General Shell Element Research Articles

Enhanced beam and plate finite elements with shear stress continuity for compressible sandwich structures

This paper concerns and presents new improved beam and plate finite elements for analysing sandwich structures from the structural and modal points of view. In this study, the material behaviour is supposed to be linear elastic and orthotropic, but the method could be extended for nonlinear problems as well. The current models are able to treat the compressibility of the soft core materials, besides the interlaminar continuity conditions and the zero shear stresses on the shear-free surfaces are granted by the Lagrange multiplier method. In order to provide stress continuity along the element boundaries, the Hermite interpolation is employed for the elementary formulations that yields subparametric finite elements. On the contrary, the semi-analytical solutions of these enhanced beam and plate models are also discussed here for particular boundary conditions. The verification of the new finite elements is carried out by two case studies. The test examples are modelled and solved in commercial finite element software (ANSYS) with general shell and solid elements, too. The results derived from different models are subjected to a comprehensive comparison. Also, the structural and modal analyses of the test examples are presented during the case studies. The structural analysis of an overconstrained sandwich panel is also discussed to point out the significance of the proper shear stress distributions. Finally, the advantages and disadvantages of the newly developed elements and semi-analytical solutions are revealed in detail.

On the Geometrically Nonlinear Analysis of Composite Axisymmetric Shells

Composite axisymmetric shells have numerous applications; many researchers have taken advantage of the general shell element or the semi-analytical formulation to analyze these structures. The present study is devoted to the nonlinear analysis of composite axisymmetric shells by using a 1D three nodded axisymmetric shell element. Both low and higher-order shear deformations are included in the formulation. The displacement field is considered to be nonlinear function of the nodal rotations. This assumption eliminates the restriction of small rotations between two successive increments. Both Total Lagrangian Formulation and Generalized Displacement Control Method are employed for analyzing the shells. Several numerical tests are performed to corroborate the accuracy and efficiency of the suggested approach.

Finite Element Analysis of Nanoindentation and Elastic Behavior of Bi2Te3 Two-Dimensional Nanosheets

Nanoindentation and elastic behavior of two-dimensional (2D) layered bismuth telluride (Bi2Te3) were investigated using a finite element analysis (FEA) approach. A circular suspended Bi2Te3 2D nanosheet is subjected to a vertical point force load at the nanosheet center, in which the resultant vertical deflection (i.e., indentation depth) is shown to be elastic and exhibits a linear to cubic transition with increasing point force. Using the shell models in the ABAQUS package, the elastic deformation transition could be clearly captured and it was shown to be strongly dependent on nanosheet's thickness as well as the pre-stretch magnitude in 2D nanosheets. Using a general shell element S4, the pre-stretch effect and Young's modulus were individually tuned in order to assess the elastic behavior experimentally observed by atomic force microscope (AFM). Complementary to AFM-based nanoindentations, this FEA work would help to validate the experimental acquisition of Young's moduli, and could also delineate the detailed strain/stress fields underneath the indenter.

Benchmark Computations of Stresses in a Spherical Dome with Shell Finite Elements

We present a computational framework for analyzing thin shell structures using the finite element method. The framework is based on a mesh-dependent shell model which we derive from the general laws of three-dimensional elasticity. We apply the framework for the so-called Girkmann benchmark problem involving a spherical shell stiffened with a foot ring. In particular, we compare the accuracy of different reduced strain four-node elements in this context. We conclude that the performance of the bilinear shell finite elements depends on the mesh quality but reasonable accuracy of the quantities of interest of the Girkmann problem can be attained in contrast to earlier results obtained with general shell elements for the problem.

Interactive design and simulation of tubular supporting structure

This paper presents a system for design and simulation of supporting tube structure. We model each freeform tube component as a swept surface, and employ boundary control and skeletal control to manipulate its cross-sections and its embedding respectively. With the parametrization of the swept surface, a quadrilateral mesh consisting of nine-node general shell elements is automatically generated and the stress distribution of the structure is simulated using the finite element method. In order to accelerate the complex finite element simulation, we adopt a two-level subspace simulation strategy, which constructs a secondary complementary subspace to improve the subspace simulation accuracy. Together with the domain decomposition method, our system is able to provide interactive feedback for parametric freeform tube editing. Experiments show that our system is able to predict the structural character of the tube structure efficiently and accurately.

Three dimensional numerical simulation of residential building on shrink–swell soils in response to climatic conditions

SummaryShrink–swell soils can cause distresses in buildings, and every year, the economic loss associated with this problem is huge. This paper presents a comprehensive system for simulating the soil–foundation–building system and its response to daily weather conditions. Weather data include rainfall, solar radiation, air temperature, relative humidity, and wind speed, all of which are readily available from a local weather station or the Internet. These data are used to determine simulation flux boundary conditions. Different methods are proposed to simulate different boundary conditions: bare soil, trees, and vegetation. A coupled hydro‐mechanical stress analysis is used to simulate the volume change of shrink–swell soils due to both mechanical stress and water content variations. Coupled hydro‐mechanical stress‐jointed elements are used to simulate the interaction between the soil and the slab, and general shell elements are used to simulate structural behavior. All the models are combined into one finite element program to predict the entire system's behavior. This paper first described the theory for the simulations. A site in Arlington, Texas, is then selected to demonstrate the application of the proposed system. Simulation results are shown, and a comparison between measured and predicted movements for four footings in Arlington, Texas, over a 2‐year period is presented. Finally, a three‐dimensional simulation is made for a virtual residential building on shrink–swell soils to identify the influence of various factors. Copyright © 2015 John Wiley & Sons, Ltd.

P1-Nonconforming shell element and its application to topology optimization

The P1-nonconforming quadrilateral element developed by Park and Sheen [19] is employed for the formulation of a four-node degenerated shell element. The numerical stability of the P1-nonconforming quadrilateral element verified in plane elasticity problems and Stokes flow problems is investigated for the application of mitigating locking phenomena in shell problems. To facilitate the stiffness matrix computation for a non-flat general quadrilateral shell element, a nonparametric reference scheme using both affine transformation and bilinear transformation is adopted. Based on a field-consistency concept, the spurious constraints that cause locking are analyzed and an effective reduced integration scheme is found. The proposed shell element is applied to multi-physics topology optimization problems involving fluid analysis and shell analysis. For fluid analysis, the P1-nonconforming quadrilateral element is also adopted to utilize its volumetric locking-free property for incompressible materials. Vein layouts of leaves are designed by topology optimization and compared with natural vein layouts to verify the effectiveness of the proposed element.

면내회전자유도를 갖는 4절점 곡면 쉘요소

이 연구에서는 감절점쉘요소의 개념에 근거한 새로운 4절점 곡면 쉘요소를 제시하였다. 회전장이 독립변수로 도입된 범함수에 의하여 면내회전자유도를 도입함으로써 개발된 쉘요소에서는 절점당 6자유도를 갖도록 하였다. 아울러 쉘요소의 면내거동 개선을 위하여 4개의 비적합변위형에 의한 비적합변위를 면내방향의 변위성분에 추가하였으며, 면외거동 개선을 위하여 대체전단변형률장이 적용되었다. 이 연구에서의 비적합변위형의 수치적 구현에 있어서 일정한 변형률상태를 표현할 수 있도록 하기 위하여 비적합변위형의 직접 수정법이 적용되었다. 이렇게 정식화된 쉘요소 강성행렬의 수치적분에 있어서는 부피적분을 위하여 9점 적분법이 사용되었다. 개발된 쉘요소는 바람직하지 못한 영에너지모드를 갖지 않으며, 일정한 변형률 상태를 표현할 수 있음을 확인하였다. 개발된 4절점 곡면 쉘 요소에 대한 다양한 수치예제를 통한 검증 결과, 전반적으로 양호한 거동을 보여주고 있음을 확인하였다. In this study, a new 4-node general shell element with 6 DOFs per node is presented. Drilling rotational degrees of freedom are introduced by the variational principle with an independent rotation field. In formulation of the element, substitute transverse shear strain fields are used to avoid shear locking, while four nonconforming modes are applied in the in-plane displacement fields as a remedy for membrane locking. In addition, a direct modification method for nonconforming modes is employed in the numerical implementation of nonconforming modes to represent constant strain states. A 9-points integration rule is adopted for volume integration in the computation of the element stiffness matrix. With the combined use of these techniques, the developed shell element has no spurious zero energy modes, and can represent a constant strain state. Several numerical tests are carried out to evaluate the performance of the new element developed. The test results show that the behavior of the elements is satisfactory.

Towards improving the MITC6 triangular shell element

Effective triangular shell elements are of utmost interest in engineering practice, and the MITC6a element – a 6 node quadratic general shell element of the MITC family – has been shown to significantly reduce the locking phenomena arising in bending dominated behaviours. However, for some specific combinations of midsurface geometry and boundary conditions, the MITC6a element features some non-physical displacement modes with vanishing membrane strain energy. This phenomenon is thoroughly analyzed, and a remedy based on a stabilized bilinear form is proposed. Detailed numerical tests are included and the results demonstrate the good performance of the proposed method both for membrane and bending dominated problems.

The treatment of “pinching locking” in3D-shell elements

We consider a family of shell finite elements with quadratic displacements across the thickness. These elements are very attractive, but compared to standard general shell elements they face another source of numerical locking in addition to shear and membrane locking. This additional locking phenomenon - that we call ''pinching locking'' - is the subject of this paper and we analyse a numerical strategy designed to overcome this difficulty. Using a model problem in which only this specific source of locking is present, we are able to obtain error estimates independent of the thickness parameter, which shows that pinching locking is effectively treated. This is also confirmed by some numerical experiments of which we give an account.

Recent advances in local–global FE analysis of shells of revolution

A local–global finite element technique, suitable for the analysis of shells of revolution with localized non-axisymmetric effects such as cracks, cutouts and column supports, is presented. Both material and geometrical nonlinearities are considered. The model combines, in a single analysis, rotational shell elements, general shell elements, and column elements. Rotational shell elements are employed in the axisymmetric portion of the shell, where the nonaxisymmetric behaviour in loading and deformation is accounted for by including appropriate Fourier harmonics. In the local zone, where deviations from axisymmetry are contained, a general isoparametric shell element is employed. Continuity of displacements between the rotational and general shell elements is achieved either by a layer of transitional elements or by a direct coordinate transformation. If the shell is supported on a discrete system of columns , a standard 3-D beam element with six degrees-of-freedom per node is deployed. This local–global approach has been applied to a wide variety of shell problems.

Some new results and current challenges in the finite element analysis of shells

This article, a companion to the article by Philippe G. Ciarlet on the mathematical modelling of shells also in this issue of Acta Numerica, focuses on numerical issues raised by the analysis of shells.Finite element procedures are widely used in engineering practice to analyse the behaviour of shell structures. However, the concept of ‘shell finite element’ is still somewhat fuzzy, as it may correspond to very different ideas and techniques in various actual implementations. In particular, a significant distinction can be made between shell elements that are obtained via the discretization of shell models, and shell elements – such as the general shell elements – derived from 3D formulations using some kinematic assumptions, without the use of any shell theory. Our first objective in this paper is to give a unified perspective of these two families of shell elements. This is expected to be very useful as it paves the way for further thorough mathematical analyses of shell elements. A particularly important motivation for this is the understanding and treatment of the deficiencies associated with the analysis of thin shells (among which is the locking phenomenon). We then survey these deficiencies, in the framework of the asymptotic behaviour of shell models. We conclude the article by giving some detailed guidelines to numerically assess the performance of shell finite elements when faced with these pathological phenomena, which is essential for the design of improved procedures.

Optimal consistency errors for general shell elements

We obtain estimates for the consistency errors arising in general shell element procedures, which are widely used in engineering practice. These estimates improve a previous result by the same authors. Moreover, numerical experiments indicate that these new estimates are optimal. Further, we introduce a modified procedure for which nominal convergence is recovered. These results are of much practical significance.

A mixed formulation for general triangular isoparametric shell elements based on the degenerated solid approach

A mixed formulation to construct general shell elements based on the degenerated solid approach is presented. This mixed formulation can be understood as a modification of the discrete potential energy of the shell in which the membrane and shear energy terms are obtained as a combination of the usual terms, i.e., directly obtained from the displacement and geometry interpolation assumptions, and membrane and shear terms obtained from assumed strain fields. This energy splitting idea was first introduced in the context of the Naghdi shell theory by Arnold and Brezzi. This mixed formulation is used to study quadratic general triangular shell elements based on the degenerated solid approach. Although considerable success has been achieved in the formulation of reliable, free of locking, general quadrilateral shell elements, the attempts of formulating general triangular shell elements have not yet produced an element that satisfies the locking free requirements. Therefore, there is a great interest in investigating new approaches, as the one discussed in this paper, that can lead to triangular elements that do not suffer from locking difficulties. We present in this paper the formulation of specific elements and their numerical assessment through the use of well established benchmark shell problems.

The mathematical shell model underlying general shell elements

We show that although no actual mathematical shell model is explicitly used in ‘general shell element’ formulations, we can identify an implicit shell model underlying these finite element procedures. This ‘underlying model’ compares well with classical shell models since it displays the same asymptotic behaviours—when the thickness of the shell becomes very small—as, for example, the Naghdi model. Moreover, we substantiate the connection between general shell element procedures and this underlying model by mathematically proving a convergence result from the finite element solution to the solution of the model. Copyright © 2000 John Wiley & Sons, Ltd.

Performance of shell elements in modeling spot-welded joints

The numerical performance of general shell elements in modeling spot-welded joints is investigated. Finite element meshes composed of purely three-dimensional (3D) elements and purely general shell elements are used to analyze the stress and deformation fields in a symmetric coach-peel spot-welded specimen and their solutions are compared in detail. It is found that a properly refined finite element mesh of general shell elements can produce stress and deformation solutions comparable to those generated by a similarly refined mesh of 3D solid elements. The findings of this paper have direct implications to the analysis of sheet metal structures with spot-welded joints (such as automotive structures).

An evaluation of the MITC shell elements

Based on fundamental considerations for the finite element analysis of shells, we evaluate in the present paper the performance of the MITC general shell elements. We give the results obtained in the analysis of judiciously selected test problems and conclude that the elements are effective for general engineering applications.

A new one-point quadrature, general non-linear quadrilateral shell element with physical stabilization

A new one-point quadrature, general non-linear quadrilateral shell element with physical stabilization

A local–global FE model for nonlinear analysis of column-supported shells of revolution

A local–global finite element model suitable for the analysis of column-supported shells of revolution is presented. The model combines, in a single analysis, axisymmetric shell elements, general shell elements, and column elements. Axisymmetric or rotational shell elements which can accommodate geometric nonlinearities are employed in the axisymmetric portion of the shell. In this region, the nonaxisymmetric behavior in loading and deformation is accounted for by including appropriate Fourier harmonics. The column-supported area is modeled using individual column elements. The local zone is identified as the region consisting of the column elements and the physically and geometrically nonlinear general shell elements inserted between the column-support area and the axisymmetric shell portion. In addition, either deviations from the axisymmetric geometry of the shell, such as imperfections or cut-outs, and/or nonlinear materials can be easily included in the local zone. Examples which illustrate the proposed procedure are presented.

Dynamic and acoustic characteristics of bell type structure using finite element method

Dynamic and acoustic characteristics of the bell type structure were analyzed numerically. The finite element method with 3-D general shell element was used to identify the natural frequencies and mode shapes of the structure. Mode shapes and stress distributions of a transient dynamic analysis were effectively displayed by using computer graphic technique. The results of this numerical study were in good agreement with those obtained from the experimental test of fast Fourier transform analyer. Vibrational modes, which effect the acoustic characteristics of the typical bell-type structure were found to be the first flexural mode (4-0 mode) and the second flexural mode (6-0 mode). Asymmetric effects by Dangjwas and acoustic holes gave rise to beat frequencies, and the Dangjwa was found to be most effective. When the impact load was applied to the bell, the stress concentration occured at the rim part of the bell. It was found that the bell type structure should be designed thickly at the rim part in order to prevent failure from impact loads.