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.

In this study, we instrument the foundations and towers for two onshore shallow wind turbine generators (WTGs) to evaluate foundation response, quantify in-service loads, and assess the assumptions behind WTG foundation design calculations. Measurements of pressure at the soil-foundation interface, soil strain just below foundation level, and tower moments over long periods provide insights into the operational moments experienced by the tower and the load transfer mechanisms to the foundation system. The results of this study have implications for design practices in three distinct ways: (1) the assessment of rotational stiffness calculation assumptions, (2) the evaluation of pressure distribution used in the bearing capacity formulation, and (3) the estimation of tower loads used in the tower and anchor bolt design. Our observations show that the induced overturning moments correlate well with incipient wind speeds and directions and the associated soil pressure and strain responses. The overturning moments and the response parameters relate linearly within the spectrum of measured magnitudes. However, the pressure distribution across the foundation footprint does not monotonically increase or decrease with distance from the neutral axis of the foundation base (e.g., the pressure sensed at the foundation's center close to the foundation is between 1.5 and 2 times greater than the pressures sensed at the edges). In addition, the measured soil strain as a function of cyclic moments shows that the in-service cyclic shear strains are less than 1.4 x 10-5 (i.e., two orders of magnitude smaller than the assumed design strain level). Finally, the spectrum of cyclic moments follows a semilog trend, thus indicating that operational and nonoperational loads dominate the fatigue load spectrum. Our study suggests that adequately designed WTG foundations on competent fine-grained soil result in very low operational soil shear stresses and strains, which might indicate that the current design practices are too conservative in nature. Field measurements establish load spectrums for cyclic fatigue loads for the long-term operational conditions of WTGs.