International Journal of Structural Stability and Dynamics

In this paper, a procedure based on the transfer matrix method for obtaining the exact solution to the equations of free vibration of damaged frame structures, considering the effects of axial extension, shear deformation, rotatory inertia, and all compliance components arising due to the presence of a crack, is presented. The crack is modeled by a rotational and/or translational spring based on the concept of linear elastic fracture mechanics. Only the in-plane motion of planar structures is considered. The formulation is validated through some examples existing in the literature. Additionally, the mode shapes and natural frequencies of a frame with pitched roof are provided. The variation of natural frequencies with respect to the crack location is presented. It is concluded that considering the axial compliance, and axial-bending coupling due to the presence of a crack results in different dynamic characteristics, which should be considered for problems where high precision is required, such as for the crack identification problems.

Structures designed against earthquake loads based on using control systems may experience explosions during their lifetime. In this paper, the performance of a hybrid control system composed of a low-damping base isolation and a supplemental magneto-rheological (MR) damper under external explosion has been studied. Base isolation system has the ability of decreasing the maximum structural response under blast loadings by shifting the period of the structure. In addition, MR damper improves the base isolation system performance by controlling the base drift of the structure. Hence, in this paper, the capability of a hybrid base isolation system equipped with an MR damper at the base has been evaluated in reducing the maximum structural response and base drift under external blast loadings. To determine the voltage of the semi-active MR damper, the H₂/Linear Quadratic Gaussian (LQG) and clipped-optimal control algorithms have been applied. For numerical simulations, a 10-storey shear frame subjected to blast loadings applied on different floors has been considered and the performance of the hybrid isolation system and MR damper has been studied. The results have proven the effectiveness of the hybrid control system in controlling the maximum response and base drift of the isolated structure against spherical external explosion. Furthermore, comparing the performance of the hybrid passive and semi-active base isolation systems indicates that the semi-active hybrid base isolation system is more effective in reducing the root-mean-square (RMS) value of the base drift. Similarly, it has been found that the semi-active hybrid base isolation system also performs better than the high-damping base isolation system.

Stochastic dynamic analysis of structures with random parameters continues to be an open question in the field of civil engineering. As a newly developed method, the probability density evolution method (PDEM) can provide the probability density function (PDF) of the dynamic responses of highly nonlinear structures. In this paper, a new method based on PDEM and the kriging surrogate model, named the K-PDEM, is proposed to study the stochastic response of a structure. Being an exact interpolation method, the Gaussian process regression or the so-called kriging method is capable of producing highly accurate results. Unlike the traditional PDEM numerical method whose numerical precision is strongly influenced by the number of representative points, the K-PDEM employs the kriging method at each instant to generate additional time histories. Then, the PDEM, which is capable of capturing the instantaneous PDF of a dynamic response and its evolution, is employed in nonlinear stochastic dynamic systems. Because of the decoupling properties of the K-PDEM, the numerical precision of the result is improved by the enrichment of the generalized density evolution equations without increasing the computation time. The result shows that the new method is capable of calculating the stochastic response of structures with efficiency and accuracy.

Free vibration analysis of a sandwich plate with viscoelastic material core and functionally graded material (FGM) constraining layer under centrifugal force field is investigated herein. One edge of the sandwich plate is fixed to a rotating hub. The first-order shear deformation theory (FSDT) is used in the finite element modeling of the problem. The effects of strains due to the longitudinal and transverse deformations are also considered in addition to the shear deformation of the core. Various parametric studies are carried out to examine the effects of volume fraction index, setting angle, hub radius and rotational speed on the vibration characteristics of the sandwich plate. It is found that the fundamental frequency of the plate decreases with an increase in the volume fraction index of the FGM layer, viscoelastic core thickness and setting angle. The first mode loss factor increases with respect to the increasing volume fraction index. Increase in rotational speed and hub radius lead to an increase in the natural frequencies and a decrease in the modal loss factors.

Current inelastic seismic spectra suffer from a conceptual limitation: they are significant only on the maximum demand of ductility and they do not include any influence of the number of response cycles, yield excursions, stiffness and strength degradation and damage potential to structures. This paper presents a stochastic approach for obtaining damage-based inelastic seismic spectra. In order to consider the cumulative damage phenomenon in structural systems under strong ground motions, the authors adopt the Park and Ang damage model that includes the displacement ductility and the hysteretic energy. The novelty is that the peak theory of random processes is adopted to achieve damage-based seismic spectra. This approach has some advantages compared with the standard statistical approaches based on a large number of recorded accelerograms. First, it drastically reduces the computational effort, while allowing us to typify the seismic motion by some parameters such as the frequency content, peak acceleration, energy content and strong motion duration, i.e. all the parameters that affect the structural response under seismic actions. Besides, it allows to obtain the following stochastic inelastic seismic response spectra: damage-based (i) displacement and acceleration inelastic spectra, (ii) response modification factor spectra, (iii) yield strength demand spectra, and (iv) damage-based inelastic displacement ratio spectra.

Nonlinear resonance response of an electrostatic-actuated nanobeam is controlled by using a delayed axial electrostatic force with near-half the natural frequency. A graphene sensor pasted on the surface of the nanobeam is used to extract the vibration voltage signal. An axial-delayed capacitive controller is designed to produce delayed axial force to control the nonlinear vibration of the nanobeam. The vibration voltage signal from the graphene sensor is input to the axial-delayed capacitive controller to attenuate the nonlinear vibration of the nanobeam. The dynamic response of the resonator is investigated by using the method of multiple scales directly. The sufficient conditions of guaranteeing the system stability and the saddle-node bifurcation are studied. The attenuation ratio is defined as the ratio of the peak amplitude of the nonlinear vibration system with control to that without control. A critical feedback gain is given, which can shift the frequency–amplitude curves from the nonlinear vibration to a linear vibration. An optimal method in which the attenuation ratio is taken as objective function and the aforementioned sufficient conditions as the constraint conditions is given to calculate the optimal feedback gains. Numerical simulations are conducted for uniform nanobeams.

The dynamic pull-in instability of a thermoelectromechanically loaded micro-beam is investigated based on the “symmetric stress” gradient elasticity theory and Euler–Bernoulli beam theory. The beam is subjected to the combined action of an electric voltage, axial static force and a uniform temperature change. By employing Galerkin’s method, the nonlinear partial differential governing equation is decoupled into a set of nonlinear ordinary differential equations, which are then solved using Runge–Kutta method. Numerical results show that compared with the size-dependent micro-beam model, the classical elasticity theory in which the size effect is ignored underestimates the pull-in voltage. The effects of size, temperature change, axial force, geometric nonlinearity, fringe effect, initial gap, beam length and width on the pull-in instability of the micro-beam are discussed in detail.

Recently, a conditionally stable explicit time integration scheme using the cubic B-spline function was proposed for solving the structural dynamic problems. This paper presents a new unconditionally stable version of the previous algorithm, based on the uniform cubic B-spline piecewise polynomial approximations and collocation method. First, the method is implemented to solve the differential equation of motion for the single-degree-of-freedom (SDOF) systems. Then, it is generalized for the multi-degree-of-freedom (MDOF) systems. In this paper, a simple step-by-step algorithm is presented for the proposed method, with the stability and accuracy analyses carried out. The unconditional stability of the method is achieved through use of an adjustable collocation parameter $\alpha$. The computational accuracy and efficiency of the proposed method are demonstrated in three numerical examples.

The vibration of a four-span continuous plate with two rails on top and four extra elastic supports excited by a moving model car is studied through numerical simulations and experiments. Modal testing is carried out to identify the Young’s moduli of the plate material and the rail material. Shell elements and beam elements are adopted for the plate and the rails of their Finite Element (FE) model, respectively. An offset is required to connect the rails and the plate in the FE model and the offset ratio of the shell element is updated to bring the numerical frequencies of the structure (plate with rails) closest to its experimental frequencies. Modal Superposition (MS) method with numerical modes of the structure and an iterative method are combined to predict the vibration of the structure subjected to the moving car. The displacements of four points of the plate are measured during the crossing of the car and compared with predicted results. The two sets of results agree well, which validates the model of the system. Parametric analysis is then made using the validated system model.

This paper presents a new technique for input reconstruction based on the explicit fourth-order Runge–Kutta (RK4) method. First, the state-space representation of the dynamic system is discretized by the explicit RK4 method under the assumption of linear interpolation for the dynamic load, leading to a recurrence equation between the current state and the previous state. Then, the mapping from the sequences of input to output is established through the recursive operation of the system equation and observation equation. Finally, the stabilized force information is recovered using the Tikhonov regularization method. This approach makes use of the good stability and high precision of the RK4 method; in addition, the computational efficiency is enhanced by avoiding the computation of the inverse stiffness matrix. The proposed method is numerically illustrated and validated with various excitations on a simple four-story shear building and a more complicated 2D truss structure, along with a detailed parametric study. The simulation studies show that the external loads can be reconstructed with high efficiency and accuracy under a low noise environment.

Displacement- and reliability-based designs of tuned mass damper (TMD) for a shear building are studied herein. Different sources of uncertainties such as earthquake records and their peak ground accelerations (PGA), masses of floors, cross-sectional dimensions of structural members, damping of the structure and modulus of elasticity are considered. Monte Carlo simulation (MCS) is used for evaluating the performance of the designed TMD. A method for generating artificial earthquake record by using wavelet packet transform (WPT) and particle swarm optimization (PSO) is proposed to generate artificial records for areas without sufficient strong ground motion records. An illustrative example is used to study the displacement- and reliability-based designs of TMD, which are related to minimizing the structural displacement and maximizing the performance of TMD, respectively. In addition, the performance of TMD on mitigating the response of structure and its reliability under uncertain parameters of loading and structural properties are investigated. The results show that a displacement-based designed TMD could reduce the lateral displacement of a structure. Furthermore, it illustrates that the reliability-based designed TMD has a better performance in real condition of loading and structural parameters.

Member failure due to low cycle fatigue may occur for steel structures under strong earthquakes. In this paper, the seismic response of single-layer latticed domes composed of welded round pipes is analyzed by the software Open System for Earthquake Engineering Simulation (OpenSees) incorporating low cyclic fatigue, in which the equations of initial camber for members are obtained from the buckling coefficient curve of compression members in Chinese codes. Single-layer latticed domes composed of welded round pipes with different parameters are modeled, and the seismic responses of the domes with and without material fatigue are compared by the incremental dynamic analysis. The results show that under strong earthquakes, the seismic responses including the maximum displacements and plastic development of the domes with fatigue are larger than those of the domes without fatigue. The collapsed PGAs decrease by 10–25% if low cycle fatigue is incorporated. Therefore, the low cycle fatigue of material should be taken into account in the seismic analyses of single-layer latticed domes under strong earthquakes.

This paper is concerned with the temperature effect on the vibration of a cable-stayed beam. The thermal effect is considered by using two non-dimensional factors for the cable tension force and sag. The nonlinear in-plane and out-of-plane vibration equations of motion of the cable-stayed beam with thermal effect are derived by using the extended Hamilton’s principle. Eigenvalue analysis is performed to obtain closed-form eigenvalue solutions. It is shown that the effect of temperature variation plays a dominant role on the vibration behavior of the cable-stayed beam, and the effect is closely related with the initial tension force and the stiffness ratio. As to the in-plane motion, both positive and negative correlations between the temperature variations and natural frequencies are found, which depend on the mode order and the stiffness ratio of the cable-stayed beam. However, as to the out-of-plane motion, there only exist negative correlations between the temperature variation and natural frequencies, and the effect of temperature dropping condition seems more obvious. Moreover, both for the in-plane and out-of-plane motions, the locations of veering points between two natural frequencies are shifted under the thermal effect, which can significantly affect the internal resonance between different modes of the cable-stayed beam.

The galloping or vortex-induced vibration of transmission lines will lead to a periodic excitations to the masts of the cross-rope suspension tower (CRST). The mast of the CRST is modeled as a straight beam with an elastic support subjected to a pulsating axial force on the top, which will change the stiffness of the mast, thereby resulting in produce harmonic excitation and instability. The dynamic characteristics of the system are investigated, which show that the bending frequency of the CRST decreases linearly with increase in axial static load, while it increases nonlinearly with the increase in boundary stiffness. Then, the method of multiple scales is adopted to analyze the vibration. It is found that the wind load on the mast brings primary resonance, but has no effects on instability. In addition, the steady state solution of the primary resonance is obtained by the polar form of the reduced amplitude modulation equations (RAMEs), with the effects of the following parameters on the vibration amplitude of the mast studied: the prestressing load in the guy, magnitude of the dynamic force, detuning parameter and wind load. Finally, the instability regions of two cases ($\mathrm{\Omega }$ near $\omega$ and $\mathrm{\Omega }$ near $2\omega$) are studied by the Cartesian form of the RAMEs, with focus on the influence of the axial harmonic load produced by the galloping of the transmission lines on the instability area. It is observed that the magnitude of excitation frequency of the dynamic force in the range of instability region becomes larger until the spring stiffness is increased up to a certain value.

Generalized Beam Theory (GBT), intended to analyze the structural behavior of prismatic thin-walled members and structural systems, expresses the member deformed configuration as a combination of cross-section deformation modes multiplied by the corresponding longitudinal amplitude functions. The determination of the latter, usually the most computer-intensive step of the analysis, is almost always performed by means of GBT-based conventional 1D (beam) finite elements. This paper presents the formulation, implementation and application of the so-called “exact element method” in the framework of GBT-based linear buckling analyses. This method, originally proposed by Eisenberger (1990), uses the power series method to solve the governing differential equation and obtains the buckling eigenvalue problem from the boundary terms. A few illustrative numerical examples are presented, focusing mainly on the comparison between the combined accuracy and computational effort associated with the determination of buckling solutions with the exact and standard GBT-based (finite) elements. This comparison shows that the GBT-based exact element method may lead to significant computational savings, particularly when the buckling modes exhibit larger half-wave numbers.

This paper proposes the discrete analytical method (DAM) to determine exactly and efficiently the fully nonstationary random responses of rectangular Kirchhoff plates under temporally and spectrally nonstationary acceleration excitation of earthquake ground motions. First, the fully nonstationary power spectral density (PSD) model is suggested by replacing the filtered frequency and damping of Gaussian filtered white-noise model with the time-variant ones. The exact solutions of free vibration of thin plates with two opposite edges simply supported boundary conditions are introduced. Then, the full analytical procedure for random vibration analysis of the plate is established by using a pseudo excitation method (PEM) that can consider all modal auto-correlation and cross-correlation terms. Owing to involving a series of Duhamel time integrals of single degree of freedom systems, it is difficult to fully analytically evaluate the PSD of time-variant responses such as the transverse deflection, velocity, acceleration and stress components. Thus, DAM that combines the PEM with precise integration technique is developed to enhance the computational efficiency. Finally, comparison of the results by the DAM with Monte Carlo simulations and the analytical stationary random vibration analysis demonstrates the high efficiency and accuracy of DAM. Moreover, the fully nonstationary excitation imposes a remarkable effect on the response PSD of rectangular Kirchhoff plates.