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.     

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.     

Generalized Differential Quadrature for Frequency of Rotating Multilayered Conical Shell

This paper uses the generalized differential quadrature (GDQ) method to study the influence of boundary conditions on the natural frequency of a rotating thin truncated circular multilayered conical shell. The governing equations of motion include the effects of initial hoop tension and centrifugal and Coriolis accelerations due to rotation. The GDQ method is applied to the discrete grid points in the meridional direction. Results are obtained to study the influence of boundary conditions on the frequency at different circumferential wave numbers, rotating speeds, and geometric properties. The influences of the cone angle and layered configuration on the variation of frequency with rotating speed also are presented. To validate the accuracy and efficiency of the GDQ method, comparisons are made with those available in the open literature and very good agreements are achieved.     

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.     

Buckling of Columns with Intermediate Elastic Restraint

This paper presents a comprehensive set of exact stability criteria for Euler columns with an intermediate elastic restraint. A subset of this class of problem is the buckling problem of columns with an intermediate rigid support where the elastic restraint takes on an infinite stiffness. Also, this study reiterates the existence of a critical elastic restraint stiffness in which the buckled mode switches to a higher-buckling mode of the corresponding column without an intermediate support. It is clear that this critical stiffness value exists only when the restraint is placed at the node of the higher-buckling mode and the buckling load associated with this critical stiffness value is the maximum achievable value that can be attained with an intermediate elastic restraint.     

Plastic Buckling Transition Modes in Moderately Thick Cylindrical Shells

Moderately thick perfect cylindrical shells under axial compression first exhibit an axisymmetric buckling mode, where a localization of buckling patterns, referred to as an elephant foot bulge, is caused by the first plastic bifurcation. However, the transition from the axisymmetric buckling mode to a nonaxisymmetric buckling mode, referred to as a diamond buckling mode, may occur due to the next bifurcation if we continue the loading under displacement control. Herein, this phenomenon is examined, based on a rigorous plastic bifurcation analysis. As a result, it is observed that the circumferential wave number of the diamond buckling mode increases with the decrease of the wall thickness. The boundary conditions also considerably influence the occurrence of diamond buckling. It is found that the strain concentration is intensified for the diamond buckling modes, compared with the axisymmetric modes.     

Effect of Normal Strains in Buckling of Thick Orthotropic Shells

An improved elasticity solution to the problem of buckling of orthotropic cylindrical shells subjected to external pressure is presented. The 2D axisymmetric cylindrical shell is studied (ring approximation). Specifically, in the development of the governing equations and boundary conditions for the buckling state, the solution includes the terms with the prebuckling normal strains and stresses as coefficients (i.e., the terms ekk0σij′ and σkk0eij′, which were neglected in the earlier work as being too small compared to the terms σij′ and σkk0ωj′, respectively). The formulation results in a two-point boundary eigenvalue problem for ordinary differential equations in r, with the external pressure p as the parameter. The results show that the effect of including the normal strains and stresses is to further decrease the critical load. This decrease (versus the earlier elasticity solution without these terms) depends on the shell thickness and is generally moderate, and in no event comparable with the (quite large) decrease of the elasticity versus the shell theory prediction. This decrease depends also on the degree of orthotropy, and it is smaller for the isotropic case. Finally, a formula is derived for the critical pressure based on a first-order shear deformation formulation, and the comparison shows an improvement versus the classical shell for thick shells, but still the elasticity solution is noticeably lower than the first-order shear deformation prediction.     

Closed-Form Solution of Axisymmetric Slender Elastic Toroidal Shells

Toroidal shells are widely used in structural engineering. The governing equations of toroidal shells are very complicated because of its variable coefficients with singularity. To find their analytical solution, traditionally, the complex form governing equations were proposed and some useful solutions were obtained. Unfortunately, no any closed-form solution has even been obtained for either general or slender toroidal shells. This paper focus on a special case of toroidal shells, i.e., slender symmetrical toroidal shells. For the first time, the closed-form solution of this kind of shell has been successfully obtained from displacement form governing equations. The closed-form solution is demonstrated for the example of thermal compensation devices. The correction of well-known Dahl formula for slender toroidal shell has been proposed based on the solution obtained in this paper.     

Anisotropic Nonlinear Elastic Model for Particulate Materials

This paper presents the development of an elastic model for particulate materials based on micromechanics considerations. A particulate material is considered as an assembly of particles. The stress–strain relationship for an assembly can be determined by integrating the behavior of the interparticle contacts in all orientations and using a static hypothesis which relates the average stress of the granular assembly to a mean field of particle contact forces. Hypothesizing a Hertz–Mindlin law for the particle contacts leads to an elastic nonlinear behavior of the particulate material, we were able to determine the elastic constants of the granular assembly based on the properties of the particle contacts. The numerical predictions, compared to the results obtained during experimental studies on different granular materials, show that the model is capable of taking into account both the influence of the inherent anisotropy and the influence of the stress-induced anisotropy for different stress conditions.     

Buckling of Column with Base Attached to Elastic Half-Space

Buckling of a heavy elastic column loaded by a concentrated force at the top is analyzed. It is assumed that the base of the column is fixed to a rigid circular plate that is positioned on a homogeneous, isotropic, linearly elastic half-space. The plate has adhesive contact with the half-space. The constitutive equations for the column are assumed in the form that allows axial compressibility and takes into account the influence of shear stresses. It is shown that eigenvalues of the linearized equations determine the bifurcation points of the full nonlinear system of equilibrium equations. The type of bifurcation at the lowest eigenvalue is examined and is shown that it could be super- or subcritical. The postcritical shape of the column is determined by numerical integration of the equilibrium equations.     

Stochastic Finite-Element Analysis for Elastic Buckling of Stiffened Panels

The variability of the random buckling loads of beams and plates with stochastically varying material and geometric properties is studied in this paper using the concept of the variability response function. The elastic modulus, moment of inertia, and thickness are assumed to be described by homogeneous stochastic fields. The variance of the buckling load is expressed as the integral of the auto- and cross-spectral density functions characterizing the stochastic fields multiplied by the deterministic variability response functions. Using this expression spectral-distribution-free upper bounds of the buckling load variability are established. Further, the buckling load variability for prescribed forms of the spectral density functions is calculated. Using a local average approach, the commercial finite-element package ABAQUS is incorporated into the analysis of these random buckling loads. The technique is applied to study variability of the critical buckling load of a stiffened steel plate used in experiments to model a barge deck.     

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.     

Localized Buckling of a Bilinear Elastic Ring under External Pressure

A fully nonlinear finite element analysis for prediction of localization in moderately thick imperfect rings under applied hydrostatic pressure is presented. The present nonlinear finite element solution methodology includes all the nonlinear terms in the kinematic equations and utilizes the total Lagrangian formulation in the constitutive equations and incremental equilibrium equations. A curved six-node element, based on an assumed quadratic displacement field (in the circumferential coordinate), employs a two-dimensional hypothesis, known as linear displacement distribution through thickness theory, to capture the effect of the transverse shear/normal (especially, shear) deformation behavior. The driving factor behind this analysis is to determine the onset of localization arising out of the bilinear material behavior of the ring with modal imperfection. Numerical results suggest that material bilinearity is primarily responsible for the appearance of a limit or localization (peak pressure) point on the postbuckling equilibrium path of an imperfect ring.     

Buckling of a Weakened Infinite Beam on an Elastic Foundation

An infinite beam attached to an elastic foundation is buckled by an axial force. The beam is weakened by one or more joints or partial cracks. The governing equations are solved analytically and an exact nonlinear characteristic equation gives the buckling criterion. It is found that the buckling force depends on the foundation stiffness and the rotational resistance of the joints. The buckling modes are complex, and may be either antisymmetric or symmetric.     

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.     

Analysis of Finite Strain Anisotropic Elastoplastic Fracture in Thin Plates and Shells

A finite element methodology for analyzing fracture in thin shells in the large strain elastoplastic regime is presented. The postlocalization constitutive model is based on a cohesive surface dissipation mechanism. We employ a Kirchhoff-Love shell model (and the corresponding discretization by finite elements) and make use of the extended finite element technique in the (implicit) form of midsurface displacement and director field discontinuities. Applications showing the possibilities of this technique are shown, and the effect of plastic anisotropy in the crack pattern is numerically inspected.     

Effect of Longitudinal Stress Gradients on Elastic Buckling of Thin Plates

This paper analyzes the effect of longitudinal stress gradients on the elastic buckling of thin isotropic plates. Two types of thin plates are considered: (1) a plate simply supported on all four edges and rotationally restrained on two longitudinal edges; and (2) a plate simply supported on three edges with one longitudinal edge free and the opposite longitudinal edge rotationally restrained. These two cases illustrate the influence of longitudinal stress gradient on stiffened and unstiffened elements, respectively. A semianalytical method is derived and presented herein to calculate the elastic-buckling stress of both types of rectangular thin plates subjected to nonuniform applied longitudinal stresses. Finite-element analysis using ABAQUS is employed to validate the semianalytical model for plates with fixed and/or simple supports. Empirical formulas are produced to calculate the buckling coefficients of plates with fixed and/or simple supports under longitudinal stress gradients. The results help establish a better understanding of the effect of longitudinal stress gradients on the elastic buckling of thin plates and are intended to aid in the development of design provisions to include these effects in the strength prediction of thin-walled beams under moment gradients.     

Time-Dependent Response of an Axially Loaded Elastic Bar in a Multilayered Poroelastic Medium

This paper considers load transfer from an axially loaded long elastic bar into a multilayered poroelastic half-space. The problem is analyzed by decomposing the bar-half-space system into an extended half-space governed by Biot’s theory of poroelasticity and a one-dimensional fictitious bar. The interaction problem is formulated in the Laplace transform domain. Vertical displacement of the bar is approximated by an exponential series with a set of arbitrary functions. The arbitrary functions are determined by using a variational method. The vertical displacement influence function of a multilayered half-space subjected to a buried uniform vertical patch load is required in the variational formulation. The required influence function is obtained by employing a previously developed exact stiffness matrix method. Time domain solutions are computed by using a numerical Laplace inversion scheme. Selected numerical results are presented to portray the influence of the bar length–radius ratio, layer configuration, poroelastic material parameters, and loading time history on the time dependent response of a bar.     

Initial Compressive Buckling of Clamped Plates Resting on Tensionless Elastic or Rigid Foundations

The unilateral contact buckling problem of thin plates resting on tensionless foundations is investigated. Three different plate models are considered. For a plate of limited length on a tensionless elastic foundation, the plate is first simplified to a one-dimensional mechanical model by assuming a buckling mode in terms of transverse coordinates, after which a new method is employed to determine the initially unknown boundaries of the areas in contact. Based on the continuity condition on the borderline between contact and noncontact regions, the buckling mode displacements of the whole plate may be expressed through the critical load coefficient and the first half-wavelength, reducing the buckling problem to two nonlinear algebraic equations with two unknowns. This procedure has been named the transfer function method. For a very long plate with a symmetric buckling mode, an infinite plate model with two half-waves is presented. For a plate on a rigid foundation, a single half-wave buckling model is shown to be appropriate. Comparison of limiting cases with exact solutions and with ABAQUS results showed good agreement. Finally, the influences of aspect ratio and foundation stiffness are presented.