Bending-Moment-Eigenvector Expansion of Free End Displacement of Cantilever Rigid Frame

摘要

This paper is Bending-Moment-eigenvector expansion of end-displacement of cantilever rigid frame of which be made of several poles of stiffness be constant. Those are from two parts. First are properties of Bending-Moment-eigenvector and of bar-end-displacement bending moment diagram [2]. And second is basic physical relationship between internal force and end displacement of straight cantilever beam that is out from relationship of end-displacement and internal force of beam [1,3]. The shown of free end displacement of frame is in terms of Bending-Moment-eigenvector of each pole. In a pole it needs only one Bending-Moment-eigenvector of which at characteristic-point of pole. The parameters of expansion are line-stiffness of pole and location of pole standing to free end of frame. And shear force has been expunction from equation. The characteristic-point of pole is determined by the position projection of the free-end on the pole. The characteristic-point position on range [1/3,1/2] of pole, if projection of free-end of frame is on direction prolong of pole. The characteristic-point position on [0, 1/3] and/or [1/2, 1] respectively, if the projection is on opposite and it is divided by region [-1/3, -2/3]. There is no characteristic-point and no Bending-Moment-eigenvector for need of this expansion, if the projection is in (-1/3, -2/3) of pole. There is a straight beam of having a static equilibrium of several concentrated force. In which force equals to ratio of bending-moment-eigenvector to stiffness of pole. When each force is at midpoint of each segment of the beam the calculation of end bending moment is dual with calculation of expansion.