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In this paper, free vibration analysis of moderately thick smart FG rectangular plate is presented on the basis of Mindlin plate theory. This structure is composed of a host FG plate and two bonded piezoelectric layers. The plate has two opposite edges simply supported (i.e., Levy-type rectangular plates). The closed circuit piezoelectric layers can be used as an actuator. According to a power-law distribution of the volume fraction of the constituents, material properties vary continuously through the thickness of host plate. Using Hamilton's principle and Maxwell electrostatic equation, six complex coupled equations are introduced. These equations are exactly solved introducing the new potential and auxiliary functions as well as using separation of variables method. The accuracy of the frequencies is verified by the available literature and the finite element method. The present exact solution can accurately predict not only the out of plane, but also the in-plane modes of FG plate. Finally, the effects of various parameters such as boundary conditions, gradient index and thickness of piezoelectric layers on the natural frequency are investigated.

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In this paper, results of  inelastic displacement ratios based on earthquake ground motions of Iran are presented. These ratios are calculated for single degree of freedom systems with elastic perfectly plastic behavior model and various strength reduction factors subjected to 204 earthquake ground motion records. These records are recorded on firm soil sites of Iran and have following characteristics 1) Recorded in earthquakes with seismic moment magnitude larger than 5,  2) At least one of the two horizontal components of records has peak ground acceleration larger than 50 cm/s², 3) Recorded in free field stations so that potential soil-structure interaction effects omitted,  4) Records with hypo central distance larger than 15 km so that near fault effects omitted,  5) Recorded on soil conditions 1, 2 and 3 according to spectral ratio H/V of ground motions,  6) Records in which band pass range in correction process were at least between 0.33 to 20 Hz.  204 acceleration time histories including 70, 30 and 104 acceleration time histories related to soil condition 1, 2 and 3 respectively are used. In this statistical study, 422688 inelastic displacement ratios (related to 204 acceleration time histories, 296 period of vibration and 7 strength reduction factors) from response of SDOF systems with elastic perfectly plastic behavior model are calculated. Mean values of inelastic displacement ratios for each period of vibration and each strength reduction factor subjected to all 204 earthquake records and their dispersions are presented. The influence of period of vibration, strength reduction factor, soil condition, earthquake magnitude and hypocentral distance on inelastic displacement ratios are evaluated. This ratio in short period of vibration is larger than unit (maximum inelastic displacement larger than maximum elastic displacement) and for long period of vibration is close to unit (maximum inelastic displacement nearly equal to maximum elastic displacement). Soil condition effects on inelastic displacement ratio, for period of vibration larger than 1.5 second is very small and neglectable but neglecting of soil condition in periods between 0.4 and 1.5 second cause error up to 20 percent and for periods smaller than 0.4 second cause error up to 40 percent in estimation of inelastic displacement ratio. Neglecting of distance effects to focal of earthquakes in estimation of inelastic displacement ratio, for period of vibration smaller than 1 second, causes error up to maximum 20 percent. This error increase when strength reduction factor increases. For period of vibration larger than 1 second, neglecting of distance effect doesn’t make considerable error. Neglecting of magnitude effects of earthquakes in estimation of inelastic displacement ratio, for period of vibration smaller than 1 second, causes error up to maximum 60 percent. This error increase when strength reduction factor increases. For period of vibration larger than 1 second, neglecting of distance effect doesn’t make considerable error. By using of nonlinear regression analysis, a simplified equation based on mean results is calculated and Maximum inelastic displacement of single degree of freedom systems on Iran firm sites could be estimated using proposed equation and maximum elastic displacement demand. Finally, proposed equation is compared with C₁ coefficient of target displacement in FEMA440.

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In this paper, exact closed-form solutions in explicit forms are presented to investigate small scale effects on the buckling of Lévy-type rectangular nanoplates based on the Reddy’s nonlocal third-order shear deformation plate theory. Two other edges may be restrained by different combinations of free, simply supported, or clamped boundary conditions. Hamilton’s principle is used to derive the nonlocal equations of motion and natural boundary conditions of the nanoplate. Two comparison studies with analytical and numerical techniques reported in literature are carried out to demonstrate the high accuracy of the present new formulation. Comprehensive benchmark results with considering the small scale effects on buckling load ratios and non-dimensional buckling loads of rectangular nanoplates with different combinations of boundary conditions are tabulated for various values of nonlocal parameters, aspect ratios and thickness to length ratios. Due to the inherent features of the present exact closed-form solution, the present findings will be a useful benchmark for evaluating the accuracy of other analytical and numerical methods, which will be developed by researchers in the future. Also, the present study may be useful for static and dynamic analysis of thicker nano scale plate-like structures, multi-layer graphene and graphite as composite or sandwich structures.

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In this paper, the cyclic behavior and available ductility of batten columns subjected to constant axial and cyclic lateral load (seismic condition) and their failure mode are evaluated numerically using nonlinear finite element analysis. The column specimens were steel I-shape sections and were analyzed as an equivalent cantilever column. Batten columns are compression members composed of two or more similar longitudinal components (chords) that are connected at points along their length with batten plates as transverse connectors. These connectors ensure that the column behaves as one integral unit to achieve maximum axial capacity. In the past decades, many research activities were conducted on the buckling problem of batten columns. When a batten column is subjected to lateral load or bending moment about its hollow axis (axis perpendicular to battens) in addition to axial compression, the additional internal actions will be imposed to its members (chords and battens). In this case, it is expected that the batten column will have different behavior and failure modes. If the lateral load or displacement is due to seismic actions, more complexities will exist in the column behavior due to nonlinearities and its post-failure response. Few researches were reported about the behavior of batten columns in seismic conditions and their ductility. In this research, the backbone curves for batten columns have been also developed based on their cyclic response. The component backbone curve represents the nonlinear behavior of component in plastic hinge locations and was used in the nonlinear pushover analysis. The backbone curve for some structural components has been found in many standard and guidelines of seismic evaluation like as FEMA356. Using the backbone curve, the available ductility of column considering its post failure response under cyclic lateral loads, could be evaluated. The backbone curve for batten columns does not exist in any guideline or research reports. Because of differences between behavior and failure modes of batten and solid web columns under seismic action, it was expected that their backbone curves had been substantially different. In this research, cyclic response of batten columns with different geometries have been investigated subjected to cyclic lateral and 3 level of constant axial load. Using cyclic curves, the backbone curves of considered batten columns have been developed. The results show that the available ductility of batten columns is considerably low compared with solid web columns. The failure mode of batten columns is local buckling of bottom chords (in flanges and web) in combination with overall buckling of these chords symmetrically. It is also shown that the backbone curves of batten columns are different from solid web columns. The backbone curves of batten columns are semi-ductile (Type 2) based on FEMA356 classification and don’t have any residual strength. Finally, a conservative backbone curve has been proposed for engineering applications.

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In this paper, for the free vibration analysis of bilayer graphenes with interlayer shear effect the sandwich beam model is introduced. Because of the similarity between the bilayer graphene and the sandwich structures, in which at the top and the bottom of the bilayer graphene there is a single layer graphene and between them there is Vander walls bindings, the bilayer graphene is modeled as a sandwich beam and its free vibration is investigated for free-clamp end condition. To obtain the governing equations, each graphene layer is modeled based on the Euler-Bernoulli theory and in-plane displacements are also considered in addition to the transverse displacement. It is also assumed that the graphene layers do not have relative displacement during vibration. The effect of the Vander walls bindings is introduced in the governing equations as the shear modulus. The results obtained by the sandwich beam model, presented in this paper for the first time, include the first five natural frequencies of the bilayer graphenes with 7 to 20 nanometer lengths. These results are validated by the molecular dynamic and the Multi-Beam-Shear model results.

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In this paper, exact closed-form solutions in explicit forms are presented to investigate small scale effects on the transverse vibration behavior of Lévy-type rectangular nanoplates based on the Reddy’s nonlocal third-order shear deformation plate theory. Two other edges may be restrained by different combinations of free, simply supported, or clamped boundary conditions. Hamilton’s principle is used to derive the nonlocal equations of motion and natural boundary conditions of the nanoplate. Two comparison studies with analytical and numerical techniques reported in literature are carried out to demonstrate the high accuracy of the present new formulation. Comprehensive benchmark results with considering the small scale effects on frequency ratios and non-dimensional fundamental natural frequencies of rectangular nanoplates with different combinations of boundary conditions are tabulated for various values of nonlocal parameters, aspect ratios and thickness to length ratios. Due to the inherent features of the present exact closed-form solution, the present findings will be a useful benchmark for evaluating the accuracy of other analytical and numerical methods, which will be developed by researchers in the future. Also, the present study may be useful for static and dynamic analysis of thicker nano scale plate-like structures, multi-layer graphene and graphite as composite or sandwich structures.

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In this study, by modeling van der Waals (vdWs) interactions based on the Lennard-Jones potential function interlayer tensile-compressive and shear moduli of bilayer graphene sheets are analytically calculated. To this end, by varying potential depth parameter which shows the strength of vdWs interactions a new model is presented for calculating interlayer in-plane and out-of-plane moduli for two different stacking patterns. In order to determine the interlayer vdWs moduli, a small flake of monolayer graphene is sliding on a large monolayer graphene substrate and accordingly variations of vdWs forces as well as the interlayer shear and normal strains are recorded. The relative displacements of layers cause linear strain and stress. In the model, bilayer graphene geometry (being armchair or zigzag, and stacking pattern) and potential depth parameter are two important parameters for determination of vdWs moduli. The accuracy of the method is verified by comparing the present results with those reported in literatures. Finally, close-form relations for interlayer tensile-compressive and shear moduli of vdWs interactions versus the depth potential parameter are presented for ABA and AAA stacking patterns as well as zigzag and armchair directions. It is observed that the interlayer moduli have linear relation with the potential depth parameter.