Ritz analysis of discontinuous beams using local trigonometric functions

Abstract

The objective of the current paper is to present a Ritz-type analytical model for predicting the behavior of discontinuous beams such as thin-walled beams with cracks and multiply-stepped beams. The beam is discretized in the cracked as well as the un-cracked domains for a cracked thinwalled beam and in uniform beams for a multiple-stepped beam. A set of local trigonometric trial functions is used to define the twist angle for the cracked domain and the un-cracked domains, as well as to define the displacement field for uniform domains. A global equation system of unknown Ritz coefficients is derived by minimizing the Lagrangian functional or the total potential energy. In the present Ritz model, the interface continuity conditions between sub-domains are investigated and enforced into the global equation system using the condensation procedure or the Lagrange multipliers. Examples are presented to illustrate the effectiveness of the current model for free vibration and torsional analysis. Results obtained from the current model are found to agree well with those obtained using a detailed Present Address: finite element method or with existing results in literature. The proposed model offers an efficient approach to reduce the modeling efforts and computational time required to analyze complex beams with cracks or multiple steps.

Convergence Study of Local Hierarchical Functions for Free Vibration Analysis with Application to Multi-Step Beams

“…The results are shown in Figures 11 and 12. Table 2 shows the comparison of the present results with those obtained by Dang et al [5], with the error calculated using the exact result obtained by Elishakoff et al [6]. A monotonic convergence is observed for all sets under p-refinements.…”

“… From Table 2, it is noted that the use of trigonometric functions does not yield results obtained by Dang et al [5]. Although in both the studies, the convergence begins from an upper bound, the frequency obtained by Dang et al is found to go below the exact value.…”

“…A modification in the trigonometric set is proposed to enable the satisfaction of global natural boundary conditions to improve the accuracy of initial modes when using the trigonometric functions. The results are compared with those obtained by Dang et al [5] using the trigonometric functions and exact results by Elishakoff et al [6].…”

Natural frequency analyses of segmented Timoshenko–Euler beams using the Rayleigh–Ritz method

“…The function used is valid only for cantilever beam and contains multiplied trigonometric and hyperbolic terms that may cause increase of solution time. A set of local trigonometric trial functions, which also contains multiplied trigonometric terms, is employed in the method to determine vibration of multi-stepped Euler–Bernoulli beam (Dang et al., 2011). Bhat (1997) presents variational formulation of the R–R method and investigates the stationarity of the natural frequencies with respect to the arbitrary coefficients of the assumed trigonometric deflection shape of Euler–Bernoulli beams.…”

“…Han et al (1999) presents the analyses of models on uniform beams. Most of the works take into account the fundamental beam theory known as ''Euler-Bernoulli beam'' that neglects the effects of rotary inertia and shear deformation (Dang et al, 2011;Janaina and Jose Juliano, 2014;Jaworski and Dowell, 2008;Naguleswaran, 2002Naguleswaran, , 2003Popplewell and Chang, 1996;Pradhan and Chakraverty, 2013;Yamada et al, 1994;Zhou and Cheung, 2000). This theory assumes that energy transmission occurs at the speed approaching infinite for the flexural vibration of thin beams.…”