Large Rotation Theory Research Articles

Large deflections of magneto-electro-elastic cylindrical shells reinforced with functionally graded carbon nanotubes

This article deals with large deflection analysis of functionally graded carbon nanotube-reinforced magneto-electro-elastic (MEE) cylindrical shells using fully geometrically nonlinear finite element methods. The first-order shear deformation theory and strain–displacement relations of large rotations are employed to obtain the governing equations. The nonlinear dynamic models are derived from Hamilton’s principle. Four distribution patterns of carbon nanotubes embedded in the piezoelectric matrix are considered. The credibility of the proposed fully geometrically nonlinear model is firstly verified by comparing the displacements of MEE laminated plates and shells in the published literature. Further, the influence of various parameters on the static and dynamic performance of the structure has been studied. The study believed that the large rotation theory can obtain more accurate results when the structures undergo large deformation.

Piezoelectric Co-Rotational 3-Node Shell Element

Geometrically nonlinear FE-analysis of active piezoelectric thin-walled shell structures, based on a Co-Rotational (CR) formulation and the Cell-Smoothed Discrete-Shear-Gap 3-node shell element (CS-DSG3) is presented in this study. The electric potential across the thickness of piezolayers is approximated by a linear function. The basic assumption of the applied Co-Rotational (CR) FEM formulation is that translations and rotations are finite, whereas the strains are small. Large rotation theory is used to update the rotational degrees of freedom. The proposed formulation is tested by means of the originally developed program. A set of piezoelectric coupled static linear and nonlinear cases are studied and the obtained results are compared with those available in the literature. A very good agreement of the results is noted, whereby the numerical effort could be reduced and numerical stability improved compared to other geometrically nonlinear FEM formulations.

Efficient Co-Rotational 3-Node Shell Element

Geometric nonlinear FE-analysis of thin-walled shell structures, using a co-rotational formulation and the Cell Smoothed Discrete Shear Gap triangular shell element (CS-DSG3) is presented in this study. The CS-DSG3 element formulation uses the Mindlin-Reissner kinematic hypothesis to include transverse shear effects. In order to avoid locking effects Discrete Shear Gap (DSG) method is applied. In addition, cell based smoothing technique is adopted in order to improve accuracy and stability of the element. For the purpose of comparison, the Discrete Kirchhoff-Constant strain-Triangle (DKT-CST) is also implemented and studied in the linear static analysis. In the framework of the co-rotational FE-analysis rotations and displacements are adopted as finite, while strains are infinitesimal. Large rotation theory has been utilized to take into account the non-vectorial characteristic of rotations. Several static linear and nonlinear benchmark examples are presented and compared with commercial FE software Abaqus and analytical results. The presented approach, using CS-DSG3 element in co-rotational nonlinear analysis, illustrates very good results compared to reference solution and Abaqus results. The numerical effort can be reduced compared to Lagrange formulation with a similar accuracy for the studied cases. The formulation (including CS-DSG3 shell element) has been implemented into a test program.

Active shape and vibration control for piezoelectric bonded composite structures using various geometric nonlinearities

This paper deals with simulations of the static and dynamic response, including shape and vibration control, for piezoelectric bonded smart structures using various geometrically nonlinear shell theories based on the first-order shear deformation (FOSD) hypothesis. The nonlinear theories include refined von Kármán nonlinear shell theory (RVK5), moderate rotation shell theory (MRT5), fully geometrically nonlinear shell theory with moderate rotations (LRT5), and fully geometrically nonlinear shell theory with large rotations (LRT56). The structures simulated are mainly comprised of cross-ply or angle-ply laminated thin-walled master structures bonded with isotropic piezoelectric layers or patches that are considered as actuators. Nonlinear finite element (FE) models are constructed for shape and vibration control of structures undergoing large displacements and rotations. Various plates and shells are validated by comparison with those reported in the literature, and then simulated for the current shape and vibration control. A widely used negative proportional velocity feedback control is adopted for the active vibration control. From the simulations, it can be concluded that large rotation theory should be considered for structures undergoing deformations beyond the range of moderate rotations. Additionally, the results show that by applying an appropriate voltage, a desired shape can be achieved, as well as the vibration can be significantly suppressed.

Static and dynamic FE analysis of piezoelectric integrated thin-walled composite structures with large rotations

A geometrically nonlinear finite element (FE) model based on large rotation shell theory is developed for static and dynamic analysis of piezoelectric integrated thin-walled structures with cross-ply or angle-ply laminates. The implemented large rotation theory has six kinematic parameters expressed by five nodal degrees of freedom (DOFs) based on first-order shear deformation (FOSD) hypothesis. An eight-node shell element with five mechanical DOFs and one electrical DOF is employed. Due to the assumption of small strains and weak electric potential, linear constitutive equations and constant electric field through the thickness are considered. The large rotation piezoelectric coupled FE model is validated by one static benchmark problem and afterwards applied to the static and dynamic analysis of piezoelectric integrated smart plates and shells composed of cross-ply or angle-ply laminates.

Large rotation theory for static analysis of composite and piezoelectric laminated thin-walled structures

A fully geometrically nonlinear finite element (FE) model is developed using large rotation shell theory for static analysis of composite and piezoelectric laminated thin-walled structures. The proposed large rotation theory is based on the first-order shear deformation (FOSD) hypothesis. It has six independent kinematic parameters which are expressed by five mechanical nodal degrees of freedom (DOFs). Linear electro-mechanically coupled constitutive equations with a constant electric field distribution through the thickness of each smart material layer are considered. Eight-node quadrilateral plate/shell elements with five mechanical DOFs per node and one electrical DOF per smart material layer are employed in the FE modeling. The present large rotation FE model is implemented into static analysis of both composite and piezoelectric laminated plates and shells. The equilibrium equation is solved by Newton–Raphson algorithm with system matrices updated in every iteration. The results are compared with those presented in the literature and others calculated by various simplified nonlinear shell theories. They indicate that large rotation theory has to be considered for the calculation of displacements and sensor output voltages of smart structures undergoing large deflections, since other simplified nonlinear theories fail to predict the static response precisely in many cases.

Large rotation FE transient analysis of piezolaminated thin-walled smart structures

A geometrically nonlinear large rotation shell theory is proposed for dynamic finite element (FE) analysis of piezoelectric integrated thin-walled smart structures. The large rotation theory, which has six independent kinematic parameters but expressed by five nodal degrees of freedom (DOFs), is based on first-order shear deformation (FOSD) hypothesis. The two-dimensional (2D) FE model is constructed using eight-node quadrilateral shell elements with five mechanical DOFs per node and one electrical DOF per piezoelectric material layer with linear constitutive equations. The linear and nonlinear dynamic responses are determined by the central difference algorithm (CDA) and the Newmark method. The results are compared with those obtained by simplified nonlinear theories, as well as those reported in the literature. It is shown that the present large rotation theory yields considerable improvement if the structures undergo large displacements and rotations.

Nonlinear Dynamic Finite Element Analysis of Composite Cylindrical Shells Considering Large Rotations

A new nonlinear finite element formulation is developed for the nonlinear dynamics of shell structures. Using a vectorial approach, the kinematics that can be used to describe very large displacements and large rotations are derived. Green strain measures and second Piola Kirchhoff stress measures are used in the determination of the total potential. A displacement-based eight-noded cylindrical shell element with a total of 36 degrees of freedom is developed. Several numerical examples are dynamically analyzed to observe the characteristics of large displacements and large rotations. A convergence study was carried out to establish the numerical accuracy of the model. The most efficient and accurate model was used. HE nonlinear vibration analysis of shells has been the focus of research for the past few years. Thin shells subjected to dynamic loads could encounter deflections of the order of the thick- ness of the shell. Dynamic response of thin shells also could lead to the phenomena of dynamic snapping or dynamic buckling. Be- cause these kinds of responses cannot be determined accurately using small displacement and small rotation theories, large defor- mation and large rotation theories are required. The complex nature of these theories requires solving numerous simultaneous, nonlin- ear, differential equations. Even with simplifying assumptions, these equations remain complex and extensive. As background, a few relevant past efforts are reviewed. Earlier work regarding the nonlinear vibrations for isotropic shell structures can be attributed to Carr, 1 Yeh,2 Clough and Wilson,3 Belytschko and Marchertas,4 and Belytschko and Tsay.5 These researchers used flat plate elements. Saigal and Yang6 developed a curved shell element for isotropic shells. For composite shell structures, Simitses7 provided an elegant analytical solution of thin laminated shells subjected to sudden loads. He ignored through-the-thickness shear effects and used Donnell-type kinematic relations. Raouf and Palazotto8'9 developed a perturbation procedure to derive a set of asymptotically consistent nonlinear equations of motion for an arbi- trary laminated composite cylindrical shell in cylindrical bending. They also neglected transverse shear effects. Simo and Tarnow10 de- veloped an energy and momentum conservating algorithm for shell analysis to analyze shells undergoing large rigid-body motions. Palazotto and Dennis11 presented the simplified large rotation (SLR) theory to analyze shells experiencing moderately large ro- tations. The SLR theory determines the equilibrium path of or- thotropic shells using a total Lagrangian approach and includes a parabolic transverse shear stress distribution. This approach cap- tures the appropriate kinematics through displacement polynomi- als. Green strain displacement relations are used, and all of the final displacement functions are carried into the expression for the total potential without making any attempt at separating the rigid- body movement. Using this approach, the features classically pre- sented for, including through-the-thickness shear (for example), can be exploited.12 Smith and Palazotto13 discussed eight higher-order shear theories to the nonlinear finite element analysis of compos- ite shells. They also used the Green-Lagrange strains and second Piola Kirchhoff stresses. These theories were successful for large displacements and moderately large rotations. A reason for the fail- ure of these theories in the modeling of large rotation can be at- tributed to the approximations used in the kinematics. Gummadi and Palazotto14 modified the kinematics incorporated in the SLR theory and successfully carried out the solutions to problems ex- hibiting the characteristics of very large displacements and rota- tions. In their work, a vectorial approach was followed to derive the appropriate kinematics that can capture the characteristics of large displacements and large rotations. Using appropriate assumptions, the kinematics for the SLR theory can be obtained as a special case. The large rotation kinematics was introduced through the use of sine and cosine functions. A 36-degree-of-freedom composite shell element was developed based on this theory. Based on the SLR theory, Tsai and Palazotto15 formulated a nonlinear dynamic finite element analysis procedure for composite cylindrical shells. In their work, a linear mass matrix was incorpo- rated along with the nonlinear stiffness matrix that was originally developed in SLR theory. This theory again has the same drawbacks as the SLR theory in terms of realizing the large displacements and large rotations. In the present paper, a theory that can capture the large displacement and large rotations is discussed. The present work is an extension of the authors' earlier work on large rotation theory for static analysis. A new mass matrix is generated using Hamilton equations. The mass matrix thus generated is a nonlin- ear mass matrix. A fi-M method15 is used to solve the nonlinear algebraic equations resulting from the finite element technique. Se- lected numerical examples are solved to determine the validity of the developed theory.

Nonlinear finite element analysis of composite beams and arches using a large rotation theory

Today's aerospace industry has advanced to the point of using optimum design techniques in virtually all applications. In structural elements, orthotropic fiber composite materials have emerged as lighter, stronger and a more easily manufactured solution to the material application aspect of design. Composites have the distinct advantage of being designed and built to many different desired specifications by varying materials, amount of matrix/fiber and orientation. As with many high performance applications, the analysis techniques of fiber composites are more complicated than for the simpler counterparts. This research has been directed to capturing large cross-sectional rotation incorporating a geometrically nonlinear finite element composite arch model. The model was derived and simplified from a (two-dimensional) 2-D shell theory that has been demonstrated to be accurate for large displacements and only moderate rotations. The current effort uses a similar potential energy based finite element model with through-the-thickness shear. Large rotation kinematics are derived in a vector format which includes a tangent function in the in-plane displacement relationship. Previously published research (Creaghan and Palazotto, 1994; Palazotto and Dennis, 1992) used a small angle approximation to simplify stiffness expressions, limiting those theories to moderate cross-sectional rotation angles. This tangent function is modeled by a series representation of the angle and thereby preserving the existing degrees of freedom. The approach decomposes the Green strain components into convenient forms for inclusion in the potential energy function which is then extended to a nonlinear finite element solution method. The potential energy is simplified by substituting the new rotation function for the previous rotation angle. Riks and displacement control are used to show solutions to several nonlinear arch problems. Other published analytical and experimental results are compared with the current research. This work is a simple extension of a previously published large displacement/moderate rotation theory (Creaghan and Palazotto, 1994), but the results show significant improvement when large cross sectional bending angles are present.

A co-rotational, updated Lagrangian formulation for geometrically nonlinear finite element analysis of shell structures

A co-rotational, updated Lagrangian formulation for geometrically nonlinear analysis of shells is presented. In this finite element procedure, a standard updated Lagrangian formulation is employed to generate the tangent stiffness matrix, and a co-rotational theory is used for updating element strain, stress and internal force vectors during the Newton-Raphson iterations. Large rotation theory has been accommodated to take into account the nonvectorial characteristic of the rotational degrees-of-freedom. The present procedure is ideally suited for implementation in existing linear finite element programs and its effectiveness has been demonstrated by a number of numerical examples.

Elastic postbuckling analysis via finite element and perturbation techniques. Part 1: Formulation

AbstractThe main equations for the equilibrium, stability and critical state analysis of discrete elastic systems are presented following the works of Thompson, but in such a way that the original set of generalized coordinates and loads are preserved in the Total Potential Energy. This introduces differences in the resulting equations in bifurcation analysis but does not introduce any new feature regarding the physics of the problem. The new formulation is approximated by means of a standard finite element approach based on interpolation of displacements, in which the derivatives of the potential energy are approximated. The terms retained are those of moderately large rotation theory. The energy analysis is finally related to the more conventional finite element notation in terms of stiffness matrices, and it is shown how in such a way it can be included in present day codes.Part 2 of the paper deals with applications to the analysis of shells of revolution using a semi‐analytical approximation. Two cases are presented in detail: bifurcation in axisymmetric and in asymmetric modes, and the results show good correlation with those of other authors. The influence of load and geometric imperfections is evaluated.

Stability characteristics of laminated cylindrical panels under transverse loading

Static and dynamic approaches for the stability analysis of a laminated composite arch, a laminated composite cylindrical panel, and an elastic isotropic cylindrical panel subjected to transverse loading are discussed. The inspiration of this stability analysis during the snapping process is based on the phenomenon of energy transfer from potential energy to kinetic energy as the applied load reaches the critical value. Some important features in dynamic analysis characterizing the snapping motion, which are not observed in static analysis, are shown. Different snapping phenomena in dynamic analyses are presented in terms of variations in depth and effects of anisotropy. HE stability analysis of laminated panels under trans- verse loading merits a great deal of attention due to its occurrence in practical applications. Thin laminated panels under transverse loads could encounter deflections of the or- der of the shell thickness or greater. Laminated cylindrical panels subjected to transverse static loading could also result in the snapping through phenomenon, which characterizes an instantaneous dynamic instability. Some researchers applied static analysis for such instantaneous instability problems, whereas others argue that the dynamic analysis should be employed to cope with the complex transient phenomenon the snapping process. Nevertheless, both static and dynamic approaches dealing with the stability problem in the occur- rence of snapping and/or buckling have been studied in the past.1 The nature of the stability for elastic media was dis- cussed2 thoroughly in terms of the energy criterion. Static and dynamic buckling behaviors of elastic structures were investi- gated,3 in which definitions of dynamic buckling loads, esti- mates of the buckling loads, and assessments of the accuracy of these estimates were presented. The objective of this re- search is to investigate the fundamental concept and basic issues of instantaneous instability for geometrical nonlinear laminated panels when snapping occurs and to identify some important features in both static and dynamic analyses. Because of the high ratio of in-plane elastic modulus to transverse shear modulus in laminated composite structures, transverse shear deformation plays a much more important role in reducing the effective flexural stiffness of laminated plates and shells.4 Snapping stability response of these kinds cannot be predicted correctly by using the small or intermedi- ate displacement theory. In addition, the transverse shear deformation, which is usually neglected in the classical shell theory, cannot be ignored in the stability analysis of laminated composite shell structures subjected to transverse loading. As a result, the need to include large displacement, large rotation theory in studying the stability characteristics of thin lami- nated panels is of great importance. In the present study, the kinematics of large displacements and rotations for small

Mixed finite element formulation for frictionless contact problems

Simple mixed finite element models and a computational procedure are presented for the solution of frictionless contact problems. The analytical formulation is based on a form of Reissner's large-rotation theory with the effects of transverse shear deformation included. The contact conditions are incorporated into the formulation by using a perturbed Lagrangian approach with the fundamental unknowns consisting of the internal forces (stress-resultants), the generalized displacements, and the Lagrange multipliers associated with the contact conditions. The elemental arrays are obtained by using a modified form of the two-field, Hellinger-Reissner mixed variational principle. The internal forces and the Lagrange multipliers are allowed to be discontinuous at interelement boundaries. The Newton-Raphson iterative scheme is used for the solution of the nonlinear algebraic equations, and for the determination of the contact region and the contact pressures. Two numerical examples, axisymmetric deformations of a hemispherical shell and planar deformations of a circular ring, are presented. Both structures are pressed against a rigid plate. Detailed information about the response of both structures is presented. These examples demonstrate the high accuracy of the mixed models and the effectiveness of the computational procedure developed.