Free vibration analysis of civil engineering structures by component-wise models

Higher-order beam models are used in this paper to carry out free vibration analysis of civil engineering structures. Refined kinematic fields are developed using the Carrera Unified Formulation (CUF), which allows for the implementation of any-order theory without the need for ad hoc formulations. The principle of virtual displacements in conjunction with the finite element method (FEM) is used to formulate stiffness and mass matrices in terms of fundamental nuclei. The nuclei depend neither on the adopted class of beam theory nor on the FEM approximation along the beam axis. This paper focuses on a particular class of CUF models that makes use of Lagrange polynomials to discretize cross-sectional displacement variables. This class of models are referred to as component-wise (CW) in recent works. According to the CW approach, each structural component (e.g. columns, walls, frame members, and floors) can be modeled by means of the same 1D formulation. A number of typical civil engineering structures (e.g. simple beams, arches, truss structures, and complete industrial and civil buildings) are analyzed and CW results are compared to classical beam theories (Euler–Bernoulli and Timoshenko), refined beam models based on Taylor-like expansions of the displacements on the cross-section, and classical solid/shell FEM solutions from the commercial code MSC Nastran. The results highlight the enhanced capabilities of the proposed formulation. It is in fact demonstrated that CW models are able to replicate 3D solid results with very low computational efforts.     

Computations and evaluations of higher-order theories for free vibration analysis of beams

This paper deals with higher-order theories for the analysis of free vibration of beam structures. Refined theories are implemented by the application of the Unified Formulation by the first author which allows one to introduce any-order expansions of the displacement unknowns over the beam sections. The selection of the most appropriate theory is made by using a so-called axiomatic–asymptotic approach which permits one to retain only those terms of the displacement expansion which have been established to be significant with respect to an assigned control parameter. The finite element method is used to provide numerical solutions. Various beam sections as well as boundary conditions are considered. Depending on the vibration modes (bending, torsion, etc.), quite different theories are selected. In general, the number of the effective terms of the resultant theories is much lower than the full expansion case amount. The nature of these terms can differ very much as different beam geometries and boundary conditions are considered. It has been concluded that the method proposed appears to be suitable and convenient to establish the most appropriate beam theory for a given problem; it leads, in fact, to the cheapest computational model for a given accuracy.     

Exact dynamic stiffness elements based on one-dimensional higher-order theories for free vibration analysis of solid and thin-walled structures

In this paper, an exact dynamic stiffness formulation using one-dimensional (1D) higher-order theories is presented and subsequently used to investigate the free vibration characteristics of solid and thin-walled structures. Higher-order kinematic fields are developed using the Carrera Unified Formulation, which allows for straightforward implementation of any-order theory without the need for ad hoc formulations. Classical beam theories (Euler–Bernoulli and Timoshenko) are also captured from the formulation as degenerate cases. The Principle of Virtual Displacements is used to derive the governing differential equations and the associated natural boundary conditions. An exact dynamic stiffness matrix is then developed by relating the amplitudes of harmonically varying loads to those of the responses. The explicit terms of the dynamic stiffness matrices are also presented. The resulting dynamic stiffness matrix is used with particular reference to the Wittrick–Williams algorithm to carry out the free vibration analysis of solid and thin-walled structures. The accuracy of the theory is confirmed both by published literature and by extensive finite element solutions using the commercial code MSC/NASTRAN^®$\mathrm{MSC}/{\mathrm{NASTRAN}}^{\circledR }$.     

Dynamic stiffness matrix of thin-walled composite I-beam with symmetric and arbitrary laminations

For the spatially coupled free vibration analysis of thin-walled composite I-beam with symmetric and arbitrary laminations, the exact dynamic stiffness matrix based on the solution of the simultaneous ordinary differential equations is presented. For this, a general theory for the vibration analysis of composite beam with arbitrary lamination including the restrained warping torsion is developed by introducing Vlasov's assumption. Next, the equations of motion and force–displacement relationships are derived from the energy principle and the first order of transformed simultaneous differential equations are constructed by using the displacement state vector consisting of 14 displacement parameters. Then explicit expressions for displacement parameters are derived and the exact dynamic stiffness matrix is determined using force–displacement relationships. In addition, the finite-element (FE) procedure based on Hermitian interpolation polynomials is developed. To verify the validity and the accuracy of this study, the numerical solutions are presented and compared with analytical solutions, the results from available references and the FE analysis using the thin-walled Hermitian beam elements. Particular emphasis is given in showing the phenomenon of vibrational mode change, the effects of increase of the modulus and the bending–twisting coupling stiffness for beams with various boundary conditions.     

Optimal design of thin walled I beams for extreme natural frequency of torsional vibrations

The optimal design of thin-walled I beams so as to extremize the natural frequency of torsional vibration is considered. It is assumed that only one dimension of the cross-section, except for the web height, may be variable in given limits, along the axis of the beam. The optimality condition for the variable dimension is settled by means of Pontryagin's maximum principle. The effect of the constant, axial loads is also included. the solution of the problem formulated is generally found in an iterative way. Some numerical examples of optimization of the I beam with variable widt of flanges are given.     

Optimal design of thin-walled I beams for a given natural frequency of torsional vibrations

A method of extremum weight design of thin-walled I beams for a given natural frequency of torsional vibrations is presented. The effects of warping stresses and constant axial loads are taken into account. The optimality condition for only one (except for the web height) dimension of the cross-section, variable along the axis of the beam, is derived by using Pontryagin's maximum principle. The solution of the problem formulated, with account also taken of the additional geometrical conditions, is obtained in an iterative way. Some numerical examples of optimal design of an I beam with variable flange width, for a specified fundamental frequency, are given.     

Three dimensional dynamics of pretwisted beams: A spectral-Tchebychev solution

This paper presents the application of the spectral-Tchebychev (ST) technique for solution of three-dimensional dynamics of unconstrained pretwisted beams with general cross-section (including both straight and curved cross-sections). In general, the dynamic response of pretwisted beams presents three-dimensional (3D) motions, including coupled bending–bending–torsional–axial motions. As such, accurately solving pretwisted beam dynamics requires a 3D solution approach. In this work, the integral boundary value problem based on the 3D linear elasticity equations is solved numerically using the 3D-ST approach. To simplify evaluation of the volume integrals, the boundaries are simplified by applying two coordinate transformations to render the pretwisted beam with curved cross-section into an equivalent straight beam with rectangular cross-section. Three sample pretwisted beam problems with rectangular, curved, and airfoil cross-sections at different twist rates are solved using the presented approach. In each case, the convergence of the solution is analyzed, and non-dimensional natural frequencies and mode shapes are compared to those from a finite-element (FE) solution. Furthermore, cross-sectional stress and displacements are obtained from the 3D-ST solution. Lastly, the non-dimensional natural frequencies from the 3D-ST and a 1D/2D solutions are compared. It is concluded that the 3D-ST solution can capture the three-dimensional dynamic behavior of pretwisted beams as accurately as an FE solution, but for a fraction of the computational cost. Furthermore, it is shown that 1D/2D solution can lead to significant errors at high twist rates, and thus, the 3D-ST solution should be preferred.     

Non-linear free torsional vibrations of thin-walled beams with bisymmetric cross-section

A finite element method for studying non-linear free torsional vibrations of thin-walled beams with bisymmetric open cross-section is presented. The non-linearity of the problem arises from axial loads generated at moderately large amplitude torsional vibrations due to immovability of end supports. The derivation of the fundamental differential equation of the problem is based on the classical assumption of a thin-walled beam with a non-deformable cross-section. The non-linear eigenvalue problem is solved iteratively by series of linear eigenvalue problems until the required accuracy is obtained. Non-linear frequencies, fundamental mode shapes and axial loads computed for various amplitude of torsional vibrations of thin-walled I beams are included.     

Numerical investigation of elastic modes of propagation in helical waveguides

Steel multi-wire cables are widely employed in civil engineering. They are usually made of a straight core and one layer of helical wires. In order to detect material degradation, nondestructive evaluation methods based on ultrasonics are one of the most promising techniques. However, their use is complicated by the lack of accurate cable models. As a first step, the goal of this paper is to propose a numerical method for the study of elastic guided waves inside a single helical wire. A finite element (FE) technique is used based on the theory of wave propagation inside periodic structures. This method avoids the tedious writing of equilibrium equations in a curvilinear coordinate system yielding translational invariance along the helix centerline. Besides, no specific programming is needed inside a conventional FE code because it can be implemented as a postprocessing step of stiffness, mass and damping matrices. The convergence and accuracy of the proposed method are assessed by comparing FE results with Pochhammer-Chree solutions for the infinite isotropic cylinder. Dispersion curves for a typical helical waveguide are then obtained. In the low-frequency range, results are validated with a helical Timoshenko beam model. Some significant differences with the cylinder are observed.     

Triply coupled vibrational band gap in a periodic and nonsymmetrical axially loaded thin-walled Bernoulli–Euler beam including the warping effect

The propagation of triply coupled vibrations in a periodic, nonsymmetrical and axially loaded thin-walled Bernoulli–Euler beam composed of two kinds of materials is investigated with the transfer matrix method. The cross-section of the beam lacks symmetrical axes, and bending vibrations in the two perpendicular directions are coupled with torsional vibrations. Furthermore, the effect of warping stiffness is included. The band structures of the periodic beam, both including and excluding the warping effect, are obtained. The frequency response function of the finite periodic beam is simulated with the finite element method. These simulations show large vibration-based attenuation in the frequency range of the gap, as expected. By comparing the band structure of the beam with plane wave expansion method calculations that are available in the literature, one finds that including the warping effect leads to a more accurate simulation. The effects of warping stiffness and axial force on the band structure are also discussed.     

A new approach for free vibration of axially functionally graded beams with non-uniform cross-section

This paper studies free vibration of axially functionally graded beams with non-uniform cross-section. A novel and simple approach is presented to solve natural frequencies of free vibration of beams with variable flexural rigidity and mass density. For various end supports including simply supported, clamped, and free ends, we transform the governing equation with varying coefficients to Fredholm integral equations. Natural frequencies can be determined by requiring that the resulting Fredholm integral equation has a non-trivial solution. Our method has fast convergence and obtained numerical results have high accuracy. The effectiveness of the method is confirmed by comparing numerical results with those available for tapered beams of linearly variable width or depth and graded beams of special polynomial non-homogeneity. Moreover, fundamental frequencies of a graded beam combined of aluminum and zirconia as two constituent phases under typical end supports are evaluated for axially varying material properties. The effects of the geometrical and gradient parameters are elucidated. The present results are of benefit to optimum design of non-homogeneous tapered beam structures.     

Investigating the non-classical boundary conditions relevant to strain gradient theories

In the present study, two classes of non-classical constitutive equations consisting of the first and the second order strain gradients theories (FSG and SSG) were applied in order to develop the governing equations of static and free vibrational behavior of beam structures. The governing equations in orders of six and eight were constructed for FSG and SSG theories, respectively. Therefore, higher order or in other words non-classical boundary conditions (HOBCs or NCBCs) came into play in addition to the classical ones (CBCs). Some explanations were presented about the concept of the non-classical boundary conditions. Analytical and finite element (FE) approaches were employed to solve the governing equations. The analytical solutions were utilized in validation and convergence study of FE results. Comparisons were made with the relevant data reported in the open literature; however, to the best of the authors’ knowledge, few references have been published on SSG theory and HOBCs. In the numerical studies, the effects of applying different combinations of CBCs and HOBCs to the static and free vibration behaviors of the beam were investigated. Moreover, the impacts of non-classical elastic constants and the beam size on its behavior were also studied.     

ANALYTICAL AND NUMERICAL STUDY ON SPATIAL FREE VIBRATION OF NON-SYMMETRIC THIN-WALLED CURVED BEAMS

For spatial free vibration of non-symmetric thin-walled circular curved beams, an accurate displacement field is introduced by defining all displacement parameters at the centroidal axis and three total potential energy functionals are consistently derived by degenerating the potential energy for the elastic continuum to that for thin-walled curved beams. The closed-form solutions are newly obtained for in-plane and out-of-plane free vibration analysis of monosymmetric curved beams respectively. Also, two thin-walled curved beam elements are developed using the third and fifth order Hermitian polynomials. In order to illustrate the accuracy and the practical usefulness of the present method, analytical and numerical solutions by this study are presented and compared with previously published results or solutions by ABAQUS' the shell element. Particularly, effects of the thickness curvature as well as the inextensional condition are investigated on free vibration of curved beams with monosymmetric and non-symmetric cross-sections.     

Assessment of nanotube structures under a moving nanoparticle using nonlocal beam theories

Dynamic analysis of nanotube structures under excitation of a moving nanoparticle is carried out using nonlocal continuum theory of Eringen. To this end, the nanotube structure is modeled by an equivalent continuum structure (ECS) according to the nonlocal Euler-Bernoulli, Timoshenko and higher order beam theories. The nondimensional equations of motion of the nonlocal beams acted upon by a moving nanoparticle are then established. Analytical solutions of the problem are presented for simply supported boundary conditions. The explicit expressions of the critical velocities of the nonlocal beams are derived. Furthermore, the capabilities of various nonlocal beam models in predicting the dynamic deflection of the ECS are examined through various numerical simulations. The role of the scale effect parameter, the slenderness ratio of the ECS and velocity of the moving nanoparticle on the time history of deflection as well as the dynamic amplitude factor of the nonlocal beams are scrutinized in some detail. The results show the importance of using nonlocal shear deformable beam theories, particularly for very stocky nanotube structures acted upon by a moving nanoparticle with low velocity.     

Numerical and experimental analysis of the vibration and band-gap properties of elastic beams with periodically variable cross sections

The band-gap properties of non-uniform periodic beams are analyzed using numerical and experimental methods. The flexural wave equations are established based on the Euler–Bernoulli and Timoshenko beam theories. The beams with periodically variable cross sections are investigated. The transfer matrix method is used to explore the dynamic behaviors of the periodic beams, that is, the natural frequencies of the finite periodic beams with different cross-section ratios between the adjacent sub-cells and the band-gaps of the infinite periodic beams based on the Bloch theory. The validity and accuracy of the band-gaps acquired by the present method are verified by comparing the results with those obtained from the finite element method and the vibration experiments. The effects of the different lengths of adjacent sub-cells on the band-gap properties are then investigated. The research results and conclusions should be useful in the study of vibration control applications.     

Exact dynamic and static stiffness matrices of shear deformable thin-walled beam-columns

Total potential energy of non-symmetric thin-walled beam-columns in the general form is presented by introducing the displacement field based on semitangential rotations and deriving transformation equations between displacement and force parameters defined at the arbitrary axis and the centroid-shear center axis, respectively. Next, governing equations and force-deformation relations are derived from the total potential energy for a shear-deformable, uniform beam element and a system of linear eigenproblem with non-symmetric matrices is constructed based on 14 displacement parameters. And then explicit expressions for displacement parameters are derived and exact dynamic stiffness matrices are determined using force-deformatin relationships. In addition, the modified numerical method to eliminate multiple zero eigenvalues and to evaluate the exact static stiffness matrix is developed for spatial stability analysis. Finally, in order to demonstrate the validity and the accuracy of this study, the spatially coupled natural frequencies and buckling loads are evaluated and compared with analytical solutions or results analyzed by thin-walled beam elements and ABAQUS's shell elements.     

Mathematical Models for the Propagation of Stress Waves in Elastic Rods: Exact Solutions and Numerical Simulation

In this work, the Bishop and Love models for longitudinal vibrations are adopted to study the dynamics of isotropic rods with conical and exponential cross-sections. Exact solutions of both models are derived, using appropriate transformations. The analytical solutions of these two models are obtained in terms of generalised hypergeometric functions and Legendre spherical functions respectively. The exact solution of Love model for a rod with exponential cross-section is expressed as a sum of Gauss hypergeometric functions. The models are solved numerically by using the method of lines to reduce the original PDE to a system of ODEs. The accuracy of the numerical approximations is studied in the case of special solutions.     

Nonlinear finite element model updating of an infilled frame based on identified time-varying modal parameters during an earthquake

A model updating methodology is proposed for calibration of nonlinear finite element (FE) models simulating the behavior of real-world complex civil structures subjected to seismic excitations. In the proposed methodology, parameters of hysteretic material models assigned to elements (or substructures) of a nonlinear FE model are updated by minimizing an objective function. The objective function used in this study is the misfit between the experimentally identified time-varying modal parameters of the structure and those of the FE model at selected time instances along the response time history. The time-varying modal parameters are estimated using the deterministic–stochastic subspace identification method which is an input–output system identification approach. The performance of the proposed updating method is evaluated through numerical and experimental applications on a large-scale three-story reinforced concrete frame with masonry infills. The test structure was subjected to seismic base excitations of increasing amplitude at a large outdoor shake-table. A nonlinear FE model of the test structure has been calibrated to match the time-varying modal parameters of the test structure identified from measured data during a seismic base excitation. The accuracy of the proposed nonlinear FE model updating procedure is quantified in numerical and experimental applications using different error metrics. The calibrated models predict the exact simulated response very accurately in the numerical application, while the updated models match the measured response reasonably well in the experimental application.     

Damage Identification of Truss Structures Based on Force Method

An computationally efficient damage identification technique for the planar and space truss structures is presented based on the force method and the micro genetic algorithm. For this purpose, the general equilibrium equations and the kinematic relations in which the reaction forces and the displacements at nodes are take into account, respectively, are formulated. The compatibility equations in terms of forces are explicitly presented using the singular value decomposition (SVD) technique. Then governing equations with unknown reaction forces and initial elongations are derived. Next, the micro genetic algorithm (MGA) is used to properly identify the site and extent of multiple damage cases in truss structures. In order to verify the accuracy and the superiority of the proposed damage detection technique, the numerical solutions are presented for the planar and space truss models. The numerical results indicate that the combination of the force method and the MGA can provide a reliable tool to accurately and efficiently identify the multiple damages of the truss structures.