linearization. An incremental iterative approach is used for estimation of the critical load. Different combinations of simply supported, clamped and free boundary conditions are considered. The effects of plate aspect ratio, lamination scheme, number of layers and material properties on the critical loads 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.     

Exact Buckling Solutions For Rectangular Plates Under Intermediate and End Uniaxial Loads

This paper is concerned with the elastic buckling of rectangular plates subjected to both intermediate and end uniaxial loads. The rectangular plates have two simply supported opposite edges that are perpendicular to the in-plane load direction, while the other two plate edges can have free, simply supported, or clamped edges. The solution procedure involves the use of the Levy approach, the domain decomposition technique, and the state-space concept. The method furnishes exact stability criteria; samples of which are presented in a graphical form for plates with various boundary conditions. These results will be useful to engineers who design plates (or walls) that support intermediate floors.     

Effects of Random Material Properties on Buckling of Composite Plates

Composites exhibit a considerable amount of variation in their material properties because their fabrication∕manufacturing process involves a large number of parameters that cannot be controlled effectively. In the present study, the material properties have been modeled as random variables for better prediction of the system behavior. The classical laminate theory and first-order and higher-order shear deformation theories have been employed in deriving the governing equations for buckling of laminated rectangular plates. A mean-centered first-order perturbation technique has been used to find the second-order statistics of the buckling load. The approach has been validated by comparison with results of Monte Carlo simulation. Typical results have been presented for a plate with all edges simply supported. The effectiveness of the theories in predicting the buckling load dispersions has been examined. The sensitivity of buckling loads to change in the standard deviation of random material properties and to system parameters—side-to-thickness ratio and aspect ratio—has been examined for cross-ply symmetric and antisymmetric laminates.     

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.     

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.