Hierarchical beam element is proposed for the matrix analysis of space framed structures. The element is formulated with C⁰ continuous polynomials based on Timoshenko beam theory and an expanded warping theory considered shear deformation, and with nodal displacements at the both ends and higherorder internal modes. The generalized displacements, i. e. translations, twist, rotations and warping are interpolated independently, so that the well-known ‘shear locking’ phenomenon appears in the analysis of Bernoulli-Euler beams. Therefore, an effective reduced integration technique is developed for the bending and warping analysis of slender beams.
It is found by the numerical calculations that the proposed hierarchical elements yield superior accuracy for warping problems, bending problems of a beam with non-uniform cross section.