International Journal of Structural Stability and Dynamics

A three-dimensional (3D) method of analysis is presented for determining the free vibration frequencies of joined hemispherical–cylindrical shells of revolution with a top opening. Unlike conventional shell theories, which are mathematically two-dimensional (2D), the present method is based upon the 3D dynamic equations of elasticity. Displacement components u_r, u_θ and u_z in the radial, circumferential, and axial directions, respectively, are taken to be periodic in θ and in time, and algebraic polynomials in the r and z directions. Potential (strain) and kinetic energies of the joined shells are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies. Natural frequencies are presented for different boundary conditions. The frequencies from the present 3D method are compared with those from 2D thin shell theories.

The vibrations of axially loaded composite beams with partial interaction are considered. The equations of motion and the pertaining boundary conditions are derived from Hamilton's principle. Mainly free — but also forced — vibrations are considered. The natural frequencies are calculated as a function of the axial force and the stiffness of the interaction between the layers. The analytical result obtained for the eigenfrequencies of the simply supported beam is taken as a starting point for an approximation of the eigenfrequencies of beams subject to other boundary conditions. The proposed approximation — which has no numerical fitting parameters — is shown to be in good agreement with the exact solutions, especially for the case of a beam clamped at both ends.

Based on Hamilton's principle, a new accurate solution methodology is developed to study the torsional bifurcation buckling of functionally graded cylindrical shells in a thermal environment. The effective properties of functionally graded materials (FGMs) are assumed to be functions of the ambient temperature as well as the thickness coordinate of the shell. By applying Donnell's shell theory, the lower-order Hamiltonian canonical equations are established, from which the eigenvalues and eigenvectors are solved as the critical loads and buckling modes of the shell of concern, respectively. The effects of various aspects, including the combined in-plane and transverse boundary conditions, dimensionless geometric parameters, FGM parameters and changing thermal surroundings, are discussed in detail. The results reveal that the in-plane axial edge supports do have a certain influence on the buckling loads. On the other hand, the transverse boundary conditions only affect extremely short shells. With increasing thermal loads, the material volume fraction has a different influence on the critical stresses. It is concluded that the optimized FGM mixtures to withstand thermal torsional buckling are Si₃N₄/SUS304 and Al₂O₃/SUS304 among the materials studied in this paper.

Tensegrity structures under certain conditions may be prone to snap-through buckling. The temporary loss of equilibrium due to snap-through normally results in a dynamic force being applied to each node associated with the snap-through. This paper presents a numerical study on the progressive collapse behavior of tensegrity structures due to the buckling of struts. Emphasis is given to the dynamic nature of the coupled member and nodal snap-through effects on the overall structural behavior. Member buckling is taken into consideration by carefully following the buckling load–displacement response of the member. It is assumed that the structure is subjected only to static gravitational load. Results of the present study allow one to assess the effects of various design parameters such as self-stress levels, effective-length factor of struts and damping characteristics on the propagation of snap-through buckling in these structures. The conclusions, drawn from this study, lead to the suggestion of some guidelines and recommendations for the design of such structures.

The use of a pseudodynamic testing method to obtain a response from an impulsive loading may lead to an incorrect test result even though the test setup is properly adjusted and the time step is appropriately chosen based on general considerations. The cause to this problem is analytically explored and a pseudodynamic technique is proposed to overcome this difficulty. The technique is to perform a single small time step immediately upon termination of the applied impulse, while the other time steps can still be conducted by the time step determined from general considerations. The feasibility of this technique is analytically verified and confirmed by a series of pseudodynamic tests for both linear elastic systems and inelastic systems.

Presented herein is a computationally efficient 2D theoretical model for simulating the steady response of a floating slab track-tunnel-soil system. The track-tunnel coupled system is simplified as a beam-spring system and embedded in soil layers. The tunnel is modeled by a Timoshenko beam with its interaction with the soil layers accounted for by two transfer matrices, with each derived for the soil layer above and beneath the tunnel. The approach as proposed herein has been referred to as the Timoshenko beam-transfer matrix method (TTMM), that allows one to analyze the response of the coupled system, including the tunnel motion and soil stresses. The results obtained were compared with those furnished by the pipe-in-pipe (PIP) approach, and were found to be consistent for exciting frequencies smaller than the tunnel second-mode cut-on frequency. The origin of discrepancies was investigated by the dispersion characteristic analysis, which is attributed to the absence of several in-plane modes when the tunnel is simplified as a Timoshenko beam.

Once a RC structure is laterally restrained, both the static and dynamic load resistances will be enhanced due to the membrane action. Despite this known advantage, the apparent lack of systemic and efficient methods of analysis poses a drawback in the design and assessment of blast-resistant RC structures. First, a simplified membrane action theory was presented by modifying the maximum membrane force design method (MMFM) for predicting the total static resistance-deflection curves of restrained beam-slab RC structures. Second, a series of constrained beams were tested to validate the new theory, for which better agreement was observed between the test data, the results predicted by the proposed theory and those by MMFM. The results show that the static load carrying capacity and membrane force increase with increasing restraint stiffness, and the smaller the reinforcement ratio is, the larger the load carrying capacity increases. Third, based on the improved compressive static membrane action theory, a new analytical method was developed to investigate the dynamic responses of restrained RC structures subjected to blast loads, using an equivalent single degree of freedom system that combines the three-parameter elasto-viscoplastic rate-sensitive material model with the proposed static theory. Good agreement is observed between the test data and the analytical results. Finally, it is demonstrated that the dynamic resistance capacity increases with increasing load rate and restraint stiffness and with decreasing tensile reinforcement ratio, but the larger the dynamic resistance is, the larger the plastic deformation of the structure.

In this paper, we analyze the transverse vibration of a circular plate loaded by uniform pressure along its edge. The plate is supported by an elastic ring support being coaxial with the plate. At its edge the plate is clamped but the radial displacement is allowed. Apart from this problem, the heated plate clamped at its edge, but without the possibility of radial displacement, is also analyzed. The analytical solution of governing equation is obtained in the form of Bessel's functions. Using the analytical solution, the frequencies of transverse vibrations depending on loads, elastic ring stiffness and the location of ring are obtained. The results show that the lowest frequencies vibrations can be either symmetric or asymmetric having one or two nodal diameters. It is also shown that multiple vibration frequencies can occur for special values of load and ring stiffness.

It has been observed that undulating periodic patterns formed on an initial flat annular plate that model the leaves of plants are a physical response to the expansion of the surface under a lateral restraint. They are not visible at the beginning but become apparent only when the plants continue to grow. The behavior can be explained via the inhomogeneous deformation of gel materials that behave in a similar manner to hyperelastic materials in solid mechanics.This paper compares the stability of thin annular plates clamped along the inner edge and free along the outer periphery using numerical simulations of the swelling of thin gel annular plate held along the inner edge as well as analysis of a similar class of structures by solid mechanics concept via energy principle. The trends of results from both approaches compare favorably. The buckling patterns of annular plates with various values of inner radius to outer radius ratio illustrate the relationship between the geometry of the annular plate and the inhomogeneous deformation of gels or buckling patterns of solid mechanics materials. The undulating patterns on leaves such as those of flowering cabbage can thus be explained via the buckling behavior of annular plates, which can be regarded as thin soft materials adhered to a stiffer core. The study can also be extended to cover other stimuli under different environmental conditions and the outcome may bring further insights into the evolution of plants.

This paper is concerned with the nonlinear damped forced vibration problem of pre-stressed orthotropic membrane structure under impact loading. The governing equations of motion were derived based on the von Kármán large deflection theory and D'Alembert's principle, and solved by using the Bubnov–Galerkin method and the Krylov–Bogolubov–Mitropolsky (KBM) perturbation method. The asymptotic analytical solutions of the frequency and lateral displacement of rectangular orthotropic membrane with fixed edges were obtained. In the computational example, the frequency results were compared and analyzed. Meanwhile, the vibration mode of the membrane and the displacement and time curves of each feature point on the membrane surface were analyzed. The results obtained herein provide a simple and convenient approach to calculate the frequency and lateral displacement of the nonlinear forced vibration of rectangular orthotropic membranes with low viscous damping under impact loading. In addition, the results provide some computational basis for the vibration control and dynamic design of membrane structures.

This paper investigates the free vibration of nanocomposite beams reinforced by single-walled carbon nanotubes (SWCNTs). The distribution of the SWCNTs may vary through the thickness of a beam and are aligned along the beam axial direction. The virtual strain and kinetic energies of the carbon nanotube (CNT) composite beam are obtained using the classical variational method of Hamilton's principle, and the geometric nonlinearity of von Kármán sense is also included. The eigenvalue equation for free vibration of the beam is derived by the p-Ritz method. Vibration frequency parameters for the uniformly distributed (UD) and functionally graded (FG) CNT beams based on the first-order and third-order beam theories are presented and the effects of CNT filler volume fraction, distribution, beam length to depth ratio and end support conditions on the nonlinear free vibration characteristics of the beams are discussed. Comparison studies for UD-CNT and FG-CNT beams based on the first-order and the third-order beam theories are also performed and the differences in vibration frequencies and the nonlinear to linear frequency ratios between these two theories are highlighted.