Unveiling the Efficiency of the Moment Distribution Method: A Geometric Series Perspective

The moment distribution method, originally introduced by Hardy Cross in his influential publication “Analysis of Continuous Frames by Distributing Fixed‐End Moments” (published in Transactions of the ASCE, 1932;96(1793):1–10), is widely recognized as a fundamental aspect of both structural engineering and global education. This iterative technique aims to improve solution accuracy at each step of distribution and has stood the test of time. In our study, we delve into the mathematical foundation of this method and make a crucial discovery: the terms generated during iterations follow a geometric series pattern . By leveraging this property, we propose a streamlined formula for efficiently calculating the sum of all the terms produced in each step of distribution iteration. Through comparative analysis with alternative methods, we highlight the method’s superior precision and robustness in structural analysis. Our investigation primarily focuses on demonstrating the geometric series nature of the terms generated through iterative moment distribution, providing specific theorems as references. Ultimately, our objective is to present concise, closed‐form mathematical solutions that showcase the practical effectiveness of this method in various applications within the field of structural engineering.