scholarly journals On the Shear Buckling of Clamped Narrow Rectangular Orthotropic Plates

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Seyed Rasoul Atashipour ◽  
Ulf Arne Girhammar
Keyword(s):  
Closed Form ◽  
Plate Theory ◽  
Exact Analytical Solution ◽  
Differential Quadrature Method ◽  
Thin Plates ◽  
Orthotropic Plates ◽  
Classical Plate Theory ◽  
Aspect Ratios ◽  
Wide Range ◽  
Shear Buckling

This paper deals with stability analysis of clamped rectangular orthotropic thin plates subjected to uniformly distributed shear load around the edges. Due to the nature of this problem, it is impossible to present mathematically exact analytical solution for the governing differential equations. Consequently, all existing studies in the literature have been performed by means of different numerical approaches. Here, a closed-form approach is presented for simple and fast prediction of the critical buckling load of clamped narrow rectangular orthotropic thin plates. Next, a practical modification factor is proposed to extend the validity of the obtained results for a wide range of plate aspect ratios. To demonstrate the efficiency and reliability of the proposed closed-form formulas, an accurate computational code is developed based on the classical plate theory (CPT) by means of differential quadrature method (DQM) for comparison purposes. Moreover, several finite element (FE) simulations are performed via ANSYS software. It is shown that simplicity, high accuracy, and rapid prediction of the critical load for different values of the plate aspect ratio and for a wide range of effective geometric and mechanical parameters are the main advantages of the proposed closed-form formulas over other existing studies in the literature for the same problem.

Effect of Shear Deformations on the Bending of Rectangular Plates

1960 ◽  
Vol 27 (1) ◽  
pp. 54-58 ◽  
Author(s):  
V. L. Salerno ◽  
M. A. Goldberg
Keyword(s):  
Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Plate Theory ◽  
Order Partial Differential Equation ◽  
Rectangular Plates ◽  
Partial Differential ◽  
Classical Plate Theory ◽  
Simply Supported ◽  
Wide Range

The three partial differential equations derived by Dr. E. Reissner2, 3 have been reduced to a fourth-order partial differential equation resembling that of the classical plate theory and to a second-order differential equation for determining a stress function. The general solution for the two partial differential equations has been applied to a simply supported plate with a constant load p and to a plate with two opposite edges simply supported and the other two edges free. Numerical calculations have been made for the simply supported plate and the results compared with those of classical theory. The calculations for a wide range of parameters indicate that the deviation is small.

Vibration analysis of a nano-turbine blade based on Eringen nonlocal elasticity applying the differential quadrature method

2016 ◽  
Vol 23 (19) ◽  
pp. 3247-3265 ◽  
Author(s):  
Majid Ghadiri ◽  
Navvab Shafiei
Keyword(s):  
Boundary Conditions ◽  
Nonlocal Elasticity ◽  
Plate Theory ◽  
Differential Quadrature Method ◽  
Differential Quadrature ◽  
Small Scale ◽  
Quadrature Method ◽  
Bending Vibrations ◽  
Classical Plate Theory ◽  
Axial Forces

This study investigates the small-scale effect on the flapwise bending vibrations of a rotating nanoplate that can be the basis of nano-turbine design. The nanoplate is modeled as classical plate theory (CPT) with boundary conditions as the cantilever and propped cantilever. The axial forces are also included in the model as the true spatial variation due to the rotation. Hamilton’s principle is used to derive the governing equation and boundary conditions for the classic plate based on Eringen’s nonlocal elasticity theory and the differential quadrature method is employed to solve the governing equations. The effect of the small-scale parameter, nondimensional angular velocity, nondimensional hub radius, setting angle and different boundary conditions in the first four nondimensional frequencies is discussed. Due to considering rotating effects, results of this study are applicable in nanomachines such as nanomotors and nano-turbines and other nanostructures.

scholarly journals Buckling and Vibration of Non-Homogeneous Rectangular Plates Subjected to Linearly Varying In-Plane Force

2013 ◽  
Vol 20 (5) ◽  
pp. 879-894 ◽  
Author(s):  
Roshan Lal ◽  
Renu Saini
Keyword(s):  
Differential Equation ◽  
Plate Theory ◽  
Differential Quadrature Method ◽  
Three Dimensional ◽  
Axial Direction ◽  
Mode Shapes ◽  
Rectangular Plates ◽  
Plate Material ◽  
Classical Plate Theory ◽  
Simply Supported

The present work analyses the buckling and vibration behaviour of non-homogeneous rectangular plates of uniform thickness on the basis of classical plate theory when the two opposite edges are simply supported and are subjected to linearly varying in-plane force. For non-homogeneity of the plate material it is assumed that young's modulus and density of the plate material vary exponentially along axial direction. The governing partial differential equation of motion of such plates has been reduced to an ordinary differential equation using the sine function for mode shapes between the simply supported edges. This resulting equation has been solved numerically employing differential quadrature method for three different combinations of clamped, simply supported and free boundary conditions at the other two edges. The effect of various parameters has been studied on the natural frequencies for the first three modes of vibration. Critical buckling loads have been computed. Three dimensional mode shapes have been presented. Comparison has been made with the known results.

scholarly journals SHEAR BUCKLING EVALUATION IN STEEL PLATE GIRDERS WITH RIGID END POSTS SUBJECTED TO ELEVATED TEMPERATURES

2016 ◽  
Author(s):  
André Reis ◽  
Nuno Lopes ◽  
Paulo Vila Real
Keyword(s):  
Elevated Temperatures ◽  
Steel Plate ◽  
Numerical Study ◽  
Ultimate Shear Strength ◽  
Plate Girders ◽  
Aspect Ratios ◽  
Wide Range ◽  
Geometrical Imperfections ◽  
Shear Buckling ◽  
Steel Grades

The current study intends to analyse the behaviour of steel plate girders with rigid end posts subjected to elevated temperatures, aiming the assessment of the ultimate shear strength in case of fire. A parametric numerical study was performed involving a wide range of cross-section’s dimensions, plate girders’ aspect ratios and steel grades. Plate girders were numerically tested at both normal and elevated temperature, being considered three different uniform temperatures. The influence of the geometrical imperfections, as well as the residual stresses, was taken into account. Finally, the numerical results were compared to the Eurocode 3 (EC3) prescriptions, adapted to fire situation by the direct application of the reduction factors for the stress-strain relationship of steel at elevated temperatures. It was shown that the EC3 design rules should be improved because they are not conservative, conducting to unsafe results.

Thermal buckling load optimization of laminated general quadrilateral and trapezoidal thin plates

2013 ◽  
Vol 20 (1) ◽  
pp. 87-94 ◽  
Author(s):  
Umut Topal
Keyword(s):  
Plate Theory ◽  
Mathematical Formulation ◽  
Thermal Buckling ◽  
Laminated Plates ◽  
Buckling Load ◽  
Thin Plates ◽  
Load Optimization ◽  
Aspect Ratios ◽  
Feasible Direction Method ◽  
Optimization Routine

AbstractThis paper deals with thermal buckling load optimization of symmetrically laminated angle-ply general quadrilateral and trapezoidal thin plates. The objective function is to maximize the critical temperature capacity of the quadrilateral and trapezoidal laminated plates and the fiber orientation is considered as a design variable. The mathematical formulation is based on the classical laminated plate theory for the frequency analysis. The modified feasible direction method is used as the optimization routine. Therefore, a program based on FORTRAN is used. Finally, the significant effects of aspect ratios, boundary conditions, taper ratios and unsymmetric trapezoidal plates on the optimal results are investigated and the results are compared.

Nonlinear thermo-nonlocal vibration and instability analysis of an embedded single-layered graphene sheet conveying nanoflow using differential quadrature method

2012 ◽  
Vol 227 (9) ◽  
pp. 2104-2117 ◽  
Author(s):  
A Ghorbanpour Arani ◽  
MJ Maboudi ◽  
H Haghighi ◽  
R Kolahchi
Keyword(s):  
Graphene Sheet ◽  
Plate Theory ◽  
Differential Quadrature Method ◽  
Nonlocal Theory ◽  
Differential Quadrature ◽  
Small Scale ◽  
Quadrature Method ◽  
Classical Plate Theory ◽  
Instability Analysis ◽  
Instability Characteristic

In this study, transverse nonlinear vibration and instability analysis of a viscous-fluid-conveyed single-layered graphene sheet (SLGS) subjected to thermal gradient are investigated. The small-size effects on bulk viscosity and slip boundary conditions of nanoflow through Knudsen number ( Kn), as a small size parameter is considered. Viscopasternak model is considered to simulate the interaction between the graphene sheet and the surrounding elastic medium. Continuum orthotropic plate model and relations of classical plate theory are used. The nonlocal theory of Eringen is employed to incorporate the small-scale effect into the governing equations of the graphene sheet. Differential quadrature method is employed to solve the governing differential equations for simply supported edges. The convergence of the procedure is shown and the effects of flow velocity, temperature change and aspect ratio on the frequency of the single-layered graphene sheet are investigated. Moreover, the critical flow velocities and the instability characteristic are determined. It is evident from the results that the natural frequency of nanosheet increases with rising temperature.

scholarly journals Novel Semi-Analytical Solutions for the Transient Behaviors of Functionally Graded Material Plates in the Thermal Environment

Materials ◽  
2019 ◽  
Vol 12 (24) ◽  
pp. 4084 ◽  
Author(s):  
Zeng Cao ◽  
Xu Liang ◽  
Yu Deng ◽  
Xing Zha ◽  
Ronghua Zhu ◽  
...  
Keyword(s):  
Temperature Change ◽  
Thermal Environment ◽  
Plate Theory ◽  
Differential Quadrature Method ◽  
Functionally Graded ◽  
Fourier Series Expansion ◽  
Equilibrium Equations ◽  
Classical Plate Theory ◽  
Peak Displacement ◽  
Sampling Points

The primary objective of this article is to present a semi-analytical algorithm for the transient behaviors of Functionally Graded Materials plates (FGM plates) considering both the influence of in-plane displacements and the influence of temperature changes. Based on the classical plate theory considering the effect of in-plane displacements, the equilibrium equations of the motion system are derived by Hamilton’s principle. Here, we propose a novel, accurate, and efficient semi-analytical method that incorporates the Fourier series expansion, the Laplace transforms, and its numerical inversion and the Differential Quadrature Method (DQM) to simulate the transient behaviors. This paper validates the proposed method by comparisons with semi-analytical natural frequency results and those from the literature. Expressly, the results of dynamic response also agree well with those generated by the Navier’s method and Finite Element Method (FEM). A convergence study that utilizes the different numbers of sampling points shows that the process can converge quickly, and a few sampling points can achieve high accuracy. The effects of various boundary conditions at the ends, material graded index, and temperature change are further investigated. From the detailed parametric study, it is seen that the peak displacement increases as the edge degrees of freedom, the gradient index of the material, and temperature change increase.

scholarly journals On a Novel Approximate Solution to the Inhomogeneous Euler–Bernoulli Equation with an Application to Aeroelastics

Aerospace ◽  
2021 ◽  
Vol 8 (11) ◽  
pp. 356
Author(s):  
Dominique Fleischmann ◽  
László Könözsy
Keyword(s):  
Finite Difference ◽  
Exact Analytical Solution ◽  
Homogeneous Equation ◽  
Inhomogeneous Equation ◽  
Bernoulli Equation ◽  
Applied Loading ◽  
Aspect Ratios ◽  
Aeroelastic Analysis ◽  
Wide Range ◽  
Beam Bending

This paper focuses on the development of an explicit finite difference numerical method for approximating the solution of the inhomogeneous fourth-order Euler–Bernoulli beam bending equation with velocity-dependent damping and second moment of area, mass and elastic modulus distribution varying with distance along the beam. We verify the method by comparing its predictions with an exact analytical solution of the homogeneous equation, we use the generalised Richardson extrapolation to show that the method is grid convergent and we extend the application of the Lax–Richtmyer stability criteria to higher-order schemes to ensure that it is numerically stable. Finally, we present three sets of computational experiments. The first set simulates the behaviour of the un-loaded beam and is validated against the analytic solution. The second set simulates the time-dependent dynamic behaviour of a damped beam of varying stiffness and mass distributions under arbitrary externally applied loading in an aeroelastic analysis setting by approximating the inhomogeneous equation using the finite difference method derived here. We compare the third set of simulations of the steady-state deflection with the results of static beam bending experiments conducted at Cranfield University. Overall, we developed an accurate, stable and convergent numerical framework for solving the inhomogeneous Euler–Bernoulli equation over a wide range of boundary conditions. Aircraft manufacturers are starting to consider configurations with increased wing aspect ratios and reduced structural weight which lead to more slender and flexible designs. Aeroelastic analysis now plays a central role in the design process. Efficient computational tools for the prediction of the deformation of wings under external loads are in demand and this has motivated the work carried out in this paper.

Closed-form Analytical Solutions for Free Vibration of Rectangular Functionally Graded Thin Plates in Thermal Environment

2018 ◽  
Vol 10 (03) ◽  
pp. 1850025 ◽  
Author(s):  
Y. F. Xing ◽  
Z. K. Wang ◽  
T. F. Xu
Keyword(s):  
Free Vibration ◽  
Closed Form ◽  
Analytical Solutions ◽  
Thermal Environment ◽  
Plate Theory ◽  
Normal Modes ◽  
Functionally Graded ◽  
Thin Plates ◽  
Numerical Comparison ◽  
Fg Plates

Based on classical small deflection plate theory, the governing equation of functionally graded (FG) plates in thermal environment is derived by using Hamilton’s principle. Closed-form analytical solutions are obtained via separation-of-variable method for free vibration of rectangular FG thin plates with simply supported, clamped and guided edges, especially for the plates with two adjacent clamped edges in thermal environment. The normal modes and frequencies are in an elegant and explicit closed form. Comprehensive numerical comparison and results in dimensionless form validate the present method and reveal a few physical phenomena.

Sign in / Sign up

Export Citation Format

Share Document