Triangular Shell Finite Element Research Articles

Çeşitli Kenar Sınır Koşullarına Sahip İnce İzotropik Düz Plakaların ve Eğri Plakaların Panel Çarpıntı Sayısal Çalışması

In this article, supersonic panel flutter analysis of flat plates and curved plates with different edge boundary conditions are studied, using efficient, high precision triangular shallow shell finite elements. The fluid on the underside of the plate was is assumed to be stationary. The linear piston theory can be applied to the top surface of the plate. The linear piston theory was used to evaluate the aerodynamic loads. The solution of a complex eigenvalue problem was formulated according to Hamilton’s principle. Lagrange’s equation of motion was obtained using standard methods for finding eigenvalues. Current finite element analysis ignores aerodynamic damping. For panels, the theory of thin and small deformed shells was taken into account. To validate the developed finite element code, the results of a square and rectangular flat-panels with simply supported edges (S-S-S-S), a square plate with four fixed edges (C-C-C-C), and a square plate with the length side clamps (C-S-C-S) were compared with the published data. The flutter results of other edge boundary conditions (S-C-S-C, C-S-C-S, and C-C-C-C) for square and rectangular flat panels are evaluated for which literature data is limited. It has been found that the fixed condition in the cross-flow direction (S-C-S-C) has a significant effect on the critical flutter pressure parameters and flutter frequencies. Further, to study the aforementioned effect, the current finite element (FE) has been extended to curved plates with S-C-S-C(constrained in the cross-flow direction and exposed to supersonic flow), SS-S-S boundary conditions to find flutter results.

Explicit dynamics of shells with a flat‐facet triangular finite element

AbstractThe use of a flat‐facet finite element with three nodes and six degrees of freedom per node for modeling of linear wave‐propagation events is discussed. The triangular shell finite element was presented in a preceding publication. The main novelty of the current approach is the treatment of the drilling rotations that decouples the drilling degree of freedoms on the global level, and hence makes it possible to eliminate negative effects of these degrees of freedom on the explicit time‐stepping algorithms employed to compute the dynamic response, such as deterioration in accuracy or artificially reduced time step. The element is shown to be robust for wave propagation simulations, and its performance is illustrated with examples including guided‐wave scenarios.

A co-rotational triangular finite element for large deformation analysis of smooth, folded and multi-shells

A six-node co-rotational curved triangular shell finite element with a novel rotation treatment for folded and multi-shell structures is presented. Different from other co-rotational triangular element formulations, rotations are not represented by axial (pseudo) vectors, but by components of polar (proper) vectors, of which additivity and commutativity lead to symmetry of the tangent stiffness matrices in both local and global coordinate systems. In the co-rotational local coordinate system, the two smallest components of the shell director are defined as the nodal rotational variables. Similarly, the two smallest components of each director in the global coordinate system are adopted as the global rotational variables for nodes located either on smooth shells or away from non-smooth shell intersections. At intersections of folded and multi-shells, global rotational variables are defined as three selected components of an orthogonal triad initially oriented along the global coordinate system axes. As such, the vectorial rotational variables enable simple additive update of all nodal variables in an incremental-iterative procedure, resulting in significant enhancement in computational efficiency for large deformation analysis. To alleviate membrane and shear locking phenomena, an assumed strain method is employed in obtaining the element tangent stiffness matrices and the internal force vector. The effectiveness of the presented co-rotational triangular shell element formulation is verified by analyzing several benchmark problems of smooth, folded and multi-shell structures undergoing large displacements and large rotations.

The use of the strain approach to develop a new consistent triangular thin flat shell finite element with drilling rotation

Dans la presente communication, nous presentons un nouvel element fini a facette plane. Il est le resultat de lacombinaison d'un element de membrane et d'un element de plaque en flexion, tous deux bases sur l'approche endeformation. Nous avouons que les elements classiques de membrane de classe C° fournissent une deflexion et unecontrainte mediocre pour les problemes ou le mode de flexion est dominant. De plus, ils rencontrent des problemes decontinuite et de conformite lorsqu'ils sont connectes a des elements de flexion (poutre et plaque) de la classe C1. L'objectifde la presente communication est de surmonter ces problemes lorsqu'un element de membrane est couple a un element deplaque mince afin de construire un element de coque. L'element membranaire utilise est un element triangulaire a quatrenoeuds, trois noeuds aux sommets du triangle et le quatrieme en son centre. Chaque noeud possede trois degres de liberte,deux translations et une rotation autour de la normale (Drilling rotation). Les coefficients lies aux degres de liberte du noeudinterne sont supprimes de la matrice de raideur elementaire en utilisant la technique de condensation statique. Les fonctionsd'interpolation des champs de deformations, de deplacements et de contraintes sont etablies a partir des conditionsd'equilibre. L'element de plaque utilise pour la construction du present element de coque est un element de plaque mincetriangulaire a quatre noeuds base sur : la theorie des plaques minces de Kirchhoff, l'approche en deformation, le quatriemenoeud fictif, la condensation statique et l'integration analytique. L’element de coque resultat de cette combinaison estrobuste, competitif et efficace. MOTS-CLES : coque, plaque, modele en deformation, condensation, drilling rotation

The use of the strain approach to develop a new consistent triangular thin flat shell finite element with drilling rotation

In the present paper, we offer a new flat shell finite element. It is the result of the combination of a membrane element and a bending element, both based on the strain-based formulation. It is known that Co plane membrane elements provide poor deflection and stress for problems where bending is dominant. In addition, they encounter continuity and compliance problems when they connect to C1 class plate elements. The reach of the present work is to surmount these problems when a membrane element is coupled with a thin plate element in order to construct a shell element. The membrane element used is a triangular element with four nodes, three nodes at the vertices of the triangle and the fourth one at its barycenter. Each node has three degrees of freedom, two translations and one rotation around the normal. The coefficients related to the degrees of freedom at the internal node are subsequently removed from the element stiffness matrix by using the static condensation technique. The interpolation functions of strain, displacements and stresses fields are developed from equilibrium conditions. The plate element used for the construction of the present shell element is a triangular four-node thin plate element based on Kirchhoff plate theory, the strain approach, the four fictitious node, the static condensation and the analytic integration. The shell element result of this combination is robust, competitive and efficient.

A consistent triangular thin flat shell finite element with drilling rotation based on the strain approach

The paper offers a new flat shell finite element. It results from the combination of a membrane and a bending elements, both founded on the strain based formulation. The used membrane element is the T43_Eq. It is a triangular element with four nodes, three nodes at the vertices and the fourth one at the barycenter. Each node has three degrees of freedom, two translations and one rotation around the normal. The coefficients related to the degrees of freedom at the internal node are subsequently removed from the element stiffness matrix by using the static condensation procedure. The interpolation functions of strain, displacements and stresses fields are developed from equilibrium conditions. Then, the plate element used for the construction of the shell element is a triangular four-node thin plate element based on: the deformation approach, the four fictitious node, the static condensation and the analytic integration. The shell element result of this combination is robust, competitive and efficient. The shell element result of this combination is robust, competitive and efficient.

A consistent triangular thin flat shell finite element with drilling rotation based on the strain approach

The paper offers a new flat shell finite element. It results from the combination of a membrane and a bending elements, both founded on the strain based formulation. The used membrane element is the T43_Eq. It is a triangular element with four nodes, three nodes at the vertices and the fourth one at the barycenter. Each node has three degrees of freedom, two translations and one rotation around the normal. The coefficients related to the degrees of freedom at the internal node are subsequently removed from the element stiffness matrix by using the static condensation procedure. The interpolation functions of strain, displacements and stresses fields are developed from equilibrium conditions. Then, the plate element used for the construction of the shell element is a triangular four-node thin plate element based on: the deformation approach, the four fictitious node, the static condensation and the analytic integration. The shell element result of this combination is robust, competitive and efficient. The shell element result of this combination is robust, competitive and efficient.

NUMERICAL MODELING OF THE BUCKLING RESISTANCE OF RULED HELICOIDAL SHELLS

The paper concerns the buckling analysis of thin shells of right helicoid form. The buckling analysis was performed by the means of finite element software. Shells with variable pitch number and same contour radiuses and height were compared, their straight edges fixed and the curvilinear contours free. Was used for the analysis triangular shell finite elements (No. 42). The total number of nodal unknowns was the same in each of the considered tasks and was 16 206. Numerical investigation of the stability was performed by the finite element method in the software package Lira-Sapr 2017. The number of nodes in each task was the same. The loading includes combination of gravity (dead load) and vertical equally distributed load. The buckling mode and stability factor for every case is calculated. Boundary conditions - elastic built in shells along the bottom and top generatrices. To plot the midsurface of each shell were used parametric equations in rectangular coordinates. Of particular interest is the study of natural oscillations of the shells considered. To define the frequencies and forms of free vibrations is taken into account only the own weight of the helicoidal shells.

A corotational triangular facet shell element for geometrically nonlinear analysis of thin piezoactuated structures

A simple and effective corotational triangular facet shell finite element is proposed for the geometric nonlinear analysis of thin piezoactuated structures. The structures under investigation are laminated shells composed of elastic layers and sensory/active piezoelectric layers, perfectly bonded to each other. A polar decomposition based corotational framework is adopted. By filtering out large rigid body motions from structural displacements, this framework allows (i) to account for arbitrarily large displacements/rotations, provided strains are small, and (ii) to use existing and high-performance linear elements as core-elements. Within the classical laminated plate theory, the small strain core element formulation based on superposition of OPT membrane and DKT plate element is used, along with a layer-wise constant interpolation of the transversal component of the electric field. Numerical simulations dealing with benchmark problems and applications of technological interest prove accuracy and effectiveness of the proposed formulation.

Strain-Based Finite Element Analysis of Stiffened Cylindrical Shell Roof

A new triangular cylindrical shell finite element is used to obtain an elastic linear analysis of a cylindrical shell roof. The element’s displacement fields are in terms of curvilinear coordinates, satisfy the exact requirement of rigid body modes of deformation and have five degrees of freedom at each of the three corner nodes. The efficiency of the developed element is first tested and then applied to analyze stiffened cylindrical shell roof. The results of the displacement and stress resultant along the stiffener are presented and a parametric study is carried out to find the effect of the varying geometry of the stiffener and the effect of axis rotation of the beam on the level of the internal forces.

Nonlinear dynamic analysis of creased shells

Recent studies analyze the behavior of advanced shell structures, like foldable, multistable or morphing shell structures. Simulating a thin foldable curved structure is not a trivial task: the structure may go through many snapping transitions from a stable configuration to another. Then, one could claim arc-length methods or use a dynamic approach to perform such simulations. This work presents a geometrically exact shell model for nonlinear dynamic analysis of shells. An updated Lagrangian framework is used for describing kinematics. Several numerical examples of folding a thin dome are presented, including creased shells. The triangular shell finite element used offers great flexibility for the generation of the unstructured curved meshes, as well as great results.

The MITC3+ shell element and its performance

In this paper, we present an effective new 3-node triangular shell finite element, called the MITC3+ element. The new shell element is based on the concepts earlier published for the MITC3 shell element (Lee and Bathe, 2004) [1] but is enriched by a cubic bubble function for the rotations. A new assumed transverse shear strain field is developed for the element. The shell element passes the three basic tests (the isotropy, patch and zero energy mode tests) and shows excellent convergence behavior in basic and encompassing convergence tests.

The MITC3 shell finite element enriched by interpolation covers

In this paper, we develop a scheme to enrich the 3-node triangular MITC shell finite element by interpolation cover functions. The MITC method is used for the standard and enriched displacement interpolations. The enriched 3-node shell finite element not only captures higher gradients but also decreases inter-elemental stress jumps. In particular, the enrichment scheme increases the solution accuracy without any traditional local mesh refinement. Convergence studies considering a fully clamped square plate problem, cylindrical shell problems, and hyperboloid shell problems demonstrate the good predictive capability of the enriched MITC3 shell finite element, even when distorted meshes are used. We evaluate the effectiveness of the method, and also illustrate the use of the enrichment scheme applied only locally through the solution of two additional shell problems: a shaft–shaft interaction problem and a monster shell problem.

Buckling analysis of I-section portal frames with stochastic imperfections

This work examines the effect of random geometric imperfections in the buckling response of I-profile steel beam–column members as well as portal frame structures. Geometric imperfections are assumed to be non-homogeneous Gaussian random fields. Samples of these fields are generated using the spectral representation method with evolutionary power spectra derived from experimental measurements using the method of separation. A number of samples of random imperfect geometries are generated and simulated with detailed discretization with triangular shell finite elements. The stochastic buckling loads are determined in the context of brute force Monte Carlo simulation by repeated material and geometrically nonlinear finite element analyses. Single beam–column members are subjected to pure axial compression while the portal frames are tested for lateral loading. Various types of boundary conditions are implemented and histograms of bucking loads are derived for each case examined leading to useful conclusions for this type of structures, regarding imperfection triggered buckling response and buckling load variability.

Analysis of multibody systems with flexible plates using variational graph-theoretic methods

This work treats the problem of modelling multibody systems with structural flexibility. By combining linear graph theory with the principle of virtual work and finite elements, a dynamic formulation is obtained that extends graph-theoretic (GT) modelling methods to the analysis of thin flexible plates for multibody systems. The system is represented by a linear graph, in which nodes represent reference frames on flexible plates, and edges represent components that connect these frames. To generate the equations of motion with elastic deformations, the flexible plates are discretized using a triangular thin shell finite element based on the discrete Kirchhoff criterion and can be used to discretize bidirectional bodies such as satellite panels, flatbed trailers, and mechanisms with plates. Three flexible systems with plates are analyzed to illustrate the performance of this new variational graph-theoretic formulation and its ability to generate directly a set of motion equations for flexible multibody systems (FMS) without additional user input.

Large Nonlinear Responses of Spatially Nonhomogeneous Stochastic Shell Structures Under Nonstationary Random Excitations

A novel approach for determining large nonlinear responses of spatially homogeneous and nonhomogeneous stochastic shell structures under intensive transient excitations is presented. The intensive transient excitations are modeled as combinations of deterministic and nonstationary random excitations. The emphases are on (i) spatially nonhomogeneous and homogeneous stochastic shell structures with large spatial variations, (ii) large nonlinear responses with finite strains and finite rotations, (iii) intensive deterministic and nonstationary random disturbances, and (iv) the large responses of a specific spherical cap under intensive apex nonstationary random disturbance. The shell structures are approximated by the lower order mixed or hybrid strain based triangular shell finite elements developed earlier by the author and his associate. The novel approach consists of the stochastic central difference method, time coordinate transformation, and modified adaptive time schemes. Computed results of a temporally and spatially stochastic shell structure are presented. Computationally, the procedure is very efficient compared with those entirely or partially based on the Monte Carlo simulation, and it is free from the limitations associated with those employing the perturbation approximation techniques, such as the so-called stochastic finite element or probabilistic finite element method. The computed results obtained and those presented demonstrate that the approach is simple and easy to apply.

Detailed reliability assessment of triangular MITC elements for thin shells

Effective shell finite elements are of utmost interest in engineering practice, and more particularly triangular shell elements are often needed to handle complex geometries. The quadrilateral elements of the MITC family have been shown to be effective for general asymptotic behaviors, but a triangular MITC shell finite element of comparable performance remains to be found. The main difficulty is to reduce the locking phenomenon arising in bending dominated situations without destroying the ability of the procedure to accurately capture membrane dominated and mixed behaviors. In this context, we have designed a simple numerical rank test aimed at assessing the reliability of MITC triangular elements. This test led to selecting some specific MITC6 tying schemes, which have been further assessed using some meaningful test problems. The assessment results showed that the MITC6a shell element is a most promising candidate, although it may feature some non-physical membrane spurious modes in some specific shell problems.

Optimal control of random vibration in plate and shell structures with distributed piezoelectric components

The state covariance assignment (SCA) method of Skelton and associates has been extended to the optimal random vibration control of large-scale complicated shell structures with distributed piezoelectric components and under nonstationary random excitations. It provides a direct approach for achieving performance goals stated in terms of the root-mean-square (RMS) values which are common in many engineering system designs. The large-scale shell structures with distributed piezoelectric components of complicated geometrical configurations are approximated by the hybrid strain or mixed formulation based lower order triangular shell finite elements developed in the present investigation. Each of these shell finite elements has three nodes. Every node has seven degrees of freedom (dof) which include three translational, three rotational and one electric potential dof. These elements are better alternatives to those based on the displacement formulation and that hinged on the truly hybrid strain formulation in that they eliminate locking problems commonly associated with displacement based lower order shell finite elements. Computed results applying the proposed approach for a plate and a cylindrical shell structures with distributed piezoelectric components and under stationary random excitations are included to demonstrate its simplicity of use and efficiency of implementing the proposed approach.

Insight into 3-node triangular shell finite elements: the effects of element isotropy and mesh patterns

In this paper, we study the convergence characteristics of some 3-node triangular shell finite elements. We review the formulations of three different isotropic 3-node elements and one non-isotropic 3-node element. We analyze a clamped plate problem and a hyperboloid shell problem using various mesh topologies and present the convergence curves using the s-norm. Considering simple bending tests, we also study the transverse shear strain fields of the shell finite elements. The results and insight given are valuable for the proper use and the further development of triangular shell finite elements.

Geometrically nonlinear analysis of shell structures using a flat triangular shell finite element

This paper presents a state of the art review on geometrically nonlinear analysis of shell structures that is limited to the co-rotational approach and to flat triangular shell finite elements. These shell elements are built up from flat triangular membranes and plates. We propose an element comprised of the constant strain triangle (CST) membrane element and the discrete Kirchhoff (DKT) plate element and describe its formulation while stressing two main issues: the derivation of the geometric stiffness matrix and the isolation of the rigid body motion from the total deformations. We further use it to solve a broad class of problems from the literature to validate its use.