• Composites Research
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• v.13 no.5
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• pp.37-43
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• 2000

In this paper an analytical investigation pertaining to the elastic shear buckling behavior of transversely stiffened orthotropic plate under in-plane shear forces is presented. All edges of plate are assumed to be simply supported and the evenly placed stiffener is considered as a beam element neglecting its torsional rigidity. For the solution of the problem Rayleigh-Ritz method is employed. Using the derived equation, the limit of buckling stress of transversely stiffened plate is suggested as a graphical form. Based on the limit of buckling stress of stiffened plate, graphical form of results for finding the required stiffener rigidity is presented when one and two stiffeners are located, respectively.

• Proceedings of the KSME Conference
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• 2008.11a
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• pp.967-972
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• 2008

This paper presents the free vibration analysis of a circular plate with a concentric square hole. The present problem deals with the numerical calculation of the natural frequencies and mode shapes of vibration of the structure by means of Independent Coordinate Coupling Method (ICCM). In this study, the boundary condition is the edge of the square hole is free and the outer circular plate is simply supported. Due to the geometric abnormality, this analysis does not permit an exact solution. Since the ICCM employs coordinate systems corresponding to each domain independently, the kinetic and potential energy expressions necessary for the Rayleigh-Ritz method can be easily obtained. Lastly, the kinematic relation is imposed. In this way, the eigenvalue problem can be easily set up. The numerical results show the efficacy of the ICCM and changes in natural frequencies and modes due to the square hole size.

• Journal of the Korean Society of Safety
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• v.14 no.1
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• pp.199-207
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• 1999

The differential quadrature method (DQM) is applied to computation of eigenvalues of the equations of motion governing the free in-plane and out-of-plane vibrations for circular curved beams. Fundamental frequencies are calculated for the members with various end conditions and opening angles. The results are compared with existing exact solutions and numerical solutions by other methods (Rayleigh-Ritz, Galerkin or FEM) for cases in which they are available. The differential quadrature method gives good accuracy even when only a limited number of grid points is used.

• Steel and Composite Structures
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• v.33 no.1
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• pp.133-142
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• 2019

In the present study, microstructure-dependent static stability analysis of inhomogeneous tapered micro-columns is performed. It is considered that the micro column is made of functionally graded materials and has a variable cross-section. The material and geometrical properties of micro column vary continuously throughout the axial direction. Euler-Bernoulli beam and modified couple stress theories are used to model the nonhomogeneous micro column with variable cross section. Rayleigh-Ritz solution method is implemented to obtain the critical buckling loads for various parameters. A detailed parametric study is performed to examine the influences of taper ratio, material gradation, length scale parameter, and boundary conditions. The validity of the present results is demonstrated by comparing them with some related results available in the literature. It can be emphasized that the size-dependency on the critical buckling loads is more prominent for bigger length scale parameter-to-thickness ratio and changes in the material gradation and taper ratio affect significantly the values of critical buckling loads.

• Structural Engineering and Mechanics
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• v.70 no.6
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• pp.661-669
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• 2019

Buckling analysis of shape memory alloy (SMA) rectangular plates subjected to uniform and linearly distributed inplane loads is the main objective in the present paper. Brinson's model is developed to express the constitutive characteristics of SMA plate. Using the classical plate theory and variational approach, stability equations are derived. In addition to external inplane mechanical loads, the plate is subjected to the pre-stresses caused by the recovery stresses that are generated during martensitic phase transformation. Ritz method is used for solving the governing stability equations. Finally, the effects of conditions on the edges, thickness, aspect ratio, temperature and pre-strains on the critical buckling loads of SMA plate are investigated in details.

• Structural Engineering and Mechanics
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• v.12 no.4
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• pp.363-376
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• 2001

Sandwich plates are widely used in lightweight design due to their high strength and stiffness to weight ratio. Due to the heterogeneous structure of sandwich plates, they can exhibit local instabilities (wrinkling), which lead to a sudden loss of stiffness in the structure. This paper presents an analytical solution to the wrinkling problem of sandwich plates. The solution is based on the Rayleigh-Ritz method, by assuming an appropriate deformation field. In contrast to the other approaches up to now, this model takes arbitrary and different orthotropic face layers, finite core thickness and orthotropic core material into account. This approach is the first to cover the wrinkling of unsymmetric sandwiches and sandwiches composed of orthotropic FRP face layers, which are most common in advanced lightweight design. Despite the generality of the solution, the computational effort is kept within bounds. The results have been verified using other analytical solutions and unit cell 3D FE calculations.

• Structural Engineering and Mechanics
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• v.31 no.6
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• pp.625-638
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• 2009

Applications of membrane mechanisms are widely found in nano-devices and nano-sensor technologies nowadays. An alternative approach for large deflection analysis of the orthotropic, elliptic membranes - subject to gravitational, uniform pressures often found in nano-sensors - is described in this paper. The material properties of membranes are assumed to be orthogonally isotropic and linearly elastic, while the principal directions of elasticity are parallel to the coordinate axes. Formulating the potential energy functional of the orthotropic, elliptic membranes involves the strain energy that is attributed to inplane stress resultant and the potential energy due to applied pressures. In the solution method, Rayleigh-Ritz method can be used successfully to minimize the resulting total potential energy generated. The set of equilibrium equations was solved subsequently by Newton-Raphson. The unparalleled model formulation capable of analyzing the large deflections of both circular and elliptic membranes is verified by making numerical comparisons with existing results of circular membranes as well as finite element solutions. The results are found in excellent agreements at all cases. Then, the parametric investigations are given to delineate the impacts of the aspect ratios and orthotropic elasticity on large static tensions and deformations of the orthotropic, elliptic membranes.

• Steel and Composite Structures
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• v.27 no.3
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• pp.273-283
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• 2018

Nonlinear low velocity impact response of sandwich beam with laminated composite face sheets and soft core is studied based on Extended High Order Sandwich Panel Theory (EHSAPT). The face sheets follow the Third order shear deformation beam theory (TSDT) that has hitherto not reported in conventional EHSAPT. Besides, the two dimensional elasticity is used for the core. The nonlinear Von Karman type relations for strains of face sheets and the core are adopted. Contact force between the impactor and the beam is obtained using the modified Hertz law. The field equations are derived via the Ritz based applied to the total energy of the system. The solution is obtained in the time domain by implementing the well-known Runge-Kutta method. The effects of boundary conditions, core-to-face sheet thickness ratio, initial velocity of the impactor, the impactor mass and position of the impactor are studied in detail. It is found that each of these parameters have significant effect on the impact characteristics which should be considered. Finally, some low velocity impact tests have been carried out by Drop Hammer Testing Machine. The contact force histories predicted by EHSAPT are in good agreement with that obtained by experimental results.

• Advances in Computational Design
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• v.3 no.2
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• pp.165-190
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• 2018

In this article, the static behavior of non-prismatic sandwich beams composed of functionally graded (FG) materials is investigated for the first time. Two types of beams in which the variation of elastic modulus follows a power-law form are studied. The principle of minimum total potential energy is applied along with the Ritz method to derive and solve the governing equations. Considering conventional boundary conditions, Chebyshev polynomials of the first kind are used as auxiliary shape functions. The formulation is developed within the framework of well-known Timoshenko and Reddy beam theories (TBT, RBT). Since the beams are simultaneously tapered and functionally graded, bending and shear stress pushover curves are presented to get a profound insight into the variation of stresses along the beam. The proposed formulations and solution scheme are verified through benchmark problems. In this context, excellent agreement is observed. Numerical results are included considering beams with various cross sectional types to inspect the effects of taper ratio and gradient index on deflections and stresses. It is observed that the boundary conditions, taper ratio, gradient index value and core to the thickness ratio significantly influence the stress and deflection responses.

• Structural Engineering and Mechanics
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• v.36 no.5
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• pp.545-560
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• 2010

Two or more distinct materials are combined into a single functionally graded material (FGM) where the microstructural composition and properties change gradually. Thermal post-buckling behavior of uniform slender FGM beams is investigated independently using the classical Rayleigh-Ritz (RR) formulation and the versatile Finite Element Analysis (FEA) formulation developed in this paper. The von-Karman strain-displacement relations are used to account for moderately large deflections of FGM beams. Bending-extension coupling arising due to heterogeneity of material through the thickness is included. Simply supported and clamped beams with axially immovable ends are considered in the present study. Post-buckling load versus deflection curves and buckled mode shapes obtained from both the RR and FEA formulations for different volume fraction exponents show an excellent agreement with the available literature results for simply supported ends. Response of the FGM beam with clamped ends is studied for the first time and the results from both the RR and FEA formulations show a very good agreement. Though the response of the FGM beam could have been studied more accurately by FEA formulation alone, the authors aim to apply the RR formulation is to find an approximate closed form post-buckling solutions for the FGM beams. Further, the use of the RR formulation clearly demonstrates the effect of bending-extension coupling on the post-buckling response of the FGM beams.