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The statics of fully deformable parabolic arches affected by a small crack at opposite sides of a damaged cross section is studied. The finite governing equations are linearized; the mechanical response for ‘small’ displacements and rotation is assumed. The effect of the crack is modelled by springs with stiffnesses calculated through linear elastic fracture mechanics. Closed-form exact static solutions are found under suitable boundary and continuity conditions. The effects of the crack position along the arch axis, its depth, and location on the cross section for different loading and boundary conditions are investigated and commented. The possibility of using these solutions in structural identification is discussed.

A structural damage identification technique hinged on the combination of orthogonal empirical mode decomposition and modal analysis is proposed. The output-only technique is based on the comparison between pre- and post-damage free structural vibrations signals. The latter are either kinematic (displacements, velocities or accelerations) or deformation measures (strains or curvatures). The response data are decomposed by means of the orthogonal empirical mode decomposition to derive a finite set of orthogonal intrinsic mode functions; the latter are used as a multi-frequency and data-driven basis to build pseudo-modal shapes. A new damage index, the so-called pseudo-mode index, is introduced to compare the response obtained for the two states of the structural system and detect potential damages. The performance of the devised index in detecting a localised damage is shown through numerical and experimental tests on two structural models, namely a 4-degrees-of-freedom system and a two-hinged parabolic arch.

Dissipative properties of a structural system are difficult to be characterized in real structure. Nevertheless, damping features may be dominant in several operating conditions of railway bridges influencing fatigue life or passenger comfort during train passage. Observations treating real data acquired in operational condition on steel and concrete railway bridges belonging to the Italian network permits to highlight dissipative sources and features. Consequently, linearized modal damping ratios are evaluated through a recursive process on the acceleration signals acquired before, during and after train passages and/or in environmental conditions. Stochastic Subspace Identification has been used to identify state-space dynamical models able to reproduce the vibrations. Through these models, characterized by an increasing number of state-space variables, it is possible to extract modal damping ratios. A mechanical interpretation of damping characteristics is pursued through the evaluation of the differences with respect to a classical Rayleigh proportional damping matrix of the viscous matrix belonging to the identified state-space models determined through the system spectral features. A non-proportional damping index is presented as a basis to determine the influence of different sources of non-proportionality in the damping matrix (as the ballast layer under the track) and to justify the high value of damping observed in specific experimental campaigns.

Structural monitoring plays a central role in civil engineering and, in particular, optimal sensor positioning is essential for correct monitoring both in terms of usable data and for optimizing the cost of the set-up sensors. In this context we focus our attention on the identification of the dynamic response of beam-like structures with uncertain damages. In particular, the non-localized damage is described using a Gaussian distributed random damage parameter. Furthermore, a procedure for selecting an optimal number of sensor placements has been presented based on the comparison among the probability of damage occurrence and the probability to detect the damage, where the former can be evaluated from the known distribution of the random parameter, whereas the latter is evaluated exploiting the closed-form asymptotic solution provided by a perturbation approach. The presented case study shows the capability and reliability of the proposed procedure for detecting the minimum number of sensors such that the monitoring accuracy (estimated by an error function measuring the differences among the two probabilities) is not greater than a control small value.

The seismic vulnerability assessment of masonry buildings reinforced with FRP was investigated by means of random sampling of their mechanical properties and repeated nonlinear static analyses. To this aim, the Monte Carlo simulation method was adopted to account for the uncertainty of the mechanical parameters characterizing the analytical model representative of each structural prototype. In particular, a two-step procedure was proposed. First, to ensure an appropriate calibration of the model, was investigated the mechanical behavior of two case-studies structures: an unreinforced masonry panel and a masonry panel externally reinforced with FRP composite strips applied with a grid configuration and anchored properly at their ends. Then, the effectiveness of a FRP reinforcement designed for the main facade of the historical Camponeschi Palace (located in the city of L’Aquila, Italy) was estimated. The proposed methodology was used to determine the probability of occurrence of several damage states and to estimate the relevant probability density functions corresponding to different levels of ground motion. The novelties of the modeling procedure proposed, together with the benefits provided by the FRP reinforcement, were presented and discussed in detail.

This paper is dedicated to the identifiability of vibrating beam structures with uncertain damages. The probability of damage occurrence is computed assuming a Gaussian distributed random damage parameter. Then, we propose a technique for selecting an optimized solution of sensors placement based on the comparison among the probability of damage occurrence and the probability to detect the damage, where the latter is evaluated exploiting the closed-form asymptotic solution provided by a perturbation approach. This comparison must be intended as an investigation on the minimum number of sensors beyond which monitoring accuracy (estimated by an error function measuring the differences among the two probabilities) increases less than a ‘small’ predetermined threshold. The capabilities and efficiency of the technique are shown through a parametric analysis on a sample case study: a simply supported beam with a random parameter ruling the evolution of a non-localized damage. The relevant results are presented and discussed, showing which conditions (sensors network) properly characterizes the beam dynamics.

This paper addresses the problem of identifying structural damage affecting one element of a steel truss. The purpose is to detect damages in relation to their magnitude, location and extension. A planar model of a damaged steel truss is used to illustrate the procedure. The direct problem is addressed by FEM, proposing the local stiffness of a damaged truss element. A modal identification is first implemented to obtain dynamic characteristics: natural frequencies, damping ratios and mode shapes. Damage is described as a reduction of the truss cross section, and defined in terms of its magnitude, extent and position. Moreover, a damaged element stiffness matrix has been derived and implemented within of a classical FEM procedure to obtain a numerical model used to generate a pseudo-experimental dynamic structural response under environmental noise. Then, Stochastic Subspace Identification (SSI), is carried out to extract and identify the main modal parameters related to both damage and undamaged truss system. A damage index is defined to evaluate the effectiveness of the proposed procedure.

This work investigates the capabilities of two different approaches for the analysis of thin-walled structures, both based on enriched beam theories that include out-of-plane cross-section warping, being the in-plane deformations neglected.

We propose a mixed variational principle for deducing the generalized Marguerre–von Karman equations, governing the relatively large deflections of thin elastic shallow shells. These equations account for both non-flat stress-free configurations of the shell and inelastic strains. We implement this formulation by using $$C^0$$ interior penalty methods within the UFL language provided by the FEniCS project. We present two numerical examples, with the aim to discuss the role of the shallowness and the inelastic strain, comparing the results with the fully non-linear shell model a la Naghdi and the classical displacement formulation.

A mixed variational principle is proposed for deducing the Foppl–von Karman equations governing the large deflections of thin elastic plates or shallow shells. Proper boundary conditions are found for the case of applied in-plane tractions and displacements, and simple mechanical interpretations are achieved. Numerical implementation is carried out, along with examples and comparisons with the classical formulation in terms of displacements.

Damage detection and localization in structural systems is experimentally studied. A novel pseudo-modal approach, recently proposed by some of the authors, is adopted. It is based on the comparison between free vibrations of the undamaged and damaged states, and aims to maximize the damage signature embedded in the data by exploiting the energy content of the vibration signals. Towards this goal, the latter signals are analyzed by means of the Orthogonal Empirical Mode Decomposition (OEMD) technique in order to derive a data-driven damage index, the so called Pseudo Modal Index (PMI). In this paper the results of an experimental campaign on a small-scale shear type steel frame structure are presented and discussed. The tested structure is modeled as a four degree-of-freedom dynamical system and the damage is represented by a localized stiffness reduction. A filtering technique applied to the Intrinsic Mode Functions is also proposed in order to tackle the presence of noise polluted data.

The papers focusing on dynamic identification of structural damages usually rely on the comparison of two or more responses of the structure; the measure of damage is related to the differences of the vibration signals. Almost all literature methods assume damping proportionality to mass and stiffness; however, this is acceptable for new, undamaged structures, but not for existing, potentially damaged structures, especially when localised damages occur. It is well-known that in non-proportionally damped systems the modes are no longer the same of the undamped system: thus, some authors proposed to use modal complexity as a damage indicator. This contribution presents a perturbation approach that can easily reveal such a modal complexity.

We present the use of piezoelectric disk buzzers, usual in stringed musical instruments to acquire sound as a voltage signal, for experimental modal analysis. These transducers helped in extracting natural frequencies and mode shapes of an aluminium beam and a steel arch in the laboratory. The results are compared with theoretical predictions and experimental values obtained by accelerometers and a laser displacement transducer. High accuracy, small dimensions, low weight, easy usage, and low cost, make piezoelectric pickups an attractive tool for the experimental modal analysis of engineering structures.

The hybridisation of fibres reinforced laminates, i.e., the combined use of two or more families of fibres, is an effective technique to achieve a pseudo-ductile response and overcome the inherent brittleness which limits the wider use of composite materials. In this paper, a one-dimensional analytical model for unidirectional hybrid laminates is derived. The model considers two elastic–brittle layers bonded together by a cohesive elasto–plastic–brittle interface. This formulation is applied to the study of the debonding and fracture of laminates under uniaxial loading and the results compared to experiments available from the open literature. This study shows that the proposed model provides a close fit to the experimental data and it is able to match accurately the crack patterns seen in the experiments. The model predicts four different failure mechanisms and is able to discriminate among them according to the geometrical and mechanical properties of the layers.

This paper presents the formulation of a three-dimensional beam finite element (FE) that accounts for cross-section warping and dynamic inertia effects. The model is the extension of an existing mixed formulation, originally developed for the static analysis of thin-walled beams, to the case of dynamic loading conditions. Four independent fields are considered to derive the element governing equations, i.e. material rigid displacements, strains and stresses and an additional displacement field, describing the out-of-plane warping displacement of the beam cross-sections. The latter is independently interpolated in the element volume by including additional degrees of freedom (DOF) to the nodal translations and rotations classically considered in beam formulations. To obtain a consistent form of the element mass matrix, the cross-section displacement shape functions are computed, relating the generalized cross-section displacement fields to the element nodal variables. In mixed FE formulations, these are not assigned a priori, as in displacement-based approaches, but are derived on the basis of material stiffness and element geometry, together with compatibility conditions. Thus, the Unit Load method is applied to deduce the expressions of the shape functions consistent with the force-based approach, assuming the simply-supported beam as reference element configuration. As opposed to the original FE model, the additional warping DOFs are not condensed-out with the definition of the element quantities but are treated as additional global unknowns. This permits a correct description of the inertia effects and ensures continuity of the warping displacement fields between adjacent FEs. Correlation studies are presented to validate the proposed model and investigate the effects of cross-section warping on the dynamic behavior of thin-walled structures. For selected specimens, the studies compare solutions obtained adopting the proposed beam element with those resulting from shell or brick FE models. Modal decompositions and time-history analyses are conducted, assuming both linear elastic and nonlinear constitutive behavior for the latter.

This paper deals with the identification of linear structural systems with random parameters. The stiffness matrix of a four-storey shear frame structure is assumed to be linearly dependent on a random parameter ruling the damage evolution of the columns. The evaluation of natural frequencies and the mode-shapes is in the context of random eigenvalue problems in structural dynamics. Using a Taylor expansion of the mass and stiffness matrices, a perturbation technique is first applied to derive the asymptotic solution up to the second order. Then, the evaluation of the statistic of the frequencies and mode-shapes is derived up to the second order. Finally a stochastic identification technique is proposed to characterize the statistics of the random parameter.

We report about the experimental identification of viscoelastic constitutive models for frequencies ranging within 0–10 Hz. Dynamic moduli data are fitted for several materials of interest to medical applications: liver tissue ( Chatelin et al., 2011 ), bioadhesive gel ( Andrews et al., 2005 ), spleen tissue ( Nicolle et al., 2012 ) and synthetic elastomer ( Osanaiye, 1996 ). These materials actually represent a rather wide class of soft viscoelastic materials which are usually subjected to low frequencies deformations. We also provide prescriptions for the correct extrapolation of the material behavior at higher frequencies. Indeed, while experimental tests are more easily carried out at low frequency, the identified viscoelastic models are often used outside the frequency range of the actual test. We consider two different classes of models according to their relaxation function: Debye models, whose kernel decays exponentially fast, and fractional models, including Cole–Cole, Davidson–Cole, Nutting and Havriliak–Negami, characterized by a slower decay rate of the material memory. Candidate constitutive models are hence rated according to the accurateness of the identification and to their robustness to extrapolation. It is shown that all kernels whose decay rate is too fast lead to a poor fitting and high errors when the material behavior is extrapolated to broader frequency ranges.

A finite differences procedure is used to study the buckling of non-trivial equilibrium solutions for open thin-walled beams in a dynamic setting. A direct one-dimensional model with a coarse descriptor of warping is adopted. The algorithm describes non-trivial equilibrium paths by integrating discretized field equations, suitably written in terms of velocities. Some benchmark cases under conservative loading are discussed. Known results for the first critical loads are found to validate the procedure. New results are found accounting for non-trivial equilibrium paths, thus providing an estimate for the error made by linearizing around trivial equilibrium paths. The effect of warping on the critical loads is also investigated.

Uncertainty characterization plays a key role in the safety assessment of many engineering structures. Nevertheless, the inverse problem for structures with uncertain parameters has received less attention than the relevant direct one, since one deals with stochastic structural system identification. This paper discusses the dynamic identification of linear structural systems with random stiffness parameters. Following a perturbation approach, recently proposed by the authors in a discrete framework, an identification technique for transversely vibrating 1-D uncertain continua is proposed. Results for a paradigmatic case, a simply supported beam, are presented and discussed.

Fading memory is a distinguishing characteristic of viscoelastic solids. Its assessment is often achieved by measuring the stress due to harmonic strain histories at different frequencies: from the experimental point of view, the storage and loss moduli are, hence, introduced. On the other side, the mathematical modeling of viscoelastic materials is usually based on the consideration of a kernel function whose decay rate is sufficiently fast. For several different solid materials, we have collated experimental evidence showing an high sensitivity to frequency variations of both the storage and loss moduli. By contrast, we prove that the commonly employed viscoelastic kernels (Prony series, continuous kernel, etc.) cannot reproduce this experimental behavior, as the resulting frequency sensitivity of the storage modulus is always zero when assessed at low frequency. This leads to identification problems of the material parameters which are strongly ill conditioned. However, we identify the specific kernel property which is responsible for this misbehavior: the long-term material memory must not decrease too fast. Some viscoelastic kernels, showing the correct memory’s rate of decay, are introduced and their improved ability to match the experimental data analyzed.

Carbon black-filled rubber and soft biological tissues are only two examples of materials which undergo large deformation processes and exhibit relevant dissipation and hysteresis losses. Nonlinear viscoelasticity encompass a wide class of constitutive models aimed at describing the behavior of such materials. The main goal of the present paper is to draw a comparison between the many viscoelastic constitutive relations recently proposed. To this aim the current stress value is expressed through a single hereditary integral of the deformation history; this choice, although generalizable, yet permits the introduction of an unifying formulation in which hereditary, strain rate and fractional-derivatives models can all be included. The models comparison is based on several compression tests with cylindrical carbon black-filled rubber specimens carried out at Dipartimento di Ingegneria Chimica e Materiali di Sapienza Universita di Roma and in cooperation with the Bridgestone Technical Center Europe SpA. In particular relaxation, creep, loading–unloading cycles at different speeds were considered. For each test case and for each model under consideration, a nonlinear optimization problem is solved to identify the optimal constitutive parameters; the robustness of the identified parameters to perturbations in the experimental data is also computed. The predictive capabilities of the models are then compared in terms of the stress response and the energy dissipation. Finally, the ability of each model in the cross-prediction of the relaxation/creep behavior when identified via cyclic loading and viceversa is examined.

The paper deals with the problem of the determination of the in-plane behavior of periodic masonry material. The macromechanical equivalent Cosserat medium, which naturally accounts for the absolute size of the constituents, is derived by a rational homogenization procedure based on the Transformation Field Analysis. The micromechanical analysis is developed considering a Cauchy model for masonry components. In particular, a linear elastic constitutive relationship is considered for the blocks, while a nonlinear constitutive law is adopted for the mortar joints, accounting for the damage and friction phenomena occurring during the loading history. Some numerical applications are performed on a Representative Volume Element characterized by a selected commonly used texture, without per- forming at this stage structural analyses. A comparison between the results obtained adopting the proposed procedure and a nonlinear micromechanical Finite Element Analysis is presented. Moreover, the substantial differences in the nonlinear behavior of the homogenized Cosserat material model with respect to the classical Cauchy one, are illustrated.

A technique for finding the stiffnesses and shear flows in multi-cell thin-walled girders subjected to linear elastic torsion is proposed. The girder is thought as the superposition of elementary closed tracks, just like open girders are the juxtaposition of thin strips. For each track there is a uniform value of the stress flow function, found by means of a linear system of compatibility equations resembling those for the redundant reactions in statically indeterminate frames. The connection between the proposed approach and fundamental properties of graphs is discussed; the advantages with respect to the standard procedures are also enlightened referring to two sample cross-sections.

Piezoelectric disk buzzers are commonly used on stringed musical instruments to acquire the sound in the form of a voltage signal. Aim of the present investigation is to assess the possibility of using these transducers for experimental modal analysis. Piezoelectric disks were therefore used in the laboratory to extract the natural vibration frequencies and mode shapes of an aluminum cantilever beam and of a steel arch. The results are compared with theoretical predictions and with other experimental values obtained using a laser displacement transducer and accelerometers. Due to their high accuracy, small dimensions, low weight, easy usage, and low cost, piezoelectric disks seem to be an attractive tool for experimental modal analysis of engineering structures.

The detection of structural damping is a crucial point in structural identification. Classic techniques usually refer to deterministic systems, since the assumption of randomness in the mechanical quantities implies non-trivial analytical difficulties in the development of both the direct and the inverse problem. In some recent works, starting from the statistics of mode-shapes and (undamped) frequencies, a perturbative approach has been introduced by the authors for the estimation of mean and variance of uncertain mass and stiffness. Here dissipative structures are considered; in detail, the method is applied for the stochastic structural identification of classically damped linear dynamic systems, dependent on a random parameter, assumed to be Gaussian. A numerical validation of the technique is then presented and discussed.

Different solution strategies to the relaxed Saint-Venant problem are presented and comparatively discussed from a mechanical and computational point of view. Three approaches are considered; namely, the displacement approach, the mixed approach, and the modified potential stress approach. The different solution strategies lead to the formulation of two-dimensional Neumann and Dirichlet boundary-value problems. Several solution strategies are discussed in general, namely, the series approach, the reformulation of the boundary-value problems for the Laplace's equations as integral boundary equations, and the finite-element approach. In particular, the signatures of the finite-element weak solutions—the computational costs, the convergence, the accuracy—are discussed considering elastic cylinders whose cross sections are represented by piece-wise smooth domains.

Free wave propagation patterns for general three-coupled periodic structures are investigated by means of the transfer matrix approach. It is shown that an exhaustive description of the propagation domains requires spaces that are stratified in homogeneous regions, whose dimension is given by the number of invariants of the transfer matrix characteristic equation and whose boundaries are represented by codimension-one manifolds. Three types of three-coupled periodic mechanical models characterized by constitutive elastic and/or inertial coupling between mono- and bi-coupled dynamics, namely pipes, thin-walled beams and truss beams, are considered. From the design standpoint, an adequate representation of the propagation domains pattern is obtained through a nonlinear mapping from the space of the invariants to the physical parameters plane. The analyzed models give rise to equations of motion where the three-coupled nature stems from the coupling between transversal (bi-coupled) and longitudinal (mono-coupled) dynamics for the pipes and truss beams, whilst coupling occurs between transversal and torsional (mono-coupled) dynamics when it comes to the thin-walled beams. A mechanical interpretation associated with the bounding frequencies of the propagation regions is given and the evolution of the propagation properties when coupling parameters tend to vanish is discussed.

A mechanical model describing finite motions of nonshallow cables around the initial catenary configurations is proposed. An exact kinematic formulation accounting for finite displacements is adopted, whereas the material is assumed to be linearly elastic. The nondimensional mechanical parameters governing the motions of nonshallow cables are obtained via a suitable nondimensionalization, and the regions of their physically plausible values are portrayed. The spectral properties of linear unforced undamped vibrations around the initial static configurations are investigated via a Galerkin-Ritz discretization. A classification of the modes is obtained on the basis of their associated energy content, leading to geometric modes, elastostatic modes (with prevalent transverse motions and appreciable stretching), and elastodynamic modes (with prevalent longitudinal motion). Moreover, an extension of Irvine’s model to moderately nonshallow cables is proposed to determine the frequencies and mode shapes in closed form.

The stability of a cantilever elastic beam with rectangular cross-section under the action of a follower tangential force and a bending conservative couple at the free end is analyzed. The beam is herein modeled as a non-linear Cosserat rod model. Non-linear, partial integro-differential equations of motion are derived expanded up to cubic terms in the transversal displacement and torsional angle of the beam. The linear stability of the trivial equilibrium is studied, revealing the existence of buckling, flutter and double-zero critical points. Interaction between conservative and non-conservative loads with respect to the stability problem is discussed. The critical spectral properties are derived and the corresponding critical eigenspace is evaluated.

Free undamped in-plane vibrations of shear undeformable beams around their highly buckled configurations are investigated neglecting rotary inertia effects. The beams are inertially nonuniform since a lumped mass is rigidly clamped along the span. Two mechanical models are considered depending on the boundary conditions in the post-buckling phases. First, the beam is considered inextensible because it is hinged at one end and is acted upon by an axial compressive force on the other end, a roller support, both in the buckling and post-buckling phases. In the second model, the beam is extensible in the post-buckling phase because the roller support boundary is changed into a fixed hinged end. Free undamped vibrations are governed, in the first case, by a homogeneous integral-partial-differential equation and, in the second case, by two coupled partial-differential equations with variable coefficients. The solutions of the associated eigenvalue problems are found employing two approaches: a semi-analytical method based on Galerkin discretization and a finite element method. A close agreement in the outcomes is found. The leading differences relating to the natural frequencies and linear normal modes of the two pre-stressed curved beam models are discussed; in particular, the occurrence of the veering phenomenon and the crossovers are outlined.

International audience; Classes of motion of general multiresonant systems are derived through a geometrical algorithm based on a set representation. First, the elementary classes existing under simple resonance conditions are evaluated; rules governing the interaction between elementary classes belonging to different resonance conditions are then drawn up as applications of a unique theorem. Illustrative examples are given. The method also permits a hierarchical ordering of the amplitudes of the resonant modes, according to their participation in the classes; it also makes it possible to ascertain in advance the existence of a standard form for the amplitude modulation equations. The stability analysis of incomplete steady solutions is then addressed. Three classes of perturbation are distinguished, namely: in-class perturbations, out-of-class resonant perturbations, and out-of-class nonresonant perturbations. The structure of the Jacobian variational matrix is studied. The Jacobian matrix is shown to comprise three diagonal blocks associated with the three perturbation classes, so that the stability equations are uncoupled. Further possible uncouplings of one of the blocks are analyzed in relation to some of the geometrical properties of the classes.

International audience; A general multiresonant system is considered, in which the linear frequency and, possibly, a forcing frequency are involved in a set of linear conditions. The nature of the resonances is first discussed, by distinguishing independent and dependent equations, and both the analysis and design problems of the system are addressed. Rules are then given to construct the qualitative form of the AMEs to any desired order. Two families of terms are identified: improper resonant terms (not associated with any resonance conditions) and proper resonant terms (depending on the specific conditions), sub-divided into primary (of lower order) and secondary (of higher order). Theorems are proved to show that both improper and secondary resonant terms have no qualitative but only quantitative effects on classes of motion; reference is therefore made to reduced equations. An algebraic algorithm is illustrated to determine classes of motion, using only the integer resonance coefficients. The concept of degree of constraint of a given resonance condition is introduced, entailing a hierarchic order among the resonance conditions, the implications of which are discussed. Finally, some numerical simulations are shown to test the robustness of classes of motion to higher-order terms not accounted for in the asymptotic analysis.

Multiple-zero bifurcation of a general multiparameter dynamic system is analyzed using the multiple-scale method and exploiting the close similarities with eigensolution analysis for defective systems. Because of the coalescence of the eigenvalues, the Jacobian matrix at the bifurcation is nilpotent. This entails using timescales withfractionalpowersoftheperturbationparameter.Thereconstitution methodisemployedto obtainanordinary differential equation of order equal to the algebraic multiplicity of the zero eigenvalue, in the unique unknown amplitude. When the algorithm is applied to a double-zero eigenvalue, Bogdanova ‐Arnold’ s normal form for the bifurcation equation is recovered. A detailed step-by-step algorithm is described for a general system to obtain the numerical coefe cients of the relevant bifurcation equation. The mechanical behavior of a nonconservative two-degree-of-freedom system is studied as an example.

We investigate the effects of warping on the dynamic stability of non-trivial equilibrium configurations for non-symmetric open thin-walled beams. We use a direct one-dimensional model coarsely describing warping; the rest of the kinematics is exact. Dynamic derives from the balance of power; constitutive relations are non-linear, hyper-elastic, and distinguish the roles of the centroids and shear centres; inertial actions account for warping, too. By centred finite differences, the warping inertial action is found ineffective on the natural angular frequencies. Then, we follow non-trivial equilibrium paths and investigate their Ljapounov stability, by examining the small superposed oscillations. Results for generic, non-symmetric cross-sections are presented and discussed, showing the effects of warping and of coupling constitutive coefficients.

Localization of damages becomes rather challenging when the associated stiffness reduction is small in presence of structural uncertainties. This work presents a sensitivity analysis and an improvement of a novel pseudo-modal approach, recently proposed by the authors. Starting from free vibrations of the undamaged and damaged states, the method aims to maximize the damage signature embedded in the data exploiting the peculiar features of the Orthogonal Empirical Mode Decomposition technique. The role of the length of the signals and the boundary effects are here investigated; a cut-off rule useful for reducing the latter issue is also proposed.Copyright © 2015 by ASME

The complex amplitude modulation equations of a discrete dynamicalsystem are derived under general conditions of simultaneous internal andexternal resonances. Alternative forms of the real amplitude and phaseequations are critically discussed. First, the most popular polar formis considered. Its properties, known in literature for a multitude ofspecific problems, are here proven for the general case. Moreover, thedrawbacks encountered in the stability analysis of incomplete motions(i.e. motions containing some zero amplitudes) are discussed as aconsequence of the fact the equations are not in standard normal form.Second, a so-called Cartesian rotating form is introduced, which makesit possible to evaluate periodic solutions and analyze their stability,even if they are incomplete. Although the rotating form calls for theenlargement of the space and is not amenable to analysis of transientmotions, it systematically justifies the change of variables sometimesused in literature to avoid the problems of the polar form. Third, amixed polar-Cartesian form is presented. Starting from the hypothesisthat there exists a suitable number of amplitudes which do not vanish inany motion, it is proved that the mixed form leads to standard formequations with the same dimension as the polar form. However, if suchprincipal amplitudes do not exist, more than one standard form equationshould be sought. Finally, some illustrative examples of the theory arepresented.

The governing equations for dynamic transverse motion of a cable-stayed beam are obtained by means of a classical variational formulation. The analytical model permits a parametric investigation of linear and non-linear behaviour in a family of cable-stayed beam systems. Analytical eigensolutions of the linearized equations are used to investigate how the mechanical characteristics influence the occurrence of global, local and coupled modes. The exact eigenfunctions are assumed to describe the forced harmonic motion in the neighbourhood of a selected frequency. The frequency–amplitude relationship, obtained by the use of the multiple scale method, permits the description of softening and hardening behaviour in the global, local and coupled classes of motion. Copyright © 2002 John Wiley & Sons, Ltd

International audience; A general, multiparameter system admitting a double-zero eigenvalue at a critical equilibrium point is considered. A sensitivity analysis of the critical eigenvalues is performed to explore the neighborhood of the critical point in the parameter space. Because the coalescence of the eigenvalues implies that the Jacobian matrix is defective (or nilpotent), well-suited techniques of perturbation analysis must be employed to evaluate the eigenvalues and the eigenvector sensitivities. Different asymptoticmethods are used, based on perturbations both of the eigenvalue problem and the characteristic equation. The analysis reveals the existence of a generic (nonsingular) case and of a nongeneric (singular) case. However, even in the generic case, a codimension-1 subspace exists in the parameter space on which a singularity occurs. By the use of the relevant asymptotic expansions, linear stability diagrams are built up, and different bifurcationmechanisms (divergence-Hopf, double divergence, double divergence-Hopf, degenerate Hopf) are highlighted.The problem of Ž nding a unique expression uniformly valid in the whole space is then addressed. It is found that a second-degree algebraic equation governs the behavior of the critical eigenvalues. It also permits clariŽ cation of the geometrical meaning of the unfolding parameters, which has been discussed in literature for the Takens-Bogdanova bifurcation. Finally, a mechanical system loaded by nonconservative forces and exhibiting a double-zero bifurcation is studied as an example.

Higher-order multiple-scale methods for general multiparameter mechanical systems are studied. The role played by the control and imperfection parameters in deriving the perturbative equations is highlighted. The definition of the codimension of the problem, borrowed from the bifurcation theory, is extended to general systems, excited either externally or parametrically. The concept of a reduced dynamical system is then invoked. Different approaches followed in the literature to deal with reconstituted amplitude equations are discussed, both in the search for steady-state solutions and in the analysis of stability. Four classes of methods are considered, based on the consistency or inconsistency of the approach, and on the completeness or incompleteness of the terms retained in the analysis. The four methods are critically compared and general conclusions drawn. Finally, three examples are illustrated to corroborate the findings and to show the quantitative differences between the various approaches.

International audience; The multiple time-scale method is adapted to study the post-critical behavior of general non-conservative symmetric systems, possibly affected by imperfections, for which divergence and Hopf bifurcations interact. The procedure illustrated makes it possible to elude the computational burden related to the application of the center manifold reduction. It also furnishes explicit expressions of the coefficients of the standard normal form bifurcation equations in terms of the coefficients of the original system. As an example, the method is applied to a two-degree-of-freedom rigid bar subjected to axial load (Augusti's model) and transversal flow. The critical and post-critical scenarios are analyzed in detail, for both the perfect and imperfect systems.