Nonlinear Kinematic Relations Research Articles

Thermoelastic stability analysis of imperfect functionally graded cylindrical shells

Elastic buckling analysis of imperfect FGM cylindrical shells under axial compression in thermal environments is carried out, using two different models for geometrical imperfections. The material properties of the functionally graded shell are assumed to vary continuously through the thickness of the shell according to a power law distribution of the volume fraction of the constituent materials, also temperature dependency of the material properties is considered. Derivation of equations is based on classical shell theory using the Sanders nonlinear kinematic relations. The stability and compatibility equations for the imperfect FGM cylindrical shell are obtained, and the buckling analysis of shell is carried out using Galerkin’s method. The novelty of the present work is to obtain closed form solutions for critical buckling loads of the imperfect FGM cylindrical shells, which may be easily used in engineering design applications. The effects of shell geometry, volume fraction exponent, magnitude of initial imperfections, and environment temperature on the buckling load are investigated. The results reveal that initial geometrical imperfections and temperature dependency of the material properties play major roles in dictating the bifurcation point of the functionally graded cylindrical shells under the action of axial compressive loads. Also results show that for a particular value of environment temperature, critical buckling load is almost independent of volume fraction exponent.

Linear ensemble-coding in midbrain superior colliculus specifies the saccade kinematics

Recently, we proposed an ensemble-coding scheme of the midbrain superior colliculus (SC) in which, during a saccade, each spike emitted by each recruited SC neuron contributes a fixed minivector to the gaze-control motor output. The size and direction of this ‘spike vector’ depend exclusively on a cell’s location within the SC motor map (Goossens and Van Opstal, in J Neurophysiol 95: 2326–2341, 2006). According to this simple scheme, the planned saccade trajectory results from instantaneous linear summation of all spike vectors across the motor map. In our simulations with this model, the brainstem saccade generator was simplified by a linear feedback system, rendering the total model (which has only three free parameters) essentially linear. Interestingly, when this scheme was applied to actually recorded spike trains from 139 saccade-related SC neurons, measured during thousands of eye movements to single visual targets, straight saccades resulted with the correct velocity profiles and nonlinear kinematic relations (‘main sequence properties– and ‘component stretching’) Hence, we concluded that the kinematic nonlinearity of saccades resides in the spatial-temporal distribution of SC activity, rather than in the brainstem burst generator. The latter is generally assumed in models of the saccadic system. Here we analyze how this behaviour might emerge from this simple scheme. In addition, we will show new experimental evidence in support of the proposed mechanism.

Thermal Buckling of Simply Supported Piezoelectric FGM Cylindrical Shells

A thermal buckling analysis is presented for functionally graded cylindrical shells that are integrated with surface-bonded piezoelectric actuators and are subjected to the combined action of thermal load and constant applied actuator voltage. The material properties are assumed to vary as a power form of the thickness coordinate. Derivation of the equations is based on the higher-order shear deformation shell theory using the Sanders nonlinear kinematic relations. Results for the buckling temperatures are obtained in the closed form solution. The effects of the applied actuator voltage, shell geometry, and volume fraction exponent of functionally graded material on the buckling temperature are investigated. The results for simpler states are validated with known data in the literature.

Geometrically nonlinear analysis of elastic membranes with embedded rigid inclusions

In this paper the deformation of membranes containing rigid inclusions is analyzed. These rigid inclusions can significantly change the entire stress distribution in the membrane and therefore create major difficulties for the design. The initially flat membrane, which may be prestretched by boundary in-plane tractions or displacements, is subjected to externally applied loads and to the weight of the rigid inclusions. The composite system is examined in cases where its deformation reaches a state for which the undeformed and deformed shapes are substantially different. In such cases large deflections of membranes are considered, which result from nonlinear kinematic relations. The three coupled nonlinear equations in terms of the displacements governing the response of the membrane are solved using the analog equation method, which reduces the problem to the solution of three uncoupled Poisson's equations with fictitious domain source densities. The problem is strongly nonlinear [Katsikadelis JT, Nerantzaki MS. The boundary element method for nonlinear problems. Eng Anal Boundary Elements 1999;23:365–73]. In addition to the geometrical nonlinearity, the problem is itself nonlinear, since the membrane's reactions on the boundary of the rigid inclusions are not a priori known as they depend on the produced deflection surface. Iterative schemes are developed for calculation of deformed membrane's configuration, which converge to the final equilibrium state of the membrane with the given external applied loads. Several example problems are presented, which illustrate the method and demonstrate its accuracy and efficiency. The method employed for the solution is boundary only with all the advantages of the pure BEM.

Thermal Buckling of Imperfect Functionally Graded Cylindrical Shells Based on the Wan–Donnell Model

ABSTRACT A thermal buckling analysis of an imperfect functionally graded cylindrical shell is considered using the Wan–Donnell model for initial geometrical imperfections. Derivation of the equations is based on the first-order classical shell theory using the Sanders nonlinear kinematic relations. Results for the buckling loads are obtained in the closed form. The effects of shell geometry and volume fraction exponent of functionally graded material on the buckling load are investigated. The results are validated with known data in the literature.

The relations of deformation theory in the quadratic approximation and the problems of constructing improved versions of the geometrically non-linear theory of laminated structures

Two versions of the approximate relations of the deformation theory of continuous media, known as the complete version (see, for example [1]) and the incomplete version (see, for example [2, 3]) of the quadratic approximation of the non-linear theory are analysed. It is shown that the relations of the complete version, which define the elongation deformation, and the relations of the incomplete versions, which define the shear deformations, are incorrect, since, when solving specific problems, they lead to the occurrence of false bifurcation points. For small elongation deformation and medium shear deformation a non-contradictory version of the kinematic relations is constructed in the quadratic approximation, representing a combination of the relations of the complete and incomplete versions. The simplest examples of its application, connected with the reduction of the two-dimensional non-linear problem of the deformation of a strip in the form of a rod to homogenous equations and their subsequent use to detect possible forms of loss of stability for characteristic forms of loading them are considered. Essentially new results are obtained connected with the investigation of forms of loss of stability of a rod under uniform transverse compression and pure shear. In the first case the behaviour of the load turns out to be important: if it remains normal to the deformation axis of the rod, bifurcation is only possible with respect to the shear form, if it retains its direction, and then, in addition to bifurcation with respect to the shear form, a bending form of loss of stability is possible, which is identical in form with the Euler form, for which there are no shears. In the second case, i.e. when there is a load which causes pure shear of the rod, to investigate its bifurcation values, it is necessary to describe the shear deformation by non-linear kinematic relations in the complete quadratic version, whereas when there are no subcritical shear stresses one can use the simplified relations. An example of the investigation of the forms of loss of stability of a circular ring when acted upon by a uniform external pressure having zero variability in the circumferential direction is also considered.

On the non-linear high-order theory of unidirectional sandwich panels with a transversely flexible core

The paper presents a general geometrically non-linear high-order theory of sandwich panels that takes into account the high-order geometrical non-linearities in the core as well as in the face sheets and is based on a variational approach. The formulation, which yields a set of rather complicated governing equations, has been simplified in two different approaches and has been compared with FEA results for verification. The first formulation uses the kinematic relations of large displacements with moderate rotations for the face sheets, non-linear kinematic relations for the core and it assumes that the distribution of the vertical normal stresses through the depth of the core are linear. The second approach uses the general formulation to the non-linear high-order theory of sandwich panels (HSAPT) that considers geometrical non-linearities in the face sheets and only linear high-order effects in the core. The numerical results of the two formulations are presented for a three point bending loading scheme, which is associated with a limit point behavior. The results of the two formulations are compared in terms of displacements, bending moments and shear stresses and transverse (vertical) normal stresses at the face–core interfaces on one hand, and load versus these structural quantities on the other hand. The results have compared well with FEA results obtained using the commercial codes ADINA and ANSYS.

Nonlinear (buckling) effects in RC beams strengthened with composite materials subjected to compression

The nonlinear behavior of reinforced concrete beams strengthened with externally bonded fiber reinforced composite materials in cases where the bonded laminate is subjected to compression is investigated. An analytical nonlinear model is derived using the concepts of the high order theory along with the large deflections and moderate rotations nonlinear kinematic relations, equilibrium conditions, and compatibility requirements. The governing equations take the form of a 14th order set of nonlinear differential equations with variable coefficients and are solved numerically via the nonlinear multiple points shooting method. The numerical investigation of a beam strengthened with externally bonded composite strips but subjected to loads that yield compressive stresses in the bonded laminate is presented. The influence of localized damage accumulated in the adhesive layer on the nonlinear response of the strengthened beam is also investigated. Results in terms of deflections, stresses, and stress resultants reveal the buckling and wrinkling phenomenon and their influence on the localized and overall behavior of the structure. They also throw light on the development of stresses and stress concentrations in critical locations along the beam, and provide a quantitative description of its pre- and post-buckling behavior.

Ponding on floating membranes

Membranes subjected to ponding loads and floating on a liquid are analyzed. The initially flat membrane, which may be prestressed by edge in-plane tractions or displacements, is subjected to the weight of a liquid (e.g. rain water) filling the space created by the deflection of the membrane. Large deflections of membranes are considered, which result from nonlinear kinematic relations. The three coupled nonlinear equations in terms of the displacements governing the response of the membrane are solved using the analog equation method, which reduces the problem to the solution of three uncoupled Poisson's equations with fictitious sources. The problem is strongly nonlinear. In addition to the geometrical nonlinearity, the ponding problem is itself nonlinear, because the ponding load and the liquid reaction are not a priori known as they depend on the produced deflection surface. Iterative schemes are developed which converge to the equilibrium state of the membrane. Example problems are presented, which illustrate the method and demonstrate its efficiency. The method has all the advantages of the pure BEM.

The ponding problem on elastic membranes: an analog equation solution

In this paper, the nonlinear response of elastic membranes with arbitrary shape under partial and full ponding loads has been analyzed. Large deflections are considered, which result from nonlinear kinematic relations. The problem is formulated in terms of the displacements components and the three coupled nonlinear governing equations are solved using the analog equation method (AEM). The membrane may be prestressed either by prescribed boundary displacements or tractions. Using the concept of the analog equation the three coupled nonlinear equations are replaced by three uncoupled Poisson's equations with fictitious sources under the same boundary conditions. Subsequently, the fictitious sources are established using a procedure based on the BEM and the displacement components as well as the stress resultants at any point of the membrane are evaluated from their integral representations. In addition to the geometrical nonlinearity, the ponding problem is itself nonlinear, because the ponding load depends on the deflection surface that it produces. Iterative schemes are developed which converge to the equilibrium state of the membrane under the ponding loads. Several membranes are analyzed which illustrate the method and demonstrate its efficiency and accuracy. The method has all the advantages of the pure BEM, since the discretization and integration is limited only to the boundary.

THERMOELASTIC BUCKLING OF THIN SPHERICAL SHELLS

Thermoelastic stability of thin perfect spherical shells based on deep and shallow shell theories is presented. To derive the equilibrium and stability equations according to deep shell theory, Sanders's nonlinear kinematic relations are substituted into the total potential energy function of the shell and the results are extremized by the Euler equations in the calculus of variation. The same equations are also derived based on quasi-shallow shell theory. An improvement is obtained for equilibrium and stability equations related to the deep shell theory in comparison with the same equations related to shallow shell theory. Approximate one-term solutions that satisfy the boundary conditions are assumed for the displacement components. The Galerkin-Bubnov method is used to minimize the errors due to this approximation. The eigenvalue solution of the stability equations is obtained using computer programs. For several thermal loads it is found that the deep shell theory results are slightly more stable as compared to the shallow shell theory results under the same thermal loads. The results are compared with the Algor finite element program and other known data in the literature.

Effect of thickness on the large elastic deformation behavior of laminated shells

A serendipity type surface-parallel cubic (24-node) isoparametric element for analysis of thick deep imperfect laminated shells is developed. The element is capable of accurately modeling the curved geometry of a laminated shell by taking advantage of general tensorial formulation and using the surface-parallel curvilinear coordinates of non-Euclidean geometry. The present nonlinear finite element solution methodology is based on the hypothesis of layerwise linear displacement distribution through thickness (LLDT) and the total Lagrangian formulation, which accounts for fully nonlinear kinematic relations so that stable equilibrium paths in the advanced nonlinear regime can be accurately computed. An important computational feature is the successful implementation of the BFGS (Broyden-Fletcher-Goldfarb-Shanno) iterative scheme, used to solve the resulting nonlinear equations. First, the large strain behavior of a two-dimensional rubber sheet, made of Mooney-Rivlin type hyperelastic material, under tension is evaluated for the purpose of comparison with available experimental results. Then, thin/shallow clamped cylindrical cross-ply [ 0 ° 90 ° ] panels, subjected to radial pressure loading, are investigated to test the convergence of the present element. A new concept of relative(-to-linear) nonlinear membrane-to-shear factor, defined to be the ratio of normalized deflections computed using the nonlinear and linear analyses, is introduced to determine the relative roles of interlaminar shear/normal deformation and surface parallel membrane effects in thin to thick laminated perfect shell regimes, subjected to radial pressure loading.

Imperfection Sensitivity of Moderately Thick Laminated Cylindrical Shells

The instability problem of imperfect, laminated, moderately thick, circular cylindrical shells under the action of individual and combined application of uniform axial compression and external lateral pressure is investigated. The analysis is based on nonlinear kinematic relations, where the effect of transverse shear deformation is taken into account. The buckling is assumed to be elastic and the geometry to have initial geometric imperfections. A solution methodology has been developed and employed in generating results. The imperfection sensitivity is investigated. The results obtained indicate that geometric imperfections have small effect on the limit point load for moderately thick cylindrical shells. In addition, limited parametric studies have been made in order to assess the effect of length-to-radius ratio, radius-to-thickness ratio, and stacking sequence on the critical loads.

Imperfection sensitivity of shear deformable moderately thick laminated cylindrical shells

The problem of instability of imperfect, laminated, circular cylindrical shells under the action of uniform axial compression is investigated. The analysis is based on nonlinear kinematic relations where the effect of transverse shear deformation is taken into account. The buckling is assumed to be elastic and the geometry to have initial geometric imperfections. The kinematic relations, governing equations and the related boundary conditions for the nonlinear analysis are derived and presented. A solution methodology has been developed and employed in generating results. The imperfection sensitivity is investigated. The results obtained indicate that geometric imperfections have little effect on the limit point load for moderately thick cylindrical shells.

Geometrically nonlinear analysis of cylindrical shells using surface-parallel quadratic elements

A moderately thick cylindrical shell isoparametric element that is capable of accurately modeling cylindrically curved geometry, while also incorporating appropriate through-thickness kinematic relations is developed. The analysis accounts for fully nonlinear kinematic relations so that stable equilibrium paths in the advanced nonlinear regime can be accurately predicted. The present nonlinear finite element solution methodology is based on the hypothesis of linear displacement distribution through thickness (LDT) and the total Lagrangian formulation. A curvilinear side 16-node element with eight nodes on each of the top and bottom surfaces of a cylindrical shell has been implemented to model the transverse shear/normal deformation behavior represented by the LDT. The BFGS iterative scheme is used to solve the resulting nonlinear equations. A thin-shallow clamped cylindrical panel is investigated to test the convergence of the present element, and also to compare the special case of the present solution based on the KNSA (von Karman strain approximation) with those computed using the available faceted elements, discrete Kirchhoff constraint theory (DKT) and classical shallow shell finite elements, spanning the entire computed equilibrium path.

Full and von Karman geometrically nonlinear analyses of laminated cylindrical panels

A total Lagrangian-type nonlinear analysis for prediction of large deformation behavior of thick laminated composite cylindrical shells and panels is presented. The analysis, based on the hypothesis of layerwise linear displacement distribution through thickness, accounts for fully nonlinear kinematic relations, in contrast to the commonly used von Karman nonlinear strain approximation, so that stable equilibrium paths in the advanced nonlinear regime can be accurately predicted. The resulting degenerated surface-parallel quadratic (16-node) layer element, with 8 nodes on each of the top and bottom surfaces of each layer, has been implemented in conjunction with full and reduced numerical integration schemes to efficiently model both thin and thick shell behavior. The modified Newton-Raphson iterative scheme with Aitken acceleration factors is used to obtain hitherto unavailable numerical results corresponding to fully nonlinear behavior of the analyzed panels. A two-layer [0/90] thin/shallow clamped cylindrical panel is investigated to assess the convergence rate for full and reduced integration schemes and to check the accuracy of the present degenerate cylindrical shell layer element. Accuracy of the von Karman nonlinear approximation, currently employed in many investigations on buckling/postbuckling behavior of thin shells, is assessed, in the case of laminated thin cylindrical panels, by comparing the numerical results obtained using this approximation with those due to fully nonlinear kinematic relations, especially in the advanced stable postbuckling regime.

Buckling of Moderately Thick Composite Cylindrical Shells under Destabilizing Loads

The instability of perfect, laminated, circular cylindrical shells with fixed ends under the action of uniform axial compression, lateral pressure, and torsion is investigated. The analysis is based on nonlinear kinematic relations where the effect of transverse shear deformation is taken into account. Three theories (higher order shear deformation theory, first-order shear deformation theory, and classical) are compared to determine their range of applicability in predicting critical conditions for moderately thick cylindrical shells under the action of various loads. Also, the effects of stacking sequence, radius-to-thickness ratio, and length-to-radius ratio are assessed. The results obtained indicate that the first-order shear deformation theory with a proper shear correction factor is a good candidate for the analysis of moderately thick laminated cylindrical shells.

Nonlinear analysis of laminated non-circular cylindrical shells

The nonlinear analysis of laminated initially imperfect non-circular cylindrical shells is presented. The analytical model is based on Donnell's nonlinear kinematic relations. The equations are derived via the Hu-Washizu mixed formulation, and are expressed in terms of the transverse displacement and the Airy stress function. The curvature of the non-circular cross-section is expanded into a Fourier series, allowing for representation of arbitrary closed cross-sections. The solution procedure is based on expansion of the variables into truncated trigonometric series in the circumferential direction and a finite difference scheme in the longitudinal one. Errors introduced by the truncated series are minimized by the Galerkin procedure and the equations are linearized by the Newton-Raphson method. Solutions beyond the limit point are obtained by Riks' constant arc-length algorithm. Results of both isotropic and laminated, axially loaded oval and elliptic shells are presented. The non-circular configurations are found to be less imperfection sensitive than the circular ones, and for largely eccentric cross-sections the shells are insensitive to initial imperfections.

Thermoelastoviscoplastic Buckling Behavior of Cylindrical Shells

The thermoelastoviscoplastic buckling behavior of cylindrical shells under axial compression is investigated. The analysis is based on nonlinear kinematic relations and nonlinear rate-dependent unified constitutive equations. Bodner-Partom's model is employed to represent the thermoelastoviscoplastic material behavior. The material model does not separate the inelastic deformation into time-dependent (creep) and time-independent (plastic) deformations. It can cover elastic and inelastic material behavior, and temperature effects simultaneously. A finite-element approach which with a two-phase solution scheme for the unified constitutive equations is employed to predict the inelastic buckling behavior of the structure. Numerical examples are given to demonstrate the change of critical load, deformation mode, and the load-carrying capability in the postbuckling stage. The effects of several parameters, which include temperature, small initial imperfections, and the thickness of the shell are assessed. The creep buckling is also studied as an example of the time-dependent deformation.

Thermo-Elastoviscoplastic Buckling Behavior of Plates

The thermo-elastoviscoplastic buckling behavior of plates is investigated. The analysis is based on nonlinear kinematic relations and nonlinear rate-dependent unified constitutive equations which include both Bodner-Partom’s and Walker’s material models. A finite element approach is employed to predict the inelastic buckling behavior. Numerical examples are given to demonstrate the effects of several parameters, which include temperature, small initial imperfections, and the thickness of the plate. Comparisons of buckling responses for the two models, Bodner-Partom’s and Walker’s, are also presented.