Flat Shell Element Research Articles

Analysis of the cylindrical shell model using a new four node quadrilateral element compared to ABAQUS elements

Shell models and loadings are very complicated in practice, especially in civil and mechanical engineering, due to their lightweight properties and ability to resist loads. Then; numerical tests are very important for this type of structure. In this work, numerical investigations of a cylindrical shell under concentrated loads are presented. Two finite elements used for this study, the new element ACM-Q4SBE1 and the S4R element of the code “ABAQUS”. The results obtained by the two elements, ACM-Q4SBE1 and S4R ABAQUS, are in agreement. The superiority and efficiency of the flat shell element approach for thin shell structures has been confirmed.

Development of nonlinear flat shell element with nonlinear thickness variation for highly flexible wind turbine blade

Large deformation from continuously varied thicknesses on existing and wind turbine blade shell structures is investigated. Various structure models having complicated outer shapes and thickness variations are considered and validated in this paper. A co-rotational method is developed to analyze geometric nonlinearity with a triangular shell structure. The updated stiffness matrix from interpolated thickness function and integral formulation is suggested to consider complicated geometry and thickness variation. Moreover, the co-rotational flat shell element based on the updated stiffness matrix is developed to analyze various nonlinear shell structures, which are the main feature of this research. The numerical analysis for structure deformations show well matched trends in static load analysis. Furthermore, the deformation of a simplified flexible wind turbine blade having tapered geometry and various thickness cases is adopted as a practical case. Moreover, modal analysis for composite beam structure is performed to verify the dynamic problems. The structural behaviors of the nonlinear thickness shell are validated against the solid and the shell models in commercial software, ABAQUS.

Geometrically nonlinear analysis of plates and shells by a cell-based smoothed CS-MITC18+ flat shell element with drilling degrees of freedom

A development of a 3-node triangular flat shell element to analyze geometric nonlinearity of plate and shell structures is presented in this study. The proposed flat shell element has 6 degrees of freedom per node, including the drilling rotational ones, by using the Allman-type approximations for the in-plane displacements and the bending displacement approximations enriched by a cubic shape function at the bubble node located at the element's centroid. The geometrically nonlinear behaviors are described by the von Karman's large deflection assumption. The membrane and bending strains of the presented flat shell elements are averaged over sub-triangular elements based on the cell-based smoothed (CS) technique. To attenuate the shear locking phenomenon, the transverse shear strains are re-interpolated following the mixed interpolation of tensorial components (MITC) technique designed for the 3-node triangular degenerated shell elements enriched by a cubic bubble function (MITC3+). Combined with the Newton-Raphson iterations and load increments predicted by the arc-length method, the suggested 3-node triangular flat shell elements, namely the CS-MITC18+ element, can analyze moderately geometrical nonlinearity, including the snap-thought and snap-back behaviors, of various plates and shells with the different shapes, thickness, and boundaries. The numerical investigations show that the load-displacement curves provided by the CS-MITC18+ flat shell elements well agree with other references.

A CS-MITC18+ flat shell element for static and frequency analyses of laminated composite plate and shell structures

A CS-MITC18+ flat shell element for static and frequency analyses of laminated composite plate and shell structures

Development of Higher Order Stiffened Shell Element (HOST9) for the Static Analysis of Stiffened Laminated Plates

Abstract A new higher order stiffened shell element (HOST9) is developed comprising of a 9 noded flat shell element and a 3 noded stiffener element for analysis of stiffened laminated plates. The shell element is a lagrangian element. The stiffener is a beam element. The stiffness of the stiffener is computed separately and added to that of the shell element at appropriate locations. As a result, the stiffener can be located arbitrarily in the shell element. Static analysis of stiffened laminated plate is carried out using the newly developed element and is validated with the existing literature. It is seen that the proposed element gives comparable results and can be used for the analysis.

Alternative Low-Order Triangular Elements for Thick and Thin Shell Analysis

In this paper, the authors discuss the development of six low-order triangular elements for the linear static analysis of thick and thin shell structures. The elements combine two triangular plane stress elements with three triangular thick plate bending elements to produce three-node flat shell elements with 18 degrees of freedom. The two plane stress components are based on the free formulation and the template formulation, while the three plate bending components are based on the free formulation, the deviatoric strain formulation and the hybrid-Trefftz formulation. The theoretical backgrounds and formulations of all elements are briefly discussed to provide a concise summary, while further details are given in the quoted references. The elements are used to analyze five standard shell benchmark examples, and the results obtained are compared to previous work in this area. The elements provide accurate and robust results for the benchmarks, with only small differences for the results as the meshes are refined. While all the proposed elements can be recommended for general shell analysis, the combination of the template plane stress element and the hybrid-Trefftz plate bending element generally provides the best results.

Linear composite curvature MITC3+ flat shell elements

This paper introduces a novel linear composite curvature MITC3+ flat shell element, named κMITC3+, that combines the advantages of the MITC3+ theory for flexural (bending and shear) behavior and the Allman-like triangular element (which ensures a true rotation of drilling degrees of freedom and eliminates a spurious energy mode) for membrane behavior. To enhance the bending behavior of the MITC3+ approach, we propose an assumed composite curvature field that is constructed using a polynomial projection technique derived from the orthogonality condition of the compatibility equation in the Hu-Washizu principle. Since two additional internal cross-section rotations are interpolated by cubic bubble functions, a linear polynomial function of the admissible space is chosen that provides a significant enhancement of bending behavior. The κMITC3+ element successfully passes all critical tests and exhibits superior performance compared to the original MITC3+ element and the recent CS-MITC18+ element.

Analysis of shell structures by an improved 3-node triangular flat shell element with a bubble function and cell-based strain smoothing

An enhanced 3-node triangular flat shell element with 18 degrees of freedom according to the Reissner–Mindlin theory is presented in this paper. New cell-based smoothed (CS) bending strains were derived from the displacement approximations enriched by a cubic shape function for the rotations at the bubble node located at the centroid of the triangular element. The true drilling degrees of freedom of the proposed element, which is independent from user-defined coefficients, was achieved by cell-based smoothing the Allman’s membrane strains. The membrane and bending stiffness matrices were then efficiently determined by the line integration of the smoothing domains’ boundaries. The shear-locking removal MITC3+ technique was employed to interpolate the transverse shear strains. The outstanding performance of the developed 3-node triangular flat shell element, so-called the CS-MITC18+ shell element, was demonstrated through the static and free vibration analyses of several benchmark shell structures. Numerical results shown that the CS-MITC18+ flat shell element has the spatially isotropic and zero-energy mode properties, and provides competitive accuracy, convergence, and computational cost in comparison to other conventional and strain-smoothed 3-node triangular flat shell elements.

Large Displacement Analysis With Material Nonlinear of Shell Structure Using New Flat Shell Elements

In this study, we propose a numerical algorithm for a nonlinear large displacement analysis method using flat shell elements in the hybrid-type penalty method (HPM). First, we describe the flat plate elements of HPM using the local coordinate system. Even with a flat plate, the adjacent elements have an angle at which a large displacement occurs. Next, we propose the relative displacement in this case. The “step-by-step method” is used as the algorithm for material nonlinear analysis and large displacement analysis. Finally, the accuracy of the solution of the proposed method is verified using a simple numerical example.

A four node quadrilateral shell element for free vibration response of functionally graded spherical shell panels

Abstract The present work involves the free vibration analysis of functionally graded spherical shells using a four-node quadrilateral shell element. This four-node, shell element has seven DOF (Degrees of Freedom) per node namely; three displacements, two rotations of mid-plane, and two transverse shear strain com-ponents. The element is developed by modifying the discrete Kirchhoff quadri-lateral shell element based on Reddy’s third-order theory developed earlier by the authors for the cylindrical shell. By transforming all the degrees of freedom from local to global coordinates using a transformation matrix converts the plate bending element into a flat shell element, resulting in the present formulation being suitable to give results for thin shells as well as for thick shells. In the present work functionally graded spherical shell panels with simply supported as well as clamped boundary condition, with various radius to span ratios and with different volume fraction indices are analyzed for free vibration response. The power law property variation through the thickness is considered in this study. To assess the performance of the developed element, the results, of non-dimen-sionalised frequencies are compared with the results presented in the literature. Comparison of results shows that the developed element yields quite accurate results even with the coarser mesh, which indicates the computational efficiency. It is also seen that the percentage difference between the present results and the other results available in the literature is less than 2 in most of the cases.

An edge center based strain-smoothing element with discrete shear gap for the analysis of Reissner–Mindlin shell

In this paper, an edge center based strain-smoothing element with discrete shear gap method (ECSSE-DSG) based on triangular Reissner–Mindlin flat shell element is proposed to solve shell problems. In the proposed ECSSE, the strains for three edge centers were calculated by the strain-smoothing method, which performed strain-smoothing operation directly on the three edges of a linear finite element instead of requiring an extra smoothing domain. In this way, not only could a linear strain field be constructed within an element, but strain consistency at the center of the element’s intersecting edges could also be ensured. The proposed ECSSE was adopted to manage membrane and bending strains. Meanwhile, the discrete shear gap (DSG) method was organically combined with the ECSSE for processing the shear strain to eliminate shear-locking. The ECSSE-DSG showed outstanding performance in a variety of tests, including those for plates, folded plates and curved shells; static, free vibration, dynamic, and buckling analyses; benchmark and engineering examples. High accuracy was obtained for both displacement and strain energy, there was no appearance of shear-locking, a smoother stress field was produced, and there was temporal stability. The excellent performance of the ECSSE-DSG when employed for engineering problems indicated its promising practical application prospects.

WITHDRAWN: A quadrilateral flat-shell element for the analysis of composite and sandwich cylindrical, globular and conical shell panels

This article has been withdrawn at the request of the author(s) and/or editor. The Publisher apologizes for any inconvenience this may cause. The full Elsevier Policy on Article Withdrawal can be found at https://www.elsevier.com/about/our-business/policies/article-withdrawal .

Static and free vibration analysis of sandwich shell panels using quadrilateral flat shell finite element

A quadrilateral flat-shell element with nine DOF (Degrees of Freedom) per node is developed for the static (deflections and stresses) and free vibration analysis (natural frequencies and accompanying mode shapes) of sandwich cylindrical and globular shell panels. The element is based on zigzag theory and is augmented using the discrete Kirchhoff quadrilateral bending plate element having seven DOF per node. It is vital to convert the rotations and translations degrees of freedom from local to global direction for the plate bending element to transform into a flat shell element. DKZigTS is developed by adding two fictitious degrees of freedom and has nine local and global DOF per node. The transformation matrix of size 36 × 36 is used for DKZigTS. It does not face the problem of the ill-conditioned stiffness matrix. The developed element is judged for its performance for simply supported and clamped shell panels. The DKZigTS element’s result are compared with results gotten using the 3 D finite element (FE) results of ANSYS (Mechanical APDL). To establish the confidence the comparison is also done with those 2 D analytical, 2 D numerical and 3 D elasticity results available in the literature. The numerical evaluation with DKZigTS element exhibits a very satisfactory performance for the static and free vibration response of moderately thick and thick sandwich spherical and cylindrical shell panels.

A quadrilateral flat-shell element for the static and dynamic analysis of composite and sandwich cylindrical, spherical and conical shell panels

Abstract A quadrilateral flat-shell element is developed for analysing the deflections, stresses and natural frequencies along with their allied mode shapes of cylindrical, spherical, and conical shell panels made up of layered composite and sandwich material. The developed element (DKZigTS1) is based on zigzag theory and has seven local as well as global DOF per node. The concept of obtaining transformation matrix is used for transforming actions and reactions from local to global direction to convert the plate bending into a flat-shell element. The two separate coordinate systems are used to transform rotational and translational degrees-of-freedom (DOF), from local to global direction. The local translational DOF are transformed to global Cartesian coordinates (x, y, z) and the local rotational DOF are transformed to the surface coordinate system (ξ 1, ξ 2, ξ 3), in which ξ 3 is perpendicular to the surface. The DKZigTS1 element gives fairly accurate results that align with the 2D analytical and the 3D elasticity solutions, reported in the literature for moderately thick and thick shell panel. The present results are also in good agreement with the 3D finite element solutions for shallow and deep shell panels having various material properties, boundary restrained environments, and geometrical shapes considered in this study.

Nonlinear analysis of reinforced concrete structures using thin flat shell elements

abstract: This paper presents the development of a nonlinear finite element analysis program for reinforced concrete structures, subject to monotonic loading, using thin flat shell finite elements. The element thickness is discretized in concrete and steel layers. It is adopted the Newton-Raphson method, considering a secant stiffness approach for the Material Nonlinear Analysis, based on the Modified Compression Field Model (MCFT), unlike the usual tangent stiffness approach. The original formulation was expanded to also consider the Geometric Nonlinear Analysis, through a Total Lagrangian Formulation. The program was validated through comparison with experimental results, for different structures. It was observed good agreement, besides adequate computational cost.

Numerical formulation for nonlinear analysis of concrete and steel shells with deformable connections

A two-dimensional interface finite element capable of associating flat shell elements positioned one above the other was developed. The implemented interface element can physically simulate the contact between the flat shell elements and connect the reference planes of the shell elements above and below it. The formulation presented allows consideration of nonlinear behavior for the deformable connection as well as for the concrete and steel materials that make up the shell structure. One of the practical applications analyzed in this research is the numerical simulation of composite floors formed by a reinforced concrete slab connected to steel beams through a deformable connection. In this case, the concrete slab and the steel beams are discretized by flat shell elements and the deformable connection is discretized by two-dimensional interface elements. Experimental and numerical results from literature were used to validate the implemented elements. In the two examples analyzed, the results obtained for the displacements were close, with the difference, in the first case, being associated with uncertainties during the experimental test and in the second, the difference in theories used in the formulation of the implemented elements.

Development of Flat Shell Elements for Arbitrary-shaped Shell Structures by using HPM

Development of Flat Shell Elements for Arbitrary-shaped Shell Structures by using HPM

A corotational mixed flat shell finite element for the efficient geometrically nonlinear analysis of laminated composite structures

AbstractA corotational flat shell element for the geometrically nonlinear analysis of laminated composite structures is presented. The element is obtained from the Hellinger–Reissner variational principle with assumed stress and displacement fields. The stress interpolation is derived from the linear elastic solution for symmetric composite materials. The element is isostatic, namely the stress interpolation is ruled by the minimum number of parameters. Displacement and rotation fields are only assumed along the contour of the element. As such, all the operators are efficiently obtained through analytical contour integration. The geometrical nonlinearity is introduced by means of a corotational formulation. The proposed finite element, named MISS‐4c, proves to be locking free and shows no rank defectiveness. A multimodal Koiter's algorithm is used to obtain the initial postbuckling response. Results show good accuracy and high convergence rate in the geometrically nonlinear analysis of composite shell structures.

Shape‐free arbitrary polygonal hybrid stress/displacement‐function flat shell element for linear and geometrically nonlinear analyses

AbstractA high‐performance shape‐free arbitrary polygonal hybrid stress/displacement‐function flat shell finite element method is proposed for linear and geometrically nonlinear analyses of shells. First, an arbitrary polygonal Mindlin–Reissner plate element and an arbitrary polygonal membrane element with drilling degrees of freedom are constructed based on hybrid displacement‐function and hybrid stress‐function methods, respectively. Both elements have only two corner nodes along each edge. Second, by assembling the plate and the membrane elements, an arbitrary polygonal flat shell element is constructed. Third, based on the corotational method, a proper best‐fit corotated frame for geometrically nonlinear polygonal elements is designed. By updating the analytical trial functions of shell element in each increment step, the original linear flat shell element is generalized to a geometrically nonlinear model. Numerical examples show that the new element possesses excellent performance for both linear and geometrically nonlinear analyses, and possesses outstanding flexibility in dealing with complex loading distributions and mesh shapes.

A Combined Strain Element in Static, Frequency and Buckling Analyses of Laminated Composite Plates and Shells

This paper deals with numerical analyses of laminated composite plate and shell structures using a new four-node quadrilateral flat shell element, namely SQ4C, based on the first-order shear deformation theory (FSDT) and a combined strain strategy. The main notion of the combined strain strategy is based on the combination of the membrane strain and shear strain related to tying points as well as bending strain with respect to cell-based smoothed finite element method. Many desirable characteristics and the enforcement of the SQ4C element are verified and proved through various numerical examples in static, frequency and buckling analyses of laminated composite plate and shell structures. Numerical results and comparison with other reference solutions suggest that the present element is accuracy, efficiency and removal of shear and membrane locking.