Postbuckling of laminated anisotropic panels

A two-part study of the buckling and postbuckling of laminated anisotropic plates with bending-extensional coupling is presented. The first part involves the development and application of a modified Rayleigh-Ritz analysis technique. Modifications made to the classical technique can be grouped into three areas. First, known symmetries of anisotropic panels are exploited in the selection of approximation functions. Second, a reduced basis technique based on these same symmetries is applied in the linear range. Finally, geometric boundary conditions are enforced via an exterior penalty function approach, rather than relying on choice of approximation functions to satisfy these boundary conditions. Numerical results are presented for both the linear and nonlinear range, with additional studies made to determine the effect of variation in penalty parameter and number of basis vectors. In the second part, six panels possessing anisotropy and bending-extensional coupling are tested. Detailed comparisons are made between experiment and finite element results in order to gain insight into the postbuckling and failure characteristics of such panels. The panels are constructed using two different lamination sequences, and panels with three different aspect ratios were constructed for each lamination sequence

Morphing shell structures:A generalised modelling approach

AbstractMorphing shells are nonlinear structures that have the ability to change shape and adopt multiple stable states. By exploiting the concept of morphing, designers may devise adaptable structures, capable of accommodating a wide range of service conditions, minimising design complexity and cost. At present, models predicting shell multistability are often characterised by a compromise between computational efficiency and result accuracy. This paper addresses the main challenges of describing the multistable behaviour of thin composite shells, such as bifurcation points and snap-through loads, through the development of an accurate and computationally efficient energy-based method. The membrane and the bending components of the total strain energy are decoupled by using the semi-inverse formulation of the constitutive equations. Transverse displacements are approximated by using Legendre polynomials and the membrane problem is solved in isolation by combining compatibility conditions and equilibrium equations. This approach provides the strain energy as a function of curvature only, which is of particular interest, as this decoupled representation facilitates efficient solution. The minima of the energy with respect to the curvature components give the multiple stable configurations of the shell. The accurate evaluation of the membrane energy is a key step in order to correctly capture the multiple configurations of the structure. Here, the membrane problem is solved by adopting the Differential Quadrature Method (DQM), which provides accurate results at a relatively small computational cost. The model is benchmarked against three exemplar case studies taken from the literature

Computational nonlinear vibration analysis for distributed geometrical nonlinearities in structural dynamics

The demand to reduce the impact of aviation on the environment is leading jet engine manu- facturers to increase the fuel and propulsion efficiency of the engines. This in turn is pushing materials to their physical limits by undergoing increasingly higher thermo-mechanical loads. In this regime, blades and other engine components are subjected to large deforma- tions generating nonlinearities that activate new failure mechanisms not dealt with before. Therefore, vibration analysis is essential to develop new methodologies for the accurate prediction of components’ behaviour. This research focuses on investigating the effect of the distributed geometric nonlinearities and rotational speed on the dynamic behaviour of three-dimensional structures. The Green-Lagrange strain measures are employed in this research to express the nonlinear relationship between the displacement and the strain. The nonlinear algorithms used for the numerical simulations are developed based on the Finite Element Method combined with the Harmonic Balance method. The complex geometries are discretised by using the geometric exact three-dimensional solid elements. The forced response functions and the backbone curves for the steady-state response of the nonlinear system can be computed. The research aims to develop and validate methodologies for the identification and control of undesired vibration modes which will inform new design choices. Finite element modelling of the blades generally involves an immense number of degree-of-freedoms, which could be infeasible to compute. The reduced order modelling (ROM) techniques are crucial for achieving an accurate prediction of the nonlinear behaviour in an efficient way. Detailed computation strategies for the intrusive ROM methods are delivered. ROM techniques based on the linear and nonlinear mapping between the full model and the reduced basis are presented. The capabilities and limitations of both methods are assessed. The projection method based on the linear eigenmodes only has a slow converge to the full system. On the other hand, the quadratic manifold method with the static modal derivatives involved in the reduced coordinates provides a fast convergence.Open Acces

Mixed Models and Reduction Techniques for Large-Rotation, Nonlinear Analysis of Shells of Revolution with Application to Tires

An effective computational strategy is presented for the large-rotation, nonlinear axisymmetric analysis of shells of revolution. The three key elements of the computational strategy are: (1) use of mixed finite-element models with discontinuous stress resultants at the element interfaces; (2) substantial reduction in the total number of degrees of freedom through the use of a multiple-parameter reduction technique; and (3) reduction in the size of the analysis model through the decomposition of asymmetric loads into symmetric and antisymmetric components coupled with the use of the multiple-parameter reduction technique. The potential of the proposed computational strategy is discussed. Numerical results are presented to demonstrate the high accuracy of the mixed models developed and to show the potential of using the proposed computational strategy for the analysis of tires

Semi-analytical methods for vibration and stability analysis of pressurized and rotating toroidal shells based on the energy approach

Prikazane su polu-analitičke metode za analizu vibracija torusnih ljuski izloženih tlaku, koje rotiraju oko svoje osi simetrije. Ovisnost deformacija rastezanja i savijanja o pomacima ljuske izvedena je iz općih izraza za rotacijske ljuske. Izrazi za deformacijsku (potencijalnu) i kinetičku energiju izvedeni su za rotirajući polarni koordinatni sustav. Potencijalna energija je najprije formulirana za slučaj velikih deformacija, a zatim je rastavljena na linearni i nelinearni dio, koji je zatim lineariziran. Korištena je nelinearna Green-Lagrangeova formulacija. Kinetička energija osim vibracijske komponente uključuje centrifugalni i Coriolisov dio. Za promjenu pomaka u, v i w u cirkularnom smjeru postavljeni su točni harmonijski izrazi. Pomaci u meridijalnom smjeru su pretpostavljeni u obliku Fourierovih redova. Korištena je Rayleigh-Ritzova metoda za minimiziranje ukupne energije. To je rezultiralo općom matricom krutosti, geometrijskom matricom krutosti uslijed prednaprezanja, te trima matricama masa vezanim za kvadrat prirodne frekvencije, umnožak prirodne frekvencije i brzine vrtnje, te kvadrat brzine vrtnje. Primjena razvijenog postupka ilustrirana je na primjeru zatvorene i otvorene torusne ljuske i tankostijenog torusnog prstena. Dobiveni rezultati (prirodne frekvencije i oblici vibriranja) uspoređeni su s rezultatima dobivenim metodom konačnih elemenata i uočeno je dobro podudaranje. Prednost prikazanog postupka je u znatno skraćenom vremenu obrade problema na računalu. U nastavku je razvijena metoda vrpčastih elemenata za analizu istih problema.Za deformacijsku i kinetičku energiju korišteni su ranije postavljeni izrazi u okviru Rayleigh-Ritzove metode. Ljuska je u meridijalnom smjeru modelirana nizom dvočvornih vrpčastih elemenata. Promjena pomaka u, v i w u meridijalnom smjeru unutar svakog elementa aproksimirana je štapnim i grednim funkcijama oblika. Minimiziranjem ukupne energije vrpčastog elementa formirane su matrice krutosti i matrice masa. U svrhu ubrzanja konvergencije rješenja razvijen je vrpčasti element višeg reda s tri čvora. Prikazanom metodom riješen je problem zatvorene torusne ljuske. Dobiveni rezultati uspoređeni su s rezultatima Rayleigh- Ritzove metode i metode konačnih elemenata. Nadalje, razmatrane su fleksijske i torzijske vibracije rotirajućeg prstena. Fleksijske vibracije se sprežu sa rasteznim vibracijama, a torzijske sa savojnim vibracijama. Odgovarajuće jednadžbe gibanja izvedene su iz teorije vibracija torusne ljuske. U prvom slučaju prsten je promatran kao vršni segment torusne ljuske, a u drugom slučaju kao bočni segment. Pokazano je da rotacija prstena dovodi do bifurkacije fleksijskih prirodnih frekvencija, a ne i torzijskih frekvencija. Teorija vibracija prstena ocjenjena je usporedbom rezultata analize vibracija jednog prstena s rezultatima metode konačnih elemenata i metode vrpčastih elemenata, te izmjerenim vrijednostima dostupnim u literaturi.In this self-contained paper, free vibrations of a pressurised toroidal shell, rotating around its axis of symmetry, are considered. Extensional and bending strain-displacement relationships are derived from general expressions for a thin shell of revolution. The strain and kinetic energies are determined in the co-rotating reference frame. The strain energy is first specified for large deformations, and then split into a linear and a nonlinear part. The non-linear part, which is subsequently linearized, is necessary in order to take into account the effects of centrifugal and pressure pre-tensions. The Green-Lagrange non-linear strains are considered. The kinetic energy is formulated taking into account the centrifugal and the Coriolis terms. The variation of displacements u, v and w in the circumferential direction is described exactly. The dependence of the displacements on the meridional coordinate is described through the Fourier series. The Rayleigh-Ritz method is applied to determine the Fourier coefficients. As a result thereof, an ordinary stiffness matrix, a geometric stiffness matrix due to pressurisation and centrifugal forces, and three inertia matrices incorporating squares of natural frequencies, products of rotational speed and natural frequencies and squares of the rotational speed, are derived. The application of the developed procedure is illustrated in cases of a closed and open toroidal shell and a thin-walled toroidal ring. The obtained results are compared with FEM results, and a very good agreement is observed. The advantage of the proposed semi-analytical method is high accuracy and low CPU time-consumption. Additionally, a finite strip for vibration analysis of rotating toroidal shells subjected to internal pressure is developed. The expressions for strain and kinetic energies are taken from the previous Rayleigh-Ritz method. The variation of displacements u, v and w with the meridional coordinate is modelled through a discretization with a number of finite strips. The finite strip properties, i.e. the stiffness matrix, the geometric stiffness matrix and the mass matrices are defined by employing bar and beam shape functions, and by minimizing the strain and kinetic energies. In order to improve the convergence of the results, the strip of a higher order is developed too. The application of the finite strip method is illustrated in case of closed toroidal shell. The obtained results are compared with those determined by the Rayleigh-Ritz method and the finite element method. The rigorous formulae for natural frequencies of in-plane and outof- plane free vibrations of a rotating ring are derived. An in-plane vibration mode of the ring is characterised by coupled flexural and extensional deformations, whereas an out-of-plane mode is distinguished by coupled flexural and torsional deformations. For the in-pane vibrations, the ring is considered to be a short top segment of a toroidal shell. The expressions for the ring strain and kinetic energies are deduced from the corresponding expressions for the torus. It is shown that the ring rotation causes the bifurcation of natural frequencies for the in-plane vibrations only. The bifurcation of natural frequencies of the out-of-plane vibrations does not occur. The derived analytical results are validated by a comparison with FEM and FSM (Finite Strip Method) results, as well as with experimental results available in the literature

Finite element modeling and analysis of tires

Predicting the response of tires under various loading conditions using finite element technology is addressed. Some of the recent advances in finite element technology which have high potential for application to tire modeling problems are reviewed. The analysis and modeling needs for tires are identified. Reduction methods for large-scale nonlinear analysis, with particular emphasis on treatment of combined loads, displacement-dependent and nonconservative loadings; development of simple and efficient mixed finite element models for shell analysis, identification of equivalent mixed and purely displacement models, and determination of the advantages of using mixed models; and effective computational models for large-rotation nonlinear problems, based on a total Lagrangian description of the deformation are included

Vibration analysis of cracked aluminium plates

This research is concerned with analytical modelling of the effects of cracks in structural plates and panels within aerospace systems such as aeroplane fuselage, wing, and tail-plane structures, and, as such, is part of a larger body of research into damage detection methodologies in such systems. This study is based on generating a so-called reduced order analytical model of the behaviour of the plate panel, within which a crack with some arbitrary characteristics is present, and which is subjected to a force that causes it to vibrate. In practice such a scenario is potentially extremely dangerous as it can lead to failure, with obvious consequences. The equation that is obtained is in the form of the classical Duffing equation, in this case, the coefficients within the equation contain information about the geometrical and mass properties of the plate, the loading and boundary conditions, and the geometry, location, and potentially the orientation of the crack. This equation has been known for just over a century and has in the last few decades received very considerable attention from both the analytical dynamics community and also from the dynamical systems researchers, in particular the work of Ueda, Thompson, in the 1970s and 1980s, and Thomsen in the 1990s and beyond. An approximate analytical solution is obtained by means of the perturbation method of multiple scales. This powerful method was popularized in the 1970s by Ali H.Nayfeh, and discussed in his famous books, ‘Perturbation Methods’ (1974) and ‘Nonlinear Oscillations’ (1979, with D.T.Mook), and also by J.Murdock (1990), and M.P.Cartmell et al. (2003) and has been shown to be immensely useful for a wide range of nonlinear vibration problems. In this work it is shown that different boundary conditions can be admitted for the plate and that the modal natural frequencies are sensitive to the crack geometry. Bifurcatory behaviour of the cracked plate has then been examined numerically, for a range of parameters. The model has been tested against experimental work and against a Finite Element model, with good corroboration from both. In all events, this is a significant new result in the field and one that if implemented within a larger damage detection strategy, could be of considerable practical use