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

Abstract

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.

1. 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.

The models for the smallest contact area with the ground have been developed for circular, square and rectangular isolated footings under biaxial bending [1,2,3,4,5,6,7,8,9,10,11]. Likewise, models have been presented for combined footings for the smallest contact area with the ground under biaxial bending in each column of L-shaped or corner [12], strap [13] and T-shaped [14] footing.

Several models have been developed for the comprehensive design of reinforced concrete for circular, square and rectangular isolated footings subjected to biaxial bending [15,16,17,18,19,20]. Likewise, models for combined footings have been investigated for the complete design of reinforced concrete under biaxial bending in each column of trapezoidal-shaped [21], rectangular [22], L or corner [23], strap [24] and T [25] footing.

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.

2. Materials and Methods

The available allowable load capacity of the soil q_aa is obtained by the following equation [44]:

$$q_{aa}=q_a-{\gamma }_{wf}-{\gamma }_{ws},$$

(1)

$$q_{aa}=q_a-{\gamma }_{cd}\left(d+r\right)-{\gamma }_{sd}\left(H-d-r\right),$$

(2)

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/m²); γ_wf = the self-weight of the footing (kN/m²); γ_ws = the self-weight of the soil fill (kN/m²); γ_cd = concrete density (24 kN/m³); γ_sd = soil density (kN/m³); 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. Coordinates (x, y) of the strap combined footing at each vertex.

Pressures below Footing	q1	q2	q3	q4	q5	q6	q7	q8	q9	q10	q11	q12
Coordinates	x1	x2	x3	x4	x5	x6	x7	x8	x9	x10	x11	x12
	a1/2	−a1/2	a1/2	c/2	−c/2	−a1/2	b1/2	c/2	−c/2	−b1/2	b1/2	−b1/2
	y1	y2	y3	y4	y5	y6	y7	y8	y9	y10	y11	y12
	yt	yt	yt − a2	yt − a2	yt − a2	yt − a2	b2− yb	b2− yb	b2− yb	b2− yb	−yb	−yb

where: q₁, q₂, q₃, q₄, q₅, q₆, q₇, q₈, q₉, q₁₀, q₁₁, q₁₂ are the pressures below footing at each vertex (kN/m²).

shows the coordinates (x, y) of the pressures below a footing at each vertex with respect to the X and Y axes.

Table 2. Coordinates (x₁, y₁) of footing 1 at each corner.

Pressures below Footing 1	q1	q2	q3	q6
Coordinates	x1	x2	x3	x6
	a1/2	−a1/2	a1/2	−a1/2
	y1	y2	y3	y6
	a2/2	a2/2	−a2/2	−a2/2

presents the coordinates (x₁, y₁) of the pressures below footing 1 at each vertex with respect to the X₁ and Y₁ axes.

Table 3. Coordinates (x₂, y₂) of footing 2 at each corner.

Pressures below Footing 2	q7	q10	q11	q12
Coordinates	x7	x10	x11	x12
	b1/2	−b1/2	b1/2	−b1/2
	y7	y10	y11	y12
	b2/2	b2/2	−b2/2	−b2/2

shows the coordinates (x₂, y₂) of the pressures below footing 2 at each vertex with respect to the X₂ and Y₂ axes.

2.1. Minimum Contact Area on the Soil for Strap Combined Footings

The objective function for smallest contact area with the ground “A_min” is [13]:

$$A_{min}=a_1{a}_2+\left(b-a_2-b_2\right)c+b_1{b}_2,$$

(3)

The constraint functions are:

$$q_n=\frac{R}{A}+\frac{M_{xT}{y}_n}{I_x}+\frac{M_{yT}{x}_n}{I_y},$$

(4)

$$R=P_1+P_2,$$

(5)

$$M_{xT}=M_{x1}+M_{x2}+R\left(y_t-e\right)-P_{2}L,$$

(6)

$$M_{yT}=M_{y1}+M_{y2},$$

(7)

$$y_t=\frac{{a_2}^2\left(a_1-c\right)+c{\left(b-b_2\right)}^2+b_1{b}_2\left(2b-b_2\right)}{2\left[a_1{a}_2+\left(b-a_2-b_2\right)c+b_1{b}_2\right]},$$

(8)

$$y_b=\frac{a_2\left(2b-a_2\right)\left(a_1-c\right)+c\left(b^2-{b_2}^2\right)+b_1{b_2}^2}{2\left[a_1{a}_2+\left(b-a_2-b_2\right)c+b_1{b}_2\right]},$$

(9)

$$\begin{gathered} I_x=\frac{a_1{a_2}^3+c{\left(b-a_2-b_2\right)}^3+b_1{b_2}^3}{12}+c\left(b-a_2-b_2\right){\left[a_2+\frac{\left(b-a_2-b_2\right)}{2}-y_t\right]}^2 \\ +a_1{a}_2{\left(y_t-\frac{a_2}{2}\right)}^2+b_1{b}_2{\left(b-y_t-\frac{b_2}{2}\right)}^2, \end{gathered}$$

(10)

$$I_y=\frac{a_2{a_1}^3+\left(b-a_2-b_2\right)c^3+b_2{b_1}^3}{12},$$

(11)

$$0\le \left\{\begin{array}{l}{q}_1\\ q_2\\ q_3\\ q_4\\ q_5\\ q_6\\ q_7\\ q_8\\ q_9\\ q_{10}\\ q_{11}\\ q_{12}\end{array}\right\}\le q_{aa},$$

(12)

$$e+L+f=b,$$

(13)

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 (m²), 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 (m⁴), I_y = the moment of inertia with respect to the Y axis (m⁴).

2.2. 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 X₁ axis is considered and, for footing 2, the X₂ axis is considered.

The general equation for any footing under biaxial bending using factored loads and moments in the Y direction is [24]:

$$q_u\left(x, y\right)=\frac{R_u}{A}+\frac{M_{uxT}y}{I_x}+\frac{M_{uyT}x}{I_y},$$

(14)

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 X₁ direction on footing 1 is:

$$q_{u1}\left(x_1, y_1\right)=\frac{P_{u1}}{A_1}+\frac{\left[M_{ux1}+P_{u1}\left(e-a_2/2\right)\right]y}{I_{x1}}+\frac{M_{uy1}x}{I_{y1}},$$

(15)

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 X₁ axis (kN-m), M_uy1 = the factored moment around the Y₁ axis (kN-m), A₁ = a₁a₂, I_x1 = a₁ a₂³/12, I_y1 = a₁³a₂/12.

The general equation for any footing under biaxial bending using factored loads and moments in the X₂ direction on footing 2 is:

$$q_{u2}\left(x_2, y_2\right)=\frac{P_{u2}}{A_2}+\frac{\left[M_{ux2}+P_{u2}\left(f-b_2/2\right)\right]y}{I_{x2}}+\frac{M_{uy1}x}{I_{y2}},$$

(16)

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 X₂ axis (kN-m), M_uy2 = the factored moment around the Y₂ axis (kN-m), A₂ = b₁b₂, I_x2 = b₁ b₂³/12, I_y2 = b₁³b₂/12.

2.2.1. 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:

$$M_{ua}=-{\int }_{-a_2/2}^{a_2/2}{\int }_{c_1/2}^{a_1/2}{q}_{u1}\left(x_1, y_1\right)\left(x-\frac{c_1}{2}\right)dxdy,$$

(17)

$$M_{ua}=-\frac{a_2\left\{{6P}_{u1}{I}_{y1}{\left(a_1-c_1\right)}^2+M_{uy1}{A}_1\left({c_1}^3+{a_1}^2\left[2a_1-3c_1\right]\right)\right\}}{48A_1{I}_{y1}},$$

(18)

$$M_{ub}=-{\int }_{-b_2/2}^{b_2/2}{\int }_{c_3/2}^{b_1/2}{q}_{u2}\left(x_2, y_2\right)\left(x-\frac{c_3}{2}\right)dxdy,$$

(19)

$$M_{ub}=-\frac{b_2\left\{{6P}_{u2}{I}_{y2}{\left(b_1-c_3\right)}^2+M_{uy2}{A}_2\left[{c_3}^3+{b_1}^2\left(2b_1-3c_3\right)\right]\right\}}{48A_2{I}_{y2}},$$

(20)

$$M_{uc}=-{\int }_{y_t-e+c_2/2}^{y_t}{\int }_{-a_1/2}^{a_1/2}{q}_u\left(x, y\right)\left(y-y_t+e-\frac{c_2}{2}\right)dxdy,$$

(21)

$$M_{uc}=-\frac{a_1{\left(c_2-2e\right)}^2\left[{6R}_u{I}_x+M_{uxT}A\left(6y_t+c_2-2e\right)\right]}{48A{I}_x},$$

(22)

$$M_{ud}=\frac{P_{u1}{c}_2}{2}+M_{ux1}-{\int }_{y_t-e-c_2/2}^{y_t}{\int }_{-a_1/2}^{a_1/2}{q}_u\left(x, y\right)\left(y-y_t+e+\frac{c_2}{2}\right)dxdy,$$

(23)

$$M_{ud}=\frac{P_{u1}{c}_2}{2}+M_{ux1}-\frac{a_1{\left(c_2+2e\right)}^2\left[{6R}_u{I}_x+M_{uxT}A\left(6y_t-c_2-2e\right)\right]}{48A{I}_x},$$

(24)

$$M_{ue}=P_{u1}\left(a_2-e\right)+M_{ux1}-{\int }_{y_t-a_2}^{y_t}{\int }_{-a_1/2}^{a_1/2}{q}_u\left(x, y\right)\left(y-y_t+a_2\right)dxdy,$$

(25)

$$M_{ue}=P_{u1}\left(a_2-e\right)+M_{ux1}-\frac{a_1{a_2}^2\left[{3R}_u{I}_x+M_{uxT}A\left(3y_t-a_2\right)\right]}{6A{I}_x},$$

(26)

For y_t − e ≥ y_m ≥ y_t − a₂

$$M_{uy}=P_{u1}\left(y_t-e-y_m\right)+M_{ux1}-{\int }_{y_m}^{y_t}{\int }_{-a_1/2}^{a_1/2}{q}_u\left(x, y\right)\left(y-y_m\right)dxdy,$$

(27)

For y_t − a₂ ≥ y_m ≥ y_t − b + b₂

$$M_{uy}=P_{u1}\left(y_t-e-y_m\right)+M_{ux1}-{\int }_{y_t-a_2}^{y_t}{\int }_{-a_1/2}^{a_1/2}{q}_u\left(x, y\right)\left(y-y_m\right)dxdy-{\int }_{y_m}^{y_t-a_2}{\int }_{-c/2}^{c/2}{q}_u\left(x, y\right)\left(y-y_m\right)dxdy,$$

(28)

For y_t − b + b₂ ≥ y_m ≥ y_t − b + f

$$\begin{gathered} M_{uy}=P_{u1}\left(y_t-e-y_m\right)+M_{ux1}-{\int }_{y_t-a_2}^{y_t}{\int }_{-a_1/2}^{a_1/2}{q}_u\left(x, y\right)\left(y-y_m\right)dxdy-{\int }_{y_t-b+b_2}^{y_t-a_2}{\int }_{-c/2}^{c/2}{q}_u\left(x, y\right)\left(y-y_m\right)dxdy \\ -{\int }_{y_m}^{y_t-b+b_2}{\int }_{-b_1/2}^{b_1/2}{q}_u\left(x, y\right)\left(y-y_m\right)dxdy, \end{gathered}$$

(29)

$$\begin{gathered} M_{ug}=P_{u1}\left(b-e-b_2\right)+M_{ux1}-{\int }_{y_t-a_2}^{y_t}{\int }_{-a_1/2}^{a_1/2}{q}_u\left(x, y\right)\left(y-y_t+b-b_2\right)dxdy \\ -{\int }_{y_t-b+b_2}^{y_t-a_2}{\int }_{-c/2}^{c/2}{q}_u\left(x, y\right)\left(y-y_t+b-b_2\right)dxdy, \end{gathered}$$

(30)

$$\begin{gathered} M_{ug}=P_{u1}\left(b-e-b_2\right)-\frac{\left[R_u{I}_x+M_{uxT}A{y}_t\right]\left[c{\left(a_2-b+b_2\right)}^2-a_1{a}_2\left(a_2-2b+2b_2\right)\right]}{2A{I}_x} \\ -\frac{M_{uxT}\left[{a_2}^2\left(a_1-c\right)\left(2a_2-3b+3b_2\right)-c{\left(b-b_2\right)}^3\right]}{6I_x}+M_{ux1}, \end{gathered}$$

(31)

$$\begin{gathered} M_{uh}=P_{u1}\left(b-e-f-\frac{c_4}{2}\right)+M_{ux1}-{\int }_{y_t-a_2}^{y_t}{\int }_{-a_1/2}^{a_1/2}{q}_u\left(x, y\right)\left(y-y_t+b-f-\frac{c_4}{2}\right)dxdy \\ -{\int }_{y_t-b+b_2}^{y_t-a_2}{\int }_{-c/2}^{c/2}{q}_u\left(x, y\right)\left(y-y_t+b-f-\frac{c_4}{2}\right)dxdy \\ \hspace{1em}\hspace{1em}-{\int }_{y_t-b+f+c_4/2}^{y_t-b+b_2}{\int }_{-b_1/2}^{b_1/2}{q}_u\left(x, y\right)\left(y-y_t+b-f-\frac{c_4}{2}\right)dxdy, \end{gathered}$$

(32)

$$\begin{gathered} M_{uh}=P_{u1}\left(b-e-f-\frac{c_4}{2}\right)-\frac{\left[R_u{I}_x+M_{uxT}A{y}_t\right]\left[b_1{c}_4\left(c_4+4f-4b_2\right)-4a_1{a}_2\left(a_2+c_4-2b+2f\right)\right]}{8A{I}_x} \\ \hspace{1em}\hspace{1em}\hspace{1em}-\frac{\left[R_u{I}_x+M_{uxT}A{y}_t\right]\left[b_1{\left(f-b_2\right)}^2+c\left(a_2-b_2-b+c_4+2f\right)\left(a_2+b_2-b\right)\right]}{2A{I}_x} \\ -\frac{M_{uxT}\left\{4{a_2}^2\left(a_1-c\right)\left[4a_2+3\left(c_4+2f-2b\right)\right]+b_1{\left(c_4+2f-2b\right)}^3\right\}}{48I_x} \\ \hspace{1em}-\frac{M_{uxT}\left(b_1-c\right)\left[4{b_2}^3-3{\left(b-b_2\right)}^2\left(c_4+2f\right)+2b^3-6b{b_2}^2\right]}{12I_x}+M_{ux1}, \end{gathered}$$

(33)

$$\begin{gathered} M_{ui}=P_{u1}\left(b-e-f+\frac{c_4}{2}\right)+\frac{P_{u2}{c}_4}{2}+M_{ux1}+M_{ux2}-{\int }_{y_t-a_2}^{y_t}{\int }_{-a_1/2}^{a_1/2}{q}_u\left(x, y\right)\left(y-y_t+b-f+\frac{c_4}{2}\right)dxdy \\ -{\int }_{y_t-b+b_2}^{y_t-a_2}{\int }_{-c/2}^{c/2}{q}_u\left(x, y\right)\left(y-y_t+b-f+\frac{c_4}{2}\right)dxdy \\ \hspace{1em}\hspace{1em}-{\int }_{y_t-b+f-c_4/2}^{y_t-b+b_2}{\int }_{-b_1/2}^{b_1/2}{q}_u\left(x, y\right)\left(y-y_t+b-f+\frac{c_4}{2}\right)dxdy, \end{gathered}$$

(34)

$$\begin{gathered} M_{ui}=P_{u1}\left(b-e-f+\frac{c_4}{2}\right)+\frac{P_{u2}{c}_4}{2}-\frac{\left[R_u{I}_x+M_{uxT}A{y}_t\right]\left[b_1{c}_4\left(c_4-4f+4b_2\right)-4a_1{a}_2\left(a_2-c_4-2b+2f\right)\right]}{8A{I}_x} \\ \hspace{1em}\hspace{1em}\hspace{1em}-\frac{\left[R_u{I}_x+M_{uxT}A{y}_t\right]\left[b_1{\left(f-b_2\right)}^2+c\left(a_2-b_2-b-c_4+2f\right)\left(a_2+b_2-b\right)\right]}{2A{I}_x} \\ -\frac{M_{uxT}\left\{4{a_2}^2\left(a_1-c\right)\left[4a_2-3\left(c_4+2b-2f\right)\right]-b_1{\left(c_4+2b-2f\right)}^3\right\}}{48I_x} \\ \hspace{1em}\hspace{1em}\hspace{1em}-\frac{M_{uxT}\left(b_1-c\right)\left[4{b_2}^3+3{\left(b-b_2\right)}^2\left(c_4-2f\right)+2b^3-6b{b_2}^2\right]}{12I_x}+M_{ux1}+M_{ux2}, \end{gathered}$$

(35)

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.

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.

2.2.2. 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:

$$V_{uj}={\int }_{-a_2/2}^{a_2/2}{\int }_{c_1/2+d}^{a_1/2}{q}_{u1}\left(x_1, y_1\right)dxdy,$$

(36)

$$V_{uj}=\frac{a_2\left(a_1-c_1-2d\right)\left[{4P}_{u1}{I}_{y1}+M_{uy1}{A}_1\left(a_1+c_1+2d\right)\right]}{8A_1{I}_{y1}},$$

(37)

$$V_{uk}={\int }_{-b_2/2}^{b_2/2}{\int }_{c_3/2+d}^{b_1/2}{q}_{u2}\left(x_2, y_2\right)dxdy,$$

(38)

$$V_{uk}=\frac{b_2\left(b_1-c_3-2d\right)\left[{4P}_{u2}{I}_{y2}+M_{uy2}{A}_2\left(b_1+c_3+2d\right)\right]}{8A_2{I}_{y2}},$$

(39)

$$V_{ul}={\int }_{y_t-e+c_2/2+d}^{y_t}{\int }_{-a_1/2}^{a_1/2}{q}_u\left(x, y\right)dxdy,$$

(40)

$$V_{ul}=\frac{a_1\left(2e-c_2-2d\right)\left[{4R}_u{I}_x+M_{uxT}A\left(4y_t-2e+c_2+2d\right)\right]}{8A{I}_x},$$

(41)

$$V_{um}={\int }_{y_t-e-c_2/2-d}^{y_t}{\int }_{-a_1/2}^{a_1/2}{q}_u\left(x, y\right)dxdy-P_{u1},$$

(42)

$$V_{um}=\frac{a_1\left(2e+c_2+2d\right)\left[{4R}_u{I}_x+M_{uxT}A\left(4y_t-2e-c_2-2d\right)\right]}{8A{I}_x}-P_{u1},$$

(43)

$$V_{un}={\int }_{y_t-a_2}^{y_t}{\int }_{-a_1/2}^{a_1/2}{q}_u\left(x, y\right)dxdy-P_{u1},$$

(44)

$$V_{un}=\frac{a_1{a}_2\left[{2R}_u{I}_x+M_{uxT}A\left(2y_t-a_2\right)\right]}{2A{I}_x}-P_{u1},$$

(45)

$$V_{uo}={\int }_{y_t-a_2}^{y_t}{\int }_{-a_1/2}^{a_1/2}{q}_u\left(x, y\right)dxdy+{\int }_{y_t-b+b_2}^{y_t-a_2}{\int }_{-c/2}^{c/2}{q}_u\left(x, y\right)dxdy-P_{u1},$$

(46)

$$V_{uo}=\frac{a_2\left(a_1-c\right)\left[{2R}_u{I}_x+M_{uxT}A\left(2y_t-a_2\right)\right]}{2A{I}_x}+\frac{c\left(b-b_2\right)\left[{2R}_u{I}_x+M_{uxT}A\left(2y_t-b+b_2\right)\right]}{2A{I}_x}-P_{u1},$$

(47)

$$V_{up}={\int }_{y_t-a_2}^{y_t}{\int }_{-a_1/2}^{a_1/2}{q}_u\left(x, y\right)dxdy+{\int }_{y_t-b+b_2}^{y_t-a_2}{\int }_{-c/2}^{c/2}{q}_u\left(x, y\right)dxdy+{\int }_{y_t-b+f+c_4/2+d}^{y_t-b+b_2}{\int }_{-b_1/2}^{b_1/2}{q}_u\left(x, y\right)dxdy-P_{u1},$$

(48)

$$\begin{gathered} V_{up}=\frac{\left[R_u{I}_x+M_{uxT}A{y}_t\right]\left[2a_2\left(a_1-c\right)+2c\left(b-b_2\right)-b_1\left(c_4+2d+2f-2b_2\right)\right]}{2A{I}_x} \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\frac{M_{uxT}{b}_1\left[4{\left(f+d\right)}^2+4{\left(b-d\right)}^2-{\left(c_4-2d\right)}^2+2\left(c_4+2f\right)\left(c_4-2b\right)\right]}{8I_x} \\ -\frac{M_{uxT}\left[{a_2}^2\left(a_1-c\right)-\left(b_1-c\right){\left(b-b_2\right)}^2\right]}{2I_x}-P_{u1}, \end{gathered}$$

(49)

$$V_{uq}={\int }_{y_t-a_2}^{y_t}{\int }_{-a_1/2}^{a_1/2}{q}_u\left(x, y\right)dxdy+{\int }_{y_t-b+b_2}^{y_t-a_2}{\int }_{-c/2}^{c/2}{q}_u\left(x, y\right)dxdy+{\int }_{y_t-b+f-c_4/2-d}^{y_t-b+b_2}{\int }_{-b_1/2}^{b_1/2}{q}_u\left(x, y\right)dxdy-P_{u1}-P_{u2},$$

(50)

$$\begin{gathered} V_{uq}=\frac{\left[R_u{I}_x+M_{uxT}A{y}_t\right]\left[2a_2\left(a_1-c\right)+2c\left(b-b_2\right)+b_1\left(c_4+2d-2f+2b_2\right)\right]}{2A{I}_x} \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\frac{M_{uxT}{b}_1\left[4{\left(d-f\right)}^2+4{\left(b+d\right)}^2-{\left(c_4-2d\right)}^2+2\left(c_4+2b\right)\left(c_4-2f\right)\right]}{8I_x} \\ -\frac{M_{uxT}\left[{a_2}^2\left(a_1-c\right)-\left(b_1-c\right){\left(b-b_2\right)}^2\right]}{2I_x}-P_{u1}-P_{u2}. \end{gathered}$$

(51)

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).

2.2.3. Punching Shears

The critical punching shear sections appear at a distance “d/2” starting from the junction of the column with the footing in two directions, as shown in Figure 4.

The general equations for each punching shear that acts on the footing due to each column are:

$$V_{up1}=P_{u1}-{\int }_{y_t-e-\left(c_2+d\right)/2}^{y_t-e+c_2/2+s_1}{\int }_{-\left(c_1+d\right)/2}^{\left(c_1+d\right)/2}{q}_u\left(x, y\right)dxdy,$$

(52)

$$V_{up1}=P_{u1}-\frac{\left\{4\left[R_u{I}_x+M_{uxT}A\left(y_t-e\right)\right]\left(d+2c_2+2s_1\right)-M_{uxT}A\left[{\left(c_2+d\right)}^2-{\left(c_2+2s_1\right)}^2\right]\right\}\left(c_1+d\right)}{8A{I}_x},$$

(53)

$$V_{up2}=P_{u2}-{\int }_{y_t-b+f-c_4/2-s_2}^{y_t-b+f+\left(c_4+d\right)/2}{\int }_{-\left(c_3+d\right)/2}^{\left(c_3+d\right)/2}{q}_u\left(x, y\right)dxdy,$$

(54)

$$V_{up2}=P_{u2}-\frac{\left\{4\left[R_u{I}_x+M_{uxT}A\left(y_t-b+f\right)\right]\left(d+2c_4+2s_2\right)+M_{uxT}A\left[{\left(c_4+d\right)}^2-{\left(c_4+2s_2\right)}^2\right]\right\}\left(c_3+d\right)}{8A{I}_x},$$

(55)

where: V_up1 and V_up2 are the factored punching shears that act on each column (kN), e ≥ c₂/2 + d/2 → s₁ = d/2 and e ≤ c₂/2 + d/2 → s₁ = s₁, f ≥ c₄/2 + d/2 → s₂ = d/2 and f ≤ c₄/2 + d/2 → s₂ = s₂.

2.2.4. Objective Function for Lower Cost

The lowest cost is presented based on the cost of steel and the cost of concrete (the costs include materials and manpower). The lowest cost of the strap combined footing is [38]:

$$C_{min}=V_c{C}_c+{V_s\gamma }_s{C}_s,$$

(56)

where: C_min = lower cost (dollars); C_c = the cost of concrete (dollars/m³); C_s = the cost of steel (dollars/kN); V_c = concrete volume (m³); V_s = steel volume (m³); γ_s = Steel density (76.94 kN/m³).

The volumes for the strap combined footings are:

$$\begin{gathered} V_s=\left(A_{syf1B}+A_{syf1T}\right)a_2+\left(A_{sxf1B}+A_{sxf1T}\right)a_1+\left(A_{syf2B}+A_{syf2T}\right)b_2+\left(A_{sxf2B}+A_{sxf2T}\right)b_1 \\ +\left(A_{sbT}+A_{sbB}\right)\left(\frac{c_2}{2}+L+\frac{c_4}{2}\right)+2n{A}_{sv}\left(c+d_1-3r\right), \end{gathered}$$

(57)

$$\begin{array}{ll}{V}_c=& \left[a_1{a}_2+\left(b-a_2-b_2\right)c+b_1{b}_2\right]\left(d+r\right)+\left(\frac{c_2}{2}+L+\frac{c_4}{2}\right)c\left(d_1-d-r\right)-\left(A_{sxf1B}+A_{sxf1T}\right)a_1\\ & \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\left(A_{syf2B}+A_{syf2T}\right)b_2-\left(A_{sxf2B}+A_{sxf2T}\right)b_1-\left(A_{sbT}+A_{sbB}\right)\left(\frac{c_2}{2}+L+\frac{c_4}{2}\right)\\ & \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-2n{A}_{sv}\left(c+d_1-3r\right),\end{array}$$

(58)

where: A_syf1B = the steel area at the bottom of footing 1 in the Y direction with a width of a₁ (m²); A_syf1T = the steel area at the top of footing 1 in the Y direction with a width of a₁ (m²); A_sxf1B = the steel area at the bottom of footing 1 in the X direction with a width of a₂ (m²); A_sxf1T = the steel area at the top of footing 1 in the X direction with a width of a₂ (m²); A_syf2B = the steel area at the bottom of footing 2 in the Y direction with a width of b₁ (m²); A_syf2T = the steel area at the top of footing 2 in the Y direction with a width of b₁ (m²); A_sxf2B = the steel area at the bottom of footing 2 in the X direction with a width of b₂ (m²); A_sxf2T = the steel area at the top of footing 2 in the X direction with a width of b₂ (m²); A_sbB = the steel area at the bottom of the beam in the Y direction with a width of c (m²); A_sbT = the steel area at the top of the beam in the Y direction with a width of c (m²); A_sv = the steel area due to beam shear in the transverse direction; d₁ = the effective depth of the beam (m); n = the number of stirrups.

Note: A_sv is the number of times the steel area of the stirrup crosses the neutral axis of the beam and, in this case, it will be twice the steel area of the stirrup; the length of each stirrup will be 2c + 2d₁ − 6r.

By substituting Equations (57) and (58) into Equation (56) we obtained:

$$\begin{array}{ll}{C}_{min}=& C_c\left\{\right[a_1{a}_2+\left(b-a_2-b_2\right)c+b_1{b}_2]\left(d+r\right)+\left(\frac{c_2}{2}+L+\frac{c_4}{2}\right)c\left(d_1-d-r\right)-\left(A_{sxf1B}+A_{sxf1T}\right)a_1\\ & \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\left(A_{syf2B}+A_{syf2T}\right)b_2-\left(A_{sxf2B}+A_{sxf2T}\right)b_1-\left(A_{sbT}+A_{sbB}\right)\left(\frac{c_2}{2}+L+\frac{c_4}{2}\right)\\ & \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-2n{A}_{sv}\left(c+d_1-3r\right)\}\\ & \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\gamma }_s{C}_s[\left(A_{syf1B}+A_{syf1T}\right)a_2+\left(A_{sxf1B}+A_{sxf1T}\right)a_1+\left(A_{syf2B}+A_{syf2T}\right)b_2\\ & \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\left(A_{sxf2B}+A_{sxf2T}\right)b_1+\left(A_{sbT}+A_{sbB}\right)\left(\frac{c_2}{2}+L+\frac{c_4}{2}\right)+2n{A}_{sv}\left(c+d_1-3r\right)].\end{array}$$

(59)

Now, substituting γ_sC_s = αC_c into Equation (57) is presented:

$$\begin{array}{ll}{C}_{min}=& C_c\left\{\right[\left(A_{syf1B}+A_{syf1T}\right)a_2+\left(A_{sxf1B}+A_{sxf1T}\right)a_1+\left(A_{syf2B}+A_{syf2T}\right)b_2+\left(A_{sxf2B}+A_{sxf2T}\right)b_1\\ & \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\left(A_{sbT}+A_{sbB}\right)\left(\frac{c_2}{2}+L+\frac{c_4}{2}\right)+2n{A}_{sv}\left(c+d_1-3r\right)]\left(\alpha -1\right)\\ & \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\left[a_1{a}_2+\left(b-a_2-b_2\right)c+b_1{b}_2\right]\left(d+r\right)+\left(\frac{c_2}{2}+L+\frac{c_4}{2}\right)c\left(d_1-d-r\right)\},\end{array}$$

(60)

where: α = γ_sC_s/C_c (the relationship between the cost of steel and the cost of concrete).

2.2.5. Constraint Functions

The moment equations are [43]:

$$\left|M_{ua}\right|,\left|M_{ub}\right|,\left|M_{uc}\right|,\left|M_{ud}\right|,\left|M_{ue}\right|,\left|M_{uf}\right|,\left|M_{ug}\right|,\left|M_{uh}\right|,\left|M_{ui}\right|\le {Ø}_f{f}_{y}d{A}_s\left(1-\frac{0.59A_s{f}_y}{b_{w}d{f^{\prime }}_c}\right),$$

(61)

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 a₂, for M_ub it is b₂, for M_uc and M_ud it is a₁, for M_ue, M_uf and M_ug it is c, for M_uh and M_ui it is b₁; 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]:

$$\left|V_{uj}\right|,\left|V_{uk}\right|,\left|V_{ul}\right|,\left|V_{um}\right|,\left|V_{up}\right|,\left|V_{uq}\right|\le 0.17{\varnothing }_v\sqrt{{f\prime }_c}{b}_{ws}d,$$

(62)

$$\left|V_{un}\right|,\left|V_{uo}\right|\le 0.17{\varnothing }_v\sqrt{{f^{\prime }}_c}{b}_{ws}d+\frac{{\varnothing }_v{A}_{sv}{f}_{y}d}{s},$$

(63)

$$n\ge \frac{\left(c_2+2L+c_4\right)}{2s},$$

(64)

$$s\le \frac{d_1}{2},$$

(65)

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 a₂, for V_uk it is b₂, for V_ul and V_um it is a₁, for V_un and V_uo it is c, for V_up and V_uq it is b₁.

The punching shear equations are [43]:

$$\left|V_{up1}\right|,\left|V_{up2}\right|\le \left\{\begin{array}{c}0.17{\varnothing }_v\left(1+\frac{2}{{\beta }_c}\right)\sqrt{{f\prime }_c}{b}_{0}d\\ 0.083{\varnothing }_v\left(\frac{{\alpha }_{s}d}{b_0}+2\right)\sqrt{{f\prime }_c}{b}_{0}d\\ 0.33{\varnothing }_v\sqrt{{f\prime }_c}{b}_{0}d\end{array}\right.,$$

(66)

where: β_c = the long side of the column divided by the short side of the column; b₀ = 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]:

$${\rho }_{xf1B},{\rho }_{xf2B},{\rho }_{ybB},{\rho }_{ybT},{\rho }_{yf1B},{\rho }_{yf2B}\le 0.75\left[\frac{0.85{\beta }_1{f\prime }_c}{f_y}\left(\frac{600}{600+f_y}\right)\right],$$

(67)

$${\rho }_{xf1B},{\rho }_{xf2B},{\rho }_{ybB},{\rho }_{ybT},{\rho }_{yf1B},{\rho }_{yf2B}\ge \left\{\begin{array}{c}\frac{0.25\sqrt{{f^{\prime }}_c}}{f_y}\\ \frac{1.4}{f_y}\end{array}\right.,$$

(68)

$$0.65\le {\beta }_1=\left(1.05-\frac{{f^{\prime }}_c}{140}\right)\le 0.85.$$

(69)

The reinforcing steel areas are [43]:

$$A_{sxf1B}\ge {\rho }_{xf1B}{a}_{2}d,$$

(70)

$$A_{sxf1T}\ge 0.0018a_2\left(d+r\right),$$

(71)

$$A_{syf1B}\ge {\rho }_{xf1B}{a}_{1}d,$$

(72)

$$A_{syf1T}\ge 0.0018a_1\left(d+r\right),$$

(73)

$$A_{sxf2B}\ge {\rho }_{xf1B}{b}_{2}d,$$

(74)

$$A_{sxf2T}\ge 0.0018b_2\left(d+r\right),$$

(75)

$$A_{syf2B}\ge {\rho }_{xf1B}{b}_{1}d,$$

(76)

$$A_{syf2T}\ge 0.0018b_1\left(d+r\right),$$

(77)

$$A_{sbB}\ge {\rho }_{ybB}c{d}_1,$$

(78)

$$A_{sbT}\ge {\rho }_{ybT}c{d}_1,$$

(79)

$$A_{sv}\ge \frac{0.35b_{ws}s}{f_y},$$

(80)

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).

Figure 6 shows the use of software (Maple 15) for the optimal design of strap combined footings (flowchart).

3. Numerical Examples

Table 4. Minimum area of strap combined footings.

Concept	Free Ends		Equal Sides for the Footings 1 and 2		Equal Sides for the Footing 1		Equal Sides for the Footing 2
	Theoretical	Practical	Theoretical	Practical	Theoretical	Practical	Theoretical	Practical
Ix (m4)	139.75	143.19	179.49	189.20	150.79	154.13	116.58	120.65
Iy (m4)	44.03	44.54	9.94	11.09	19.49	19.89	39.41	40.66
MxT (kN-m)	−623.79	−553.30	−572.81	−611.62	753.88	611.46	−585.50	−475.07
MyT (kN-m)	480	480	480	480	480	480	480	480
R (kN)	2300	2300	2300	2300	2300	2300	2300	2300
a1 (m)	8.07	8.10	3.11	3.20	2.64	2.70	7.72	7.80
a2 (m)	1.00	1.00	3.11	3.20	2.64	2.70	1.00	1.00
b (m)	9.63	9.70	9.68	9.75	8.52	8.55	8.12	8.15
b1 (m)	1.00	1.00	2.24	2.30	5.70	5.70	1.85	1.90
b2 (m)	2.63	2.70	2.24	2.30	1.00	1.00	1.85	1.90
c (m)	0.40	0.40	0.40	0.40	0.40	0.40	0.40	0.40
e (m)	0.20	0.20	1.56	1.60	1.32	1.35	0.20	0.20
f (m)	2.43	2.50	1.12	1.15	0.20	0.20	0.92	0.95
yt (m)	2.71	2.74	4.09	4.12	4.43	4.40	2.73	2.78
σ1 (kN/m2)	207.44	207.29	201.97	189.41	211.90	204.09	206.88	205.35
σ2 (kN/m2)	119.48	120.00	51.93	50.95	146.83	138.92	112.79	113.27
σ3 (kN/m2)	211.90	211.15	211.90	199.76	198.69	193.37	211.90	209.29
σ4 (kN/m2)	170.10	169.67	146.44	139.18	171.08	165.62	167.29	165.61
σ5 (kN/m2)	165.74	165.36	127.12	121.87	161.23	155.96	162.42	160.89
σ6 (kN/m2)	123.94	123.87	61.66	61.29	133.62	128.21	117.82	117.21
σ7 (kN/m2)	200.15	196.08	204.74	194.03	211.90	210.34	202.61	195.14
σ8 (kN/m2)	196.88	192.85	160.24	152.92	146.69	146.38	193.79	186.28
σ9 (kN/m2)	192.52	188.54	140.92	135.61	136.84	136.72	188.92	181.56
σ10 (kN/m2)	189.25	185.31	96.41	94.50	71.63	72.76	180.09	211.90
σ11 (kN/m2)	211.90	206.52	211.90	201.46	206.90	206.37	211.90	202.62
σ12 (kN/m2)	201.00	195.74	103.57	101.94	66.93	68.79	189.38	180.19
Amin (m2)	13.10	13.20	16.44	17.23	14.63	14.93	13.25	13.51

where: a₁, a₂, b, b₁, b₂, e and f are data adjusted to practical values, these practical values are rounded to multiples of five or ten for falsework situations, since the formwork is sold in units of the English system.

and

Table 5. Minimum-cost design for strap combined footings.

Concept	Free ends		Equal Sides for the Footings 1 and 2		Equal Sides for the Footing 1		Equal Sides for the Footing 2
	Theoretical	Practical	Theoretical	Practical	Theoretical	Practical	Theoretical	Practical
d (cm)	90.92	92.00	39.57	42.00	41.86	42.00	90.17	92.00
d1 (cm)	141.90	142.00	63.56	67.00	67.62	72.00	97.31	102.00
n	21.24	22	23.28	24	21.89	21	21.15	22
s (cm)	34.83	34	31.78	31	33.81	36.00	34.99	35
AsbB (cm2)	20.63	25.35 (5Ø1”)	8.47	10.14 (2Ø1”)	9.01	10.14 (2Ø1”)	12.96	15.21 (3Ø1”)
AsbT (cm2)	18.90	20.28 (4Ø1”)	25.40	30.42 (6Ø1”)	30.47	35.49 (7Ø1”)	30.92	35.49 (7Ø1”)
Asxf1B (cm2)	54.13	55.77 (11Ø1”)	46.46	50.70 (10Ø1”)	37.64	40.56 (8Ø1”)	52.46	55.77 (11Ø1”)
Asxf1T (cm2)	17.81	20.28 (4Ø1”)	27.40	30.42 (6Ø1”)	24.23	25.35 (5Ø1”)	17.67	20.28 (4Ø1”)
Asxf2B (cm2)	81.74	86.19 (17Ø1”)	30.31	35.49 (7Ø1”)	73.70	76.05 (15Ø1”)	57.05	60.84 (12Ø1”)
Asxf2T (cm2)	48.07	50.70 (10Ø1”)	19.70	25.35 (5Ø1”)	8.98	10.14 (2Ø1”)	33.57	35.49 (7Ø1”)
Asyf1B (cm2)	245.23	248.43 (49Ø1”)	42.17	45.63 (9Ø1”)	37.64	40.56 (8Ø1”)	243.21	243.36 (48Ø1”)
Asyf1T (cm2)	144.22	147.03 (29Ø1”)	27.40	30.42 (6Ø1”)	24.23	25.35 (5Ø1”)	137.83	141.96 (28Ø1”)
Asyf2B (cm2)	32.17	35.49 (13Ø1”)	34.93	35.49 (7Ø1”)	79.46	81.12 (16Ø1”)	57.05	60.84 (12Ø1”)
Asyf2T (cm2)	17.81	20.28 (4Ø1”)	19.70	25.35 (5Ø1”)	51.16	55.77 (11Ø1”)	33.57	35.49 (7Ø1”)
Asv (cm2)	1.42	1.42 (2Ø3/8”)	1.42	1.42 (2Ø3/8”)	1.42	1.42 (2Ø3/8”)	1.42	1.42 (2Ø3/8”)
ρxf1B	0.00595	0.00606	0.00367	0.00377	0.00333	0.00358	0.00582	0.00606
ρxf2B	0.00333	0.00347	0.00333	0.00367	0.01761	0.01811	0.00333	0.00348
ρybB	0.00363	0.00446	0.00333	0.00378	0.00333	0.00352	0.00333	0.00373
ρybT	0.00333	0.00357	0.00999	0.01135	0.01126	0.01232	0.00794	0.00870
ρyf1B	0.00333	0.00333	0.00333	0.00340	0.00333	0.00358	0.00333	0.00339
ρyf2B	0.00354	0.00386	0.00384	0.00367	0.00333	0.00339	0.00333	0.00348
Cmin	28.79Cc	29.89Cc	17.60Cc	19.26Cc	19.36Cc	20.35Cc	27.98Cc	29.53Cc

where: d, d₁, n, s, b₂, 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 d₁ 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.

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 general data for the footings of the four examples are: c₁ = c₂ = c₃ = c₄ = c = 0.40 m, r = 0.08 m, L = 7.00 m, H = 2.50 m, P_D1 = 800 kN, P_L1 = 500 kN, M_Dx1 = 200 kN-m, M_Lx1 = 150 kN-m, M_Dy1 = 140 kN-m, M_Ly1 = 100 kN-m, P_D2 = 600 kN, P_L2 = 400 kN, M_Dx2 = 150 kN-m, M_Lx2 = 100 kN-m, M_Dy2 = 140 kN-m, M_Ly2 = 100 kN-m, q_a = 260 kN/m², γ_cd = 24 kN/m³, γ_sd = 15 kN/m³, f’_c = 28 MPa, f_y = 420 MPa and α = 90.

The final total thickness of the footing after making several iterations is 1.10 m. Now, substituting the corresponding values into Equation (2) gives: q_aa = 212.60 kN/m². The unfactored loads and moments acting on the ground are: P₁ = 1300 kN, M_x1 = 350 kN-m, M_y1 = 240 kN-m, P₂ = 1000 kN, M_x2 = 250 kN-m and M_y2 = 240 kN-m. The factored loads and moments acting on the ground are: P_u1 = 1760 kN, M_ux1 = 480 kN-m, M_uy1 = 328 kN-m, P_u2 = 1360 kN, M_ux2 = 340 kN-m and M_uy2 = 328 kN-m.

Table 4 presents the minimum area of the four examples. Table 5 shows the minimum-cost design of the four examples.

4. Results

The verification process is carried out by substituting in the known points for moments and bending shears, and for the punching shears they are compared with the equations proposed by Yañez-Palafox [24].

The verification of the proposed model is presented below.

• For moments:
  • If c₁/2 is replaced by a₁/2 in Equation (17) to find the moment about the axis formed by q₁ and q₃, the moment Mq₁q₃ = 0.
  • If c₁/2 is replaced by −a₁/2 and the addition of P_u1 a₁/2 + M_uy1 into Equation (17) to obtain the moment about the axis formed by q₂ and q₆, the moment Mq₂q₆ = 0.
  • If c₃/2 is replaced by b₁/2 in Equation (19) to find the moment about the axis formed by q₇ and q₁₁, the moment Mq₇q₁₁ = 0.
  • If c₃/2 is replaced by −b₁/2 and the addition of P_u2b₁/2 + M_uy2 into Equation (19) to obtain the moment about the axis formed by q₁₀ and q₁₂, the moment Mq₁₀q₁₂ = 0.
  • If c₂/2 and e are eliminated from Equation (21) to obtain the moment about the axis formed by q₁ and q₂, the moment Mq₁q₂ = 0.
  • If c₄/2 and f are eliminated in the integrals and replaced by P_u2f and P_u2c₄/2 in Equation (34) to find the moment about the axis formed by q₁₁ and q₁₂, the moment Mq₁₁q₁₂ = 0.
• For bending shears:
  • If c₁/2 + d is replaced by a₁/2 in Equation (36) to find the bending shear about the axis formed by q₁ and q₃, the bending shear Vq₁q₃ = 0.
  • If c₁/2 + d is replaced by −a₁/2 and the addition of −P_u1 into Equation (36) to find the bending shear about the axis formed by q₂ and q₆, the bending shear Vq₂q₆ = 0.
  • If c₃/2 + d is replaced by b₁/2 in Equation (38) to find the bending shear about the axis formed by q₇ and q₁₁, the bending shear Vq₇q₁₁ = 0.
  • If c₃/2 + d is replaced by −b₁/2 and the addition of −P_u2 into Equation (38) to find the bending shear about the axis formed by q₁₀ and q₁₂, the bending shear Vq₁₀q₁₂ = 0.
  • If e, c₂/2 and d are eliminated from Equation (40) to obtain the bending shear about the axis formed by q₁ and q₂, the bending shear Vq₁q₂ = 0.
  • If c₄/2, d and f are eliminated from Equation (50) to obtain the bending shear about the axis formed by q₁₁ and q₁₂, the bending shear Vq₁₁q₁₂ = 0.
• For moments and bending shears:
  • If Equations (27)–(29) are developed and differentiated with respect to y_m and set equal to zero, the position of the maximum moment is obtained. Now, if y_t − e − c₂/2 − d is replaced by y_m in Equation (42), y_t − b − b₂ is replaced by y_m in Equation (46) and y_t − b + f + c₄/2 + d is replaced by y_m in Equation (48) and the equations are developed and set equal to zero, the position of the maximum moment is obtained. The positions of the maximum moment y_m through the moment and bending shear equations are the same.
• For punching shears:
  • If s₁ = 0 and e = c₂/2 are substituted into Equation (53), the punching shear V_up1 is obtained with footing 1 limited in the Y direction. This equation is presented in equation (63) by Yañez-Palafox [24].
  • If b = c₂/2 + L + b₂/2, s₂ = d/2 and f = b₂/2 are substituted into Equation (55), the punching shear V_up2 is obtained with footing 1 limited in the Y direction and footing 2 is square and column 2 is located in the center of the footing. This equation is presented in equation (65) by Yañez-Palafox [24].

Table 4 (the minimum area for strap combined footings) shows the following: When 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.

5. 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, c₁, c₂, c₃, c₄, P₁, M_x1, M_y1, P₂, M_x2, M_y2 and q_aa, and the dependent variables are: A_min, a₁, a₂, b, b₁, b₂, e and f. Once the dimensions of the footing are known, the work continues with step two. Step two is to obtain the minimum cost; the independent variables are: a₁, a₂, b, b₁, b₂, c, c₁, c₂, c₃, c₄, e, f, P_u1, M_ux1, M_uy1, M_uxT, P_u2, M_ux2, M_uy2, M_uyT, R_u and A_sv, and the dependent variables or decision variables are: C_min, d, d₁, n, s, ρ_xf1B, ρ_yf1B, ρ_xf2B, ρ_yf2B, ρ_ybB, ρ_ybT, A_sxf1B, A_sxf1T, A_syf1B, A_sxf2B, A_sxf2T, A_syf1T, A_syf2B, A_syf2T, A_sbB and A_sbT.

The main contributions are:

• 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.
• The proposed model is not limited to d = d₁ or to one or two property lines, as presented by some authors [13,24].
• The equations for the moments, bending shears and punching shears are verified by equilibrium (see Section 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.
• 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.
• This model can be used for T-shaped combined footings by substituting c with b₁ and b₂ with b − a₂ in all equations.
• This model can be used for rectangular combined footings by substituting c with a₁, b₁ with a₁, b₂ with b and a₂ with b in all equations.

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).