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In this paper elastoplastic buckling of thin rectangular plates are analyzed with deformation theory (DT) and incremental theory (IT) and the results are investigated under different loads and boundary conditions. Load is applied in plane and in uniform tension and compression form. The used material is AL7075T6 and the plate geometry is . The Generalize Differential Quadrature method is employed as numerical method to analyze the problem. The influences of loading ratio, plate thickness and various boundary conditions on buckling factor were investigated in the analysis using both incremental and deformation theories. In thin plates the results obtained from both plasticity theories are close to each other, however, with increasing the thickness of plates a considerable difference between the buckling loads obtained from two theories of plasticity is observed. The results are compared with those of others published reports. Moreover, for some different situations new results are presented. Some new consequences are achieved regarding the range of validation of two theories.

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The main purpose of this study is to investigate nonlinear bending and buckling analysis of radially functionally graded annular plates subjected to uniform in-plane compressive loads by Dynamic Relaxation method. The mechanical properties of plates assumed to vary continuously along the radial direction by the Mori–Tanaka distribution. The nonlinear formulations are based on first order shear deformation theory (FSDT) and large deflection von Karman equations. The dynamic relaxation (DR) method combined with the finite difference discretization technique is employed to solve the equilibrium equations. Due to the lack of similar research for the bending and buckling of functionally graded annular plates with material variation in the radial direction, some results are compared with the ones obtained by the Abaqus finite element software. Furthermore, some comparison study is carried out to compare the current solution with the results reported in the literature for annular isotropic plates. The achieved good agreements between the results indicate the accuracy of the present numerical method. Finally, numerical results for the maximum displacement and critical buckling load for various boundary conditions, effects of grading index, thickness-to-radius ratio and inner radius -to-outer radius ratio are presented.

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In this study, nonlinear bending analysis of ring-stiffened annular laminated composite plates is studied. A discretely stiffened plate theory for elastic large deflection analysis of uniformly distributed loaded is introduced. The governing equations are derived based on a first-order shear deformation plate theory (FSDT) and large deflection von Karman equations. The numerical results are obtained using the dynamic relaxation (DR) method combined with the central finite difference discretization technique. For this purpose, a FORTRAN computer program is developed to generate the numerical results. In order to verify the accuracy of the present method the results are compared with those available in the literatures and ABAQUS finite element package as well. The computer code can handle symmetric, unsymmetrical and general theta-ply schemes. The effects of the plate thicknesses, different ratio of outer to inner radius, depth of stiffener, boundary condition and laminates lay-up are studied in detail.

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In this paper, the governing equations for a vibratory beam with moderately large deflection are derived using the first order shear deformation theory. These equations which are a system of nonlinear partial differential equations with constant coefficients are solved analytically with the perturbation technique and the natural frequencies and the buckling load of the system are determined. A parametric study is performed and the effects of the geometrical and material properties on the natural frequency and buckling load are investigated and the effect of normal transverse strain and axial load on natural frequency are examined. Some results based on the first order shear deformation theory are consistent with classic theories of beams and some yield different results. Formulation presented to calculate the transverse frequency, determines the axial frequency too. Also, the natural frequencies and buckling load are calculated with the finite elements method by applying one and three-dimensional elements and the results are compared with the analytical solution.

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The objective of this paper is representation an analytical solution to calculate sound transmission loss (TL) of infinite thick transverse-isotropic cylindrical shell immersed in a fluid medium with an uniform external airflow and contains internal fluids where external sidewall of the shell excited by an oblique plane wave. In order to derive the governing equations the third-order shear deformation theory (TSDT) is used. Also, equation of motion of shell is obtained using Hamilton's principle. With solving shell vibration equations and acoustic wave equations simultaneously, the exact solution for TL is obtained. Transmission loss resultant from this solution is compared with those of other authors. The results also indicate that TSDT is more powerful than FSDT and CST, especially in high frequency and less R/h.

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In this paper, the buckling of rectangular plates subjected to non-uniform in-plane loading is investigated. At first the equilibrium equations of plate based on the first order shear deformation theory have been extracted. The kinematic relations have been assumed based on the von-Karman model and the Hook’s law has been considered as the constitutive equations. The adjacent equilibrium method has been used for deriving the stability equations. The equilibrium equations which are related to the prebuckling stress distribution, have been solved using the differential equations theory. To determine the buckling load of a simply supported plate, the Galerkin method has been used for solving the stability equations which are a system of differential equations with variable coefficients. In this paper, four types of in-plane loading, including the uniform, parabolic, cosine and triangular loading, have been considered and the effects of the plate aspect ratio and thickness on the buckling load has been investigated and the results have been compared with the finite element method and the classical plate theory. The comparison of the results show that for all loading cases, the buckling load computed by the classical plate theory is higher than the value obtained based on first order shear deformation theory.

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In this paper, free vibration of two-dimensional functionally graded (2D-FG) annular sectorial plate surrounded by Winkler-Pasternak elastic foundation has been investigated. It is assumed that the plate properties vary continuously through its both circumference and thickness according to power law distribution of the volume fraction. Primarily, we calculate the forces and resultant moments and then the total potential energy of system. Then, by applying the Hamilton’s principal any by regarding the first order shear deformation plate theory (FSDT) the governing differential equations have been derived. The numerical differential quadrature method, (DQM), has been employed for solving the motion equations. Two different boundary conditions such as simply supported and clamp-simply supported are considered. Initially, the obtained results were verified against those given in the literature and by ANSYS software and we confident from the obtain results. The effects of geometrical and elastic foundation parameters along with FG power indices effects on the natural frequencies have been studied. The study of results shows that, elastic foundation and FG parameters have significant effects on natural frequencies. By doing this research for 2D-FG materials the characteristic vibration of structure can be controlled by more parameters than 1D-FG materials.

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Performance-based design optimization (PBDO) is a relatively new concept in structural seismic design optimization. One of the PBDO methods which has been introduced in recent years is the optimization based on the uniform deformation theory. This method is quite different from other optimization techniques and formed based on the concept of structural performance and uniform distribution of deformation demands in the structure subjected to the seismic excitation. The aim of this method is to assign specific sections to elements such that all of the elements can reach their allowable deformation capacity during the earthquake. According to this theory, inefficient material is gradually shifted from the strong to weak areas leads to a uniform deformation (ductility) state at the end of a repetitive process. Although the base of this theory and proposed algorithm is to attain a uniform state of deformation in the whole structure, but the allowable limit of deformation values defined in PBD codes is not constant for all of structural elements. Additionally, in these codes, some actions of structural elements may be controlled by deformation and some controlled by force. Therefore, by considering the acceptance criteria of PBD codes, it is not possible to reach a uniform deformation state in the whole structure. Hence, in this paper uniform distribution of demand capacity ratio (DCR) is considered instead of uniform state of deformation. Historical review of applying this methodology shows that researchers mostly have used it to the optimum design of the structures under the earthquake records separately. Since earthquakes are random by nature, it is unlikely that the same earthquake ground motion will be repeated at some future time. This reveals that design based only one earthquake is insufficient and it is necessary to consider several earthquakes in checking the dynamic responses of a building. This paper presents an algorithm to PBDO of steel moment frames under set of ground motion records using the basic concepts of the uniform deformation theory. The proposed method consists of two phases. In the first phase of the search, to enhance the convergence rate, the search space of design variables is assumed to be continuous. Additionally in this phase of the search, only the deformation-controlled elements may vary. In the Second phase of the search, first for each structural element groups, the nearest discrete section to the imaginary section achieved in the first phase is identified and selected and then the structure is analyzed again and the DCRs are controlled. In this phase, acceptance criteria for both deformation and forced-controlled elements are supposed to be satisfied. Efficiency of the proposed algorithm is demonstrated in the optimum design of two baseline steel moment frames under a set of ground motion records. Results indicate that the proposed algorithm has a high speed to reach the optimum solution. The results are also compared with the optimum designs obtained by pushover analysis. It is shown that the optimization based on the pushover analysis results higher frame weight than time history analysis.

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In this article, the buckling of multilayer rectangular thick plate made of functionally graded, transversely isotropic and piezoelectric materials in both closed and open circuit conditions are investigated. Based on the shear and normal higher-order deformation theory, the governing equilibrium equations of plate are obtained using the principle of minimum total potential energy and Maxwell’s equation. Using an analytical approach, the governing stability equations of functionally graded rectangular plates with piezoelectric layers have been presented in terms of displacement components and electric potentials. In order to obtain the stability equations, the adjacent equilibrium criterion is used. The stability equations are then solved analytically, assuming simply support boundary condition along all edges. Finally after ensuring the validation of the results, the effects of different parameters such as different loading conditions, functionally graded power law index, thickness-to-length ratio and aspect ratio, on the critical buckling loads of plates are studied in details. Furthermore, the effect of piezoelectric thickness on the plate critical buckling loads has been studied. The results present better accuracy in comparison with the classic and third order shear theories.

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Nowadays, availability, durability, reliability, weight and strength, as the most important factors in optimum engineering design, are responsible for the widespread application of plates in the industry. Buckling in the elastic or elastoplastic regim is one of the phenomena that can be occurred in the axial compressive loading. Using Galerkin method on the basis of trigonometric shape functions, the elastoplastic dynamic buckling of a thin rectangular plate with different boundary conditions subjected to compression exponetiail pulse functions is investigated in this paper. Based on two theories of plasticity: deformation theory of plasticity (DT) with Hencky constitutive relations and incremental theory of plasticity (IT) with Prandtl-Reuss constitutive relations the equilibrium, stability and dynamic elastoplastic buckling equations are derived. Ramberg-Osgood stress-strain model is used to describe the elastoplastic material property of plate. The effects of symmetrical and asymmetrical boundary conditions, geometrical parameters of plate, force pulse amplitude, and type of plasticity theory on the velocity and deflection histories of plate are investigated. According to the dynamic response of plate the results obtained from DT are lower than those predicted through IT. The resistance against deformation for plate with clamped boundary condition is more than plate with simply supported boundary condition. Consequently, the adjacent points to clamped boundary condition have a lower velocity field than adjacent points to simply supported boundary condition.

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In this paper, elastic-plastic buckling of a thick rectangular plate has been investigated based on both Incremental (IT) and Deformation (DT) plasticity theories. Uniform biaxial edge traction was assumed as the plate loading while simply supported as the boundary conditions. Integral uniqueness criterion has been minimized to determine the critical buckling traction. Based on Rayleigh-Ritz method, a linear combination of polynomial base functions, which satisfy the geometrical boundary conditions, has been used as the trial functions for rotations and transverse deflection. To validate the analysis, the results for the Mindlin plate theory have been compared with the previously published results and a very close agreement has been observed. Then the effects of thickness ratio, aspect ratio and also different biaxial traction ratios on the buckling traction have been investigated. The results show that for the problem considered here, very close critical buckling traction is predicted by the both Mindlin and sinusoidal plate theories. This implies that Mindlin plate theory is sufficiently accurate to predict critical buckling traction in this problem. Moreover when the loading is gradually changed from biaxial into uniaxial compression or when the thickness-ratio is increased, the difference between the two theories is also increased. Also for compression-tension loading case, the critical buckling traction predicted by deformation theory is much less than the incremental theory.

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In this paper, the nonlocal buckling behavior of a biaxially loaded graphene sheet with piezoelectric layers based on an isotropic smart nanoplate model is studied. The equilibrium equations are derived with the von Karman-type geometrical nonlinearity by considering the small scale effect. The buckling of multilayer smart nanoplate made of graphene and piezoelectric materials in open circuit conditions is investigated. Based on the nonlocal elasticity and shear and normal deformation theories, the governing equilibrium equations are obtained using the principle of minimum total potential energy and Maxwell’s equation.
Using an analytical approach, the governing stability equations of smart nanoplate have been presented in terms of displacement components and electrical potential. In order to obtain the stability equations, the adjacent equilibrium criterion is used. The stability equations are then solved analytically, assuming simply supported boundary condition along all edges. To validate the results, the critical buckling load values have been compared with available resources. Finally, following validation of the results, numerical results for intelligent nanoplate are presented.
Also, the effects of different parameters such as nanoplate length, different nonlocal parameter, piezoelectric layers thickness, the graphene thickness to length ratio, the piezoelectric layer thickness to graphene thickness ratio and type of Piezoelectric material on the critical buckling loads of intelligent nanoplate are studied in detail. Furthermore, the effect of the mentioned parameters on the critical buckling loads have been presented in some figures.

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In this paper, the elastoplastic buckling of rectangular plates over the Pasternak foundation has been analyzed with the fixed and simply supported boundary conditions. Associated with the uniform loading conditions on the plate by the in- plane compression and tension, the influence of the elastic foundation is investigated in terms of two stiffness parameters; including the Winkler spring and the Pasternak shear coefficients. In order to extract governing equations, two theories are used from the plasticity: deformation theory (DT) with the Hencky constitutive relations and the incremental theory (IT) based on the Prandtl-Reuss constitutive relations. By implementing the generalized differential quadrature method to discrete the differential equations, influences of loading ratio, length to width ratio, plate thickness, and the elastic foundation characters are studied. By comparing the obtained results with the data reported in references, the accuracy of the model is verified. Consideration of results shows that applying the elastic foundation causes to increase critical buckling load. In addition, enhancing the elastic foundation parameters leads to amplifying the difference between buckling loads obtained from two theories, especially in the larger thicknesses. Moreover, according to increasing the plate thickness in the tensile state of the loading, application of the elastic foundation causes to reach plate stress to a value more than the ultimate stress of the specimen.

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Viscoelasticity is a property of materials that exhibit both viscous and elastic characteristics. In linear viscoelasticity, the stress is linearly related to the history function of strain. This paper discusses vibration analysis of functionally graded viscoelastic rectangular plate. The viscoelastic behavior of the plate is modeled using the Zener three-parameter model. Also, the material properties of the plate are graded through the thickness according to the volume fraction model. The maximum stress and strain are calculated based on the linear first-order shear deformation theory and the simply support boundary conditions is assumed at all four edges of the plate. A code is prepared using the Mathematica software to obtain the frequency values and effect of inherent and geometric characteristics of the sheet on natural frequency of the plate. These effects are studied using the tables and graphs represented in the results and discussion section of the paper. The results obtained in this paper are simplified to a functionally elastic plate to compare with those given in the literature search. The comparison of results shows good agreement against data given in literature for both cases.

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In this study, based on the third-order shear deformation theory the equations of motion are obtained to analyses the deformation of a long and slender composite beam. The beam has initial geometric imperfection and subjected to impact load. The impact procedures are applied by rigid body with a specific speed, off-center and at a certain distance from the beam's surface. Hamilton’s principle and the von-Karman nonlinear strain-displacement relationship are used to obtain the equations of motion that they are based on displacement and in a set of coupled nonlinear partial differential equations in dynamic mode. The generalized differential quadrature Method (GDQM) is used to discretize the obtained equations and convert them into a set of ordinary differential equations. Newton-Raphson iterative scheme is employed to solve the resulting system of nonlinear algebraic equations. Then, by solving the equations of the system, the effects of initial geometric imperfection on the beam’s deflection have been studied. Also the effects of mass and the initial velocity of the impactor on the beam’s deformation are investigated. The results of this research show that an increase in the amount of the initial velocity and mass of the impactor entail an increase in the beam deformation.

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In this study, an improved third order shear deformation theory is used to analyze the thermoelastic buckling of a functionally graded rectangular plate. The plate is assumed to be under two types of thermal loading, namely, uniform temperature rise across the thickness and linear temperature change across the thickness of the plate. Moreover, the material properties of the functionally graded plate vary linearly through the thickness and simply supported are considered for all edges of the plate. First, the nonlinear strain-displacement relations are considered based on improved third order theory and then the equilibrium and stability equations are derived. In continue, displacements and the pre-buckling forces are calculated using the equilibrium equations. The temperature difference relation of buckling is obtained by solving the stability equations. To obtain the critical temperature difference, the recent relation is minimized with respect to the number of half wave parameters. Resulting equations are compared with the literature. The results show that, the values of temperature difference buckling obtained based on improved third order shear deformation theory, are lower compared with the classical plate theory, first and third order shear deformation theories. Moreover, the value of critical temperature difference under linear temperature change is bigger compared with the uniform temperature rise across the thickness, and the difference between the two values will be bigger with increasing the thickness of the plate.

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Designing explosion of gas pipelines, gun tubes, pulse detonation engine tubes and etc, all related to problem of cylindrical shell subjected to dynamic internal loads. In this paper, dynamic response of the thick cylindrical shell subjected to dynamic internal load with considering the high order shear deformation theory (HODT) is investigated and compared with the first order shear deformation theory of Mirsky- Hermann (FSDT). The effects of transverse shear deformation and rotatory inertia were included in the governing equations of the dynamic system. First, the equations of motion have been derived by using Hamilton’s principle then by changing variables the obtained partial differential equations have been converted to ordinary differential equations. With this method, the problem can be solved for various mechanical moving pressure loads without considering the effect of boundary conditions with long length assumption. The results of the present analytical method have been verified by comparing with finite element results, by using software. The comparison of the results with the finite element method shows that the high order theory and first order Mirsky-Hermann theory can predict the dynamic response of the thick cylindrical shell and the high order theory in areas away from the middle layer is more successful.

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In this paper, axial buckling of a composite cylindrical shell with and without a rectangular cutout is studied based on the first-order shear deformation theory. The equations are derived in a general form and can be converted to Donnell`s, Love`s, and Sanders` theories. To investigate the perforated shell, a physical domain is decomposed into several elements with uniform boundary and loading conditions in each element edges. In each element, the governing equations are discretized in both longitudinal and circumferential directions by the use of generalized differential quadrature method (GDQM). By assembling these discretized relations, a system of algebraic equations is generated. The boundary conditions at the shell and cutout edges, and the compatibility conditions at the interface boundaries of adjacent elements are also discretized by GDQM. Finally, the buckling load is calculated by an eigenvalue solution. To validate the presented method, the results of GDQM are compared with the available ones in the literature and also with ABAQUS finite element model. Then a parametric analysis is performed to investigate the effects of different parameters on the buckling behavior of the shells with and without cutouts. This study illustrates that the shell layup has a great effect on the buckling load of a shell. In addition, the influence of increasing the cutout size is not identical for different layups. However, the buckling behavior is independent of the shell material. Moreover, it was concluded that the shell with a square cutout has higher critical load than the one with a rectangular opening.

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This study aims to investigate the transverse vibration of single- and double-layered graphene sheets embedded in an elastic medium based on the third-order shear deformation theory considering the axial force effect within the framework of Eringen’s nonlocal elasticity theory, where the governing equations of motion are obtained using Hamilton’s principle. The superiority of the studied non-local continuum model to its local counterpart is to consider the effect of size on the mechanical behavior of the structure. The results from a natural frequency analysis are obtained for different conditions such as the effect of size and aspect ratio, axial force, nonlocal coefficient, and change in the stiffness properties of the surrounding elastic medium by using the Navier-type solution for simply supported boundary conditions. Given that in a double-layered graphene sheet, the system has an in-phase vibrational mode and anti-phase vibrational mode with 180-degrees phase difference, the effect of van der Waals force on both vibrational modes is attempted to be investigated and it is shown that the van der Waals force has no effect on in-phase vibrational mode and by increasing it, the anti-phase frequency increases. It is also demonstrated that the nonlocal parameter is not a constant parameter but its value depends on the size and atomic structure, like chiral and zigzag configurations, and even on the type of boundary conditions.

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