A Curvature Effect in the Free Vibration of Naturally Curved and Twisted Beams

This study investigates the critical role of curvature expressions in the analysis of naturally curved and twisted beams, focusing on two distinct curvature formulations: the Frenet-Serret and Darboux frames. Whereas the Frenet-Serret frame is inherently aligned with the curve's trajectory, the Darboux frame is affixed to the cross-section of the curved beam. For elastic beams, the preference is to employ the Darboux frame to establish the cross-sectional rigidity about the beam's reference line. Within the Darboux frame framework, we derive a curvilinear strain tensor, underpinning the assumption of the Euler-Bernoulli beam theory. Subsequently, we utilize this beam strain measure to formulate a three-noded finite element tailored for the analysis of naturally curved and twisted beams. The element is also designed to be free from membrane-locking issues. We apply this formulation to analyze the free vibration behavior of beams with varying opening and twist angles. Our results exhibited excellent agreement with those obtained from commercial finite element software, both qualitatively and quantitatively.