The purpose of the work presented in this thesis is to implement co-rotational beam elements and branch-switching procedures in order to analyse elastic and elastoplasticinstability problems. For the 2D beam elements, the co-rotational framework is taken from Crisfield [23]. The main objective is to compare three different local elasto-plastic elements. The 3D co-rotational formulation is based on the work of Pacoste and Eriksson [73],with new items concerning the parameterisation of the finite rotations, the definitionof the local frame, the inclusion of warping effects through the introduction of aseventh nodal degree of freedom and the consideration of rigid links. Differenttypes of local formulations are considered, including or not warping effects. It isshown that at least some degree of non-linearity must be introduced in the localstrain definition in order to obtain correct results for certain classes of problems. Within the present approach any cross-section can be modelled, and particularly, the centroid and shear center are not necessarily coincident.Plasticity is introduced via a von Mises material with isotropic hardening. Numericalintegration over the cross-section is performed. At each integration point, theconstitutive equations are solved by including interaction between the normal andshear stresses. Concerning instabilities, a new numerical method for the direct computation of elasticcritical points is proposed. This is based on a minimal augmentation procedure asdeveloped by Eriksson [32–34]. In elasto-plasticity, a literature survey, mainly concernedwith theoretical aspects is first presented. The objective is to get a completecomprehension of the phenomena and to give a basis for the two branch-switchingprocedures presented in this thesis.A large number of examples are used in order to assess the performances of the elements and the path-following procedures. / QC 20100512

The purpose of this thesis is to develop corotational beam elements for the nonlinear dynamic analyse of flexible beam structures. Whereas corotational beam elements in statics are well documented, the derivation of a corotational dynamic formulation is still an issue. In the first journal paper, an efficient dynamic corotational beam formulation is proposed for 2D analysis. The idea is to adopt the same corotational kinematic description in static and dynamic parts. The main novelty is to use cubic interpolations to derive both inertia terms and internal terms in order to capture correctly all inertia effects. This new formulation is compared with two classic formulations using constant Timoshenko and constant lumped mass matrices. In the second journal paper, several choices of parametrization and several time stepping methods are compared. To do so, four dynamic formulations are investigated. The corotational method is used to develop expressions of the internal terms, while the dynamic terms are formulated into a total Lagrangian context. Theoretical derivations as well as practical implementations are given in detail. Their numerical accuracy and computational efficiency are then compared. Moreover, four predictors and various possibilities to simplify the tangent inertia matrix are tested. In the third journal paper, a new consistent beam formulation is developed for 3D analysis. The novelty of the formulation lies in the use of the corotational framework to derive not only the internal force vector and the tangent stiffness matrix but also the inertia force vector and the tangent dynamic matrix. Cubic interpolations are adopted to formulate both inertia and internal local terms. In the derivation of the dynamic terms, an approximation for the local rotations is introduced and a concise expression for the global inertia force vector is obtained. Four numerical examples are considered to assess the performance of the new formulation against two other ones based on linear interpolations. Finally, in the fourth journal paper, the previous 3D corotational beam element is extended for the nonlinear dynamics of structures with thin-walled cross-section by introducing the warping deformations and the eccentricity of the shear center. This leads to additional terms in the expressions of the inertia force vector and the tangent dynamic matrix. The element has seven degrees of freedom at each node and cubic shape functions are used to interpolate local transversal displacements and axial rotations. The performance of the formulation is assessed through five examples and comparisons with Abaqus 3D-solid analyses. / <p>QC 20131017</p>

Damage identification is an important topic in structural assessment and structural health monitoring (SHM). Vibration-based identification techniques use modal data to identify the existence, location and severity of possible damages in structures, often via a numerical model updating procedure. Among other factors influencing the practicality and reliability of a damage identification approach, two are of primary interest to this study. The first one concerns the amount and quality of modal data that can be used as ‘response’ data for the model updating. It is generally recognised that natural frequencies can be measured with relatively high accuracy; however, their number is limited. Mode shapes, on the other hand, are susceptible to larger measurement errors. Seeking additional modal frequency data is therefore of significant value. The second one concerns the errors at the numerical (finite element) model level, particularly in the representation of the effect of damage on the dynamic properties of the structure. An inadequate damage model can lead to inaccurate and even false damage identification. The first part of the thesis is devoted to enhancing the modal dataset by extracting the so called ‘artificial boundary condition’ (ABC) frequencies in a real measurement environment. The ABC frequencies correspond to the natural frequencies of the structure with a perturbed boundary condition, but can be generated without the need of actually altering the physical support condition. A comprehensive experimental study on the extraction of such frequencies has been conducted. The test specimens included steel beams of relatively flexible nature, as well as thick and stiffer beams made from metal material and reinforced concrete, to cover the typical variation of the dynamic characteristics of real-life structures in a laboratory condition. The extracted ABC frequencies are subsequently applied in the damage identification in beams. Results demonstrate that it is possible to extract the first few ABC frequencies from the modal testing in different beam settings for a variety of ABC incorporating one or two virtual pin supports. The inclusion of ABC frequencies enables the identification of structural damages satisfactorily without the necessity to involve the mode shape information. The second part of the thesis is devoted to developing a robust model updating and damage identification approach for beam cracks, with a special focus on thick beams which present a more challenging problem in terms of the effect of a crack than slender beams. The priority task has been to establish a crack model which comprehensively describes the effect of a crack to reduce the modelling errors. A cracked Timoshenko beam element model is introduced for explicit beam crack identification. The cracked beam element model is formulated by incorporating an additional flexibility due to a crack using the fracture mechanics principles. Complex effects in cracked thick beams, including shear deformation and coupling between transverse and longitudinal vibrations, are represented in the model. The accuracy of the cracked beam element model for predicting modal data of cracked thick beams is first verified against numerically simulated examples. The consistency of predictions across different modes is examined in comparison with the conventional stiffness reduction approach. Upon satisfactory verification, a tailored model updating procedure incorporating an adaptive discretisation approach is developed for the implementation of the cracked beam element model for crack identification. The updating procedure is robust in that it has no restriction on the location, severity and number of cracks to be identified. Example updating results demonstrate that satisfactory identification can be achieved for practically any configurations of cracks in a beam. Experimental study with five solid beam specimens is then carried out to further verify the developed cracked beam element model. Both forward verification and crack damage identification with the tested beams show similar level of accuracy to that with the numerically simulated examples. The cracked beam element model can be extended to crack identification of beams with complex cross sections. To do so the additional flexibility matrix for a specific cross-section type needs to be re-formulated. In the present study this is done for box sections. The stress intensity factors (SIF) for a box section as required for the establishment of the additional flexibility matrix are formulated with an empirical approach combining FE simulation, parametric analysis and regression analysis. The extended cracked beam element model is verified against both FE simulated and experimentally measured modal data. The model is subsequently incorporated in the crack identification for box beams. The successful extension of the cracked beam element model to the box beams paves the way for similar extension to the crack identification of other types of sections in real-life engineering applications.

L’objectif de cette thèse est de proposer des éléments finis poutres corotationnels 2D et 3D pour l’analyse du comportement dynamique non-linéaire des structures à barres. La contribution majeure de cette thèse est l’utilisation de fonctions d’interpolations cubiques à la fois pour la détermination de l’expression des efforts internes mais aussi celle des termes d’inertie. En négligeant le carré du déplacement transversal dans le repère local, une expression analytique des termes dynamiques en 2D est obtenue. Sur base d’une étude comparative approfondie sur la paramétrisation des rotations et les algorithmes d’intégration temporelle et d’une approximation des rotations locales, nous proposons deux éléments finis poutre 3D précis et robustes. Contrairement au premier, le second élément 3D prend en compte les déformations de gauchissement et l'excentricité du centre de cisaillement. Les diverses comparaisons réalisées démontrent la pertinence des formulations proposées. / The purpose of this thesis is to propose several corotational beam formulations for both 2D and 3D nonlinear dynamic analyse of flexible structures. The main novelty of these formulations is that the cubic interpolation functions are used to derive not only the internal force vector and the tangent stiffness matrix but also the inertial force vector and the dynamic matrix. By neglecting the quadratic terms of the local transversal displacements, closed-form expressions for the inertial terms are obtained for 2D problems. Based on an extensive comparative study of the parameterizations of the finite rotations and the time stepping method, and by adopting an approximation of the local rotations, two consistent and effective beam formulations for 3D dynamics are developed. In contrast with the first formulation, the second one takes into account the warping deformations and the shear center eccentricity. The accuracy of these formulations is demonstrated through several numerical examples.

The dissertation deals with solution of beam and frame structure considering material nonlinearity. The finite elements method (FEM) was used as calculation method. The objective of the dissertation was to develop and test finite beam element considering material nonlinearity. A detailed analysis of the problem provided a group of formulations of beam element. Part of the dissertation´s results has been, in several forms, implemented in commercial software. Beam element focused on solution of stress over solid cross sections is introduced in the theoretical part. In terms of topology it is a classical prismatic beam element with two nodes. Six degrees of freedom for translations and rotations are defined in each node plus 7-th degree of freedom for warping function from torsion. Load form axial force, bending moments, primary torsion, shear forces and eventually bimoment for secondary torsion were considered in cross section. Several variants of formulations were created according to inclusion of loading components into material nonlinearity and according to numeric integration method. Inclusion of geometric nonlinearity and fire resistance calculation are discussed in the dissertation. The above mentioned formulations were tested on prototypes as described in the application part which also provides information on the general procedure, architecture and technologies used for implementation of knowledge from the theoretical part into commercial FEM software. The dissertation shows implementation of plasticity for shell, solutions of cross section characteristics and stress on cross section, implementation of beam element with material nonlinearity and module for fire resistance of column. The above mentioned implementations of theoretical conclusions are the main outputs of the dissertation. These implementations are available in thousands of installations throughout Europe where being used at projection of significant constructions.

Examensarbetet behandlar ett brotvärsnitt som inte entydigt kan betraktas som ett balktvärsnitt eller plattvärsnitt. Med de måttdefinitioner som används vid broprojektering ska en plattkonstruktion ha en bredd som är fem gånger höjden, annars ska konstruktionen ses som en balk där även balkens längd definieras att vara större än tre gånger höjden. Brotvärsnittet som studeras i detta examensarbete kan alltså definieras både som ett plattvärsnitt och som ett balktvärsnitt. Målet med arbetet är att undersöka om det är möjligt att finna en metod att konstruera denna typ av tvärsnitt som befinner sig i gränslandet mellan två definitioner. Skillnaderna mellan en plattas och en balks verkningssätt ligger i att plattan antas bära last i två riktningar medan en balk enbart bär last i en riktning. Examensarbetet är genomfört i sammarbete med WSP Bro- och vattenbyggnad i Örebro, som konstruerade en bro med just detta tvärsnitt. Bro 344 över parkstråk i trafikplats Rinkeby å ramp mot Ärvinge, är 181 m lång bro i 9 spann och finns belägen vid trafikplats Rinkeby som är en del utav Trafikverkets projekt, E18 Hjulsta – Kista. Lasterna som används i analyserna är betongens egentyngd, utbredd last av beläggning och vertikala trafiklaster. I ett första skede i arbetet analyseras modellerna med rörliga trafiklaster. Det framkom dock under arbetets gång att förenklingar vad gäller trafiklasterna måste göras då arbetet skulle bli för omfattande annars. En statisk boggilast placeras ut i ett spann mitt i mellan dess tredjedelspunkt och halva spannlängden. Beräkningar utförs i en mjukvara där modellen både byggs upp av skalelement som en långsträckt platta där snittkrafter kommer ut som enhet per meter och med balkelement som en halvinspänd balk där snittkrafter kommer ut i enhet per balk. Mjukvaran som används är ett tredimensionellt finit element program, SOFISTIK, som likt många andra FE-program erbjuder användarvänliga modelleringsmiljöer, hanterar rörliga laster och har en mängd inbyggda moduler och funktioner. Beräkningarna som sedan utvärderas och jämförs är dels SOFISTIKs olika resultat för skalmodellen och balkmodellen. Där dimensionerande armeringsmängder beräknas för max fältmoment och max stödmoment. Dessa resultat från SOFISTIKs skalmodell respektive balkmodell jämförs också med resultat från de mjukvaror som användes vid dimensioneringen från början, vilket var för skalmodellanalysen Brigade Standard och för balkanalysen Strip Step 3. Armeringsmängderna jämförs slutligen genom att studera tre fall: •   Skalmodell SOFISTIK - Brigade Standard •   Balkmodell SOFISTIK - Strip Step 3 •   SOFISTIK skalmodell – balkmodell Jämförelserna visar att både skalmodellerna från de olika programmen (SOFISTIK – Brigade Standard) och balkmodellerna från de olika programmen (SOFISTIK – Strip Step 3) ger likvärdiga armeringsmängder vilket ger en trygg verifiering av modellerna. Vidare visar jämförelse mellan skal- och balkmodell i SOFISTIK att balkmodellen ger avsevärt högre armeringsmängder, både i fält och över stöd. ar / The aim of this Master thesis is to study a cross section of a bridge that cannot be unambiguously considered to be defined as a beam cross-section or a slab cross-section. With the given definitions used in bridge engineering, a slab construction has to have a width wider than five times the height, otherwise it should be regarded as a beam construction. The length of a beam construction is also defined to be three times longer than the height. The cross section in this thesis can thus be treated as both a slab cross-section and as a beam cross-section. The aim of this work is to investigate whether it is possible to find a method to construct this type of cross-section that falls within both these two definitions. The difference in mode of action between a plate and a beam is that the plate is assumed to carry loads in two directions while a beam only carries load in one direction. The work done in this report has been performed in cooperation with the consulting company WSP Bridge &amp; Hydraulic Design in Örebro who has constructed a bridge with the studied section, Bro 344 över parkstråk i trafikplats Rinkeby å ramp mot Ärvinge. This bridge is 181 m long in 9 spans and are located at the traffic interchange Rinkeby which is part of the Swedish Transport Administration project, E18 Hjulsta - Kista. The loads, which are discussed and analyzed are the deadweight of the concrete, distributed load of road surface and vertical traffic loads. In the first stage of the work the models are being analyzed with moving traffic loads, it appears, however, during the process that simplifications in terms of the moving traffic loads must be made, when the work would be too wide otherwise. A static bogie-load is deployed in one of the spans, in between the third point and half the span length. Calculations are performed using a computer software, in which the bridge is modeled both by shell elements and by beam elements. The shell-model is created as an elongated plate section in which the force comes out as unit per meter. The beam-model is considered as a semi-restrained beam in which the section forces come out in unit for the whole beam. Software used is a three-dimensional finite element program, SOFISTIK. As many other FEprograms SOFISTIK provide a user-friendly modeling workspace, it handles variable and moving loads and has a variety of embedded modules and functions. The calculations which are being evaluated and compared, is at the first hand the different results in between the shell-model and the beam-model from the models made in SOFISTIK. The amounts of designing reinforcement are calculated for the maximum bending moment and for the minimum bending moment. Those results, also compares with results from other software, the software used in the design from the beginning, which for the shell-analyze the software Brigade Standard and for the beam-analyze the software Strip Step 3. The amounts of design reinforcement are finally compared by studying three cases: •   The Shell-model from SOFISTIK - Brigade Standard •   Beam-model from SOFISTIK - Strip Step 3 •   SOFISTIK the shell-model – the beam-model The comparisons show that both the shell-models from the two different programs (SOFISTIK and Brigade Standard) and the beam-models from the different two programs (SOFISTIK - Strip Step 3) give equivalent amounts of reinforcement which provides a secure verification of the models. Furthermore the comparison between the shell-model and the beam-model, in SOFISTIK , shows that the beam-model provides significantly higher amounts of reinforcement in both the field and at the support.

Flexible beam structures are popular in civil and mechanical engineering. Many of these structures undergo large displacements and finite rotations, but with small deformations. Their dynamic behaviors are usually investigated using finite beam elements. A well known method to derive such beam elements is the corotational approach. This method has been extensively used in nonlinear static analysis. However, its application in nonlinear dynamics is rather limited. The purpose of this thesis is to investigate the nonlinear dynamic behavior of flexible beam structures using the corotational method. For the 2D case, a new dynamic corotational beam formulation is presented. The idea is to adopt the same corotational kinetic description in static and dynamic parts. The main novelty is to use cubic interpolations to derive both inertia terms and internal terms in order to capture correctly all inertia effects. This new formulation is compared with two classic formulations using constant Timoshenko and constant lumped mass matrices. This work is presented in the first appended journal paper. For the 3D case, update procedures of finite rotations, which are central issues in development of nonlinear beam elements in dynamic analysis, are discussed. Three classic and one new formulations of beam elements based on the three different parameterizations of the finite rotations are presented. In these formulations, the corotational method is used to develop expressions of the internal forces and the tangent stiffness matrices, while the dynamic terms are formulated into a total Lagrangian context. Many aspects of the four formulations are investigated. First, theoretical derivations as well as practical implementations are given in details. The similarities and differences between the formulations are pointed out. Second, numerical accuracy and computational efficiency of these four formulations are compared. Regarding efficiency, the choice of the predictor at each time step and the possibility to simplify the tangent inertia matrix are carefully investigated. This work is presented in the second appended journal paper. To make this thesis self-contained, two chapters concerning the parametrization of the finite rotations and the derivation of the 3D corotational beam element in statics are added. / QC 20120521

The purpose of this thesis is to propose several corotational beam formulations for both 2D and 3D nonlinear dynamic analyse of flexible structures. The main novelty of these formulations is that the cubic interpolation functions are used to derive not only the internal force vector and the tangent stiffness matrix but also the inertial force vector and the dynamic matrix. By neglecting the quadratic terms of the local transversal displacements, closed-form expressions for the inertial terms are obtained for 2D problems. Based on an extensive comparative study of the parameterizations of the finite rotations and the time stepping method, and by adopting an approximation of the local rotations, two consistent and effective beam formulations for 3D dynamics are developed. In contrast with the first formulation, the second one takes into account the warping deformations and the shear center eccentricity. The accuracy of these formulations is demonstrated through several numerical examples.

Dans une problématique d'analyse de la durabilité des structures en béton armé, la quantification de la localisation des déformations et des propriétés des fissures sont deux points clés. Ce travail présente une méthode permettant, dans le cadre de la mécanique des milieux continus, d'améliorer la description de l'évolution de la localisation des déformations. En se basant sur une approche continue du problème, l'évolution des nonlinéarités dans le béton est décrite au travers d'un modèle d'endommagement régularisé. Pour améliorer la description de la localisation des déformations, une modification de la méthode de régularisation nonlocale intégrale sur les variables internes est proposée. L'influence de l'état de contrainte sur les interactions nonlocales est introduite dans la régularisation, afin de prendre en compte la dégradation de la structure ainsi que l'influence des conditions aux limites sur les interactions nonlocales. Cette méthode, implantée dans le code aux éléments finis Cast3M, est validée sur différents cas tests analysant l'évolution des nonlinéarités de l'enclenchement de l'endommagement jusqu'à la rupture et permet notamment de résoudre des pathologies identifiées pour la méthode nonlocale originale. La comparaison avec des résultats expérimentaux montre également la capacité du modèle à décrire l'évolution de la fissuration dans une structure. Le modèle développé est ensuite utilisé pour analyser le comportement de structures en béton armé et sert de base pour introduire une description de la fissuration dans une modélisation simplifiée de type poutre multifibre. A partir de calcul 3D sur des éléments en béton armé utilisant le modèle développé, une loi uniaxiale est identifiée pour déterminer la fissuration dans une fibre en fonction de l'énergie dissipée par le modèle d'endommagement. Une comparaison avec des résultats expérimentaux est effectuée et montre la capacité de cette approche simplifiée à estimer la fissuration. / For the durability analysis of reinforced concrete structures, the modelling of strain localization and the estimation of cracking properties are hot topics. This work introduces a method allowing, in the framework of continuous mechanics, to improve the description of the evolution of strain localization. Based on a continuous description of the problem, the evolution of nonlinearities in concrete is described with a regularized damage model. In order to improve the description of strain localization, a modification of the nonlocal integral regularization method is proposed. The influence of the stress state on the nonlocal interactions is introduced in the regularization method, in order to take into account the degradation of the structure (decrease of the bearing capacities) as well as the influence of free boundary conditions. This method, implemented in the finite element code Cast3M, is validated against several cases of study, by analyzing the evolution of nonlinearities from damage initiation up to failure. It allows solving several pathologies pointed out for the original nonlocal method. The comparison with experimental results shows also the capacity of the proposed model to describe the evolution of cracking in a structure. Then, the model developed is used to analyse the behaviour of reinforced concrete structures and to develop a method to quantify cracking in a multifiber beam element modelling. From 3D calculation on reinforced concrete element with the new nonlocal model developed, a uniaxial law is identified in order to estimate cracking as a function of the energy dissipated by the damage model. A comparison with experimental data is performed and shows the potentiality of this simplified approach to estimate cracking.

The thesis deals with computational modelling of ceramic foams and analysis of influence of structural manufacturing defects (like broken struts, closed pores and material clumps) have on foam strength. Model of foam geometry was discretized using beam elements in order to decrease computational cost. In place where several struct join, rigid beam element was used so that the increased stiffness is better modelled. Closed walls of pores were modelled and discretised by shell elements. Influence of loading direction was analysed on foams containing no defects and then influence of amount of defects in foam on strength was further analysed. Highest strength show foams created by cells whose structs are oriented in direction of loading. These were losing strength most rapidly. Foam with structure of rhombic dodecahedral cell was least influenced by presence of closed pore defects but it also showed lowest strength even without defects. Cells with struts oriented in direction of loading experienced biggest drop in strength. Kelvin cell is a compromise. It was shown that difference in strength of strut with constant and varying cross-section is tenths of percent.