Quadrilateral Shell Element Research Articles

Deployment Simulation of Foldable Membranes Using Absolute Nodal Coordinate Formulation and Validation

It has become a trend to use foldable membranes to construct large spacecraft structures such as solar sails, antennas, and drag sails with the advantages of being lightweight and having high packing efficiency. To enhance the existing numerical simulation of membrane deployment, the nonlinear behavior of the crease is finely described based on experiments and integrated into the numerical model via the principle of virtual work. Specifically, a foldable membrane is modeled as a multibody system (MBS) based on the absolute nodal coordinate formulation (ANCF), where the flexible facet is meshed with both ANCF triangular and quadrilateral thin shell elements and the crease is treated as a virtual torsional spring with special constraints along the fold line. The MBS modeling method is validated via the deployment experiment of a Z-folding membrane. The deployment of a four-unit Miura-ori membrane is further analyzed to show the capability of the approach in modeling foldable membranes with complicated configurations. Good agreement is obtained on the membrane-deformed configurations between the simulation and experiment. Additionally, the driving force at the corners is obtained. This research is expected to provide a more accurate simulation to facilitate the design and optimization of the space deployable membrane structure.

Derivation and implementation of one-point quadrature quadrilateral shell element with MITC4+ method (MITC4+R)

In this work, a novel degenerated shell element called MITC4+R is developed, which is based on the MITC4+ (an improved MITC quadrilateral shell element) and reduced integration with one point quadrature. The assumed natural strain method is employed to construct the membrane and shear strains. A physical stabilization term is derivated after the assumed natural strains are decomposed into the linear and a constant term with respect to the natural coordinates. The hourglass modes owing to one point quadrature is controlled in the computation of deformation by introducing this physical stabilization term. The developed degenerated shell element is successfully implemented into the implicit static and explicit dynamic analyses. In order to verify its computational accuracy and efficiency, several standard benchmarks and practical problems are performed. These results show that the MITC4+R element eliminates various locking problems common to shell elements, and it significantly improves the computational efficiency of the original MITC4+ element. Meanwhile, the MITC4+R owns the same accuracy and highly consistent convergence as the original MITC4+ element, making it more promising for practical applications.

Towards composite suspension control arm: Conceptual design, structural analysis, laminate optimization, manufacturing, and experimental testing

The automotive industry needs composite materials to decrease the weight of new-generation vehicles whilst increasing their strength. In this study, one of the critical (load bearing) components of automobiles, i.e., the suspension control arm made of steel, is fully redesigned for its suitable manufacturing using composite materials. To this end, innovative mechanical simulation methods are developed and coupled to perform the design, analysis, and optimization of the automotive suspension control arm. The main design/optimization criteria are set to reduce no less than 75% weight of the metal control arm and increase its safety by at least 60% by using composite materials and a new geometry suitable for mass production. To predict the deformation-stress state of the control arm, a four-node quadrilateral shell element is implemented based on the kinematics of refined zigzag theory (RZT). Once verified numerically, the computer implementation of the RZT is combined with the optimization algorithm to achieve the optimum laminate stacking sequence of the control arm. Accordingly, prototypes of the composite control arms with optimum lamination plans are manufactured and then experimentally tested under the loading and constraint conditions defined at the conceptual design stage. The numerical and full-scale experimental results are compared, and the RZT models are comprehensively validated. Hence, the advantages of the overarching design-analysis-optimization strategy presented herein are revealed for redesigning and manufacturing automobile parts from composite materials.

A corotational isogeometric assumed natural strain shell element in updated lagrangian formulation for general geometric nonlinear analysis of thin-walled structures

A new general geometric nonlinear curved quadrilateral shell element, which can handle large displacements and large rotations, is presented in this paper. It is formulated based on the incremental virtual work equation in updated Lagrangian formulation for general large displacement analysis. The kinematics of shell element is described using non-uniform B-spline patch, which results in the element displacement field with higher order continuity and the element performance more resistive to shear locking and membrane locking. In order to perfectly avoid such membrane locking and shear locking behaviors, the assumed in-plane membrane strain field and the assumed transverse shear strain field for the incremental covariant Green-Lagrange strain are constructed as linear combinations of non-uniform B-spline basis functions with degrees lower than those adopted by the original displacement field interpolation. In addition, based on Euler parameters, the corotational procedure available for high order isogeometric shell element is developed to extract the strain-producing deformational displacements and rotations and to update the element stresses and realistic internal force vectors. The accuracy and robustness of proposed shell element are demonstrated through the solutions of various benchmark problems.

Vibration mode shifting due to scale-dependency for cylindrical, conical, and spherical microshells

Microshells structures are broadly utilized in micro-electro-mechanical systems (MEMS) such as microresonator, microgyroscope, microsensor, microactuator, microantenna, and microgripper. Taking into consideration the scale-dependency in such a scale and the importance of vibration characteristics in design of such microstructures, the current study aims to analyze the role of scale-decency in mode shifting of common types of thin microshells in MEMS. A scale-dependent quadrilateral finite shell element based on the modified couple stress theory is developed to compute the frequencies and mode shapes of the system. Verification studies are presented to display the accuracy of the established numerical model for the prediction of the vibrational response of microshells with and without scale-dependency. A comprehensive examination is then carried out to disclose the impact of material length scale parameters together with geometrical parameters and boundary conditions on the mode shifting of microscale cylindrical, conical, and spherical shells. It is believed that the results of the present study are handy in the design of the microshells when the scale-decency is of great importance.

A Universal Quadrilateral Shell Element for the Absolute Nodal Coordinate Formulation

Abstract A simple quadrilateral shell element is proposed in this work to study large deformations and large rotations of membrane/plate/shell structures. There are three merit characters in this element: locking-free; immune to mesh distortions; and robust to surface tessellations. Numerical issues in plates/shell elements such as shear-locking and thickness-locking problems are resolved, and quadrilateral area coordinates are adopted to solve the mesh distortion issues. This element can be adopted to curved shell structures, and warped deformations can be well described. Moreover, even if a shell structure cannot be easily tessellated by high quality quadrilateral polygons, it can still be discretized by a mesh consisting of high-quality triangular and quadrilateral elements, then this element can work together with a corresponding triangular element to provide accurate results on this combined mesh, and the degree-of-freedom for the discretized system is no more than several times of the number of nodes. Numerical tests validate the effectiveness, efficiency, and universality of this element in engineering scenarios.

Geometrically nonlinear analysis of functionally graded composite shells using MITC4 and MITC9 elements

In this paper, the geometrically nonlinear behaviors of functionally graded composite shells under large displacements and large rotations are analyzed. Instead of constructing conventional layer-wised models, equivalent single-layer composite shell elements based on general displacement fields are developed to model the inhomogeneity of composite materials. The mixed interpolation of tensorial components technique is used to eliminate membrane locking and shear locking phenomena. The assumed covariant strain fields are initially constructed before being used to evaluate the second Piola–Kirchhoff stress and construct stiffness matrices in local Cartesian coordinates. The standard full Newton–Raphson method is utilized to solve a system of nonlinear equations. Geometrically nonlinear analyses are performed on different thin-walled composite structures including thin cantilever beams, slit annular plates, pinched cylindrical shells, and hemispherical shells. Obtained results demonstrate good convergence characteristics and modeling capability of the developed quadrilateral composite shell elements in analyzing thin-walled composite structures.

A linear quadrilateral shell element for laminated composites

A linear quadrilateral shell element for laminated composites

Investigation on free vibration behavior of laminated angle ply shell with numerical validation

This article attempts an investigation on the free vibration behavior of the laminated composite angle ply shell experimentally and the obtained results are validated by laminated shell model based on modified higher order zigzag theory. The angle ply shell is modeled by placing six layers of 45° glass fiber laminae. The mechanical properties such as Poisson’s ratio, Young’s modulus and shear modulus are determined experimentally. The experimental procedure is performed using a vibration testing instrument by Bruel and Kjaer. The theoretical laminated shell is modeled by taking thickness coordinate to radius of curvature of shell ratio in the strain displacement relationship with shear-free conditions at free surfaces. For the theoretical computations of natural frequencies, a C0 finite element model with nine-noded quadrilateral shell element with seven degrees of freedom at each node has been chosen. Theoretical findings have been compared with experimental results and also with relevant published results in the existing literature. Moreover, extensive parametric studies have been performed to investigate the effect of geometric configurations and lamination schemes with different boundary properties on free vibration behavior of laminated shells, which will serve as benchmark solutions for future researchers.

Investigation on out-of-plane displacement measurements of thin films via a mechanical constraint-based 3D-DIC technique

Thin film is functioning as a key material in surface engineering, and accurate measurement of its deformation are particularly important for the reliability and stability assessment of coating-substrate structures. However, the displacement field measured by the conventional three-dimensional subset-based digital image correlation (3D-DIC) method has no spatial continuity; thus, it cannot precisely reflect the true deformation of thin films. To improve the measurement accuracy, a 3D-DIC method considering mechanical constraints (3DMC-DIC) was proposed in this paper. In this method, a mesh with 8-noded quadrilateral shell elements satisfying spatial continuity was employed to determine the out-of-plane displacements of all nodes based on trusted points, which are the speckle points with higher measurement accuracy obtained by the conventional subset-DIC method. The whole out-of-plane displacement field was then inversely obtained based on the optimized displacements of these nodes. A blister test was performed, and the results show that the out-of-plane displacement field of thin film with spatial continuity and high accuracy on specimen edge can be obtained by the 3DMC-DIC method.

Locking-free 6-noded triangular shell elements based on hierarchic optimisation

Conforming lower-order shell elements based on Reissner-Mindlin plate theory generally exhibit an over-stiff response under loading, typically manifested through various forms of locking. A recently developed hierarchic optimisation approach addresses locking by enriching the conforming strains with hierarchic strain terms towards an objective ‘smoother’ strain distribution afforded by the element, which has proven to be effective in relieving shear, membrane and distortion locking in 9-noded quadrilateral shell elements. Nevertheless, in some practical structural problems that involve complex geometry, triangular shell elements are required to avoid a highly distorted mesh of quadrilateral elements. This paper presents a family of 6-noded Reissner-Mindlin triangular shell elements based on the hierarchic optimisation approach. The proposed curved triangular shell elements not only effectively alleviate inaccuracies arising from locking, but also embrace the desirable characteristics of spatial isotropy and insensitivity to element distortion. The family of 6-noded triangular elements have been incorporated within a co-rotational framework to allow large displacement analysis of thin to moderately thick plates and shells. Several numerical examples are finally presented to demonstrate the effectiveness and accuracy of the proposed 6-noded shell element formulation as well as its superior locking-free performance compared to existing shell elements. • Novel family of 6-noded locking-free curved shell elements. • Mitigation of shear, membrane and distortion locking via hierarchic optimisation. • Objective and corrective assumed-strain variants including spatial isotropy. • Incorporation within co-rotational framework for large displacement analysis. • Element variant H3O6 offers superior performance compared to MITC6 element.

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.

Finite Element Modelling for Bending and Vibration Analysis of Composite and Sandwich Spherical Shells

Finite element modelling is used to illustrate the bending and vibration response of composite and sandwich spherical shells. ABAQUS is used to model the shell as a three-dimensional deformable solid part. Composites are modelled using mixed modelling technique. The quadrilateral continuum shell element, which has eight nodes, is used for finite element meshing. For a three-layered composite spherical shell with simply supported boundary conditions, the accuracy of the employed element is first evaluated. After that, three potential end conditions are evaluated for further analysis: cantilever (CFFF), two opposing sides clamped (CFCF), and all sides clamped (CCCC). Bending response is produced by applying two types of pressure loads to the exterior surfaces of composite and sandwich spherical shells: uniformly distributed load (UDL) and sinusoidal load (SINO). For various span to thickness ratios, free vibration response and forced vibration response (Time History) are provided under step pressure (UDL and SINO) loading. The element type being used is found to be quite precise and sturdy.

Thermo-mechanical analysis of laminated composites shells exposed to fire

This paper describes the research performed within the scope of H2020 project FIBRESHIP in the development and validation of a thermo-mechanical model to assess the fire performance of composite structures. A one-dimensional thermal model with pyrolysis is used to obtain the temperature profile across the thickness, and later, introduced in the thermo-mechanical model with a quadrilateral shell element approach. The composite constitutive model employed is the so-called Serial/Parallel Rule of Mixtures (SPROM) which has been modified to introduce the effect of the thermal deformation. A set of experimental tests are then used to validate the correctness of the numerical method proposed. The experimental data used to validate the thermal model is the classic Henderson experimental test. The thermo-mechanical coupling is validated against an original vertical furnace test of an FRP ship’s bulkhead, following on the 2010 FTP Code standards. These validations demonstrate the correctness and accuracy of the proposed decoupled thermo-mechanical formulation.

Study on the Vector Quadrilateral Isoparametric Shell Element with Shear Deformation

Based on the linear superposition of the quadrilateral isoparametric membrane element and quadrilateral Mindlin plate element, basic vector formulas for the quadrilateral isoparametric shell element with the shear deformation is derived. Two different coordinate modes for updating the particle displacement and the element node internal force are put forward. The uniform numerical solution for an irregular quadrilateral element is realized by the isoparametric integral scheme along the element plane. Then, the bilinear elastoplastic constitutive of material is introduced into the vector finite shell element, and the nonlinear effect of the material is realized based on the integral method along the thickness. On this basis, the nonlinear calculation and analysis program of the quadrilateral isoparametric shell element is developed. Numerical example results show that the static and dynamic analysis, large deformation and large rotation analysis, and buckling analysis can all be well performed for the shell structures by the developed program, which verifies the validity of theoretical derivation and computer program.

Level set-based topology optimization for compliance and stress minimization of shell structures using trimmed quadrilateral shell meshes

In this paper, a novel level set-based topology optimization method for shell structures is presented to minimize both compliance and stress under a volume constraint. During the optimization process, trimmed quadrilateral shell meshes are generated by cutting a background quadrilateral shell mesh with the zero-isolines of a level set function. Polygonal shell elements with assumed strains are used for the trimmed quadrilateral shell elements created along the boundaries of shell structures. A new sensitivity field for the evolution of a level set function is proposed to minimize both compliance and stress of shell structures. A p -norm function is employed to aggregate local stresses into a single global stress measure in the sensitivity analysis. Several numerical examples are carried out to show the validity and effectiveness of the proposed method.

Static and dynamic analyses of FGPM cylindrical shells with quadratic thermal gradient distribution

Functionally graded material (FGM) has great potential in the application of thin-walled structures, since delamination failure can be suppressed. Functionally graded piezoelectric material (FGPM) is a special kind of FGM, which couples with elastic and electric fields. The modeling of FGPM coupled smart structures under nonlinear thermal load is still a challenge task. This paper develops an electro-thermo-elastic finite element (FE) model of FGPM integrated smart structures based on the first-order shear deformation hypothesis. The FE model proposes two different nonlinear configurations of thermal gradient distribution through the thickness, which are compared with linear thermal gradient distribution. An eight-node quadrilateral shell element is developed for finite element analysis. The present FE model is firstly validated by a PVDF bimorph beam under nonlinear thermal loads. Later, the model is applied to the static and dynamic simulation of FGPM cylindrical shells. The results show that thermal distribution configurations have great impact on structural response, which should be carefully chosen in the simulation of thermal coupled problems.

Analytical study of the sectional behavior and the effective width of T-shaped reinforced concrete walls

In this article, the sectional behavior (strain and stress) and the effective width in T-shaped reinforced concrete wall are analytically studied. For the parametric study, a nonlinear quadrilateral thin flat layered shell element with 6 degrees of freedom (DOF) per node for the modeling of reinforced concrete wall structures developed by Rojas et al. (2019) is used, varying the axial load (AL), longitudinal steel ratio of the web (LSRW) and flange (LSRF) boundary, aspect ratio (AR) relative to the length of the web of the wall (lw) and height of the wall (hw), and length of the flange (Lf). The parametric study is used to compare the element response that couple the axial, flexion and shear behavior, with a model that only incorporates the coupling of flexion and axial, maintaining plane sections (flexural model). Based on the parametric study, it is calibrated an analytic model of the strain profile of the tension flange of T-shaped walls. Additionally, by including the effect of shear in the model, amplification of the maximum compression strain and a reduction of the maximum tensile strain in the web is observed, mainly in squat walls. Besides, the parametric study is also used to calculate the effective width of T-shaped walls with the flange in tension, finding that it depends mainly on the length of the flange and the roof drift. After initiation of yielding in the flange, the effective width grows quickly, concluding that independent of the geometry and the slenderness, the effective width of the flange in tension covers the entire section if yielding is expected.

Free vibration analysis of laminated composite hybrid and GFRP shells based on higher order zigzag theory with experimental validation

Free vibration response behavior of laminated composite hybrid and Glass Fiber Reinforced Plastic (GFRP) shells is analyzed using a modified higher order zigzag theory (HOZT). The laminated shell model is developed by considering the thickness co-ordinate to radius of curvature ratio (z/R) as well as zero shear conditions at free surfaces of the shell. The C0 finite element formulations with nine-noded quadrilateral isoparametric shell element having seven degrees of freedom per node are adopted to compute the natural frequency of the shells. The hybrid laminate is modeled by placing carbon fibers in the outermost layers and glass fibers in the rest to improve the flexural behavior of the shell. Experimental studies are performed to find out the mechanical properties of hybrid and GFRP laminates such as Young modulus, shear modulus and poison's ratio. The experimental procedure of free vibration is carried out by Bruel & Kjaer (B&K) modal analysis instruments along with RT Pro modal analysis software and the results are compared with the theoretical findings. Moreover the present theoretical results have been examined by comparing with 2D solutions as well as 3D elasticity solutions available in the published literature to verify the accuracy of the present HOZT. An extensive parametric study on laminated composite hybrid and GFRP shells with various laminations is undertaken to investigate the effect of reinforcement scheme on the free vibration behavior of such laminated shells.

Topology optimization of ultra high resolution shell structures

This work presents a high performance computing framework for ultra large scale, shell-element based topology optimization. The shell elements are formulated using a linear elastic, small strain assumption and are of the solid type, meaning that each quadrilateral shell element is extruded and assigned 24 degrees of freedom. The resulting linear system is solved using a fully parallelized multigrid preconditioned Krylov method, tailored specifically for unstructured quadrilateral shell meshes. The multigrid approach is shown to have good parallel scaling properties and is able to efficiently handle the ill-conditioning arising from the ‘Solid Interpolation of Material Properties’ (SIMP) method. For the optimization, the classical minimum compliance design problem with multiple load cases, prescribed minimum length scale and a local volume constraint is investigated. The latter is implemented through efficient PDE-filtering in contrast to usual local image filtering based implementations. Finally, the framework is demonstrated on two idealized examples from civil and aerospace engineering, solving shell optimization problems with up to 11 million shell elements on 800 cores. As an example, this resolution corresponds to a minimum feature size of 1.5 cm on a high-riser of height 80 m.