Mathematical Modeling of the Optimal Cost for the Design of Strap Combined Footings

: 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.


Introduction
The main objective of a foundation is to transfer the loads of columns or walls to the resistant strata of the soil, in such a way ensuring stability and permissible settlements according to site standards.
Foundations generally fall into two categories: shallow and deep.
The choice of shallow foundations may vary depending on the type of soil, the layout, the magnitude and nature of the loads, among other things.
The different types of reinforced concrete footings are: isolated footings, combined footings, strap footings and raft or mat foundations.
The isolated footings can be: square, rectangular or circular.The combined footings can have the following shape: rectangular, trapezoidal, strap, L (corner) and T.
Many mathematicians have explored mathematical models for the optimal design of reinforced concrete foundations.
Several researchers have studied optimization following the minimum-cost criterion for square or rectangular isolated footings using different algorithms such as the hybrid big bang-big crunch [26], multi-objective genetic [27], global-local gravitational search [28], mixed-integer nonlinear programming [29], binary-coded genetic and unified particle swarm optimization [30], genetic according to ACI [31], generalized reduced gradient method [32] and computational methods [33].Some researchers have presented optimization based on a practical tool called metaheuristic [34,35], other researchers have taking into account the eccentricity in the footings [36][37][38], including for circular isolated footings subjected to generalized loadings using the sequential unconstrained minimization technique [39].Other researchers have proposed models for the lowest cost design for rectangular combined footings using the modified complex box method [40], using algorithms based on swarm intelligence [41] and, for trapezoidal combined footings, using nonlinear programming methods [42].
According to the bibliographic review, the documents closest to the topic presented in this work are: Aguilera-Mancilla et al. [13] developed a mathematical model for strap combined footings under biaxial bending in each column for the contact area with the soil, but it is restricted to one or two sides in the longitudinal direction, i.e., the case of free sides is not presented.Yáñez-Palafox et al. [24] presented a mathematical model for strap combined footings under biaxial bending in each column for the complete design (areas of steel and thickness) from the known dimensions of the footing, but it is limited to the thickness of footings and beam being equal, one or two property lines, and does not show the lower-cost design.Therefore, this work is complete because it presents the minimum area for the four possible conditions and for the minimum-cost design it shows the four possible conditions and takes into account different thicknesses for the footings and beam.
This paper presents a new mathematical model to obtain the minimum-cost design for the reinforced concrete strap combined footings under biaxial bending, with each column using a genetic algorithm and the pressure being assumed to be linearly distributed along the contact area.This work was carried out in two steps: step one shows the smallest area in contact with the soil and step two presents the lowest cost.The moments, bending shears and punching shears were obtained by integration according to the ACI standard of the American Concrete Institute (318-14) [43].A numerical example is shown to find the minimum cost for the design of strap combined footings taking into account four different conditions: (1) when the footing in the Y direction is not restricted at the ends; (2) when the two footings are square and the columns are located in the centers of the footings; (3) when footing 1 is square and column 1 is located in the center of the footing and footing 2 is not restricted at the ends; (4) when footing 2 is square and column 1 is located in the center of the footing and footing 1 is not restricted at the ends.

Materials and Methods
The available allowable load capacity of the soil q aa is obtained by the following equation [44]: where: q a = allowable load capacity of the soil-this must be determined by principles of soil mechanics in accordance with the general building code (kN/m 2 ); γ wf = the self-weight of the footing (kN/m 2 ); γ ws = the self-weight of the soil fill (kN/m 2 ); γ cd = concrete density (24 kN/m 3 ); γ sd = soil density (kN/m 3 ); d = effective depth of the footing (m); r = the coating of the footing (m); H = the depth of the footing measured from the base of the footing to the ground-free surface (m).
Figure 1 shows a strap combined footing that supports two rectangular columns of different dimensions under an axial load and with two orthogonal moments in each column.

Materials and Methods
The available allowable load capacity of the soil qaa is obtained by the following equation [44]: where: qa = allowable load capacity of the soil-this must be determined by principles of soil mechanics in accordance with the general building code (kN/m 2 ); γwf = the self-weight of the footing (kN/m 2 ); γws = the self-weight of the soil fill (kN/m 2 ); γcd = concrete density (24 kN/m 3 ); γsd = soil density (kN/m 3 ); d = effective depth of the footing (m); r = the coating of the footing (m); H = the depth of the footing measured from the base of the footing to the ground-free surface (m).
Figure 1 shows a strap combined footing that supports two rectangular columns of different dimensions under an axial load and with two orthogonal moments in each column.Table 1 shows the coordinates (x, y) of the pressures below a footing at each vertex with respect to the X and Y axes.Table 2 presents the coordinates (x1, y1) of the pressures below footing 1 at each vertex with respect to the X1 and Y1 axes.Table 1 shows the coordinates (x, y) of the pressures below a footing at each vertex with respect to the X and Y axes.
Table 1.Coordinates (x, y) of the strap combined footing at each vertex.
Table 2 presents the coordinates (x 1 , y 1 ) of the pressures below footing 1 at each vertex with respect to the X 1 and Y 1 axes.

Pressures below Footing 1
q 1 q 2 q 3 q 6 Coordinates Table 3 shows the coordinates (x 2 , y 2 ) of the pressures below footing 2 at each vertex with respect to the X 2 and Y 2 axes.The objective function for smallest contact area with the ground "A min " is [13]: The constraint functions are: where: e = the distance from the center of column 1 to the free end of the footing in the Y direction (m), f = the distance from the center of column 2 to the free end of the footing in the Y direction (m), R = resultant force (kN), M xT = resultant moment around the X axis (kN-m), M yT = resultant moment around the Y axis (kN-m), x n = the distance in the X direction measured from the Y axis to the fiber under study (m), y n = the distance in the Y direction measured from the X axis to the fiber under study (m), A = the area of the footing (m 2 ), y t = the distance to the center of gravity of the footing in the positive Y direction (m), y b = the distance to the center of gravity of the footing in the negative Y direction (m), I x = the moment of inertia with respect to the X axis (m 4 ), I y = the moment of inertia with respect to the Y axis (m 4 ).

Minimum-Cost Design for Strap Combined Footings
The moments, bending shears and punching shears that act on the footing are obtained by integration.
The analysis in the Y direction is carried out with all loads and moments, and the analysis in the X direction is carried out individually for each footing, i.e., for footing 1, the X 1 axis is considered and, for footing 2, the X 2 axis is considered.
The general equation for any footing under biaxial bending using factored loads and moments in the Y direction is [24]: where: q u = the soil pressure under the footing due to the factored loads and moments, R u = the factored resultant force (kN), M uxT = the factored resultant moment around the X axis (kN-m), M uyT = the factored resultant moment around the Y axis (kN-m).
The general equation for any footing under biaxial bending using factored loads and moments in the X 1 direction on footing 1 is: where: q u1 = the soil pressure under footing 1 due to the factored loads and moments of column 1, P u1 = the factored axial load of footing 1 (kN), M ux1 = the factored moment around the X 1 axis (kN-m), M uy1 = the factored moment around the Y 1 axis (kN-m), A 1 = a 1 a 2 , I x1 = a 1 a 2 3 /12, I y1 = a 1 3 a 2 /12.The general equation for any footing under biaxial bending using factored loads and moments in the X 2 direction on footing 2 is: where: q u2 = the soil pressure under footing 2 due to the factored loads and moments of column 2, P u2 = the factored axial load of footing 2 (kN), M ux2 = the factored moment around the X 2 axis (kN-m), M uy2 = the factored moment around the Y 2 axis (kN-m),

Moments
Critical sections for moments appear in sections a, b, c, d, e, f, g, h and i, as shown in Figure 2.
The general equations for each moment on their respective axes are: For For For where: M ua , M ub , M uc , M ud , M ue , M uf , M ug , M uh and M ui are the factored moments acting on each axis (kN-m); y m is the location of the maximum moment; and R u , M uxT and M uyT are obtained by Equations ( 5)-( 7) using factored loads and moments.

Moments
Critical sections for moments appear in sections a, b, c, d, e, f, g, h and i, as shown in Figure 2. The general equations for each moment on their respective axes are: Note: The moment "M uf " is obtained using Equations ( 27)-( 29) depending on where "y m " is located.The steps to obtain the maximum moment "M uf " are: (1) Equations ( 27)-( 29) are developed and subsequently differentiated with respect to y m and set equal to zero to obtain the position of the maximum moment; (2) the equation that should be used to obtain the maximum moment depends on the location of y m .

Bending Shears
The critical bending shear sections appear in sections j, k, l, m, n, o, p and q, as shown in Figure 3.The general equations for each bending shear on their respective axes are: The general equations for each bending shear on their respective axes are: q u1 (x 1 , y 1 )dxdy, (36) q u (x, y)dxdy − P u1 , (46) q u (x, y)dxdy − P u1 − P u2 , (50) where: V uj , V uk , V ul , V um , V un , V uo , V up and V uq are the factored bending shears that act on each axis (kN) and d = the effective depth of the footing (m).
where: α = γ s C s /C c (the relationship between the cost of steel and the cost of concrete).

Constraint Functions
The moment equations are [43]: where: Ø f = the bending strength reduction factor (0.90); f y = the specified yield strength of the reinforcement of steel (MPa); f' c = the specified compressive strength of the concrete at 28 days (MPa); the width of the study surface for moment b w for M ua is a 2 , for M ub it is b 2 , for M uc and M ud it is a 1 , for M ue , M uf and M ug it is c, for M uh and M ui it is b 1 ; the area of steel for moment A s ,for M ua is A sxf1B , for M ub it is A sxf2B , for M uc and M ud it is A syf1B , for M uf it is A sbT , for M ue and M ug it is A sbB , for M uh and M ui it is A syf2B .The bending shear equations are [43]: where: Ø v = the shear strength reduction factor (0.85); s = the space between stirrups; the width of the study surface for bending shear b ws for V uj is a 2 , for V uk it is b 2 , for V ul and V um it is a 1 , for V un and V uo it is c, for V up and V uq it is b 1 .
The punching shear equations are [43]: where: β c = the long side of the column divided by the short side of the column; b 0 = the critical punching shear perimeter (m); α s = 30 for edge columns, α s = 20 for corner columns, and α s = 40 for interior columns.The percentages of reinforcing steel are [43]: The reinforcing steel areas are [43]: A sbB ≥ ρ ybB cd 1 , (78) where: A sxf1T , A syf1T , A sxf2T and A syf2T = the steel areas by temperature, these areas of steel are provided when the thickness is greater than 30 cm and there is no bending [43].Figure 5 presents the algorithm for the optimal design process of strap combined footings (flowchart).

Numerical Examples
Tables 4 and 5 show the four examples of obtaining optimal designs for the reinforced concrete strap combined footings under biaxial bending (one axial load and two orthogonal moments in each column).Example 1: When the footing in the Y direction is not restricted at the ends.Example 2: When the two footings are square, and the columns are located in the center of the footings.Example 3: When footing 1 is square and column 1 is located in the center of the footing, and footing 2 is not restricted at the ends.Example 4: When footing 2 is square and column 1 is located in the center of the footing, and footing 1 is not restricted at the ends. The

Numerical Examples
Tables 4 and 5 show the four examples of obtaining optimal designs for the reinforced concrete strap combined footings under biaxial bending (one axial load and two orthogonal moments in each column).Example 1: When the footing in the Y direction is not restricted at the ends.Example 2: When the two footings are square, and the columns are located in the center of the footings.Example 3: When footing 1 is square and column 1 is located in the center of the footing, and footing 2 is not restricted at the ends.Example 4: When footing 2 is square and column 1 is located in the center of the footing, and footing 1 is not restricted at the ends.the sides of the two footings are not limited in the Y direction (Example 1), the smallest area is presented.Table 5 (the minimum-cost design for strap combined footings) presents the following: When one side of the two footings is not located on the face of the column in the Y direction and the footings are square (Example 2), the lowest cost occurs.
Figure 7 shows in detail the dimensions and area of reinforcing steel for a strap combined footing.

Conclusions
This paper presents an analytical model to determine the minimum cost for the design of strap combined footings under biaxial bending due to the force exerted by each column; the footings are supported on elastic soil, and the analysis assumes a linear distribution of soil pressure.
This work is presented in two steps.
Step one is to find the minimum area; the independent variables are: L, c, c 1 , c 2 , c 3 , c 4 , P 1 , M x1 , M y1 , P 2 , M x2 , M y2 and q aa , and the dependent variables are: A min , a The main contributions are: 1. Some engineers use trial and error to find the dimensions of strap combined footings under biaxial bending, and the design is obtained by assuming maximum and uniform pressure along the bottom of the footing.

2.
The proposed model is not limited to d = d 1 or to one or two property lines, as presented by some authors [13,24].

3.
The equations for the moments, bending shears and punching shears are verified by equilibrium (see Section 4).

4.
The minimum area does not guarantee that it is the lowest cost, since the smallest area is presented in example 1 and the lowest cost appears in example 2.

5.
The proposed model in this paper can be used for any other building code, taking into account the equations that resist the moments, the bending shears and the punching shears, and the equations to obtain the steel areas of the footings and beams.Future papers may be the minimum-cost design for a strap combined footings under biaxial bending, assuming that the contact area of the footing with the ground works partially in compression (one part of the footing contact area is under compression and the other part has no pressure or zero pressure).

Figure 1 .
Figure 1.Isometric view of a strap combined footing.

Figure 1 .
Figure 1.Isometric view of a strap combined footing.

Figure 2 .
Figure 2. Critical sections for moments that correspond to the axes a, b, c, d, e, f, g, h and i.

Figure 2 .
Figure 2. Critical sections for moments that correspond to the axes a, b, c, d, e, f, g, h and i.

Mathematics 2024 , 22 Figure 3 .
Figure 3. Critical bending shear sections that correspond to the axes j, k, l, m, n, o, p and q.

Figure 3 .
Figure 3. Critical bending shear sections that correspond to the axes j, k, l, m, n, o, p and q.

Mathematics 2024 , 22 Figure 5 .
Figure 5. Process for the optimal design of reinforced concrete strap combined footings.

Figure 5 .
Figure 5. Process for the optimal design of reinforced concrete strap combined footings.

Figure 6 .
Figure 6.Process of using software (Maple 15) for the optimal design of reinforced concrete strap combined footings.

Figure 7 .
Figure 7. Diagram of a reinforced concrete strap combined footing.Figure 7. Diagram of a reinforced concrete strap combined footing.

Figure 7 .
Figure 7. Diagram of a reinforced concrete strap combined footing.Figure 7. Diagram of a reinforced concrete strap combined footing.
be used for T-shaped combined footings by substituting c with b 1 and b 2 with b − a 2 in all equations.7.This model can be used for rectangular combined footings by substituting c with a 1 , b 1 with a 1 , b 2 with b and a 2 with b in all equations.

Author
Contributions: A.L.-R.contributed to the original idea of the article, the mathematical development of the new model and coordinated the work in general.V.M.M.-L.contributed to the verification of the model and the discussion of results.G.S.-H.contributed to the programming of the MAPLE 15 software.F.J.O.-C.contributed to the verification of the new model.L.D.L.-L.contributed to the application of the proposed model (examples).E.R.D.-G.contributed to the elaboration of the bibliographic review, figures and tables.All authors have read and agreed to the published version of the manuscript.Funding: The research described in this work was funded by the Universidad Autónoma de Coahuila and Universidad Autónoma del Estado de Hidalgo, Mexico.

Table 4 .
Minimum area of strap combined footings.

Table 4
presents the minimum area of the four examples.Table5shows the minimumcost design of the four examples.

Table 5 .
Minimum-cost design for strap combined footings.A sbB , A sbT , A sxf1B , A sxf1T , A sxf2B , A sxf2T , A syf1B , A syf1T , A syf2B and A syf2T are data adjusted to practical values, the practical values d and d 1 are rounded to multiples of five or ten for falsework situations, and the steel bars adjust to diameters found on the market.A sv is proposed to obtain n and s.