• Smart Structures and Systems
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• v.20 no.6
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• pp.709-728
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• 2017

This disquisition proposes a nonlocal strain gradient beam theory for thermo-mechanical dynamic characteristics of embedded smart shear deformable curved piezoelectric nanobeams made of porous electro-elastic functionally graded materials by using an analytical method. Electro-elastic properties of embedded curved porous FG nanobeam are assumed to be temperature-dependent and vary through the thickness direction of beam according to the power-law which is modified to approximate material properties for even distributions of porosities. It is perceived that during manufacturing of functionally graded materials (FGMs) porosities and micro-voids can be occurred inside the material. Since variation of pores along the thickness direction influences the mechanical and physical properties, so in this study thermo-mechanical vibration analysis of curve FG piezoelectric nanobeam by considering the effect of these imperfections is performed. Nonlocal strain gradient elasticity theory is utilized to consider the size effects in which the stress for not only the nonlocal stress field but also the strain gradients stress field. The governing equations and related boundary condition of embedded smart curved porous FG nanobeam subjected to thermal and electric field are derived via the energy method based on Timoshenko beam theory. An analytical Navier solution procedure is utilized to achieve the natural frequencies of porous FG curved piezoelectric nanobeam resting on Winkler and Pasternak foundation. The results for simpler states are confirmed with known data in the literature. The effects of various parameters such as nonlocality parameter, electric voltage, coefficient of porosity, elastic foundation parameters, thermal effect, gradient index, strain gradient, elastic opening angle and slenderness ratio on the natural frequency of embedded curved FG porous piezoelectric nanobeam are successfully discussed. It is concluded that these parameters play important roles on the dynamic behavior of porous FG curved nanobeam. Presented numerical results can serve as benchmarks for future analyses of curve FG nanobeam with porosity phases.

• Advances in concrete construction
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• v.8 no.4
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• pp.285-294
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• 2019

In the present paper, the fiber theory has been employed to model the reinforced concrete (RC) deep beams (DBs) considering the reinforcing steel bar-concrete interaction. To simulate numerically the behavior of materials, the uniaxial materials' constitutive laws have been employed for reinforcements and concrete and the bond stress-slip between the reinforcing steel bars and surrounding concrete are taken into account. Because of the high sensitivity of DBs to shear deformations, the Timoshenko beam theory has been applied. The shear stress-strain (S-SS) relationship has been defined by the modified compression field theory (MCFT) model. By modeling about 300 RC panels and employing a produced numerical database, a study has been carried out to show the sensitivity of the MCFT model. This is performed based on the multiple linear regression (MLR) models. The results of this research also illustrate how different parameters such as characteristic compressive strength of concrete, yield strength of reinforcements and the percentages of reinforcements in different directions get involved in the shear behavior of RC panels without applying complex theories. Based on the results obtained from the analysis of the MCFT S-SS model, a relatively simplified numerical S-SS model has been proposed. Application of the proposed S-SS model in modeling and analyzing the considered samples indicates that there is a good agreement between the simulated and the experimental test results. The comparison between the proposed S-SS model and the MCFT model indicates that in addition to the advantage of better accuracy, the main advantage of the proposed method is simplicity in application.

• Smart Structures and Systems
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• v.23 no.2
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• pp.141-153
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• 2019

This research deals with wave propagation of the functionally graded (FG) nano-beams based on the nonlocal elasticity theory considering surface and flexoelectric effects. The FG nano-beam is resting in Winkler-Pasternak foundation. It is assumed that the material properties of the nano-beam changes continuously along the thickness direction according to simple power-law form. In order to include coupling of strain gradients and electrical polarizations in governing equations of motion, the nonlocal non-classical nano-beam model containg flexoelectric effect is used. Also, the effects of surface elasticity, dielectricity and piezoelectricity as well as bulk flexoelectricity are all taken into consideration. The governing equations of motion are derived using Hamilton principle based on first shear deformation beam theory (FSDBT) and also considering residual surface stresses. The analytical method is used to calculate phase velocity of wave propagation in FG nano-beam as well as cut-off frequency. After verification with validated reference, comprehensive numerical results are presented to investigate the influence of important parameters such as flexoelectric coefficients of the surface, bulk and residual surface stresses, Winkler and shear coefficients of foundation, power gradient index of FG material, and geometric dimensions on the wave propagation characteristics of FG nano-beam. The numerical results indicate that considering surface effects/flexoelectric property caused phase velocity increases/decreases in low wave number range, respectively. The influences of aforementioned parameters on the occurrence cut-off frequency point are very small.

• Structural Engineering and Mechanics
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• v.11 no.2
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• pp.123-132
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• 2001

An improved version of the Element-free Galerkin method (EFGM) is presented here for addressing the problem of transverse shear locking in shear-deformable beams with a high length over thickness ratio. Based upon Timoshenko's theory of thick beams, it has been recognized that shear locking will be completely eliminated if the rotation field is constructed to match the field of slope, given by the first derivative of displacement. This criterion is applied directly to the most commonly implemented version of EFGM. However in the numerical process to integrate strain energy, the second derivative of the standard Moving Least Square (MLS) shape functions must be evaluated, thus requiring at least a $C^1$ continuity of MLS shape functions instead of $C^0$ continuity in the conventional EFGM. Yet this hindrance is overcome effortlessly by only using at least a $C^1$ weight function. One-dimensional quartic spline weight function with $C^2$ continuity is therefore adopted for this purpose. Various numerical results in this work indicate that the modified version of the EFGM does not exhibit transverse shear locking, reduces stress oscillations, produces fast convergence, and provides a surprisingly high degree of accuracy even with coarse domain discretizations.

• Structural Engineering and Mechanics
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• v.35 no.2
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• pp.141-173
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• 2010

In this paper a boundary element method is developed for the general flexural-torsional buckling analysis of Timoshenko beams of arbitrarily shaped cross section. The beam is subjected to a compressive centrally applied concentrated axial load together with arbitrarily axial, transverse and torsional distributed loading, while its edges are restrained by the most general linear boundary conditions. The resulting boundary value problem, described by three coupled ordinary differential equations, is solved employing a boundary integral equation approach. All basic equations are formulated with respect to the principal shear axes coordinate system, which does not coincide with the principal bending one in a nonsymmetric cross section. To account for shear deformations, the concept of shear deformation coefficients is used. Six coupled boundary value problems are formulated with respect to the transverse displacements, to the angle of twist, to the primary warping function and to two stress functions and solved using the Analog Equation Method, a BEM based method. Several beams are analysed to illustrate the method and demonstrate its efficiency and wherever possible its accuracy. The range of applicability of the thin-walled theory and the significant influence of the boundary conditions and the shear deformation effect on the buckling load are investigated through examples with great practical interest.

• Structural Engineering and Mechanics
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• v.66 no.6
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• pp.729-736
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• 2018

This paper studies thermal buckling and post-buckling behaviors of functionally graded materials (FGM) tubes subjected to a uniform temperature rise and resting on elastic foundations via a refined beam model. Compared to the Timoshenko beam theory, the number of unknowns of this model are the same and no correction factors are required. The material properties of the FGM tube vary continuously in the radial direction according to a power function. Two ends of the tube are assumed to be simply supported and in-plane boundary conditions are immovable. Energy variation principle is employed to establish the governing equations. A two-step perturbation method is adopted to determine the critical thermal buckling loads and post-buckling paths of the tubes with arbitrary radial non-homogeneity. Through detailed parametric studies, it can be found that the tube has much higher buckling temperature and post-buckling strength when it is supported by an elastic foundation.

• Structural Engineering and Mechanics
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• v.49 no.3
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• pp.373-393
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• 2014

A finite element model for predicting the entire nonlinear behavior of reinforced high-strength concrete continuous beams is described. The model is based on the moment-curvature relations pre-generated through section analysis, and is formulated utilizing the Timoshenko beam theory. The validity of the model is verified with experimental results of a series of continuous high-strength concrete beam specimens. Some important aspects of behavior of the beams having different tensile reinforcement ratios are evaluated. In addition, a parametric study is carried out on continuous high-strength concrete beams with practical dimensions to examine the effect of tensile reinforcement on the degree of moment redistribution. The analysis shows that the tensile reinforcement in continuous high-strength concrete beams affects significantly the member behavior, namely, the flexural cracking stiffness, flexural ductility, neutral axis depth and redistribution of moments. It is also found that the relation between the tensile reinforcement ratios at critical negative and positive moment regions has great influence on the moment redistribution, while the importance of this factor is neglected in various codes.

• Journal of the Society of Naval Architects of Korea
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• v.53 no.4
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• pp.307-314
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• 2016

Energy flow analysis(EFA) is a representative method that can predict the statistical energetics of structures at high frequencies. Generally, as the frequency increases, the shear distortion and rotatory inertia effects in the out-of-plane motion of beams or plates become important. Therefore, to predict the out-of-plane energetics of coupled structures in the high frequency range, the energy flow analyses of Timoshenko beam and Mindlin plate are required. Unlike the energy flow model of Kirchhoff plate, the energy flow model of Mindlin plate is composed of three kinds of energy governing equations(out-of-plane shear wave, bending dominant flexural wave, and shear dominant flexural wave). This paper performed the energy flow finite element analysis(EFFEA) of coplanar coupled Mindlin plates. For EFFEA of coplanar coupled Mindlin plates, the energy flow finite element formulation of out-of-plane energetics in the Mindlin plate was performed. The general EFFEA program was implemented by MATLAB® language. For the verification of EFFEA of Mindlin plate, the various numerical applications were done successfully.

• Transactions of the Korean Society for Noise and Vibration Engineering
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• v.25 no.7
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• pp.510-517
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• 2015

The free vibration and stability analysis of a spinning composite shaft modelled as a thin-walled closed beam is performed for several design parameters, such as ply angle, aspect ratio, and spin speed. The governing equations of spinning shafts based on the Timoshenko beam theory are derived via Hamilton's variational principle. Coriolis acceleration and anisotropy of constituent materials are incorporated in the derivation. The equations of motion are then transformed to the standard form of an eigenvalue problem for free vibration and stability analysis. Analytical results both for uniform circular cylindrical shaft and rectangular cross-section shaft are obtained by using extended Galerkin method, and the results are compared with those from FEM ANSYS analysis for a verification.

• Journal of the Korean Society for Precision Engineering
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• v.32 no.2
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• pp.117-125
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• 2015

This paper presents a dynamic modeling method for the indeterminate spindle-bearing system supported by multiple bearings of different types. A spindle-bearing system supported by ball and cylindrical roller bearings is considered. The de Mul's bearing model is extended for calculating ball and cylindrical roller bearing stiffness matrices with inclusion of centrifugal force and gyroscopic moment. The dependence between spindle shaft reaction forces and bearing stiffness is effectively resolved using an iterative approach. The spindle rotor dynamics is established with the Timoshenko beam theory based finite elements. The spindle reaction forces, bearings stiffness and spindle natural frequencies are obtained with taking into account spindle radial load, ball bearing axial preload and rotational speed effects. The developed method is verified by comparing the simulation results with those from a commercial program.