Multi-scale Modelling and Design of Composite Structures

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A free vibration analysis of three-dimensional sandwich beams using hierarchical one-dimensional finite elements

This paper presents a family of beam higher-orders finite elements based on a hierarchical one-dimensional unified formulation for a free vibration analysis of three-dimensional sandwich structures. The element stiffness and mass matrices are derived in a nucleal form that corresponds to a generic term in the displacement field approximation over the cross-section. This fundamental nucleus does not depend upon the approximation order nor the number of nodes per element that are free parameters of the formulation. Higher-order beam theories are, then, obtained straightforwardly. Timoshenko's classical beam theory is obtained as a special case. Short and slender beams are investigated. Simply supported, cantilevered and clamped-clamped boundary conditions are considered. Several natural frequencies as well as the corresponding modes are investigated. Results are validated in terms of accuracy and computational costs towards three-dimensional finite element solutions. The proposed hierarchical models, upon an appropriate choice of approximation order, yield accurate results with a reduced computational cost

Hierarchical one-dimensional finite elements for the thermal stress analysis of three-dimensional functionally graded beams

In this work, the thermoelastic response of functionally graded beams is studied. To this end, a family of advanced one-dimensional finite elements is derived by means of a Unified Formulation that is not dependent on the order of approximation of the displacements upon the beam cross-section. The temperature field is obtained via a Navier-type solution of Fourier’s heat conduction equation and it is considered as an external load within the mechanical analysis. The stiffness matrix of the elements is derived via the Principle of Virtual Displacements. Numerical results in terms of temperature, displacements and stresses distribution are provided for different beam slenderness ratios and type of material gradation. Linear, quadratic and cubic elements are used. Results are validated through comparison with three-dimensional finite elements solutions obtained by the commercial software ANSYS. It is shown that accurate results can be obtained with reduced computational costs

A multi-scale model of fibre reinforced beams using hierarchical one-dimensional finite elements

A multi-scale model of fibre reinforced beams using hierarchical one-dimensional finite elements A multi-scale analysis of fibre reinforced composite beams was proposed by this presentation. At structural level, several higher-order refined beam theories can be easily implemented on the basis of Carrera's unified formulation (CUF) by deriving a fundamental nucleus that does not depend upon the approximation order nor the number of nodes per element (they are free parameters of the formulation). Under the framework of FE2 method, the effective properties of the fibre-reinforced composite material are found by numerical homogenization over representative volume elements, that is, the unknown constitutive relationship at the macro-scale is obtained by solving a local finite element problem at the micro-scale. Consequently, a coupled two-scale problem is obtained for linear cases. Results are validated in terms of accuracy and computational costs towards FEM solutions. Numerical investigations show that accurate results can be obtained with reduced computational costs

Multi-scale Modelling and Design of Composite Structures

Questa dissertazione propone un nuovo paradigma per l'analisi multi-scala delle strutture di tipo trave, utilizzando la formulazione unificata di Carrera ("Carrera's Unified Formulation", CUF). La modellizzazione multi-scala collega la micromeccanica e la teoria strutturale macroscopica. I risultati del processo di esplorazione possono essere riassunti nei due aspetti seguenti: un modello macroscopico di trave, contenente non-linearità geometriche, ed un modello di trave multi-scala, anch'esso non-lineare da un punto di vista geometrico. L'indagine qui presentata inizia con uno studio della modellazione macroscopica non lineare. Il modello strutturale viene stabilito accoppiando la CUF non-lineare con il Metodo Numerico Asintotico ("Asymptotic Numerical Method", ANM). Questo modello di trave, basato sulla CUF contenente non-linearità geometriche, è realizzato in collaborazione con G. De Pietro: si tratta di uno dei primi studi che estende i modelli CUF unidimensionali accoppiati con il metodo ANM. Si presentano analisi non lineari statiche, post-buckling e snap-through delle strutture traviformi e se ne valutano le corrispondenti curve carico-spostamento e carico-sforzo. I risultati sono confrontati con soluzioni agli elementi finiti bidimensionali. Si dimostra che, per i casi considerati, una descrizione quadratica attraverso lo spessore garantisce accurati spostamenti a componente di tensione assiale. Inoltre, si necessita di un ordine di espansione più elevato al fine di prevedere con precisione la componente di sollecitazione di taglio. Nell'analisi post-buckling considerata, i modelli CUF di ordine basso rilevano il punto di biforcazione in modo accurato. Tuttavia, al fine di ottenere risultati accurati riguardanti la sollecitazione di taglio, si richiede un modello di ordine superiore. Nell'analisi di snap-through, è necessaria una teoria raffinata per tracciare accuratamente il percorso di equilibrio. Per affrontare problemi contenenti non-linearità geometriche provenienti da scale diverse, un modello di trave multi-scala, basato sulla CUF contenente non-linearità geometriche, viene derivato accoppiando il modello macroscopico proposto ed il framework agli Elementi Finiti Multilivello (noto anche come FE$^2$). La soluzione consiste in un'analisi macroscopico/strutturale e un'analisi microscopica/materiale. Alla scala macroscopica, la legge costitutiva incognita è calcolata attraverso un'omogeneizzazione numerica di un Elemento di Volume Rappresentativo ("Representative Volume Element", RVE). Viceversa, il gradiente di deformazione microscopico è calcolato tramite modello macroscopico. Il sistema matematico non-lineare risultante è risolto attraverso il metodo ANM, il quale risulta essere più affidabile e meno dispendioso dal punto di vista dei tempi di calcolo, rispetto ai metodi iterativi classici. La metodologia proposta viene utilizzata per studiare l'effetto delle imperfezioni alla scala microscopica (fibre di carbonio non perfettamente diritte) sulla risposta macroscopica (instabilità). I risultati vengono analizzati in termini di accuratezza e costi computazionali, rispetto alle soluzioni FEM. Tre fattori sono identificati per un'analisi parametrica di sensibilità alle imperfezioni: lunghezza d'onda ed, ampiezza della imperfezione e dimensione dell'RVE

A hygro-thermal stress finite element analysis of laminated beam structures by hierarchical one-dimensional modelling

A hygro-thermal stress finite element analysis of laminated beam structures by hierarchical one-dimensional modelling Composite structure operating under severe temperature conditions and/or wet environments are very common is several engineering fields such as aeronautics, space and transportation. Hygro-thermal solicitation of beam-like structures results in a three- dimensional response that classical one-dimensional models are not always capable of describe effectively. An accurate prediction calls, then, for refined higher-order theories making this subject of research relevant and up-to-date. In this work, laminated composite Several beam models are hierarchically derived by means of a unified formulation [1, 2] that allows for atheoretical derivation of the finite elements independent from the displacements polynomial approximation order over the cross-section as well as the number of nodes per element. Elements stiffness matrix are derived in a compact form (“fundamental nucleus”) via the Principle of Virtual Displacements. As a result, a family of several one-dimensional finite elements accounting for transverse shear deformations and cross section in- and out-of-plane warping can be obtained. Temperature and humid-ity profiles are obtained by directly solving the corresponding diffusion equation(Fourier’s heat conduction equation for temperature and Fick’s law for moisture). These fields are, then, accounted as sources terms in the elastic analysis through Hooke’s law. Simply supported and cantilever configurations are considered. Numerical results in terms of temperature, moisture, displacement and stress distributions are provided for different beam slenderness ratios. Three-dimensional finite element solutions developed within the commercial code Ansys are presented for validation. The numerical investigations show that the hygro-thermo-elastic problem presents a complex three-dimensional stress state that can be efficiently obtained by a suitable choice of approximation order over the cross section: the accuracy is comparable to the reference solutions whereas the computational costs can beconsiderably reduce

Detecting the Spatially Non-Stationary Relationships between Housing Price and Its Determinants in China: Guide for Housing Market Sustainability

Given the rapidly developing processes in the housing market of China, the significant regional difference in housing prices has become a serious issue that requires a further understanding of the underlying mechanisms. Most of the extant regression models are standard global modeling techniques that do not take spatial non-stationarity into consideration, thereby making them unable to reflect the spatial nature of the data and introducing significant bias into the prediction results. In this study, the geographically weighted regression model (GWR) was applied to examine the local association between housing price and its potential determinants, which were selected in view of the housing supply and demand in 338 cities across mainland China. Non-stationary relationships were obtained, and such observation could be summarized as follows: (1) the associations between land price and housing price are all significant and positive yet having different magnitudes; (2) the relationship between supplied amount of residential land and housing price is not statistically significant for 272 of the 338 cities, thereby indicating that the adjustment of supplied land has a slight effect on housing price for most cities; and (3) the significance, direction, and magnitude of the relationships between the other three factors (i.e., urbanization rate, average wage of urban employees, proportion of renters) and housing price vary across the 338 cities. Based on these findings, this paper discusses some key issues relating to the spatial variations, combined with local economic conditions and suggests housing regulation policies that could facilitate the sustainable development of the Chinese housing market