Sandwich constructions are the most commonly used structures in aircraft and navy industries, traditionally. These structures are made up of the face sheets and the core, where the face sheets will be taking the load and is connected to other structural members, while the soft core material, will be used to absorb energy during impact like situation. Thus, sandwich constructions are mainly employed in light weight structures where the high energy absorption capability is required. Generally the face sheets will be thin, made up of either metallic or composite material with high stiffness and strength, while the core is light in weight, made up of soft material. Cores generally play very crucial role in achieving the desired properties of sandwich structures, either through geometric arrangement or material properties or both. Foams are in extensive use nowadays as core material due to the ease in manufacturing and their low cost. They are extensively used in automotive and industrial field applications as the desired foam density can be fabricated by adjusting the mixing, curing and heat sink processes. Modeling of sandwich beams play a crucial role in their design with suitable finite elements for face sheets and core, to ensure the compatibility between degrees of freedom at the interfaces. Unless the mathematical model simulates the physics of the model in terms of kinematics, boundary and loading conditions, results predicted will not be accurate. Accurate models helps in obtaining an efficient design of sandwich beams. In Structural Health Monitoring studies, the responses under the impact loading will be captured by carrying out the wave propagation analysis. The loads applied will be for a shorter duration (in the orders of micro seconds), where higher frequency modes will be excited. Wavelengths at such high frequencies are very small and hence, in such cases, very fine mesh generally is employed matching the wavelength requirement of the propagating wave. Traditional Finite element softwares takes enormous time and computational e ort to provide the solution. Various possible models and modeling aspects using the existing Finite element tools for wave propagation analysis are studied in the present work. There exists a huge demand for an accurate, efficient and rapidly convergent finite elements for the analysis of sandwich beams. E orts are made in the present work to address these issues and provide a solution to the sandwich user community. Super convergent and Spectral Finite sandwich Beam Elements with metallic or composite face sheets and soft core are developed. As a philosophy, the sandwich beam finite element is constructed with the combination of two beams representing the face sheets (top and bottom) at their neutral axis. The core effects are captured at the interface boundaries in terms of shear stress and normal transverse stress. In the case of wave propagation analysis, the equations are coupled in time domain and spatial domain and solving them directly is a difficult task. In Spectral Finite Element Method(SFEM), the displacement functions are derived by solving the transformed governing equations in the frequency domain. By transforming them and forces from time domain to frequency domain, the coupled partial differential equations will become coupled ordinary differential equations. These equations in frequency domain, can be solved exactly as they are normally ordinary differential equation with constant coefficients with frequency entering as a parameter. These solutions will be used as interpolating functions for spectral element formulation and in this respect it differs from conventional FE method wherein mostly polynomials are used as interpolating functions. In addition, SFEM solutions are expressed in terms of forward and backward moving waves for all the degrees of freedom involved in the formulations and hence, SFEM provides faster and efficient solutions for wave propagation analysis. In the present work, strong form of the governing differential equations are derived for a given system using Hamilton's principle. Super Convergent elements are developed by solving the static part of the governing differential equations exactly and hence the stiffness matrix derived is exact for point static loads. For wave propagation analysis, as the mass is not exactly represented, these elements are required in the optimal numbers for getting good results. The number of these elements required are generally much lesser than the number of elements required using traditional finite elements since the stiffness distribution is exact. Spectral elements are developed by solving the governing equations exactly in the frequency domain and hence the dynamic stiffness matrix derived is exact for the dynamic loads. Hence, one element between any two joints is enough to solve the whole system under impact loads for simple structures. Developing FE for sandwich beams is quiet challenging. Due to small thickness, the face sheets can be modeled using 1D idealization, while modeling of large core requires 2-D idealization. Hence, most finite or spectral elements requires stitching of these two idealizations into 1-D idealization, which can be accomplished in a variety of ways, some of which are highlighted in this thesis. Variety of finite and spectral finite elements are developed considering Euler and Timoshenko beam theories for modeling the sandwich beams. Simple element models are built with rigid core in both the theories. Models are also developed considering the flexible core with the variation of transverse displacements across depth of the core. This has direct influence on shear stress variation and also transverse normal stress in the core. Simple to higher order models are developed considering different variations in shear stress and transverse normal stress across depth of the core. Development of super convergent finite Euler Bernoulli beam elements Eul4d (4 dof element), Eul10d (10 dof element) are explained along with their results in Chapter 2. Development of different super convergent finite Timoshenko beam elements namely Tim4d (4 dof), Tim7d (7 dof), Tim10d (10 dof) are explained in Chapter 3. Validation of Euler Bernoulli and Timoshenko elements developed in the present work is carried out with test cases available in the open literature for displacements and free vibration frequencies are presented in Chapter 2 and Chapter 3. The results indicates that all developed elements are performing exceedingly well for static loads and free vibration. Super convergence performance for the elements developed is demonstrated with related examples. Spectral elements based on Timoshenko theory STim7d, STim6d, STim6dF are developed and the wave propagation characteristics studies are presented in Chapter 4. Euler spectral elements are derived from Timoshenko spectral elements by enforcing in finite shear rigidity, designated as SEul7d, SEul6d, SEul6dF and are presented. E orts were made in this present work to model the horizontal cracks in top or bottom face sheets using the spectral elements and the methodology is presented in Chapter 4. Wave propagation analysis using general purpose software N AST RAN and the super convergent as well as spectral elements developed in this work, are discussed in detail in Chapter 5. Modeling aspects of sandwich beam in N AST RAN using various combination of elements available and the performance of four possible models simulated were studied. Validation of all four models in N AST RAN, Super convergent Euler, Timoshenko and Spectral Timoshenko finite elements was carried out by simulating a homogenous I beam by comparing the longitudinal and transverse responses. Studies were carried out to find out the response predictions of a sandwich beam with soft core and all the predictions were compared and discussed. The responses in case of cracks in top or bottom face sheets under the longitudinal and transverse loading were studied in this chapter. In Chapter 6, Parametric studies were carried out for bringing out the sensitiveness of the important specific parameters in overall behaviour and performance of a sandwich beam, using Super convergent and Spectral elements developed. This chapter clearly brings out the various aspects of design of sandwich beam such as material selection of core, geometrical configuration of overall beam and core. Effects of shear modulus, mass density on wave propagation characteristics, effects of thick or thin cores with reference to the face sheets and dynamic effects of core are highlighted. Wave propagation characteristics studies includes the study of wave numbers, group speeds, cut off frequencies for a given configuration and identification of frequency zone of operations. The recommendations for improvement in design of sandwich beams based on the parametric studies are made at the end of chapter. The entire thesis, written in seven Chapters, presents a unified treatment of sandwich beam analysis that will be very useful for designers working in the area.

Esta investigación se enfoca en el análisis de vibración libre de vigas Timoshenko utilizando el método de elementos finitos. Se desarrolla el modelo utilizando el principio de Hamilton y la teoría de vigas Timoshenko que incluye deformaciones por corte. Se asume interpolaciones de alto orden para la aproximación de las variables fundamentales. Los materiales para emplear son isotrópicos. Se implementa un programa para estos materiales en MATLAB. Se comparan resultados con otros obtenidos en la literatura para validar el modelo. Se realiza un estudio paramétrico con una misma longitud y diferentes esbelteces. Se verifica que la formulación sea bastante precisa con resultados muy satisfactorios. / This research focuses on the free vibration analysis of Timoshkenko beams using the finite element method. The model is developed using the Hamilton principle and the Timoshenko beam theory that includes shear deformations. high order interpolations are assumed for the approximation of the fundamental variables. The materials to be used are isotropic. A program for these materials is implemented in MATLAB. Results are compared with others obtained in the literature to validate the model. A parametric study is carried out with the same length and different slenderness. It is verified that the formulation is quite precise with satisfactory results to the investigation. / Trabajo de investigación

[pt] Uma análise não linear geométrica de estruturas, utilizando o Método dos Elementos Finitos (MEF), depende de cinco aspectos: a teoria de flexão, da descrição cinemática, das relações entre deformações e deslocamentos, da metodologia de análise não linear e das funções de interpolação de deslocamentos. Como o MEF é uma solução numérica, a discretização da estrutura fornece grande influência na resposta dessa análise. Contudo, ao se empregar funções de interpolação correspondentes à solução homogênea da equação diferencial do problema, obtêm-se o comportamento exato da estrutura para uma discretização mínima, como ocorre em uma análise linear. Assim, este trabalho visa a integrar as soluções para o problema da não linearidade geométrica, de maneira a tentar reduzir essa influência e permitir uma discretização mínima da estrutura, considerando ainda grandes deslocamentos e rotações. Então, utilizando-se a formulação Lagrangeana atualizada, os termos de ordem elevada no tensor deformação, as teorias de flexão de Euler-Bernoulli e Timoshenko, os algoritmos para solução de problemas não lineares e funções de interpolação, que consideram a influência da carga axial, obtidas da solução da equação diferencial do equilíbrio de um elemento infinitesimal na condição deformada, desenvolve-se um elemento de pórtico espacial com uma formulação completa. O elemento é implementado no Framoop e sua resposta, utilizando-se uma discretização mínima da estrutura, é comparada com formulações usuais, soluções analíticas e com o programa Mastan2 v3.5. Os resultados evidenciam a eficiência da formulação desenvolvida para prever a carga crítica de estruturas planas e espaciais utilizando uma discretização mínima. / [en] A structural geometric nonlinear analysis, using the finite element method (FEM), depends on the consideration of five aspects: the bending theory, the kinematic description, the strain-displacement relations, the nonlinear solution scheme and the interpolation (shape) functions. As MEF is a numerical solution, the structure discretization provides great influence on the analysis response. However, applying shape functions calculated from the homogenous solution of the differential equation of the problem, the exact behavior of the structure is obtained for a minimum discretization, as for a linear analysis. Thus, this work aims to integrate the solutions for the formulations of the geometric nonlinearity problem, in order to reduce this influence and allow a minimum discretization of the structure, also considering, large displacements and rotations. Then, using an updated Lagrangian kinematic description, considering a higher-order Green strain tensor, The Euler-Bernoulli and Timoshenko beam theories, the nonlinear solutions schemes and the interpolation functions, that includes the influence of axial force, obtained directly from the solution of the equilibrium differential equation of an deformed infinitesimal element, a spatial bar frame element is developed using a complete formulation. The element was implemented in the Framoop, and their results, for a minimum discretization, were compared with conventional formulations, analytical solutions and with the software Mastan2 v3.5. Results clearly show the efficiency of the developed formulation to predict the critical load of plane and spatial structures using a minimum discretization.

In der vorliegenden Arbeit werden ein Berechnungs- und Sicherheitskonzept für Verbundbauteile in Holzverbundbauweise erarbeitet und validiert. Damit ist eine globale Dimensionierung bzw. Bemessung dieser Bauteile möglich. Der Fokus der Konzepte liegt dabei auf Anwendungen und Anforderungen im Maschinenbau. Hierzu werden zunächst Grundlagen zu Holzwerkstoffen und dem Lastfall Dreipunktbiegung erarbeitet. Darauf aufbauend werden in Dreipunktbiegeversuchen die Biege- und Schubeigenschaften ausgewählter Sperrhölzer ermittelt. Im nächsten Schritt wird ein Berechnungskonzept zur Vorhersage der Kraft-Verformungs-Kurve von Bauteilen in Holzbauweise unter Verwendung der Materialkennwerte der Einzelelemente erarbeitet und validiert. Die Validierung erfolgt anhand ausgewählter Versuchsmuster in Holzbauweise. Anschließend wird ein an die Sicherheitsanforderungen im Maschinenbau angepasstes semiprobabilistisches Sicherheitskonzept erarbeitet. Abschließend werden anhand praxisnaher Beispiele die Anwendbarkeit der Konzepte validiert und Möglichkeiten bzw. Grenzen aufgezeigt.:1 Einleitung 14 1.1 Motivation 14 1.2 Präzisierung der Aufgabenstellung 16 1.3 Lösungsansätze und Abgrenzung der Arbeit 17 1.4 Aufbau der Arbeit 18 2 Grundlagen 20 2.1 Ingenieurtechnische Grundlagen 20 2.1.1 Begriffsdefinition 20 2.1.2 Zusammenhang zwischen Kraft- und Verformungsgrößen 21 2.1.3 Materialkennwerte 23 2.1.4 Verbundbauteile 24 2.2 Grundlagen der Dreipunktbiegung 25 2.2.1 Allgemeine Modellannahmen 27 2.2.2 Timoshenko-Balkentheorie 27 2.2.3 Einfluss des Stützweiten-Höhen-Verhältnisses 30 2.2.4 Schubkorrekturfaktor 32 2.3 Grundlagen zum Werkstoff Holz 33 2.3.1 Struktureller Aufbau 33 2.3.2 Inhomogenität von Holz 35 2.3.3 Anisotropie des Holzes 35 2.3.4 Mechanische Eigenschaften 39 2.3.5 Äußere Einflussfaktoren auf die Materialeigenschaften von Holz 41 2.4 Holzfurnierlagenverbundwerkstoffe (WVC) 42 2.4.1 Einteilung 42 2.4.2 Struktureller Aufbau von Sperrholz 44 2.4.3 Mechanische Eigenschaften von Sperrholz 44 2.4.4 Spannungszustand von Sperrholz bei Dreipunktbiegebeanspruchung 46 3 Materialcharakterisierung 49 3.1 Stand der Technik 49 3.1.1 Literaturkennwerte, Materialdatenblätter und Leistungserklärungen 49 3.1.2 Aktuelle Normung 49 3.1.3 Kritik am Stand der Technik 51 3.1.4 Ableitung von Anforderungen an Materialversuche und Kennwerte 53 3.1.5 Ziele der Materialcharakterisierung 54 3.2 Grundlagen der Datenanalyse 55 3.2.1 Statistische Lage- und Streumaße 55 3.2.2 Graphische Darstellung empirischer Daten 56 3.2.3 Normalverteilung 57 3.2.4 Statistische Testverfahren 58 3.3 Material und Methoden 60 3.3.1 Material 60 3.3.2 Stützweitenversuch – Versuchssetup und Auswertemethodik 62 3.3.3 Kurzbiegeversuche – Versuchssetup und Auswertemethodik 67 3.4 Kennwertermittlung 68 3.4.1 Materialcharakterisierung von WVC-01 68 3.4.2 Materialcharakterisierung von WVC-02 75 3.4.3 Universelle Spannungs-Dehnungs-Kennwerte 77 4 Berechnungskonzept für Verbundbauteile in Holzbauweise 82 4.1 Annahmen und Eingrenzung 82 4.2 Aktueller Stand der Technik 82 4.2.1 Berechnung der Bauteilsteifigkeit von Verbundbauteilen 82 4.2.2 Berechnung von Versagenspunkten 83 4.2.3 Kritik am Stand der Technik 86 4.2.4 Zielstellung 88 4.3 Berechnungskonzept für Verbundbauteile aus Holzwerkstoffen 88 4.3.1 Prinzipielles Vorgehen 88 4.3.2 Berechnung der Bauteilmodulkennwerte 89 4.3.3 Berechnung der Versagenspunkte 91 4.3.4 Berechnung der Geometrieparameter bei gegebener Mindesttragfähigkeit 92 4.4 Validierung des Berechnungskonzeptes 94 4.4.1 Methodik 94 4.4.2 Aufbau und Geometrie 94 4.4.3 Experimentelle Ergebnisse der Bauteiltests 99 4.4.4 Berechnung der Tragfähigkeit mit Materialkennwerten 100 4.4.5 Berechnung mit universellen Normaldehnungen 105 4.4.6 Diskussion 106 5 Sicherheitskonzept für Holzwerkstoffe im Maschinenbau 108 5.1 Stand der Technik 108 5.1.1 Sicherheit – Definition und Arten 108 5.1.2 Sicherheit im Maschinenbau 113 5.1.3 Sicherheit in der Kunststofftechnik 113 5.1.4 Sicherheit im Ingenieurholzbau – EUROCODE 5 114 5.1.5 Kritik am Stand der Technik 117 5.1.6 Ziel des Sicherheitskonzeptes für Holzwerkstoffe im Maschinenbau 119 5.2 Entwicklung des Sicherheitskonzeptes 120 5.2.1 Analyse der Teilsicherheitsbeiwerte des EUROCODE 5 120 5.2.2 Teilsicherheitsbeiwerte für statische Lastfälle des Maschinenbau 122 5.2.3 Beiwert zur Berücksichtigung der Kennwertstreuung 125 5.2.4 Sicherheitsbezogene Klassifizierung von Maschinenbauanwendungen 126 5.2.5 Ableitung globaler Sicherheitsfaktoren 129 5.2.6 Zusammenfassung 132 5.3 Validierung 133 5.3.1 Bemessung nach EUROCODE 5 133 5.3.2 Bemessung nach Sicherheitskonzept für Maschinenbau 135 5.3.3 Vergleich EUROCODE 5 und Sicherheitskonzept für Maschinenbau 136 6 Anwendbarkeitsstudie 139 6.1 Bemessung von Hohlprofilen 139 6.2 Dimensionierung von Hohlprofilen 142 6.3 Globale Bemessung komplexer Bauteile 144 6.4 Anschließende Bemessungsaufgaben 147 7 Zusammenfassung und Ausblick 149 7.1 Zusammenfassung 149 7.2 Ausblick 151 8 Verzeichnisse 152 9 Anhang 169 9.1 Anhang zu Kapitel 2 169 9.2 Anhang zu Kapitel 3 171 9.3 Anhang zu Kapitel 4 194 9.4 Anhang zu Kapitel 5 207 9.5 Anhang zu Kapitel 6 212 / In the present work, a calculation and safety concept for composite components in wood composite construction is developed and validated. This enables a global dimensioning of these components. The focus of the concepts is on applications and requirements in mechanical engineering. For this purpose, the basics of wood-based materials and the three-point bending load case are first elaborated. Based on this, the bending and shear properties of selected plywood are determined in three-point bending tests. In the next step, a calculation concept for predicting the force-deformation-curve of components in timber construction using the material parameters of the individual elements will be developed and validated. The validation is based on selected test components. A semi-probabilistic safety concept adapted to the safety requirements in mechanical engineering is then developed. Finally, using practical examples, the applicability of the concepts is determined and possibilities and limits are shown.:1 Einleitung 14 1.1 Motivation 14 1.2 Präzisierung der Aufgabenstellung 16 1.3 Lösungsansätze und Abgrenzung der Arbeit 17 1.4 Aufbau der Arbeit 18 2 Grundlagen 20 2.1 Ingenieurtechnische Grundlagen 20 2.1.1 Begriffsdefinition 20 2.1.2 Zusammenhang zwischen Kraft- und Verformungsgrößen 21 2.1.3 Materialkennwerte 23 2.1.4 Verbundbauteile 24 2.2 Grundlagen der Dreipunktbiegung 25 2.2.1 Allgemeine Modellannahmen 27 2.2.2 Timoshenko-Balkentheorie 27 2.2.3 Einfluss des Stützweiten-Höhen-Verhältnisses 30 2.2.4 Schubkorrekturfaktor 32 2.3 Grundlagen zum Werkstoff Holz 33 2.3.1 Struktureller Aufbau 33 2.3.2 Inhomogenität von Holz 35 2.3.3 Anisotropie des Holzes 35 2.3.4 Mechanische Eigenschaften 39 2.3.5 Äußere Einflussfaktoren auf die Materialeigenschaften von Holz 41 2.4 Holzfurnierlagenverbundwerkstoffe (WVC) 42 2.4.1 Einteilung 42 2.4.2 Struktureller Aufbau von Sperrholz 44 2.4.3 Mechanische Eigenschaften von Sperrholz 44 2.4.4 Spannungszustand von Sperrholz bei Dreipunktbiegebeanspruchung 46 3 Materialcharakterisierung 49 3.1 Stand der Technik 49 3.1.1 Literaturkennwerte, Materialdatenblätter und Leistungserklärungen 49 3.1.2 Aktuelle Normung 49 3.1.3 Kritik am Stand der Technik 51 3.1.4 Ableitung von Anforderungen an Materialversuche und Kennwerte 53 3.1.5 Ziele der Materialcharakterisierung 54 3.2 Grundlagen der Datenanalyse 55 3.2.1 Statistische Lage- und Streumaße 55 3.2.2 Graphische Darstellung empirischer Daten 56 3.2.3 Normalverteilung 57 3.2.4 Statistische Testverfahren 58 3.3 Material und Methoden 60 3.3.1 Material 60 3.3.2 Stützweitenversuch – Versuchssetup und Auswertemethodik 62 3.3.3 Kurzbiegeversuche – Versuchssetup und Auswertemethodik 67 3.4 Kennwertermittlung 68 3.4.1 Materialcharakterisierung von WVC-01 68 3.4.2 Materialcharakterisierung von WVC-02 75 3.4.3 Universelle Spannungs-Dehnungs-Kennwerte 77 4 Berechnungskonzept für Verbundbauteile in Holzbauweise 82 4.1 Annahmen und Eingrenzung 82 4.2 Aktueller Stand der Technik 82 4.2.1 Berechnung der Bauteilsteifigkeit von Verbundbauteilen 82 4.2.2 Berechnung von Versagenspunkten 83 4.2.3 Kritik am Stand der Technik 86 4.2.4 Zielstellung 88 4.3 Berechnungskonzept für Verbundbauteile aus Holzwerkstoffen 88 4.3.1 Prinzipielles Vorgehen 88 4.3.2 Berechnung der Bauteilmodulkennwerte 89 4.3.3 Berechnung der Versagenspunkte 91 4.3.4 Berechnung der Geometrieparameter bei gegebener Mindesttragfähigkeit 92 4.4 Validierung des Berechnungskonzeptes 94 4.4.1 Methodik 94 4.4.2 Aufbau und Geometrie 94 4.4.3 Experimentelle Ergebnisse der Bauteiltests 99 4.4.4 Berechnung der Tragfähigkeit mit Materialkennwerten 100 4.4.5 Berechnung mit universellen Normaldehnungen 105 4.4.6 Diskussion 106 5 Sicherheitskonzept für Holzwerkstoffe im Maschinenbau 108 5.1 Stand der Technik 108 5.1.1 Sicherheit – Definition und Arten 108 5.1.2 Sicherheit im Maschinenbau 113 5.1.3 Sicherheit in der Kunststofftechnik 113 5.1.4 Sicherheit im Ingenieurholzbau – EUROCODE 5 114 5.1.5 Kritik am Stand der Technik 117 5.1.6 Ziel des Sicherheitskonzeptes für Holzwerkstoffe im Maschinenbau 119 5.2 Entwicklung des Sicherheitskonzeptes 120 5.2.1 Analyse der Teilsicherheitsbeiwerte des EUROCODE 5 120 5.2.2 Teilsicherheitsbeiwerte für statische Lastfälle des Maschinenbau 122 5.2.3 Beiwert zur Berücksichtigung der Kennwertstreuung 125 5.2.4 Sicherheitsbezogene Klassifizierung von Maschinenbauanwendungen 126 5.2.5 Ableitung globaler Sicherheitsfaktoren 129 5.2.6 Zusammenfassung 132 5.3 Validierung 133 5.3.1 Bemessung nach EUROCODE 5 133 5.3.2 Bemessung nach Sicherheitskonzept für Maschinenbau 135 5.3.3 Vergleich EUROCODE 5 und Sicherheitskonzept für Maschinenbau 136 6 Anwendbarkeitsstudie 139 6.1 Bemessung von Hohlprofilen 139 6.2 Dimensionierung von Hohlprofilen 142 6.3 Globale Bemessung komplexer Bauteile 144 6.4 Anschließende Bemessungsaufgaben 147 7 Zusammenfassung und Ausblick 149 7.1 Zusammenfassung 149 7.2 Ausblick 151 8 Verzeichnisse 152 9 Anhang 169 9.1 Anhang zu Kapitel 2 169 9.2 Anhang zu Kapitel 3 171 9.3 Anhang zu Kapitel 4 194 9.4 Anhang zu Kapitel 5 207 9.5 Anhang zu Kapitel 6 212