Present Shell Element Research Articles

A weak form quadrature element formulation of geometrically exact strain gradient shells

A geometrically exact strain gradient shell model and its weak form quadrature element formulation are proposed. The governing equations, boundary conditions and corresponding variational form of the shell model are presented. The generalized differential quadrature analogue is employed for derivative approximations of displacements and rotations with C1 continuity conditions brought about by the second-order strain gradient terms. A novel scheme based on auxiliary vectors is developed to represent and update the derivatives of constrained rotations on element edges for the numerical approach. The present quadrature shell element is feasible to incorporate strain gradient terms and free from shear and membrane locking problems due to its adjustable high-order approximation property in nonlinear shell analysis. To demonstrate the potential of this formulation for nonlinear analysis of shells with size effects, five benchmark numerical examples are presented. The results verify the feasibility of this formulation and illustrate the strain gradient effects on elastic shells undergoing large displacements and rotations.

Vibration characteristics of plates and shells with functionally graded pores imperfections using an enhanced finite shell element

The novelty of the current study is the development of an enhanced finite shell model using the first order shear deformation theory (FSDT) to study the vibration characteristics of imperfect functionally graded (FGM) plates and shells including porosities. Unlike standard (FSDT) theory, the proposed model presents an improvement of the FSDT via the introduction of a quadratic shear function allowing the exact representation of the parabolic distribution of the shear strains and avoiding the utilization of shear correction factors which are computationally costs. The finite element method is used for the discretization of the equations of motion and the assumed natural transverse shear strain (ANS) method is employed to avoid shear locking problems. The effect of porosity parameter and its distributions, namely even and uneven forms, is investigated through different numerical illustrative examples. The present finite shell element model provides better agreement with exact solutions in comparison by other theories. Finally, effects of different factors on vibrational behavior of imperfect FGM porous plates and shells are examined in detail.

A higher-order quadrilateral shell finite element for geometrically nonlinear analysis

In this paper, we present a geometrically nonlinear formulation for a nine-node shell finite element and demonstrate its performance. The total Lagrangian formulation is utilized to allow for large displacements and large rotations. The mixed interpolation of tensorial components (MITC) technique is applied to effectively reduce the membrane and shear locking phenomena without refining mesh, introducing additional nodes, and using reduced integrations. Many-core accelerators with GPU-compatible libraries are used to achieve the highest speed-up for the solution process. The performance of the present nine-node shell element is compared with those of the four-node and six-node shell elements in several benchmark problems, including Cook's skew beam, cantilever plate, pinched cylindrical shell, slit annular plate, and hemisphere shell. The present element achieves excellent performance even when the coarse mesh is used.

Thermal free vibration analysis of functionally graded plates and panels with an improved finite shell element

This article deals with thermal free vibration analysis of functionally graded material (FGM) plates and panels using an improved first-order shear deformable (I-FSDT) shell model. The Effective material properties of the FGMs are estimated according to a power law distribution with temperature dependency. The FGM structure can be subjected to uniform or nonlinear temperature rise. The results obtained from the proposed formulation are validated with those available in the literature. Natural frequencies of FGM plates and cylindrical panels are computed for different power law indexes, thermal loadings, boundary conditions and aspect ratios. The present (I-FSDT) finite shell element model is much like the standard (FSDT) model but does not need shear correction factors and is of high accuracy. This work is the first application of the formulation to free vibration analysis of FG plate and shell structures including thermal effects.

Residual capacity of cold-formed steel rack uprights following stomping during rocking

When selective pallet racks are allowed to rock in the cross-aisle direction during an earthquake, the uprights are subjected to short duration high axial forces at stomping. In this paper, the amplitude of the stomping force needed to compromise the upright's residual capacity is assessed for 59 configurations using nonlinear inelastic static and dynamic analyses. Parametric studies are performed to investigate the effects of upright length, bracing pitch, section slenderness, torsional restraints and multiple impulses on the residual capacity of an upright due to rocking. Uprights that fail in the flexural-torsional buckling mode perform better than those that fail by local-distortional buckling as the stomping causes permanent local-distortional deformations rather than sweep (torsional deformation). A rack upright that has a greater length, greater thickness and lower torsional restraint tends to have a higher residual capacity (relative to the undamaged capacity). A typical cold-formed steel rack upright can sustain a 0.1 s stomping force that is at least 15% greater than its static ultimate capacity without significant reduction in residual capacity. An implication is that an unanchored upright that survives an earthquake through rocking may double its storage load during the post-earthquake emergency period. The present shell element analysis results can be used to plan an experimental program for optimising the resilience of storage rack uprights against stomping.

Classical, higher-order, zig-zag and variable kinematic shell elements for the analysis of composite multilayered structures

Abstract In the present work, a shell finite element with a variable kinematic field based on a new zig-zag power function is proposed for the analysis of laminated shell structures . The kinematic field is written by using an arbitrary number of continuous piecewise polynomial functions. The polynomial expansion order of a generic subdomain is a combination of zig-zag power functions depending on the shell thickness coordinate. As in the classical layer-wise approach, the shell thickness can be divided into a variable number of mathematical subdomains. The expansion order of each subdomain is an input parameter of the analysis. This feature enables the solution to be locally refined over generic regions of the shell thickness by enriching the kinematic field. The advanced finite shell elements with variable kinematics are formulated in the framework of the Carrera Unified Formulation. The finite element arrays are formulated in terms of fundamental nuclei, which are invariants of the theory approximation order and the modelling technique (Equivalent-Single-Layer, Layer-Wise). In this work, the attention is focused on linear static stress analyses of composite laminated shell structures. The governing equations are obtained by applying the Principle of Virtual Displacements, and they are solved using the Finite Element method . Furthermore, the Mixed Interpolated Tensorial Components (MITC) method is employed to contrast the shear locking phenomenon . Several numerical investigations are carried out to validate and demonstrate the accuracy and efficiency of the present shell element.

Electro-mechanical analysis of composite and sandwich multilayered structures by shell elements with node-dependent kinematics

ABSTRACTIn this work, a new class of finite elements for the analysis of composite and sandwich shells embedding piezoelectric skins and patches is proposed. The main idea of models coupling is developed by presenting the concept of nodal dependent kinematics where the same finite element can present at each node a different approximation of the main unknowns by setting a node-wise through-the-thickness approximation base. In a global/local approach scenario, the computational costs can be reduced drastically by assuming refined theories only in those zones/nodes of the structural domain where the resulting strain and stress states, and their electro-mechanical coupling present a complex distribution. Several numerical investigations are carried out to validate the accuracy and efficiency of the present shell element. An accurate representation of mechanical stresses and electric displacements in localized zones is possible with reduction of the computational costs if an accurate distribution of the higher-order kinematic capabilities is performed. On the contrary, the accuracy of the solution in terms of mechanical displacements and electric potential values depends on the global approximation over the whole structure. The efficacy of the present node-dependent variable kinematic models, thus, depends on the characteristics of the problem under consideration as well as on the required analysis type.

A 4-node thin shell element based on co-rotational for laminated composite structures

The finite formulation of the 4-node thin shell element based on Co-rotational (CR) and Timoshenko`s theory was presented for the analysis of laminated composite structures. Based on the Timoshenko`s theory the present thin shell element avoids shear locking behavior, and a bilinear in-plane displacement field is introduced for the coupling of in-plane and bending actions, and such element performs simple formulation and highly efficienct. A highly efficient CR formulation was also established to define the motion of the element with large displacement. The characteristics of the element are more pronounced when shells get thin, each node of the 4-node element has five degrees of freedom. An incremental iterative method based on the Newton Raphson method was used to solve the nonlinear equilibrium equations. A number of numerical examples were given to verify that the formula is computationally efficient and the results showed good agreement.

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.

Parallel computation for three-dimensional shell analysis of curved configuration based on domain decomposition method

In this paper, a parallel computational algorithm is developed based on finite element tearing and interconnecting (FETI) method, specifically, localized Lagrange multipliers. The proposed FETI method decomposes large-size structures into non-overlapping subdomains via localized Lagrange multipliers. To consider the curved configuration of large-size structures, a facet shell element created by combining an optimal triangle membrane and discrete Kirchhoff triangle bending plate (OPT-DKT) is suggested and used by introducing rotational operators. Moreover, practical performance of the present OPT-DKT facet shell element is evaluated through static and dynamic analysis. Finally, parallel computation is implemented for the proposed approach using message passing interface (MPI).

Shell elements with through-the-thickness variable kinematics for the analysis of laminated composite and sandwich structures

Abstract Several efforts have been made in the last years to improve the efficiency and the effectiveness of structural models for the analysis of laminated shell structures. Among the others, many recent and past works in the literature have been aimed at formulating theories of structures that maximize the accuracy of analysis meanwhile reducing the computational costs. In this paper, this objective is pursued by implementing advanced shell theories with through-the-thickness variable kinematic capabilities. By employing the Carrera Unified Formulation (CUF), the proposed shell model is obtained by expressing the displacement field as an arbitrary and, eventually, hierarchical expansion of the primary unknowns along the thickness. Thus, Equivalent-Single-Layer (ESL), Layer-Wise (LW) models as well as variable kinematic models which combine ESL and LW approaches within the shell thickness can be obtained in a straightforward and unified manner. After the unified shell model is formulated, the governing equations and the related finite element arrays are obtained by employing the principle of virtual work. A nine node finite element is implemented to approximate the solution field, and the Mixed Interpolation of Tensorial Components (MITC) method is used to contrast the membrane and shear locking phenomena. Some numerical examples are discussed, including three- and ten-layered cross-ply shells under bi-sinusoidal load and simply-supported boundary conditions, a multilayered spherical panel subjected to bi-sinusoidal load and a sandwich cylinder undergoing bi-sinusoidal pressure. Moreover, various thickness and radius-to-thickness ratios are considered. Whenever possible, the results are compared with those from the literature and from exact elasticity solutions. The analysis of the results clearly shows the enhanced capabilities of the present variable-kinematic shell element, which allows the analyst to opportunely reduce the computational costs and enhance the accuracy of the model only in those regions of the thickness domain where an accurate evaluation of the stress/strain field is needed.

Free Vibrations of Plates and Shells with an Isogeometric RM Shell Element

Free vibration analysis of plates and shells is carried out by using isogeometric approach. For this purpose, an isogeometric shell element based on Reissner-Mindlin (RM) shell theory is developed. Non-uniform rational B-spline surface (NURBS) definition is introduced to represent the geometry of shell and it is also used to derive all terms required in the isogeometric element formulation. New anchor positions are proposed to calculate the shell normal vector. Gauss integration rule is used for the formation of stiffness and mass matrices. The proposed shell element is then used to examine vibrational behaviours of plate and shell structures. From numerical results, it is found to be that reliable natural frequencies and associated mode shapes can be predicted by the present isogeometric RM shell element.

Postbuckling analysis of laminated composite shells under shear loads

The postbuckling behavior of laminated composite plates and shells, subjected to various shear loadings, is presented, using a modified 8-ANS method. The finite element, based on a modified first-order shear deformation theory, is further improved by the combined use of assumed natural strain method. We analyze the influence of the shell element with the various location and number of enhanced membrane and shear interpolation. Using the assumed natural strain method with proper interpolation functions, the present shell element generates neither membrane nor shear locking behavior even when full integration is used in the formulation. The effects of various types of lay-ups, materials and number of layers on initial buckling and postbuckling response of the laminated composite plates and shells for various shear loading have been discussed. In addition, the effect of direction of shear load on the postbuckling behavior is studied. Numerical results and comparisons of the present results with those found in the literature for typical benchmark problems involving symmetric cross-ply laminated composites are found to be excellent and show the validity of the developed finite element model. The study is relevant to the simulation of barrels, pipes, wing surfaces, aircrafts, rockets and missile structures subjected to intense complex loading.

Shape control of piezolaminated thin‐walled structures using nonlinear piezoelectric actuators

AbstractIn the present paper a 2D‐shell finite element model is proposed to carry out static analysis of piezolaminated composite shells by incorporating nonlinear constitutive relations in order to describe the electromechanical coupling under strong electric fields. The present shell element has 5 mechanical DOFs and 3 electrical DOFs per node. The developed composite piezolaminated shell element is employed to study the static behavior and shaping of spherical antenna reflector laminated with piezo‐patches under both weak and strong electric fields. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

등기하 퇴화쉘 요소를 이용한 판과 쉘의 기하학적 비선형해석

In this study, geometrically nonlinear analysis of plate and shell structures is carried out by using a degenerated shell element. The present shell element is formulated by using the isogeometric concept. Reissner-Mindlin assumption is adopted in the element formulation where total Lagrange formulation is used. In particular, the positions of control points is consistently used to create the normal vector for the mapping between control points and associated points on real surface. The arc-length method is employed to handle the snap-through behaviour of shells. Several benchmark tests are tackled to verify the performance of the present shell element. From numerical tests, the present shell element can remove locking phenomena with the only use of refinement technique and it performs satisfactorily for both thin and thick plate and shell structures under geometrically nonlinear situations.

A refined element-based Lagrangian shell element for geometrically nonlinear analysis of shell structures

For the solution of geometrically nonlinear analysis of plates and shells, the formulation of a nonlinear nine-node refined first-order shear deformable element-based Lagrangian shell element is presented. Natural co-ordinate-based higher order transverse shear strains are used in present shell element. Using the assumed natural strain method with proper interpolation functions, the present shell element generates neither membrane nor shear locking behavior even when full integration is used in the formulation. Furthermore, a refined first-order shear deformation theory for thin and thick shells, which results in parabolic through-thickness distribution of the transverse shear strains from the formulation based on the third-order shear deformation theory, is proposed. This formulation eliminates the need for shear correction factors in the first-order theory. To avoid difficulties resulting from large increments of the rotations, a scheme of attached reference system is used for the expression of rotations of shell normal. Numerical examples demonstrate that the present element behaves reasonably satisfactorily either for the linear or for geometrically nonlinear analysis of thin and thick plates and shells with large displacement but small strain. Especially, the nonlinear results of slit annular plates with various loads provided the benchmark to test the accuracy of related numerical solutions.

A 6-node co-rotational triangular elasto-plastic shell element

A 6-node co-rotational triangular elasto-plastic shell element is developed. The local coordinate system of the element is defined by the vectors directing from one vertex to the other two vertices and their cross product. Based on such a co-rotational framework, the element rigid-body rotations are excluded in calculating the local nodal variables from the global nodal variables. The two smallest components of each nodal orientation vector are defined as rotational variables, resulting in the desired additive property for all nodal variables in a nonlinear incremental solution procedure. Different from other existing co-rotational finite element formulations, both the element tangent stiffness matrices in the local and in the global coordinate systems are symmetric owing to the commutativity of the nodal variables in calculating the second derivatives of the strain energy with respect to the local nodal variables and, through chain differentiation, with respect to the global nodal variables. For elasto-plastic analysis, the Maxwell---Huber---Hencky---von Mises yield criterion is employed together with the backward-Euler return-mapping method for the evaluation of the elasto-plastic stress state, where a consistent tangent modulus matrix is employed. To overcome locking problems, the assumed linear membrane strains and shear strains are obtained by using the line integration method proposed by MacNeal, and the assumed higher-order membrane strains are obtained by enforcing the stationarity of the mixed displacement-strain canonical functional, these assumed strains are then employed to replace the corresponding conforming strains. The reliability and convergence of the present 6-node triangular shell element formulation are verified through two elastic plate patch tests as well as two elastic and five elasto-plastic plate/shell problems undergoing large displacements and large rotations.

등기하 퇴화쉘 요소를 이용한 연속체 쉘의 선형해석

In this study, linear static analysis of continuum shell structures is conducted by using a degenerated shell element which is developed by using isogeometric approach. Notably, the present isogeomeric shell element uses the positions of control points to create the normal vector via the mapping between control points and associated points on real surface. A series of benchmark tests are performed to verify the performance of the present isogeometric degenerated shell element. From numerical test, the present shell element can overcome locking phenomena with only use of refinements and therefore any special treatment such as reduced integration, assumed strains, non-conforming modes are not required in the formulation.

Static and dynamic finite rotation FE-analysis of thin-walled structures with piezoelectric sensor and actuator patches or layers

This paper deals with static and dynamic analysis of thin-walled structures with integrated piezoelectric layers as sensors and actuators in the geometrically nonlinear range of deformations. A variational formulation is derived by using the Reissner–Mindlin first-order shear deformation (FOSD) hypothesis and full geometrically nonlinear strain-displacement relations accounting for finite rotations. The finite rotations are treated by Rodriguez parameterization. In order to enhance the accuracy of a four-node shell element, a combination of an assumed natural strain (ANS) method for the shear strains, an enhanced assumed strain (EAS) method for the membrane strains and an enhanced assumed gradient (EAG) method for the electric field are employed. The present shell element has five mechanical degrees of freedom (DOFs) and three electrical DOFs per node. The Newton–Raphson method for static analysis and the Newmark method for dynamic analysis are used to perform linear and nonlinear simulations. In comparison to the results obtained by simplified nonlinear models reported in the existing literature, the finite-element simulations performed in this paper show the importance of the present model, precisely for structures undergoing finite deformations and rotations.

Free Vibrations of Thin Shells with Isogeometric Approach

Free vibration analysis of thin shells is carried out by using isogeometric approach. For this purpose, a thin shell element based on Kirchhoff-Love shell theory is developed. Non-uniform rational B-spline surface (NURBS) definition is introduced to represent the geometry of shell and also used to derive all terms required in the isogeometric element formulation. Gauss integration rule is used for stiffness and mass matrices. The present shell element is then applied to examine vibrational behaviours of thin plate and shell structures. From numerical results, it is found be that reliable natural frequencies and associated mode shapes of thin shell structures can be predicted by the present isogeometric shell element.