• Structural Engineering and Mechanics
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• 제72권4호
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• pp.491-502
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• 2019

This paper explores the analytical buckling solution of rectangular thin plates by the finite integral transform method. Although several analytical and numerical developments have been made, a benchmark analytical solution is still very few due to the mathematical complexity of solving high order partial differential equations. In solution procedure, the governing high order partial differential equation with specified boundary conditions is converted into a system of linear algebraic equations and the analytical solution is obtained classically. The primary advantage of the present method is its simplicity and generality and does not need to pre-determine the deflection function which makes the solving procedure much reasonable. Another advantage of the method is that the analytical solutions obtained converge rapidly due to utilization of the sum functions. The application of the method is extensive and can also handle moderately thick and thick elastic plates as well as bending and vibration problems. The present results are validated by extensive numerical comparison with the FEA using (ABAQUS) software and the existing analytical solutions which show satisfactory agreement.

• Structural Engineering and Mechanics
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• 제86권1호
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• pp.69-83
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• 2023

Analytical solutions to problems are crucial because they provide high-quality comparison data for assessing the accuracy of numerical solutions. Benchmark analytical solutions for the vibrations of cracked functionally graded material (FGM) plates are not available in the literature because of the high level of complexity of such solutions. On the basis of first-order shear deformation plate theory (FSDT), this study proposes analytical series solutions for the vibrations of FGM rectangular plates with side or internal cracks parallel to an edge of the plates by using Fourier cosine series and the domain decomposition technique. The distributions of FGM properties along the thickness direction are assumed to follow a simple power law. The proposed analytical series solutions are validated by performing comprehensive convergence studies on the vibration frequencies of cracked square plates with various crack lengths and under various boundary condition combinations and by performing comparisons with published results based on various plate theories and the theory of three-dimensional elasticity. The results reveal that the proposed solutions are in excellent agreement with literature results obtained using the Ritz method on the basis of FSDT. The paper also presents tabulations of the first six nondimensional frequencies of cracked rectangular Al/Al2O3 FGM plates with various aspect ratios, thickness-to-width ratios, crack lengths, and FGM power law indices under six boundary condition combinations, the tabulated frequencies can serve as benchmark data for assessing the accuracy of numerical approaches based on FSDT.

• 대한토목학회논문집
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• 제26권1A호
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• pp.117-132
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• 2006

3변 고정 1변 자유 직사각형 얇은 판에 대한 기존 해석해를 무차원식으로 유도하고 특성을 분석하였다. Timoshenko와 Woinowsky-Krieger의 방법(1959)은 변장비가 1보다 작은 경우에만 제한적으로 해가 존재하여 처짐 특성에 대한 실용적인 해가 되지 않음을 밝혔다. 굴정(堀井)와 본(本)의 방법(1968)에 수치안정을 위한 항을 추가하여 최대 150개 항까지 구성된 급수해를 구하였고 이로부터 계산한 휨 모멘트의 수렴을 분석하였다. 수정 굴정(堀井)와 본(本)의 방법은 모든 변장비에 대한 처짐 특성을 구할 수 있으나 고정단과 자유단이 접하는 교차점에서의 모멘트 계산은 자유단 경계조건을 만족하지 않으며 그 원인을 분석하였다.

• Steel and Composite Structures
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• 제30권1호
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• pp.57-67
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• 2019

This paper deals with the static analysis of axially functionally graded rectangular plates. It is assumed that the flexural rigidity of the plate varies exponentially along one of the plate's in-plane dimensions. Both an analytical approach and a numerical method are utilized to solve the problem. The analytical solution is obtained by using the Green's function method. To employ this approach, the adjoint boundary value problem is established. Then, exact solutions for deflection of the plate for different boundary conditions are found. In another way, a finite element formulation for the problem is developed. In order to demonstrate the validity of the Authors' formulation, the results obtained via both mentioned schemes are compared with each other for functionally graded plates and with results of previously published works for homogeneous plates. The effect of plate parameters on the response of the plate is also investigated. To remind the research background, a brief review on the application of Green's function method in plates' analysis and functionally graded plates is also presented.

• Structural Engineering and Mechanics
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• 제75권2호
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• pp.157-174
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• 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.

• International Journal of Aeronautical and Space Sciences
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• 제12권2호
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• pp.200-209
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• 2011

An analytical solution for rectangular laminated composite plates was obtained via a formal asymptotic method. From threedimensional static equilibrium equations, the microscopic one-dimensional and macroscopic two-dimensional equations were systematically derived by scaling of the thickness coordinate with respect to the characteristic length of the plate. The onedimensional through-the-thickness analysis was performed by applying a standard finite element method. The derived twodimensional plate equations, which take the form of recursive equations, were solved under sinusoidal loading with simplysupported boundary conditions. To demonstrate the validity and accuracy of the present method, various types of composite plates were studied, such as cross-ply, anti-symmetric angle-ply and sandwich plates. The results obtained were compared to those of the classical laminated plate theory, the first-order shear deformation theory and the three-dimensional elasticity. In the present analysis, the characteristic length of each composite was dependent upon the layup configurations, which affected the convergence rate of the method. The results shown herein are promising that it can serve as an efficient tool for the analysis and design of laminated composite plates.

• Structural Engineering and Mechanics
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• 제60권2호
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• pp.271-299
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• 2016

In this article, based on the higher-order shear deformation plate theory, buckling analysis of a rectangular plate made of functionally graded piezoelectric materials and its effective parameters are investigated. Assuming the transverse distribution of electric potential to be a combination of a parabolic and a linear function of thickness coordinate, the equilibrium equations for the buckling analysis of an FGP rectangular plate are established. In addition to the Maxwell equation, all boundary conditions including the conditions on the top and bottom surfaces of the plate for closed and open circuited are satisfied. Considering double sine solution (Navier solution) for displacement field and electric potential, an analytical solution is obtained for full simply supported boundary conditions. The accurate buckling load of FGP plate is presented for both open and closed circuit conditions. It is found that the critical buckling load for open circuit is more than that of closed circuit in all loading conditions. Furthermore, it is observed that the influence of dielectric constants on the critical buckling load is more than those of others.

• Structural Engineering and Mechanics
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• 제62권6호
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• pp.675-693
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• 2017

In this paper, based on the first-order shear deformation plate theory, buckling analysis of piezoelectric coupled transversely isotropic rectangular plates is investigated. By assuming the transverse distribution of electric potential to be a combination of a parabolic and a linear function of thickness coordinate, the equilibrium equations for buckling analysis of plate with surface bonded piezoelectric layers are established. The Maxwell's equation and all boundary conditions including the conditions on the top and bottom surfaces of the plate for closed and open circuited are satisfied. The analytical solution is obtained for Levy type of boundary conditions. The accurate buckling load of laminated plate is presented for both open and closed circuit conditions. From the numerical results it is found that, the critical buckling load for open circuit is more than that of closed circuit in all boundary and loading conditions. Furthermore, the critical buckling loads and the buckling mode number increase by increasing the thickness of piezoelectric layers for both open and closed circuit conditions.

• 대한조선학회논문집
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• 제28권1호
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• pp.169-181
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• 1991

본 연구에서는 2축방향압축력(軸方向壓縮力)을 받는 주변단순지지평판(周邊單純支持平板)의 좌굴전(挫屈前), 좌굴강도(挫屈强度), 좌굴후(挫屈後) 거동(擧動), 최종강도(最終强度) 및 최종강도후거동(最終强度後擧動)에 대한 해석해(解析解)를 도출한다. 또한 본 연구에서는 도출한 해석해(解析解)를 적용하여 판(板)의 종횡비(縱橫比)와 세장비(細長比) 및 하중성분비(荷重成分比)의 변화에 따른 시리즈해석을 수행한다. 본(本) 해석해(解析解)는 각종 수치해석(數値解析) 및 실험결과(實驗結果)에 대한 기준치(基準値)로서 널리 사용될 수 있으리라 기대된다.

• Advances in nano research
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• 제14권3호
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• pp.261-271
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• 2023

The present study follows three main goals. First, an analytical solution with high accuracy is developed to assess the effects of embedding pre-strained shape memory alloy (SMA) wires on the critical buckling temperatures of rectangular sandwich plates made of soft core and graphite fiber/epoxy (GF/EP) face sheets based on piecewise low-order shear deformation theory (PLSDT) using Brinson's model. As the second goal, this study compares the effects of SMAs on the thermal buckling of sandwich plates with those of carbon nanotubes (CNTs). The glass transition temperature is considered as a limiting factor. For each material, the effective ranges of operating temperature and thickness ratio are determined for real situations. The results indicate that depending on the geometric parameters and thermal conditions, one of the SMAs and CNTs may outperform the other. The third purpose is to study the thermal buckling of sandwich plates with advanced hybrid SMA/CNT/GF/EP composite face sheets. It is shown that in some circumstances, the co-incorporation of SMAs and CNTs leads to an astonishing enhancement in the critical buckling temperatures of sandwich plates.