International Journal of Structural Stability and Dynamics

This technical note presents a numerical study on the stability of single degree of freedom (SDOF) systems with asymmetric bi-linear hysteretic restoring force, subjected to earthquake excitations. The aim is to report: (a) the existence of an unstable behavior in the response of such systems, under a specific ground motion, given small modifications of the yielding conditions of the hysteresis model, and (b) the introduction of a novel three-dimensional graphic visualization of the problem. The modifications of the yielding conditions were introduced via the symmetry-breaking produced by very small variations of the static equilibrium position of the system, equivalent to having an initial position and restoring force different from zero and symmetric yielding. The concise study comprises of nonlinear dynamic analyses of three system cases, one of them with symmetric (reference) and two with asymmetric yielding conditions. The results show that the system presented a stable response and severe ratcheting toward the weakest yielding direction for the symmetric and asymmetric cases, respectively. Differences as large as $\pm 2800$% between the asymmetric and reference cases were obtained for the residual displacement of the systems, due to variations as small as $\pm 7$% in the static-equilibrium position, and consequent $\pm 7$% variations of the positive/negative yielding displacements and forces. In turn, negligible variations of the velocity between the three cases were predicted. To conclude, the paper introduces novel three-dimensional representations of the solution-curve and of the hysteresis cycles of the systems, deepening the discussion on the identified bifurcation. The 3D hysteresis curve, in particular, can be of much use for seismic engineering and mechanical studies, either numerical or experimental, as it allows visualizing the sequence of events in the hysteresis plots in a much clearer fashion compared to the traditional two-dimensional counterparts.

In this paper, the nonlinear vibrational behavior of a sandwich plate with embedded viscoelastic material is studied through the use of constitutive equations with fractional derivatives. The studied sandwich structure is consisted of a viscoelastic core that is located between the faces of functionally graded magneto-electro-elastic (FG-MEE). In order to determine the frequency-dependent feature of the viscoelastic layer, four-parameter fractional derivative model is utilized. The material properties of FG-MEE face sheets have been distributed considering the power law scheme along the thickness. In addition, for derivation of the governing equations on the sandwich plate, first-order shear deformation plate theory along with von Karman-type of kinematic nonlinearity are implemented. The derived partial differential equations (PDEs) have been transformed to the ordinary differential equations (ODEs) through the Galerkin method. After that, the nonlinear vibration equations for the sandwich plate have been solved by multiple time scale perturbation technique. Moreover, for evaluating the effect of different parameters such as electric and magnetic fields, fractional order, the ratio of the core-to-face thickness and the power low index on the nonlinear vibration characteristics of sandwich plates with FG-MEE face sheets, the parametric analysis has been performed. The obtained results revealed the enhanced nonlinear natural frequency through an increment in the fractional order. Furthermore, the prominent influence of fractional order on the nonlinear frequency of sandwich plate was declared at the negative electric potential and positive magnetic potential.

In this paper, the natural vibration characteristics of elastically supported functionally graded material plate are investigated using the dynamic stiffness method (DSM). Power-law functionally graded (P-FG) plate, the material properties of which vary smoothly along the thickness direction following the power-law function, that has been used for the analysis. Classical plate theory and Hamilton’s principle are used for deriving the governing differential equation of motion and associated edge conditions for P-FG plate supported by elastic foundation. During the formulation of dynamic stiffness (DS) matrix, the concepts of rotary inertia and neutral surface are implemented. Wittrick–Williams (W-W) algorithm is used as a solving technique for the DS matrix to compute eigenvalues. The results thus obtained by DSM for the isotropic, P-FG plate, and the P-FG plate with elastic foundation compare well with published results that are based on different analytical and numerical methods. The comparisons indicate that this approach is very accurate. Furthermore, results are provided for elastically supported P-FG plate under four different considerations in order to see the differences in frequencies with the inclusion or exclusion of neutral surface and/or rotary inertia. It is noticed that the inclusion of rotary inertia and neutral surface influences the eigenvalues of P-FG plate, and that cannot be discounted. The study also examines the influence of plate geometry, material gradient index, edge conditions, and elastic foundation modulus on the natural frequency of P-FG plate.

Within the framework of a variational mixed formation and higher-order shear deformation theory (HSDT), a numerical approach is developed in this research to investigate the buckling and post buckling behaviors of variously-shaped plates made of functionally graded graphene platelet-reinforced composites (FG-GPLRCs) taking the effect of porosity into account. By the proposed approach, which can be named as VDQ-FEM, thick and moderately thick plate-type structures with different shapes (e.g. rectangular, skew, or quadrilateral) with arbitrary-shaped cutout (e.g. circular or rectangular) can be studied. Various types for porosity distribution scheme and GPL dispersion pattern including uniform and different functionally graded patterns are considered along the thickness of plate. In the computation of material properties, the closed-cell Gaussian Random field scheme and Halpin–Tsai micromechanical model are utilized. One of the key novelties of proposed approach is developing an efficient way according to the mixed formulation to accommodate the continuity of first-order derivatives on the common boundaries of elements for the used HSDT model. Several numerical examples are given to analyze the influences of porosity coefficient/distribution pattern, GPL weight fraction/dispersion pattern, cutout and boundary conditions on the buckling and postbuckling characteristics of FG-GPLR porous composite plates.

This paper presents a computational methodology to compute the critical buckling stress of built-up cold-formed steel columns joined with discrete fasteners. The fasteners are modeled as three-dimensional beam elements, and their effect is integrated into the spline finite strip framework, evolving the compound strip methodology. Although this technique has been presented in the literature, this paper presents yet another robust framework for the buckling load evaluation of compound cold-formed steel columns with arbitrarily located fasteners. The proposed framework is applied to study the effect of fasteners on the formation of local, distortional, and global buckling modes of built-up section and a comparison is drawn with the buckling behavior of a single section. In this study, the proposed formulations are also used to get insights into the stability behavior of single-span and multi-span compound cold-formed steel columns in the presence of (i) fasteners with varied spacings with respect to span and (ii) the presence of the additional restraining system such as wall panels. For different buckling modes, a significant increment in buckling stress for a built-up section from a single section is observed when the fastener spacing is kept less than the critical buckling half-wavelength of the respective buckling modes. The study on the effect of wall panels shows that in comparison to unsheathed wall studs, the sheathed wall studs that produce additional constraints lead to the elimination of the global buckling deformations. The proposed formulations are simple, yet rigorous and have been validated using finite element-based numerical results.

The dynamic vibrations of airport runways induced by moving aircraft loads are semi-analytically studied in this paper. The airport runway consists of an infinite Kirchhoff plate, an elastic base course, and an unsaturated poroelastic half-space. The aircraft loads are modeled according to the mechanical properties of the main landing gear of the A380 civil airliner. The governing equations of the whole system are solved in the wavenumber domain using the double Fourier transform. Then the results in the spatial domain are obtained by applying the inverse double Fourier transform. Various parameters including the observation location, soil saturation, load speed, load frequency, and pavement rigidity on the vibration characters of the whole system are investigated. It is found that all these effects are crucial, and the increase of soil saturation leads to a larger maximum vertical displacement and lower critical speed.

Pounding tuned mass dampers (PTMD), which relies on impact to dissipate vibration energy, have shown remarkable performance in suppressing structural vibrations with its modest design. However, the optimum design of PTMDs is computationally expensive due to non-smooth contact-impact behavior. This study provides a computationally efficient approach to determine the optimal parameters of single-sided PTMDs used in vibration control of structural systems. An optimization strategy is used to minimize the maximum response of the controlled structure. As is well-known, the calculation of the dynamic response of a structure could be cumbersome when conventional time-stepping techniques are used in each iteration of the optimization routine. Hence, an exact analytical solution of the steady-state vibration is used to calculate the response for different excitation frequencies, which substantially decreases the computational burden. The adopted method is computationally very inexpensive with respect to the conventional time-stepping techniques used to solve the nonlinear equations of motion to obtain response quantities. The exact solution only requires the solution of the system of five nonlinear equations in order to evaluate the steady-state response per each excitation frequency of harmonic force. A four-storey shear building is used to evaluate the optimally-tuned PTMD by the proposed procedure. In addition, simplified design equations for the coefficient of restitution and frequency ratio are provided using curve and surface fitting for preliminary design. It was shown that the effect of damping ratio of the primary structure on the optimal coefficient of restitution value is not considerable, while it has significant influence on the optimal frequency ratio. It was also realized that the objective function used in optimum parameter design has only one local optimum, which is suitable for the application of gradient-based optimization methods.

The Trigonometric Zigzag theory is utilized in this research for analytically evaluating the forced vibration responses of smart multilayered laminated composite plates with piezoelectric actuators and sensors. This theory, as the name suggests, incorporates a trigonometric function, namely the secant function for describing the nonlinear behavior of transverse shear stresses through the thickness of the smart composite plates. The kinematics for the in-plane displacement components are obtained by superposing a globally varying nonlinear field through the thickness of the plate structure on a piecewise linearly varying zigzag field with slope discontinuities at the layer interfaces. The model also satisfies the inter-laminar continuity conditions of tractions at the interfaces of the multilayered plate. The equations of motion are derived using Hamilton’s principle, and the separation of the variables technique is extended to assume the solutions for the primary variables in space and time and solved analytically using Navier’s solution technique along with Newmark’s time integration scheme. A detailed analytical investigation of the dynamic behavior of the smart laminated plate coupled with piezoelectric materials like PVDF and piezoelectric fiber-reinforced composite (PFRC) is carried out by considering several forms of the time-dependent electromechanical excitations and also covering different geometrical and material features of the smart plate structure. The responses are found to be in close agreement with the elasticity solutions and some new results are also presented to show the dynamic controlling capacity of the piezoelectric layers.

Power spectrum density transmissibility (PSDT) functions have attracted widespread attention in operational modal analysis (OMA) because of their robustness to excitations. However, the selection of the peaks and stability axes are still subjective and requires further investigation. To this end, this study took advantage of PSDT functions and support-vector machines (SVMs) to propose a two-stage automated modal identification method. In the first stage, the automated identification of peaks is achieved by introducing the peak slope (PS) as a critical index and determining its threshold using the SVM classifier. In the second stage, the automated identification of the stability axis is achieved by introducing the relative difference coefficients (RDCs) of the modal parameters as indicators and determining their thresholds using the SVM classifier. To verify its feasibility and accuracy, the proposed method was applied to an ASCE-benchmark structure in the laboratory and in a high-rise building installed with a structural health monitoring system (SHMS). The results showed that the automated identification method could effectively eliminate spurious modes and accurately identify the closely spaced modes. The proposed method can be automatically applied without manual intervention, and it is robust to noise. It is promising for application to the real-time condition evaluation of civil structures installed with SHMSs.

Adaptive structures have the ability to modify their shapes in different operational conditions. Multi-stable structures are one of the methods of making adaptive structures. In the bi-stable square laminates, due to geometric symmetry and equality of strain energy between stable states, it is possible to continue actuating between stable states, specially when using dynamic or thermal load. The bi-stability of the hybrid square laminated structure with the stacking sequence of [0/90/Al] is asymmetric. This leads to inequality of strain energy in stable states and therefore, development of an effective method to control and avoid automatic actuating. In this paper, the deformation and strain energy of the bi-stable hybrid square laminated structure is investigated. To show the effect of elastic boundary condition, a similar section with the stacking sequence of [0/0/Al] is connected to the mentioned hybrid laminate. The effects of the temperature, the presence of an aluminum layer and its thickness on the potential multiple shapes are also studied. To check the accuracy, the bi-stability behavior is investigated using finite element analysis and the results are compared with the experimental data.

Pitch angle is one of the most important parameters of wind turbine blade. This study is aimed to investigate the effect of the pitch angle on the deformation of a VAWT. Lagrangian mechanics and Euler’s beam theory are used to derive the motion equations of linear structural vibration for straight blade vertical axis wind turbine blade with the pitch angle $\alpha$. The complete equations of motion take account of the 4-DOF deformation of flexural–flexural–torsion–extension as well as the material damping. Vibration analysis of generalized displacement about the equilibrium state (GDAES) is carried out with respect to the displacement of the equilibrium state (DOES), which is separated from the motion of vibration. After simplifying the equilibrium equation of 4-DOF into 1-DOF system, the exact solution of displacement $u_1^{\mathrm{\text{E}}}$ of the equilibrium state is derived. The correction ${\mathrm{\Delta }}_{\mathrm{\Omega }}^{(\alpha )}$ of $max{u}_1^{\mathrm{\text{E}}}$ due to the pitch angle and the characteristics of $max{u}_1^{\mathrm{\text{E}}}$ with constant linear speed are analyzed. Furthermore, we investigate the coupling effect of lateral bending and axial extension of the blade on $max{u}_1^{\mathrm{\text{E}}}$ is analyzed. Finally, the exact solution of $max{u}_1^{\mathrm{\text{E}}}$ is verified by the central difference method.

Inflated membranes are popularly used in civil and aerospace engineering. They are flexible and their behaviors are featured by the interaction between the inner air pressure and deformation of the enveloping membrane (air–membrane interaction) which has not yet received attention in the literature. This paper aims at studying the air–membrane interaction and its influence on the static and dynamic properties of an inflated membrane by numerically analyzing a square ETFE (ethylene–tetrafluoroethylene) cushion. To account for the air–membrane interaction, the inner air was regarded as a linear potential fluid in developing the governing equations. The finite element model was derived from the discretized equations and verified through comparison with experimental results and those in the literature. Thereafter, the air–membrane interaction and its variation with influencing factors were investigated in the static and dynamic analysis by comparing results from the verified finite element model with the numerical solutions where the inner air was treated as the traction boundary conditions of the enveloping membrane. Results of this study indicate that (1) air–membrane interaction becomes more prominent with increasing external load and is gradually weakened with a rise in the frequency order; (2) air–membrane interaction makes the top membrane joined with the bottom membrane in the deformation and vibration; and (3) air–membrane interaction is strengthened with an increase in the initial inner pressure or geometric dimensions, but weakened when the membrane thickness or rise–span ratio increases. The present research is helpful to the understanding of the role the inner air plays in the behavior of inflated membranes, and may therefore improve the accuracy in analysis and the rationality in the design.

The response of monolithic steel plates subjected to blast was extensively studied, and dozens of dimensionless numbers were proposed to predict the response. While the existing dimensionless numbers are not convenient to use in some scenarios with relatively complicated conditions, the dimensionless number proposed for blast loaded steel plates based on dimensional analysis extends the range of application. Different from other dimensionless numbers, the properties of medium with which the blast load transmits are incorporated to extend the application range to more general scenarios. The responses of the plate subjected to both near-field blast with non-uniform load and far-field blast with uniform load, i.e. both the external and internal blasts (for steel boxes), are reasonably predicted. The physical implication of the proposed dimensionless number is clear in that the media properties, geometrical features, characteristics of material, and loading are incorporated, which are readily available in test. A variety of test data are used to validate the applicability and versatility of the proposed dimensionless number.

This paper further analyzes three Bathe algorithms ($\gamma$-Bathe, ${\beta }_1∕{\beta }_2$-Bathe and ${\rho }_{\infty }$-Bathe) with their unknown properties revealed. The analysis shows firstly that three Bathe algorithms can cover two common integration schemes, trapezoidal rule and backward Euler formula, and that the second-order ${\beta }_1∕{\beta }_2$-Bathe algorithm is algebraically identical to the ${\rho }_{\infty }$-Bathe algorithm. Via formulation of the generalized two-sub-step Newmark algorithm, it is shown that the common Newmark method cannot be considered as a special case of the ${\rho }_{\infty }$-Bathe algorithm. For wave propagation problems, optimal Courant–Friedrichs–Lewy (CFL) numbers for reducing dispersion errors are found for the three Bathe algorithms by considering spatial and temporal discretizations simultaneously, while the modified integration rules are used for the element mass and stiffness matrices to reduce the anisotropy in wave propagating directions. The recommended optimal algorithmic parameters are given for the three Bathe algorithms to help users effectively solve various dynamic and wave propagation problems.

In this investigation, the nonlinear flexural free vibration characteristics of size dependent curved isotropic nano/microbeams are studied using the nonlocal elasticity theory along with the sinusoidal shear deformation theory (SSDT). Based on Hamilton’s principle, the governing equations are derived in terms of generalized displacements using finite element approach. The formulation by extending the von-Karman model accounts for both large deflections and rotations in the strain displacement definition. Direct iterative procedure is used to solve the nonlinear problem through eigenvalue procedure. This investigation aims to study the effect of various design parameters like slenderness ratio of the beam, curved beam included angle, nonlocal parameter, and boundary conditions on the amplitude–frequency response obtained from the free vibration behavior of curved nano/microbeams. This study is very useful to designers for optimal design of nano/microsystem involving structural elements of the beam type.