Free vibration of arches using a curved beam element based on a coupled polynomial displacement field

The performance of a curved beam finite element with coupled polynomial distributions for normal displacement (w) and tangential displacement (u) is investigated for in-plane flexural vibration of arches. A quartic polynomial distribution for u is derived from an assumed cubic polynomial field for w using force–moment equilibrium equations. The coupling of these displacement fields makes it possible to express the strain field in terms of only six generalized degrees of freedom leading to a simple two-node element with three degrees of freedom per node. Numerical performance of the element is compared with that of the other curved beam elements based on independently assumed field polynomials. The formulation is shown to be free from any spurious constraints in the limit of inextensional flexural vibration modes and hence does not exhibit membrane locking. The resulting well-conditioned stiffness matrix with consistent mass matrix shows excellent convergence of natural frequencies even for very thin deep arches and higher vibrational modes. The accuracy of the element for extensional flexural motion is also demonstrated.