• Structural Engineering and Mechanics
• /
• v.69 no.5
• /
• pp.511-525
• /
• 2019

This paper presents an analytical study of wave propagation in simply supported graduated functional plates resting on a two-parameter elastic foundation (Pasternak model) using a new theory of high order shear strain. Unlike other higher order theories, the number of unknowns and governing equations of the present theory is only four unknown displacement functions, which is even lower than the theory of first order shear deformation (FSDT). Unlike other elements, the present work includes a new field of motion, which introduces indeterminate integral variables. The properties of the materials are assumed to be ordered in the thickness direction according to the two power law distributions in terms of volume fractions of the constituents. The wave propagation equations in FG plates are derived using the principle of virtual displacements. The analytical dispersion relation of the FG plate is obtained by solving an eigenvalue problem. Numerical examples selected from the literature are illustrated. A good agreement is obtained between the numerical results of the current theory and those of reference. A parametric study is presented to examine the effect of material gradation, thickness ratio and elastic foundation on the free vibration and phase velocity of the FG plate.

• Structural Engineering and Mechanics
• /
• v.82 no.6
• /
• pp.725-734
• /
• 2022

This work presents a simple exponential shear deformation theory for the flexural and free vibration responses of thick bridge deck. Contrary to the existing higher order shear deformation theories (HSDT) and the first shear deformation theory (FSDT), the proposed model uses a new displacement field which incorporates undetermined integral terms and involves only two variables. Governing equations and boundary conditions of the theory are derived by the principle of virtual work. The simply supported thick isotropic square and rectangular plates are considered for the detailed numerical studies. Results of displacements, stresses and frequencies are compared with those of other refined theories and exact theory to show the efficiency of the proposed theory. Good agreement is achieved of the present results with those of higher order shear deformation theory (HSDT) and elasticity theory. Moreover, results demonstrate that the developed two variable refined plate theory is simple for solving the flexural and free vibration responses of thick bridge deck and can achieve the same accuracy of the existing HSDTs which have more number of variables.

• Smart Structures and Systems
• /
• v.26 no.2
• /
• pp.253-262
• /
• 2020

In this paper, a new higher order shear deformation model is developed for static analysis of functionally graded beams with considering porosities that may possibly occur inside the functionally graded materials (FGMs) during their fabrication. The model account for higher-order variation of transverse shear strain through the depth of the beam and satisfies the zero traction boundary conditions on the surfaces of the beam without using shear correction factors. The present work aims to study the effect of the distribution forms of porosity on the bending of simply supported FG beam. Based on the present higher-order shear deformation model, the equations of motion are derived by the principle of virtual works. Navier type solution method was used to obtain displacement and stresses, and the numerical results are compared with those available in the literature. A comprehensive parametric study is carried out to assess the effects of volume fraction index, porosity fraction index, and geometry on the bending of imperfect FG beams. It can be concluded that the proposed model is simple and precise for the resolution of the behavior of flexural FGM beams while taking into account the shape of distribution of the porosity.

• Proceedings of the Korean Society of Marine Engineers Conference
• /
• 2001.11a
• /
• pp.21-27
• /
• 2001

Springs are widely utilized in machine element. To find out stiffness of frustum-shaped coil spring, the space beam theory using the finite element method is adopted in this paper In three dimensional space, a space frame element is a straight bar of uniform cross section which is capable of resisting axial forces, bending moments about two principal axes in the plane of its cross section and twisting moment about its centroidal axis. The corresponding displacement degrees of freedom are twelve. To find out load vector of coil spring subjected to distributed compression, principle of virtual work is adapted The displacements of nodal points due to small increment of force are calculated by the finite element method and the calculated nodal displacements are added to coordinates of nodal points. The new stiffness matrix of the system using the new coordinates of nodal points is adopted to calculate the another increments of nodal displacements, that is, the step by step method is used in this paper. The results of the finite element method are fairly well agreed with those of various experiments. Using MATLAB program developed in this paper, spring constants and stresses can be predicted by input of few factors.

• Advances in materials Research
• /
• v.8 no.3
• /
• pp.155-177
• /
• 2019

In this paper, a new displacement based high-order shear deformation theory is introduced for the static response of functionally graded sandwich plate with new definition of porosity distribution taking into account composition and the scheme of the sandwich plate. Unlike any other theory, the number of unknown functions involved is only four, as against five in case of other shear deformation theories. The theory presented is variationally consistent, has strong similarity with classical plate theory in many aspects, does not require shear correction factor, and gives rise to transverse shear stress variation such that the transverse shear stresses vary parabolically across the thickness satisfying shear stress free surface conditions. Material properties of FGM layers are assumed to vary continuously across the plate thickness according to either power-law or sigmoid function in terms of the volume fractions of the constituents. The face layers are considered to be FG across each face thickness while the core is made of a ceramic homogeneous layer. Governing equations are derived from the principle of virtual displacements. The closed-form solution of a simply supported rectangular plate subjected to sinusoidal loading has been obtained by using the Navier method. Numerical results are presented to show the effect of the material distribution, the sandwich plate geometry and the porosity on the deflections and stresses of FG sandwich plates. The validity of the present theory is investigated by comparing some of the present results with other published results.

• Steel and Composite Structures
• /
• v.27 no.4
• /
• pp.525-536
• /
• 2018

This paper focuses on the application of the first-order shear deformation theory (FSDT) to thermo-elastic static problems of functionally graded carbon nanotubes reinforced composite (FG-CNTRC) cylindrical pressure vessels. A symmetric displacement field is considered as unknown function along the longitudinal direction, whereas a linear distribution is assumed along the thickness direction. The cylindrical pressure vessels are subjected to an inner and outer pressure under a temperature increase. Different patterns of reinforcement are applied as distribution of CNTs. The effective material properties of FG-CNTRC cylindrical pressure vessels are measured based on the rule of mixture, whereas the governing equations of the problem are here derived through the principle of virtual works. A large parametric investigation studies the effect of some significant parameters, such as the pattern and volume fraction of CNTs, on the longitudinal distribution of deformation, strain and stress components, as useful tool for practical engineering applications.

• Structural Engineering and Mechanics
• /
• v.75 no.2
• /
• pp.157-174
• /
• 2020

In the present paper, a simple analytical model is developed based on a new refined parabolic shear deformation theory (RPSDT) for free vibration and buckling analysis of orthotropic rectangular plates with simply supported boundary conditions. The displacement field is simpler than those of other higher-order theories since it is modeled with only two unknowns and accounts for a parabolic distribution of the transverse shear stress through the plate thickness. The governing differential equations related to the present theory are obtained from the principle of virtual work, while the solution of the eigenvalue problem is achieved by assuming a Navier technique in the form of a double trigonometric series that satisfy the edge boundary conditions of the plate. Numerical results are presented and compared with previously published results for orthotropic rectangular plates in order to verify the precision of the proposed analytical model and to assess the impacts of several parameters such as the modulus ratio, the side-to-thickness ratio and the geometric ratio on natural frequencies and critical buckling loads. From these results, it can be concluded that the present computations are in excellent agreement with the other higher-order theories.

• Steel and Composite Structures
• /
• v.6 no.3
• /
• pp.237-255
• /
• 2006

This paper presents a novel analytical formulation for the analysis of composite beams with partial shear interaction stiffened by a bolted longitudinal plate accounting for time effects, such as creep and shrinkage. The model is derived by means of the principle of virtual work using a displacement-based formulation. The particularity of this approach is that the partial interaction behaviour is assumed to exist between the top slab and the joist as well as between the joist and the bolted longitudinal stiffening plate, therefore leading to a three-layered structural representation. For this purpose, a novel finite element is derived and presented. Its accuracy is validated based on short-and long-term analyses for the particular cases of full shear interaction and partial shear interaction of two layers for which solutions in closed form are available in the literature. A parametric study is carried out considering different stiffening arrangements to investigate the influence on the short-and long-term behaviour of the composite beam of the shear connection stiffness between the concrete slab and the steel joist, the stiffness of the plate-to-beam connection, the properties of the longitudinal plate and the concrete properties. The values of the deflection obtained from the finite element simulations are compared against those calculated using the effective flexural rigidity in accordance with EC5 guidelines for the behaviour of elastic multi-layered beams with flexible connection and it is shown how the latter well predicts the structural response. The proposed numerical examples highlight the ease of use of the proposed approach in determining the effectiveness of different retrofitting solutions at service conditions.

• International Journal of Aerospace System Engineering
• /
• v.3 no.1
• /
• pp.10-16
• /
• 2016

A trigonometric Layerwise higher order shear deformation theory (TLHSDT) is developed and implemented for free vibration and buckling analysis of laminated composite and sandwich plates by analytical and finite element formulation. The present model assumes parabolic variation of out-plane stresses through the depth of the plate and also accomplish the zero transverse shear stresses over the surface of the plate. Thus a need of shear correction factor is obviated. The present zigzag model able to meet the transverse shear stress continuity and zigzag form of in-plane displacement continuity at the plate interfaces. Hence, botheration of shear correction coefficient is neglected. In the case of analytical method, the governing differential equation and boundary conditions are obtained from the principle of virtual work. For the finite element formulation, an efficient eight noded $C^0$ continuous isoparametric serendipity element is established and employed to examine the dynamic analysis. Like FSDT, the considered mathematical model possesses similar number of variables and which decides the present models computationally more effective. Several numerical predictions are carried out and results are compared with those of other existing numerical approaches.

• Transactions of the Korean Society for Noise and Vibration Engineering
• /
• v.19 no.2
• /
• pp.208-216
• /
• 2009

In this paper, we present the equations of motion by which the natural vibration of a rotating annular disk can be analyzed accurately. These equations are derived from the theory of finite deformation and the principle of virtual work. The radial displacements of annular disk at the steady state where the disk is rotating at a constant angular velocity are determined by non-linear static equations formulated with 1-dimensional finite elements in radial direction. The linearlized equations of the in-plane vibrations at the disturbed state are also formulated with 1-dimensional finite elements in radial direction along the number of nodal diameters. They are expressed as in functions of the radial displacements at the steady state and the disturbed displacements about the steady state. In-plane static deformation modes of an annular disk are used as the displacement functions for the interpolation functions of the 1-dimensional finite elements. The natural vibrations of an annular disk with different boundary conditions are analyzed by using the presented model and the 3-dimensional finite element model to verify accuracy of the presented equations of motion. Its results are compared and discussed.