• 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.

• Structural Engineering and Mechanics
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• v.44 no.5
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• pp.681-704
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• 2012

The dynamic response of Euler-Bernoulli beams to resonant harmonic moving loads is analysed. The non-dimensional form of the motion equation of a beam crossed by a moving harmonic load is solved through a perturbation technique based on a two-scale temporal expansion, which permits a straightforward interpretation of the analytical solution. The dynamic response is expressed through a harmonic function slowly modulated in time, and the maximum dynamic response is identified with the maximum of the slow-varying amplitude. In case of ideal Euler-Bernoulli beams with elastic rotational springs at the support points, starting from analytical expressions for eigenfunctions, closed form solutions for the time-history of the dynamic response and for its maximum value are provided. Two dynamic factors are discussed: the Dynamic Amplification Factor, function of the non-dimensional speed parameter and of the structural damping ratio, and the Transition Deamplification Factor, function of the sole ratio between the two non-dimensional parameters. The influence of the involved parameters on the dynamic amplification is discussed within a general framework. The proposed procedure appears effective also in assessing the maximum response of real bridges characterized by numerically-estimated mode shapes, without requiring burdensome step-by-step dynamic analyses.

• 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.

• Transactions of the Korean Society for Noise and Vibration Engineering
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• v.15 no.11 s.104
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• pp.1318-1323
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• 2005

In this paper, the vibration analysis for the Euler-Bernoulli complete and truncate wedge beams by differential Transformation method(DTM) was investigated. The governing differential equation of the Euler-Bernoulli complete and truncate wedge beams with regular singularity is derived and verified. The concepts of DTM were briefly introduced. Numerical calculations are carried out and compared with previous published results. The usefulness and the application of DTM are discussed.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 2005.11a
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• pp.507-512
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• 2005

This paper investigated the vibration analysis fer the Euler-Bernoulli complete and truncate wedge beams by Differential Transformation Method(DTM). The governing differential equation of the Euler-Bernoulli complete and truncate wedge beams with regular singularity is derived and verified. The concepts of DTM were briefly introduced. Numerical calculations are carried out and compared with previous published results. The usefulness and the application of DTM are discussed.

• 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.42 no.1
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• pp.95-119
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• 2012

In this paper, forced vibration differential equations of motion of Euler-Bernoulli beams with different boundary conditions and dynamic loads are solved using differential transform method (DTM), analytical solutions. Then, the modal deflections of these beams are obtained. The calculated modal deflections using DTM are represented in tables and depicted in graphs and compared with the results of the analytical solutions where a very good agreement is observed.

• Smart Structures and Systems
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• v.4 no.1
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• pp.67-84
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• 2008

The present contribution is concerned with the static and dynamic stability of a piezo-laminated Bernoulli-Euler beam subjected to an axial compressive force. Recently, an inconsistent derivation of the equations of motions of such a smart structural system has been presented in the literature, where it has been claimed, that an axial piezoelectric actuation can be used to control its stability. The main scope of the present paper is to show that this unfortunately is impossible. We present a consistent theory for composite beams in plane bending. Using an exact description of the kinematics of the beam axis, together with the Bernoulli-Euler assumptions, we obtain a single-layer theory capable of taking into account the effects of piezoelectric actuation and buckling. The assumption of an inextensible beam axis, which is frequently used in the literature, is discussed afterwards. We show that the cited inconsistent beam model is due to inadmissible mixing of the assumptions of an inextensible beam axis and a vanishing axial displacement, leading to the erroneous result that the stability might be enhanced by an axial piezoelectric actuation. Our analytical formulations for simply supported Bernoulli-Euler type beams are verified by means of three-dimensional finite element computations performed with ABAQUS.

• Steel and Composite Structures
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• v.15 no.4
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• pp.439-449
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• 2013

In this study simply supported nonlinear Euler-Bernoulli beams resting on linear elastic foundation and subjected to the axial loads is investigated. A new kind of analytical technique for a non-linear problem called He's Energy Balance Method (EBM) is used to obtain the analytical solution for non-linear vibration behavior of the problem. Analytical expressions for geometrically non-linear vibration of Euler-Bernoulli beams resting on linear elastic foundation and subjected to the axial loads are provided. The effect of vibration amplitude on the non-linear frequency and buckling load is discussed. The variation of different parameter to the nonlinear frequency is considered completely in this study. The nonlinear vibration equation is analyzed numerically using Runge-Kutta $4^{th}$ technique. Comparison of Energy Balance Method (EBM) with Runge-Kutta $4^{th}$ leads to highly accurate solutions.

• Magazine of the Korean Society of Agricultural Engineers
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• v.41 no.1
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• pp.106-114
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• 1999

This paper explores the geometrical non-linear behavior of the simply supported tapered beams subject to the trapezoidal distributed load and end moments. In order to apply the Bernoulli -Euler beam theory to this tapered beam, the bending moment equation on any point of the elastical is obtained by the redistribution of trapezoidal distributed load. On the basis of the bending moment equation and the BErnoulli-Euler beam theory, the differential equations governging the elastical of such beams are derived and solved numerically by using the Runge-Jutta method and the trial and error method. The three kinds of tapered beams (i.e. width, depth and square tapers) are analyzed in this study. The numerical results of non-linear behavior obtained in this study from the simply supported tapered beams are appeared to be quite well according to the results from the reference . As the numerical results, the elastica, the stress resultants and the load-displacement curves are given in the figures.