Marcello Vasta - Profile on Academia.edu (original) (raw)
Papers by Marcello Vasta
DAMAGE IDENTIFICATION OF MONUMENTAL MASONRY STRUCTURES: THE CASE OF FOSSANOVA GOTHIC CHURCH
ABSTRACT In the present paper, the seismic behavior of a physical 1:5.5 scaled model of the churc... more ABSTRACT In the present paper, the seismic behavior of a physical 1:5.5 scaled model of the church of the Fossanova Abbey is investigated by means of numerical and experimental analyses. Aiming at defining the seismic vulnerability of such a structural typology a wide experimental campaign was carried out. The achieved experimental results lead to the definition of a refined FE model reproducing the dynamic behavior of the whole structural complex. Then, the central transversal three-central bays of the church, as it mostly influences the seismic vulnerability of the Abbey, was investigated in a more detail by means of a shaking table test on a 1:5.5 scaled physical model in the Laboratory of the Institute for Earthquake Engineering and Engineering Seismology in Skopje. In the present paper a brief review of the numerical activity related to the prediction of the shaking table test response of the model is first proposed. Then, the identification of frequency decay during collapse is performed through decomposition of the measured power spectral density matrix. Finally, the localization and evolution of damage in the structure is analyzed. The obtained results shown that a very good agreement is achieved between the experimental data and the predictive/interpretative numerical analyses..
We used speleothem vulnerability for assessing the seismic ground shaking threshold probably expe... more We used speleothem vulnerability for assessing the seismic ground shaking threshold probably experienced or not in caves located within Apennines seismogenic areas. On the southern flank of the Pollino Range, which is considered a current seismic gap, cave observations suggest the lack of significant earthquake-related deformation in the last ~5-7 ka, but the widespread occurrence of creep. In the central Apennines, homogenously-trending collapses within the Grotta Cola (Petrella Liri, AQ) evidence a significant co-seismic deformation event radiometrically bracketed at ~4-5 ka. We supplemented the cave observations with numerical modeling and static test analysis of the speleothem mechanical behaviour to stipulate probability density functions for the bending stress leading to rupture, and established different vulnerability curves for speleothems according to their shapes. Model results are consistent with the co-seismic deformation at Grotta Cola as being caused by the Liri or by ...
Hyperelastic modeling of biological tissues with distributed collagen fibers
A perturbation approach for the identification of uncertain structures
International Journal of Dynamics and Control, 2015
ABSTRACT This paper deals with the identification of linear structural systems with random parame... more ABSTRACT 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.
Approximate method for probability density of the response of a linear oscillator to a non-Poisson impulse process
III European Conference on Computational Mechanics, 2006
State vector of a dynamic system under a Poisson train of impulses is a non-diffusive Markov proc... more State vector of a dynamic system under a Poisson train of impulses is a non-diffusive Markov process and its joint probability density function satisfies an integro-differential generalized Fokker-Planck- Kolmogorov equation which is also called Kolmogorov-Feller equation. If the train of impulses is driven by non-Poisson, for example renewal, counting processes, the state vector is not a Markov process. Non-Markov pulse
Journal of Computational and Nonlinear Dynamics, 2009
A generalized damped Beck's column under pulsating actions is considered. The nonlinear partial i... more A generalized damped Beck's column under pulsating actions is considered. The nonlinear partial integrodifferential equations of motion and the associated boundary conditions, expanded up to cubic terms, are tackled through a perturbation approach. The multiple scales method is applied to the continuous model in order to obtain the bifurcation equations in the neighborhood of a Hopf bifurcation point in primary parametric resonance. This codimension-2 bifurcation entails two control variables, namely, the amplitude of the static and dynamic components of the follower force, playing the role of detuning and bifurcation parameters, respectively. In the postcritical analysis bifurcation diagrams and relevant phase portraits are examined. Two bifurcation paths associated with specific values of the follower force static component are discussed and the birth of new stable period-2 subharmonic motion is observed.
Probabilistic Engineering Mechanics, 2014
We describe three-dimensional and planar models of hyperelastic fiber reinforced materials charac... more We describe three-dimensional and planar models of hyperelastic fiber reinforced materials characterized by statistical distribution of the fiber orientation. Our models are based on a second order approximation of the strain energy density in terms of the fourth pseudo-invariant I 4 , typically employed in the description of fiber reinforced materials. For a particular choice of the strain energy density associated to the fiber reinforcement, it is possible to derive the explicit expression of the material and spatial stress tensors and of the stress covariance tensors. The mechanical behavior of the models is assessed through uniaxial, biaxial and shear tests.
The paper tackles the dynamic identification and the damage detection carried out by a spectral-b... more The paper tackles the dynamic identification and the damage detection carried out by a spectral-based method on the well-known Z24 bridge, a three-span prestressed concrete bridge located in Switzerland. Before being destroyed, the bridge was progressively damaged and tested in the framework of the Brite Euram project SIMCES. Starting from this benchmark, the presented spectral-based identification technique is validated and the usefulness of this method as a non-destructive tool able to catch the dynamic behavior of a structure and locate the damage is widely discussed. Firstly, a FE model of the bridge was built and calibrated in order to analyze its response to different excitation types (free vibration, triangular pulse, swept sine, shaker and random vibrations) and several damage scenarios. Secondly, aiming at identifying both the modal parameters and the damage of the bridge, the spectral-based method is applied making use of the power spectral matrix decomposition. Finally, a proper index is defined and applied to this case-study in order to locate the damage.
DAMAGE IDENTIFICATION OF MONUMENTAL MASONRY STRUCTURES: THE CASE OF FOSSANOVA GOTHIC CHURCH
ABSTRACT In the present paper, the seismic behavior of a physical 1:5.5 scaled model of the churc... more ABSTRACT In the present paper, the seismic behavior of a physical 1:5.5 scaled model of the church of the Fossanova Abbey is investigated by means of numerical and experimental analyses. Aiming at defining the seismic vulnerability of such a structural typology a wide experimental campaign was carried out. The achieved experimental results lead to the definition of a refined FE model reproducing the dynamic behavior of the whole structural complex. Then, the central transversal three-central bays of the church, as it mostly influences the seismic vulnerability of the Abbey, was investigated in a more detail by means of a shaking table test on a 1:5.5 scaled physical model in the Laboratory of the Institute for Earthquake Engineering and Engineering Seismology in Skopje. In the present paper a brief review of the numerical activity related to the prediction of the shaking table test response of the model is first proposed. Then, the identification of frequency decay during collapse is performed through decomposition of the measured power spectral density matrix. Finally, the localization and evolution of damage in the structure is analyzed. The obtained results shown that a very good agreement is achieved between the experimental data and the predictive/interpretative numerical analyses..
Stationary and Nontationary Response Probability Density Function of a Beam under Poisson White Noise
In this paper an approximate explicit probability density function for the analysis of external o... more In this paper an approximate explicit probability density function for the analysis of external oscillations of a linear and geometric nonlinear simply supported beam driven by random pulses is proposed. The adopted impulsive loading model is the Poisson White Noise , that is a process having Dirac’s delta occurrences with random intensity distributed in time according to Poisson’s law. The
Wind-tunnel tests on high-rise buildings: wind modes and structural response
ABSTRACT The evaluation of pressure fields acting on slender structures under wind loads is curre... more ABSTRACT The evaluation of pressure fields acting on slender structures under wind loads is currently performed in experimental aerodynamic tests. For wind-sensitive structures, in fact, the knowledge of global and local wind actions is crucial for design purpose. This paper considers a particular slender structure under wind excitation, representative of most common high-rise buildings, whose experimental wind field on in-scale model was measured in the CRIACIV boundary-layer wind tunnel (University of Florence) for several angles of attack of the wind. It is shown that an efficient reduced model to represent structural response can be obtained by coupling the classical structural modal projection with the so called blowing modes projection, obtained by decomposing the covariance or power spectral density (PSD) wind tensors. In particular, the elaboration of experimental data shows that the first few blowing modes can effectively represent the wind-field when eigenvectors of the PSD tensor are used, while a significantly larger number of blowing modes is required when the covariance wind tensor is used to decompose the wind field.
A new energy-based system identification method is developed, applicable in situations where the ... more A new energy-based system identification method is developed, applicable in situations where the dynamic response of a structure is measurable but the excitation is unmeasurable and describable only in terms of a stochastic process. It is shown that, in the case of a non-linear single degree of freedom system subjected to purely parametric, non-white random excitation, the power spectrum of the excitation can be identified through an estimation of the diffusion coefficient relating to the energy envelope of the response process. Through an estimation of the drift coefficient an identification of the system damping is also possible. The method is validated through application to simulated data relating to a Duffing oscillator with non-linear damping.
Response of non-linear oscillators to non-white random excitation using an energy based method
It is shown that, for a non-linear oscillator responding to non-white random excitation, the ener... more It is shown that, for a non-linear oscillator responding to non-white random excitation, the energy envelope of the response can be modelled approximately as a one-dimensional Markov process. Using a self-consistent asymptotic analysis, for small damping, simple expressions for the drift and diffusion coefficients of this Markov process are derived: these involve Fourier coefficients derived from the solution for free,
O tema da identificação de dano (deteção e localização) em estruturas através de métodos não dest... more O tema da identificação de dano (deteção e localização) em estruturas através de métodos não destrutivos recebeu, ao longo das últimas décadas, um interesse acrescido em resultado de um significativo número de estruturas monitorizadas, tais como pontes ou edifícios históricos de elevado interesse patrimonial. Vários métodos de identificação de dano têm sido sugeridos com base em alterações dos parâmetros dinâmicos (e.g. frequências naturais ou modos de vibração), sempre que a matriz de rigidez ou de massa seja alterada no sistema. Contudo, o problema da identificação de dano é complexo, não havendo, até ao momento, um método ou um conjunto de métodos que resolva o problema. O presente artigo apresenta um método de identificação baseado na utilização da totalidade da informação espetral dentro de uma gama de frequências de interesse, obtida por ensaios de vibração ambiente (sem medir as excitações). O método é aplicado a três casos de estudo. Os resultados demostram que o método baseado na utilização da totalidade da informação espectral poderá ser um importante contributo para a identificação do dano em estruturas.
The present work deals with the damage identification of a historical masonry chimney located in ... more The present work deals with the damage identification of a historical masonry chimney located in Guimarães (Portugal), including a detailed survey, inspection and diagnosis. The chimney was object of a continuous monitoring campaign carried out to catch the evolution of the modal parameters and evaluate the success of the rehabilitation works planned after a lightning accident. Based on the dynamic features extracted from the OMA data, a damage identification analysis was performed making use of different damage identification techniques. Considering the explicit dependence of output-only power spectral densities on frequency contents, a spectral-based identification method was used to detect the damage. Finally, an appropriate localization index was defined combining evolutionary complex eigenvectors obtained from the decomposition of the power spectral density matrix. The results allow to conclude that the spectral-based dynamic identification method is a nondestructive tool able to capture the global behavior of a structure and may reveal itself of great help for exploring damage at an early stage in historical constructions.
Conservation of ancient built heritage plays a leading role for modern societies.
Non Linear Systems Under Complex α-Stable Lévy White Noise
Applied Mechanics and Biomedical Technology, 2003
ABSTRACT
Time-Variant Spectral Characteristics of Non-Stationary Processes
Proceedings of the 6th International Conference on Computational Stochastic Mechanics(CSM-6), 2011
Extended Entropy Functional for Nonlinear Systems in Stochastic Dynamics
PAMM, 2002
ABSTRACT
International Journal of Non-Linear Mechanics, 2006
The stability of a cantilever elastic beam with rectangular cross-section under the action of a f... more 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 integrodifferential 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. ᭧
DAMAGE IDENTIFICATION OF MONUMENTAL MASONRY STRUCTURES: THE CASE OF FOSSANOVA GOTHIC CHURCH
ABSTRACT In the present paper, the seismic behavior of a physical 1:5.5 scaled model of the churc... more ABSTRACT In the present paper, the seismic behavior of a physical 1:5.5 scaled model of the church of the Fossanova Abbey is investigated by means of numerical and experimental analyses. Aiming at defining the seismic vulnerability of such a structural typology a wide experimental campaign was carried out. The achieved experimental results lead to the definition of a refined FE model reproducing the dynamic behavior of the whole structural complex. Then, the central transversal three-central bays of the church, as it mostly influences the seismic vulnerability of the Abbey, was investigated in a more detail by means of a shaking table test on a 1:5.5 scaled physical model in the Laboratory of the Institute for Earthquake Engineering and Engineering Seismology in Skopje. In the present paper a brief review of the numerical activity related to the prediction of the shaking table test response of the model is first proposed. Then, the identification of frequency decay during collapse is performed through decomposition of the measured power spectral density matrix. Finally, the localization and evolution of damage in the structure is analyzed. The obtained results shown that a very good agreement is achieved between the experimental data and the predictive/interpretative numerical analyses..
We used speleothem vulnerability for assessing the seismic ground shaking threshold probably expe... more We used speleothem vulnerability for assessing the seismic ground shaking threshold probably experienced or not in caves located within Apennines seismogenic areas. On the southern flank of the Pollino Range, which is considered a current seismic gap, cave observations suggest the lack of significant earthquake-related deformation in the last ~5-7 ka, but the widespread occurrence of creep. In the central Apennines, homogenously-trending collapses within the Grotta Cola (Petrella Liri, AQ) evidence a significant co-seismic deformation event radiometrically bracketed at ~4-5 ka. We supplemented the cave observations with numerical modeling and static test analysis of the speleothem mechanical behaviour to stipulate probability density functions for the bending stress leading to rupture, and established different vulnerability curves for speleothems according to their shapes. Model results are consistent with the co-seismic deformation at Grotta Cola as being caused by the Liri or by ...
Hyperelastic modeling of biological tissues with distributed collagen fibers
A perturbation approach for the identification of uncertain structures
International Journal of Dynamics and Control, 2015
ABSTRACT This paper deals with the identification of linear structural systems with random parame... more ABSTRACT 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.
Approximate method for probability density of the response of a linear oscillator to a non-Poisson impulse process
III European Conference on Computational Mechanics, 2006
State vector of a dynamic system under a Poisson train of impulses is a non-diffusive Markov proc... more State vector of a dynamic system under a Poisson train of impulses is a non-diffusive Markov process and its joint probability density function satisfies an integro-differential generalized Fokker-Planck- Kolmogorov equation which is also called Kolmogorov-Feller equation. If the train of impulses is driven by non-Poisson, for example renewal, counting processes, the state vector is not a Markov process. Non-Markov pulse
Journal of Computational and Nonlinear Dynamics, 2009
A generalized damped Beck's column under pulsating actions is considered. The nonlinear partial i... more A generalized damped Beck's column under pulsating actions is considered. The nonlinear partial integrodifferential equations of motion and the associated boundary conditions, expanded up to cubic terms, are tackled through a perturbation approach. The multiple scales method is applied to the continuous model in order to obtain the bifurcation equations in the neighborhood of a Hopf bifurcation point in primary parametric resonance. This codimension-2 bifurcation entails two control variables, namely, the amplitude of the static and dynamic components of the follower force, playing the role of detuning and bifurcation parameters, respectively. In the postcritical analysis bifurcation diagrams and relevant phase portraits are examined. Two bifurcation paths associated with specific values of the follower force static component are discussed and the birth of new stable period-2 subharmonic motion is observed.
Probabilistic Engineering Mechanics, 2014
We describe three-dimensional and planar models of hyperelastic fiber reinforced materials charac... more We describe three-dimensional and planar models of hyperelastic fiber reinforced materials characterized by statistical distribution of the fiber orientation. Our models are based on a second order approximation of the strain energy density in terms of the fourth pseudo-invariant I 4 , typically employed in the description of fiber reinforced materials. For a particular choice of the strain energy density associated to the fiber reinforcement, it is possible to derive the explicit expression of the material and spatial stress tensors and of the stress covariance tensors. The mechanical behavior of the models is assessed through uniaxial, biaxial and shear tests.
The paper tackles the dynamic identification and the damage detection carried out by a spectral-b... more The paper tackles the dynamic identification and the damage detection carried out by a spectral-based method on the well-known Z24 bridge, a three-span prestressed concrete bridge located in Switzerland. Before being destroyed, the bridge was progressively damaged and tested in the framework of the Brite Euram project SIMCES. Starting from this benchmark, the presented spectral-based identification technique is validated and the usefulness of this method as a non-destructive tool able to catch the dynamic behavior of a structure and locate the damage is widely discussed. Firstly, a FE model of the bridge was built and calibrated in order to analyze its response to different excitation types (free vibration, triangular pulse, swept sine, shaker and random vibrations) and several damage scenarios. Secondly, aiming at identifying both the modal parameters and the damage of the bridge, the spectral-based method is applied making use of the power spectral matrix decomposition. Finally, a proper index is defined and applied to this case-study in order to locate the damage.
DAMAGE IDENTIFICATION OF MONUMENTAL MASONRY STRUCTURES: THE CASE OF FOSSANOVA GOTHIC CHURCH
ABSTRACT In the present paper, the seismic behavior of a physical 1:5.5 scaled model of the churc... more ABSTRACT In the present paper, the seismic behavior of a physical 1:5.5 scaled model of the church of the Fossanova Abbey is investigated by means of numerical and experimental analyses. Aiming at defining the seismic vulnerability of such a structural typology a wide experimental campaign was carried out. The achieved experimental results lead to the definition of a refined FE model reproducing the dynamic behavior of the whole structural complex. Then, the central transversal three-central bays of the church, as it mostly influences the seismic vulnerability of the Abbey, was investigated in a more detail by means of a shaking table test on a 1:5.5 scaled physical model in the Laboratory of the Institute for Earthquake Engineering and Engineering Seismology in Skopje. In the present paper a brief review of the numerical activity related to the prediction of the shaking table test response of the model is first proposed. Then, the identification of frequency decay during collapse is performed through decomposition of the measured power spectral density matrix. Finally, the localization and evolution of damage in the structure is analyzed. The obtained results shown that a very good agreement is achieved between the experimental data and the predictive/interpretative numerical analyses..
Stationary and Nontationary Response Probability Density Function of a Beam under Poisson White Noise
In this paper an approximate explicit probability density function for the analysis of external o... more In this paper an approximate explicit probability density function for the analysis of external oscillations of a linear and geometric nonlinear simply supported beam driven by random pulses is proposed. The adopted impulsive loading model is the Poisson White Noise , that is a process having Dirac’s delta occurrences with random intensity distributed in time according to Poisson’s law. The
Wind-tunnel tests on high-rise buildings: wind modes and structural response
ABSTRACT The evaluation of pressure fields acting on slender structures under wind loads is curre... more ABSTRACT The evaluation of pressure fields acting on slender structures under wind loads is currently performed in experimental aerodynamic tests. For wind-sensitive structures, in fact, the knowledge of global and local wind actions is crucial for design purpose. This paper considers a particular slender structure under wind excitation, representative of most common high-rise buildings, whose experimental wind field on in-scale model was measured in the CRIACIV boundary-layer wind tunnel (University of Florence) for several angles of attack of the wind. It is shown that an efficient reduced model to represent structural response can be obtained by coupling the classical structural modal projection with the so called blowing modes projection, obtained by decomposing the covariance or power spectral density (PSD) wind tensors. In particular, the elaboration of experimental data shows that the first few blowing modes can effectively represent the wind-field when eigenvectors of the PSD tensor are used, while a significantly larger number of blowing modes is required when the covariance wind tensor is used to decompose the wind field.
A new energy-based system identification method is developed, applicable in situations where the ... more A new energy-based system identification method is developed, applicable in situations where the dynamic response of a structure is measurable but the excitation is unmeasurable and describable only in terms of a stochastic process. It is shown that, in the case of a non-linear single degree of freedom system subjected to purely parametric, non-white random excitation, the power spectrum of the excitation can be identified through an estimation of the diffusion coefficient relating to the energy envelope of the response process. Through an estimation of the drift coefficient an identification of the system damping is also possible. The method is validated through application to simulated data relating to a Duffing oscillator with non-linear damping.
Response of non-linear oscillators to non-white random excitation using an energy based method
It is shown that, for a non-linear oscillator responding to non-white random excitation, the ener... more It is shown that, for a non-linear oscillator responding to non-white random excitation, the energy envelope of the response can be modelled approximately as a one-dimensional Markov process. Using a self-consistent asymptotic analysis, for small damping, simple expressions for the drift and diffusion coefficients of this Markov process are derived: these involve Fourier coefficients derived from the solution for free,
O tema da identificação de dano (deteção e localização) em estruturas através de métodos não dest... more O tema da identificação de dano (deteção e localização) em estruturas através de métodos não destrutivos recebeu, ao longo das últimas décadas, um interesse acrescido em resultado de um significativo número de estruturas monitorizadas, tais como pontes ou edifícios históricos de elevado interesse patrimonial. Vários métodos de identificação de dano têm sido sugeridos com base em alterações dos parâmetros dinâmicos (e.g. frequências naturais ou modos de vibração), sempre que a matriz de rigidez ou de massa seja alterada no sistema. Contudo, o problema da identificação de dano é complexo, não havendo, até ao momento, um método ou um conjunto de métodos que resolva o problema. O presente artigo apresenta um método de identificação baseado na utilização da totalidade da informação espetral dentro de uma gama de frequências de interesse, obtida por ensaios de vibração ambiente (sem medir as excitações). O método é aplicado a três casos de estudo. Os resultados demostram que o método baseado na utilização da totalidade da informação espectral poderá ser um importante contributo para a identificação do dano em estruturas.
The present work deals with the damage identification of a historical masonry chimney located in ... more The present work deals with the damage identification of a historical masonry chimney located in Guimarães (Portugal), including a detailed survey, inspection and diagnosis. The chimney was object of a continuous monitoring campaign carried out to catch the evolution of the modal parameters and evaluate the success of the rehabilitation works planned after a lightning accident. Based on the dynamic features extracted from the OMA data, a damage identification analysis was performed making use of different damage identification techniques. Considering the explicit dependence of output-only power spectral densities on frequency contents, a spectral-based identification method was used to detect the damage. Finally, an appropriate localization index was defined combining evolutionary complex eigenvectors obtained from the decomposition of the power spectral density matrix. The results allow to conclude that the spectral-based dynamic identification method is a nondestructive tool able to capture the global behavior of a structure and may reveal itself of great help for exploring damage at an early stage in historical constructions.
Conservation of ancient built heritage plays a leading role for modern societies.
Non Linear Systems Under Complex α-Stable Lévy White Noise
Applied Mechanics and Biomedical Technology, 2003
ABSTRACT
Time-Variant Spectral Characteristics of Non-Stationary Processes
Proceedings of the 6th International Conference on Computational Stochastic Mechanics(CSM-6), 2011
Extended Entropy Functional for Nonlinear Systems in Stochastic Dynamics
PAMM, 2002
ABSTRACT
International Journal of Non-Linear Mechanics, 2006
The stability of a cantilever elastic beam with rectangular cross-section under the action of a f... more 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 integrodifferential 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. ᭧