Exact Solution Using Sextic–Quintic Equations for Rectangular and Circular Footing Design

Abstract

The determination of footing dimensions subjected to axial load and bending moment is commonly performed using a trial-and-error procedure in conventional structural design. This approach often requires iterative calculations and may lead to inefficient design processes. This study presents an analytical formulation for determining the dimensions of rectangular and circular footings by transforming the governing design equations into polynomial forms. The derivation results in sextic equations for rectangular footings and quintic equations for circular footings. These nonlinear polynomial equations are solved using numerical solution techniques to obtain the exact footing dimensions that satisfy the structural design requirements. The proposed formulation eliminates the need for repetitive trial calculations and provides a systematic computational approach for footing design under combined loading conditions. The results demonstrate that the developed analytical–numerical approach can efficiently determine the required footing dimensions while maintaining the design safety criteria. This method provides an alternative computational tool that may assist engineers in performing more accurate and efficient footing design analysis.