Superelement Research Articles

Static and stability analyses of arbitrary straight-sided quadrilateral thin plates by DQM

A differential quadrature (DQ) methodology is employed for the static and stability analysis of irregular quadrilateral straight-sided thin plates. A four-noded super element is used to map the irregular physical domain into a square computational domain. Second order transformation schemes with relative ease and low computational effort are employed to transform the fourth order governing equations of thin plates between the domains. Within the domain, the displacements are the only degrees of freedom whereas, along the boundaries, the displacements as well as the second order derivatives of the displacements with respect to the associated normal coordinate variables in the computational domain are the two sets of degrees of freedom. The implementation procedures for different boundary conditions including free-edge boundaries are formulated. To demonstrate the accuracy, convergency and stability of the methodology, detailed studies of skewed and trapezoidal plates for different geometries under different boundary and loading conditions are made. Good agreement is achieved between the results of the present methodology and those of other DQ methodologies or other comparable numerical algorithms.

A novel hybrid finite element analysis of bimaterial wedge problems

An assumed hybrid-stress finite element model together with a new super singular wedge-tip element with numerical eigensolutions is developed to study the bimaterial wedge/notch problems. The establishment of the super wedge-tip element consists of (1) a finite element method-based eigenanalysis is developed and applied to determine the order of the stress singularity and the angular dependences of the stress and displacement fields, (2) these fields are subsequently used to develop a finite element surrounding the wedge tip. To demonstrate the validity of the method, three types of the bimaterial wedge problems are examined and compared with analytical/referenced solutions. The high accuracy and general applicability of the present technique are shown.

Passive Vibration Suppression in Truss-Type Structures with Tubular Members

A method for vibration suppression using viscous damping material has been developed for truss structures consisting of axisymmetric tubular members of circular cross section. The vibration damping scheme calls for the construction of a structural member with two or more tubes by overlapping the end portion of adjacent tubes and bonding them together by means of a thin layer of viscoelastic material. The equations of motion for a e nite length section consisting of concentric layers of an inner tube, viscous material damping layer, and outer tube are obtained using Hamilton’ s principle. Only axial motion is considered. An exact solution to the equation is obtained and used to develop the dynamic stiffness matrix for the section. A scheme has been presented to develop a super element that consists of two or more such damping sections. The super elements have then been used to model members in a truss-type structure to study the vibration suppression in the structure. The vibration suppression is studied for a range of frequencies. A parametric study has been performed to establish the effects of thickness, loss factor, and length of overlap. The study has shown that the damping effectiveness is dependent on the coupled effects of these parameters.

Vibrations of complex shells of revolution using B-spline finite elements

This paper presents and evaluates a local–global B-spline finite element method for the modal analysis of joined thin shells of revolution. Each shell component is modelled by a super element. For smooth meridian, local displacements of the shell middle surface and their first derivatives are matched at the junction. However, for discontinuous shells, the C 1 continuity of tangential displacements is relaxed and a local to global frame transformation is applied for normal and longitudinal displacements. The super element formulation in local–global frame reference and the minimum number of degrees of freedom (three nodal displacements augmented by the bending rotation at the joints only), provided by the uniform cubic B-spline interpolation, are the main originalities of the present work. Efficiency and accuracy of the proposed technique are evaluated through various well known two-branch shell combinations. Results are compared to analytical, experimental and numerical solutions found in the literature. A very satisfactory behaviour of the approach was observed.

Combination of fracture and damage mechanics for numerical failure analysis

A new approach for the analysis of crack propagation in brittle materials is proposed, which is based on a combination of fracture mechanics and continuum damage mechanics within the context of the finite element method. The approach combines the accuracy of singular crack-tip elements from fracture mechanics theories with the flexibility of crack representation by softening zones in damage mechanics formulations. A super element is constructed in which the typical elements are joined together. The crack propagation is decided on either of two fracture criteria; one criterion is based on the energy release rate or the J-integral, the other on the largest principal stress in the crack-tip region. Contrary to many damage mechanics methods, the combined fracture⧹damage approach is not sensitive to variations in the finite element division. Applications to situations of mixed-mode crack propagation in both two- and three-dimensional problems reveal that the calculated crack paths are independent of the element size and the element orientation and are accurate within one element from the theoretical (curvilinear) crack paths.

Neocortical development in IL-9 transgenic mouse brain

Failure assessment diagram (FAD) has great potentials to be a powerful tool to assess the integrity of hull structures. However, the current methodology to obtain the fracture ratio and the load ratio, two axes of FAD, for hull structures is tedious and burden which is one of the major obstacles to advance the further application of FAD. In this paper, a super element is introduced to obtain those two ratios simultaneously in the framework of linear elastic analysis within a single step. Besides, the virtual section closure technique is proposed to compute the reference stress under the inspiration of virtual crack closure technique to compute the stress intensity factor. The capacity of the proposed super element has been assessed by two classic examples of a plate with central crack and a cylinder with circumferential crack. The results obtained from the super element are compared to the corresponding analytical solutions. Finally, the cracked Nishihara specimen (NST-3) was examined and the facture ratio and the load ratio obtained by the super element were compared to those obtained by the global analyses. The comparisons indicate that the proposed super element is accurate. No convergence troubles were encountered. Therefore, the methodology developed in this paper could be a very useful addition to perform the direct analysis on the failure of hull structures due to large crack extensions based on the failure assessment diagram.

Sectional forming analysis by membrane finite elements considering bending effects

The sectional forming analysis considering bending effects from the geometrically deformed shape of two linear membrane finite elements(called super element) was performed under plane strain assumption for analyzing forming processes of an arbitrarily shaped draw-die. For the evaluation of bending effects, the bending equivalent forces are calculated from the bending moment computed using the changes in the interior angle at the middle node of super element, and are agumented to the membrane stretch forces. In order to verify the validity of the bending formulation, the simulation results for the stretch, draw, and bend sections were compared with membrane analysis results and measurements.

Dynamical Lie algebra and conditional operators for many fermion Green functions

A method is introduced to calculate thermodynamic Green functions. A powerful theorem by Masson is the motivation for expressing the Fourier-time transform of the Green function as a super space matrix element of the resolvent of the Liouville operator. The eigenvalues of the Liouville operator are then expressed in a form first suggested by Judd for atomic systems and which are shown to be members of a Lie group.

Impact analysis of laminated composite plates and shells by super finite elements

A super finite element method that exhibits coarse-mesh accuracy is used to predict the transient response of laminated composite plates and cylindrical shells subjected to non-penetrating impact by projectiles. The governing equations are based on the classical theories of thin laminated plates and shells taking into account the von Karman kinematics assumptions for moderately large deflections. A non-linear Hertzian-type contact law accounting for curvatures of the colliding bodies is adopted to calculate the impact force . The theoretical basis of the present finite element model is verified by analysing impact-loaded laminated composite plate and shell structures that have previously been studied through analytical or other numerical procedures. The predictive capability of the present numerical approach is successfully demonstrated through comparisons between experimentally-measured and computed force-time histories for impact of carbon fibre-reinforced plastic (CFRP) plates. The current computational model offers a relatively simple and efficient means of predicting the structural impact response of laminated composite plates and shells.

Saint-Venant decay rates: A procedure for the prism of general cross-section

A procedure previously developed for the determination of decay rates for self-equilibrated loadings at one end of a pin-jointed framework consisting of repeated identical cells, wherein the decay factors are the eigenvalues of the single cell transfer matrix, is here further developed and applied to a prismatic continuum beam of general cross-section. A sectional length of beam is treated within ANSYS finite element code as a super element; nodes at both ends of the section are treated as master nodes and the stiffness matrix relating forces and displacements at these master nodes is constructed within ANSYS. Manipulation of this stiffness matrix within MATLAB gives the transfer matrix from which the eigenvalues and eigenvectors may be readily determined. Accuracy of the method is assessed by treating the plane strain strip, the plane strain sandwich strip, and the rod of circular cross-section, representing a selection of the examples for which exact analytical solutions are available, and is found to be very good in all cases.

Nonlinear analysis of orthogonally stiffened cylindrical shells by a super element approach

The super finite element method is extended to the nonlinear static and dynamic analysis of orthogonally stiffened cylindrical shells. Both geometric and material nonlinearities are included and the temporal equations are solved by the implicit Newmark-β method with Newton-Raphson subiterations. The new formulation has been tested in various applications and a few illustrative examples are presented.

Infinitely small element for the dynamic problem of a cracked beam

A super element formulated by using an infinitely small element for analyzing the dynamic behavior of a cracked beam is proposed. It is based on the concept of matrix condensation, using the characteristic properties of the stiffness for the plane elements. A group of infinitely small elements is generated layer by layer around the point of singularity. The subelement around the crack tip can be subdivided to be arbitrarily small to model the stress singularity with sufficient accuracy. The construction of a super element using this approach does not involve special treatment. Therefore, the present method can also be applied to any kind of singularity problems.

Stability of cold-formed members

A finite element method capable of predicting the buckling capacity of arbitrarily shaped thin-walled structural members under any general load and boundary conditions is presented. A rectangular thin plate element with 30 degrees of freedom is used. The linear and geometric stiffness matrices for this element are derived explicitly using symbolic manipulation, thereby eliminating the need for the expensive process of numerical integration. Further, the explicit form of the stiffness matrices makes it easier to interpret the physical significance of the various stiffness terms. For sections composed of rigid flanges and flexible web, a lower-order plate element is used in combination with a general beam-column element to form a super element for predicting distortional buckling modes. Formex formulation is used for the automatic generation of the data necessary for the analysis. Numerical examples of thin-walled structural members involving local, distortional and flexural-torsional buckling modes are presented to demonstrate the accuracy, efficiency and versatility of the method.

Vibration Analysis of Orthogonally Stiffened Cylindrical Shells Using Super Finite Elements

An efficient numerical technique for the free vibration analysis of orthogonally stiffened cylindrical shell structures is presented. The new technique is developed within the framework of a super finite element method and involves formulation of C1 cylindrical shell and curved beam elements. The displacement functions used in the present formulation combine the usual polynomials with some analytical functions, carefully chosen to provide a good approximation of basic vibration modes in conjunction with coarse grid modelling strategy. Detailed results of vibration application of these elements are presented.

A super element model for non‐linear analysis of stiffened box structures

AbstractA numerical model for non‐linear static and dynamic analysis of stiffened box structures is presented. The model is based on a new super element formulation which provides complete C1 continuity for both plate and beam elements. Geometric and material non‐linearities are included and the temporal equations are solved by the implicit Newmark‐β method with Newton‐Raphson subiteration. The new formulation has been applied to the static, vibration and transient analysis of various structures such as flat plates, folded plates and rectangular boxes. Both isotropic and beam stiffened structures are considered and the results obtained are compared with other available solutions. It is observed that the new super element formulation can provide reasonable solutions to both linear and non‐linear problems of stiffened box structures. The mathematical formulation of the model is presented in this paper, while the numerical verifications are given in the companion paper.1

Applications of a super element model for non‐linear analysis of stiffened box structures

AbstractNumerical investigations are carried out to verify the new super element model developed in Reference 1 for non‐linear analysis of stiffened box structures. The results presented here are obtained from a variety of examples which are chosen to illustrate the accuracy and convergence of the model in various applications. Good agreement is obtained for most of the problems considered and the present formulation is found to be accurate and economical for preliminary engineering design.

New design-analysis techniques for blast loaded stiffened box and cylindrical shell structures

This paper presents a brief review of some new design-analysis techniques for blast loaded engineering structures. The super finite element method, previously developed for stiffened plates, has been extended to more complex and larger structures such as orthogonally stiffened rectangular boxes and cylindrical shells. Both geometric and material nonlinearities are included in the modelling. The mathematical formulation of the new super elements is briefly outlined and some illustrative examples are presented.

Research developments in analyses of plates and shells

Abstract The results presented in this paper are based on research collaborations between the three institutions, namely: (1) Nanyang Technological University, (2) The University of Queensland and (3) National University of Singapore. A software code named ‘SEPSA’, an acronym for super element for plate and shell analysis, has been developed. The software is applied to solve bending, buckling and vibration problems. The formulations and method of analysis are briefly discussed. Several advantages of the present method over other approximate discretization methods are also addressed. To demonstrate the capability of SEPSA, some plate and shell solutions are presented.

Stability of thin-walled members having arbitrary flange shape and flexible web

A finite element method is presented for analysing thin-walled structural members comprising a flexible web connected to one or two rigid flanges of arbitrary shape. A general thin-walled beam-column element is used to model the flanges while a thin plate element is used to model the web. Based on the derived total potential energy functional, explicit linear and geometric stiffness matrices for the two types of element are obtained. Using static condensation and appropriate transformations, the beam-column element and the plate element are combined to yield a super element with 22 degrees of freedom capable of modelling the flexural, torsional, web distortional and coupled web and flange local buckling modes of a general thin-walled member. The technique may be used to predict the elastic buckling load of members under any loading and boundary conditions. Several numerical examples are presented to demonstrate the accuracy, efficiency and versatility of the method.

Crack propagation in two-belt beams

The FEM was used to construct a detailed model of contact interaction of a fastening element and the web of a two-belt beam. The super element approach was used as a base for evaluating the SIF in separation and SIF in transverse shearing for cracks in the zone of the fastening hole of the longitudinal joint and in the belt. The results show that inclined cracks (type A) are situated in the normal opening conditions, whereas the vertical (type C) and horizontal (type D) cracks are in the conditions with mainly transverse shear. The nature of crack propagation depends strongly on the material of the web of the beam. For example, in type D16 material in which cracks can grow by the shear mechanism in a wide range of the values of SIF, the type C and D cracks propagate linearly. The predicted rate of their growth is considerably (15–60 times) higher than the rate of growth of opening cracks. This is caused both by a lower-resistance of the material to propagation of fatigue cracks by the shear mechanism and by adding up all the values of kII in the opposite loading half cycles. However, if the wall is made of V95-type material in which straight growth of shear cracks takes place only in pure shear conditions the type C and D cracks rotate during growth and degenerate to a separation crack propagating at a considerably low rate.