Vibrations And Stability Of Bernoulli-Euler And Timoshenko Beams On Two-Parameter Elastic Foundation

Abstract

The vibration and stability analysis of uniform beams supported on two-parameter elastic foundation are performed. The second foundation parameter is a function of the total rotation of the beam. The effects of axial force, foundation stiffness parameters, transverse shear deformation and rotatory inertia are incorporated into the accurate vibration analysis. The work shows very important question of relationships between the parameters describing the beam vibration, the compressive force and the foundation parameters. For the free supported beam, the exact formulas for the natural vibration frequencies, the critical forces and the formula defi ning the relationship between the vibration frequency and the compressive forces are derived. For other conditions of the beam support conditional equations were received. These equations determine the dependence of the frequency of vibration of the compressive force for the assumed parameters of elastic foundation and the slenderness of the beam.

Contrasting three alternative versions of Timoshenko‐Ehrenfest theory for beam on Winkler elastic foundation – simply supported beam

“…T A B L E 1 9 Comparison of natural frequency with paper of Obara [28] We also confirm that solutions presented in this paper nearly coincide with results reported by Obara [28] except for a very difference in the third digits probably due to roundoff error. Indeed, by following, [28] we utilize the following values = 0.5 m, ℎ = 0.8 m, = 31 ⋅ 10 9 Pa, = 24.52 ⋅ 10 9 , = 0.25, = 1.2 and √ ∕ 2 = 10, obtaining results presented in Table 19. The percentagewise difference is less than 0.03% for the first frequency,0.04% for the second frequency and 0.06% for the third one due to possibly roundoff error.…”

“…We then investigate parameter 1 observing that its denominator is positive. Therefore, we move the attention to the numerator's sign by solving the following equation stemming from Equation 28:…”

“…Hence, since the value in the square root is positive, we impose that the sign of the right side has to be positive. Making use of Equation 28 we obtain the following inequality:…”

A Kerr‐type elastic foundation model for the buckling analysis of a beam bonded on an elastic layer

“…The buckling of a beam on an elastic foundation is of significant concern in various engineering applications, [1][2][3][4] therefore attracts considerable interests for decades. To mention a few, the buckling of a beam on soils is visited in [5][6][7], the buckling of a stiff thin beam on a compliant substrate is investigated in [3,8,9], the buckling of a functionally graded material beam on an elastic foundation is studied in [1,4,10], the wrinkling of skin on an elastic foundation is analyzed in [2]. In addition, the buckling of a fiber in matrix can be also analyzed as a beam on an elastic foundation.…”

“…To overcome this weakness, the two-parameter models such as Pasternak's model [10,17,18] and Wieghardt model, [19] and the threeparameter models such as Reissner's model [20] and Kerr's model [21] (which is different from the present Kerr-type model [22] ), were employed to define the constitutive relation of the elastic layer beneath the beam. [4,6,7,11,17,[23][24][25][26][27][28][29][30][31][32][33][34] Dillard proposed an incompressible elastomeric foundation model, [35] which was recently employed in the analysis of the buckling of a laterally supported beam. [36] To obtain a precise critical buckling force theoretically, it is essential to find a suitable elastic foundation model which is able to accurately describe the mechanical response of the elastic layer.…”