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.     

Vibration and Stability of Thick Plates on Elastic Foundations

Natural frequencies and buckling stresses of a thick isotropic plate on two-parameter elastic foundations are analyzed by taking into account the effect of shear deformation, thickness change, and rotatory inertia. Using the method of power series expansion of the displacement components, a set of fundamental dynamic equations of a two-dimensional, higher-order theory for thick rectangular plates subjected to in-plane stresses is derived through Hamilton's principle. Several sets of truncated approximate theories are used to solve the eigenvalue problems of a simply supported thick elastic plate. To assure the accuracy of the present theory, convergence properties of the minimum natural frequency and the buckling stress are examined in detail. The distribution of modal transverse stresses are obtained by integrating the three-dimensional equations of motion in the thickness direction. The present approximate theories can accurately predict the natural frequencies and buckling stresses of thick plates on elastic foundations as compared with Mindlin plate theory and classical plate theory.     

Postbuckling of Pressure-Loaded Functionally Graded Cylindrical Panels in Thermal Environments

A postbuckling analysis is presented for a functionally graded cylindrical panel of finite length subjected to lateral pressure in thermal environments. Material properties 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. The governing equations of a functionally graded cylindrical panel are based on Reddy’s higher-order shear deformation shell theory with von Kármán–Donnell-type of kinematic nonlinearity and include thermal effects. The two straight edges of the panel are assumed to be simply supported and two curved edges are either simply supported or clamped. The nonlinear prebuckling deformations and initial geometric imperfections of the panel are both taken into account. A boundary layer theory of shell buckling, which includes the effects of nonlinear prebuckling deformations, large deflection in the postbuckling range, and initial geometric imperfections of the shell, is extended to the case of functionally graded cylindrical panels. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling behavior of simply supported, pressure-loaded, perfect and imperfect, functionally graded cylindrical panels with two constituent materials under different sets of thermal environments. The influences played by temperature rise, volume fraction distributions, transverse shear deformation, panel geometric parameters, as well as initial geometric imperfections, are studied.     

In-Plane Nonlinear Buckling Analysis of Deep Circular Arches Incorporating Transverse Stresses

Large discrepancies exist among current classical theories for the in-plane buckling of arches that are subjected to a constant-directed radial load uniformly distributed around the arch axis. Discrepancies also exist between the classical solutions and nonlinear finite-element results. A new theory is developed in this paper for the nonlinear analysis of circular arches in which the nonlinear strain-displacement relationship is based on finite displacement theory. In the resulting variational equilibrium equation, the energy terms due to both nonlinear shear and transverse stresses are included. This paper also derives a set of linearized equations for the elastic in-plane buckling of arches, and presents a detailed analysis of the buckling of deep circular arches under constant-directed uniform radial loading including the effects of shear and transverse stresses, and of the prebuckling deformations. The solutions of the new theory agree very well with nonlinear finite-element results. Various assumptions often used by other researchers, in particular the assumption of inextensibility of the arch axis, are examined. The discrepancies among the current theories are clarified in the paper.     

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.     

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.     

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.     

Stability of Anisotropic Laminated Rings and Long Cylinders Subjected to External Hydrostatic Pressure

The problem of buckling of rings under external pressure has attracted interest since the late 1950s; however, the formulations developed, to date, to obtain the critical pressure are limited to special cases of orthotropic laminated construction. In this work, analytical and numerical treatments are carried out to provide results on the buckling of thin and moderately thick anisotropic rings and long cylinders. A generalized closed-form analytical formula for the buckling of thin anisotropic laminated rings is developed. Standard energy-based formulation and classical lamination theory are used to obtain the equilibrium equations assuming an intermediate class of deformation. The constitutive equations are statically condensed, in terms of the ring’s boundary conditions, to produce the effective axial, coupling, and flexural rigidities. In addition, a three-dimensional (3D) tube finite-element model is developed for nonlinear analysis of anisotropic laminated composite rings or long cylinders. The element accounts for prebuckling ring twist and first-order shear deformations. Fourier series expansions are used to express the in-plane and out-of-plane components of deformation and geometry at the three nodes of the cylindrical element. Isoparametric quadratic shape functions are used to interpolate the displacement field in?between. Comparisons of the analytical and numerical results show excellent agreement for thin rings. Parametric studies are also conducted to address the effects of lamination, shell thickness, and initial out-of-roundness imperfection on the external buckling pressure.     

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.     

3D Finite-Element Simulation of a Square Silo with Flexible Walls

Most of the studies that have been published to evaluate stresses on silo walls during filling or discharge stages are based on a rigid wall assumption. In a 2D approach, the wall flexibility can be approximately modeled by using a corrective factor applied to the whole pressure distribution. It has been shown that such models are in agreement with experimental measurements on circular silos. But in square or rectangular silos, the variation of the stiffness of the wall due to vertical or horizontal stiffeners produces nonuniform wall deformations that are not taken into account in the previous axisymmetric 2D model. The aim of this paper is to present a full 3D modeling of the filling and discharge stages using a nonlinear finite-element method. The bulk material behavior is based on an elastoplastic law. Contact elements using a Mohr-Coulomb criterion simulate the interaction between the wall and the bulk material, and the flexibility of the silo structure is modeled with beam and shell elements. A detailed analysis of the numerical results computed for a square silo filled with wheat and discharged through a central outlet is presented and discussed.     

Plastic Buckling of Simply Supported, Polygonal Mindlin Plates

This paper is concerned with the plastic buckling of Mindlin plates of polygonal plan shape and whose straight edges are simply supported. The plates are subjected to a uniform in-plane compressive stress. Two well-known competing theories of plasticity are considered here: the incremental theory of plasticity (with the Prandtl–Reuss constitutive relations) and the deformation theory of plasticity (with the Hencky constitutive relation). Based on an analogy approach, the plastic buckling stresses of such Mindlin plates are expressed in terms of their corresponding elastic buckling stresses of Kirchhoff (classical thin) plates, albeit in a transcendental form. Using this buckling stress relationship and the readily available elastic buckling solutions, one may deduce the plastic buckling stresses of the corresponding Mindlin plates. Tabulated herein are some buckling stress factors for various polygonal shaped plates with material properties defined by the Ramberg–Osgood relation.     

Axisymmetric Wrinkling of Cylinders with Finite Strain

A simple analytical solution for the bifurcation buckling of a cylinder under axial loading is provided including finite-strain effects. Thus, the small strain theory result of Batterman is generalized. In addition to the thin shell theory solution (excluding shear deformations), a solution including shear deformation effects is also given. All solutions can be evaluated for either the flow or deformation theory of plasticity. The finite-strain constitutive theory used is one in which small strain type relationships apply between the Jauman rate of the Kirchhoff stress tensor and the deformation rate tensor. The analytical results are compared to finite-element analyses to test the validity of the assumptions made. The solutions are explicit. Starting with a point on the stress-strain curve, one calculates explicitly the diameter-to-thickness ratio D∕t for a cylinder that will buckle at that level of stress and strain (repeating this as necessary to generate a plot of wrinkling strains as a function of D∕t). Unless the tangent modulus at bifurcation is large compared to the stress, the results clearly indicate that finite strains have an important stabilizing effect, leading to higher bifurcation strains.     

Stability of Composite and Sandwich Struts by Mixed Formulation

A unified mixed, higher-order analytical formulation is presented to evaluate the buckling of laminated composite struts. The formulation can also be used to evaluate the overall buckling and wrinkling loads of a general multilayer, multicore sandwich strut having any arbitrary sequence of stiff layers and cores. The usual assumptions of thin stiff layers and antiplane core are advantageously eliminated. Displacements as well as transverse stress continuities are enforced in the formulation by incorporating them as the degrees of freedom, thus avoiding separate calculations of the modal transverse stresses. Two sets of mixed models are proposed, based on individual layer and equivalent single layer theories, by selectively incorporating nonlinear components of Green’s strain tensor. Limitations of the equivalent single-layer theories and typical simplifying assumptions are highlighted. A parametric investigation is presented, concerning the influence of the material and geometric properties on the buckling behavior of a sandwich strut. A few recommendations are also made for the stability analysis of laminated composite struts.     

Buckling Experiments on Cone-Cylinder Intersections under Internal Pressure

Cone-cylinder intersections are used in many shell structures including tanks, silos, pressure vessels, and piping, and internal pressurization is often an important loading condition. For the intersection of the large end of a cone and a cylinder, internal pressurization causes large circumferential compressive stresses in the intersection. These stresses can lead to failure of the intersection by either axisymmetric collapse or nonsymmetric buckling. This paper presents the first carefully conducted experimental study on these intersections. Following a brief summary of the experimental setup, the experimental results are presented together with finite-element predictions. Both experimental and numerical results show that the postbuckling behavior of internally pressurized cone-cylinder intersections is stable, but the postbuckling growth of deformations and associated strains can cause rupture failure at welds. The experimental buckling load can be closely approximated by the nonlinear bifurcation load of the perfect geometry, indicating that the effect of initial imperfections is very limited.     

Plastic-Buckling of Rectangular Plates under Combined Uniaxial and Shear Stresses

This paper is concerned with the plastic-buckling of rectangular plates under uniaxial compressive and shear stresses. In the prediction of the plastic-buckling stresses, we have adopted the incremental theory of plasticity for capturing the inelastic behavior, the Mindlin plate theory for the effect of transverse shear deformation, the Ramberg-Osgood stress–strain relation for the plate material, and the Ritz method for the bifurcation buckling analysis. The interaction curves of the plastic uniaxial buckling stress and the plastic shear buckling stress for thin and thick rectangular plates are presented for various aspect ratios. The effect of transverse shear deformation is examined by comparing the interaction curves obtained based on the Mindlin plate theory and the classical thin plate theory.     

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.     

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.     

Nonlinear Vibration of Composite Shell Subjected to Resonant Excitations

Analysis of the vibration of a shallow, simply supported, nonsymmetric unbalanced cross-ply laminated, circular cylindrical composite shell is presented. The subject is particularly relevant, considering the widespread use of cylindrical shell structures in engineering applications. This research applies the discretized Lagrangian∕method of multiple scales solution technique. The Donnell shallow shell strain-displacement relations and the single-mode displacement field from the linear eigenvalue problem are applied. The system Lagrangian is developed and integrated over the spatial domain and then substituted into Lagrange's equation. The resulting equation of motion is a second-order temporally nonlinear ordinary differential equation in the form of the Duffing oscillator. The natural frequency, the coefficient of the cubic nonlinearity, and the strength of the nonlinearity are investigated. The method of multiple scales is applied to the nonlinear equation of motion in order to analyze the frequency response. Primary resonance, subharmonic resonance, and superharmonic resonance are analyzed.     

Tension and Compression Stability and Second-Order Analyses of Three-Dimensional Multicolumn Systems: Effects of Shear Deformations

The stability and second-order analyses of three-dimensional (3D) multicolumn systems including the effects of shear deformations along the span of each column are presented in a condensed manner. This formulation is an extension to an algorithm presented recently by the writer in 2002 and 2003 by which the critical load of each column, the total critical load, and the second-order response of a 3D multicolumn system with semirigid connections can be determined directly. The proposed solution includes not only the combined effects of flexural deformations and shear distortions along the columns in their two principal transverse axes, but also the effect of the shear forces along each member induced by the applied end axial force as the columns deform and deflect (as suggested by Haringx in 1947 and explained by Timoshenko and Gere in 1961) in their two principal transverse axes. The extended characteristic transcendental equations (corresponding to multicolumn systems with sidesway and twist uninhibited, partially inhibited, and totally inhibited) that are derived and discussed in this publication find great applications in the stability and second-order analyses of 3D multicolumn systems made of materials with relatively low shear stiffness such as orthotropic composite materials (fiber reinforced plastic) and multilayer elastomeric bearings used for seismic isolation of buildings. The phenomenon of buckling under axial tension in members with relatively low shear stiffness (observed by Kelly in 2003 in multilayer elastomeric bearings, and recently discussed by the writer in 2005) is captured by the proposed method. Tension buckling must not be ignored in the stability analysis of multicolumn systems made of columns in which the shear stiffness GAs is of the same order of magnitude as π2EI/h2.     

Elastic Buckling of Multilayered Anisotropic Conical Shells

On the basis of 3D elasticity, asymptotic solutions for buckling analysis of multilayered anisotropic conical shells under axial compression are presented. By means of proper nondimensionalization, asymptotic expansion, and successive integration, the classical shell theory is derived as a first-order approximation to the 3D theory. Because the governing equations for various orders consist of partial differential equations with variable coefficients, the use of analytical techniques is restricted. The method of differential quadrature is adopted in the present study. The modifications of the buckling loads and associated buckling modes can be determined in a consistent and hierarchic manner by considering the solvability and normalization conditions for various orders. The critical loads of cross-ply conical shells with simply supported–simply supported boundary conditions are studied to demonstrate the performance of the present asymptotic theory.