Analytische Untersuchung von Spannungsfeldern in Windturbinenblättern

The thesis aims at shedding light on the difference in the stress distribution between prismatic and non-prismatic beams. The approach strategy has being that of drastically simplifying the initial problem in the attempt to identify the relevant governing parameters. The geometry of a real wind turbine blade has been first drastically simplified: from the multi-cellular airfoil to a box girder and to a single web panel. Once solved the simplest problems, the intention has been to gradually reintegrate the complexity of the problem with the aim of applying the obtained results to a real wind turbine blade. The thesis consists of three parts. In the first part some brief information on the technology of wind energy and wind turbines are given. Also, the state of the art of the modeling of non-prismatic beams is discussed, with a particular focus on tapered beams. After some first remarkable results obtained at the beginning of the 20th century, the literature survey shows that only in recent years it has been an increase of interest on the subject. In the second part all the theoretical details that lead to the approach that is subject of the present thesis are deepened. In the third part the results of the application of the strategy proposed to both elementary and more complex geometries are shown, and the final remarks on the 10MW wind turbine blade are discussed.Die Arbeit zielt darauf ab, den Unterschied in der Spannungsverteilung zwischen prismatischen und nicht prismatischen Strahlen zu beleuchten. Die Herangehensweise besteht darin, das ursprüngliche Problem bei dem Versuch, die relevanten maßgeblichen Parameter zu identifizieren, drastisch zu vereinfachen. Die Geometrie eines echten Windturbinenblatts wurde zunächst drastisch vereinfacht: vom mehrzelligen Tragflügel über einen Kastenträger bis hin zu einem einzelnen Stegpaneel. Nachdem die einfachsten Probleme gelöst worden waren, bestand die Absicht darin, die Komplexität des Problems schrittweise wieder zu integrieren, um die erhaltenen Ergebnisse auf ein echtes Windturbinenblatt anzuwenden. Die Arbeit besteht aus drei Teilen. Im ersten Teil werden einige kurze Informationen zur Technologie von Windenergie und Windkraftanlagen gegeben. Außerdem wird der Stand der Technik bei der Modellierung nicht-prismatischer Strahlen mit besonderem Schwerpunkt auf sich verjüngenden Strahlen erörtert. Nach einigen ersten bemerkenswerten Ergebnissen zu Beginn des 20. Jahrhunderts zeigt die Literaturübersicht, dass das Interesse an diesem Thema erst in den letzten Jahren zugenommen hat. Im zweiten Teil werden alle theoretischen Details vertieft, die zu dem Ansatz führen, der Gegenstand der vorliegenden Arbeit ist. Im dritten Teil werden die Ergebnisse der Anwendung der vorgeschlagenen Strategie auf elementare und komplexere Geometrien gezeigt und die abschließenden Bemerkungen zum 10 MW-Windturbinenblatt erörtert

Measurement of cohesive laws from mixed bending-tension tests

The mixed bending-tension (MBT) test was proposed by Macedo et al. (2012) to assess the mode I interlaminar fracture toughness of composite laminates with very low bending stiffness and strength. Specimens obtained from such laminates may fail in bending prior to delamination growth, when tested using the double cantilever beam test (ASTM D5528-13). In the MBT test, the specimen with a pre-implanted delamination is adhesively bonded to two metal bars and then loaded in opening mode. Bennati et al. (2015) developed a mechanical model of the MBT test, where the two separating parts of the specimen are connected by a cohesive interface with bilinear traction-separation law. Accordingly, the specimen response can be subdivided into three stages: (i) linearly elastic behaviour, (ii) progressive material damage, and (iii) crack propagation. The theoretical predictions were in good agreement with the experimental results by Macedo et al. (2012) in the linearly elastic stage. Instead, only qualitative agreement was obtained for the subsequent stages. Here, we upgrade the previous model by introducing a piece-wise linear, discontinuous tractionseparation law for the cohesive zone (Valvo et al., 2015). We show how the global response of the specimen depends on the cohesive law parameters. Besides, we present an operative procedure to determine the cohesive law parameters based on the test measures

Experimental validation of the enhanced beam-theory model of the mixed-mode bending test

We present the results of an experimental campaign on a set of specimens manufactured from a typical carbon/epoxy unidirectional laminate. Preliminary tests are performed to evaluate the elastic properties of the base laminate. Then, double cantilever beam (DCB) and end-notched flexure (ENF) tests are conducted to assess the delamination toughness in pure fracture modes I and II, respectively, and evaluate the elastic interface constants. Afterwards, mixed-mode bending (MMB) tests are carried out with three values of the lever-arm length. The outcomes of the preliminary and pure fracture mode tests are used as an input to a previously developed enhanced beam theory (EBT) model of the MMB test. Lastly, theoretical predictions and exper-imental results are compared

An elastic interface model of the mixed bending-tension (MBT) test

The mixed bending-tension (MBT) test has been introduced by Macedo et al. to assess the interlaminar fracture toughness of laminates with low bending stiffness and strength in the longitudinal direction. In the experimental setup, the delaminated specimen is adhesively bonded to two pin-loaded metal beams. We have developed a mechanical model of the test, where the specimen is modelled as an assemblage of two beams connected by an elastic interface, while the metal beams are modelled as rigid beams. An analytical solution has been obtained by applying classical beam theory. Furthermore, to better describe the experimental results, we have developed also a cohesive zone model based on a bilinear traction-separation law

Modellazione meccanica della frattura interlaminare di provini in composito non simmetrici

La delaminazione, o frattura interlaminare, è una delle più pericolose modalità di danneggiamento dei laminati compositi fibrorinforzati. E' ormai consolidata l'abitudine di studiare questo fenomeno attraverso le tecniche della meccanica della frattura. In particolare per poter prevedere la condizione di iniziazione e propagazione di una fessura, applicando un opportuno criterio di crisi, è necessario conoscere con la miglior accuratezza possibile il valore del tasso di rilascio dell'energia elastica critico: migliori livelli di confidenza permettono di minimizzare i coefficienti di sicurezza in fase di progetto. Il valore critico del tasso di rilascio dell'energia è una proprietà del materiale che è possibile ricavare per via indiretta da prove sperimentali su provini pre-fessurati. Alla base di questo procedimento vi è l'applicazione di un opportuno modello meccanico che permetta di risalire alla proprietà del materiale attraverso le grandezze misurate durante la prova di delaminazione (in particolare forze e spostamenti). Sebbene per lo studio dei laminati unidirezionali esista una consolidata letteratura e le procedure siano state codificate nella forma di normative per le prove standard (ASTM e JISC), per lo studio di laminati non simmetrici non esistono ancora metodologie universalmente accettate come valide. Quando la delaminazione propaghi in modo non simmetrico rispetto al piano medio del laminato, o quando il laminato sia costituito da più materiali diversi, la modalità di frattura è in generale di tipo misto, e è importante conoscere il contributo dei singoli modi (apertura, o modo I e scorrimento, o modo II). Negli ultimi anni sono stati sviluppati, da un gruppo di ricerca presso l'Università di Pisa, una serie di modelli analitici per la descrizione delle differenti prove sperimentali di tenacità a frattura. Tutti questi modelli si basano sull'idea comune che un provino possa essere assimilato ad un sistema di due travi sovrapposte e collegate tra loro da una interfaccia elastica. Questa Tesi nasce con lo scopo di generalizzare questo modello meccanico, fornendo la soluzione generale per un generico elemento di laminato caricato solo da azioni ai suoi estremi, in corrispondenza dei quali non vengono fissate particolari limitazioni alle condizioni di vincolo. Questo elemento è supposto costituito da due travi laminate di Timoshenko che possono presentare sequenze di impilamento generiche caratterizzate anche da accoppiamenti flesso-estensionali. Applicando a questo modello meccanico le opportune condizioni al bordo, che specializzano l'analisi al particolare problema in esame, è possibile descrivere la risposta alle sollecitazioni di un qualsiasi provino, comunque esso sia caricato e comunque esso sia fissato alla macchina di prova. A titolo di esempio viene mostrato come poter applicare questo modello per la riduzione dei dati di provini simmetrici e asimmetrici. [English summary Delamination, or interlaminar fracture, is one of the most dangerous failures for fiber-reinforced composite laminates. It's a well established habit of study this phenomenon through the techniques of fracture mechanics. In particular, in order to predict the conditions of initiation and propagation of a crack, applying an appropriate crisis criterion, it is necessary to asses, with the best possible accuracy, the value of the strain energy release rate: in the conceptual design phase, best confidence in the values of this material property allows the minimization of the safety coefficients. The critical value of the energy release rate can be evaluated indirectly from experimental tests on pre-cracked specimens. An appropriate mechanical model have to be applied, allowing to estimate this property of the material through the quantities measured during the delamination test (in particular forces and displacements). Although the characterization of unidirectional laminates is supported by a well-established literature and the procedures have been codified in the form of regulations for standard tests (ASTM and JISC), for the study of not symmetric laminates there are not yet universally accepted methodologies. When the delamination propagates in a unsymmetrical way with respect to the middle plane of the laminate, or when the laminate consists of different materials, the mode of fracture is generally of mixed mode, and it is important to properly separate the contribution of the single modes (opening, or mode I, and sliding or mode II). Over the past decade, a research group of the University of Pisa have proposed a series of analytical models for the description of different experimental tests. All of these models are based on the common idea that a specimen could be assimilated to a system of two overlapping beams connected together by a elastic interface . This Thesis is born with the aim to generalize this mechanical model, providing the general solution for a generic element of laminate. This element is consisting of two laminated Timoshenko's beams with generic stacking sequences characterized also by bending-extension coupling stiffnesses. The laminated element may be loaded only by concentrated forces acting at its ends, in correspondence of which there are not specific limitations on the bonding conditions. By imposing the appropriate set of boundary conditions to the model (specializing the analysis to the particular problem), it's possible to describe the response to the test of any kind of specimen, indipendently on how it's loaded and bonded to the test machine. As an example it is shown how to apply this mechanical model to the data reduction of symmetric and asymmetric specimen tests.

Correction: Bennati et al. An Elastic Interface Model for the Delamination of Bending-Extension Coupled Laminates. Appl. Sci. 2019, 9, 3560

We, the authors, wish to make the following corrections to our paper [...

An Elastic Interface Model for the Delamination of Bending-Extension Coupled Laminates

The paper addresses the problem of an interfacial crack in a multi-directional laminated beam with possible bending-extension coupling. A crack-tip element is considered as an assemblage of two sublaminates connected by an elastic-brittle interface of negligible thickness. Each sublaminate is modeled as an extensible, flexible, and shear-deformable laminated beam. The mathematical problem is reduced to a set of two differential equations in the interfacial stresses. Explicit expressions are derived for the internal forces, strain measures, and generalized displacements in the sublaminates. Then, the energy release rate and its Mode I and Mode II contributions are evaluated. As an example, the model is applied to the analysis of the double cantilever beam test with both symmetric and asymmetric laminated specimens

On shear stresses in tapered beams

We analyse the effects of taper on the stress distribution in a constantly tapered cantilever beam loaded at its free end by concentrated axial force, shear force, and bending moment. First, an exact analytical solution based on classical elasticity theory is deduced. Then, a simplified solution based on the approximate shear theory is proposed. Both solutions are validated against finite element analyses of the problem

Correction: Bennati et al. An elastic interface model for the delamination of bending-extension coupled laminates. Appl. Sci. 2019, 9, 3560

We, the authors, wish to make the following corrections to our paper [1]. Equations (26), (27), (28), and (41) were affected by some typos and should be substituted by the following ones (corrections are colored in red): Furthermore, we observe that the constant terms in the shear stress expressions (18) and (32) (corresponding to Jourawski's solution for an unbroken beam) should not contribute to the Mode II energy release rate GII. Thus, the peak values of the shear interfacial stress entering Equation (44) should be computed as τ0=τ(0)-f7/B and τ0=τ(0)-g7/B in the balanced and unbalanced cases, respectively. As a consequence, Equations (45) and (46) should be replaced by the following ones: The corrections do not affect the results and scientific conclusions of the paper. We apologize for any inconvenience caused