Static, Vibration, and Transient Dynamic Analyses by Beam Element with Adaptive Displacement Interpolation Functions

An approach to analyzing structures by using beam elements is developed with adaptive displacement interpolation functions. First, the element stiffness matrix and equivalent nodal loads are derived on the basis of the equilibrium between nodal forces and section forces rather than the compatibility between nodal deformations and section deformations, which avoids discretization errors caused by the limitation of conventional polynomial interpolation functions. Then, six adaptive element displacement interpolation functions are derived and extended to include several cases, such as beams with variable cross-section, variable material properties, and many different steps in cross-section and/or material properties. To make the element usable in dynamic analyses, consistent mass matrix CMM and diagonally lumpedmass matrix LMM are constructed using the presented adaptive displacement interpolation functions. All these features have made the element elegant, which is tested with a number of simple static, vibration, and dynamic examples to show its accuracy.