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The transfer matrix method based on the Euler-Bernoulli beam theory is employed in order to originally achieve some exact analytical formulas for elastically supported beams under a point force together with uniformly distributed force and uniformly distributed couple moments. Those closed-form formulas can be used in a variety of engineering applications especially at the pre-design stage to get an insight into the response of the structure. Contrary to the classical boundary conditions, it is also observed that the Euler-Bernoulli solutions of a beam with elastic supports are sensitive to the ratio of length to thickness (L/h).
In this study the fixed-fixed column subjected to axial Euler’s load has been investigated. The load is placed between the fixed ends of the structure and its location can be changed along column’s length. The boundary problem of free vibrations of the mentioned system has been formulated on the basis of Bernoulli – Euler theory and taking into account non-linear axial deformation relationship. Due to non-linear expressions the solution of the problem was done with small parameter method. In the paper the change of the first vibration frequency in relation to location and magnitude of the loading force was obtained. The relationship between natural vibration frequency and the amplitude is also discussed.
A numerical technique for solving the linear problems of the calculus of variations is presented in this paper. Multiwavelets and multiwavelet packets of Legendre functions are used as basis functions in the Ritz method of formulation. An operational matrix of integration of multiwavelets and multiwavelet packets is introduced and is used to reduce the calculus of variation problem to the solution of the system of algebraic equations. The algorithm is applied to the analysis of mechanic problems which are formulated as func-tionals. Two examples are considered in this paper. The first example concerns the stability problem of a Euler–Bernoulli beam and the second one presents the calculation of the extreme value of the functional which defines the potential energy of an elastic string. The presented method yields the approximate solutions which are convergent to accurate results.
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