On the Dynamic Response of Uniform and Non-Uniform Deep Beams to Moving Concentrated Loads.

Topics: Elementary algebra, Differential equation, Vibration Pages: 21 (4423 words) Published: February 14, 2011


B. Omolofe
Department of Mathematical Sciences, Federal University
of Technology, Akure Ondo State, Nigeria.
e-mail: babatope_omolofe@yahoo.com
Mobile phone: +2348035643019


The Dynamic response of Timoshenko beams of uniform and non-uniform cross-sections resting on elastic foundation whose rigidity is of a linear function and subjected to fast traveling concentrated loads of constant and varying magnitude is scrutinized using an elegant analytical approach. In particular, the spectral generalized Galerkin’s method in conjunction with Integral transform method is used to treat the problem of the coupled system of partial differential equations describing the transverse motion of the vibrating system. The specific aim is to compare the dynamic stability of Timoshenko beams of uniform and non-uniform cross-sections when under the actions of moving concentrated loads of constant and varying magnitudes. The closed-form solutions for both beam problems are obtained. Analytical and numerical Results show that as the foundation parameter K0 increases the response amplitudes for both beam decrease. Results also show that foundation parameter K0 produces a more noticeable effect on the deflection of a non-uniform beam than on a uniform beam when subjected to variable magnitude loads. It is equally established that, for higher values of foundation modulus, the risk of resonance is sufficiently reduced for both uniform and non-uniform Timoshenko beams resting on variable elastic foundation and under the actions of concentrated moving loads of any magnitude.

Key Words:Dynamic response, Dynamic Stability, Foundation parameter, Transverse motion, Timoshenko beam, Concentrated loads, Galerkin’s method, Resonance.


Owing to its practical relevance, technological and economical importance the problem of assessing the transverse motions of elastic structures under the actions of either concentrated or distributed loads has been given a considerable attention by researchers in the field of Engineering sciences, Mathematical Physics and Applied Mathematics, hence numerous scholarly studies have been published in this area of study in recent years [1-11].

In most of the existing practical and analytical studies involving beam-load interactions, the classical Bernoulli-Euler beam model is the model generally employed in investigating the response of beam subjected to moving loads [12]. However, it is well known that Timoshenko beam models are developed to achieve more precise description of the dynamic behaviour of beams. This is because the effects due to the rotatory inertia and the deviation of the beam cross-section after deformation are taken into account in the governing equations [13]. Traditionally, there are two types of beams one can consider when studying the response of beams subjected to moving loads. The beam whose parameters (e.g mass per unit length and the moment of inertia) do not vary along the span L of the beam this is known as a beam of uniform cross-section. This class of beams has been extensively dealt with in literatures see for example [8, 14] and the references therein. There are also Beams whose parameters vary along the span L and this type is known as beam with non-uniform cross-sections. Studies on this class of beams are very meager in literatures and even when such beam problems are considered, the approach often employed is numerical simulations, comparatively few studies concentrate on analytical developments [15].

This paper therefore assesses the transverse motions of prismatic and non-prismatic deep beams resting on elastic foundation and subjected to fast accelerating loads using a robust analytical technique. Essentially, the responses of beams of uniform and non-uniform cross sections to moving concentrated loads are...
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