Discussion: “Vibration Analysis of Composite Beams With End Effects Via the Formal Asymptotic Method” (Kim, J.-S., and Wang, K. W., 2010, ASME J. Vib. Acoust., 132(4), p. 041003)

Abstract

In their paper [1], Kim and Wang presented the vibration analysis of composite beams by using a finite element-based formal asymptotic expansion method. After a detailed mathematical treatment of the formal asymptotic method-based beam analysis (FAMBA), they considered various examples of composite beams with solid and closed thin-walled cross sections for verification purposes and the numerical results were compared with those from 3D FEM and other methods, such as the variational-asymptotic beam section analysis (VABS) [2].One particular case, the 15 deg CUS thin-walled composite box beam (whose ply material properties are listed in Table 3 of [1]), was analyzed with VABS by taking the stiffness model given in [2] and applying it to a typical Rankine–Timoshenko beam model. The normalized first and second eigenvalues computed by 3D FEM, FAMBA, VABS, etc. were plotted against the beam length-to-height ratio in Figs. 6 and 7. While it is clear that FAMBA-second predictions are reasonably close to those of 3D FEM, the authors (Kim and Wang) also show, and hence claimed, that results of VABS deviate significantly from those of 3D FEM, even qualitatively so for the second eigenvalue.The normalized first, second, and eighth mode shapes in Figs. 8 and 9 and 10 in [1] also show VABS as being unable to match 3D FEM predictions, with the deviation most obvious in Figs. 8(b), 9(a), and 10(b) in [1]. This is solely attributed by the authors to VABS inaccurately capturing the bending-shear coupling of the box beam. These claims call for the following comments:1. The stiffness model given in [2] (as calculated by versions of VABS till v.3.1), when used in tandem with a 1D beam analysis code based on a geometrically exact intrinsic beam theory (GEBT), provide results that are both qualitatively and quantitatively in close agreement to 3D FEM unlike the results reported in [1]. This can be seen in Figs. 1–5 which have now been recalculated by the authors (Kim and Wang). We can safely assume that the fault lies neither with the Rankine–Timoshenko model that the authors used nor the geometrically exact nonlinear beam theory (since the former seemed to have worked with other beam models and the latter is designed to capture all the geometrical nonlinearities obtainable by a beam model). Therefore, it seems logical to conclude that the source of discrepancy was an error in applying the given stiffness model correctly to the macroscopic analysis.2. The discrepancy has been examined by the authors (Kim and Wang). It is found that the sign convention used in VABS [2] was not consistently applied to the calculation of VABS results in [1]. The vibration analysis results from VABS and others have been corrected and replotted, as shown in Figs. 1–5.In conclusion, we find that although there are minor differences in the prediction of the bending-shear coupling for the CUS box beam between VABS [2] and FAMBA [1], both stiffness models yield very acceptable results in terms of vibration analysis. Specifically, VABS does not significantly deviate, qualitatively or quantitatively, from 3D FEM predictions. Additionally, the authors (Kovvali, Hodges, and Yu) would like to note that since the publication of [1], VABS has been updated and now provides results that are still further improved for some problems; see [3].