Rectangular Shell Element Research Articles

A novel [formula omitted] continuity finite element based on Mindlin theory for doubly-curved laminated composite shells

As a widely used structural form, doubly-curved composite shells have been applied in aviation and other engineering fields. With a refined mechanical model, structural performances can be accurately predicted to help designers choosing best geometrical and material parameters. This work introduces a novel rectangular finite element for doubly-curved laminated composite shells based on a new set of strain-based shape functions. The governing equations are established on the Mindlin shell theory (a type of first order shear deformable shell theory), which incorporates a rectification of shear correction factor for laminates and von Kármán type nonlinearity. Based on strain approach, the shape functions for rectangular element are assumed as polynomial of 28 parameters to consider the influence of shear effect. Apart from 20 geometrical relations of four element nodes, shear force equilibrium equations are introduced to offer the additional eight equations to derive a new set of shape functions for finite element model. Using shape functions, a novel rectangular shell element is proposed for doubly curved laminated shell, which also maintains a compatibility with Kirchhoff shells. Numerical results for linear static bending, dynamic vibration and nonlinear bending cases of flat plate, cylindrical shell and doubly-curved laminated shell are compared with the available results in literatures and ABAQUS simulation for the sake of validating the present method.

Technical Stability of Very Slender Rectangular Columns Compressed by Ball-And-Socket Joints without Friction

Stability of load-bearing members, as known, is a challenging issue and several tools are available for designers. Disregarding the material properties in use, the avoidance of possible stability troubles is a mandatory and challenging step of the overall design process. In this study, a theoretical model is presented for the of loss of stability in elastic states of very slender rectangular shell elements axially compressed through ball-and-socket joints without friction. According to this theory, as shown, a loss of carrying capacity of very slender columns in elastic states occurs when the line of force leaves a critical transverse cross-section. The critical transverse cross-section, moreover, progressively moves because of the superposition of bending and pure compression. The theory allows to determine the governing differential equations of curved central lines and their slopes, as well as the critical stresses of columns, in the form of surface function in dependence on slenderness ratios and cross-sectional areas. The graphs for the elastic deflected central line y(x), slope dy/dx, dependence yL/2(P), stresses and strains for a rectangular column made of steel and compressed by ball-and-socket joints without friction, as well as the corresponding surface graphs of critical stress, are presented in this study. The obtained surface graphs of critical stresses are then discussed and compared with Euler’s formulation.

A Shell Element for Buckling Analysis of Thin-Walled Composite-Laminated Members

This paper introduces a new shell element formulation and investigates the buckling behavior of thin-walled beams composed of fiber-reinforced polymer composite-laminates, which is a primary design concern for thin-walled beams composed of fiber-reinforced polymer composite-laminates due to their slenderness. Although global buckling behavior can be captured using beam-column type two-node simple finite element formulations, shell-type more sophisticated elements are needed in order to be able to capture the effects due to cross-sectional deformations. Pursuit of an efficient shell element formulation continues to date and in this study, a new flat rectangular shell element formulation is developed for the buckling analysis of thin-walled composite-laminated members. The plate component of the shell is locking-free and based on the twist-Kirchhoff theory. For the membrane component of the shell element, variational formulation employing drilling degrees of freedom is adopted. Convergence studies were presented to illustrate the numerical performances of the element. A broad class of problems including distortional as well as global buckling cases were solved and compared with solutions from the literature to validate the use of the developed shell element for the buckling analysis of thin-walled composite-laminated members.

A rectangular shell element formulation with a new multi-resolution analysis

A multi-resolution rectangular shell element with membrane-bending based on the Kirchhoff-Love theory is proposed. The multi-resolution analysis (MRA) framework is formulated out of a mutually nesting displacement subspace sequence, whose basis functions are constructed of scaling and shifting on the element domain of basic node shape functions. The basic node shape functions are constructed from shifting to other three quadrants around a specific node of a basic element in one quadrant and joining the corresponding node shape functions of four elements at the specific node. The MRA endows the proposed element with the resolution level (RL) to adjust the element node number, thus modulating structural analysis accuracy accordingly. The node shape functions of Kronecker delta property make the treatment of element boundary condition quite convenient and enable the stiffness matrix and the loading column vectors of the proposed element to be automatically acquired through quadraturing around nodes in RL adjusting. As a result, the traditional 4-node rectangular shell element is a mono-resolution one and also a special case of the proposed element. The accuracy of a structural analysis is actually determined by the RL, not by the mesh. The simplicity and clarity of node shape function formulation with the Kronecker delta property, and the rational MRA enable the proposed element method to be implemented more rationally, easily and efficiently than the conventional mono-resolution rectangular shell element method or other corresponding MRA methods.

Response of an open curved thin shell to a random pressure field arising from a turbulent boundary layer

A method is presented to predict the root mean square displacement response of an open curved thin shell structure subjected to a turbulent boundary-layer-induced random pressure field. The basic formulation of the dynamic problem is an efficient approach combining classic thin shell theory and the finite element method, in which the finite elements are flat rectangular shell elements with five degrees of freedom per node. The displacement functions are derived from Sanders’ thin shell theory. A numerical approach is proposed to obtain the total root mean square displacements of an open curved thin structure in terms of the cross spectral density of random pressure fields. The cross spectral density of pressure fluctuations in the turbulent pressure field is described using the Corcos formulation. Exact integrations over surface and frequency lead to an expression for the total root mean square displacement response in terms of the characteristics of the structure and flow. An in-house program based on the presented method was developed. The total root mean square displacements of a curved thin blade subjected to turbulent boundary layers were calculated and illustrated as a function of free stream velocity and damping ratio. A numerical implementation for the vibration of a cylinder excited by fully developed turbulent boundary layer flow was presented. The results compared favorably with those obtained using software developed by Lakis and Païdoussis (J. Sound Vib. 25 (1972) 1–27) using cylindrical elements and a hybrid finite element method.

Experimental and Numerical Study of Damaged Cantilever

The introduction of a crack in a steel structure will cause a local change in the stiffness and damping capacity. The change in stiffness will lead to a change of some of the natural frequencies of the structure and a discontinuity in the associated mode shapes. This paper contains a presentation of the results from experimental and numerical tests with hollow section cantilevers containing fatigue cracks. Two different finite-element (FE) models have been used to estimate the modal parameters numerically. The first FE model consists of beam elements. The second FE model consists of traditional rectangular shell elements and one rectangular shell element with a transverse, internal, open crack. The analytical results from the numerical models are compared with data obtained from experimental tests. The numerical models give good agreements with the experimental data. The beam model takes into account only the first mode of the crack evaluation. In the shell model all three modes of the crack growth are taken into account. Nevertheless, the results obtained for both models are satisfactory because the beam is subjected to bending. It can be concluded that it is sufficient to use crack models for calculating natural frequencies in bending, taking into account the first mode of the crack extension only.

FREE VIBRATION OF LONGITUDINALLY STIFFENED CURVED PANELS WITH CUTOUT

Free vibration characteristics of longitudinally stiffened square panels with symmetrical square cutouts are investigated by using the finite-element method. A rectangular shell element with seven degrees of freedom per node together with a beam element of seven degrees of freedom per node is used for the analysis. Studies are carried out on unstiffened as well as stiffened panels keeping the size, weight and boundary conditions the same. Numerical results indicate that the frequency of the stiffened panel is higher. Maximum increase in fundamental frequency is observed on the panel with the smallest radius of curvature.

Discussion of “Stiffness Matrix for Shallow Rectangular Shell Element”

Discussion of “Stiffness Matrix for Shallow Rectangular Shell Element”