GENERAL THREE-DIMENSIONAL BEAM ELEMENT [时] 信管县息 CLOSING REMARKS Frame structures analysis Members in a frame are considered to be rigidly connected.Both forces and moments 44 (a) Nodal displacements in the element coordinate system. (b) Nodal displacements in the global coordinate system. Reorder the element stiffness matrix: The element displacements are written in terms of the global displacements as: in matrix form: it is readily shown that the 6 × 6 element stiffness matrix in the global system is given by: where [R] is the transformation matrix that relates element displacements to global displacements. A GENERAL THREE-DIMENSIONAL BEAM ELEMENT (a) Three-dimensional beam element. (b) Nodal displacements in element xz plane. The element equilibrium equations for a two-plane bending element with axial stiffness are written in matrix form as: Adding the torsional characteristics to the general beam element, the element equations become: CLOSING REMARKS In this chapter, finite elements for beam bending are formulated using elastic flexure theory from elementary strength of materials. The resulting elements are very useful in modeling frame structures in two or three dimensions. A general three-dimensional beam element including axial, bending, and torsional effects is developed by, in effect, superposition of a spar element, two flexure elements, and a torsional element. In development of the beam elements, stiffening of the elements owing to tensile loading, the possibility of buckling under compressive axial loading, and transverse shear effects have not been included. In most commercial finite element software packages, each of these concerns is an option that can be taken into account at the user’s discretion. Frame structures analysis Members in a frame are considered to be rigidly connected. Both forces and moments can be transmitted through their joints. We need the general beam element (combination of bar and simple beam elements) to model frames