George C. Tsiatas

Non-linear dynamic analysis of beams with variable stiffness

Abstract In this paper the analog equation method (AEM), a BEM-based method, is employed to the non-linear dynamic analysis of a Bernoulli–Euler beam with variable stiffness undergoing large deflections, under general boundary conditions which maybe non-linear. As the cross-sectional properties of the beam vary along its axis, the coefficients of the differential equations governing the dynamic equilibrium of the beam are variable. The formulation is in terms of the displacements. The governing equations are derived in both deformed and undeformed configuration and the deviations of the two approaches are studied. Using the concept of the analog equation, the two coupled non-linear hyperbolic differential equations with variable coefficients are replaced by two uncoupled linear ones pertaining to the axial and transverse deformation of a substitute beam with unit axial and bending stiffness, respectively, under fictitious time-dependent load distributions. A significant advantage of this method is that the time history of the displacements as well as the stress resultants are computed at any cross-section of the beam using the respective integral representations as mathematical formulae. Beams with constant and varying stiffness are analyzed under various boundary conditions and loadings to illustrate the merits of the method as well as its applicability, efficiency and accuracy.

A Microstructure-dependent Orthotropic PlateModel Based On A Modified Couple Stress Theory

In this paper a modified couple stress model containing only one material length scale parameter is developed for the static analysis of orthotropic micro-plates with arbitrary shape. The proposed model is capable of handling plates with complex geometries and boundary conditions. From a variational procedure the governing equilibrium equation of the micro-plate and the most general boundary conditions are derived, in terms of the deflection, using the principle of minimum potential energy. The resulting boundary value problem is of the fourth order (instead of existing gradient theories which is of the sixth order) and it is solved using the Analog Equation Method (AEM), which is a boundary-type meshless method. Several plates of various shapes, aspect and Poisson’s ratios are analyzed to illustrate the applicability of the developed micro-plate model and to reveal the differences between the current model and the classical plate model. Moreover, useful conclusions are drawn from the micron-scale response of this new orthotropic plate model.

The analog equation method for large deflection analysis of heterogeneous orthotropic membranes: a boundary-only solution

In this paper, the analog equation method (AEM) is applied to nonlinear analysis of heterogeneous orthotropic membranes with arbitrary shape. In this case, the transverse deflections influence the in-plane stress resultants and the three partial differential equations governing the response of the membrane are coupled and nonlinear with variable coefficients. The present formulation, being in terms of the three displacement components, permits the application of geometrical in-plane boundary conditions. The membrane may be prestressed either by prescribed boundary displacements or by tractions. Using the concept of the analog equation, the three coupled nonlinear equations are replaced by three uncoupled Poissons equations with fictitious sources under the same boundary conditions. Subsequently, the fictitious sources are established using a procedure based on the BEM and the displacement components as well as the stress resultants are evaluated from their integral representations at any point of the membrane. Several membranes are analyzed which illustrate the method, and demonstrate its efficiency and accuracy. Moreover, useful conclusions are drawn for the nonlinear response of heterogeneous anisotropic membranes. The method has all the advantages of the pure BEM, since the discretization and integration are limited only to the boundary.

Nonlinear dynamic analysis of heterogeneous orthotropic membranes by the analog equation method

In this paper the analog equation method, a BEM-based method, is employed to analyze the dynamic response of flat heterogeneous orthotropic membranes of arbitrary shape, undergoing large deflections. The problem is formulated in terms of the three displacement components. Due to the heterogeneity of the membrane, the elastic constants are position dependent and consequently the coefficients of the partial differential equations governing the dynamic equilibrium of the membrane are variable. Using the concept of the analog equation, the three-coupled nonlinear second order hyperbolic partial differential equations are replaced with three uncoupled Poissons quasi-static equations with fictitious time dependent sources. The fictitious sources are represented by radial basis functions series and are established using a BEM-based procedure. Both free and forced vibrations are considered. Membranes of various shapes are analyzed to illustrate the merits of the method as well as its applicability, efficiency and accuracy. The proposed method is boundary-only in the sense that the discretization and the integration are restricted on the boundary. Therefore, it maintains all the advantages of the pure BEM.

Elastic-plastic analysis of functionally graded bars under torsional loading

Abstract In this paper a new integral equation solution to the elastic-plastic problem of functionally graded bars under torsional loading is presented. The formulation is general in the sense that it can be applied to an arbitrary cross-section made of any type of elastoplastic material. In material science the Functionally Graded Material (FGM) is a non-homogeneous composite which performs as a single-phase material, by unifying the best properties of its constituent phase material. The nonlinear elastic-plastic behavior is mathematically described by the deformation theory of plasticity. According to this theory, the material constants are assumed variable within the cross section, and are updated through an iterative process so as the equivalent stress and strain at each point coincide with the uniaxial material curve. In this investigation a new straightforward nonlinear procedure is introduced in the deformation theory of plasticity which simplifies the solution method. At each iteration step, the warping function is obtained by solving the torsion problem of a non-homogeneous isotropic bar using the Boundary Element Method (BEM) in conjunction with the Analog Equation Method (AEM). Without restricting the generality, the FGM material is comprised of a ceramic phase and a metal phase. The ceramic is assumed to behave linearly elastic, whereas the metal is modeled as an elastic – linear hardening material. Furthermore, the TTO homogenization scheme for estimating the effective properties of the two-phase FGM was adopted. Several bars with various cross-sections and material types are analyzed, in order to validate the proposed model and exemplify its salient features. Moreover, useful conclusion are drawn from the elastic-plastic behavior of functionally graded bars under torsional loading.

Dynamic Analysis and Seismic Response of Planar Circular Arches with Variable Cross-Section

The aim of this paper is to investigate the dynamic response of planar circular arches with variable cross-section subjected to seismic ground motions. Arches have a wide range of application (e.g. bridges, roofs) thanks to their capacity to span large areas by resolving vertical actions into compressive stresses and confining tensile stresses. The full understanding of their dynamic response is a challenging technical and computational problem, especially when seismic loading is considered. For example, the assumption of axial inextensibility simplifies the differential equations but overestimates the vibration frequencies, especially those of shallow arches since axial forces are of paramount importance (as opposed to beams). In lieu of the above, our formulation incorporates the effect of axial extension, and the arches are modeled using a new generic curved beam model that includes both axial (tangential) and transverse (normal) to the arch centerline deformations, and is able to account for variable mass and stiffness properties, as well as elastic support or restraint. The resulting dynamic governing equations of the circular arch are formulated in terms of the displacements, and solved using an efficient integral equation method. Three circular arches with variable rectangular cross-section are analyzed in order to investigate their dynamic properties and seismic performance. Using both time history and modal analysis useful conclusions are drawn with regard to the contribution of each mode on the calculation of different response quantities.

A Simple Rate-Independent Uniaxial Shape Memory Alloy (SMA) Model

In this paper, three specific uniaxial phenomenological models commonly used for the description of a Shape Memory Alloy (SMA) behavior are examined in detail. In particular, the models examined are the Graesser-Cozzarelli model, the Wilde-Gardoni-Fujino model, and the Zhang-Zhu model. The pertinent model parameters are examined with respect to their physical representation, if any. Based on this analysis, a new simple rate-independent model is proposed which addresses all issues in a unified manner. Finally, powerful metaheuristics are employed for system identification, producing excellent fit with experimental data while revealing valuable information regarding the relative sensitivity of the proposed model parameters.

PARAMETER OPTIMIZATION OF THE KDAMPER CONCEPT IN SEISMIC ISOLATION OF BRIDGES USING HARMONY SEARCH ALGORITHM

In this paper a novel design methodology for seismic isolation of bridges is presented employing the KDamper passive system and the Harmony Search (HS) optimization algorithm. The implementation of the KDamper concept to the absorption of seismic excitation of bridge structures requires the solution of a linear dynamic problem during which the basic parameters of KDamper design must be predefined. As this selection appears to be a quite complex procedure, the need for an optimization process is of paramount importance. In our work, an emerging metaheuristic algorithm, the Harmony Search algorithm, which has been successfully applied to several engineering problems, is employed to obtain optimum design parameters for a typical bridge structure under seismic excitation. The optimal solution is obtained for a set of 5 earthquake excitation records using as objective function the Root Mean Square of the displacement ratio. Comparative results of the dynamic response between the initial and the isolated structure are presented to verify the effectiveness, validity and reliability of the proposed design method. 37 Available online at www.eccomasproceedia.org Eccomas Proceedia COMPDYN (2017) 37-51

A Layered Boundary Element Nonlinear Analysis of Beams

This work aims to introduce a new layered approach to the nonlinear analysis of initially straight Euler-Bernoulli beams by the Boundary Element Method (BEM). The beam is studied in the context of both geometrical and material nonlinearity. The governing differential equations, derived by applying the principle of minimum total potential energy, are coupled and nonlinear, while the boundary conditions are the most general and may include elastic support or restraint. The boundary value problem, regarding the axial and transverse displacements, is solved using the Analog Equation Method (AEM), a BEM based method, together with an iterative procedure. Although a direct solution to the geometrical nonlinear problem has already been presented, in this work an alternative layered analysis is proposed. The discretization is applied in both the longitudinal direction and the cross-sectional plane, and an iterative process is commenced. First, initial fictitious load distributions are assumed at beam’s each cross-section, and the displacements, as well as their derivatives, are computed using the AEM. Second, the two stress resultants, i.e., the axial force and bending moment, are evaluated by appropriate integration over the cross-section. In the end, the derivatives of the stress resultants are evaluated, and the equilibrium of the governing equations is checked. If the equilibrium is satisfied, the process is terminated. Otherwise, the fictitious load distributions are updated, and the procedure starts over again. Several representative examples are studied, and the results are compared with those presented in the literature, validating the reliability and effectiveness of the proposed method.

A new Hysteretic Nonlinear Energy Sink (HNES)

Abstract The behavior of a new Hysteretic Nonlinear Energy Sink (HNES) coupled to a linear primary oscillator is investigated in shock mitigation. Apart from a small mass and a nonlinear elastic spring of the Duffing oscillator, the HNES is also comprised of a purely hysteretic and a linear elastic spring of potentially negative stiffness, connected in parallel. The Bouc-Wen model is used to describe the force produced by both the purely hysteretic and linear elastic springs. Coupling the primary oscillator with the HNES, three nonlinear equations of motion are derived in terms of the two displacements and the dimensionless hysteretic variable, which are integrated numerically using the analog equation method. The performance of the HNES is examined by quantifying the percentage of the initially induced energy in the primary system that is passively transferred and dissipated by the HNES. Remarkable results are achieved for a wide range of initial input energies. The great performance of the HNES is mostly evidenced when the linear spring stiffness takes on negative values.