• Advances in aircraft and spacecraft science
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• 제11권1호
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• pp.55-81
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• 2024

This paper presents the exact solutions and closed forms for of nonlinear stability and vibration behaviors of straight and curved beams with nonlinear viscoelastic boundary conditions, for the first time. The mathematical formulations of the beam are expressed based on Euler-Bernoulli beam theory with the von Karman nonlinearity to include the mid-plane stretching. The classical boundary conditions are replaced by nonlinear viscoelastic boundary conditions on both sides, that are presented by three elements (i.e., linear spring, nonlinear spring, and nonlinear damper). The nonlinear integro-differential equation of buckling problem subjected to nonlinear nonhomogeneous boundary conditions is derived and exactly solved to compute nonlinear static response and critical buckling load. The vibration problem is converted to nonlinear eigenvalue problem and solved analytically to calculate the natural frequencies and to predict the corresponding mode shapes. Parametric studies are carried out to depict the effects of nonlinear boundary conditions and amplitude of initial curvature on nonlinear static response and vibration behaviors of curved beam. Numerical results show that the nonlinear boundary conditions have significant effects on the critical buckling load, nonlinear buckling response and natural frequencies of the curved beam. The proposed model can be exploited in analysis of macrosystem (airfoil, flappers and wings) and microsystem (MEMS, nanosensor and nanoactuators).

• Structural Engineering and Mechanics
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• 제37권2호
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• pp.149-162
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• 2011

The aim of this research is a comprehensive review and evaluation of beam theories resting on elastic foundations that used to model mode-I delamination in multidirectional laminated composite by DCB specimen. A compliance based approach is used to calculate critical strain energy release rate (SERR). Two well-known beam theories, i.e. Euler-Bernoulli (EB) and Timoshenko beams (TB), on Winkler and Pasternak elastic foundations (WEF and PEF) are considered. In each case, a closed-form solution is presented for compliance versus crack length, effective material properties and geometrical dimensions. Effective flexural modulus ($E_{fx}$) and out-of-plane extensional stiffness ($E_z$) are used in all models instead of transversely isotropic assumption in composite laminates. Eventually, the analytical solutions are compared with experimental results available in the literature for unidirectional ($[0^{\circ}]_6$) and antisymmetric angle-ply ($[{\pm}30^{\circ}]_5$, and $[{\pm}45^{\circ}]_5$) lay-ups. TB on WEF is a simple model that predicts more accurate results for compliance and SERR in unidirectional laminates in comparison to other models. TB on PEF, in accordance with Williams (1989) assumptions, is too stiff for unidirectional DCB specimens, whereas in angle-ply DCB specimens it gives more reliable results. That it shows the effects of transverse shear deformation and root rotation on SERR value in composite DCB specimens.

• Smart Structures and Systems
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• 제18권2호
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• pp.355-374
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• 2016

This paper presents and compares a one-dimensional (1D) bending theory for piezoelectric thin beam-type structures with resistive-inductive electrodes to ANSYS$^{(R)}$ three-dimensional (3D) finite element (FE) analysis. In particular, the lateral deflections and vibrations of slender piezoelectric beams are considered. The peculiarity of the piezoelectric beam model is the modeling of electrodes in such a manner that is does not fulfill the equipotential area condition. The case of ideal, perfectly conductive electrodes is a special case of our 1D model. Two-coupled partial differential equations are obtained for the lateral deflection and for the voltage distribution along the electrodes: the first one is an extended Bernoulli-Euler beam equation (second-order in time, forth order in space) and the second one the so-called Telegrapher's equation (second-order in time and space). Analytical results of our theory are validated by 3D electromechanically coupled FE simulations with ANSYS$^{(R)}$. A clamped-hinged beam is considered with various types of electrodes for the piezoelectric layers, which can be either resistive and/or inductive. A natural frequency analysis as well as quasi-static and dynamic simulations are performed. A good agreement between the extended beam theory and the FE results is found. Finally, the practical relevance of this type of electrodes is shown. It is found that the damping capability of properly tuned resistive or resistive-inductive electrodes exceeds the damping performance of beams, where the electrodes are simply linked to an optimized impedance.

• Structural Engineering and Mechanics
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• 제31권4호
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• pp.453-475
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• 2009

The literature regarding the free vibration analysis of Bernoulli-Euler and Timoshenko beams on elastic soil is plenty, but the free vibration analysis of Reddy-Bickford beams on elastic soil with/without axial force effect using the Differential Transform Method (DTM) has not been investigated by any of the studies in open literature so far. In this study, the free vibration analysis of axially loaded Reddy-Bickford beam on elastic soil is carried out by using DTM. The model has six degrees of freedom at the two ends, one transverse displacement and two rotations, and the end forces are a shear force and two end moments in this study. The governing differential equations of motion of the rectangular beam in free vibration are derived using Hamilton's principle and considering rotatory inertia. Parameters for the relative stiffness, stiffness ratio and nondimensionalized multiplication factor for the axial compressive force are incorporated into the equations of motion in order to investigate their effects on the natural frequencies. At first, the terms are found directly from the analytical solutions of the differential equations that describe the deformations of the cross-section according to the high-order theory. After the analytical solution, an efficient and easy mathematical technique called DTM is used to solve the governing differential equations of the motion. The calculated natural frequencies of one end fixed and the other end simply supported Reddy-Bickford beam on elastic soil using DTM are tabulated in several tables and figures and are compared with the results of the analytical solution where a very good agreement is observed and the mode shapes are presented in graphs.

• 한국전산구조공학회논문집
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• 제28권5호
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• pp.459-465
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• 2015

항공기는 목적에 따라서 민간 항공기, 무인항공기, 전투기, 헬리콥터 등 다양한 항공기가 존재한다. 이 각각의 항공기는 특정한 목적에 맞게 형상 및 설계가 된다. 특히 항공기 개발과정에서 중요한 해석과정 중 하나가 구조해석이다. 하지만 항공기 구조가 복잡해지고 3차원 모델로 구조해석을 하게 되면 시간과 비용이 크게 증가하게 된다. 따라서 해석 효율성을 위해서 1차원 등가 보나 2차원 평면 응력 조건을 이용하여 실제 구조를 보다 간단하게 모델링한다. 하지만 이런 모델링은 실제 구조와 차이가 있으므로 실제 구조를 잘 반영할 수 있는 적절한 모델링이 필요하다. 따라서 구조형태에 따라서 1차원 등가 보와 2차원 평면응력 조건을 적절하게 선택하여야 한다. 본 논문에서는 EDISON에 업로드 된 구조해석 프로그램을 이용하여 1차원 구조해석과 2차원 구조해석을 검증하고 구조형태에 따라서 1차원 해석과 2차원 해석을 각각 3차원 MSC NASTRAN 구조해석과 비교하여 적절한 해석방법을 찾고자 한다. 비교결과 길이 대 높이 비가 증가할수록 1차원 해석과 3차원 해석의 오차가 급격히 줄어들었으며 이 비율이 18보다 증가하였을 때는 1차원 해석이 2차원 해석보다 3차원 해석의 결과와 일치하였다.

• 대한기계학회논문집A
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• 제40권4호
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• pp.381-387
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• 2016

본 논문에서는 생브낭의 원리가 근본적으로 구조물의 거동 예측에 잠재적으로 적용되어 있다는 점에 착안하여, 응력해석에 국한되어 있던 방법론을 자유진동 및 좌굴 문제 등에 적용하여 고전 보 이론의 정확도를 고차이론 수준으로 개선한다. 먼저 생브낭의 원리를 소개하고, 고전 보 이론에 의한 자유진동 그리고 좌굴해석 정식화를 진행하였다. 고전 보 이론의 변위장에 워핑함수와 섭동항을 추가하고, 합응력 등가(즉, 생브낭의 원리)를 적용하여 섭동항을 찾는다. 여기서 워핑함수들은 응력 평형방정식을 통하여 계산하였으며, 이 워핑함수들은 추가된 섭동항에 의하여 보의 응력 평형을 만족하게 된다. 제안된 방법론을 외팔보와 단순지지 보 문제에 적용하여 주파수 및 좌굴하중을 개선하였으며, 전단수정계수의 도입 없이 예측을 개선할 수 있음을 보였다.

• Steel and Composite Structures
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• 제36권2호
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• pp.143-161
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• 2020

In nanosized structures as the surface area to the bulk volume ratio increases the classical continuum mechanics approaches fails to investigate the mechanical behavior of such structures. In perforated nanobeam structures, more decrease in the bulk volume is obtained due to perforation process thus nonclassical continuum approaches should be employed for reliable investigation of the mechanical behavior these structures. This article introduces an analytical methodology to investigate the size dependent, surface energy, and perforation impacts on the nonclassical bending behavior of regularly squared cutout nanobeam structures for the first time. To do this, geometrical model for both bulk and surface characteristics is developed for regularly squared perforated nanobeams. Based on the proposed geometrical model, the nonclassical Gurtin-Murdoch surface elasticity model is adopted and modified to incorporate the surface energy effects in perforated nanobeams. To investigate the effect of shear deformation associated with cutout process, both Euler-Bernoulli and Timoshenko beams theories are developed. Mathematical model for perforated nanobeam structure including surface energy effects are derived in comprehensive procedure and nonclassical boundary conditions are presented. Closed forms for the nonclassical bending and rotational displacements are derived for both theories considering all classical and nonclassical kinematics and kinetics boundary conditions. Additionally, both uniformly distributed and concentrated loads are considered. The developed methodology is verified and compared with the available results and an excellent agreement is noticed. Both classical and nonclassical bending profiles for both thin and thick perforated nanobeams are investigated. Numerical results are obtained to illustrate effects of beam filling ratio, the number of hole rows through the cross section, surface material characteristics, beam slenderness ratio as well as the boundary and loading conditions on the non-classical bending behavior of perforated nanobeams in the presence of surface effects. It is found that, the surface residual stress has more significant effect on the bending deflection compared with the corresponding effect of the surface elasticity, Es. The obtained results are supportive for the design, analysis and manufacturing of perforated nanobeams.

• 한국전산구조공학회논문집
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• 제33권5호
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• pp.279-286
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• 2020

본 연구에서는 점근해석 및 논로컬 이론에서 요구하는 4차 이상의 고차 미분근사를 수행하기 위하여 계방정식에 혼합변분이론을 적용하여 MLS 차분법으로부터 구해지는 고차 미분근사의 정확도를 큰 폭으로 향상시킨다. 또한, MLS 차분법에 존재하는 세 가지 조건변수에 따른 고차미분근사의 정확도를 비교·분석한다. 혼합변분이론의 합응력을 후처리하여 변위의 미분을 근사할 경우 기존의 변위장 기반 계방정식의 차분 결과에 비해 미분 차수가 2차 낮아진다. 해석 범위내 절점 수가 과도하게 많거나 기저 차수가 클 경우 MLS 차분법의 영향영역 내에서 과적합(overfitting)이 발생한다. 또한 영향영역이 최적 범위 이상으로 넓어질 경우 근사의 정확도가 떨어진다. 위 내용을 사인 하중을 받는 단순지지보 수치예제로부터 확인하였다.