Mathematical Development for the Minimum Cost of Elliptical Combined Footings

This work shows the mathematical development for the minimum cost of ECF (elliptical combined footings) subjected to biaxial bending due to the two columns, assuming that the distribution of soil pressure below the footing is linear and that the footing rests on elastic soil. There are no similar contributions on the subject of this article, as it is an innovative contribution in terms of its form. This work is developed in two parts: first, determine the minimum area in contact with the soil below the footing, and then the minimum cost is obtained. The formulation of the development by integration is shown to determine the moments, unidirectional shears, and punching shears acting on the critical sections, according to the ACI (American Concrete Institute) design code, and then the flowchart algorithm is applied to determine the solution using Maple Software, which is the main contribution of this article. Some authors show studies on the combined footings of various shapes such as rectangular, trapezoidal, strap, corner or L, and T, but there are none for ECF. Two numerical studies are shown with different length: the first with free ends in the longitudinal direction and the second with ends limited in the longitudinal direction to estimate the minimum cost of ECF under biaxial bending. A third numerical study is shown, with different allowable bearing capacities of the ground and with free ends in the longitudinal direction. Also, a comparison is developed between ECF and RCF (rectangular combined footings). The model for the design of ECF shows a savings of 7.17% with limited ends and a savings of 1.67% with free ends for the minimum area, and for the minimum cost, it shows a savings of 23.95% with limited ends and a savings of 9.14% with free ends rather than RCF. Therefore, the proposed development will be of great help to structural engineers specializing in foundations, as it represents significant savings.