The vibration results shown in Figures 14(a)-(c) concern laminated plate lipped channel beams (members under uniform major axis bending) with (i) walls formed by three equally thick orthotropic layers made of identical FRP materials (epoxy resin reinforced with e-glass fibres) and exhibiting a cross-ply configuration [0°, 90°, 0°], (ii) locally/globally pinned and free-to-warp end sections and (iii) uniform mass density p = 2.1 gcm-3 (Silvestre and Camotim, 2005a, b). The curves in Figure 14(a) show how the fundamental natural frequency rnf, M varies with the beam length L (logarithmic scale) and bending moment ratio M/Mcr – the upper curve is related to the load-free member (M = 0) and is always associated with single-wave vibration modes (&>f.0 = &>i.0).4 The remaining seven curves correspond, in descending order, to increasing M/Mcr value – each

4 Due to space limitations, the results concerning the buckling behaviour and load-free vibration behaviour of the beams cannot be presented here. The interested reader is referred to the recent works by Silvetre and Camotim (2005a, b).

curve is associated with a fixed percentage of Mcr, which varies with the beam length. Because Mcr corresponds to buckling modes exhibiting several waves for 55 cm < L < 180 cm, it is interesting to look more closely at the curves ш/,M(L) in this length range – thus, Figure 14(b) shows these curves for 40 cm < L < 200 cm. Moreover, the modal participation diagrams shown in Figure 14(c) supply additional information about the contribution of the GBT deformation modes (see Figure 2) to the fundamental vibration modes of beams with nine M/Mcr values.[5] Note that the presence of none, one, two or three bars underneath a mode number indicates a single, two, three or four-wave contribution to the vibration mode. The analysis of the results displayed in Figures 14(a)-(c) prompts following conclusions and/or comments:

(i) For L < 55 cm or L > 180 cm, the beams buckle in single-wave critical buckling modes (see footnote 4), which means that one has ш/,M(L) = m1.m(L) and all curves have fairly similar shapes. Moreover, it is worth pointing out that, although the beam fundamental vibration modes always exhibit a single wave, their shapes differ from those of the beam (i1) critical buckling modes and (i2) load-free fundamental vibration modes – indeed, the beam fundamental vibration mode shape “travels” between them as the ratio M/Mcr increases (see Figure 14(c)). For instance, note that the participation of mode 2, which contributes significantly to the load – free member flexural-torsional vibration modes, continuously decreases as M/Mcr grows, until it vanishes for M = Mcr – mode 2 is absent from the beam critical buckling mode (it appears in its pre-buckling path).

(ii) For 55 < L < 180 cm, the beam critical buckling modes have more than one wave (see foot­note 4) and the shapes of the curves ш/,M(L) become visibly different as the value of M/Mcr increases, as clearly illustrated in Figure 14(b) – these curves (iii) cease to decrease monotonic – ally and, for large enough M/Mcr values, (ii2) they are no longer “smooth”, exhibiting sudden and quite pronounced slope reversals. Moreover, Figure 14(c) shows that the beam fundamental vibration mode wave number varies between one and the number of waves appearing in the beam critical buckling mode, as M/Mcr grows – for M > 0.90Mcr, the fundamental vibration mode successively exhibits 2, 3 and 4 waves (Figure 14(b) identifies very well the L — M/Mcr combinations associated with each case). Apparently, the M/Mcr value that triggers a non­single wave number depends on the percentage difference between Mcr and Mb1 (bifurcation moment leading to a single-wave buckling mode), i. e., a Mcr/Mb.1 decrease lowers the M/Mcr value corresponding to the change (Silvestre and Camotim, 2005a, b).

(iii) Figure 14(c) shows that a beam with L = 100 cm vibrates in (iii1) a flexural-torsional – distortional mode (2 + 4 + 6) for M = 0, (iii2) another flexural-torsional-distortional mode (2 + 4 + 6 + 5) for 0 < M/Mcr < 0.3, (iii3) a single-wave distortional mode (5 + 6) for 0.3 < M/Mcr < 0.8 and (Ш4) similar two or three-wave distortional modes for 0.8 < M/Mcr < 0.9 and 0.9 < M/Mcr < 1, respectively. Since the combination of modes 5 and 6 attenuates the rotation of the tensioned flange-lip assembly and increases that of the compressed one (see Figure 2), it is possible to conclude that, for increasing M/Mcr values, the amplitudes of the tensioned and compressed flange-lip motions tend to decrease and increase, respectively. Therefore, for moderate-to-high M/Mcr values (0.3 < M/Mcr < 1), only the compressed flange-lip and the web upper half (due to compatibility) vibrate – they exhibit either 1, 2 or 3 waves (see the last two modes in Figure 15).

In order to validate the GBT-based results, some shell finite element analyses were carried out using the code Abaqus (HKS, 2002) – they all involved beams with L = 100 cm and the results

Figure 15. FEM-based beam fundamental frequencies and vibration mode shapes (L = 100 cm).

determined concerned M/Mb.1 = 0.271, 0.407, 0.488, 0.516, 0.537, 0.543 (the last ratio means M = Mcr). The GBT and FEM-based ш/,M values read (i) ш/,M = 502,463, 409, 325, 146, 0 s-1 (GBT) and (ii) ш/,M = 508, 487, 463, 418, 315, 268 s-1 (FEM), thus making it obvious that they become increasingly apart as M/Mb.1 grows. In particular, note that, for M = Mcr, GBT yields the (theoretically expected) null ш/,M, while the FEM-based ш/,M value is about 50% of ш/,0 = 538 s-1 (it only becomes null for M = 1.0134Mcr). It was subsequently found that this rather surprising discrepancies stem from the fact that the ABAQUS shell FEA incorporate the stiffening effect due to the primary (first-order) bending deflections, which is not taken into account by the developed GBT formulation. This fact was overcome by performing the FEM vibration analyses in “adequately pre-cambered beams” (Silvestre and Camotim, 2005a, b), thus enabling a meaningful comparison between the two sets of ш/,M values. Figure 15 shows the FEM-based fundamental vibration mode shapes and frequency values for beams with L = 100 cm acted by M = 0; 0.5;

0. 9Mcr – the FEM values ш/,0 = 530 s-1, ш/,0.5 = 504 s-1 and ш/,0.9 = 330 s-1 agree very well with the GBT ones ш/.0 = 538 s-1, ш/.0.5 = 502 s-1 and ш/.0.9 = 325 s-1 (differences of 1.5, 0.3 and 1.5%).