International Journal of Structural Stability and Dynamics

The dynamics of a thin rod under the action of compressive force is considered. The compressive force increases abruptly and then remains constant. The compression of the rod is assumed to take place instantly throughout its length. The motion of the rod is studied, depending on the magnitude and time of action of the compressive force defined by the phase of the initial free oscillation of the rod. Only the initial stage of the process is investigated, for which the linear bending theory is valid. The friction forces are not taken into account. An essential relationship is shown to exist between the dynamics of the rod and the initial conditions determined by the phase of the bending oscillations at the instant of impact. Relatively weak impact leads to excitation of the bending oscillations. Rearrangement of harmonics develops with time under specified initial maximum deflection from a straight line and zero velocity. The fastest buckling takes place at some initial deflection with the velocity directed toward the increasing bend.

The natural frequencies and mode shapes of enclosed shell typed structures with variable thickness (hemispherical-cylindrical-hemispherical shells and complete hollow spherical shells) are determined by the Ritz method using a three-dimensional (3D) analysis. However, in the conventional shell analysis, mathematically two-dimensional (2D) thin shell theories or higher order thick shell theories are often employed, which adopt limiting assumptions about the displacement variation through the shell thickness. While most researchers have adopted the 3D shell coordinates that are normal and tangential to the shell mid-surface, the present analysis is based upon the circular cylindrical coordinates. By the Ritz method, the Legendre polynomials, which are mathematically orthonormal and minimal, are used as the admissible functions, instead of the ordinary algebraic polynomials. The strain and kinetic energies of the combined shell structures are formulated, and upper bound solutions of the frequencies are obtained by minimizing the solution for frequencies. As the degree of the Legendre polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies. The frequencies from the present 3D method are compared with those from other 3D approach and 2D thin and thick shell theories existing in the literature. The present 3D analysis is applicable to both very thick shells and very thin shells.

In this paper, nonlinear stochastic optimal control of multi-degree-of-freedom (MDOF) partially observable linear systems subjected to combined harmonic and wide-band random excitations is investigated. Based on the separation principle, the control problem of a partially observable system is converted into a completely observable one. The dynamic programming equation for the completely observable control problem is then set up based on the stochastic averaging method and stochastic dynamic programming principle, from which the nonlinear optimal control law is derived. To illustrate the feasibility and efficiency of the proposed control strategy, the responses of the uncontrolled and optimal controlled systems are respectively obtained by solving the associated Fokker–Planck–Kolmogorov (FPK) equation. Numerical results show the proposed control strategy can dramatically reduce the response of stochastic systems subjected to both harmonic and wide-band random excitations.

The effectiveness of the semi-active control strategies using fluid viscous dampers (SAFVDs) for seismic mitigation of reinforced concrete (RC) elevated liquid storage tanks is investigated. Three control algorithms are employed for regulating the damping coefficient of the SAFVDs: (1) Passive-OFF, (2) Passive-ON, and (3) Clipped Optimal Control (COC). The uncontrolled response of the tank is compared with those installed with SAFVD of different control algorithms. Focus is also placed on various positions of the dampers, viz., dampers installed at alternate levels (Configurations I, II, IV, and V) and at all levels (Configurations III and VI) of the staging. A discrete two-mass model for the liquid and multi-degree-of-freedom system for the staging, installed with the dampers, is developed for the RC elevated liquid storage tanks. The response of the broad and slender tanks is studied, for which the ratios of the height of the liquid to the radius of the container are 0.5 and 2.0, respectively. The time-history response of the elevated tank is evaluated for eight different earthquake ground motions, including near- and far-field earthquakes. A MATLAB code was developed to solve the coupled differential equations of motion of the system using the state-space approach. Key parameters, viz., convective displacement, rigid mass displacement, base shear, overturning moment, and damper force, are evaluated. The results show that all the control systems considered herein are beneficial in reducing the seismic responses. The frequency response function for the uncontrolled and semi-actively controlled liquid storage tank in frequency domain exhibits significant response reduction, highlighting the effectiveness of the SAFVDs. The structural response is effectively controlled using the SAFVDs with Passive-OFF (valve closed) and COC algorithms. The COC algorithm employed in this study is a promising candidate for the seismic mitigation of RC elevated liquid storage tanks using the semi-active control.

The actual dead load of an arch dam should be applied gradually through staged construction and sequenced grouting. However, the cantilever- and integral-type dead loads commonly used in the analysis of arch dams represent simplified versions of the actual loading. In this paper, these two types of dead loads, i.e. cantilever and integral types, are presented based on the Lagrange multiplier method considering the nonlinear behaviors of contraction joints. Based on the finite element method and an appropriate contact model together with artificial viscoelastic boundary conditions, a dynamic analysis model of a dam–foundation–reservoir system is established in consideration of the interactions between the arch dam and foundation, the opening and closing of contraction joints, and the radiation damping effect of the far-field boundary. Taking a 300 m high arch dam in the strong earthquake area of West China as an example, a fine mesh finite element model with a total of approximately 3.5 million degrees of freedom is established. The separate effects of the cantilever and integral dead loads on the static and dynamic responses of the dam are studied. The results demonstrate that the distribution and magnitude of the contraction joint opening width and maximum tensile stress are different under the two different dead load simplifications.

In the present paper, a practical superposition method is proposed for complex load-dependent Ritz (CLDR) vectors for use in the dynamic analysis of nonclassically damped systems. In particular, an algorithm for CLDR vector generation is developed and the CLDR vectors are calculated in the physical space, instead of the state space, to reduce the computational effort and storage space, while improving the stability of the algorithm. Moreover, single CLDR vector (i.e. using only one starting vector) and block CLDR vector (i.e. using multi-starting vectors) generation procedures are introduced for the uni and multidirectional loading patterns respectively, and the latter is applied to the system with repeated natural frequencies. In addition, a criterion, which is based on the spatial load distribution, is proposed to determine a proper number of the CLDR vectors prior to their use in the dynamic analysis. Two numerical examples are provided to illustrate the accuracy and efficiency of the proposed method. Also, the performance of the cut-off criterion is presented and 10% error or less in the participation loading distribution is recommended for practical applications.

Quantifying the higher-mode effects on the seismic demands of buildings may benefit not only the awareness of characteristics of the seismic responses of buildings, but also the development of rapid/simplified methods for the seismic assessment of buildings. This study proposes an approach that is applicable for quantifying the aforementioned effects, covering the full range of building heights and deformation types. The vehicle used in this proposed approach is the generalized building model, which has been modified from the conventional cantilever beam model. In addition to building height and deformation type, the strength ratio of each vibration mode and the site class of buildings are the parameters considered in this study. The higher-mode effects on floor displacements, inter-story drift ratios, floor accelerations, and base shears with relation to the aforementioned parameters are investigated. Finally, the proposed approach is verified via the investigation of the higher-mode effects of a 20-story exemplar building.

Two higher-order analytical models based on a new higher-order theory for sandwich plates with flexible cores are developed considering the effect of the core material density and skin-to-core-stiffness-ratio (SCSR). The main difference between the two models is the role of the flexible core in the dynamic response of sandwich plates with cores of different stiffnesses. Firstly, the governing equations of a simply supported sandwich plate with a flexible core are derived based on the two models, and the analytical solutions are determined by using Navier’s approach. Then, the free vibration, static, dynamic bending and stress field characteristics of the sandwich plates with different SCSRs are investigated. The results obtained by the proposed method are compared with other published results. In particular, an accuracy assessment of the present dynamic models is conducted for different SCSRs. Finally, conclusions on the applicability of the proposed method and other theories on sandwich plates with different SCSRs are drawn.

This paper studies the vibration of a suspended cable with small sag excited by a second excitation (normal) mode causing combination resonance in the three modes, i.e. tangent, normal and bi-normal modes. The displacement response spectra in the time domain and phase portraits are provided as evidence of the transition from the unstable steady-state motion to the stable one for nonlinear cable oscillation. A nondimensional equation for the cable tension is established with its variation evaluated for the unstable zone. The half-normalized sensitivity analysis of cable parameters, such as damping coefficient, excitation amplitude, arc length parameter and initial conditions, for their influence on cable tension is conducted in the time domain by a direct integration method. Also, the characteristics of the dynamic sensitivity of cable tension to the parameters are discussed. As a result, a sensitivity ranking chart is prepared based on the sensitivity analyses for the parameters considered. It is clearly revealed that cable tension is very sensitive to tangent and normal initial displacements in spite of their small values, whereas the same is not true for the arc length parameter and bi-normal damping coefficient. To verify the sensitivity analysis algorithm, a forced Rayleigh oscillator is introduced. A feasibility study using the oscillator shows that the numerical results obtained are in good qualitative agreement with the analytical predictions, implying that the associated algorithm works well.

Cylindrical shells containing flowing fluid have wide applications in various industries. They can be enhanced as smart structures through inclusion of piezoelectric layers, of which the dynamic behavior, however, has not been fully understood. In this paper, the vibration and dynamic analysis of a laminated composite hollow cylinder with piezoelectric layers, subjected to an internal incompressible fluid flow is investigated. It is assumed that the shell is simply supported and the fluid is inviscid and irrotational. The differential equations of the elastic layers, piezoelectric layers, and flowing fluid are derived by the three-dimensional (3D) theory of elasticity, theory of piezoelectricity, and potential flow theory, respectively. A well-known recursive method is applied and extended for the first time to solve the fluid-conveying pipes using 3D theory. This approach makes it possible for the solutions to converge to the exact ones with reasonable computational cost. After validating the results against those available in the literature, the vibrational behavior of the system is examined for various cases with the effect of each parameter investigated. Also, the influence of fluid on the vibration and stability of the shell has been analyzed. The present method can be used to analyze and design hybrid shells conveying fluid with high accuracy and low computational cost.

This paper focuses on development of a new mathematical model and its analytical solution for buckling analysis of elastic columns weakened simultaneously with transverse open cracks and partial longitudinal delamination. Consequently, the analytical solution for buckling loads is derived for the first time. The critical buckling loads are calculated using the proposed analytical model. A parametric study is performed to investigate the effects of transverse crack location and magnitude, length and degree of partial longitudinal delamination, and different boundary conditions on critical buckling loads of weakened columns. It is shown that the critical buckling loads of weakened columns can be greatly affected by all the analyzed parameters. Finally, the presented results can be used as a benchmark solution.

In the present study, free vibration and buckling characteristics of a sandwich functionally graded material (FGM) plate resting on the Pasternak elastic foundation have been investigated. The formulation is based on non-polynomial higher-order shear deformation theory with inverse hyperbolic shape function. A new modified sigmoid law is presented to compute the effective material properties of sandwich FGM plate. The governing equilibrium equations have been derived using Hamilton’s principle. Non-dimensional frequencies and critical buckling loads are evaluated by considering different boundary conditions based on admissible functions satisfying the desired primary and secondary variables. Comprehensive parametric studies have been performed to analyze the influence of geometric configuration, volume fraction exponent, elastic medium parameter, and non-dimensional load parameter on the non-dimensional frequency and critical buckling load. These parametric studies have been done for various boundary conditions and different configurations of the sandwich plate. The computed results can be used as a benchmark for future comparison of sandwich S-FGM plates.

Developing a model for the dynamic force generated by a pedestrian’s foot on a supporting structure (single footfall trace model) is crucial to advanced numerical analysis and vibration serviceability assessment of the structure. A reliable model needs to reflect the inter-subject and intra-subject randomness of human walking. This paper introduces a stochastic single footfall trace model in the form of a Fourier series in which body weight, walking frequency, and the first eight harmonics are treated as random variables. An experiment used 73 test subjects, walking at a range of pacing frequencies, to record force time histories and the corresponding gait parameters. Two variability descriptors were used to indicate inter-subject and intra-subject randomness. Further statistical analysis identified the relationships between key parameters as well as the probability distribution functions of random variables. In the final step, an application of the proposed single footfall trace model was developed and tested. The proposed model represented the experimental data well in both time and frequency domains.

Treated herein is an elastic beam that is subjected to a constant load that travels continuously (back and forth) along its span. The dynamic deflection of the beam is investigated analytically so as to predict the critical and cancellation speeds. Closed-form solutions are obtained for the damped system for each load condition and superimposed to determine the total solution of the response. Unlike the classical case of load traveling at constant speed in one direction, multiple resonance peaks are observed for reciprocating load at speeds that are lower than the classical critical speed. It is also observed that resonance may not exist at the classical critical speed for simply supported beams due to the symmetry of the beam. The dynamic deflection is examined for simply supported beams to determine the speeds that cause amplification or cancellation of the free response. The current load condition may have possible application in view of its potential use for vibration suppression, as a moving vibration absorber, or for magnification, in energy harvesting. The results are interpreted in order to understand the variation of dynamic deflection and to estimate the critical speeds for different load conditions.

This paper presents a novel approach for determining the critical lateral-torsional buckling loads of beams subjected to arbitrarily transverse loads. This new method is developed based on the classical energy method. However, the difference of the present method from the traditional energy methods is the formulation of potential energy of external loads, which is expressed in terms of the internal bending moment and internal shear force in the pre-buckling stage regardless of the type of loading. Compared to the traditional formulations of the potential energy of external loads, not only is the present formula simple in the form, easy and convenient in the calculation, but it also provides a unified form for calculating accurate critical load of lateral-torsional buckling of the beams.

Proposed herein is a new theory for the anisotropic deformation of fiber-reinforced hydrogels. This new model takes into account the real fabrication of the fiber-reinforced hydrogels, in which the hydrogels are polymerized with fibers and polymer solutions. The new free energy function is established by adding the anisotropic free energy component contributed by fibers into the Flory–Rehner model. The proposed model is implemented through a user-defined material subroutine (UMAT) in the finite element software package ABAQUS. In particular, the consistent tangent modulus is derived in detail. Then, several illustrative examples with analytical and numerical results are demonstrated. In order to study deformation behavior of natural materials, we design some simple bilayer structures to mimic the opening of seedpods and the closure of flowers, in which the buckling behavior of fiber-reinforced hydrogels have been demonstrated. We hope that the proposed approach may help to study more complex deformation phenomena in hydrogel structures.

The free and forced vibration characteristics of three-layered sandwich plates with thin isotropic faces and Leptadenia pyrotechnica rheological elastomer (LPRE) core are studied in this investigation. The LPRE core is fabricated and experimented to determine its shear storage modulus and loss modulus. It is observed that the stiffness and damping characteristics of the LPRE core is significantly higher than those of the room-temperature vulcanized silicone rubber elastomer (RTVE) core. The governing equation of motion for the sandwich plate is derived by the Lagrange principle and given in finite element form. The natural frequencies and loss factors of the three-layered sandwich plate are studied by varying the thicknesses of the core and the constraining isotropic layer, and material of the constraining layer with different boundary conditions. The results are compared with those of similar structures with different core materials and boundary conditions. In addition, a LPRE-based sandwich plate is fabricated and its fundamental frequency is determined experimentally and compared with the finite element result. The forced vibration response of the three-layered sandwich plate is also explored under a harmonic excitation force. This study provides supports for the application of the LPRE-based sandwich plates potentially to the passive vibration suppression of structures.

This paper investigates the nonlinear flexural vibration of sandwich beams with functionally graded (FG) negative Poisson’s ratio (NPR) honeycomb core in thermal environments. The novel constructions of sandwich beams with three FG configurations of re-entrant honeycomb cores through the beam thickness direction are proposed. The temperature-dependent material properties of both face sheets and core of the sandwich beams are considered. 3D full-scale finite element analyses are conducted to investigate the nonlinear vibration, and the variation of effective Poisson’s ratio (EPR) of the sandwich beams in the large deflection region. Numerical simulations are carried out for the sandwich beam with FG-NPR honeycomb core in different thermal environmental conditions, from which results for the same sandwich beam with uniform distributed NPR honeycomb core are obtained as a basis for comparison. The effects of FG configurations, temperature changes, boundary conditions, and facesheet-to-core thickness ratios on the nonlinear vibration ratio curves and EPR–deflection curves of sandwich beams are discussed in detail.