• Title/Summary/Keyword: elastic foundation beam method

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Influence of Partial Elastic Foundations on Dynamic Stability of a Cantilevered Timoshenko Beam with a Tip Mass under a follower force (끝단 질량을 갖고 종동력을 받는 외팔 Timoshenko 보의 동적안정성에 미치는 부분 탄성기초의 영향)

  • Shin, Kwang-Bok;Kim, Hyo-Jun;Ryu, Bong-Jo
    • Journal of the Korean Society for Precision Engineering
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    • v.22 no.10 s.175
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    • pp.65-71
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    • 2005
  • This paper presents the dynamic stability of a cantilevered Timoshenko beam on partial elastic foundations subjected to a follower force. The beam with a tip concentrated mass is assumed to be a Timoshenko beam taking into account its rotary inertia and shear deformation. Governing equations are derived by extended Hamilton's principle, and finite element method is applied to solve the discretized equation. Critical follower force depending on the attachment ratios of partial elastic foundations, rotary inertia of the beam and magnitude and rotary inertia of the tip mass is fully investigated.

Assessing the effect of temperature-dependent properties on the dynamic behavior of FG porous beams rested on variable elastic foundation

  • Abdeljalil Meksi;Mohamed Sekkal;Rabbab Bachir Bouiadjra;Samir Benyoucef;Abdelouahed Tounsi
    • Structural Engineering and Mechanics
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    • v.85 no.6
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    • pp.717-728
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    • 2023
  • The effect of temperature dependent material properties on the free vibration of FG porous beams is investigated in the present paper. A quasi-3D shear deformation solution is used involves only three unknown function. The mechanical properties which are considered to be temperature-dependent as well as the porosity distributions are assumed to gradually change along the thickness direction according to defined law. The beam is supposed to be simply supported and lying on variable elastic foundation. The differential equation system governing the free vibration behavior of porous beams is derived based on the Hamilton principle. Navier's method for simply supported systems is then used to determine and compute the frequencies of FG porous beam. The results of the present formulation are validated by comparing with those available literatures. Finally, the effects of several parameters such as porosity distribution and the parameters of variable elastic foundation on the free vibration behavior of temperature-dependent FG beams are presented and discussed in detail.

Bending of steel fibers on partly supported elastic foundation

  • Hu, Xiao Dong;Day, Robert;Dux, Peter
    • Structural Engineering and Mechanics
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    • v.12 no.6
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    • pp.657-668
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    • 2001
  • Fiber reinforced cementitious composites are nowadays widely applied in civil engineering. The postcracking performance of this material depends on the interaction between a steel fiber, which is obliquely across a crack, and its surrounding matrix. While the partly debonded steel fiber is subjected to pulling out from the matrix and simultaneously subjected to transverse force, it may be modelled as a Bernoulli-Euler beam partly supported on an elastic foundation with non-linearly varying modulus. The fiber bridging the crack may be cut into two parts to simplify the problem (Leung and Li 1992). To obtain the transverse displacement at the cut end of the fiber (Fig. 1), it is convenient to directly solve the corresponding differential equation. At the first glance, it is a classical beam on foundation problem. However, the differential equation is not analytically solvable due to the non-linear distribution of the foundation stiffness. Moreover, since the second order deformation effect is included, the boundary conditions become complex and hence conventional numerical tools such as the spline or difference methods may not be sufficient. In this study, moment equilibrium is the basis for formulation of the fundamental differential equation for the beam (Timoshenko 1956). For the cantilever part of the beam, direct integration is performed. For the non-linearly supported part, a transformation is carried out to reduce the higher order differential equation into one order simultaneous equations. The Runge-Kutta technique is employed for the solution within the boundary domain. Finally, multi-dimensional optimization approaches are carefully tested and applied to find the boundary values that are of interest. The numerical solution procedure is demonstrated to be stable and convergent.

Reliability analysis-based conjugate map of beams reinforced by ZnO nanoparticles using sinusoidal shear deformation theory

  • Keshtegar, Behrooz;Kolahchi, Reza
    • Steel and Composite Structures
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    • v.28 no.2
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    • pp.195-207
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    • 2018
  • First-order reliability method (FORM) is enhanced based on the search direction using relaxed conjugate reliability (RCR) approach for the embedded nanocomposite beam under buckling failure mode. The RCR method is formulated using discrete conjugate map with a limited scalar factor. A dynamical relaxed factor is proposed to control instability of proposed RCR, which is adjusted using sufficient descent condition. The characteristic of equivalent materials for nanocomposite beam are obtained by micro-electro-mechanical model. The probabilistic model of nanocomposite beam is simulated using the sinusoidal shear deformation theory (SSDT). The beam is subjected to external applied voltage in thickness direction and the surrounding elastic medium is modeled by Pasternak foundation. The governing equations are derived in terms of energy method and Hamilton's principal. Using exact solution, the implicit buckling limit state function of nanocomposite beam is proposed, which is involved various random variables including thickness of beam, length of beam, spring constant of foundation, shear constant of foundation, applied voltage, and volume fraction of ZnO nanoparticles in polymer. The robustness, accuracy and efficiency of proposed RCR method are evaluated for this engineering structural reliability problem. The results demonstrate that proposed RCR method is more accurate and robust than the excising reliability methods-based FORM. The volume fraction of ZnO nanoparticles and the applied voltage are the sensitive variables on the reliable levels of the nanocomposite beams.

Nonlinear frequency analysis of beams resting on elastic foundation using max-min approach

  • Bayat, Mahmoud;Bayat, Mahdi;Kia, Mehdi;Ahmadi, Hamid Reza;Pakar, Iman
    • Geomechanics and Engineering
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    • v.16 no.4
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    • pp.355-361
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    • 2018
  • In this paper, nonlinear vibration of Euler-Bernoulli beams resting on linear elastic foundation is studied. It has been tried to prepare a semi-analytical solution for whole domain of vibration. Only one iteration lead us to high accurate solution. The effects of linear elastic foundation on the response of the beam vibration are considered and studied. The effects of important parameters on the ratio of nonlinear to linear frequency of the system are studied. The results are compared with numerical solution using Runge-Kutta $4^{th}$ technique. It has been shown that the Max-Min approach can be easily extended in nonlinear partial differential equations.

Derivation of Exact Dynamic Stiffness Matrix of a Beam-Column Element on Elastic Foundation (균일하게 탄성지지된 보-기둥요소의 엄밀한 동적강성행렬 유도)

  • 김문영;윤희택;곽태영
    • Journal of the Computational Structural Engineering Institute of Korea
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    • v.15 no.3
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    • pp.463-469
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    • 2002
  • The governing equation and force-displacement rotations of a beam-column element on elastic foundation we derived based on variational approach of total potential energy. An exact static and dynamic 4×4 element stiffness matrix of the beam-column element is established via a generalized lineal-eigenvalue problem by introducing 4 displacement parameters and a system of linear algebraic equations with complex matrices. The structure stiffness matrix is established by the conventional direct stiffness method. In addition the F. E. procedure is presented by using Hermitian polynomials as shape function and evaluating the corresponding elastic and geometric stiffness and the mass matrix. In order to verify the efficiency and accuracy of the beam-column element using exact dynamic stiffness matrix, buckling loads and natural frequencies are calculated for the continuous beam structures and the results are compared with F E. solutions.

The mixed finite element for quasi-static and dynamic analysis of viscoelastic circular beams

  • Kadioglu, Fethi;Akoz, A. Yalcin
    • Structural Engineering and Mechanics
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    • v.15 no.6
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    • pp.735-752
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    • 2003
  • The quasi-static and dynamic responses of a linear viscoelastic circular beam on Winkler foundation are studied numerically by using the mixed finite element method in transformed Laplace-Carson space. This element VCR12 has 12 independent variables. The solution is obtained in transformed space and Schapery, Dubner, Durbin and Maximum Degree of Precision (MDOP) transform techniques are employed for numerical inversion. The performance of the method is presented by several quasi-static and dynamic example problems.

Post-buckling responses of elastoplastic FGM beams on nonlinear elastic foundation

  • Trinh, Thanh-Huong;Nguyen, Dinh-Kien;Gan, Buntara S.;Alexandrov, S.
    • Structural Engineering and Mechanics
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    • v.58 no.3
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    • pp.515-532
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    • 2016
  • The elastoplastic response of functionally graded material (FGM) beams resting on a nonlinear elastic foundation to an eccentric axial load is investigated by using the finite element method. The FGM is assumed to be formed from ceramic and metal phases with their volume fraction vary in the thickness direction by a power-law function. A bilinear elastoplastic behavior is assumed for the metallic phase, and the effective elastoplastic properties of the FGM are evaluated by Tamura-Tomota-Ozawa (TTO) model. Based on the classical beam theory, a nonlinear finite beam element taking the shift in the neutral axis position into account is formulated and employed in the investigation. An incremental-iterative procedure in combination with the arc-length control method is employed in computing the equilibrium paths of the beams. The validation of the formulated element is confirmed by comparing the equilibrium paths obtained by using the present element and the one available in the literature. The numerical results show that the elastoplastic post-buckling of the FGM beams is unstable, and the post-buckling strength is higher for the beams associated with a higher ceramic content. Different from homogeneous beams, yielding in the FGM beam occurs in the layer near the ceramic layer before in the layer near metal surface. A parametric study is carried out to highlight the effect of the material distribution, foundation support and eccentric ratio on the elastoplastic response of the beams.

Deformation analysis of a geocell mattress using a decoupled iterative method

  • Zhang, Ling;Zhao, Minghua;Zhao, Heng
    • Structural Engineering and Mechanics
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    • v.46 no.6
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    • pp.775-790
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    • 2013
  • Deformation analysis is a major concern in many geotechnical applications. In this paper, the deformation behavior of a geocell mattress subjected to symmetric loads was studied. The mattress was idealized as an elastic foundation beam. The horizontal beam-soil interfacial shear resistances at the beam top and bottom sides were taken into account by assuming the resistances to be linear with the relative horizontal displacements. A decoupled iterative method was employed to solve the differential displacement equations derived from the force analysis of a beam element and to obtain the solutions for the deformations and internal forces of the geocell reinforcement. The validity of the present solutions was verified by the existing finite element method and power-series solutions.

Free Vibrations of Compressive Members Resting on Linear Elastic Foundation (선형 탄성지반 위에 놓인 압축부재의 자유진동)

  • 이병구;이광범;모정만;신성철
    • Magazine of the Korean Society of Agricultural Engineers
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    • v.42 no.6
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    • pp.122-129
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    • 2000
  • The purpose of this study is to investigate both the fundamental and some higher natural frequencies and mode shapes of compressive members resting on the linear elastic foundation. The model of compressive member is based on the classical Bernoulli-Euler beam theory. The differential equation governing free vibrations of such members subjected to an axial load is derived and solved numerically for calculating the natural frequencies and mode shapes. The Improved Euler method is used to integrate the differential equation and the Determinant Search method combined with the Regula-Falsi method to determine the natural frequencies, respectively. In numerical examples, the hinged-hinged, hinged-clamped, clamped-hinged and clamped-clamped end constraints are considered. The convergence analysis is conducted for determining the available step size in the Improved Euler method. The validation of theories developed herein is also conducted by comparing the numerical results between this study and SAP 90. The non-dimensional frequency parameters are presented as the non-dimensional system parameters: section ratio, modulus parameter and load parameter. Also typical mode shapes are presented.

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