This research presents a new mathematical framework for the optimal design of elliptical isolated footings under vertical load and two orthogonal moments. The suggested method considers the spatial variation of contact pressure between the footing and the supporting soil, facilitating an accurate representation of structural requirements. New formulations for bending moment, unidirectional shear, and punching shear are generated using volume integration, accurately representing the complex stress distribution beneath elliptical foundations. Lagrange multipliers are utilized to identify the crucial points of maximum and minimum contact stresses for elliptical and circular footing shapes. A thorough numerical analysis illustrates the benefits of the suggested strategy by contrasting its results with those of a conventional design methodology. The findings demonstrate that the newly created model produces more cost-effective designs while maintaining structural integrity and performance, underscoring its potential as a significant asset in engineering practice. A MATLAB code for design using new formulas is programmed and results obtained to those from literature and were more efficient and economic.

The electrical network, essential to our society, frequently encounters disruptions from lightning strikes, resulting in material damage and power blackouts. Swift diversion of lightning currents to the ground is imperative to safeguard the grid. This study proposes a proportionality coefficient (K) to effectively distribute lightning current between grounding and network flow. The optimality of this coefficient depends on the tower grounding system resistances; lower resistances facilitate optimal distribution, enabling more current to flow to the ground. In the examination of the Djiri-Ngo power line in the Republic of Congo, grounding systems were optimised based on soil types. Three electrodes were used for clayey sand, while fifteen were employed for siliceous sand. Optimal coefficients were determined to be 0.86 for clayey sand and 0.81 for siliceous sand. These coefficients denote that 86% and 81% of the lightning current were directed to the ground, in contrast to non -optimal resistances (69% and 29% with a single grounding electrode). The experiments highlight the importance of adapting grounding systems to soil characteristics, rather than adhering to a uniform approach. Efficient diversion of lightning current to the ground is paramount for grid protection.