International Journal of Structural Stability and Dynamics

Beams and beam structures are structural components commonly used in mechanical, aerospace, nuclear, and civil engineering. To meet the different engineering design limitations such as operational conditions, weight, and vibrational characteristics, these components may be made of various materials such as functionally-graded materials (FGMs), composites, and homogeneous materials. Functionally-graded (FG) beams play a key role not only in classical structural applications, but also have vast applications in thermal, electric-structural and electric-thermal-structural systems, e.g. in the form of FG beam energy harvesters, sensors and actuators. In all these applications, using new materials like FGMs can greatly improve the efficiency of the structural components and systems. Since FG beams are mostly used as moving components in engineering structures, vibration analysis of these components has been studied by numerous researchers. In order to solve the governing equation and related boundary conditions of the FG beams, powerful numerical methods with a high level of accuracy and fast rate of convergence are often required. The differential quadrature method (DQM) is a powerful and reliable numerical method which has been extensively used by researchers to perform the vibration analyses of FG structures in the last decade. In this paper, firstly various mathematical models which have been used to express the material properties of FGMs are reviewed. Secondly different elasticity theories which have been applied in vibration analysis of FG beams are summarized. In addition, a review on the DQM and its applications is presented. At the next step, a comprehensive review on free vibration analyses of FG beams based on different elasticity theories and in particular those using the DQM is performed. In continue, a brief review on the application of other numerical methods in vibration analysis of FG beams is presented. Moreover, because of the importance of nonlinear vibration analysis of FG beams, a review on the application of various numerical methods and different elasticity theories on nonlinear vibration analysis of FG beams is performed. Finally, a brief review on linear and nonlinear vibration analysis of FG microbeams, as a special type of FG beams, is presented.

Structural steel frames exhibit significantly geometric and material nonlinearities which can be captured using the second-order inelastic analysis, also known as advanced analysis. Current specifications of most modern steel design codes, e.g. American code AISC360, European code EC3, Chinese code GB50017 and Australian code AS4100 permit the use of advanced analysis methods for the direct design of steel structures to avoid tedious member capacity checks. In the past three decades, a huge number of advanced analysis and modeling methods have been developed to predict the behavior of steel and composite frames. This paper presents a comprehensive review of their developments, which focus on beam-column elements with close attention to the way to capture geometric and material nonlinearity effects. A brief outline of analysis methods and analysis tools for frames was presented in the initial part of the paper. This was followed by a discussion on the development of displacement-based, force-based and mixed beam elements with distributed plasticity and concentrated plasticity models. The modeling of frames subjected to fire and explosion was also discussed. Finally, a review of the beam-column models for composite structures including concrete-filled steel tubular (CFST) columns, composite beams and composite frames was presented.

Laminated beams made of nanocomposite materials have been used in many industrial sectors. This paper reports a study on the vibration behavior of laminated beams when experiencing the large amplitude vibration. The beams are made of perfectly bonded carbon nanotube-reinforced composite (CNTRC) layers. The novel constructions of CNTRC laminated beams with out-of-plane maximum negative Poisson’s ratio (NPR) are proposed. The volume fraction of CNT may change across the beam thickness which results in a piece-wise pattern. The material properties of the CNTRC layers are temperature-dependent and can be estimated by the extended rule of mixture model. The beams are considered to rest on a two-parameter elastic foundation and under differential thermal environmental conditions. The higher order shear deformation beam theory is applied to derive the motion equations of the nonlinear vibration of FG-CNTRC laminated beams. These equations include the influencing factors such as the geometrical nonlinearity in the von Kármán sense, the thermal effects and the beam–foundation interaction. The nonlinear vibration solutions can be obtained by employing a two-step perturbation approach. The nonlinear vibration characteristics of FG-CNTRC laminated beams under different sets of loading conditions and thermal environmental conditions are discussed in detail through a series of parametric studies. Numerical results show that the NPR has a significant effect on the large amplitude vibration characteristics of CNTRC laminated beams.

In the present research, vibration behavior is presented for a thermally postbuckled double layered graphene sheet (DLGS). The DLGS is modeled as a nonlocal orthotropic plate and contains small-scale effects. The formulations are based on Kirchhoff’s plate theory and nonlinearity of von Karman type is considered in strain–displacement relations. The thermal effects, van der Waals forces between layers and chirality are also included, and some of the material properties are assumed to be temperature-dependent. A coupled system of equations is derived and a new semi-analytical solution is obtained using multiple time-scale methods. The effects of variation of small-scale parameter on the natural frequencies, deflections and response curve of DLGS are analyzed, and the numerical results are obtained from the nonlocal plate model; also, molecular dynamics (MD) simulations are used to investigate different vibration behaviors of DLGS and calibration of the small-scale coefficient. Numerical results are compared with those of similar researches. Effects of various parameters on the postbuckled vibration of DLGS in thermal environments such as scale parameter, environmental damping, length and thermal load are presented. The stability and occurrence of internal resonance between vibration modes around a stable buckled configuration are investigated.

An effective procedure is proposed for extracting bridge frequencies including the higher modes using the vehicle collected data. This is enabled by the use of the contact-point response, rather than the vehicle response, for processing by the extreme-point symmetric mode decomposition (ESMD). The intrinsic mode functions (IMFs) so decomposed are then processed by the FFT to yield the bridge frequencies. A systematic study is conducted to compare the proposed procedure with existing ones, while assessing the effects of various parameters involved. The proposed procedure was verified in the field for a two-span bridge located at the Chongqing University campus. It was confirmed to perform better than the existing ones in extracting bridge frequencies inclusive of the higher modes. The following are the reasons: (1) the ESMD is more efficient than the EMD in that remarkably less IMFs are generated; (2) the modal aliasing problem is largely alleviated, which helps enhancing the visibility of bridge frequencies in general; and (3) the contact-point response adopted is free of the vehicle frequency, which makes the higher frequencies more outstanding and detectable.

The search for efficient explicit time integration schemes is a relevant topic in the current literature on dynamic mechanical systems. In this paper, we describe a strategy of utilizing the balance relations of mechanics in their integral form, so-called general laws of balance, where the time-evolution of the integrands is approximated by established computational techniques of the discrete-mechanics-type. In a Picard-type iteration, the outcomes are used for repeating the procedure several times, leading to an increased accuracy. The advantages of the present explicit approach are discussed in the context of linear and nonlinear motions of the mathematical pendulum. We utilize the modern symbolic procedures to obtain the time integration formulae and compare the results of our methods with exact solutions and with the results of higher-order implicit methods and also with a recent explicit formulation from the literature.

An efficient numerical technique is proposed for determining the buckling load of two-dimensional skeletal structures. The key formulation is based upon the principle of stationary total potential energy and the solution procedure follows the concept of Rayleigh–Ritz approximation. A crucial aspect of the proposed technique is to supply the adaptivity to the solution space allowing the accurate representation of the buckled shape via a simple iterative scheme. The bases of such solution space are constructed in an elementwise fashion using the exact, closed-form buckled shape. An element axial force contained in the element shape functions is chosen as an adaptive parameter and the exact buckled shape of each element is achieved when such adaptive parameter converges to the element buckling load. In this study, various effects including the lateral restraints, shear deformation, and material nonlinearity are taken into account, and this, as a result, allows plane frames with/without lateral bracings, columns resting on elastic foundations, inelastic columns, and those with shear deformation to be treated. Results from an extensive numerical study have indicated that the proposed technique yields highly accurate buckling loads, comparable to the analytical and reference solutions, without the mesh refinement. In addition, a relatively low number of iterations is required to achieve the converged buckling load.

The vibration and acoustic behaviors of both glass fiber laminated plates and carbon/glass fiber hybrid laminated plates are investigated by numerical simulation. The free vibration, forced vibration and acoustic radiation of laminated plates including glass fiber laminates and carbon/glass fiber hybrid laminates in air and water are calculated by the coupled finite element and boundary element method and compared with the corresponding test results. It was demonstrated that results obtained by the coupled finite element and boundary element method are in good agreement with the experimental ones. The effects of dispersion, outer fiber types and fiber hybrid ratio on the vibration and sound radiation of the laminates plates are also discussed.

Cantilever plate structures are widely used in civil and aerospace engineering. Here, a semi-analytical method is proposed to calculate the dynamic responses of cantilever plates subjected to moving forces. The Rayleigh–Ritz method is used to obtain the semi-analytical modal frequencies and shapes of a thin, isotropic, and rectangular cantilever plate using the assumed mode shapes that fulfill the boundary conditions of the plate. The modal superposition method is used to decouple the motion equations of the cantilever plate to obtain a series of modal equations. Then, the generalized forces are transformed into a Fourier series in terms of discrete harmonic forces. The dynamic responses of the cantilever plate are obtained by superimposing the analytical responses of a number of single-degree-of-freedom modal systems under discrete harmonic forces. The proposed semi-analytical method is verified through comparison with the numerical method. Then, the vibration of the cantilever plate under the action of moving forces is investigated based on the semi-analytical results. It is found that the contribution of the high-order modes to the dynamic responses of the plate cannot be ignored. In addition, the wavelengths of the mode shapes not only affect the magnitude of the modal forces but also the dominant frequency of the modal forces. Resonant responses of the plate are produced by the moving forces when the load interval equals the wavelength of the mode shape of a high-order mode and the exciting frequency of the moving forces equals the natural frequency of this mode.

An analytical solution is presented for evaluating the dynamic responses of pile groups subjected to vertical harmonic loads. The solution allows us to consider the effects of pile geometry on the pile head impedance of the vertically loaded pile groups by the use of a new dynamic interaction factor. To this end, the stress distributions of the soil surrounding the vertically vibrating pile is first determined for calculating the pile–pile interaction factor, instead of the classical interaction factor based on two-pile displacements in past studies. Accordingly, the impedances of the pile group are derived using the proposed pile–pile interaction factor and the superposition principle. Some selected examples are presented to demonstrate the proposed refined technique for evaluating the dynamic characteristics of the pile group.

It is often necessary to calculate the vibration of noise from multilayer structures comprising several substructures coupled with discrete connections. A dynamic flexibility method (DFM) is adopted to decouple the multilayer substructures, which allows the interface forces among the substructures to be directly solved using a linear equation of deformation compatibility. The structural vibrations and power flows into each substructure can then be calculated. To illustrate the use of the DFM, a coupled train–track–bridge system for urban rail transit traffic is investigated as a case study. Two infinite plate models are used to model the U-shaped bridge substructure to improve the computing efficiency compared with the finite element models in calculating high-frequency vibration. The applicability of the infinite plate models is discussed in terms of various rail positions on the bridge, the thickness of the rail support blocks, and multiple wheels that interface with the rail. The results show that the Mindlin plate model has similar accuracy but much greater computing efficiency than the finite element models. With the vibration results from the DFM, the associated wheel–rail noise and structure-borne noise from the bridge are then calculated together with a 2D acoustic model. Good agreement is observed between the predicted noises and the measured data.

A marine riser operated in a deep-water field could be substantially affected by large amounts of movement of the floating platform, which is more complicated and very challenging to analyze. This paper presents a mathematical model involving nonlinear dynamic response analysis of a marine riser caused by sways and heave motions at the top end, which are treated as the constraint conditions. The nonlinear equation of motion, arising from the nonlinearity of the ocean current and wave loadings, is derived and written in general matrix form using the finite element method. The excitation caused by platform movement is imposed on the riser system through the time-dependent constrained condition using the penalty method. The advantages of this method are that it is easily implemented on the nonlinear equation of motion and it requires no additional unknown variable, and thus consumes less computational time. By this method, the stiffness matrix and the force vector of the system are then modified, enforcing top-end vessel motion. The dynamic responses are evaluated by using numerical time integration based on Newmark’s method with direct iteration. The effects of the oscillation frequency of top-end vessel sway and heave motions on the nonlinear dynamic characteristics of the riser are investigated. The numerical results reveal that the riser responses to the top-end vessel excitation behave like a periodic motion, which is conformable to the characteristics of vessel movements. The increase in the oscillation frequency of the top-end vessel increases the maximum displacement amplitude for both the horizontal and vertical directions. The directional motion of the vessel also significantly influences the response amplitude of the riser.

This paper deals with the nonlinear dynamic response of elastically supported stiffened plates with initial stresses under impact loads. A stiffened plate is assumed to be composed of a plate with some stiffeners, which are treated separately. The plate is modeled by the thin plate theory, whereas the stiffeners are considered as geometrically nonlinear Euler–Bernoulli beams. First, the equations of both the kinetic energies and strain energies of the plate and stiffeners are established. Then, the dynamic equilibrium equations for the stiffened plate are derived as the Lagrange’s equation of the functional. A parametric analysis is performed to evaluate how initial stresses, initial geometric imperfections, elastic supports, impact loads and configuration of stiffeners affect the time-history responses of the stiffened plates. Some useful nonlinear dynamic properties are obtained, which serve as references for engineering design and application.

The increasing use of bridges in high-speed railway (HSR) lines raises the possibility of train derailment on bridges under seismic excitations. In this paper, the influence of random multi-point earthquakes on the safe running of a train on a long-span bridge is studied in terms of the dynamic reliability, considering spatial seismic effects, and randomness of ground motions and train locations. The equations of motion for the train and the track/bridge as time-invariant subsystems under earthquakes are established, separately. The two subsystems are connected via the wheel–rail interface, for which a nonlinear contact model and detachment are considered. The time-history samples of nonstationary multi-point random earthquakes considering wave passage effects and incoherence effects are generated by the autoregressive moving average (ARMA) model. The ground motions are imposed on the bridge support points in terms of displacement and velocity. The train location at the time of earthquake is considered a uniformly distributed random variable. The running safety reliability of a train moving on a long-span bridge under earthquakes is determined by combining subset simulation (SS) with a prediction-based iterative solution method. Under different seismic components, train speeds, apparent seismic wave velocities and seismic intensities, the most unfavorable train location intervals are determined, which provides a reference for the safety performance assessment of trains traveling on bridges under earthquakes. Numerical results show that the influence of the lateral seismic component on the wheel derailment coefficient (WDC) is greater than the vertical seismic component, and the earthquake that occurs before the train’s arrival at 70% length of the bridge will significantly reduce its running safety.

A method based on Green’s functions is proposed to study the steady-state dynamic responses of a cable-stayed beam subjected to distributed and/or concentrated loadings. The cable’s initial sag and damping effects are considered. Also, the quadratic and cubic nonlinearities due to the dynamic strain of the cable are taken into account. The nonlinear governing equations of the system are solved by the perturbation method, and the Laplace transformation is employed to derive the Green’s functions for the beam and cable with specified boundary conditions. Then the closed form solution of the linear system and the perturbation analytical solution of the nonlinear system are given in an integral form based on those Green’s functions. The difference between the Green’s functions for the linear and nonlinear systems is caused by the nonlinear boundary conditions. The natural frequencies of the cable-stayed beam calculated by the proposed method are compared with those in the literature to illustrate the validity of the present approach. Particularly, the cable’s dynamic strain effects on the symmetric property of the Green’s functions are discussed. The effects of the nonlinear terms on the amplitude of the Green’s functions and vibration are investigated, along with that of damping. The technique presented should find its applicability for other complex structures, such as cable-stayed bridges and steel structures.