8.6 Conclusions The evolution of the fundamental frequency in slender beams subject to prescribed axial end displacements was experimentally investigated: three elements presenting different initial curvatures were tested in the hinged–hinged and hinged-clamped constraint conditions. For all the analyzed cases, it can be noted that a first phase, where the fundamental frequency decreases with the axial load, is followed by a stiffening one, where the trend is reversed. The effect of the geometric imperfection (initial curvature) is to reduce the negative slope of the first branch, therefore anticipating the transition point and making this transition smoother at the same time. This type of behavior, analytically predicted by previous studies [5, 7] for other shapes of the initial imperfection and end conditions, must be well taken into account when designing new structures as well as in monitoring existing ones. For example, the possibility to foresee the transition point, where the minimum frequency (minimum stiffness) is localized, could find application in monitoring in-service structures and components in order to plan consolidating interventions. Further, numerical simulations can be helpful in interpreting in-situ measurements, as well as they may allow us to perform virtual laboratory tests. References 1. Virgin LN (2007) Vibration of axially loaded structures. Cambridge University Press, New York 2. Woinowsky-Krieger S (1950) The effect of an axial force on the vibration of hinged bars. J Appl Mech 17:35–36 3. Burgreen D (1951) Free vibrations of a pin-ended column with constant distance between pin ends. J Appl Mech 18:135–139 4. Dickinson SM (1980) The lateral vibration of slightly bent slender beams subject to prescribed axial end displacement. J Sound Vib 68(4):507–514 Fig. 8.7 (a–c) Comparison between experimental and numerical fundamental frequency vs. axial load curves for specimens B1-3 and (d) collection of numerical results for specimens B1-3 8 Fundamental Frequencies of Slender Beams Subject to Imposed Axial End Displacements 65
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