Exact frequency and mode shape formulae for studying vibration and stability of Timoshenko beam system

Vibration and Stability Analysis of Non-Uniform Timoshenko Beams Under Axial and Distributed Tangential Loads

“…The results obtained for the uniform beam are in good agreement with those of the previous studies (Figure 2 of reference [16]). It can be seen that the critical buckling loads of a uniform beam are greater than those of the non-uniform one, as expected, since the sti!ness of the uniform beam is now higher.…”

“…Farghaly [15] presented closed-form, exact frequency and mode shape solutions for a uniform cantilever Euler}Bernoulli beam with an elastically mounted end mass under axial load concentrated either at the point of attachment between the beam and the end mass or at the center of gravity of that mass. The same problem but for the Timoshenko beam, based on two di!erent sets of governing equations, has been studied by Farghaly and Shebl [16]. They showed that the di!erence between the natural frequencies is signi"cant for beams subjected to larger loads and at higher modes of vibration.…”

“…Table 3 shows the "rst four natural frequencies for the uniform and non-uniform clamped}free Timoshenko beam under the action of an axial load. For recovery of the previous results, dimensionless parameters of the uniform beam were selected in the form of "0)01, " " "0, "0)03, 2 "100, F "2 which are the same as those presented in Table 1 of reference [16]. The comparison of the results presented in Table 3 with the corresponding values taken from reference [16] shows good agreement and, therefore, veri"es both the formulation and the developed computer program.…”

“…For recovery of the previous results, dimensionless parameters of the uniform beam were selected in the form of "0)01, " " "0, "0)03, 2 "100, F "2 which are the same as those presented in Table 1 of reference [16]. The comparison of the results presented in Table 3 with the corresponding values taken from reference [16] shows good agreement and, therefore, veri"es both the formulation and the developed computer program. Results for the case of a constant width, and a second order varying thickness Timoshenko beam (q( )"s( )"(1#0)2 ), ( )"r( )"(1#0)2 )) are presented in Table 3.…”

Identification of equivalent mechanical properties for unreinforced masonry walls

“…Several research programs have been carried out over the past 20 years (Farchaly and Shebl, 1995;Esmailzadeh and Ohadi, 2000), with some focused on shear beams (Zhang et al, 2013), investigating the consequences of variable sections (Naguleswaran, 2003) or of an elastically restrained base (Hernández-Urrea and Dario Aristizábal-Ochoa, 2008).…”