Deflection of Structures using Modified Betti’s Theorem

In this paper, Betti’s theorem is modified by inclusion of constraint reactions in the set of externally applied loads. Based on this, a new methodology for calculating deflections of any structure is presented in this paper. The methodology has an advantage over the conventional methods due to the fact that deflections of a structure for any general loading and for different boundary conditions are calculated mostly by simple multiplications. It does not require the knowledge of writing internal force/ moment expressions. The methodology leads to a unique concept of reference structural element. Equation of deflected elastic curve of the reference element is utilized to obtain equation of deflected elastic line of the structure with any other type of loading and boundary conditions. Present methodology is fast and easy in its use especially when the structure is loaded by a series of point loads. Even for distributed loads a simple integration of a priory known polynomial integrands leads to the result. This methodology is applicable to any class of structure as long as a reference element is defined for that class. Here, this methodology is illustrated for beams.