Stress analysis of ribbed flanges

In this paper an attempt is made to study the role of stiffening ribs on the behaviour of flanges. Shell theory is resorted to in developing the ribbed shell equations. A shell element with stiffening ribs in the meridional direction is obtained by superposing the stress resultants of a grid shell with the stress resultants of a isotropic shell element. A case of unequal bolt loads is considered by expressing them as Fourier series in the circumferential direction. The effect of varying the rib parameters like rib height, number, etc. is studied. The finite-difference method of solution is adopted. The behaviour of a ribbed flange is found to be similar to that of a taper hub flange.     

A unified approach for the analysis of bridges,plates and axisymmetric shells using the linear mindlin strip element

In this paper a finite strip formulation which allows to treat bridges, axisymmetric shells or plate structures of constant transverse cross section in an easily and unified manner is presented. The formulation is based on Mindlin's shell plate theory. One dimensional finite elements are used to discretize the transverse section and Fourier expansions are used to define the longitudinal/circumferential behavior of the structure. The element used is the simple two noded strip element with just one single integrating point. This allows to obtain all the element matrices in an explicit and economical form. Numerical examples for a variety of straight and curve bridges, axisymmetric shells and plate structures which show the efficiency of the formulation and accuracy of the linear strip element are given.     

Gershgorin theory for stiffness and stability of evolution systems and convection-diffusion

An analysis of stiffness and stability based on Gershgorin theorems for eigenvalues is developed for initial value systems. In particular, semidiscrete formulations for evolution problems are analysed. Common techniques such as semidiscrete finite difference and finite element methods are examined using eigenvalue bounds to characterize stiffness and stability of the associated systems. The analysis is applied to a prototype convection-diffusion problem to demonstrate the arguments and clarify several current questions concerning the qualitative nature of the solution and errors, including effects of: “lumped” versus “consistent” finite element formulations; high- or low-degree bases; mesh refinement, dimensionality and differing material properties. To study general initial value systems such as those arising in consistent finite element formulations, a generalized Gershgorin theory and computable bounds in the chordal metric are utilized.     

Measuring the convergence behavior of shell analysis schemes

While shells have been analyzed abundantly for many years in engineering and the sciences, improved finite element and related analysis methods are still much desired and researched. More general and effective finite element procedures are needed for complex shell structures, including for the analysis of composite shells and the optimization of shells. In this paper we discuss how finite element methods, and other analysis techniques, should be tested in order to identify their reliability and effectiveness. We summarize some important theoretical results, present appropriate test problems and convergence measures, and we illustrate our discussion through some novel numerical results. An important conclusion is that the testing has to be performed very carefully in order to obtain relevant results, and we show how this is accomplished in detail.     

Vibration of buckled elastic structures with large rotations by an asymptotic numerical method

A methodological approach based on the finite element and perturbation methods (asymptotic numerical method) has been developed for small vibration analyses of post-buckled shells with large displacements and large rotations. The coupled non-linear static and dynamic problems based on non-linear shells theory are transformed into a sequence of linear problems. Only two linear operators have to be inverted and a large number of terms of the polynomial approximation are numerically computed. The static non-linear response, the load–displacement solution, and the load–frequency dependence are investigated for large amplitudes. The load–frequency curves are obtained for various natural frequencies at any desired load level. A continuation procedure based on Padé approximants is used to get the whole solution. Limit points, bifurcation points and stability zones are analysed. The efficiency of this procedure is tested in some benchmark problems such as rectangular plates, thin and thick cylindrical roofs and deep arches.     

A three-dimensional state space finite element solution for laminated composite cylindrical shells

On the basis of the theory of three-dimensional elasticity, this paper presents a state space finite element solution for stress analysis of cross-ply laminated composite shells. This is a continuation of the authors’ previously published work on laminated plates [Compos. Struct. 57 (1–4) (2002) 117; Comput. Methods Appl. Mech. Engrg. 191 (37–38) (2002) 4259]. Once again a state space formulation is introduced to solve for through-thickness stress distributions, while the traditional finite elements are used to approximate the in-surface variations of state variables. A three-dimensional laminated shell element is established in an arbitrary orthogonal curvilinear coordinate system, while the application of the element is shown by calculating stresses in laminated cylindrical shells. Compared with the traditional finite element method, the new solution provides accurate continuous through-thickness distributions of both displacements and transverse stresses.     

Asymmetric vibration analysis of thick composite circular cylindrical shells with variable thickness

Thick isotropic and composite shells of revolution are analyzed for vibration characteristics. The theory used is that proposed by Naghdi, which includes the thickness normal strain and shear deformation. A curved-semianalytical finite element is used to solve the problem. Parametric studies are conducted to study the effect of various parameters on the vibration behavior of shells. The effect of thickness variation along the axial direction with a constraint on mass is also studied.     

Natural vibrations and stability of a stationary or rotating circular cylindrical shell containing a rotating fluid

The finite element method is applied to analyze a stationary or rotating cylindrical shell containing a co-rotating compressible fluid. The motion of the rotating fluid is described in the framework of the potential theory. The behavior of shells is analyzed using the classical shell theory. It has been found that the loss of stability in the stationary shells occurs in the form of a flutter. It has been shown that in the case of rotating shells the loss of stability is prevented by taking into account the initial circumferential tension caused by the centrifugal forces.     

Laminated composite structures by continuum-based shell elements with transverse deformation

In this paper a first-order shear/fourth-order transverse deformation theory of laminated composite shells is presented. A nonlinear continuum-based (degenerated 3D) finite element model with a strain/stress enhancement technique is developed in such a way that the nonzero surface traction boundary conditions and the interlaminar shear stress continuity conditions are all satisfied identically. Analytical integration through the shell thickness is performed. The resultants of the stress integrations are expressed in terms of the laminate stacking sequence. Consequently, the shell laminate characteristics in the normal direction can be evaluated precisely and the computational cost of the overall analysis is reduced. The numerical results are compared with analytical solutions and other finite element solutions to demonstrate the effectiveness of the theory and the computational procedure developed herein.     

A triangular finite difference scheme for large deflection problems

A lumped triangular element formulation is developed based on a finite difference approach for the large deflection analysis of plates and shallow shells. The presented formulation is independent of the boundary condition (unlike the finite difference formulation) and uses energy principles to derive a set of nonlinear algebraic equations which are solved by using an incremental Newton-Raphson iterative procedure. A study of the large deflection behaviour of thin plates is made for various edge conditions and aspect ratios, and the results obtained are compared with those using a finite element scheme. Representative nondimensional solutions for deflections and stresses are presented in the form of graphs.     

Finite element approximation of elliptic partial differential equations on implicit surfaces

The aim of this paper is to investigate finite element methods for the solution of elliptic partial differential equations on implicitly defined surfaces. The problem of solving such equations without triangulating surfaces is of increasing importance in various applications, and their discretization has recently been investigated in the framework of finite difference methods. For the two most frequently used implicit representations of surfaces, namely level set methods and phase-field methods, we discuss the construction of finite element schemes, the solution of the arising discretized problems, and provide error estimates. The convergence properties of the finite element methods are illustrated by computations for several test problems.     

Finite element free vibration analysis of conoidal shells

Doubly curved conoidal shells are increasingly used for various industrial structures. Conoidal shells are aesthetically appealing and, being ruled surfaces, provide ease of casting. The variation of curvature is the difficulty enountered in the analysis of these shells. The finite element method is used here for the analysis of generalized doubly curved shells and is applied to truncated and full conoids of different boundary conditions, aspect ratios and degrees of truncation. An eight-noded isoparametric finite element with five degrees of freedom per node, including three translations and two rotations, is utilized. The accuracy is checked by comparing the results obtained by the present analysis with those existing in the literature. Results are presented for different conoidal shells and a set of conclusions are arrived at based on a parametric study.     

Study of elastic behaviour of plates containing cracks by finite element analysis

This work applies finite element analysis very simply to cracked plates. An infinite plate and a finite plate, both with a central crack, are considered to study their elastic behaviour and some fracture mechanics concepts, such as the geometry factor and the fracture toughness. These magnitudes are calculated by means of finite element methods and the results are in very good agreement with the established theory, which proves that the finite element approach is very appropriate. The fracture toughness fraction is defined and calculated for a finite plate to predict its failure.     

A nine-node assumed-strain finite element for composite plates and shells

A new finite element for modeling fiber-reinforced composite plates and shells is developed and its performance for static linear problems is evaluated. The element is a nine-node degenerate solid shell element based on a modified Hellinger-Reissner principle with independent inplane and transverse shear strains. Several numerical examples are solved and the solutions are compared with other available finite solutions and with classical lamination theory. The results show that the present element yields accurate solutions for the test problems presented. Convergence characteristics are good, and the solution is relatively insensitive in element distortion. The element is also shown to be free of locking even for thin laminates.     

Symmetry considerations for anisotropic shells

The different types of symmetry exhibited by anisotropic shells for various loadings and boudary conditions are identified, and a simple procedure is presented for exploiting these symmetries in the finite element analysis. Examples are given of anisotropic cylindrical and doubly-curved shells where use of symmetry can significantly reduce the number of independent degrees of freedom in their finite element models.     

Large deflection analysis of plates and shells by discrete energy method

The discrete energy method—a special form of finite difference energy approach—is presented as a suitable alternative to the finite element method for the large deflection elastic analysis of plates and shallow shells of constant thickness. Strain displacement relations are derived for the calculation of various linear and nonlinear element stiffness matrices for two types of elements into which the structure is discretized for considering separately energy due to extension and bending and energy due to shear and twisting. Large deflection analyses of plates with various edge and loading conditions and of a shallow cylindrical shell are carried out using the proposed method and the results compared with finite element solutions. The computational efforts required are also indicated.     

Elastic-plastic post-buckling behaviour of ship plating

This paper presents an investigation by the finite element method of highly loaded stiffened and unstiffened plates as they occur in marine structures. Additionally, the performance of the computational methods in the solution of larger systems is evaluated. Flat plates and plates with initial pre-deformation (shallow shells) are studied under various loadings, such as lateral pressure, edge shear, compression and tension as well as differing boundary conditions.     

A finite element model for the analysis of 3D axisymmetric laminated shells with piezoelectric sensors and actuators: Bending and free vibrations

This paper addresses the bending and free vibrations of multilayered cylindrical shells with piezoelectric properties using a semi-analytical axisymmetric shell finite element model with piezoelectric layers using the 3D linear elasticity theory. In the present 3D axisymmetric model, the equations of motion are expressed by expanding the displacement field using Fourier series in the circumferential direction. Thus, the 3D elasticity equations of motion are reduced to 2D equations involving circumferential harmonics. In the finite element formulation the dependent variables, electric potential and loading are expanded in truncated Fourier series. Special emphasis is given to the coupling between symmetric and anti-symmetric terms for laminated materials with piezoelectric rings. Numerical results obtained with the present model are found to be in good agreement with other finite element solutions.     

Finite difference and finite element methods

The relationships between and relative advantages of finite difference and finite element methods are discussed. The less familiar finite element methods are described first for equilibrium problems: it is shown how quadratic elements on right triangles lead to natural generalisations of the powerful, fourth order accurate nine-point difference scheme for the Laplacian. For evolutionary problems, the recent development of more accurate difference methods is considered together with that of Galerkin methods. It is shown how conservation properties are best preserved by the latter methods and, in particular, how the supression of non-linear instabilities in the advection equation is achieved by the Arakawa schemes. Finally, an error analysis is described which is applicable to both finite difference and finite element methods.