Optimization for trapezoidal combined footings: Optimal design

This work presents a complete optimal model for trapezoidal combined footings that support a concentric load and moments around of the "X" and "Y" axes in each column to obtain the minimum area and the minimum cost. The model presented in this article considers a pressure diagram that has a linear variation (real pressure) and the equations are not limited to some cases. The classic model takes into account a concentric load and the moment around of the "X" axis (transverse axis) that is applied due to each column, i.e., the resultant force is located at the geometric center of the footing on the "Y" axis (longitudinal axis), and when the concentric load and moments around of the "X" and "Y" axes act on the footing is considered the uniform pressure applied on the contact surface of the footing, and it is the maximum pressure. Four numerical problems are presented to find the optimal design of a trapezoidal combined footing under a concentric load and moments around of the "X" and "Y" axes due to the columns: Case 1 not limited in the direction of the Y axis; Case 2 limited in the direction of the Y axis in column 1; Case 3 limited in the direction of the Y axis in column 2; Case 4 limited in the direction of the Y axis in columns 1 an 2. The complete optimal design in terms of cost optimization for the trapezoidal combined footings can be used for the rectangular combined footings considering the uniform width of the footing in the transversal direction, and also for different reinforced concrete design codes, simply by modifying the resisting capacity equations for moment, for bending shear, and for the punching shear, according to each of the codes.