Nonlinear Analysis Of Shells Research Articles

Nonlinear Forced Vibration Analysis of FG-CNTRC Cylindrical Shells Under Thermal Loading Using a Numerical Strategy

Employing an efficient numerical strategy, the nonlinear forced vibration analysis of composite cylindrical shells reinforced with single-walled carbon nanotubes (CNTs) is carried out. It is assumed that the distribution of CNTs along the thickness direction of the shell is uniform or functionally graded and the temperature dependency of the material properties is accounted. The governing equations are presented based on the first-order shear deformation theory along with von-Karman nonlinear strain-displacement relations. The vectorized form of energy functional is derived and directly discretized using numerical differential and integral operators. By the use of variational differential quadrature (VDQ) method, discretized nonlinear governing equations are obtained. Then, the time periodic differential operators are applied to perform the discretization procedure in time domain. Finally, the pseudo-arc length continuation method is employed to solve the nonlinear governing equations and trace the frequency response curve of the nanocomposite cylindrical shell. A comparison study is first presented to verify the efficiency and validity of the proposed numerical method. Comprehensive numerical results are then given to investigate the effects of the involved factors on the nonlinear forced vibration characteristics of the structure. The results show that the changes of fundamental vibrational mode shape have considerable effects on the frequency response curves of composite cylindrical shells reinforced with CNTs.

Geometrically nonlinear analysis of shells by various dynamic relaxation methods

PurposeNumerical experiences reveal that the performances of the dynamic relaxation (DR) method are related to the structural types. This paper is devoted to compare the DR schemes for geometric nonlinear analysis of shells. To achieve this task, 12 famous approaches are briefly introduced. The differences among these schemes are between the estimation of the time step, the mass and the damping matrices. In this study, several benchmark structures are analyzed by using these 12 techniques. Based on the number of iterations and the analysis duration, their performances are graded. Numerical findings reveal the high efficiency of the kinetic DR (kdDR) approach and Underwood’s strategy.Design/methodology/approachUp to now, the performances of various DR algorithms for geometric nonlinear analysis of thin shells have not been investigated. In this paper, 12 famous DR methods have been used for solving these structures. It should be noted that the difference between these approaches is in the estimation of the fictitious parameters. The aforementioned techniques are used to solve several numerical samples. Then, the performances of all schemes are graded based on the number of iterations and the analysis duration.FindingsThe final ranking of each strategy will be obtained after studying all numerical examples. It is worth emphasizing that the number of iterations and that of convergence points of the arc length algorithms are dependent on the value of the initial arc length. In other words, a slight change in the magnitude of the arc length may lead to the wrong responses. Contrary to this behavior, the analyzer’s role in the dynamic relaxation techniques is considerably less than the arc length method. In the DR strategies when the answer approaches the limit points, the iteration number increases automatically. As a result, this algorithm can be used to analyze the structures with complex equilibrium paths.Research limitations/implicationsNumerical experiences reveal that the DR method performances are related to the structural types. This paper is devoted to compare the DR schemes for geometric nonlinear analysis of shells.Practical implicationsGeometric nonlinear analysis of shells is a sophisticated procedure. Consequently, extensive research studies have been conducted to analyze the shells efficiently. The most important characteristic of these structures is their high resistance against pressure. This study demonstrates the performances of various DR methods in solving shell structures.Originality/valueUp to now, the performances of various DR algorithms for geometric nonlinear analysis of thin shells are not investigated.

Semi-analytical postbuckling strength analysis of anisotropic shell structures

An investigation of the forms of shell buckling has been the subject of many experimental and theoretical studies. On the basis of analysing of the forms of equilibrium it is possible to determine the stability of a structure as a whole, especially if a statistical analysis is used. The numerical analysis of the shells considered is based on a semi-analytical treatment of displacement and stress field. This method is proven for static and dynamic nonlinear analysis of general shells of revolution and leads to important advantages in efficiency and accuracy compared with a common finite element analysis, especially in the case of geometrically imperfect shells. The method developed permits determination of stresses in a shell by means of an experimental deflection function. Failure criterion allows predicting the sites of fracture and maintenance of a shell upon loss of stability.

A novel numerical solution strategy for solving nonlinear free and forced vibration problems of cylindrical shells

Nonlinear vibration analysis of circular cylindrical shells has received considerable attention from researchers for many decades. Analytical approaches developed to solve such problem, even not involved simplifying assumptions, are still far from sufficiency, and an efficient numerical scheme capable of solving the problem is worthy of development. The present article aims at devising a novel numerical solution strategy to describe the nonlinear free and forced vibrations of cylindrical shells. For this purpose, the energy functional of the structure is derived based on the first-order shear deformation theory and the von–Kármán geometric nonlinearity. The governing equations are discretized employing the generalized differential quadrature (GDQ) method and periodic differential operators along axial and circumferential directions, respectively. Then, based on Hamilton's principle and by the use of variational differential quadrature (VDQ) method, the discretized nonlinear governing equations are obtained. Finally, a time periodic discretization is performed and the frequency response of the cylindrical shell with different boundary conditions is determined by applying the pseudo-arc length continuation method. After revealing the efficiency and accuracy of the proposed numerical approach, comprehensive results are presented to study the influences of the model parameters such as thickness-to-radius, length-to-radius ratios and boundary conditions on the nonlinear vibration behavior of the cylindrical shells. The results indicate that variation of fundamental vibrational mode shape significantly affects frequency response curves of cylindrical shells.

Nonlinear Bending Analysis of Composite Cylindrical Shells Reinforced by Functionally Graded Carbon Nanotube in Circumferential Direction

Nonlinear Bending Analysis of Composite Cylindrical Shells Reinforced by Functionally Graded Carbon Nanotube in Circumferential Direction

An efficient mixed variational reduced‐order model formulation for nonlinear analyses of elastic shells

SummaryThe Koiter‐Newton method had recently demonstrated a superior performance for nonlinear analyses of structures, compared to traditional path‐following strategies. The method follows a predictor‐corrector scheme to trace the entire equilibrium path. During a predictor step, a reduced‐order model is constructed based on Koiter's asymptotic postbuckling theory that is followed by a Newton iteration in the corrector phase to regain the equilibrium of forces. In this manuscript, we introduce a robust mixed solid‐shell formulation to further enhance the efficiency of stability analyses in various aspects. We show that a Hellinger‐Reissner variational formulation facilitates the reduced‐order model construction omitting an expensive evaluation of the inherent fourth‐order derivatives of the strain energy. We demonstrate that extremely large step sizes with a reasonable out‐of‐balance residual can be obtained with substantial impact on the total number of steps needed to trace the complete equilibrium path. More importantly, the numerical effort of the corrector phase involving a Newton iteration of the full‐order model is drastically reduced thus revealing the true strength of the proposed formulation. We study a number of problems from engineering and compare the results to the conventional approach in order to highlight the gain in numerical efficiency for stability problems.

Geometrical nonlinear analysis based on optimization technique

• Two triangular shapes are used to formulate the iteration procedure required for nonlinear analysis. • The areas of these shapes are considered as two variables objective functions. • These functions are minimized with respect to each variable. • Two new constraint equations are obtained, which can be utilized for the nonlinear solver. • All of numerical experiences clearly demonstrate the merits of the authors' formulations. In this paper, two triangular shapes are used to formulate iteration procedures required for nonlinear analysis. These triangles are formed by the structural load–displacement curve. The areas of these shapes are considered as two variables objective functions. To find the optimal solutions, these functions are minimized with respect to each variable. As a result, two new constraint equations are obtained, which can be utilized as the nonlinear solver. To demonstrate the merit of authors’ scheme, some geometric nonlinear analyses of shells, frames and trusses are performed. Furthermore, present formulations are compared with the cylindrical arc-length method.

Nonlinear Axisymmetric Free Vibration Analysis of Liquid-Filled Spherical Shell With Volume Constraint

Nonlinear axisymmetric free vibration analysis of liquid-filled spherical shells with volume constraint condition using membrane theory is presented in this paper. The energy functional of the shell and contained liquid can be expressed based on the principle of virtual work using surface fundamental form and is written in the appropriate forms. Natural frequencies and the corresponding mode shapes for specified axisymmetric vibration amplitude of liquid-filled spherical shells can be calculated by finite element method (FEM). A nonlinear numerical solution can be obtained by the modified direct iteration technique. The results indicate that the Lagrange multiplier is a parameter for adapting the internal pressure in order to sustain the shell in equilibrium state for each mode of vibration with the volume constraint condition. The axisymmetric mode shapes of the liquid-filled spherical shells under volume constraint condition were found to be in close agreement with those in existing literature for an empty spherical shell. Finally, the effects of support condition, thickness, initial internal pressure, bulk modulus of internal liquid, and elastic modulus on the nonlinear axisymmetric free vibration and change of pressure of the liquid-filled spherical shells with volume constraint condition were demonstrated. The parametric studies showed that the change of pressure has a major impact on the fundamental vibration mode when compared with the higher vibration modes.

Invariant-based formulation of a triangular finite element for geometrically nonlinear thermal analysis of composite shells

The paper describes an invariant-based formulation of a triangular finite element for geometrically nonlinear analysis of shear flexible composite shells subjected to thermal loads. Transverse shear deformation is taken into account using the first order shear deformation theory. The focus is on the representation of the strain energy of the shell in terms of invariant quantities which depend on the components of the strain tensor and elastic constants of the material. Based on the invariant expression for the strain energy density, algorithmic relations are derived for computing the stiffness matrix of the shell finite element. The finite element formulation is used to study stability of equilibrium configurations in the region of large thermal displacements. A positive definite second variation of the total energy is used as a sufficient criterion for stability of equilibrium configurations. A series of numerical examples are given to estimate performance of the finite element in solving nonlinear problems of composite plates and shells under uniform temperature rise. Solution of some classical problems of laminated plates and shells shows that there exist equilibrium configurations not previously reported in the literature.

A Central Point-Based Discrete Shear Gap Method for Plates and Shells Analysis Using Triangular Elements

In this paper, discrete shear gap method (DSG)-based central point of the three-node triangular element (CP-DSG3) is presented for both linear and geometrically nonlinear analysis of plates and shells. A fictitious central point is introduced as the base point of the DSG to overcome the anisotropy of the formerly proposed formulation. In the CP-DSG3, the discrete shear gap at each field node is deduced using rotations of all three nodes in local coordinate system, which makes it an isotropic triangular plate and shell element. At the same time, the CP-DSG3 inherits the advantages of shear-locking-free from the DSG. Moreover, a series of numerical examples consisting of standard patch test, linear and geometrically nonlinear problems have been investigated and the results demonstrate the excellent performance and accuracy of the proposed CP-DSG3.

Hierarchic isogeometric large rotation shell elements including linearized transverse shear parametrization

Two novel hierarchic finite element formulations for geometrically nonlinear shell analysis including the effects of transverse shear are presented. Both methods combine a fully nonlinear Kirchhoff–Love shell model with hierarchically added linearized transverse shear components. Thus, large rotations can be taken into account while circumventing the peculiar task of finding a corresponding parametrization of the rotation tensor. The two formulations differ in the way the transverse shear effects are included, either using hierarchic rotations or hierarchic displacements. The underlying assertion is that in most practical applications the transverse shear angles are small even for large deformations. This is confirmed by various numerical experiments. The hierarchic construction results in an additive strain decomposition into parts resulting from membrane and bending deformation and additional contributions from transverse shear. It requires at least C1-continuous shape functions, which can be easily established within the isogeometric context using spline based finite elements. As reported earlier, this concept is intrinsically free from transverse shear locking. In the nonlinear case it dramatically facilitates representation of large rotations in shell analysis.

Nonlinear dynamic analysis and vibration of eccentrically stiffened S-FGM elliptical cylindrical shells surrounded on elastic foundations in thermal environments

Elliptical cylindrical shell is one of shells with special shape. Up to date, there is no publication on vibration and dynamic of functionally graded elliptical cylindrical shells. Therefore, the purpose of the present study is to investigate the nonlinear dynamic response and vibration of imperfect eccentrically stiffness functionally graded elliptical cylindrical shells on elastic foundations using both the classical shell theory (CST) and Airy stress functions method with motion equations using Volmir's assumption. The material properties are assumed to be temperature - dependent and graded in the thickness direction according to a Sigmoid power law distribution (S-FGM). The S-FGM elliptical cylindrical shell with metal-ceramic-metal layers are reinforced by outside metal stiffeners. Both the S-FGM elliptical shell and metal stiffeners are assumed to be in thermal environment and both of them are deformed under temperature simultaneously. Two cases of thermal loading (uniform temperature rise and temperature variation through thickness) are considered. The nonlinear motion equations are solved by Galerkin method and Runge-Kutta method (nonlinear dynamic response, natural frequencies). The effects of geometrical parameters, material properties, elastic foundations Winkler and Pasternak, the nonlinear dynamic analysis and nonlinear vibration of the elliptical cylindrical shells are studied. The some obtained results are validated by comparing with those in the literature.

Geometric nonlinear analysis of prismatic shells using the semi-analytical finite strip method

The present study sheds light on geometric nonlinear static analysis of prismatic shells using the semi-analytical finite strip method. A new computational model, which includes a fully nonlinear compound strip with a longitudinal and transverse stiffener, has been presented. Furthermore, strips with non-uniform characteristics in the longitudinal direction have been used in nonlinear analysis. This has, to the best knowledge of authors, never been reported. Also, this paper describes the design and implementation of eighteen ideal boundary conditions using three different longitudinal and six well-known transverse displacement interpolation functions.The results of the presented study were obtained using an open-source software and multi-purpose software Abaqus. Moreover, the accuracy of the applied computational approach has been verified by comparison with results from the literature. An excellent agreement of displacement fields is achieved for large deflection analyses of plates with a hole and stiffeners as well as for shells with a stepped thickness in the longitudinal direction. Additionally, results from post-buckling analyses of thin-walled structures, a snap-through and snap-back behavior of shallow shells, are matched. The work presented here has profound implications for future studies of the finite strip deployment.

Nonlinear stability analysis of shallow cylindrical shells with initial imperfections

The present article describes numerical procedures for geometrically nonlinear stability analysis of shallow flexible cylindrical shells. Computing models are based on the theory of plates with account of transverse shear deformations. The finite difference energy method of discretization is used for reducing the continuum problem to finite dimensional problem. Solution procedures for discrete nonlinear problem are based on initial value formulation. The solution of Cauchy problem is performed using the self-correcting Runge-Kutta method with the constrained equation. For stability analysis of shells with initial imperfections small asymmetric perturbations are introduced in the initial conditions of the Cauchy problem for finding asymmetric shapes of shell equilibrium.

The geometric and material nonlinear analysis of cylindrical shells by rise span ratio.

The geometric and material nonlinear analysis of cylindrical shells by rise span ratio.

Geometrically nonlinear vibration analysis of multiferroic composite plates and shells

In this article, a layerwise shear deformation theory is incorporated for geometrically nonlinear vibration (GNV) analysis of multiferroic composite plates and doubly curved shells. The coupled constitutive equations involving ferroelastic, ferroelectric and ferromagnetic properties of multiferroic composite materials along with the total potential energy principle are utilized to derive the finite element formulation for the multiferroic or magneto-electro-elastic (MEE) plates/shells. The electric and the magnetic potentials are assumed to vary linearly in the transverse direction. The electric and magnetic potential distribution in the plate/shell is computed by using the Maxwell's electromagnetic relations. The significance of geometric nonlinearity has been considered using the von Kármán nonlinear strain-displacement relations. Importance of curvature aspect ratio, curvature ratio and the thickness aspect ratio on the nonlinear frequency ratios of the multiferroic/MEE doubly curved shells has been investigated. The backbone curves for multiferroic plates and shells have been studied by considering various aspect ratios. Impact of layer stacking sequence, boundary conditions and coupled fields on the central deflection and nonlinear frequency ratio of the multiferroic plates and shells have been investigated.

Nonlinear buckling analysis of pre-stressed ring-stiffened circular cylindrical shells under external pressure

This paper is concerned with the nonlinear buckling behavior of pre-stressed ring-stiffened thin circular cylindrical shells under external pressure. According to the geometrical nonlinearity, governing equations are derived by incorporating initial stresses based on Donnell-type shell theory. The “smeared stiffeners” approach is used for ring stiffeners. The numerical analyses are conducted by Gelerkin’s method to obtain critical buckling loads of shells. This study shows effects of initial stresses on stability of shells. Moreover, effects of initial stresses on buckling modes of shells are discussed.

Geometrically nonlinear FEM analysis of FGM shells based on neutral physical surface approach in 6-parameter shell theory

The paper presents the formulation of the elastic constitutive law for functionally graded materials (FGM) on the grounds of nonlinear 6-parameter shell theory with the 6th parameter being the drilling degree of freedom. The material law is derived by through-the-thickness integration of the Cosserat plane stress equations. The constitutive equations are formulated with respect to the neutral physical surface. The influence of the power-law exponent, micropolar characteristic length is evaluated in geometrically nonlinear FEM analyses. The results obtained with the neutral physical surface approach are compared with those computed with the middle surface approach. The influence of choice of the reference surface is observed especially in nonlinear stability analysis.

Nonlinear dynamic analysis of creased shells

Recent studies analyze the behavior of advanced shell structures, like foldable, multistable or morphing shell structures. Simulating a thin foldable curved structure is not a trivial task: the structure may go through many snapping transitions from a stable configuration to another. Then, one could claim arc-length methods or use a dynamic approach to perform such simulations. This work presents a geometrically exact shell model for nonlinear dynamic analysis of shells. An updated Lagrangian framework is used for describing kinematics. Several numerical examples of folding a thin dome are presented, including creased shells. The triangular shell finite element used offers great flexibility for the generation of the unstructured curved meshes, as well as great results.

Advantages of the mixed format in geometrically nonlinear analysis of beams and shells using solid finite elements

SummaryThe paper deals with two main advantages in the analysis of slender elastic structures both achieved through the mixed (stress and displacement) format with respect to the more commonly used displacement one: (i) the smaller error in the extrapolations usually employed in the solution strategies of nonlinear problems and (ii) the lower polynomial dependence of the problem equations on the finite element degrees of freedom when solid finite elements are used. The smaller extrapolation error produces a lower number of iterations and larger step length in path‐following analysis and a greater accuracy in Koiter asymptotic method. To focus on the origin of the phenomenon, the two formats are derived for the same finite element interpolation. The reduced polynomial dependence improves the Koiter asymptotic strategy in terms of both computational efficiency, accuracy and simplicity. Copyright © 2016 John Wiley & Sons, Ltd.