Degenerated Shell Element Research Articles

Dynamics of hybrid shafts

In this paper the dynamic performance and cross-section deformation of shafts made of metals (steel and aluminum), composites (CFRP and GFRP) and hybrids of metals and composites have been studied. A layered finite degenerated shell element with transverse shear deformation and dynamic behavior is employed. Results obtained show that improvements in dynamic performance and reduction of cross-section deformation of hybrid shafts over metallic and composite shafts are possible.

기하학적 비선형 해석을 위한 곡면 2차 삼각형 쉘 요소에 관한 연구

곡면 사각형 쉘 요소들이 다수인 것에 비해, 곡면 삼각형 요소들은 아주 소수이다. 이미 발표된, 선형 해석을 위한 6절점 2차 쉘 요소의 가정 자연 변형도 이론에 기초해, 본 연구에서는 6절점 쉘 요소의 기하학적 비선형 해석을 수행하였다. 쉘 요소는 표준 절점 자유도만으로 곡면 모델링이 가능하고, 수치해석 결과가 보여주는 바와 같이 다양한 잠김 현상들을 제거하는데 효율적인 요소임을 확인하였다. Compare to the large number of curved quadrilateral degenerated-shell elements, there are only a very few curved triangular degenerated-shell elements. Based on the assumed natural strain sampling scheme previously developed for a quadratic degenerated-shell element for linear analysis, this paper devises geometric non-linear six-node degenerated-shell element. The element can be curved and is only equipped with the standard nodal d.o.f.'s. Careful consideration has been exercised to circumvent various locking phenomena that plague degenerated-shell element. Numerical examples are presented to illustrate efficiency.

Structural intensity study of plates under low-velocity impact

The transmissions of transient energy flow and dynamic transient response of plate structures under low-velocity impact are presented. The structural intensity approach is used to study the transient dynamic characteristics of plate structures under low-velocity impact. In the dynamic impact response analysis, nine-node degenerated shell elements with assumed shear and membrane strain fields are adopted to model the target and impactor. The dynamic contact-impact algorithm and the governing equations for both the target and impactor are derived based on the updated Lagrangian approach. Explicit integration algorithm has been adopted in the time integration process. The novel structural intensity streamline representation is introduced to interpret energy flow paths for transient dynamic response of plates under low-velocity impact. The effects of plates with and without structural damping on the energy flow and energy path are discussed. Numerical results, including contact force, deflection histories and transient energy flow vectors as well as structural intensity streamlines, show that the present method and representation are an efficient approach for exploring dynamic response for plate structures subjected to low-velocity impact.

A new hierarchic degenerated shell element for geometrically non-linear analysis of composite laminated square and skew plates

This paper extends the use of the hierarchic degenerated shell element to geometric non-linear analysis of composite laminated skew plates by the p-version of the finite element method. For the geometric non-linear analysis, the total Lagrangian formulation is adopted with moderately large displacement and small strain being accounted for in the sense of von Karman hypothesis. The present model is based on equivalent-single layer laminate theory with the first order shear deformation including a shear correction factor of 5/6. The integrals of Legendre polynomials are used for shape functions with p-level varying from 1 to 10. A wide variety of linear and non-linear results obtained by the p-version finite element model are presented for the laminated skew plates as well as laminated square plates. A numerical analysis is made to illustrate the influence of the geometric non-linear effect on the transverse deflections and the stresses with respect to width/depth ratio (a/h), skew angle (b), and stacking sequence of layers. The present results are in good agreement with the results in literatures.

Refined 15‐DOF triangular discrete degenerated shell element with high performances

AbstractA refined discrete degenerated 15‐DOF triangular shell element RDTS15 with high performances is proposed. For constructing the element displacement function, the exact displacement function of the Timoshenko's beam is used as the displacement on the element boundary, and the re‐constitute method for shear strain matrix is adopted. The proposed element can be used in the analysis of both moderate thick and thin plates/shells. Numerical examples presented show that the new model indeed possesses higher accuracy in the analysis of thin and thick plates/shells, and that it can pass the patch test required for the Kirchhoff thin plate elements, and also passed the inf–sup test for free cylindrical shell problems and satisfied both the bending‐ and membrane‐dominated test. Copyright © 2004 John Wiley Sons, Ltd.

Vibration characteristics of initially twisted rotating shell type composite blades

Vibration analysis of a rotating composite blade is the main purpose of this study. A general formulation is derived for an initially twisted rotating shell structures including the effect of centrifugal force and Coriolis acceleration. In this work, the blade is assumed to be a moderately thick open cylindrical shell that includes the transverse shear deformation and rotary inertia, and is oriented arbitrarily with respect to the axis of rotation to consider the effects of disc radius and setting angle. For a thick shell, we must consider the transverse shear deformation as well as rotary inertia. Thus, based on the concept of the degenerated shell element with the Reissner–Mindlin’s assumptions, the finite element method is used for solving the governing equations. In the numerical study, effects of various parameters are investigated: initial twisting angles, thickness to radius ratios, layer lamination and fiber orientation of composite blades. Also, they are compared with the previous works and experimental data.

Curved quadratic triangular degenerated‐ and solid‐shell elements for geometric non‐linear analysis

AbstractCompared to the large number of curved quadrilateral degenerated‐ and solid‐shell elements, there are only a very few curved triangular degenerated‐ and solid‐shell elements. Based on the assumed natural strain sampling scheme previously developed for a quadratic degenerated‐shell element for linear analysis, this paper devises geometric non‐linear six‐node degenerated‐shell and twelve‐node solid‐shell elements. Both elements can be curved and are only equipped with the standard nodal d.o.f.s. Careful consideration has been exercised to circumvent various locking phenomena that plague degenerated‐ and solid‐shell elements. Numerical examples are presented to illustrate their efficacy. Copyright © 2003 John Wiley & Sons, Ltd.

Finite element modeling of smart plates/shells using higher order shear deformation theory

The finite element formulation of a degenerate shell element, using higher order shear deformation theory taking the piezo-electric effect into account is presented. An eight-noded element is used to derive global coupled electro elastic behavior of the overall structure. The model incorporates the warping of cross section due to transverse shear stresses and assumes a parabolic shear strain variation over the thickness. The static deflections of bimorph beam are compared with the literature. Active vibration control performance of the piezolaminated curved beam with distributed sensors and actuators is studied. The variation of the damping effect with different gains and actuator coverage is studied.

Materially and geometrically nonlinear analysis of laminated anisotropic plates by p-version of FEM

A p-version finite element model based on degenerate shell element is proposed for the analysis of orthotropic laminated plates. In the nonlinear formulation of the model, the total Lagrangian formulation is adopted with moderately large deflections and small rotations being accounted for in the sense of von Karman hypothesis. The material model is based on the Huber–Mises yield criterion and Prandtl–Reuss flow rule in accordance with the theory of strain hardening yield function, which is generalized for anisotropic materials by introducing the parameters of anisotropy. The model is also based on the equivalent-single layer laminate theory. The integrals of Legendre polynomials are used for shape functions with p-level varying from 1 to 10. Gauss–Lobatto numerical quadrature is used to calculate the stresses at the nodal points instead of Gauss points. The validity of the proposed p-version finite element model is demonstrated through several comparative points of view in terms of ultimate load, convergence characteristics, nonlinear effect, and shape of plastic zone.

Resolution of Defects in Degenerated Shell Elements Through Modification of Gauss Integration

In this paper, the modified Gauss integration method is developed to eliminate the shear and membrane locking phenomena of the degenerated shell element. The behavior of the element based on the Mindlin/Reissner theory in plates and shells sometimes causes a problem. In displacement-based shell elements, the full integration of stiffness matrices leads to a 'locking' or over-stiff behavior. The selective or reduced integration procedures may often overcome these difficulties, while overstiff solutions may still occur in the analysis with a highly constrained boundary. Except for the six zero-energy modes associated with shell rigid body movements, during the reduced integration of the stiffness matrices, many extra zero spurious energy modes are introduced due to reduced integration. This is a serious defect of degenerated shell element. In previous studies, several methods such as the addition of nonconforming displacement modes, an assumed strain method, and hybrid and mixed elements have been introduced in an attempt to overcome these difficulties. In this study a newly modified Gauss integration method combining both a selective and a weight-modified integration is suggested. Numerical experiments show that the new selective integration and weight-modified integration rule is very effective in eliminating the shear and membrane locking in static and modal analyses, and removes spurious zero-energy modes as well. Also, the effectiveness of proposed shell element is tested by applying it to some example problems. We solved post-buckling problem by the Riks arc-length method and dynamic problem by the Newmark's time integration method, as well as static problems.

Nonlinear finite element analysis of rubber composite shells

A finite-element solution for rubber composite shells is presented. Sandwich laminates with a rubber core have also been studied. Incompressibility of a rubber matrix and complexity of composite shells bring forth the need for a sound numerical model to describe the behavior of such engineering materials. The developed model was applied to a degenerate shell element within the limits of the first- and third-order shear deformation theories. The model allows to predict with a sufficient accuracy the nonlinear behavior of sandwich shells with composite skins and rubber cores and composite shells with a rubber matrix.

Bending Analysis of Stiffened Laminated Plates

Stiffeners are commonly attached to composite laminates to achieve higher stiffness/weight and strength/weight ratios. A layer-wise finite element formulation is developed for the bending analysis of stiffened laminated plates. The laminated plate is discretized into layers in the thickness direction and then the degenerated shell elements are employed to model each layer. The stiffener is modeled using a general 3-D beam element. Interlayer continuity relations are established for all the layers in the thickness direction. The primary advantage of the proposed finite element model lies in its application to both thin and thick laminated plates. Extensive numerical examples are presented for non-stiffened isotropic plates, non-stiffened laminates, stiffened isotropic and laminated plates. Parametric studies are conducted with different aspect ratios, plate width to thickness ratios, ply orientations, and number of layers. The results agree well with the ones reported in the literature.

Buckling behavior of stiffened laminated plates

A layerwise (zigzag) finite element formulation is developed for the buckling analysis of stiffened laminated plates. The laminated plate is discretized into layers along the thickness direction. Each layer of the laminated plate is modeled by the degenerated shell elements, and the stiffener is modeled by the general 3-D beam elements. Layers are stacked together according to the interlayer continuity. In-plane displacements are considered in the derivation of geometric stiffness matrix. The advantage of the proposed model is its applicability to both thin and thick laminated plates. The significance of this study lies in the disclosure of the interaction between the lateral buckling of the stiffener and the buckling of the laminate. The inverse iteration method is adopted to extract the lowest eigenvalue corresponding to buckling. Parametric and comparative studies are conducted for different plate aspect ratios, plate thickness to length ratios, degrees of layer orthotropy, ply orientations, and stiffener depth to plate thickness ratios.

Finite element analysis of steel members under cyclic loading

A new finite element formulation for steel members under cyclic loading was presented. The formulation was based on degenerated shell element, together with the updated Lagrange formulation to deal with the geometric nonlinearity, and the material was assumed to follow mixed hardening law. Then a program based on the formulation was completed. Finally, by performing series of examples including post-buckling, strain reserve loading, and cyclic loading, numerical experiments were carried out to verify its efficiency. In general, there was a good agreement compared with other analytical and experimental results.

An simulation technology of sheet-metal forming with trial-and-error contact algorithm

In this paper, an implicit simulation code which can simulate a general three-dimensional sheet-metal forming is presented. An 8-node degenerated shell element is employed to approximate the continuous updated Lagrangian formulation of the global mechanical problem. Emphasis is put on expressing the contact conditions and taking into account the geometrical non-linearity. The characteristics of the authors’ contact algorithm is that the contact conditions just act as the boundary conditions of the FEM equations and need not be introduced into the stiffness matrix. Finally, two numerical examples demonstrating the effectiveness of the present approach are given.

Ductile damage analysis of sheet metal forming

This work shows the feasibility of using the concept of continuous damage in the prediction of the admissible deformations in sheet metal forming processes. A constitutive law of elastoplasticity coupled with isotropic damage, derived from the thermodynamic theory of the irreversible processes, and a degenerate shell element are adopted to analyze this forming process. The total Lagrangian formulation and virtual work theory are used to derive the stiffness equation and the relations between displacements and strains. Finally, a three-dimensional contact algorithm permits to impose the restrictions by the tool to the plastic flow. An industrial application example allows to illustrate the described methodology.

Development of shear locking‐free shell elements using an enhanced assumed strain formulation

AbstractThe degenerated approach for shell elements of Ahmad and co‐workers is revisited in this paper. To avoid transverse shear locking effects in four‐node bilinear elements, an alternative formulation based on the enhanced assumed strain (EAS) method of Simo and Rifai is proposed directed towards the transverse shear terms of the strain field. In the first part of the work the analysis of the null transverse shear strain subspace for the degenerated element and also for the selective reduced integration (SRI) and assumed natural strain (ANS) formulations is carried out. Locking effects are then justified by the inability of the null transverse shear strain subspace, implicitly defined by a given finite element, to properly reproduce the required displacement patterns. Illustrating the proposed approach, a remarkably simple single‐element test is described where ANS formulation fails to converge to the correct results, being characterized by the same performance as the degenerated shell element. The adequate enhancement of the null transverse shear strain subspace is provided by the EAS method, enforcing Kirchhoff hypothesis for low thickness values and leading to a framework for the development of shear‐locking‐free shell elements. Numerical linear elastic tests show improved results obtained with the proposed formulation. Copyright © 2001 John Wiley & Sons, Ltd.

A six-node pentagonal assumed natural strain solid–shell element

In this paper, a six-node pentagonal solid–shell element is formulated. Particular attention is focused on alleviating shear, trapezoidal and thickness lockings that plagues the conventional element. While assumed natural strain method is employed to alleviate shear and trapezoidal lockings, a modified generalized laminate stiffness matrix is proposed to circumvent thickness locking. Unlike the commonly adopted plane stress assumption, the modified laminate stiffness matrix enables the element to reproduce the exact thickness stress and transverse displacement when the element is loaded by thickness stress. Numerical examples reveal that the element is close in accuracy with other state-of-the-art three-node degenerated shell elements.

Effects of initial imperfections on nonlinear behaviors of thin-walled members

The effect of the initial imperfections on the nonlinear behaviors and ultimate strength of the thin-walled members subjected to the axial loads, obtained by the finite element stability analysis, are examined. As the initial imperfections, the bucking mode shapes of the members are adopted. The buckling mode shapes of the thin-walled members are obtained by the transfer matrix method. In the finite element stability analysis, isoparametric degenerated shell element is used, and the geometrical and material nonlinearity are considered based on the Green Lagrange strain definition and the Prandtl-Reuss stress-strain relation following the von Mises yield criterion. The U-, box- and I-section members subjected to the axial loads are adopted for numerical examples, and the effects of the initial imperfections on the nonlinear behaviors and ultimate strength of the members are examined.

Postbuckling Strength of Composite Plate with a Hole

Buckling and postbuckling behaviors were analyzed numerically and experimentally for composite plates with a hole. In the finite element analysis, the updated Lagrangian formulation and the eight-node degenerated shell element were used. For the progressive failure analysis, the maximum stress criterion was applied to the average stress in each layer of all the finite elements and then stiffness and stress corresponding to the failure mode were reduced to zero for the failed layers of all the elements. In the experiments, the shadow Moiré technique was used to monitor the out-of-plane deformation. PVDF sensors were used to detect the events of internal damage. The major finding was that when a plate has low bending stiffness in the axial direction, the local instabilities caused a change of mode shapes in postbuckling range. Experiments showed good agreement with the finite element results in the buckling load and the postbuckling strength. The influence of hole sizes and stacking sequences was investigated on the compression behavior of the plate. The postbuckling strength was dominated by the value of the bending stiffness in the axial direction.