Shear Deformable Shell Element Research Articles

Isogeometric analysis for geometrically exact shell elements using Bézier extraction of NURBS with assumed natural strain method

In the present study, isogeometric analysis for geometrically exact shell considering the first-order shear deformation is studied. Since kinematics of the shell is formulated in the form of a general tensor in generalized curvilinear coordinates, isogeometric analysis of arbitrarily shaped shells is possible with a direct geometry link to NURBS. For the first-order shear deformable shell element, the assumed natural strain (ANS) method is normally used to alleviate membrane and transverse shear locking phenomenon. Due to a higher-order regularity between integration knot elements of NURBS basis function has, a locking phenomenon still appears even though the field consistent approach such as ANS is utilized. To solve this problem, we adopted Bézier extraction method which maps a piecewise C0 Bernstein polynomials basis onto NURBS basis to reduce element regularity for isogeometric shell analysis. For an arbitrarily shaped shell geometry generation using NURBS of the shell, the point data sets from the exact geometry shape can be used by the global interpolation. The test results of numerical examples using the developed method show the accuracy and robustness with a higher convergence rate of the proposed approach.

Prospects of Structural Damage Identification Using Modal Analysis and Anomaly Detection

Structural health monitoring (SHM) has significant importance in providing reliability, safety, and economical sensibility for structures. High sensitivity and precision of SHM techniques can be quite demanding. Some SHM testing procedures require structures, for example, wind turbine blades or aircraft structures, to be taken out of operation, which increases costs and causes downtime. Other techniques include sophisticated signal processing and data analysis, which require skilled personnel. An alternative is to use a form of system identification technique called Operational Modal Analysis, during which an operator performs structural vibration measurements without disrupting normal operation of structure. The aim of the current study is to develop a machine learning algorithm, which is able to identify damaged states of a structure, based on the assessment of its modal parameters. The assessment of the structural health is performed using machine learning technique called Anomaly Detection. The proposed method establishes a description of normality using features representing undamaged conditions and then test for abnormality which indicates presence of damage in structure. Structure under test is one-meter-long laminated composite beam. The finite element model of the beam is developed using eight-node shear-deformable shell elements. Damage in the beam is introduced as delamination between the plies representing 5% of the total area of the beam. Numerical modal analysis is carried out to obtain modal frequencies and corresponding mode shapes for the first three bending modes of both the healthy and damaged beams. The results of the numerical experiments show possibilities of the proposed structural health monitoring method to identify damage in composite structures under varying modal parameters.

Assessment of Collapse Pressure of Laminated Composite Subsea Shells Subjected to Hydrostatic Follower Force

A doubly curved composite shear deformable shell element has been developed for the buckling analysis of composite subsea shells which takes into consideration the effect of hydrostatic follower force. This element is based on first-order shear deformation theory considering the nonlinear curvature terms of the strain–displacement relations in order to simulate the realistic structural behaviour of thin composite shells. It has been employed for the prediction of collapse pressure, including the follower force effect of thin composite cylindrical and shallow shell panels, which form a part of subsea structures.

Free vibration and damping characteristics study of doubly curved sandwich shell panels with viscoelastic core and isotropic/laminated constraining layer

Abstract In this work, a shear deformable viscoelastic sandwich shell element is proposed for doubly curved sandwich shell panels with passive constrained layer damping (PCLD) treatment. The transverse displacements of the constraining layer and base layer are considered to be independent, besides the in-plane displacements . The transverse and in-plane displacements of the viscoelastic core are considered to be varying linearly through the thickness. Shear as well as normal deformations of the core are included in the analysis. The equation of motion of the doubly curved sandwich shell panel under free vibration is derived via Hamilton's principle . The modal frequencies and modal loss factors of the Isotropic elastic-Viscoelastic-Isotropic elastic (IVI) sandwich shell panel and Laminated elastic-Viscoelastic-Laminated elastic (LVL) sandwich panel are obtained from numerical solutions, using finite element method in conjunction with Hamilton's principle. Parametric studies are carried out to ascertain the effects of shell geometries, aspect ratio, orthotropicity of the skins, core layer thickness, constraining layer thickness, core loss factor on the natural frequencies and system loss factors under different boundary conditions.

Shear deformable shell element DKMQ24 for composite structures

In this work we propose a new 4-node DKMQ24 shell element based on Naghdi-Reissner-Mindlin shell theory with 24 degrees of freedom. This new composite shell element, which is developed from DKMQ plate and shell elements, takes into account shear deformation, coupled bending-membrane energy and warping effects. This element has no spurious mode, passes patch tests for membrane, bending and shear problems. It has also successfully passed standard benchmark tests in the case of thick and thin shells without membrane and shear locking. In this paper, we extend the predictive capabilities of the DKMQ24 shell elements to composite laminated plates and shell structures. Numerical results obtained from DKMQ24 are compared against state-of-the-art shell elements. The results obtained by the proposed element converge more rapidly towards the reference solution.

Continuum Mechanics Based Bilinear Shear Deformable Shell Element Using Absolute Nodal Coordinate Formulation

In this investigation, a continuum mechanics based bilinear shear deformable shell element is developed using the absolute nodal coordinate formulation (ANCF) for the large deformation analysis of multibody shell structures. The element consists of four nodes, each of which has the global position coordinates and the transverse gradient coordinates along the thickness introduced for describing the orientation and deformation of the cross section of the shell element. The global position field on the middle surface and the position vector gradient at a material point in the element are interpolated by bilinear polynomials. The continuum mechanics approach is used to formulate the generalized elastic forces, allowing for the consideration of nonlinear constitutive models in a straightforward manner. The element lockings exhibited in the element are eliminated using the assumed natural strain (ANS) and enhanced assumed strain (EAS) approaches. In particular, the combined ANS and EAS approach is introduced to alleviate the thickness locking arising from the erroneous transverse normal strain distribution. Several numerical examples are presented in order to demonstrate the accuracy and the rate of convergence of numerical solutions obtained by the continuum mechanics based bilinear shear deformable ANCF shell element proposed in this investigation.

A refined element-based Lagrangian shell element for geometrically nonlinear analysis of shell structures

For the solution of geometrically nonlinear analysis of plates and shells, the formulation of a nonlinear nine-node refined first-order shear deformable element-based Lagrangian shell element is presented. Natural co-ordinate-based higher order transverse shear strains are used in present shell element. Using the assumed natural strain method with proper interpolation functions, the present shell element generates neither membrane nor shear locking behavior even when full integration is used in the formulation. Furthermore, a refined first-order shear deformation theory for thin and thick shells, which results in parabolic through-thickness distribution of the transverse shear strains from the formulation based on the third-order shear deformation theory, is proposed. This formulation eliminates the need for shear correction factors in the first-order theory. To avoid difficulties resulting from large increments of the rotations, a scheme of attached reference system is used for the expression of rotations of shell normal. Numerical examples demonstrate that the present element behaves reasonably satisfactorily either for the linear or for geometrically nonlinear analysis of thin and thick plates and shells with large displacement but small strain. Especially, the nonlinear results of slit annular plates with various loads provided the benchmark to test the accuracy of related numerical solutions.

On the use of elastic middle surface approach in the large deformation analysis of moderately thick shell structures using absolute nodal coordinate formulation

In this study, a shear deformable shell element is developed based on the elastic middle surface approach using the absolute nodal coordinate formulation (ANCF) for the large deformation analysis of thin to moderately thick shell structures. The bilinear shape function is used to define the global position vector in the middle surface and the transverse gradient vector which defines the orientation and deformation of the cross section within the element. The plane stress assumption is used to remedy the Poisson’s thickness locking exhibited in the ANCF shell element formulated by the continuum mechanics approach, thus the stress distribution along the shell thickness is assumed to be constant. The cross-sectional frame is introduced to define strains of the initially curved shell element using the elastic middle surface approach. The curvature thickness and transverse shear lockings are alleviated using the assumed natural strain method, while the in-plane shear locking is removed using the enhanced assumed strain method. Several numerical examples are presented in order to demonstrate the performance of the shear deformable ANCF shell element based on the elastic middle surface approach developed in this study. The developed element is compared with the continuum mechanics-based ANCF shell element to shed light on the nature of the thickness locking exhibited in the bilinear shell element and its locking remedies.

Non-linear analysis of laminated composite and sigmoid functionally graded anisotropic structures using a higher-order shear deformable natural Lagrangian shell element

Finite element solutions based on the higher-order shear deformation theory are presented for the geometrical non-linear analysis of through-thickness functionally graded plates and shells. Natural co-ordinate-based higher-order transverse shear strains are used in present shell element. The fiber orientation of plates and shells is assumed to vary according to a sigmoid distribution in terms of the volume fractions of the constituents. The formulation of a non-linear 9-node Element-based Lagrangian shell element is presented and natural co-ordinate-based strains are used in present shell element. Using the assumed natural strain method the present shell element generates neither membrane nor shear locking behavior. Numerical results of the linear and non-linear analysis are presented to show the combined effect of the fiber orientation, loading conditions, aspect ratios and side-to-thickness ratios on the mechanical behaviors. Besides, a comparison of the non-linear results of isotropic and anisotropic functionally graded structures is investigated.

Geometrically nonlinear analysis of laminated composite thin shells using a modified first-order shear deformable element-based Lagrangian shell element

The formulation of a nonlinear composite 9-node modified first-order shear deformable element-based Lagrangian shell element is presented for the solution of geometrically nonlinear analysis of laminated composite thin plates and shells. The application limit of modified shear deformation theory is presented for the correct analysis of composite laminates. However, it is evident that it results in a parabolic distribution of the transverse shear strains and satisfies the zero transverse shear stresses requirements at the shell surfaces. It also requires insignificant modifications to be implemented in existing displacement-based first-order shell elements. Natural co-ordinate-based higher-order transverse shear strains are used in present shell element. Using the assumed natural strain method the present shell element generates neither membrane nor shear locking behavior. Numerical examples demonstrate that the present element behaves quite satisfactorily either for the linear analysis of plates and shells or for the geometrically nonlinear analysis of laminated composite thin plates and shells with large displacement but small strain.

Quadrilateral Isoparametric Shear Deformable Shell Element for Use in Soil-Structure Interaction Problems

In the finite-element analysis of soil-structure interaction problems, shell elements are important to model structural components such as tunnel linings or sheet pile walls. In this paper the development, implementation, and testing of a curved quadrilateral isoparametric shear deformable shell element are presented. An overriding concern in the analysis of soil-structure interaction problems is the C0 compatibility of the shell elements with the solid elements used to model the soil. Many recent publications on shell elements deal with the elements in a purely structural context and therefore do not require the C0 compatibility with solid elements. The development of the shell element presented here forms an extension of a new approach to curved Mindlin and axisymmetric shell elements introduced by Day and Potts in 1990. There are no generally accepted benchmarking marking problems for the use of shell elements in three-dimensional geotechnical engineering problems. Consequently, the performance of the ...

Adding transverse normal stresses to layered shell finite elements for the analysis of composite structures

This work presents a formulation developed to add capabilities for representing the through thickness distribution of the transverse normal stresses, σ z , in first and higher order shear deformable shell elements within a finite element (FE) scheme. The formulation is developed within a displacement based shear deformation shell theory. Using the differential equilibrium equations for two-dimensional elasticity and the interlayer stress and strain continuity requirements, special treatment is developed for the transverse normal stresses, which are thus represented by a continuous piecewise cubic function. The implementation of this formulation requires only C 0 continuity of the displacement functions regardless of whether it is added to a first or a higher order shell element. This makes the transverse normal stress treatment applicable to the most popular bilinear isoparametric 4-noded quadrilateral shell elements. To assess the performance of the present approach it is included in the formulation of a recently developed third order shear deformable shell finite element. The element is added to the element library of the general nonlinear explicit dynamic FE code DYNA3D. Some illustrative problems are solved and results are presented and compared to other theoretical and numerical results.

Elasticity, shell theory and finite element results for the buckling of long sandwich cylindrical shells under external pressure

The buckling of a sandwich cylindrical shell under uniform external hydrostatic pressure is studied in three ways. The simplifying assumption of a long shell is made (or, equivalently, ‘ring’ assumption), in which the buckling modes are assumed to be two-dimensional, i.e. no axial component of the displacement field, and no axial dependence of the radial and hoop displacement components. All constituent phases of the sandwich structure, i.e. the facings and the core, are assumed to be orthotropic. First, the structure is considered a three-dimensional (3D) elastic body, the corresponding problem is formulated and the solution is derived by solving a set of two linear homogeneous ordinary differential equations of the second-order in r (the radial coordinate), i.e. an eigenvalue problem for differential equations, with the external pressure, p the parameter/eigenvalue. A complication in the sandwich construction is due to the fact that the displacement field is continuous but has a slope discontinuity at the face-sheet/core interfaces, which necessitates imposing ‘internal’ boundary conditions at the face-sheet/core interfaces, as opposed to the traditional two-end-point boundary value problems. Second, the structure is considered a shell and shell theory results are generated with and without accounting for the transverse shear effect. Two transverse shear correction approaches are employed, one based only on the core, and the other based on an effective shear modulus that includes the face-sheets. Third, finite element results are generated by use of the ABAQUS finite element code. In this part, two types of elements are used: a shear deformable shell element and a solid 3D (brick) element. The results from all these three different approaches are compared.

Finite element models for laminated shells with actuation capability

The formulation of the adaptive shell finite element, developed by the authors in the recent years for the 4-node element, is here extended by setting up new 9-node and 16-node elements. An active layer––made by a piezoelectric material or a similar active medium––is assumed to be included in a stacking sequence of a laminated shell. The actuation capability of the layer is represented by an in-plane assigned strain field that can be thought of as being produced by the converse piezoelectric effect or other induced strain actuation mechanism, like the classical thermoelastic effect. In this way the actuation mechanism is included in a general formulation of shear deformable shell element based on the “mixed interpolation tensorial component” approach, developed by K.J. Bathe, that is known to result into very effective elements. 4-node, 9-node and 16-node are compared in terms of convergence for several assumed cases. Comparisons with closed form solutions, when available, and numerical results allow to verify the accuracy of the formulation and to check the predicting capability of the elements.

A co-rotational 8-node assumed strain element for large displacement elasto-plastic analysis of plates and shells

The formulation of a non-linear shear deformable shell element is presented for the solution of stability problems of stiffened plates and shells. The formulation of the geometrical stiffness presented here is exactly defined on the midsurface and is efficient for analyzing stability problems of thick plates and shells by incorporating bending moment and transverse shear resultant force. As a result of the explicit integration of the tangent stiffness matrix, this formulation is computationally very efficient in incremental nonlinear analysis. The element is free of both membrane and shear locking behaviour by using the assumed strain method such that the element performs very well in the thin shells. By using six degrees of freedom per node, the present element can model stiffened plate and shell structures. The formulation includes large displacement effects and elasto-plastic material behaviour. The material is assumed to be isotropic and elasto-plastic obeying Von Mises's yield condition and its associated flow rules. The results showed good agreement with references and computational efficiency.

Three-node macro triangular shell element based on the assumed natural strains

Finite element models with simple triangular geometry facilitate pre-processing for the finite element analysis. In the present study, a robust shear deformable triangular shell element formulation is presented for general plate and shell analysis, using three-node mesh discretization. The present formulation is developed on the basis of the assumed natural strain (ANS) formulation to attenuate the shear locking effect. Furthermore, this study proposes macro triangular element scheme by dividing a three-node triangular mesh into three parts to produce three individual ANS triangular elements and then merges these by condensing out the virtual center node. The performance of the macro ANS element is highly enhanced by reducing the number of sampling locations of three sub triangular element, but still utilizes three-node mesh discretization same as does the mesh used for three-node triangular elements. The macro ANS element has invariant stiffness, possesses no commutable zero energy mechanism. Numerical tests are presented to illustrate the high performance nature of the macro ANS element in general shell analysis. In particular, the numerical study demonstrates that the macro ANS element completely removes the shear locking effect that has been detrimental in shear deformable linear triangular elements so far.

Elasto-plastic analysis of shells with the triangular element TRIC

TRIC is a simple but sophisticated three-node shear-deformable isotropic and composite facet shell element suitable for large-scale linear and nonlinear engineering computations of thin and moderately thick anisotropic plate and complex shell structures. In the present work an elasto-plastic constitutive model based on the von Mises yield criterion with isotropic hardening is incorporated into the element. The characteristic feature of this formulation is that the nonlinear material behaviour is taken into account entirely in the natural system of the element. This is achieved by transforming quantities such as equivalent plastic strain, equivalent stress, the expression of the yield surface and the components of flow vector from the material coordinate system to the natural coordinate system. These transformations lead to simple and elegant expressions for the respective quantities in the natural system which eventually result to an efficient and cost effective treatment of the nonlinear analysis of arbitrary shells including material and geometrical nonlinearities.

Finite element analysis and distributed control of laminated composite shells using LQR/IMSC approach

In the present study a shear deformable shell element is developed based on Reissner's hypothesis for the analysis of smart laminated composite shells. The electric field is defined in the curvilinear co-ordinate system. The mathematics of arbitrary shell geometry is amenable to the flexible mapping characteristic of the finite element formulation developed in the present work. Lame's parameters and the radii of curvature are generated within the model. An IMSC based LQR control methodology is adopted for the active vibration control of laminated spherical shell with different fibre orientation and varying radius of curvature. Results are presented and are discussed.

Sandwich shell finite element for dynamic explicit analysis

AbstractThe contribution of this paper consists of new development of transverse shear stresses through the thickness and finding an expression for the critical time step for explicit time integration of layered shells. This work presents the finite element (FE) formulation and implementation of a higher‐order shear deformable shell element for dynamic explicit analysis of composite and sandwich shells. The formulation is developed using a displacement‐based third‐order shear deformation shell theory. Using the differential equilibrium equations and the interlayer requirements, special treatment is developed for the transverse shear, resulting in a continuous, piecewise quartic distribution of the transverse shear stresses through the shell thickness. Expressions are developed for the critical time step of the explicit time integration for orthotropic homogeneous and layered shells based on the developed third‐order formulation. To assess the performance of the present shell element, it is implemented in the general non‐linear explicit dynamic FE code DYNA3D. Several problems are solved and results are presented and compared to other theoretical and numerical results. Copyright © 2002 John Wiley & Sons, Ltd.

Effective Shear Deformable Shell Elements for Adaptive Laminated Structures

An effective mixed interpolation approach for shell elements is here extended to the analysis of adaptive shells consisting of a sequence of active and passive layers. An active layer – made by a piezoelectric material or a similar active medium – is assumed to be included in the stacking sequence of a laminated shell. The actuation capability of the layer is represented by a given inplane strain field that can be thought of as being produced by the converse piezoelectric effect or other induced strain actuation mechanism. In this way, the actuation mechanism is included in the formulation of shear deformable shell element that has been demonstrated not to suffer of shear locking effects. A four node element was developed. Several comparisons have been performed to verify the accuracy of the formulation and to check the predicting capability of the element in comparison with both numerical and experimental results of the recent available literature. Also an application to a real composite structure is illustrated in order to demonstrate the possible use of the proposed element in the engineering practice.