Publications

Abstract: The flexible beams carrying attachments and ends elastically restrained against rotational and translation inertia often appear in engineering structures, modal analysis of those structures is important and necessary in structural design. In case of structure with large aspect ratio of height and length the Timoshenko beam theory (TBT) is used, instead of the Euler-Bernoulli theory (EBT), since it takes both shear and rotary inertia into account. Shear effect is extremely large in higher vibration modes due to reduced mode half wave length. In this paper, the full development and analysis of TBT for transversely vibrations uniform beam are presented for elastically supported ends. A two-node beam element with two degree of freedom per node is obtained based upon Hamilton’s principle. The influence of stiffnesses of the supports on the free vibration characteristics is investigated. For this purpose, the eigenvalues of the Timoshenko beam are calculated for various rigidity values of translational and rotational springs. The results obtained are discussed and compared with results obtained by other researchers.

Abstract: Vibration analysis of a beam is an important subject of study in engineering. All real physical structures, when subjected to loads or displacements, behave dynamically. In case of structure with large aspect ratio of height and length the Timoshenko beam theory (TBT) is used, instead of the Euler-Bernoulli theory (EBT), since it takes both shear and rotary inertia into account. Shear effect is extremely large in higher vibration modes due to reduced mode half wave length. In this paper, the full development and analysis of TBT for the transversely vibrating uniform beam are presented for classical boundary condition. Finally, a finite element is developed in terms of dimensionless parameters of rotatory and shear. The stiffness and mass matrices for a two-node beam element with two degree of freedom per node is obtained based upon Hamilton’s principle. Cubic and quadratic Lagrangian polynomials are made interdependent by requiring them to satisfy both of the homogeneous differential equations associated with TBT. Numerical examples are given for some boundary conditions. The results showed that for frequencies above critical frequency, Timoshenko beams presents distinct mode shapes behavior including the presence of double eigenvalues, shear mode or remarkably modes.

Abstract: The dynamic response of structures interacting with the soil is a topic of significant interest, as vibrations propagating through the soil can compromise the performance and safety of vibration-sensitive facilities. The damaging effects of ground-borne vibration can be mitigated with the installation of surface walls, provided adequate geometric and constitutive properties are selected for the walls. The effect of constitutive and geometric parameters of surface walls in their vibration attenuation performance can be analyzed using various numerical techniques. This work presents an analysis of the vibration attenuation performance of surface walls using the Perfectly Matched Layer (PML) method. The PML method, widely used in fields such as electromagnetism, acoustics, and elastodynamics, uses a complex coordinate stretching formulation of energy-absorbing layers that can be incorporated into the boundaries of finite domains. These layers effectively absorb incident waves at any angle and frequency, thereby minimizing reflections at the model boundaries. The incorporation of these energy-absorbing layers causes finite domains to behave as unbounded domains, in the sense that they comply with Sommerfeld's radiation condition. Hence, the PML is an effective way to obtain an accurate representation of unbounded domains, like the soil, while using classical finite domain discretizations. This method is used in this paper to model the response of various configurations of surface walls interacting with the soil. Numerical results show excellent agreement with reference solutions, confirming the accuracy of the PML modeling. Additional simulations are performed to assess the influence of depth of wall embedment on ground vibration attenuation. The results demonstrate that partially buried walls enhance isolation performance and that increasing the depth of embedment shifts the insertion loss peak toward higher frequencies. The proposed formulation provides a reliable framework for analyzing dynamic soil-structure interaction problems involving vibration mitigation using embedded or surface structures.

Abstract: Surface walls have been presented as an effective strategy to attenuate ground vibration and protect vibration-sensitive facilities. This attenuation is based on properties of the walls as locally-resonant bodies, the attenuation effectiveness of which is maximized at their natural bending and compression frequencies. This paper presents a numerical analysis of the effectiveness of a series of surface walls in attenuating ground vibration and modifying wave propagation in the soil. The model uses a classical Finite Element Method (FEM) formulation to represent the surface walls, coupled with Indirect Boundary Element Method (IBEM) formulation to accurately represent the soil as an unbounded, wave-propagating medium. Seismic excitation is represented by Rayleigh waves propagating along the surface of the soil. The results show that the spacing between the walls and the number of walls significantly influence the attenuation of ground vibrations induced by Rayleigh waves.

Abstract: This paper investigates the influence of buried foundations on the ground vibration attenuation performance of surface walls. Using a coupled IBEM-FEM model, we analyze the effects of a partially buried wall subjected to surface waves. The results show that embedding the wall alters its vibration attenuation performance by affecting its ability to scatter surface waves into bulk waves. These findings provide valuable insights for designing vibration mitigation solutions in environments where ground vibration is a concern.

Abstract: Metamaterials have become a topic of considerable interest in the scientific and engineering communities in recent decades. Such artificially designed structures offer innovative solutions to complex problems related to wave propagation and vibration control, and have enormous potential for applications in various engineering fields. A topic that has received considerable attention in recent years is the ability to enhance emerging bandgaps due to Bragg scattering and local resonance, and thus improve the wave filtering ability of metamaterials. The present article proposes to analyze a locally resonant Euler-Bernoulli beam that is coupled to a large number of resonators with multiple degrees of freedom that are periodically distributed throughout the structure in order to increase the areas where wave attenuation is observed. Wave and Finite Element Method is applied in order to obtain the dispersion relation for wave propagation for the analysis of wave attenuation mechanisms. The results indicate the pronounced presence of Bragg scattering bandgaps and local resonance bandgaps. The influence of increasing the number of resonators is analyzed, resulting in an increase in the number of bandgaps. Effects of varying the length of the periodic unit on the attenuation property are also investigated. An increase of this parameter leads to an increment of the maximum attenuation coefficient and a new bandgap of the Bragg scattering is observed. In addition, the bandgap behavior is examined when the periodicity due to the presence of resonators is harmonized with the material periodicity, resulting from the use of two materials per unit cell. Finally, the vibration transmittance is determined by means of the classical Finite Element Method. The results of the vibration transmission show that the wave attenuation appears in the same frequency bands as those obtained by the Wave and Finite Element Method, which corroborates the efficient wave filtering performance of the locally resonant metamaterial beam.

Abstract: Most wind power is generated on onshore structures, but in the last ten years, offshore wind turbines have gained prominence due to their superior wind quality and abundance of space in the ocean environment. This increase in demand for offshore structures show the importance of understanding how these facilities respond to the maritime environment. Recent studies have focused on analyzing the impact of loads from wind, currents and waves, highlighting the need for a precise formulation for each component of these forces. This article focuses on the elaboration of the distributed transverse fluid load, employing the semi-empirical equation developed by Morison to model the in-plane fluid force. The out-of-plane force, exemplified by the shedding of vortices, is approached in a simplified way, taking the form of a sine wave. To achieve this objective, random waves are modeled by the Airy linear wave theory, incorporating both the Pierson-Moskowitz (P-M) spectrum, Bretschneider spectrum, while also applying the central limit theorem. The P-M spectrum consideres only one parameter, the wind velocity, which can be interpreted as the significant wave height, whereas the Bretschneider spectrum incorporates two parameters, additionally considering the duration of the wind. This approach not only enables the accurate reproduction of wave profiles in the spatial and temporal domains, but also allows the observation of striking surface effects in shallow waters, contrasting with the smoother profiles in deep waters. Is numerically demonstrated that the wave profile curve shifts for both spectra as a result of the newly added parameter. This methodology facilitates the assessment of wave speeds in relation to depth. The evaluation of the out-of-plane force revealed it to be insignificant in situations of low Reynolds numbers, but it becomes noticeable at high Reynolds numbers, leading to oscillation of the fluid close to the structure due to the vortices. Finally, improvements in predicting this oscillation aid in preventing natural frequencies synchronization with shed vortices that could lead to structural collapse.

Abstract: The dynamic behavior of systems subject to soil-structure interaction is of great importance in civil engineering. Among the applications of this multiphysics problem can be highlighted railroad tracks, highway pavement, continuously supported pipelines and strip foundations. In this context, the aim of this paper is to use the vibration analysis by finite element method to study the behavior and the second spectrum of uniform beams supported on two-parameter elastic foundation. The first foundation parameter is modeled using linear springs similar to the Winkler foundation and the second foundation parameter is modeled as a shear layer as a function of the total slope of the beam. The effects of axial force, foundation stiffness parameters, transverse shear deformation and rotatory inertia are incorporated into the accurate vibration analysis. The motion equation is derived using Hamilton’s variational principle based on the finite element Method (FEM). A finite element is developed by means of the Rayleigh-Ritz using a cubic and quadratic approximated polynomial. The finite element and the analytic solutions are compared to verify the accuracy of the method in this kind of problem. The results obtained are discussed and compared with results obtained by other researchers. The rotary inertia parameter, the axial load parameter and the foundation stiffness influence in the frequency parameter are investigated. The results showed that the axial load decrease the frequency parameter and the foundation increase the frequency parameter. The second spectrum was studied for the hinged-hinged beam presenting a tiny difference for the addition of the foundation and the axial load. In addition, the second spectrum was not verified for the clamped-clamped beam, as expected. Finally, the finite element results present well agreement with the other researches results.

Abstract: The dynamic behavior in offshore structures is of great importance in the design, conception and operation phase, in order to predict failures, guarantee stability, safety and avoid environmental and financial impacts. An offshore structure consists of a fixing or support plot that departs from the sea soil to the surface of the water and connects to superstructures such as wind turbines, oil and natural gas platforms. In this article, the free vibration of a free-clamped uniform beam, similar to the tendons of a tension leg platform, is studied. Are evaluated beams not immersed and immersed in water, the influence of the dimensions of the cross-sectional and the contribution of the deck as an axial force applied at the free end. Four beam models, Euler-Bernoulli, Rayleigh's, Shear and Timoshenko are used to evaluate natural frequencies using the finite element method. To investigate the influence of axial load, natural frequencies are calculated for various percentages of critical load. The same material parameters are considered in the numerical examples. The results demonstrate that the frequencies obtained by Euler-Bernoulli, Rayleigh and shear surpass those obtained by the Timoshenko theory. In addition, the results indicate that the reduction of the cable cross-section, the influence of water and the contribution of the deck, decrease the natural frequencies of the tendons.

Abstract: In engineering projects, specifically in the mechanical area, it is indispensable to consider how the loads act on the components, because they can present themselves in different ways and cause abrupt failures. Therefore, the study that promotes the development of safe structures and with a longer useful life, is of significant importance. In this work, using the modified Goodman criterion, fatigue damage is evaluated in a cantilever beam with a hole in the domain subjected to different load conditions. The formulation of the finite element method and the mentioned failure criterion are presented. The stress field is obtained using a linear quadrilateral element, with four nodes per element and two degrees of freedom per node. The results obtained in computer programming, show the effect of fatigue in the structure for three different load conditions, it is evidenced greater resistance to failure in application of distributed load and lesser resistance under oblique point load.

Abstract: Fatigue failure analysis becomes a constant concern when intending to produce mechanical components subjected to alternating loads and stress concentration. Therefore, the study that provide the development of safe structures and with longer fatigue life proves to be of significant importance. This article presents the formulation of the finite element method applied to the fatigue problem in continuous structures from the modified Goodman, Gerber, and ASME-elliptic criteria. The stress field is obtained using a linear quadrilateral element, with four nodes per element and two degrees of freedom per node. It is considered two structures of optimized and non-optimized shape under boundary conditions of crimping and point loading at the free end. The numerical results obtained showed the effect of fatigue on the structures and among the various fatigue failure criteria. The modified Goodman criterion was more conservative, since it presented significantly higher results for the safety conditions considered, followed by the Gerber and, ASME-elliptic criteria, respectively.

Abstract: The studies of the Euler-Bernoulli beam on an elastic foundation is the basis for analysis of soil-structure interaction. In this paper is analyzed the free vibration of an Euler-Bernoulli beam resting on two-parameter foundation subject to an axial load. The frequency equations is obtained for several boundary conditions. A finite element is developed by mean of the Rayleigh-Ritz method using a cubic approximated polynomial. Numerical comparisons with others works are done for check the method accuracy for this kind of problem. The frequency parameters of the system and the mode shapes are obtained for classical and non-classical boundary conditions. It’s verified that the axial load decrease the frequency parameter and the foundation increase the frequency parameter.

Abstract: Nanobeams are nanoscale structures extensively used in nanotechnology applications. Due to the small scale effect, this nanostructures cannot be accurately modelled by traditional elastic theory. To overcome this difficulty, several continuous models including the material length scale effect were developed, like nonlocal elasticity theory. In this paper, a nonlocal finite element model for elastically supported Euler-Bernoulli (EBT) and Timoshenko (TBT) nanobeams is developed. Nonlocal differential constitutive equations of Eringen are considered to account for the small scale effect. The stiffness and mass matrices for a two-node nonlocal beam element with two degrees of freedom per node are obtained based upon Hamilton’s principle. The influence of nonlocal parameter, slenderness ratio and support stiffness on the free vibration characteristics is investigated. Numerical results obtained are discussed and compared with results obtained by other researchers.

Abstract: This work investigates time-harmonic strain and displacement fields within trenched soils through an Indirect-BEM (IBEM) approach. The method consists of a superposition of Green’s functions for surface and buried loads. Solutions for surface loads are used to discretize a rigid plate at the surface of the soil, on top of which time-harmonic vertical loads are applied. Solutions for buried load are used together with zero-stress boundary conditions to model the presence of trenches in the soil. The soil is modeled as homogeneous isotropic or transversely isotropic half-spaces, for which classical Green’s functions are available in the literature. Stress and displacement fields within the trenched half-space containing a surface plate are related through sets of fictitious loads. Post-processing from these loads yield quantities such as the displacement field anywhere in the half-space. This work uses these post-processed quantities to study how the ground vibration propagating from the loaded plate is affected by the presence of trenches.

Abstract: Various types of engineering structures are subject to periodic loading such as offshore platform parts and wind turbine blades. One of the main causes of failure in these structures is due to the resonance effect, when the frequency of external loading coincides with some natural frequency of the structure. Therefore, the maximization of natural frequencies is an increasingly sought-after topic in the design of these components. In this paper a genetic algorithm is developed to maximize natural frequencies of Euler-Bernoulli beams. Genetic algorithms are stochastic search methods, which are based on biological concepts of adaptation, natural selection, fitness and evolution, to solve optimization problems. A beam population is created, each of them discretized in a mesh of cylindrical elements with different diameters, initially random. The natural frequencies of the beam are found by the Finite Element Method, and the one with the highest natural frequency creates a new generation of offsprings. In each offspring is applied a mutation scheme that changes the diameter of any random element, making the entire population change. So over the generations the algorithm finds out the best diameter combination that maximizes the natural frequency of the beam. Results present different shapes are obtained for several boundary conditions and different natural frequencies maximized

Abstract: Timoshenko beam is one of the most complete beam models, so studies in this field are of great importance for the practical engineering. In this paper is analyzed the influence of the elastic foundation and the axial load on the dynamic behavior of the Timoshenko beam with uniform cross-section. A Finite Element is developed through of a cubic approximated polynomial. Numerical comparisons and examples are done for check the method performance and present a good agreement. The natural frequencies are obtained for several boundary conditions. It's investigated the slenderness ratio, the axial load parameter and the foundation stiffness influence in the frequency parameter. Finally, the results shows that the axial load decrease the frequency parameter and the foundation increase the frequency parameter.

Abstract: Dynamic behavior of offshore structures is an area of extensive research, since they are widely used to support superstructures like wind turbine, offshore platforms etc. This paper, the free vibration of a continuous, elastic model of a Jack-up Platform is studied. The model is considered non-immersed and immersed in water, is under going free transverse vibration in a plane. It is modeled as a uniform Timoshenko beam (TBT) which has an tip mass on one end and is fixed at the other end. Effects of shear deformation and rotary inertia are included in the beam. Such a model is representative of numerous applications. The analytical theory for Timoshenko beam is presented, the free vibration equation is derived using Hamilton’s variational principle based on Finite Element Method (FEM), which show a good agreement in results . At the end, an parametric study is carried out which provides an insight into the dependence of natural frequency on different configurations of the geometric and parameters of stiffnesses of the supports on the free vibration characteristics is investigated.

Abstract: Offshore oil platforms are large structures used for offshore drilling that house workers and machines required to drill wells in the ocean floor and for the extraction of oil and/ or natural gas. These structures can be modeled by flexible cable or beam. This paper presents a model for vibration analysis of a tendon of a floating platform, Tension Leg Platform (TLP). The model consists of a Euler-Bernoulli beam with composite elastic supports of translational and rotational spring and a mass concentrated at the free end. The equation of motion is derived from the Hamilton principle and the frequency equation is obtained analytically. Subsequently, a beam element with two degrees of freedom per node is developed. The influence of the rigidity of the elastic supports, as well as the effect of the inclusion of the water body is investigated through numerical examples.

Abstract: In many soil-structure interaction problems, the soil medium model used is the Winkler foundation for mathematical simplicity. However, this foundation model cannot represent the behavior of foundation materials for all engineering applications. The Pasternak model can accomplish a more realistic and generalized description of the soil behavior. For structures with a large aspect ratio of height and length, Timoshenko beam theory is used, instead of Euler-Bernoulli theory, since it takes both shear and rotary inertia into account. This paper investigates the effects of the Pasternak foundation on the dynamic response of the Timoshenko beam. A finite element is developed using cubic and quadratic polynomials, which are made interdependent by requiring them to satisfy the static homogeneous differential equations associated with Timoshenko beam theory. The influence of the foundation on the second spectrum is concerned. The results showed that the presence of the foundation anticipates the second spectrum and does not have a significant influence in its frequencies. Also, the presence of the foundation permits a distinction between the two spectra for boundary conditions that do not factorize.

Abstract: Free vibration analysis of beams on the elastic foundation is necessary for an optimal design in many engineering applications. This paper analyzes the effect of Winkler and Pasternak foundations in the natural frequencies of a Timoshenko beam. For this purpose, the natural frequencies are obtained by solving the partial differential equation governing the problem. A finite element is developed using cubic and quadratic polynomials for transverse displacement and slope, respectively, for a two-node beam element with two degrees of freedom per node. The finite element and the analytic solutions are compared and discussed with some numerical examples. The results presented a high accuracy and reliability for the beam-foundation iteration problem. The presence of the Pasternak foundation increases the natural frequencies of the beam. The growth of the frequencies on Pasternak foundation is mostly due to the shear layer stiffness, and it reduces the influence of the elastic stiffness. The use of the Euler-Bernoulli beam on Winkler foundation instead of the Timoshenko beam on Pasternak foundation presents a significant inaccuracy, except for specific values of rotational inertia.

Abstract: Viscoelastic materials are widely used for passive damping in a variety of engineering structures due to the need for structural stability and durability. Beams are often used as structural elements in many of those structures, like rotor blades, transmission shafts, frames and robotic arms. In this paper, the full development and analysis of Timoshenko’s beam for transversely vibrations uniform viscoelastic beam are presented for classical boundary conditions. The governing equation of motion is obtained based upon Hamilton’s principle and constitutive relations. The viscoelastic beam material is constituted by the Kelvin-Voigt rheological model. Finally, the influence of the viscous internal friction on the natural frequencies and waves dispersion are discussed and numerically demonstrated.

Abstract: Free vibration analysis of beams on the elastic foundation is necessary for an optimal design in engineering applications, especially in the fields of transportation systems. This paper analyzes the influence of the foundation parameters and the boundary conditions configuration in the dynamic response of Euler-Bernoulli beam on Pasternak foundation. A finite element solution is developed and compared with the analytic solution to verify the reliability of the method in this kind of problem. The results showed that Pasternak foundation increases the frequency parameters of the beam. Also, the presence of the foundation reduces the normal mode amplitude and the clamped-free case is the most affected by the presence of a Pasternak foundation.

Abstract: The dynamic response of beams on elastic foundation has been extensively studied in the modern engineering, especially in the fields of transportation systems. This approach is mainly used to model a railroad track. In this paper, a finite element is developed using cubic and quadratic polynomials. The finite element and the analytic solutions are compared to verify the reliability of the method in this kind of problem. The influence of the foundation parameters and the boundary conditions configuration are taken into account. The results showed that Pasternak foundation increases the frequency parameters of the beam. Also, the presence of the foundation reduces the normal mode amplitude and the clamped-free case is the most affected by the presence of a Pasternak foundation

Abstract: In the classical Bernoulli-Euler theory flexural vibrations of beam, the effect of rotatory inertia and shear are neglected. However, the equations obtained on these assumptions are inadequate for short and thin-webbed beams and for beams where higher modes are required, considerable errors may be incurred by use of equations. When a beam is subjected to lateral vibration so that depth of the beam is a significant proportion of the distance between two adjacent nodes, rotatory inertia of beam and transverse deformation arising from the severe contortions of the beam during vibration make significant contributions to the lateral deflection. Therefore rotatory inertia and shear effects must be taken into account in the and analysis of high-frequency vibration of all beams, and in all analyses of deep beams. In this work, the effect of rotatory inertia and transverse-shear deformation on beams are introduced and analysed. Frequency equation for each case are developed in terms of dimensionless parameters of rotatory and shear. Numerical examples presented in literature are re-examined.

Abstract: The flexible beams carrying attachments and non-classical boundary conditions often appear in engineering structures, modal analysis of those structures is important and necessary in structural design. The analysis of vibrating beams with ends elastically restrained against rotation and translation or with ends carrying concentrated masses or rotational inertias is of great interest in a variety of practical cases. In this analysis, the governing differential equations of the beam, which is a partial differential equation with variable coefficients, and that of the mass-spring system, which is an ordinary differential equation, are found. The exact solution of the problem is then obtained using classical and non-classical boundary conditions and the eigenvalues and eigenfunctions are found. Finally, some cases with available results in the literature are presented and analyzed.

Abstract: Early researchers reported that for hinged-hinged beam, two distinct natural frequencies correspond to the same spatial mode shape and that it establish the existence of a second distinct frequency spectrum. This phenomenon was received the terminology of the "Second Spectrum of Timoshenko Beam". However, some investigators reported that the single frequency spectrum interpretation has the same sets of frequencies calculated from the "two spectra". The objective of this paper is to provide an analysis of the solution of Timoshenko beam equations for higher modes and discuss their complex dynamical behavior. A numerical example presented in literature is re-examined.

Abstract: Beams are structural elements frequently used for support buildings, part of airplanes, ships, rotor blades and most engineering structures. In many projects it is assumed that these elements are subjected only to static loads, however dynamic loads induce vibrations, which changes the values of stresses and strains. Furthermore, these mechanical phenomenon cause noise, instabilities and may also develop resonance, which improves deflections and failure. Therefore study these structural behavior is fundamental in order to prevent the effects of vibration. The mechanical behavior of these structural elements may be described by differential equations, four widely used beam theories are Euler-Bernouli, Rayleigh, Shear and Timoshenko. This paper presents a comparison between these four beam theories for the free transverse vibration of uniform beam. Numerical examples are shown to beam models under classical boundary condition.