Diagonal, Block-Diagonal and Higher-Order Mass Matrices for the Interior-Node Rotation-Free Quintic Hermite Element of Euler–Bernoulli Beams

A systematic study is presented toward optimizing various variants of mass matrices for the interior-node rotation-free quintic Euler–Bernoulli beam element. This Hermite type beam element consists of four nodes, where the two end nodes contain both deflectional and rotational degrees of freedom (DOFs), but the two interior nodes are solely associated with the deflections. By introducing an adjustable nodal distance parameter between the end and adjacent nodes, a nodal-distance adjustable mass matrix is devised. Accordingly, both diagonal and block-diagonal mass matrices of this four-node quintic Hermite beam element are developed with respect to different nodal distance parameters, which are determined through optimizing the corresponding frequency accuracy in the context of nodal integration. It is shown that the best performance of the diagonal mass matrix is a sub-optimal frequency convergence, while the optimal convergence can be achieved by the block-diagonal mass matrix. Subsequently, a higher-order mass matrix featured by two extra orders of frequency accuracy is formulated via optimally blending the consistent and block-diagonal mass matrices. Moreover, regarding the diagonal mass matrices, it is proved that the interior-node rotation-free four-node element yields superior frequency accuracy in comparison with the conventional three-node element where all nodes have the deflectional as well as rotational DOFs. An excellent agreement is observed between the theoretical and numerical results.