This manuscript aims to present a novel model to find the optimal area of a rectangular isolated footing with an eccentric column, taking into account that the footing is partially supported; that is, one part of the contact surface is compressed and the other part has zero pressure. The methodology is developed by integration and can also be verified using the geometric properties of a triangular-based pyramid to determine the axial load, the moments in the X and Y axes in terms of the available allowable soil pressure, the footing sides, the greatest distance on one of its sides in the X-direction where it crosses the neutral axis, the greatest distance on one of its sides in the Y-direction where it crosses the neutral axis, and the coordinates at the base of the footing. Four types of numerical problems are shown to find the optimal area of a rectangular footing with an eccentric column subjected to biaxial bending: (1) the column in the center of the footing; (2) the column on the edge of the footing in the X-direction; (3) the column on the edge of the footing in the Y-direction; and (4) the column in the corner of the footing. A comparison is presented of the new model against a model proposed by another author. The new model presents a reduction of up to 42.37% for the column in the center of the footing and up to 40.32% for the column in the corner of the footing compared to the model by the other authors. Therefore, the new model will be of great help to professionals in foundation design.

This paper presents a novel mathematical model to determine the minimum cost for the design of reinforced-concrete strap combined footings under biaxial bending, with each column using a genetic algorithm. The pressure is assumed to be linearly distributed along the contact area. This study comprises two steps: firstly, identifying the smallest ground contact area, and secondly, minimizing the cost. The methodology integrates moment, bending shear, and punching shear calculations according to the ACI standard. Some authors present a smaller area (but limited to one or two property lines) and the design considers that the thickness of the footings and beam are equal, and do not show the lower cost of a strap combined footing; generally, the beam has a greater thickness than the footings and therefore the footings would have an unnecessary thickness that would generate a higher cost. A numerical example is shown to find the lowest cost for the design of strap combined footings considering four different conditions such as square footings and other limitation at the ends of the footings. The minimum area does not guarantee that it is the lowest cost. The proposed model is versatile, applicable to T-shaped and rectangular combined footings, and is not restricted to specific property lines. The contributions include eliminating trial and error practices, accommodating various design conditions, and emphasizing equilibrium in the derived equations. The model is adaptable to different building codes, offering a comprehensive approach to achieving optimal design and cost considerations for strap combined footings.

This article shows a model for the design of circular isolated footings and the column placed anywhere in the footing under minimum cost criteria. Some designs for obtaining the diameter, effective depth, and steel areas of the footing under biaxial bending assume the maximum and uniform pressure at the bottom of the footing supported on elastic soils. All these works consider the column placed at the center of the footing. Three numerical problems are given (each problem presents four variants) to determine the lowest cost to design the circular footings under biaxial bending. Problem 1: Column without eccentricity. Problem 2: Column with eccentricity in the direction of the X axis of one quarter of the diameter of the footing. Problem 3: Column placed at the end furthest from the center of the footing on the X axis. The results are verified by the balance of moments, one-way shear or shear and two-way shear or punching. The new model shows a saving of 17.92% in the contact area with soil and of 31.15% in cost compared to the model proposed by other authors. In this way, the proposed minimum cost design model for circular footings will be of great help for the design when the column is placed on the center or edge of the footing.

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.

This paper presents an optimal model for the design of elliptical isolated footings subjected to biaxial bending under the minimum cost criterion, assuming that the footing rests on elastic soils and that the soil pressure distribution is linear. The methodology is developed in two parts. The first is used to obtain the minimum area, and the second is used to determine the minimum cost. Some authors show the equations for circular and elliptical footings for moments, bending shear, and punching shear. However, they do not present the minimum cost, and the numerical examples are presented only for circular footings and not for elliptical footings. Two numerical problems are given (each problem presents five variants), and the optimal cost design for elliptical isolated footings subjected to biaxial bending are shown. Problem 1: Modifying the moment on the Y axis. Problem 2: Modifying the axial load. In addition, a comparison is made between elliptical footings and circular footings. The results show that the minimum area is smaller for elliptical footings than for circular footings, and the minimum cost appears in elliptical footings when the footing dimensions are governed by the minimum pressure. Therefore, the new model for elliptical footings will be of great help to foundation engineering specialists.

Objective: This study aims to use the theoretical mediation of drivers and practices to support SMEs in developing capabilities related to the circular economy (CE). Through this, it seeks to overcome economic, technological, cultural, and social barriers, generating environmental and financial benefits. Theoretical Framework: Circular economy (CE) theory highlights the need for multidisciplinary approaches to address barriers. These barriers, including economic, technological, social, and cultural challenges, hinder CE adoption, particularly in SMEs. Overcoming such obstacles requires innovative solutions, collaboration with green suppliers, and advanced recycling technologies. Practices like circular design, eco-efficiency, and education are critical in driving CE capability. Moreover, mediating variables, such as drivers in circularity and sustainability-focused education, facilitate the transition to CE, creating economic opportunities and fostering sustainable development across different industries. Design/Methodology/Approach: The methodology combines qualitative and quantitative research using PLS-SEM. An analysis of six hypotheses was conducted with questionnaires applied to 280 companies, obtaining 124 responses. This innovative approach demonstrates the impact of theoretical mediation between barriers and circular economy capability (CEC), validating five of the six hypotheses. Results: The key role of mediation in overcoming barriers to the CE is highlighted. Although the direct relationship between cultural and social barriers and the CEC was not validated, the intervention of the mediator construct allowed validation. Additionally, the predictive model is strengthened with indicators such as Q2 and VAF (Variance Accounted For), showing that SMEs can improve their CE capability by overcoming key barriers. Research Limitations/Implications: In Mexico, SMEs lack a detailed analysis of their economic impact and face obstacles to adopting CE. The described methodology can be replicated in other contexts to facilitate the transition to a circular economy. Originality/Value: The novel methodology offers a way to overcome barriers to the circular economy and promote a sustainable organizational culture in SMEs. It can be said that the strategy of this study was to achieve circular economy capability (CEC), with the mediating effect serving as the key tactic.