Present Shell Element Research Articles

A Novel Shell Element for Quasi-Static and Natural Frequency Analysis of Textile Composite Structures

A shell element for analysis of textile composite structures is proposed in this paper. Based on the embedded element method and solid shell concept, the architecture, geometry, and material properties of a repeat unit cell (RUC) of textile composite are embedded in a single shell finite element. Flat and curved textile composite structures are used to apply and verify the present shell element. The deformation and natural frequency obtained by the present shell element are compared against those computed from full three-dimensional finite element analyses. It is shown that the proposed shell element is efficient, simple, and reliable for textile composite structural analysis.

Corotational nonlinear analyses of laminated shell structures using a 4-node quadrilateral flat shell element with drilling stiffness

A new 4-node quadrilateral flat shell element is developed for geometrically nonlinear analyses of thin and moderately thick laminated shell structures. The flat shell element is constructed by combining a quadrilateral area coordinate method (QAC) based membrane element AGQ6-II, and a Timoshenko beam function (TBF) method based shear deformable plate bending element ARS-Q12. In order to model folded plates and connect with beam elements, the drilling stiffness is added to the element stiffness matrix based on the mixed variational principle. The transverse shear rigidity matrix, based on the first-order shear deformation theory (FSDT), for the laminated composite plate is evaluated using the transverse equilibrium conditions, while the shear correction factors are not needed. The conventional TBF methods are also modified to efficiently calculate the element stiffness for laminate. The new shell element is extended to large deflection and post-buckling analyses of isotropic and laminated composite shells based on the element independent corotational formulation. Numerical results show that the present shell element has an excellent numerical performance for the test examples, and is applicable to stiffened plates.

Shear buckling responses of laminated composite shells using a modified 8-node ANS shell element

In this study a numerical investigation is presented for initial buckling response of laminated composite plates and shells under the combined in-plane loading using a modified 8-ANS method. The finite element, based on a modified first-order shear deformation theory, is further improved by the combined use of assumed natural strain method. We analyze the influence of the shell element with the various location and number of enhanced membrane and shear interpolation. 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. The effect of various types of lay-ups, materials and number of layers on initial buckling response is discussed. In addition, the effect of direction of shear load on the initial buckling response is studied. The interaction curves (between in-plane compression and shear for different parameters of the laminates) are presented. The numerical results obtained are in good agreement with those reported by other researchers.

Transient analysis of FGM and laminated composite structures using a refined 8-node ANS shell element

In this paper, we investigate the vibration analysis of functionally graded material (FGM) and laminated composite structures, using a refined 8-node shell element that allows for the effects of transverse shear deformation and rotary inertia. The properties of FGM vary continuously through the thickness direction according to the volume fraction of constituents defined by sigmoid function, but in this method, their Poisson’s ratios of the FGM plates and shells are assumed to be constant. The finite element, based on a first-order shear deformation theory, is further improved by the combined use of assumed natural strains and different sets of collocation points for interpolation the different strain components. We analyze the influence of the shell element with the various location and number of enhanced membrane and shear interpolation. 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. The natural frequencies of plates and shells are presented, and the forced vibration analysis of FGM and laminated composite plates and shells subjected to arbitrary loading is carried out. In order to overcome membrane and shear locking phenomena, the assumed natural strain method is used. To validate and compare the finite element numerical solutions, the reference solutions of plates based on the Navier’s method, the series solutions of sigmoid FGM (S-FGM) plates are obtained. Results of the present theory show good agreement with the reference solutions. In addition the effect of damping is investigated on the forced vibration analysis of FGM plates and shells.

An 8-Node Shell Element for Nonlinear Analysis of Shells Using the Refined Combination of Membrane and Shear Interpolation Functions

An improved 8-node shell finite element applicable for the geometrically linear and nonlinear analyses of plates and shells is presented. Based on previous first-order shear deformation theory, the finite element model is further improved by the combined use of assumed natural strains and different sets of collocation points for the interpolation of the different strain components. The influence of the shell element with various conditions such as locations, number of enhanced membranes, and shear interpolation is also identified. By using 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, to characterize the efficiency of these modifications of the 8-node shell finite elements, numerical studies are carried out for the geometrically linear and non-linear analysis of plates and shells. In comparison to some other shell elements, numerical examples for the methodology indicate that the modified element described locking-free behavior and better performance. More specifically, the numerical examples of annular plate presented herein show good validity, efficiency, and accuracy to the developed nonlinear shell element.

활절로 지지된 원통형 적층복합쉘의 기하학적 비선형 해석

In the present study, an Element-Based Lagrangian Formulation for the nonlinear analysis of shell structures is presented. The strains, stresses and constitutive equations based on the natural co-ordinate have been used throughout the Element-Based Lagrangian Formulation of the present shell element which offers an advantage of easy implementation compared with the traditional Lagrangian Formulation. The Element-Based Lagrangian Formulation of a 9-node resultant-stress shell element is presented for the anisotropic composite material. The element is free of both membrane and shear locking behavior by using the assumed natural strain method such that the element performs very well in thin shell problems. The arc-length control method is used to trace complex equilibrium paths in thin shell applications. Numerical examples for laminated composite curved shells presented herein clearly show the validity of the present approach and the accuracy of the developed shell element.

Analysis of Laminated and Sandwich Composite Structures Using Solid-like Shell Elements

A new solid-like shell element was formulated which is suitable for analysis of laminated and sandwich composite structures. Then, a multiscale analysis technique was implemented to the shell element formulation so that micro-level stresses and strains (i.e. stresses and strains in reinforcing fibers and the binding matrix) in those structures can be computed. The shell element has three displacement degrees of freedom per node like a 3-D solid element. Therefore, the shell elements can be stacked easily on top of one another like 3-D solid elements in order to represent multiple layers through the thickness of laminated and sandwich structures. The effect of a thin resin or adhesive layer in laminated and sandwich composite structures was investigated on both static and the dynamic responses of the structures using the developed shell elements. The study showed an apparent effect of the resin/adhesive layer even though it is very thin. As a result, the present shell element can be used effectively to include those thin layers in finite element analysis models of laminated and sandwich composite structures.

The effect of multi-directional stiffness degradation on the non-linear analysis of composite laminates

AbstractA formulation of element-based Lagrangian 9-node shell element based modified first-order shear deformation theory is improved for non-linear behaviors of composite laminates containing matrix cracking. Using the refined ANS (assumed natural strain) shell elements either show the optimum combination of sampling points with an excellent accuracy or remove the locking phenomenon. The multi-directional stiffness degradation caused by matrix cracking, which was proposed by Duan and Yao, is conducted. Natural coordinate based higher-order transverse shear strains are used in the present shell element. Numerical examples demonstrate that the present element behaves reasonably satisfactorily either for the linear or geometrical non-linear analysis of laminated composite structures. The results of laminated composite shells with matrix cracking may be the benchmark test for the non-linear analysis of damaged composite laminates.

A refined finite element for first-order plate and shell analysis

This paper presents an improved 8-node shell element for the analysis of plates and shells. The finite element, based on a refined first-order shear deformation theory, is further improved by the combined use of assumed natural strain method. We analyze the influence of the shell element with the different patterns of sampling points for interpolating different components of strains. 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. Further, a refined first-order shear deformation theory, 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. Numerical examples demonstrate that the present element perform better in comparison with other shell elements.

강성 저하된 적층복합판의 비선형 해석

Abstract In this study, a finite element formulation based first-order shear deformation theory is developed for non-linear behaviors of laminated composite plates containing matrix cracking. The multi-directional stiffness degradation is developed for adopting the stiffness variation induced from matrix cracking, which is proposed by Duan and Yao. The matrix cracking can be expressed in terms of the variation of material properties, such as Young's modulus, shear modulus and Possion ratio of plates, and sequently it is possible to predict the variation of the local stiffness. 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 or geometrical nonlinear analysis of laminated composite plates. The results of laminated composite plates with matrix cracking may be the benchmark test for the non-linear analysis of damaged laminated composite plates.

A finite shell element with well balanced approximation functions for piezoelectric coupling

AbstractThis contribution is concerned with a piezoelectric shell formulation. The present shell element has four nodes and bilinear interpolation functions. The nodal degrees of freedom are displacements, rotations and the electric potential on top and bottom of the shell. A 3D–material law is incorporated. In case of bending dominated problems incompatible approximation functions of the electrical and mechanical fields cause incorrect results. This effect occurs in standard element formulations, where the mechanical and electrical degrees of freedom are approximated with lowest order interpolation functions. In order to overcome this problem a mixed multi–field variational approach is introduced. It allows for approximations of the electric field and the strains independent of the bilinear interpolation functions. A quadratic approach for the shear strains and the electric field is proposed through the shell thickness. This leads to well balanced approximation functions regarding coupling of electrical and mechanical fields. A numerical example illustrates the more precise results in contrast to standard elements. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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.

A mixed finite element formulation for piezoelectric shells

AbstractA finite element formulation to analyze piezoelectric shell problems is presented. A reference surface of the shell is modelled with a four node element. Each node possesses six mechanical degrees of freedom, three displacements and three rotations, and one electric degree of freedom, which is the difference of the electric potential in thickness direction. The formulation is based on the mixed field variational principle of Hu‐Washizu. The independent fields are displacements u, electric potential φ, strains E, electric field E, stresses S and dielectric displacements D. The mixed formulation allows an interpolation of the strains and the electric field in thickness direction. Accordingly a three‐dimensional material law is incorporated in the variational formulation. It is remarked that no simplification regarding the constitutive law is assumed. The formulation allows the consideration of arbitrary constitutive relations. The normal zero stress condition and the normal zero dielectric displacement condition are enforced by the independent stress and dielectric displacement fields. They are defined as zero in thickness direction. The present shell element fulfills the important patch tests: the in‐plane, bending and shear test. Some numerical examples demonstrate the applicability of the present piezoelectric shell element. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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.

Large deformation analysis of functionally graded shells

A geometrically nonlinear analysis of functionally graded shells is presented. The two-constituent functionally graded shell consists of ceramic and metal that are graded through the thickness, from one surface of the shell to the other. A tensor-based finite element formulation with curvilinear coordinates and first-order shear deformation theory are used to develop the functionally graded shell finite element. The first-order shell theory consists of seven parameters and exact nonlinear deformations and under the framework of the Lagrangian description. High-order Lagrangian interpolation functions are used to approximate the field variables to avoid membrane, shear, and thickness locking. Numerical results obtained using the present shell element for typical benchmark problem geometries with functionally graded material compositions are presented.

A 4-node co-rotational ANS shell element for laminated composite structures

The formulation of a non-linear composite 4-node co-rotational resultant shell element is presented for the solution of bending, free vibration, critical buckling and postbuckling analysis of composite plates and shells. Using the ANS (assumed natural strain) method the present shell element generates shear locking behavior, and such element performs very well as much as shells get thin. The formulation of the geometrical stiffness presented here is exactly defined on the mid-surface and is efficient for analyzing stability problems of thin and thick laminated plates, and shells by incorporating bending moment and transverse shear resultant forces. The transverse shear stiffness is defined by an equilibrium approach instead of using the shear correction factor. The adoption of a second-order co-rotational formulation makes it computationally efficient compared to the most existing composite shell elements. The proposed formulation is computationally efficient and the test results showed good agreement.

A co-rotational quasi-conforming 4-node resultant shell element for large deformation elasto-plastic analysis

A co-rotational, quasi-conforming formulation of a 4-node stress resultant shell element is presented for non-linear analysis of plate and shell structures. The tangent stiffness matrix in this quasi-conforming formulation is explicitly integrated. This makes the element computationally efficient in incremental, non-linear analysis. It includes drilling degrees of freedom, which improves membrane behavior and allows the modeling of stiffened plates and shells. It is also free of shear locking behavior. The formulation of the geometrical stiffness is derived using the full definition of Green strain tensor. The inclusion of the bending moment and transverse shear resultant forces in the geometric stiffness allows effective analysis of stability problems of moderately thick plates and shells. The stresses are accurately taken at the nodal points without extrapolation. The plasticity is traced by applying the von Mises yield condition and Prandtl–Reuss flow rule to discrete points through the thickness. The multi-layered approach is based on equally spaced stations, including extreme fibers. A modified trapezoidal rule is used for the numerical integration of the constitutive relation in the plasticity part. Numerous tests are carried out for the non-linear validation of present 4-node shell element and the results are in good agreement with references.

Nine-Node Resultant-Stress Shell Element for Free Vibration and Large Deflection of Composite Laminates

The newly constructed formulation of an element-based resultant-stress nine-node composite shell element is presented for the solution of free vibration and large deflection problems of isotropic and composite laminates. In this paper, the effectiveness of this new formulation is investigated in the static and free vibration analysis. The strain–displacement relationship of the shell could be explained from the point of the new element-based Lagrangian finite element formulation. The newly added terms between bending strain and displacement reflect the contributions of displacements to the curvature. Natural coordinate-based strains, stresses, and constitutive equations are used throughout the element-based Lagrangian formulation of the present shell element which offers significant implementation advantages compared with the traditional Lagrangian formulation. Using the assumed natural strain method the present shell element generates neither membrane nor shear locking behavior, and such an element performs very well as much as shells become thin. The arc-length control method is used to trace complex load–displacement paths and the Lanczos method is employed in the calculation of the eigenvalues of shells. A number of numerical analyses are presented and discussed in order to explore the capabilities of the present shell element. The test results showed very good agreement.

Free and forced vibration analysis of laminated composite plates and shells using a 9-node assumed strain shell element

The natural frequencies of isotropic and composite laminates are presented. The forced vibration analysis of laminated composite plates and shells subjected to arbitrary loading is investigated. In order to overcome membrane and shear locking phenomena, the assumed natural strain method is used. To develop a laminated shell element for free and forced vibration analysis, the equivalent constitutive equation that makes the computation of composite structures efficient was applied. The Mindlin-Reissner theory which allows the shear deformation and rotary inertia effect to be considered is adopted for development of nine-node assumed strain shell element. The present shell element offers significant advantages since it consistently uses the natural co-ordinate system. Results of the present theory show good agreement with the 3-D elasticity and analytical solutions. In addition the effect of damping is investigated on the forced vibration analysis of laminated composite plates and shells.

Postbuckling analysis of laminated composite plates subjected to the combination of in-plane shear, compression and lateral loading

This study deals with postbuckling behavior of laminated composite plates under the combination of in-plane shear, compression and lateral loading using an Element-based Lagrangian formulation. Natural co-ordinate-based strains, stresses and constitutive equations are used in the present shell element. The Element-based Lagrangian formulation described in this paper, in comparison with the traditional approaches, is more attractive not only because it uses only single mapping but also it converges faster. In addition, the finite element (FE) formulation based on the assumed natural strain method for composite structures shows excellence from the standpoints of computational efficiency as well as its ability to avoid both membrane and shear locking behavior. The numerical results obtained are in good agreement with those reported by other investigators. In particular, new results reported in this paper show the influence of various types of loading, materials and number of layers on postbuckling behavior.