8.4 Experimental Results A total of six testing configurations was realized by analyzing each specimen in the following two constraint conditions: hinged–hinged and hinged-clamped. In the former case, the theoretical buckling load calculated according to Euler’s formula is about 17.5 kN, while the theoretical fundamental flexural frequency calculated according to linear dynamics is about 35.2 Hz. For the hinged-clamped specimen, the Euler’s critical load is about 37.2 kN, while the fundamental frequency is 55 Hz. The previous theoretical values are not affected by imperfections and so they hold for all specimens B1-3. First, the axial load vs. transversal displacement curves were obtained for all the tested configurations as a preliminary phase (Fig. 8.4). The axial load was intended as the specimen reaction corresponding to the axial displacement imposed to the lower beam end by the testing machine actuator, therefore its value came from the MTS. The transversal displacement was intended as the midpoint deflection of the specimen: its value was measured with the laser sensor. Figure 8.4a shows the axial load vs. transversal displacement curves for specimens B1-3 when hinged at both ends. While specimen B1 showed a bifurcation point almost corresponding to the Euler’s buckling load (17.5 kN), specimens B2 and B3 showed the typical response of beams with initial geometric imperfection. Figure 8.4b shows the analogous curves for the case in which specimens B1-3 were hinged at one end and clamped at the other. In this case larger differences between the expected and the measured results were obtained. In fact, the graph corresponding to specimen B1 showed a bifurcation in correspondence of a load equal to about 30 kN, while the theoretical Euler’s load was equal to 37.2 kN. This behavior can be interpreted as an inefficiency of the constraint condition at the upper joint, with a condition of not-perfectly-clamped end. For each testing configuration, the fundamental vibration frequency was determined under different values of the mean axial force obtained imposing different vertical displacements to the beam end. The fundamental natural frequencies were obtained by applying three different and independent procedures: (1) by identifying the resonance condition in the case of forced oscillation, i.e. analyzing the response corresponding to different frequencies of the harmonic excitation transmitted by the electromagnet; by applying the Logarithmic Decrement Method to the free response signal induced by opportune initial conditions; and, lastly, by performing the Fast Fourier Transform of the free response signal. In all the analyzed cases there was a perfect correspondence between the results given by the three adopted methods. Figure 8.5 shows the fundamental frequency vs. axial load curves for specimens B1-3, in both the analyzed constraint conditions: Fig. 8.5a refers to the hinged–hinged constraint condition, while the results related to the hinged-clamped constraint condition are collected in Fig. 8.5b. First of all, we observe that the zero-frequency condition was never reached, due to the combined effects of the geometric and material imperfections (the rectified beam was non a perfect beam in reality) and of the corresponding non-trivial equilibrium path associated to the imposed displacement. As can easily be seen, in all the analyzed cases two different trends of the frequencies with respect to the axial load were observed. In fact, the evolution of the fundamental frequency showed a decrease down to Table 8.1 Main characteristics of the laser sensor Measuring range (MR) Start of measuring range (SMR) Midrange (MR) End of measuring range (EMR) Resolution (dynamic 750 Hz) 20mm 30mm 40mm 50mm 10 μm Fig. 8.3 (a) Experimental equipment and (b) testing configuration with specimen B1 undergoing post-critical deflections 62 G. Piana et al.
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