• Structural Engineering and Mechanics
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• v.72 no.5
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• pp.617-629
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• 2019

Based on the finite element method of traditional straight Euler-Bernoulli beams and the coupled relations between linear displacement and angular displacement of a pre-twisted Euler-Bernoulli beam, the shape functions and stiffness matrix are deduced. Firstly, the stiffness of pre-twisted Euler-Bernoulli beam is developed based on the traditional straight Euler-Bernoulli beam. Then, a new finite element model is proposed based on the displacement general solution of a pre-twisted Euler-Bernoulli beam. Finally, comparison analyses are made among the proposed Euler-Bernoulli model, the new numerical model based on displacement general solution and the ANSYS solution by Beam188 element based on infinite approach. The results show that developed numerical models are available for the pre-twisted Euler-Bernoulli beam, and which provide more accurate finite element model for the numerical analysis. The effects of pre-twisted angle and flexural stiffness ratio on the mechanical property are investigated.

• Coupled systems mechanics
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• v.11 no.2
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• pp.167-198
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• 2022

In this paper we deal with classical instability problems of heterogeneous Euler beam under conservative loading. It is chosen as the model problem to systematically present several possible solution methods from simplest deterministic to more complex stochastic approach, both of which that can handle more complex engineering problems. We first present classical analytic solution along with rigorous definition of the classical Euler buckling problem starting from homogeneous beam with either simplified linearized theory or the most general geometrically exact beam theory. We then present the numerical solution to this problem by using reduced model constructed by discrete approximation based upon the weak form of the instability problem featuring von Karman (virtual) strain combined with the finite element method. We explain how such numerical approach can easily be adapted to solving instability problems much more complex than classical Euler's beam and in particular for heterogeneous beam, where analytic solution is not readily available. We finally present the stochastic approach making use of the Duffing oscillator, as the corresponding reduced model for heterogeneous Euler's beam within the dynamics framework. We show that such an approach allows computing probability density function quantifying all possible solutions to this instability problem. We conclude that increased computational cost of the stochastic framework is more than compensated by its ability to take into account beam material heterogeneities described in terms of fast oscillating stochastic process, which is typical of time evolution of internal variables describing plasticity and damage.

• Structural Engineering and Mechanics
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• v.64 no.2
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• pp.205-212
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• 2017

Based on the coupled elastic bending deformation features and relationships between the internal force and deformation of pre-twisted Euler beam, the generalized strain, the equivalent constitutive equation and the equilibrium equation of pre-twisted Euler beam are developed. Based on the properties of the dual-antisymmetric matrix, the general solution of pre-twisted Euler beam is obtained. By comparison with ANSYS solution by using straight Beam-188 element based on infinite approach strategy, the results show that the developed method is available for pre-twisted Euler beam and also provide an accuracy displacement interpolation function for the subsequent finite element analysis. The effect of pre-twisted angle on the mechanical property has been investigated.

• Journal of Mechanical Science and Technology
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• v.20 no.4
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• pp.467-472
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• 2006

A study of the natural vibrations of beam resting on elastic foundation with finite number of transverse open cracks is presented. Frequency equations are derived for beams with different end restraints. Euler-Bernoulli beam on Pasternak foundation and Euler-Bernoulli beam on Pasternak foundation are investigated. The cracks are modeled by massless substitute spring. The effects of the crack location, size and its number and the foundation constants, on the natural frequencies of the beam, are investigated.

• Coupled systems mechanics
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• v.7 no.5
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• pp.635-647
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• 2018

Dynamic problems arising from the Euler-Bernoulli beam model with inhomogeneous elastic properties are considered. The method of Green's function and perturbation theory are employed to find the deflection in the beam correct to the first-order. Eigenvalue problems appearing from transverse vibrations of inhomogeneous beams in linear and nonlinear cases are also discussed.

• Structural Engineering and Mechanics
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• v.69 no.5
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• pp.479-486
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• 2019

Based on the displacement general solution of a pre-twisted Euler-Bernoulli beam, the shape function and stiffness matrix are deduced, and a new finite element model is proposed. Comparison analyses are made between the new proposed numerical model based on displacement general solution and the ANSYS solution by Beam188 element based on infinite approach. The results show that developed numerical model is available for the pre-twisted Euler-Bernoulli beam, and that also provide an accuracy finite element model for the numerical analysis. The effects of pre-twisted angle and flexural stiffness ratio on the mechanical property are also investigated.

• Advances in nano research
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• v.7 no.2
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• pp.99-108
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• 2019

Higher-order theories are very important to investigate the mechanical properties and behaviors of nanoscale structures. In this study, a free vibration behavior of SiNW resting on elastic foundation is investigated via Eringen's nonlocal elasticity theory. Silicon Nanowire (SiNW) is modeled as simply supported both ends and clamped-free Euler-Bernoulli beam. Pasternak two-parameter elastic foundation model is used as foundation. Finite element formulation is obtained nonlocal Euler-Bernoulli beam theory. First, shape function of the Euler-Bernoulli beam is gained and then Galerkin weighted residual method is applied to the governing equations to obtain the stiffness and mass matrices including the foundation parameters and small scale parameter. Frequency values of SiNW is examined according to foundation and small scale parameters and the results are given by tables and graphs. The effects of small scale parameter, boundary conditions, foundation parameters on frequencies are investigated.

• Journal of the Korean Mathematical Society
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• v.42 no.6
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• pp.1137-1152
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• 2005

In this article we prove the existence of the solution to the mixed problem for Euler-Bernoulli beam equation with memory condition at the boundary and we study the asymptotic behavior of the corresponding solutions. We proved that the energy decay with the same rate of decay of the relaxation function, that is, the energy decays exponentially when the relaxation function decay exponentially and polynomially when the relaxation function decay polynomially.

• Journal of the Korean Mathematical Society
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• v.49 no.5
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• pp.947-964
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• 2012

In this paper, the memory type boundary stabilization for an Euler-Bernoulli beam with one end fixed and control at the other end is considered. We prove the existence of solutions using the Galerkin method and then investigate the exponential stability of solutions by using multiplier technique.

• Structural Engineering and Mechanics
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• v.51 no.5
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• pp.773-792
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• 2014

In this paper, we study the inverse mode shape problem for an Euler-Bernoulli beam, using an analytical approach. The mass and stiffness variations are determined for a beam, having various boundary conditions, which has a prescribed polynomial second mode shape with an internal node. It is found that physically feasible rectangular cross-section beams which satisfy the inverse problem exist for a variety of boundary conditions. The effect of the location of the internal node on the mass and stiffness variations and on the deflection of the beam is studied. The derived functions are used to verify the p-version finite element code, for the cantilever boundary condition. The paper also presents the bounds on the location of the internal node, for a valid mass and stiffness variation, for any given boundary condition. The derived property variations, corresponding to a given mode shape and boundary condition, also provides a simple closed-form solution for a class of non-uniform Euler-Bernoulli beams. These closed-form solutions can also be used to check optimization algorithms proposed for modal tailoring.