REFINED BEAM FINITE ELEMENT WITH A NON NODAL DEGREE OF FREEDOM

Abstract eng:
The beam element used in frame analysis has 4 degrees of freedom corresponding to the transverse displacement and the rotation at each end node. This condition allows a cubic variation of the transverse displacement along the length of the element, which is the correct one if the bending moment has linear variation, that is, no transverse load acts between the end nodes. However, this is not the usual case of beam behavior because distributed loads are commonly acting. To solve this contradiction the frame analysis is performed substituting the distributed load by fictitious equivalent nodal concentrations corresponding to the so-called fixed coordinate state. We can not get rid of these artificial fixed end moments and shear unless we establish an appropriate distributed degree of freedom, that is, the smeared coordinate of displacement along which the distributed load is acting. The purpose of this paper is to identify such non-nodal degree of freedom related to a uniform distributed load. Accordingly, a refined beam element with five degrees of freedom (four nodal and one non-nodal) is presented. Corresponding shape functions are developed and explicit expression of the 5x5 stiffness matrix is presented. The consistent load vector associated with a uniform load is discussed to show that no components exist at the end nodes. Moreover, the 5 x 5 geometric stiffness matrix is developed and successfully applied to solve second order analysis. Finally, to solve problems of beams on elastic foundation, the matrix related to the soil contribution to the refined element stiffness is explicitly given.

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