Finite-element implementation for nonlinear static and dynamic frame analysis of tapered members

Non-prismatic members are popular for civil engineering structures. This paper derives a set of unified beam column formulations for nonlinear static and dynamic analyses of the structures made of members with tapered sections, addressing the problems in engineering design practices. The element shape-functions are established upon the local-axes by extracting the rigid-body movements for simplifying mathematical expressions. To represent the variations in the stiffness gradients of tapered sections, the tapered-variability indexes are proposed. The generalized tangent stiffness and consistent mass matrices are developed based on the indexes. When analyzing non-prismatic members, the conventional method for handling member loads is inapplicable because it is derived for prismatic sections. Therefore, a new approach for converting the member loads acting on tapered members into the equivalent nodal forces is proposed based on the energy conservation principle. To consider the offsets in section axes, the eccentricity matrices are employed. For allowing large deflections, the incremental secant-stiffness method (ISM) based on Updated-Lagrangian (UL) description is proposed. Finally, extensive examples are provided for validating the accuracy and efficiency of the proposed element formulations in solving both the static and dynamic nonlinear problems.