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An asymptotic expansion-based method for a spectral approach in equivalent statistical linearization Canor, Thomas ; Blaise, Nicolas ; Denoël, Vincent in Probabilistic Engineering Mechanics (2014), 38 Equivalent linearization consists in replacing a nonlinear system with an equivalent linear one whose parameters are tuned with regard to the minimization of a suitable function. In particular, the ... [more ▼] Equivalent linearization consists in replacing a nonlinear system with an equivalent linear one whose parameters are tuned with regard to the minimization of a suitable function. In particular, the Gaussian equivalent linearization expresses the properties of an equivalent linear system in terms of the mean vector and the covariance matrix of the responses, which are the unknowns of the optimization problem in a spectral approach. Even though the system has been linearized, the resulting set of equations is nonlinear. The computational effort in this method pertains to the solution of a possibly large set of nonlinear algebraic equations involving integrals and inversions of full matrices. This work proposes to develop and apply an asymptotic expansion-based method to facilitate and to improve the statistical linearization for large nonlinear structures. The proposed developments demonstrate that for slightly to moderately coupled nonlinear systems, the equivalent linearization can be applied with an appropriate modal approach and eventually seen as a convergent series initiated with the stochastic response of a main decoupled linear system. With this method, the computational effort is attractively reduced, the conditioning of the set of nonlinear algebraic equations is improved and inversion of full transfer matrices and repeated integrations are avoided. The paper gives a formal description of the method and illustrates its implementation and performances with the computation of stationary responses of nonlinear structures subject to coherent random excitation fields. [less ▲] Detailed reference viewed: 97 (28 ULg)Calculation of third order joint acceptance function for line-like structures Blaise, Nicolas ; Canor, Thomas ; Denoël, Vincent in Proceedings of the XIII Conference of the Italian Association for Wind Engineering (2014, June 24) Detailed reference viewed: 41 (12 ULg)New perspectives on probabilistic methods for nonlinear transient dynamics in civil engineering Canor, Thomas Doctoral thesis (2014) The objective of this thesis is to develop probabilistic methods to perform nonlinear transient analysis in civil engineering applications. The manuscript is divided into two parts. For large-dimensional ... [more ▼] The objective of this thesis is to develop probabilistic methods to perform nonlinear transient analysis in civil engineering applications. The manuscript is divided into two parts. For large-dimensional structures subject to coherent random loads, an asymptotic expansion-based method is used to improve linear evolutionary spectral analysis. Then, this method is extended to equivalent statistical linearization by a multiple timescales approach. The second part of the thesis is focused on small-dimensional systems. It explores the solution of the Fokker-Planck equation with the smoothed particle hydrodynamics method. This meshless Lagrangian method particularly ensures the conservation and the positivity of the probability field. The accuracy and efficiency of both approaches are compared with Monte-Carlo simulations. [less ▲] Detailed reference viewed: 111 (33 ULg)A Fast Newton-Raphson Method in Stochastic Linearization Canor, Thomas ; Blaise, Nicolas ; Denoël, Vincent in Proceedings of the Ninth International Conference on Structural Dynamics (2014) Owing to its accessible implementation and rapidity, the equivalent linearization has become a common probabilistic approach for the analysis of large-dimension nonlinear structures, as encountered in ... [more ▼] Owing to its accessible implementation and rapidity, the equivalent linearization has become a common probabilistic approach for the analysis of large-dimension nonlinear structures, as encountered in earthquake and wind engineering. It consists in replacing the nonlinear system by an equivalent linear one, by tuning the parameters of the equivalent system, in order to minimize some discrepancy error. Consequently classical analysis tools such as the spectral analysis may be reconditioned to approximate the solution of structures with slight to moderate nonlinearities. The tuning of the equivalent parameters requires the solution of a set of nonlinear algebraic equations involving integrals. It is typically performed with the fixed-point algorithm, which is known to behave poorly in terms of convergence. We therefore advocate for the use and implementation of a Newton-Raphson approach, which behaves much better, even in its dishonest formulation. Unfortunately, this latter option requires the costly construction of a Jacobian matrix. In the approach described in this paper, this issue is answered by introducing a series expansion method that provides a fast and accurate estimation of the residual function (whose solution provides the equivalent parameters) and a fast and approximate estimation of the Jacobian matrix. An illustration demonstrate the good accuracy obtained with the proposed method. [less ▲] Detailed reference viewed: 82 (11 ULg)Vibrations of cables with bending stiffness by an asymptotic approach Canor, Thomas ; Denoël, Vincent in Zingoni, Alphose (Ed.) Fifth International Conference on Structural Engineering, Mechanics & Computation, Cape Town, 1-4 September 2013 (2013, September) Vibrations of cables with a small bending stiffness concerns many engineering applications such as the fatigue assessment of stay cables. With the finite element (FE) method, the analysis can be performed ... [more ▼] Vibrations of cables with a small bending stiffness concerns many engineering applications such as the fatigue assessment of stay cables. With the finite element (FE) method, the analysis can be performed with nonlinear truss elements, but bending effects are not taken into account. Otherwise, beam elements can be used, but the smallness of the bending stiffness may lead to numerical instability and mediocre results in boundary regions. In this context, the paper presents an alternative method to calculate the time evolution of the profile of bending moments in boundary layers of cables, avoiding heavy FE analysis. The developments combine the theory of vibrations of extensible rods with asymptotic methods. The equations are decoupled between the slow dynamics of the boundary regions and the fast dynamics of the span. Then, a composite solution is constructed by means of a matched asymptotic procedure. [less ▲] Detailed reference viewed: 67 (9 ULg)Equivalent static wind loads for structures with non-proportional damping Blaise, Nicolas ; Canor, Thomas ; Denoël, Vincent in Zingoni, A (Ed.) Fifth International Conference on Structural Engineering, Mechanics & Computation (2013, September) Detailed reference viewed: 47 (15 ULg)Uncoupled spectral analysis with non-proportional damping Canor, Thomas ; Blaise, Nicolas ; Denoël, Vincent in Proceedings of the European-African Conference on Wind Engineering (2013, July 07) The use of normal modes of vibration in the analysis of structures with non-proportional damping reduces the number of governing equations, but does not decouple them. A common practice consists in ... [more ▼] The use of normal modes of vibration in the analysis of structures with non-proportional damping reduces the number of governing equations, but does not decouple them. A common practice consists in decoupling the equations by disregarding the off-diagonal elements of the modal damping matrix. This paper proposes a method based on the asymptotic expansion of the modal transfer matrix to take into account the modal coupling in Gaussian spectral analysis. First, the mathematical background is introduced, then the relevance of the method is illustrated within the context of the analysis of a large and real structure submitted to wind loadings. [less ▲] Detailed reference viewed: 67 (8 ULg)Equivalent static wind loads for structures with non-proportional damping Blaise, Nicolas ; Canor, Thomas ; Denoël, Vincent in Proceedings of the 11th International Conference on Recent Advances in Structural dynamics (2013, July) In current practice, wind structural design is often carried out using the concept of equivalent static wind loads. The main characteristic of such loadings is to reproduce, with static analyses, the same ... [more ▼] In current practice, wind structural design is often carried out using the concept of equivalent static wind loads. The main characteristic of such loadings is to reproduce, with static analyses, the same extreme structural responses as those resulting from a formal buffeting analysis. This paper proposes a method for the computation of equivalent static wind loads for structures with slight non-proportional damping in a modal framework. Because of the smallness of the out-of diagonal terms, this method is based on recent developments related to asymptotic expansion of the modal transfer matrix of such structures. As a main benefit, the static loading is described as a perturbation of the equivalent loading that would be obtained for the uncoupled system. The main contribution of this paper is to formalize the expression of the correction terms resulting from the non-proportionality of damping. The method is presented with a detailed illustrative example. [less ▲] Detailed reference viewed: 50 (13 ULg)Transient Fokker-Planck-Kolmogorov equation solved with smoothed particle hydrodynamics method Canor, Thomas ; Denoël, Vincent in International Journal for Numerical Methods in Engineering (2013), 94(6), 535553 Probabilistic theories aim at describing the properties of systems subjected to random excitations by means of statistical characteristics such as the probability density function (pdf). The time ... [more ▼] Probabilistic theories aim at describing the properties of systems subjected to random excitations by means of statistical characteristics such as the probability density function (pdf). The time evolution of the pdf of the response of a randomly excited deterministic system is commonly described with the transient Fokker-Planck-Kolmogorov equation (FPK). The FPK equation is a conservation equation of a hypothetical or abstract fluid, which models the transport of probability. This paper presents a generalized formalism for the resolution of the transient FPK equation using the well-known mesh-free Lagrangian method, Smoothed Particle Hydrodynamics (SPH). Numerical implementation shows notable advantages of this method in an unbounded state space: (i) the conservation of total probability in the state space is explicitly written, (ii) no artifact is required to manage far- eld boundary conditions , (iii) the positivity of the pdf is ensured and (iv) the extension to higher dimensions is straightforward. Furthermore, thanks to the moving particles, this method is adapted for a large kind of initial conditions, even slightly dispersed distributions. The FPK equation is solved without any a priori knowledge of the stationary distribution; just a precise representation of the initial distribution is required. [less ▲] Detailed reference viewed: 135 (51 ULg)Dynamic Computation and Design of a World’s Record Death-Ride Denoël, Vincent ; Canor, Thomas ; in Proceedings of the 4th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (2013) A death-ride is a long cable anchored between two dead ends, allowing the crossing of wide valleys. Besides these mobility purposes, it is also operated in rescue operations, or simply for leisure. From a ... [more ▼] A death-ride is a long cable anchored between two dead ends, allowing the crossing of wide valleys. Besides these mobility purposes, it is also operated in rescue operations, or simply for leisure. From a serviceability viewpoint, a death-ride design should offer a comfortable and/or enjoyable journey to the riders, which is typically expressed in terms of maximum velocity. From a strength viewpoint, a death-ride should be designed in order to safely withstand the cable self-weight, wind forces, the rider’s load and at last but not least, the initial tension in the cable. Actually, the tensioning operation is usually subtle. Indeed, in case of a loose cable, the rider may be stopped in the in-span sagging area. Conversely a too tight cable will generate large stresses but also large outlet velocity at the end of the ride. The design of a death-ride is a tradeoff between all these aspects. The paper presents the dynamic model that was used for the design of a 1.2km long death-ride, with an only 10-mm diameter, a constraint fixed in the very first stages of this project for compatibility reasons, with the existing equipment. This death-ride, built and used in May 2010 in a Grimp competition [1], has a world record of the longest death-ride in urban area. The structural part of the dynamic model is rather common, with a co-rotational description of the cable by means of a chain-like model [2]. An interesting feature, however, concerns the interaction between the rider and the cable. In fact, because the pulley is rolling on the cable, the position of the force acting on the cable, resulting from the ride, depends on the rider’s velocity, which itself depends on the cable inclination. This coupling is solved in a staggered manner, and the details of the algorithm are given in the full paper. In some limiting cases, a comparison with analytical solutions is also provided, where these analytical solutions are available [3]. Experimental validation and observation of the operated death-ride will also be presented. [less ▲] Detailed reference viewed: 46 (1 ULg)Patching Asymptotics Solution of a Cable with a Small Bending Stiffness Denoël, Vincent ; Canor, Thomas in Journal of Structural Engineering (2013), 139 The analysis of a cable with a small bending stiffness is a problem encountered in many engineering applications such as the fatigue assessment of stay cables, the modeling of pipeline laying operation or ... [more ▼] The analysis of a cable with a small bending stiffness is a problem encountered in many engineering applications such as the fatigue assessment of stay cables, the modeling of pipeline laying operation or the determination of bending stresses in drillpipe assemblies. Because this phenomenon is modeled by a singularly perturbed equation, standard numerical techniques fail to solve these problems efficiently. As an alternative, provided the complexity of the analytical developments does not preclude their application, these problems may be tackled with appealing analytical procedures such as matching asymptotics or multiple scales. Otherwise advanced numerical simulations combining patching asymptotics within a numerical framework are the only possible approach for problems where the governing equations are too complex. Patching asymptotics also features a number of merits such as the possibility of using a boundary layer with a finite extent. Aiming at a better understanding of this latter technique, it is considered here to determine the solution of a cable with a small bending stiffness. Interesting details about patchability conditions and about how to restore higher derivatives continuity are included. The accuracy of the patching asymptotics approach is also compared with that of matched asymptotics. [less ▲] Detailed reference viewed: 62 (14 ULg)Transient Fokker-Planck equation with SPH method : Application in seismic design Canor, Thomas ; Denoël, Vincent in Papadrakakis, M; Papadopoulos, V; Plevris, V (Eds.) Proceedings COMPDYN2013 (2013) Detailed reference viewed: 22 (10 ULg)Efficient uncoupled stochastic analysis with non-proportional damping Canor, Thomas ; Blaise, Nicolas ; Denoël, Vincent in Journal of Sound & Vibration (2012), 331(2012), 52835291 The use of normal modes of vibration in the analysis of structures with nonproportional damping reduces the size of the resulting set of governing equations, but does not decouple them. A common practice ... [more ▼] The use of normal modes of vibration in the analysis of structures with nonproportional damping reduces the size of the resulting set of governing equations, but does not decouple them. A common practice consists in decoupling the equations by disregarding the o -diagonal elements in the modal damping matrix. Recently, an approximation based on an asymptotic expansion of the modal transfer matrix has been proposed in a deterministic framework to partially account for o -diagonal terms, but still with a set of uncoupled equations. This paper aims at extending this method in a stochastic context. First the mathematical background is introduced and the method is illustrated with a simple example. Then its relevance is demonstrated within the context of the structural analysis of a large and realistic structure. [less ▲] Detailed reference viewed: 112 (42 ULg)Transient Fokker-Planck Equation solved with SPH Canor, Thomas ; Denoël, Vincent in Proceedings of the 5th International Conference on Advanced Computational Methods in Engineering (2011) In many engineering matters, systems are submitted to random excitations. Probabilistic theories aim at describing the properties of a system by means of statistical properties such as probability density ... [more ▼] In many engineering matters, systems are submitted to random excitations. Probabilistic theories aim at describing the properties of a system by means of statistical properties such as probability density function (pdf). For a deterministic system randomly excited, the evolution of its pdf is commonly described with Fokker-Planck-Kolmogorov equation (FPK). The FPK equation is a conservation equation of a hypothetical fluid, which represents physically the transport of probability. To solve this equation, Smoothed Particle Hydrodynamics (SPH) are used: the system is modelled with a conservation equation for the system and a transport equation for each particle. Numerical implementation shows the superiority of this method over many other mesh-based methods: (i) the conservation of total probability in the state space is explicitly written, (ii) no specific boundary conditions must be imposed if an adaptive smoothing length is chosen and if particles are initially regularly spread out, (iii) the positivity of the pdf is ensured. Furthermore, thanks to the moving particles, this method is adapted for a large kind of initial conditions (quasi-deterministic or even discontinuous). The FPK equation can be solved without any a priori knowledge of the stationary distribution; just a precise representation of the initial distribution is required. [less ▲] Detailed reference viewed: 68 (18 ULg)On the influence of background component in resonance of cables Canor, Thomas ; Denoël, Vincent in 9th International Symposium on Cable Dynamics: ISCD 2011, Proceedings (2011) This work deals with nonlinear dynamical behaviour of cables in the context of random excitations of one of its anchors. Different studies have already been realized, that highlight different cable ... [more ▼] This work deals with nonlinear dynamical behaviour of cables in the context of random excitations of one of its anchors. Different studies have already been realized, that highlight different cable responses under random excitations such as white noise or narrow band processes. This work can be seen as an extension of these former works. It aims at modeling a more realistic random excitation by associating a background component to a resonant excitation. The background component models the effects of turbulence on the structure and the resonant component results from structural vibrations of the primary system. Any direct excitation on the cable is disregarded in this work. The first part of this paper describes the model and the random excitation generators. The second part deals with the influence of the background component on the possible resonance of a cable. Results are presented for a given level of energy in the excitation, and as a function of its dispatching between the background and resonant components. This work shows that the background component reduces the vibration in the out-of-plane mode and can avoid resonance from taking place. The on-off intermittency phenomenon is also observed for the out-of-plane mode and it depends also on the background component. [less ▲] Detailed reference viewed: 54 (10 ULg)Vandalism Prevention of a Footbridge with Cable Vibrations Canor, Thomas ; Kerschen, Gaëtan ; Denoël, Vincent in Biliszczuk, Jan; Bien, Jan; Hawryszkow, Pawel (Eds.) et al 4th International Conference Footbridge : Attractive structures at reasonable costs, Wroclaw 6-8 July 2011 (2011) This work studies an unusual way to improve comfort of a footbridge with cables. Cables can be seen as a means of dissipating energy in a structure. This complementary source of dissipation does not ... [more ▼] This work studies an unusual way to improve comfort of a footbridge with cables. Cables can be seen as a means of dissipating energy in a structure. This complementary source of dissipation does not prohibit resonance from taking place, but it is a way to limit vibrations and to impede vandals’ actions. This study is illustrated with measurements realized on a specific footbridge. This structure is a metallic arch characterized by a first natural frequency of 3.2Hz and a corresponding damping ratio of 0.55%. Intolerable accelerations (around 6m/s²) are easily reached when an ill-intentioned person is bouncing at an appropriate frequency. After installation of a single cable at a suitable location in the structure, the measured damping ratios are almost doubled and the maximum accelerations at resonance are reduced by 30%. With three cables on the footbridge, the damping ratio becomes significantly nonlinear: it reaches up to 3% for low amplitude oscillations, but drops down to 1% for moderate to high amplitudes. For higher accelerations, it does not seem to depend on the number of cables. According to these observations, a notable effect of cables is to reduce the maximum acceleration, but the main effect is to prolong the transient phase and to make the resonance frequency hardly identifiable by vandals. [less ▲] Detailed reference viewed: 47 (5 ULg)Amortissement non-linéaires des structures par des câbles Canor, Thomas Master's dissertation (2010) Ce travail traite de l'interaction câble-structure et évalue la possibilité de pouvoir intégrer des câbles comme dispositifs anti-vandalisme ou comme dispositifs de confort sur des passerelles. LE cadre ... [more ▼] Ce travail traite de l'interaction câble-structure et évalue la possibilité de pouvoir intégrer des câbles comme dispositifs anti-vandalisme ou comme dispositifs de confort sur des passerelles. LE cadre de cette étude et la passerelle de Limbourg, structure métallique hybride entre un arc et une poutre Virendeel. Un résumé de modélisation de l'action humaine sur des structures est proposé. L'action de vandales est modélisée par des sauts successifs en utilisant le modèle de Racic et Pavic. Après une description de la passerelle, des mesures réalisées sur celle-ci et des différents modèles éléments-finis exploités, l'interaction câble-structure est étudiée successivement à l'aide d'un modèle analytique, puis d'un modèle numérique. Le modèle analytique est le modèle de Ernst-Bleich. La structure est projetée dans son premier mode propre de vibration. Les simulations sont menées à l'aide de la méthode de Newmark non-linéaire. Les résultats obtenus avec ce modèle sont intéressants au regard de la dynamique non-linéaire, mais l'exacerbation de la sensibilité aux conditions initiales rend difficile toute interprétation ou prédiction. Le modèle numérique pour le câble est construit à l'aide d'éléments finis de barres non-linéaires. La structure est toujours projetée dans son premier mode propre. L'amortissement dans le câbles est intégré comme un amortissement proportionnel avec le modèle de Rayleigh. La dissipation dans le câble permet d'augmenter l'amortissement structurel équivalent de manière significative. Une étude paramétrique est menée pour dégager des paramètres pertinents afin d'utiliser des câbles comme éléments finis dissipatifs dans une structure. [less ▲] Detailed reference viewed: 112 (17 ULg)Patching asymptotic solution of a cable with a small bending stiffness Canor, Thomas ; Denoël, Vincent in Proceedings of the IVth European Conference on Computational Mechanics (2010) Detailed reference viewed: 48 (6 ULg) |
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