Optimal Surface for Elliptical Isolated Footings with Partially Compressed Contact Area

Abstract

This study shows an optimal model to estimate the minimum area in contact with the soil for an EIF (elliptical isolated footing), assuming that the partially compressed area, that is, part of the surface below the base in contact with the ground, is compressed, and the other part is not compressed (the pressure of the ground is linear). There are works that show the minimum area for an elliptical isolated footing, but the surface below the base in contact with the ground is fully compressed. The model is developed by integration to determine the equations of the axial load and the two moments (X and Y axes) for the two cases. Two numerical studies are presented: Study 1 considers that the axial load varies, and the moments are equal and remain constant; Study 2 considers that the axial load varies, and the moments are different and remain constant. Two comparisons are also made with the model proposed by other authors (fully compressed area) and the new model (partially compressed area): In the first study, it is assumed that axial load and moment about the X-axis remain constant and moment about the Y-axis is variable; in the second study, it is assumed that the two moments remain constant and the axial load is variable. The results show that significant savings of up to 59.30% can be achieved in the first study and up to 65.67% in the second study in the area of contact with the ground. Another comparison is made between rectangular isolated footings and EIFs; the results indicate that savings of up to 63.18% can be achieved using EIFs. Therefore, this article will be of great help to specialists in foundation engineering.

1. Introduction

The distribution of pressure in the soil is determined by several factors, such as the soil type, the applied load, the geometry of the surface, the relative stiffness of the soil, and the footing base.

The most common types of foundations that support a column are square, rectangular, circular, and elliptical isolated footings.

Figure 1 indicates the distribution of soil pressure below the base of a foundation as a result of the soil type for a rigid foundation. Figure 1a represents a footing on sandy soils. Figure 1b represents a footing on clay soils. Figure 1c represents a uniform distribution that is used in the current design [1].

The proposed model considers that the distribution of the soil pressure is linear for EIFs (elliptical isolated footings) under biaxial bending due to the column.

Several researchers have been developed bearing capacity studies using analytical and/or experimental methods for different types of foundations, such as: Shahin and Cheung [2], Dixit and Patil [3], ErzÍn and Gul [4], Colmenares et al. [5], Cure et al. [6], Fattah et al. [7], Uncuoğlu [8], Anil et al. [9], Khatri et al. [10], Mohebkhah [11], Zhang [12], Turedi et al. [13], Gnananandarao [14], Gör [15], Hu et al. [16], Yin et al. [17], Li et al. [18], Zhang et al. [19], Ren et al. [20], Liu et al. [21].

The most important contributions to determining the minimum surface of foundations, assuming that the contact area with the ground is partially compressed, have been investigated by several researchers, such as, for rectangular isolated footings, Özmen [22], Rodriguez-Gutierrez and Aristizabal-Ochoa [23,24], Aydogdu [25], Girgin [26], Rawat et al. [27], Lezgy-Nazargah et al. [28], Himeur et al. [29], and Vela-Moreno et al. [30]. For circular isolated footings by Soto-García et al. [31]. For rectangular combated footings by Montes-Páramo et al. [32]. For T-shaped combated footings by Luévanos-Rojas et al. [33].

The most similar papers that use elliptical bases for foundations are Elhanash et al. [34], who introduced a method to estimate the minimum cost of circular isolated footings and EIFs using Lagrange multipliers. Diaz-Gurrola et al. [35] determined an optimal design for the optimal cost of reinforced concrete EIFs. Santiago-Hurtado et al. [36] obtained a design model to estimate the thickness and reinforcing steel for reinforced concrete EIFs. All these works are developed under the criterion that the surface of the foundation is fully compressed.

Thus, the review of previous studies for EIFs shows that there are current works that assume the minimum area is fully compressed, but there are no current works that assume the minimum area is partially compressed.

This work shows a model to find the minimum area for EIFs, assuming that the area in contact with the soil is partially compressed, that is, a region of the contact surface of the footing is subjected to compression, and another part is without pressure (zero pressure). This paper shows the two possible cases of footings subjected to biaxial bending (Case I, assuming that the bottom surface of the footing is fully compressed, and Case II, assuming that the bottom surface of the footing is partially compressed, which constitutes the main contribution of this research, and it is due to its elliptical shape), and the load and the orthogonal moments on the X and Y axes are obtained by integration. This article presents two numerical studies: Study 1 is for different load values, and the moments are equal and remain constant; Study 2 corresponds to different load values, and the moments are different and remain constant. Also, a comparison is made with the model proposed by Diaz-Gurrola et al. [35] (area works completely under compression) and the new model (area works partially under compression). Another comparison is made between the RIFs (rectangular isolated footings) with the model proposed by Vela-Moreno et al. [30] and the new model of EIFs. Comparisons are developed to observe the differences.

2. Formulation of the Model

This article raises the following considerations according to Bowles [1]: the foundation is completely rigid and rests on elastic and homogeneous soil, which means that the soil pressure on the foundation acts linearly.

Figure 2 shows an EIF under an axial load and two bending moments about the X and Y axes due to the column.

The general equation to determine soil pressure in any region of the foundation base is as follows:

$$\sigma ={\displaystyle \frac{P}{A}}+{\displaystyle \frac{M_{x}y}{I_x}}+{\displaystyle \frac{M_{y}x}{I_y}},$$

(1)

where σ indicates the soil pressure at any region of the base (kN/m²); P indicates the axial load (kN); A indicates the contact area with the soil at the base (m²); M_x and M_y indicate the moments about both axes (kN-m); I_x and I_y indicate the moments of inertia about both axes (m⁴); and x and y indicate the coordinates on both axes of the base where the point of study is located (m).

The equation of the ellipse centered at the origin is as follows:

$${\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}=1\to y={\displaystyle \frac{b\sqrt{a^2-x^2}}{a}},$$

(2)

where a = semi-axis of the ellipse on the X axis, b = semi-axis of the ellipse on the Y axis.

Substituting A = πab, I_x = πab³/4, and I_y = πa³b/4 and Equation (2) into Equation (1), the soil pressure is determined:

$$\sigma ={\displaystyle \frac{P}{\pi ab}}+{\displaystyle \frac{4M_x\sqrt{a^2-x^2}}{\pi a^2{b}^2}}+{\displaystyle \frac{4M_{y}x}{\pi a^{3}b}}.$$

(3)

Differentiating Equation (3) with respect to x to determine the location of the maximum and minimum pressures, and the points are as follows:

$$x_{max}={\displaystyle \frac{ab{M}_y}{\sqrt{a^2{M_x}^2+b^2{M_y}^2}}};x_{min}=-{\displaystyle \frac{ab{M}_y}{\sqrt{a^2{M_x}^2+b^2{M_y}^2}}}.$$

(4)

Substituting Equation (4) into Equation (2) to obtain the values of y_max and y_min:

$$y_{max}={\displaystyle \frac{ab{M}_x}{\sqrt{a^2{M_x}^2+b^2{M_y}^2}}};y_{min}=-{\displaystyle \frac{ab{M}_x}{\sqrt{a^2{M_x}^2+b^2{M_y}^2}}}.$$

(5)

Now, substituting A, I_x, I_y and the coordinates (x_max, y_max) and (x_min, y_min) into Equation (1), the soil pressures on an EIF are determined:

$${\sigma }_{max1}={\displaystyle \frac{P}{\pi ab}}+{\displaystyle \frac{4{M_x}^2}{\pi b^2\sqrt{a^2{M_x}^2+b^2{M_y}^2}}}+{\displaystyle \frac{4{M_y}^2}{\pi a^2\sqrt{a^2{M_x}^2+b^2{M_y}^2}}},$$

(6)

$${\sigma }_{min2}={\displaystyle \frac{P}{\pi ab}}-{\displaystyle \frac{4{M_x}^2}{\pi b^2\sqrt{a^2{M_x}^2+b^2{M_y}^2}}}-{\displaystyle \frac{4{M_y}^2}{\pi a^2\sqrt{a^2{M_x}^2+b^2{M_y}^2}}},$$

(7)

where σ_max1 = maximum pressure that acts, and σ_min2 = minimum pressure that acts.

2.1. Case I: Fully Compressed Area

Figure 3 represents an EIF subjected to biaxial bending supported on an elastic soil; the surface is fully compressed, and the soil pressure distribution is linear.

The distance from the origin of the ellipse to the point where the maximum pressure is located is determined as follows:

$$d_m={\displaystyle \frac{ab\left(M_x+M_y\right)}{\sqrt{a^2{M_x}^2+b^2{M_y}^2}}}.$$

(8)

2.2. Case II: Partially Compressed Area

Figure 4 represents an EIF subjected to biaxial bending supported on an elastic soil; the area is partially compressed, and the soil pressure distribution is linear.

The equation of the tangent line to the ellipse at the point where the maximum pressure is located is determined below.

Differentiating Equation (2) with respect to x to determine the slope at the point where the maximum pressure appears. The slope is as follows:

$${\displaystyle \frac{dy}{dx}}=-{\displaystyle \frac{bx}{a\sqrt{a^2-x^2}}}.$$

(9)

Substituting ${x=x}_{max}={\displaystyle \frac{ab{M}_y}{\sqrt{a^2{M_x}^2+b^2{M_y}^2}}}$ into Equation (9) to determine the slope “m” at the point of maximum pressure yields the following:

$${\displaystyle \frac{dy}{dx}}=m=-{\displaystyle \frac{b^2{M}_y}{a^2{M}_x}}.$$

(10)

From the point of maximum pressure and the slope $\left[y-y_{max}=m\left(x-x_{max}\right)\right]$, the general equation of the tangent line to the ellipse passing through the point of maximum pressure is determined:

$$y=-{\displaystyle \frac{b^2{M}_{y}x}{a^2{M}_x}}+{\displaystyle \frac{b\sqrt{a^2{M_x}^2+b^2{M_y}^2}}{a{M}_x}}.$$

(11)

The following are the points on the tangent line to the ellipse where the X and Y axes intersect:

$$x_{ox}={\displaystyle \frac{a\sqrt{a^2{M_x}^2+b^2{M_y}^2}}{b{M}_y}};y_{oy}={\displaystyle \frac{b\sqrt{a^2{M_x}^2+b^2{M_y}^2}}{a{M}_x}}.$$

(12)

The distance d ($d=\sqrt{{x_{ox}}^2+{y_{oy}}^2}$) between the points where the tangent line to the ellipse intersects the axes is shown below:

$$d={\displaystyle \frac{\sqrt{\left(a^2{M_x}^2+b^2{M_y}^2\right)\left(a^4{M_x}^2+b^4{M_y}^2\right)}}{ab{M}_x{M}_y}}.$$

(13)

The angle of inclination α ($\cos \alpha =x_{ox}/d$) of the tangent line with respect to the X-axis is presented below:

$$\cos \alpha ={\displaystyle \frac{a^2{M}_x}{\sqrt{a^4{M_x}^2+b^4{M_y}^2}}}.$$

(14)

The distance y_o2 ($y_{o2}=y_0/\cos \alpha$) along the Y-axis from the origin to the point where it crosses the neutral axis is shown below:

$$y_{o2}={\displaystyle \frac{y_0\sqrt{a^4{M_x}^2+b^4{M_y}^2}}{a^2{M}_x}}.$$

(15)

The general equation of the neutral axis is determined below:

$$y=-{\displaystyle \frac{b^2{M}_{y}x}{a^2{M}_x}}+{\displaystyle \frac{y_0\sqrt{a^4{M_x}^2+b^4{M_y}^2}}{a^2{M}_x}}.$$

(16)

Substituting Equation (16) into Equation (2) to find the points where the neutral axis intersects the ellipse, the following results are obtained:

$$x_1={\displaystyle \frac{M_{y}b{y}_0\sqrt{a^4{M_x}^2+b^4{M_y}^2}+M_{x}a\sqrt{a^2{b}^2\left(a^2{M_x}^2+b^2{M_y}^2\right)-{y_0}^2\left(a^4{M_x}^2+b^4{M_y}^2\right)}}{b\left(a^2{M_x}^2+b^2{M_y}^2\right)}},$$

(17)

$$x_2={\displaystyle \frac{M_{y}b{y}_0\sqrt{a^4{M_x}^2+b^4{M_y}^2}-M_{x}a\sqrt{a^2{b}^2\left(a^2{M_x}^2+b^2{M_y}^2\right)-{y_0}^2\left(a^4{M_x}^2+b^4{M_y}^2\right)}}{b\left(a^2{M_x}^2+b^2{M_y}^2\right)}}.$$

(18)

Now, substituting Equations (17) and (18) into Equation (16) to determine the values of y₁ and y₂, the following results are obtained:

$$y_1={\displaystyle \frac{M_{x}a{y}_0\sqrt{a^4{M_x}^2+b^4{M_y}^2}-M_{y}b\sqrt{a^2{b}^2\left(a^2{M_x}^2+b^2{M_y}^2\right)-{y_0}^2\left(a^4{M_x}^2+b^4{M_y}^2\right)}}{a\left(a^2{M_x}^2+b^2{M_y}^2\right)}},$$

(19)

$$y_2={\displaystyle \frac{M_{x}a{y}_0\sqrt{a^4{M_x}^2+b^4{M_y}^2}+M_{y}b\sqrt{a^2{b}^2\left(a^2{M_x}^2+b^2{M_y}^2\right)-{y_0}^2\left(a^4{M_x}^2+b^4{M_y}^2\right)}}{a\left(a^2{M_x}^2+b^2{M_y}^2\right)}}.$$

(20)

The pressures generated by the ground on the foundation are determined using the equation of the pressure plane, starting from three known points.

The general equation of a 3-D pressure plane of the ground on the foundation is as follows:

$$Ax+By+C{\sigma }_z+D=0.$$

(21)

Any point on the plane is represented by p, whose unknown components are the Equation of the plane passing through three points (x, y, σ_z).

For Case II, the three known points of the pressure plane are as follows:

$$p_1\left(x_1, y_1, 0\right); p_2\left(x_2, y_2, 0\right); p_3\left(x_{max}, y_{max}, {\sigma }_{max}\right),$$

(22)

where p₁, p₂, and p₃ are the coordinates of the three known points of the footing (see Figure 4).

The equation of the pressure plane is obtained as follows (taking point 1 p₁, as pivot):

$$\left|\begin{array}{ccc}x-x_1& y-y_1& {\sigma }_z-0\\ x_2-x_1& y_2-y_1& 0-0\\ x_{max}-x_1& y_{max}-y_1& {\sigma }_{max}-0\end{array}\right|.$$

(23)

Solving the determinant of Equation (23) and isolating σ_z gives the pressure at any point on the footing:

$${\sigma }_z={\displaystyle \frac{{\sigma }_{max}\left[a^2{M}_{x}y+b^2{M}_{y}x-y_0\sqrt{a^4{M_x}^2+b^4{M_y}^2}\right]}{ab\sqrt{a^2{M_x}^2+b^2{M_y}^2}-y_0\sqrt{a^4{M_x}^2+b^4{M_y}^2}}}.$$

(24)

The equation of the straight line that forms the neutral axis is obtained by assuming “σ_z” equal to zero into Equation (24) (this is the definition of the neutral axis):

$$a^2{M}_{x}y+b^2{M}_{y}x-y_0\sqrt{a^4{M_x}^2+b^4{M_y}^2}=0.$$

(25)

The following are general equations that P_R, M_xR and M_yR must satisfy for Case II.

$$P_R={\int }_{x_2}^{x_1}{\int }_{-{\displaystyle \frac{b^2{M}_{y}x}{a^2{M}_x}}+{\displaystyle \frac{y_0\sqrt{a^4{M_x}^2+b^4{M_y}^2}}{a^2{M}_x}}}^{{\displaystyle \frac{b\sqrt{a^2-x^2}}{a}}}{\sigma }_{z}dydx+{\int }_{x_1}^a{\int }_{-{\displaystyle \frac{b\sqrt{a^2-x^2}}{a}}}^{{\displaystyle \frac{b\sqrt{a^2-x^2}}{a}}}{\sigma }_{z}dydx,$$

(26)

$$M_{xR}={\int }_{x_2}^{x_1}{\int }_{-{\displaystyle \frac{b^2{M}_{y}x}{a^2{M}_x}}+{\displaystyle \frac{y_0\sqrt{a^4{M_x}^2+b^4{M_y}^2}}{a^2{M}_x}}}^{{\displaystyle \frac{b\sqrt{a^2-x^2}}{a}}}{\sigma }_{z}ydydx+{\int }_{x_1}^a{\int }_{-{\displaystyle \frac{b\sqrt{a^2-x^2}}{a}}}^{{\displaystyle \frac{b\sqrt{a^2-x^2}}{a}}}{\sigma }_{z}ydydx,$$

(27)

$$M_{yR}={\int }_{x_2}^{x_1}{\int }_{-{\displaystyle \frac{b^2{M}_{y}x}{a^2{M}_x}}+{\displaystyle \frac{y_0\sqrt{a^4{M_x}^2+b^4{M_y}^2}}{a^2{M}_x}}}^{{\displaystyle \frac{b\sqrt{a^2-x^2}}{a}}}{\sigma }_{z}xdydx+{\int }_{x_1}^a{\int }_{-{\displaystyle \frac{b\sqrt{a^2-x^2}}{a}}}^{{\displaystyle \frac{b\sqrt{a^2-x^2}}{a}}}{\sigma }_{z}xdydx,$$

(28)

The solution to Equation (26) appears in Figure A1, the solution to Equation (27) appears in Figure A2, and the solution to Equation (28) appears in Figure A3 (see Appendix A).

2.3. Minimum Surface for EIFs

Minimum surface (objective function) for the two cases is as follows:

$$A_{min}=\pi ab.$$

(29)

Table 1. Constraint functions for biaxial bending.

Case	Equations
I	Equations (5) and (6), 0 ≤ σmax1 and σmin2 ≤ σmax
II	Equations (26)–(28), PR ≥ P, MxR ≥ Mx, MyR ≥ My

P_R, M_xR, M_yR indicate the load and moments that the footing must support; P, M_x, M_y indicate the load and moments acting on the footing.

indicates the constraint equations for biaxial bending in both cases.

The flowchart for the minimum area process of an EIF is represented (see Figure 5).

The flowchart using Maple software v.17 for the minimum area of an EIF is represented (see Figure 6).

3. Numerical Examples

Two numerical examples for EIFs are shown. The data for Study 1 are as follows: P = 600, 800, 1000, 1200, 1400, 1600, 1800, 2000, 2200 kN; M_x = 1000 kN-m, M_y = 1000 kN-m, σ_max = 200 kN/m². The data for Study 2 are: P₁ = 600, 800, 1000, 1200, 1400, 1600, 1800, 2000, 2200 kN; M_x = 1000 kN-m, M_y = 500 kN-m, σ_max = 200 kN/m².

Table 2. Study 1 for M_x = 1000 kN-m, M_y = 1000 kN-m.

Example	P (kN)	Case I					Case II						AminCI/AminCII
		a (m)	b (m)	σ1 (kN/m2)	σ2 (kN/m2)	AminCI (m2)	a (m)	b (m)	y0 (m)	σ1 (kN/m2)	σ2 (kN/m2)	AminCII (m2)
1	600	9.43	9.43	4.30	0	279.25	2.49	2.49	−1.17	200	0	19.49	14.33
2	800	7.07	7.07	10.19	0	157.08	2.49	2.49	−1.17	200	0	19.49	8.06
3	1000	5.66	5.66	19.89	0	100.53	2.49	2.49	−1.17	200	0	19.49	5.16
4	1200	4.71	4.71	34.38	0	69.81	2.49	2.49	−1.17	200	0	19.49	3.58
5	1400	4.04	4.04	54.59	0	51.29	2.49	2.49	−1.24	200	0	19.50	2.63
6	1600	3.54	3.54	81.49	0	39.27	2.51	2.51	−1.64	200	0	19.75	1.99
7	1800	3.14	3.14	116.02	0	31.03	2.54	2.54	−2.01	200	0	20.28	1.53
8	2000	2.83	2.83	159.15	0	25.13	2.59	2.59	−2.36	200	0	20.99	1.20
9	2200	2.63	2.63	200.00	2.30	21.75	2.68	2.68	−2.61	200	0	22.56	0.96

Note: A_minCI = minimum area for Case I, A_minCII = minimum area for Case II.

and

Table 3. Study 2 for M_x = 1000 kN-m, M_y = 500 kN-m.

Example	P (kN)	Case I					Case II						AminCI/AminCII
		a (m)	b (m)	σ1 (kN/m2)	σ2 (kN/m2)	AminCI (m2)	a (m)	b (m)	y0 (m)	σ1 (kN/m2)	σ2 (kN/m2)	AminCII (m2)
1	600	4.71	9.43	8.59	0	139.63	1.57	3.14	−0.94	200	0	15.47	9.03
2	800	3.54	7.07	20.37	0	78.54	1.57	3.14	−0.94	200	0	15.47	5.08
3	1000	2.83	5.66	39.79	0	50.27	1.57	3.14	−0.94	200	0	15.47	3.25
4	1200	2.36	4.71	68.75	0	34.91	1.57	3.15	−1.17	200	0	15.55	2.24
5	1400	2.02	4.04	109.18	0	25.65	1.60	3.19	−1.55	200	0	16.01	1.60
6	1600	1.77	3.54	162.97	0	19.63	1.60	3.33	−1.85	200	0	16.71	1.17
7	1800	1.67	3.34	200.00	5.96	17.48	1.60	3.64	−1.97	200	0	18.31	0.95
8	2000	1.71	3.41	200.00	18.71	18.29	1.73	3.75	−2.14	200	0	20.40	0.90
9	2200	1.74	3.49	200.00	30.25	19.11	1.77	4.03	−2.18	200	0	22.38	0.85

indicate the results of studies 1 and 2, respectively.

4. Results

The proposed model can be verified according to the soil pressure under the footing, as shown below.

• Substituting x = x_max and y = y_max into Equation (24) gives σ_z = σ_max.
• Substituting x = x₁ and y = y₁ into Equation (24) gives σ_z = 0.
• Substituting x = x₂ and y = y₂ into Equation (24) gives σ_z = 0.

The results of Table 2 indicate the following:

• When P in Case I increases: a, b, and A_minCI decrease; σ₁ increases; σ₂ is zero except in P = 2200 kN.
• When P in Case II increases: a, b, y₀ (absolute value) and A_minCII are the same up to P = 1200 kN, then they increase; σ₁ reaches the maximum value of 200 kN/m²; σ₂ is zero.
• In Case II, the values for P = 600, 800, 1000, 1200 kN present the same values in a, b, y₀, σ₁, σ₂, and A_minCII because P reaches a maximum value of P = 1367.83 kN.
• In all cases, the minimum area appears in Case II, except in example 9, where σ₁ reaches the maximum value of 200 kN/m² and the minimum area appears in Case I.
• In all cases, the values of a and b are equal, because the moments are equal.

The results of Table 3 indicate the following:

• When P in Case I increases: a, b, and A_minCI decrease up to P = 1800 kN, then they increase; σ₁ increases up to P = 1800 kN, then it is equal; σ₂ is zero up to P = 1600 kN, then it increases.
• When P in Case II increases: a, b, y₀ (absolute value) and A_minCII are equal up to P = 1000 kN, and then they increase; σ₁ reaches the maximum value of 200 kN/m²; σ₂ is zero.
• In Case II, the values for P = 600, 800, 1000 kN present the same values in a, b, y₀, σ₁, σ₂, and A_minCII because P reaches a maximum value of P = 1090.76 kN.
• In all cases, the minimum area appears in Case II, except in examples 7, 8, and 9, where σ₁ reaches the maximum value of 200 kN/m², and the minimum area appears in Case I.

Furthermore, a comparison is made to show the superiority of the NMPA (new model proposed in this article) for EIFs (partially compressed area) over the MPDG (model proposed by Diaz-Gurrola et al. [35]) for EIFs (fully compressed area).

The MPDG shows two studies. Study 1 provides the following data: P = 1100 kN; M_x = 500 kN-m; M_y = 900, 700, 500, 300, 150 kN-m; σ_max = 200 kN/m². Study 2 provides the following data: P = 1500, 1300, 1100, 900, 700 kN; M_x = 500 kN-m; M_y = 300 kN-m; σ_max = 200 kN/m².

Table 4. Results of Study 1 for P = 1100 kN, M_x = 500 kN-m.

Example		My (kN-m)	MPDG (Case I)					NMPA (Case II)						AminCI/AminCII
			a (m)	b (m)	σ1 (kN/m2)	σ2 (kN/m2)	AminCI (m2)	a (m)	b (m)	y0 (m)	σ1 (kN/m2)	σ2 (kN/m2)	AminCII (m2)
1	TR	900	4.63	2.57	58.84	0	37.39	2.93	1.63	−1.06	200.00	0	14.95	2.50
	PR		4.60	2.60	58.48	0.07	37.57	2.95	1.65	−1.46	200.00	0	15.29	2.46
2	TR	700	3.60	2.57	75.66	0	29.08	2.49	1.78	−1.28	200.00	0	13.87	2.10
	PR		3.60	2.60	74.61	0.21	29.41	2.50	1.80	−1.53	200.00	0	14.14	2.08
3	TR	500	2.57	2.57	105.92	0	20.77	2.01	2.01	−1.52	200.00	0	12.67	1.64
	PR		2.60	2.60	103.02	0.57	21.24	2.05	2.05	−1.86	200.00	0	13.20	1.61
4	TR	300	1.54	2.57	176.53	0	12.46	1.41	2.56	−1.62	200.00	0	11.32	1.10
	PR		1.55	2.60	173.09	0.68	12.66	1.45	2.50	−1.66	200.00	0	11.39	1.11
5	TR	150	1.25	2.57	200.00	18.32	10.08	1.36	2.75	−1.92	200.00	0	11.71	0.86
	PR		1.25	2.60	196.54	18.93	10.21	1.40	2.75	−2.07	200.00	0	12.10	0.84

and

Table 5. Results of Study 2 for M_x = 500 kN-m, M_y = 300 kN-m.

Example		P (kN)	MPDG (Case I)					NMPA (Case II)						AminCI/AminCII
			a (m)	b (m)	σ1 (kN/m2)	σ2 (kN/m2)	AminCI (m2)	a (m)	b (m)	y0 (m)	σ1 (kN/m2)	σ2 (kN/m2)	AminCII (m2)
1	TR	1500	1.57	2.62	200.00	32.47	12.90	1.59	3.07	−1.86	200.00	0	15.32	0.84
	PR		1.60	2.60	196.98	32.57	13.07	1.60	3.10	−1.88	200.00	0	15.58	0.84
2	TR	1300	1.52	2.53	200.00	15.14	12.09	1.47	2.89	−1.69	200.00	0	13.35	0.91
	PR		1.55	2.50	198.24	15.34	12.17	1.50	2.95	−1.81	200.00	0	13.43	0.91
3	TR	1100	1.54	2.57	176.53	0	12.46	1.41	2.56	−1.62	200.00	0	11.32	1.10
	PR		1.55	2.60	173.09	0.68	12.66	1.45	2.50	−1.66	200.00	0	11.39	1.11
4	TR	900	1.89	3.14	96.69	0	18.62	1.42	2.37	−1.24	200.00	0	10.61	1.75
	PR		1.90	3.15	95.50	0.24	18.80	1.45	2.40	−1.19	200.00	0	10.93	1.72
5	TR	700	2.42	4.04	45.49	0	30.77	1.41	2.34	−0.80	200.00	0	10.36	2.97
	PR		2.45	4.05	44.77	0.14	31.17	1.45	2.35	−0.69	200.00	0	10.70	2.91

represent the TR (theoretical results) and PR (practical results) of the minimum area for MPDG from studies 1 and 2.

The practical results of Table 4 indicate the following:

• When M_y, in MPDG (Case I), decreases: a and A_minCI decrease; b remains constant; σ₁ tends to increase; σ₂ practically is zero except in M_y = 150 kN-m.
• When M_y, in NMPA (Case II), decreases: a decreases; A_minCII decreases up to M_y = 300 kN-m and then increases; b increases; y₀ (absolute value) increases up to M_y = 500 kN-m, then decreases up to M_y = 300 kN-m, and then increases; σ₁ reaches the maximum value of 200 kN/m²; σ₂ is zero.
• In all cases, the minimum area appears in Case II, except in example 5, where the minimum area appears in Case I because σ₁ reaches the maximum value of 200 kN/m².
• When M_y decreases: A_minCI/A_minCII decreases.
• The highest A_minCI/A_minCII ratio appears at M_y = 900 kN-m of 2.46 times; this means that MPDG (Case I) is greater than NMPA (Case II).
• The smallest A_minCI/A_minCII ratio appears at M_y = 150 kN-m of 0.84 times; this means that NMPA (Case II) is greater than MPDG (Case I).

The practical results of Table 5 indicate the following:

• When P in Case I decreases: a, b, and A_minCI decrease up to P = 1300 kN, then they increase; σ₁ practically reaches the maximum value of 200 kN/m² up to P = 1300 kN, then it decreases; σ₂ decreases up to P = 1100 kN, then it is constant and equal to zero.
• When P in Case II decreases: a decreases up to P = 1300 kN, and then it is constant; b and y₀ (absolute value) decrease; A_minCII decreases; σ₁ reaches the maximum value of 200 kN/m²; σ₂ is zero.
• In all cases, the minimum area appears in Case II, except in examples 1 and 2, where the minimum area appears in Case I because σ₁ reaches the maximum value of 200 kN/m².
• When P decreases: A_minCI/A_minCII increases.
• The highest A_minCI/A_minCII ratio appears at P = 700 kN of 2.91 times, which means that MPDG (Case I) is greater than NMPA (Case II).
• The smallest A_minCI/A_minCII ratio appears at P = 1500 kN of 0.84 times, which means that NMPA (Case II) is greater than MPDG (Case I).

Also, a comparison is shown to demonstrate the advantages of the NMPA for EIFs over the MPVM (model proposed by Vela-Moreno et al. [30]) for RIFs (rectangular isolated footings), both with partially compressed areas.

The MPVM shows four studies that provide the following data: P = 500 kN; M_x = 250, 500, 1000, 1500 kN-m; M_y = 250 (Study 1), 500 (Study 2), 1000 (Study 3), 1500 (Study 4) kN-m; σ_max = 250 kN/m².

Table 6. Results of four studies for P = 500 kN.

Study	My (kN-m)	Mx (kN-m)	MPVM (RIFs)						NMPA (EIFs)							AminVM/AminMP
			Case	hx (m)	hy (m)	hx1 (m)	hy1 (m)	AminVM (m2)	Case	a (m)	b (m)	y0 (m)	σ1 (kN/m2)	σ2 (kN/m2)	AminMP (m2)
1	250	250	V	2.72	2.72	3.50	3.50	7.40	II	1.46	1.46	−0.69	250	0	6.67	1.11
		500	V	2.22	4.45	2.45	4.90	9.89	II	1.16	2.31	−0.69	250	0	8.40	1.18
		1000	II	1.87	7.46	1.73	6.93	13.93	II	1.00	3.38	−0.69	250	0	10.63	1.31
		1500	II	1.71	10.24	1.41	8.49	17.49	II	1.00	3.96	−0.80	250	0	12.45	1.40
2	500	250	V	4.45	2.22	4.90	2.45	9.89	II	2.31	1.16	−0.69	250	0	8.40	1.18
		500	II	3.73	3.73	3.46	3.46	13.93	II	1.84	1.84	−0.86	250	0	10.58	1.32
		1000	II	3.22	6.45	2.45	4.90	20.80	II	1.46	2.91	−0.87	250	0	13.33	1.56
		1500	II	3.00	9.00	2.00	6.00	27.00	II	1.27	3.82	−0.80	250	0	15.26	1.77
3	1000	250	II	7.46	1.87	6.93	1.73	13.93	II	3.38	1.00	−0.69	250	0	10.63	1.31
		500	II	6.45	3.22	4.90	2.45	20.80	II	2.91	1.46	−0.87	250	0	13.33	1.56
		1000	II	5.73	5.73	3.46	3.46	32.86	II	2.31	2.31	−1.09	250	0	16.80	1.96
		1500	II	5.41	8.12	2.83	4.24	43.97	II	2.02	3.03	−1.12	250	0	19.23	2.29
4	1500	250	II	10.24	1.71	8.49	1.41	17.49	II	3.96	1.00	−0.80	250	0	12.45	1.40
		500	II	9.00	3.00	6.00	2.00	27.00	II	3.82	1.27	−0.80	250	0	15.26	1.77
		1000	II	8.12	5.41	4.24	2.83	43.97	II	3.03	2.02	−1.12	250	0	19.23	2.29
		1500	II	7.73	7.73	3.46	3.46	59.78	II	2.65	2.65	−1.25	250	0	22.01	2.72

Note: A_minVM = minimum area for RIFs, A_minMP = minimum area for EIFs.

represents the results of the minimum area for MPVM for RIFs and NMPA for EIFs from four studies.

The results of Table 6 indicate the following:

• When M_x for RIFs increases: h_x and h_x1 decrease; h_y, h_y1, and A_minVM increase, and this is present in all four studies.
• When M_x for EIFs increases: a decreases; b, and A_minMP increase, this is present in all four studies; y₀ (absolute value) tends to increase in all studies except Study 2 (M_x = 1500 kN-m and M_y = 500 kN-m).
• When M_x equals M_y, square footings are generated for RIFs, and circular footings are generated for EIFs.
• The highest A_minVM/A_minMP ratio appears at M_x and M_y = 1500 kN-m, 2.72 times.
• The smallest A_minVM/A_minMP ratio appears at M_x and M_y = 250 kN-m, 1.11 times.

Figure 7 represents the graphs of the minimum areas in contact with the soil between Case I and Case II of studies 1 and 2 (see Table 2 and Table 3).

Figure 7 indicates that both studies demonstrate the advantages of using Case II (partially compressed area) compared to Case I (fully compressed area). Figure 7a shows Study 1: the greatest savings occur with 93.02% in example 1 using the NMPA, but in example 9, there is no saving. Figure 7b shows Study 2: the greatest savings occur with 88.92% in example 1 using the NMPA, but in examples 7, 8, and 9, there are no savings.

Figure 8 represents the comparison of the areas (practical dimensions) in contact with the soil between the MPDG and the NMPA.

Figure 8 indicates that both studies demonstrate savings when using NMPA (partially compressed area) compared to MPDG (fully compressed area). Figure 8a shows Study 1; the greatest savings occur with 59.30% in example 1 using the NMPA, but in example 5, there is no saving. Figure 8b shows Study 2; the greatest savings occur with 65.67% in example 5 using the NMPA, but in examples 1 and 2, there are no savings.

Figure 9 represents the comparison of the surface in contact with the soil between the MPVM for RIFs and the NMPA for EIFs.

Figure 9 indicates the advantages of using NMPA compared to MPVM. Figure 9a shows that for M_y = 250 kN-m, the greatest savings are achieved at M_x = 1500 kN-m with 28.82%. Figure 9b shows that for M_y = 500 kN-m, the greatest savings are achieved at M_x = 1500 kN-m with 43.48%. Figure 9c shows that for M_y = 1000 kN-m, the greatest savings are achieved at M_x = 1500 kN-m with 56.27%. Figure 9d shows that for M_y = 1500 kN-m, the greatest savings are achieved at M_x = 1500 kN-m with 63.18%.

5. Conclusions

The article presented in this document applies only to the minimum area of an EIF. The considerations of this work are that the footing is rigid and the soil that supports it is elastic and homogeneous, which satisfies biaxial bending; that is, the variation in soil pressure is linear.

This paper concludes the following:

• Some authors present equations to find the dimensions and minimum area of the footing, but the entire surface of the footing is fully compressed (see Case I of the two studies in Figure 7).
• The proposed model presents the minimum surface and the constraint functions for the two possible cases.
• The model can be used as a review of the allowable load capacity of the soil, taking into account the objective function σ_max and the same constraint functions presented in this article (see Figure 6).
• The two studies show significant savings using Case II compared to Case I. For Study 1 of example 1, there is a saving of 93.02%, and for Study 2 of example 1, there is a saving of 88.92% (see Figure 7). All these studies were performed under the same loads and moments for both models.
• The minimum area tends to decrease when P increases; this is because σ₂ is equal to zero and σ₁ does not reach σ_max (Case I).
• The minimum area tends to increase when P increases, and in some cases it remains constant because the load P acting is less than the load P_R that it resists (Case II).
• The two studies show significant savings using the NMPA compared to MPDG. For Study 1 of example 1, there is a saving of 59.30%, and for Study 2 of example 5, there is a saving of 65.67% (see Figure 8). All these studies were performed under the same loads and moments for both models.
• The four studies show significant savings using the NMPA for EIFs compared to MPVM for RIFs, with savings that can reach up to 63.18% (see Figure 9d). All these studies were performed under the same loads and moments for both models.
• In all cases, the minimum area appears in Case II, when σ₁ does not reach the maximum value of σ_max in Case I.
• The main advantage of this research is that it directly impacts the cost of foundation construction, since by presenting a smaller contact area with the soil, it generates a smaller volume of fill for the foundation.
• When σ₁ reaches the maximum value of 200 kN/m² in Case I, a smaller minimum area is obtained compared to Case II. This can be observed in Table 2, Table 3, Table 4 and Table 5.
• When the moments are equal, the values of “a” and “b” are equal in two cases; this means that it is a circular footing (see Table 2).

These significant savings occur when the maximum pressure acting on the footing is too low compared to the soil’s bearing capacity (Case I), and Case II ensures that the maximum pressure acting on the footing reaches the soil’s bearing capacity.

The next investigations can be the following: (1) Minimum cost design for an EIF, with partially compressed contact area; (2) Comparison of the minimum soil contact areas between rectangular and elliptical isolated footings with partially compressed surfaces.