International Journal of Structural Stability and Dynamics

This paper considers the nonlinear in-plane behaviour of a circular arch subjected to thermal loading only. The arch is pinned at its ends, with the pins being on roller supports attached to longitudinal elastic springs that model an elastic foundation, or the restraint provided by adjacent members in a structural assemblage. By using a nonlinear formulation of the strain-displacement relationship, the principle of virtual work is used to produce the differential equations of in-plane equilibrium, as well as the statical boundary conditions that govern the structural behaviour under thermal loading. These equations are solved to produce the equilibrium equations in closed form. The possibility of thermal buckling of the arch is addressed by considering an adjacent buckled equilibrium configuration, and the differential equilibrium equations for this buckled state are also derived from the principle of virtual work. It is shown that unless the arch is flat, in which case it replicates a straight column, thermal buckling of the arch in the plane of its curvature cannot occur, and the arch deflects transversely without bound in the elastic range as the temperature increases. The nonlinear behaviour of a flat arch (with a small included angle) is similar to that of a column with a small initial geometric imperfection under axial loading, while the nonlinearity and magnitude of the deflections decrease with an increase of the included angle at a given temperature. By using the closed form solutions for the problem, the influence of the stiffness of the elastic spring supports is considered, as is the attainment of temperature-dependent first yielding of a steel arch.

In this paper, we consider the nonlinear analysis of frames with shear deformation, and present a series of benchmark solutions for a variety of problems and modelling assumptions. The benchmarks enable users and developers of nonlinear analysis software to test the accuracy of their procedures when including shear deformation.

This paper intends to study the train-induced vibration of a parabolic tied-arch bridge using an analytical approach. The train loads over the bridge are regarded as a sequence of equidistant moving loads with identical weights. The tied-arch bridge considered is modeled as the combination of a parabolic flat-rise arch with two-hinged supports and a simple beam suspended by densely distributed vertical cables connected to the arch rib. Using the normal coordinate transformation method, the coupled equations of motion of the arch rib and suspended beam are converted into a set of uncoupled equations. Then, one can derive closed form solutions for the response of the tied-arch beam to successive moving loads, by which the resonant conditions of higher modes of the suspended beam can be identified. According to the present study, the critical position for the maximum acceleration on the suspended beam depends upon the vibration shape that has been excited. Moreover, the numerical results indicate that the lower the rise of the arch rib, the larger the acceleration response of the main beam suspended by the arch.

The stability behavior of a prismatic free–free column, carrying a movable mass and subjected to the action of follower end thrust and follower frictional resistance has been investigated. A finite element model of the beam is formulated using the modified Hamilton's principle. The eigenvalues of the characteristic equation of the governing differential equation of motion are extracted using the Q-R algorithm. The parameters considered in the investigation include: (i) ratio of the movable mass to the total mass, (ii) position of the movable mass, and (iii) ratio of resultant frictional drag to the magnitude of the end thrust. Some of the interesting results conducted in this study may be useful to the design engineers. Depending on the various parameters used, the column may lose stability either by divergence or by flutter. The first two frequencies vanish when the column is subjected to only an end thrust without any extra mass and/or frictional drag, indicating the occurrence of both the rigid body translation and rotation. However, only one vanishing frequency is observed when the column is carrying a movable mass and is subjected to a frictional drag (irrespective of their values), indicating the occurrence of only one rigid body motion.

In this paper, an electromechanical coupled dynamic equation of a micro beam under an electrostatic force as well as under an electromechanical coupled force is presented. The linearization of above dynamic equation is made, allowing the equation to be divided into a linear dynamic equation for dynamic displacement and a static balance equation for static displacement. Using the balance equation, the changes of the voltage along with displacement are studied. It is shown that there is a critical voltage at which the micro beam will buckle. From the linear dynamic equation, natural frequencies and vibration modes of the micro beam, and its forced responses to voltage excitation are derived. The results show that the natural frequencies and vibrating magnitudes of the micro beam are affected by mechanical and electric parameters. Smaller beam length and voltage as well as larger beam thickness and clearance should be selected in order to obtain smaller vibrating magnitudes. It is also shown that for higher vibration modes, more positions of the peak dynamic displacements occur.

The dynamics of nonlinear polar orthotropic circular plates with simply supported boundary condition are investigated. Kirchhoff strain displacement relations for thin plates plus next higher-order nonlinear terms (von Karman type geometric nonlinearity) are considered. Lagrangian density function and Hamilton's principle are utilized to derive Lagrange's equations, from which the equations of motion and associated boundary conditions are derived. Analytical solution is obtained by the perturbation techniques and numerical solution by the Runge–Kutta method. Phase diagrams, discrete Fast Fourier Transform (FFT diagrams) and time history responses are presented for studying the forced vibration behavior. The sub-harmonic and primary resonances are studied as well as the effect of adding damping foil layers. The quadratic term in the governing equation plays a softening role on the overall behavior of the plate due to its relatively large coefficient. The increase of damping tends to smooth out the unstable region (i.e. jump phenomenon) in the system.

The distribution of vibrational energy in members of a complex structure with tuned absorbers attached at the joints and subjected to dynamic loading is studied. The concept of power flows through the structure is used to determine the time-averaged energy levels of each member in the structure. The power flows are calculated using the time-averaged product of force and velocity at the input and coupling points (joints) of a general structure made of axially vibrating rods. The receptance approach is used to calculate the coupling forces and velocities in the structure. By balancing the input power against the dissipated powers, the time-averaged energy levels in members are determined. The main criteria studied here is the reduction in the frequency-averaged vibrational energy level of a member when an absorber is attached, expressed as a percentage compared to the case where there are no absorbers. The concept is first illustrated with a simple model of 2 axially vibrating rods with an absorber attached to the joint. Next, a more complex structure comprising 8 rods with arbitrary orientations and several absorbers attached to junctions is studied. The effect of activating absorbers at various locations on reducing the energy levels of certain members is examined. It is possible to estimate the usefulness of the absorber with respect to any member by determining the percentage reduction of energy level for that member.

A non-local elastic shell model is proposed for the first time in this study for considering the small-scale effect in axi-symmetric wave propagation in carbon nanotubes (CNTs). Two coupled radial and longitudinal modes, and one decoupled torsional mode, are derived from the developed non-local shell model. The small-scale effect on wave propagation is numerically studied and discussed. In addition, the cut-off frequency based on the non-local shell model is obtained, which is found to be free of the small-scale effect. It is interesting to note that only one asymptotic phase velocity exists in the CNTs using the non-local shell model, whereas two asymptotic phase velocities can be predicted using the classical or local elastic shell model. It is hoped the research presented herein can be used as a benchmark for future studies on the wave propagation of CNTs with non-local continuum models.