• Proceedings of the KSME Conference
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• 2000.04a
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• pp.351-356
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• 2000

Various transition elements are generally used for the effective analysis of a complicated mechanical structure. In this paper, a solid-to-beam transition finite element which connects a continuum element and a $c^1-continuity$ beam element each other is proposed. The shape functions of the transition finite elements, which a 8-noded hexahedral solid element fur 3D analysis and a 4-noded quadrilateral plane element fur 2D analysis are connected to a Euler's beam element, are explicitely formulated. In order to show the effectiveness and convergence characteristics of the proposed transition elements. numerical tests are performed for various examples and their results are compared with those obtained by other methods. As the result of this study. following conclusions are obtained: (1)The proposed transition finite elements show the monotonic convergence characteristics because of having used the compatible displacement folds. (2)As being used the transition element in the finite element analysis, the finite element modelings are more convenient and the analysis results are more accurate because of the formulation characteristies of the Euler's beam element.

• 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.

• Structural Engineering and Mechanics
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• v.61 no.6
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• pp.765-773
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• 2017

The quadratic B-spline finite element method yields mass and stiffness matrices which are half the size of matrices obtained by the conventional finite element method. We solve the free vibration problem of a rotating Rayleigh beam using the quadratic B-spline finite element method. Rayleigh beam theory includes the rotary inertia effects in addition to the Euler-Bernoulli theory assumptions and presents a good mathematical model for rotating beams. Galerkin's approach is used to obtain the weak form which yields a system of symmetric matrices. Results obtained for the natural frequencies at different rotating speeds show an accurate match with the published results. A comparison with Euler-Bernoulli beam is done to decipher the variations in higher modes of the Rayleigh beam due to the slenderness ratio. The results are obtained for different values of non-uniform parameter ($\bar{n}$).

• 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.

• Proceedings of the KSR Conference
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• 2011.05a
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• pp.314-316
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• 2011

This paper develops a spectral element model for the composite beams with a surface-bonded piezoelectric layer from the governing equations of motion. The governing equations of motion are derived from Hamilton's principle by applying the Bernoulli-Euler beam theory for the bending vibration and the elementary rod theory for the longitudinal vibration of the composite beams. For the PZT layer, the Bernoulli-Euler beam theory and linear piezoelectricity theory are applied. The high accuracy of the present spectral element model is evaluated through the numerical examples by comparing with the finite element analysis results.

• Structural Engineering and Mechanics
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• v.59 no.3
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• pp.431-454
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• 2016

In this paper, the nonlinear static and free vibration analysis of Euler-Bernoulli composite beam model reinforced by functionally graded single-walled carbon nanotubes (FG-SWCNTs) with initial geometrical imperfection under uniformly distributed load using finite element method (FEM) is investigated. The governing equations of equilibrium are derived by the Hamilton's principle and von Karman type nonlinear strain-displacement relationships are employed. Also the influences of various loadings, amplitude of the waviness, UD, USFG, and SFG distributions of carbon nanotube (CNT) and different boundary conditions on the dimensionless transverse displacements and nonlinear frequency ratio are presented. It is seen that with increasing load, the displacement of USFG beam under force loads is more than for the other states. Moreover it can be seen that the nonlinear to linear natural frequency ratio decreases with increasing aspect ratio (h/L) for UD, USFG and SFG beam. Also, it is shown that at the specified value of (h/L), the natural frequency ratio increases with the increasing the values amplitude of waviness while the dimensionless nonlinear to linear maximum deflection decreases. Moreover, with considering the amplitude of waviness, the stiffness of Euler-Bernoulli beam model reinforced by FG-CNT increases. It is concluded that the R parameter increases with increasing of volume fraction while the rate of this parameter decreases. Thus one can be obtained the optimum value of FG-CNT volume fraction to prevent from resonance phenomenon.

• Advances in nano research
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• v.9 no.2
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• pp.91-104
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• 2020

The bending, stability (buckling) and vibration response of nano sized beams is presented in this study based on the Eringen's nonlocal elasticity theory in conjunction with the Euler-Bernoulli beam theory. For this purpose, the bending, buckling and vibration problem of Euler-Bernoulli nanobeams are developed and solved on the basis of nonlocal elasticity theory. The effects of various parameters such as nonlocal parameter e₀a, length of beam L, mode number n, distributed load q and cross-section on the bending, buckling and vibration behaviors of carbon nanotubes idealized as Euler-Bernoulli nanobeam is investigated. The transverse deflections, maximum transverse deflections, vibrational frequency and buckling load values of carbon nanotubes are given in tables and graphs.

• Transactions of the Korean Society for Noise and Vibration Engineering
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• v.18 no.11
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• pp.1157-1169
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• 2008

This paper presents spectral element formulation which approximates Lamb wave propagation by PZT transducers bonded on a thin plate. A two layer beam model under 2-D plane strain condition is introduced to simulate high-frequency dynamic responses induced by a piezoelectric (PZT) layer rigidly bonded on a base plate. Mindlin-Herrmann and Timoshenko beam theories are employed to represent the first symmetric and anti-symmetric Lamb wave modes on a base plate, respectively. The Euler-Bernoulli beam theory and 1-D linear piezoelectricity are used to model the electro-mechanical behavior of a PZT layer. The equations of motions of a two layer beam model are derived through Hamilton's principle. The necessary boundary conditions associated with the electro-mechanical properties of a PZT layer are formulated in the context of dual functions of a PZT layer as an actuator and a sensor. General spectral shape functions of response field and the associated boundary conditions are obtained through equations of motions converted into frequency domain. Detailed spectrum element formulation for composing the dynamic stiffness matrix of a two layer beam model is presented as well. The validity of the proposed spectral element is demonstrated through numerical examples.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 2006.11a
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• pp.844-847
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• 2006

In this paper, the forced vibration of damped composite beam with I-type section was analyzed. The damping material was assumed to have complex Young's modulus. Damped composite beam structure could be modeled using equivalent beam elements with less D.O.F. rather than solid elements. Finite element method for 6 D.O.F. equivalent beam element was formulated and programmed using complex values. The results of frequency responses revealed good agreement with those of NASTRAN in both Euler beam model and Timoshenko beam model.

• Structural Engineering and Mechanics
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• v.23 no.1
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• pp.15-27
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• 2006

A locally varying temperature field or a mixture of two or more different materials can cause local variation of elasticity properties of a beam. In this paper, a new Euler-Bernoulli beam element with varying Young's modulus along its longitudinal axis is presented. The influence of axial forces according to the linearized 2nd order beam theory is considered, as well. The stiffness matrix of this element contains the transfer constants which depend on Young's modulus variation and on axial forces. Occurrence of the polynomial variation of Young's modulus has been assumed. Such approach can be also used for smooth local variation of Young's modulus. The critical loads of the straight slender columns were studied using the new beam element. The influence of position of the local Young's modulus variation and its type (such as linear, quadratic, etc.) on the critical load value and rate of convergence was investigated. The obtained results based on the new beam element were compared with ANSYS solutions, where the number of elements gradually increased. Our results show significant influence of the locally varying Young's modulus on the critical load value and the convergence rate.