... and end displacements for a variable cross-section beam. Upon simplification (series expansion and truncation), these three expressions reveal beam characteristic coefficients , which may be classified as stiffness, constraint, and energy coefficients, respectively. In Sec. 4 , we make use of two...

... as that for the simple beam—only the beam characteristic coefficients change—thus validating the generality of the BCM. The procedure is still closed-form analytical because for a given beam shape, no iterative or numerical methods are required. Furthermore, the beam shape coefficients appear as parameters...