Abstract
This research studies the vibration analysis of Euler–Bernoulli and Timoshenko beams utilizing the differential quadrature method (DQM) which has wide applications in the field of basic vibration of different components, for example, pillars, plates, round and hollow shells, and tanks. The free vibration of uniform and nonuniform beams laying on elastic Pasternak foundation will be studied under three sets of boundary conditions, that is, mixing between being simply upheld and fixed while utilizing the DQM. The natural frequencies and deflection values were produced through the examination of both beam types. Results show great concurrence with solutions from previous research studies. The impact of the nonuniform cross-section area on the vibration was contemplated. A comparison between the results from both beams is obtained. The focus of this work is on studying the deflection difference between both beam theories at different beam dimensions as well as showing the shape of rotation of the cross section while applying a nodal point load equation to simulate the moving load. The results were discussed and a general contemplation about the theories was developed.
Highlights
Timoshenko [11] contributed a theory which takes into account the shear and rotary inertia corrections that are neglected in Euler–Bernoulli’s beam theory. e solution of supported beam subjected to moving loads through using the power series expansion was studied by Timoshenko [12]
Leszek [13] studied the vibration of uniform Euler and Timoshenko beam resting on an elastic foundation using the green functions
The method of differential quadrature was applied on two beam models, that is, the Euler–Bernoulli and Timoshenko beams. e natural frequencies and deflection modes were derived from the basic constitutive equations and the numerical outputs of both problems were shown
Summary
E beam rests on an elastic Pasternak foundation. E assumptions stated by the Euler–Bernoulli beam theory are [21]. Deformed angles of rotation of the neutral axis are small e foundation of the beam is taken as a Pasternak foundation with linear stiffness: Rx,t k1wx,t − GRz2zwx2x,t,. Where Rx,t is the reaction of the foundation per unit length, wx,t is the beam vertical displacement, k1 is the first order foundation parameter (elastic stiffness), and GR is the shear deformation coefficient. Basic energy derivations for the Euler–Bernoulli beam model are z2w M EI zx. E beam bending energy Ub is given as follows:
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