Mathematical Calculations for the Design of Elliptical Isolated Foundations with Optimal Cost

Abstract

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

1. Introduction

The soil pressure below a foundation depends on the type of soil, the relative stiffness of the soil and the footing, and the depth of the foundation at the level of contact between the foundation and the soil.

Figure 1 shows the diagram of soil pressure below of the contact surface of a foundation depending on the soil type for a rigid foundation. Figure 1a shows a rigid foundation on sandy soil. Figure 1b presents a rigid foundation on clay soil. Figure 1c shows the uniform distribution used in the current design [1].

The proposed model assumes that the soil pressure distribution is linear beneath the elliptical isolated footing that is subject to biaxial bending.

The load capacity studies using analytical and/or experimental methods for different types of foundations have been investigated by several authors. For example, Shahin and Cheung [2] presented graphs for the design of strip footings. Dixit and Patil [3] obtained settlements through experimentation for square footings on finite layers of sand. ErzÍn and Gul [4] used neural networks to predict the settlement of footings on cohesionless soils based on standard penetration tests. Colmenares et al. [5] estimated the ultimate bearing capacity of the soil for conical shell foundations. Cure et al. [6] investigated the ultimate loads for eccentrically loaded strip footings close to a slope. Fattah et al. [7] studied the effect of pile group geometry in foundation slabs on the bearing capacity of the soil. Uncuoğlu [8] estimated the bearing capacity of square footings on a layer of sand over clay. Anil et al. [9] investigated the foundations of different shapes of sand experimentally and using finite element analyses. Khatri et al. [10] analyzed the pressure-settlement behavior of square and rectangular skirted footings resting on sand. Mohebkhah [11] studied the bearing capacity of soil for strip footings in a stone masonry trench in clay. Zhang [12] estimated the load-bearing behavior of reinforced concrete isolated footings. Turedi et al. [13] determined the bearing capacity of ring footings using experimentation and numerically. Gnananandarao et al. [14] obtained the bearing capacity and settlement prediction of multi-edge skirted footings resting on sand. Gör [15] analyzed the bearing capacity of shallow foundations on two-layered soil using two novel cosmology-based optimization techniques. Hu et al. [16] presented a finite element mesh optimization-based model and soil settlement prediction. Yin et al. [17] measured and analyzed of deformation of the underlying tunnel induced by the excavation of a foundation pit. Li et al. [18] examined the impact of karst caves located in front of and along a tunnel on the stability of the surrounding rock on the foundation. Zhang et al. [19] studied the overturning resistance of an isolated foundation when impact occurs during an earthquake and considered the soil–structure interaction effect. Ren et al. [20] showed a model for large deformations of earthquake-triggered landslides considering non-uniform soils with a stratigraphic inclination. Liu et al. [21] developed a model for large three-dimensional deformations of landslides in spatially variable and strain-softening soils subjected to seismic loading. Chaabani et al. [22] estimated the ultimate bearing capacity of the soil for strip footings on a reinforced sand layer over clay with voids.

The most important investigations on rectangular isolated footings for the efficient management of soil-footing interaction problems have been carried out using analytical methods [23,24,25,26,27,28,29,30,31], graphics or design aids [32,33], and analytical methods and graphics or design aids [34,35,36]. These documents are developed to determine the axial load capacity and biaxial moment of the footing or the pressure distribution in the contact area of a rigid rectangular isolated footing resting on the soil. The dimensions of the footing are obtained using an iterative procedure.

The most important contributions of the studies of combined footings have been reported by several researchers. Regarding rectangular footings, studies include those by Maheshwari and Khatri [37], Konapure and Vivek [38], Vivek et al. [39], Ravi Kumar Reddy et al. [40], and Kashani et al. [41]. For trapezoidal footings, studies include those by Al-Douri [42] and Luévanos-Rojas et al. [43]. For T-shaped footings, Moreno-Landeros et al. [44] published an important study. For corner footings, Aishwarya and Balaji [45] reported important findings. For strap footings, Luévanos-Rojas et al. [46] reported important findings.

The works closest to the subject of elliptical footings are noted as follows. Luévanos-Rojas et al. [47] presented a mathematical calculation for the minimum cost of circular footings with column placement anywhere in the footing. Elhanash et al. [48] showed a new method for the optimal design of circular and elliptical footings using Lagrange multipliers to determine the equations of moments (the moments are determined in parts, first the pressure exerted by the delimited area is obtained, the center of gravity of this area is calculated, and then the moment is determined; only the M_ua and M_ub are obtained), bending shear (the bending shears are determined by the pressure exerted by the delimited area, and only the V_ue and V_uf are obtained), and punching shear. However, the minimum cost and the numerical examples are presented only for circular footings and not for elliptical footings.

As observed in the literature review, there are no publications on minimum cost design for elliptical isolated footings, which is the main proposal of this research.

This paper shows a new model for the design of elliptical isolated footings under the minimum cost criterion, assuming that the footing rests on elastic soils and that the soil pressure distribution is linear. The first part is used to determine the minimum area, and the second part is used to obtain the minimum cost. Two numerical examples (each example shows five variants) are presented for the design of elliptical footings under minimum cost criterion subjected to biaxial bending. Example 1: Varying the moment about the Y axis. Example 2: Varying the axial load. In addition, a comparison is made between elliptical footings and circular footings, and the results show that the minimum area is smaller for elliptical footings than for circular footings. In addition, the minimum cost appears in elliptical footings when the footing dimensions are governed by the minimum pressure. Therefore, the new model for elliptical footings will be of great help to foundation engineering specialists.

2. Formulation of the Model

The load and moments in both directions are those that descend from the column to the footing. The minimum area is obtained from the unfactored axial load “P” and unfactored moments “M_x and M_y”. The minimum cost is determined from the factored axial load “P_u” and factored moments “M_xu and M_yu”.

This work makes the following considerations: the footing is completely rigid and rests on elastic and homogeneous soil, i.e., the soil pressure on the footing behaves linearly according to Bowles [1], Das et al. [49], and McCormac and Brown [50].

Figure 2 presents an elliptical footing under biaxial bending rested on an elastic soil, and the soil pressure has a linear distribution.

The general biaxial bending equation is expressed as follows:

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

(1)

where σ represents the soil pressure at any part of the base (kN/m²); P represents the unfactored axial load (kN); A represents the contact surface with the soil at the base (m²); M_x and M_y represent the unfactored moments on the X and Y axes (kN-m), respectively; I_x and I_y represent the moments of inertia on the X and Y axes (m⁴), respectively; and x and y represent the coordinates in the X and Y directions of the base (m), respectively.

Equation (1) assumes that the soil contact surface is completely compressed [1,49,50].

A resultant moment “M_R” is assumed as follows:

$$M_R=\sqrt{{M_x}^2+{M_y}^2}.$$

(2)

The angle of inclination of the resultant moment on the X axis is expressed as follows:

$$\theta =arc sin\left({\displaystyle \frac{M_x}{M_R}}\right).$$

(3)

Equation of the ellipse located at the origin is expressed as follows:

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

(4)

where a = total dimension of the elliptical on X axis, and b = total dimension of the elliptical on Y axis.

Substituting A = πab, I_x = πab³/4, and I_y = πa³b/4 into Equation (1), the soil pressure is obtained in a general way:

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

(5)

Substituting Equation (4) into Equation (5) to determine the soil pressure as a function of x, the following is obtained:

$$\sigma ={\displaystyle \frac{P}{\pi ab}}+{\displaystyle \frac{4M_x\sqrt{1-\frac{x^2}{a^2}}}{\pi a{b}^2}}+{\displaystyle \frac{4M_{y}x}{\pi a^{3}b}}.$$

(6)

Deriving Equation (6) with respect to x to obtain the location of maximum and minimum pressures, the points are expressed as follows:

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

(7)

Equation (7) is substituted into Equation (4) to determine the values of y_max and y_min:

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

(8)

Now, substituting the corresponding coordinates of each critical point into Equation (5), the soil pressures on the elliptical footing are obtained:

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

(9)

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

(10)

where σ_max and σ_min are the maximum and minimum pressures, respectively.

2.1. Minimum Surface for an Elliptical Isolated Footing

The objective function A_min (minimum area) as follows:

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

(11)

The constraint functions appear in Equations (9) and (10), where 0 ≤ σ_max and σ_min ≤ σ_aasp (allowable soil pressure available to support all loads). The allowable soil pressure available to support all loads is calculated as the allowable soil pressure minus the concrete weight and minus the soil fill weight.

2.2. Minimum Cost for an Elliptical Isolated Footing

Now, Equation (5) is shown for the factored pressure σ_u in the function of the elliptical base coordinates. The general equation is expressed as follows:

$${\sigma }_u\left(x, y\right)={\displaystyle \frac{P_u}{\pi ab}}+{\displaystyle \frac{4M_{xu}y}{\pi a{b}^3}}+{\displaystyle \frac{4M_{yu}x}{\pi a^{3}b}},$$

(12)

where P_u = factored axial load (kN); M_xu = factored moment on the X axis (kN-m); and M_yu = factored moment on the Y axis (kN-m). The factored loads and moments are obtained as follows: 1.2 the dead load or dead moment plus 1.6 the live load or live moment as the case may be, according to ACI [51].

The design of isolated footings is governed by moments, bending shear and punching shear according to ACI [51].

2.2.1. Moments

Figure 3 presents the critical sections for moments of an elliptical footing under biaxial bending according to ACI [51].

The moments parallel to the X and Y axes are developed below.

The moment with respect to the “a” axis is expressed as follows:

$$M_{ua}=-{\int }_{{\displaystyle \frac{c_x}{2}}}^a{\int }_{-b\sqrt{1-{\displaystyle \frac{x^2}{a^2}}}}^{b\sqrt{1-{\displaystyle \frac{x^2}{a^2}}}}{\sigma }_u\left(x, y\right)\left(x-{\displaystyle \frac{c_x}{2}}\right)dydx,$$

(13)

$$M_{ua}=-{\displaystyle \frac{\left[P_u{a}^2\left(8a^2+{c_x}^2\right)-M_{yu}{c}_x\left(10a^2-{c_x}^2\right)\right]\sqrt{4a^2-{c_x}^2}}{24\pi a^4}}+{\displaystyle \frac{\left(P_u{c}_x-2M_{yu}\right)}{4\pi }}\left[\pi -2\mathit{arcsin}\left({\displaystyle \frac{c_x}{2a}}\right)\right],$$

(14)

where c_x = side of the column parallel to the X axis.

The moment with respect to the “b” axis is expressed as follows:

$$M_{ub}={\displaystyle \frac{P_u{c}_x}{2}}+M_{yu}-{\int }_{-{\displaystyle \frac{c_x}{2}}}^a{\int }_{-b\sqrt{1-{\displaystyle \frac{x^2}{a^2}}}}^{b\sqrt{1-{\displaystyle \frac{x^2}{a^2}}}}{\sigma }_u\left(x, y\right)\left(x+{\displaystyle \frac{c_x}{2}}\right)dydx,$$

(15)

$$M_{ub}=-{\displaystyle \frac{\left[P_u{a}^2\left(8a^2+{c_x}^2\right)+M_{yu}{c}_x\left(10a^2-{c_x}^2\right)\right]\sqrt{4a^2-{c_x}^2}}{24\pi a^4}}+{\displaystyle \frac{\left(P_u{c}_x+2M_{yu}\right)}{4\pi }}\left[\pi -2\mathit{arcsin}\left({\displaystyle \frac{c_x}{2a}}\right)\right].$$

(16)

The moment with respect to the “c” axis is expressed as follows:

$$M_{uc}=-{\int }_{{\displaystyle \frac{c_y}{2}}}^b{\int }_{-a\sqrt{1-{\displaystyle \frac{y^2}{b^2}}}}^{a\sqrt{1-{\displaystyle \frac{y^2}{b^2}}}}{\sigma }_u\left(x, y\right)\left(y-{\displaystyle \frac{c_y}{2}}\right)dxdy,$$

(17)

$$M_{uc}=-{\displaystyle \frac{\left[P_u{b}^2\left(8b^2+{c_y}^2\right)-M_{xu}{c}_y\left(10b^2-{c_y}^2\right)\right]\sqrt{4b^2-{c_y}^2}}{24\pi b^4}}+{\displaystyle \frac{\left(P_u{c}_y-2M_{xu}\right)}{4\pi }}\left[\pi -2\mathit{arcsin}\left({\displaystyle \frac{c_y}{2b}}\right)\right],$$

(18)

where c_y = side of the column parallel to the Y axis.

The moment with respect to the “e” axis is expressed as follows:

$$M_{ue}={\displaystyle \frac{P_u{c}_y}{2}}+M_{xu}-{\int }_{-{\displaystyle \frac{c_y}{2}}}^b{\int }_{-a\sqrt{1-{\displaystyle \frac{y^2}{b^2}}}}^{a\sqrt{1-{\displaystyle \frac{y^2}{b^2}}}}{\sigma }_u\left(x, y\right)\left(y+{\displaystyle \frac{c_y}{2}}\right)dxdy,$$

(19)

$$M_{ue}=-{\displaystyle \frac{\left[P_u{b}^2\left(8b^2+{c_y}^2\right)+M_{xu}{c}_y\left(10b^2-{c_y}^2\right)\right]\sqrt{4b^2-{c_y}^2}}{24\pi b^4}}+{\displaystyle \frac{\left(P_u{c}_y+2M_{xu}\right)}{4\pi }}\left[\pi -2\mathit{arcsin}\left({\displaystyle \frac{c_y}{2b}}\right)\right].$$

(20)

2.2.2. Bending Shear

Figure 4 shows the critical sections for bending shear of an elliptical footing under biaxial bending, appearing at a distance “d” (effective footing depth) from the column faces according to ACI [51].

The bending shear with respect to the “f” axis is expressed as follows:

$$V_{uf}=-{\int }_{{\displaystyle \frac{c_x}{2}}+d}^a{\int }_{-b\sqrt{1-{\displaystyle \frac{x^2}{a^2}}}}^{b\sqrt{1-{\displaystyle \frac{x^2}{a^2}}}}{\sigma }_u\left(x, y\right)dydx,$$

(21)

$$V_{uf}={\displaystyle \frac{\left\{3P_u{a}^2\left(c_x+2d\right)-4M_{yu}\left[4a^2-{\left(c_x+2d\right)}^2\right]\right\}\sqrt{4a^2-{\left(c_x+2d\right)}^2}}{12\pi a^4}}-{\displaystyle \frac{P_u}{2\pi }}\left[\pi -2\mathit{arcsin}\left({\displaystyle \frac{c_x+2d}{2a}}\right)\right].$$

(22)

The bending shear with respect to the “g” axis is expressed as follows:

$$V_{ug}=P_u-{\int }_{-{\displaystyle \frac{c_x}{2}}-d}^a{\int }_{-b\sqrt{1-{\displaystyle \frac{x^2}{a^2}}}}^{b\sqrt{1-{\displaystyle \frac{x^2}{a^2}}}}{\sigma }_u\left(x, y\right)dydx,$$

(23)

$$V_{ug}=-{\displaystyle \frac{\left\{3P_u{a}^2\left(c_x+2d\right)+4M_{yu}\left[4a^2-{\left(c_x+2d\right)}^2\right]\right\}\sqrt{4a^2-{\left(c_x+2d\right)}^2}}{12\pi a^4}}+{\displaystyle \frac{P_u}{2\pi }}\left[\pi -2\mathit{arcsin}\left({\displaystyle \frac{c_x+2d}{2a}}\right)\right].$$

(24)

The bending shear with respect to the “h” axis is expressed as follows:

$$V_{uh}=-{\int }_{{\displaystyle \frac{c_y}{2}}+d}^b{\int }_{-a\sqrt{1-{\displaystyle \frac{y^2}{b^2}}}}^{a\sqrt{1-{\displaystyle \frac{y^2}{b^2}}}}{\sigma }_u\left(x, y\right)dxdy,$$

(25)

$$V_{uh}={\displaystyle \frac{\left\{3P_u{b}^2\left(c_y+2d\right)-4M_{xu}\left[4b^2-{\left(c_y+2d\right)}^2\right]\right\}\sqrt{4b^2-{\left(c_y+2d\right)}^2}}{12\pi b^4}}-{\displaystyle \frac{P_u}{2\pi }}\left[\pi -2\mathit{arcsin}\left({\displaystyle \frac{c_y+2d}{2b}}\right)\right],$$

(26)

The bending shear with respect to the “i” axis is expressed as follows:

$$V_{ui}=P_u-{\int }_{-{\displaystyle \frac{c_y}{2}}-d}^b{\int }_{-a\sqrt{1-{\displaystyle \frac{y^2}{b^2}}}}^{a\sqrt{1-{\displaystyle \frac{y^2}{b^2}}}}{\sigma }_u\left(x, y\right)dxdy,$$

(27)

$$V_{ui}=-{\displaystyle \frac{\left\{3P_u{b}^2\left(c_y+2d\right)+4M_{xu}\left[4b^2-{\left(c_y+2d\right)}^2\right]\right\}\sqrt{4b^2-{\left(c_y+2d\right)}^2}}{12\pi b^4}}+{\displaystyle \frac{P_u}{2\pi }}\left[\pi -2\mathit{arcsin}\left({\displaystyle \frac{c_y+2d}{2b}}\right)\right].$$

(28)

2.2.3. Punching Shear

Figure 5 shows the critical perimeter for punching shear of an elliptical footing under biaxial bending, appearing at a distance “d/2” from the column faces according to ACI [51].

The punching shear at the critical perimeter is expressed as follows:

$$V_{up}=P_u-{\int }_{-{\displaystyle \frac{c_y+d}{2}}}^{{\displaystyle \frac{c_y+d}{2}}}{\int }_{-{\displaystyle \frac{c_x+d}{2}}}^{{\displaystyle \frac{c_x+d}{2}}}{\sigma }_u\left(x, y\right)dxdy,$$

(29)

$$V_{up}={\displaystyle \frac{P_u\left[\pi ab-\left(c_x+d\right)\left(c_y+d\right)\right]}{\pi ab}}.$$

(30)

2.2.4. Objective Function

The equation to determine the minimum total cost “C_min” (dollars) is as follows:

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

(31)

where C_c = cost of concrete in USD per each cubic meter, V_c = volume of concrete in cubic meters, C_s = cost of steel in USD per each kilonewton of weight, V_s = volume of steel in cubic meters, and γ_s = steel density (78 kN/m³).

The reinforcing steel of the elliptical footing is obtained as follows. The footing is reinforced orthogonally, parallel to the main axes and an elliptical ring at a distance r (concrete cover in m) from the outer edge.

The steel area parallel to the X axis “A_sx” and the steel area parallel to the Y axis “A_sy” are expressed as follows:

$$A_{sx}={\rho }_x{b}_{wy}d;A_{sy}={\rho }_y{b}_{wx}d,$$

(32)

where A_sx is the largest area of A_sxa and A_sxb (steel areas in sections “a” and “b”), and A_sy is the largest area of A_syc and A_sye (steel areas in sections “c” and “e”). These areas are determined as follows:

$$A_{sxa}={\rho }_x{b}_{wya}d;A_{sxb}={\rho }_x{b}_{wyb}d; A_{syc}={\rho }_y{b}_{wxc}d; A_{sye}={\rho }_y{b}_{wxe}d,$$

(33)

where b_wya, b_wyb, b_wxc, and b_wxe are determined as follows:

$$b_{wya}=2b\sqrt{1-{\displaystyle \frac{{c_x}^2}{{4a}^2}}};b_{wyb}=2b\sqrt{1-{\displaystyle \frac{{c_x}^2}{{4a}^2}}};b_{wxc}=2a\sqrt{1-{\displaystyle \frac{{c_y}^2}{4b^2}}};b_{wxe}=2a\sqrt{1-{\displaystyle \frac{{c_y}^2}{4b^2}}} ,$$

(34)

The number of rods parallel to the X axis “n_x” and the number of rods parallel to the Y axis “n_y” are given as follows:

$$n_x={\displaystyle \frac{A_{sx}}{a_s}}; n_y={\displaystyle \frac{A_{sy}}{a_s}},$$

(35)

where a_s = cross-sectional steel area of the rod in the both directions.

$$s_x={\displaystyle \frac{b_{wy}{a}_s}{A_{sx}}}; s_y={\displaystyle \frac{b_{wx}{a}_s}{A_{sy}}},$$

(36)

where s_x = separation of the rods parallel to the X axis, and s_y = separation of the rods parallel to the Y axis.

The total lengths of the rods in the both directions are obtained as follows:

$$L_x=2a+4\sum _{i=1}^{\left(n_x-3\right)/2}a\sqrt{1-{\displaystyle \frac{{\left(i{s}_x\right)}^2}{b^2}}},$$

(37)

$$L_y=2b+4\sum _{j=1}^{\left(n_y-3\right)/2}b\sqrt{1-{\displaystyle \frac{{\left(j{s}_y\right)}^2}{a^2}}}.$$

(38)

Note: Equations (37) and (38) are given from the location of the rods on the X and Y axes. Then, the following expression presents the length according to the spacing of the rods. The length of the rods parallel to the X axis is $a\sqrt{1-{\displaystyle \frac{{\left(i{s}_x\right)}^2}{b^2}}}$ (length of the rods located on the positive part of the Y axis; these lengths must be multiplied by four to determine the total length of the rods located outside the X axis). The length of the rods parallel to the Y axis is $b\sqrt{1-{\displaystyle \frac{{\left(j{s}_y\right)}^2}{a^2}}}$ (length of the rods located on the positive part of the X axis; these lengths must be multiplied by four to determine the total length of the rods located outside the Y axis).

The length of the rods around the perimeter of the ellipse “L_e” is as follows:

$$L_e=2\pi \sqrt{{\displaystyle \frac{{\left(a-r\right)}^2{+\left(b-r\right)}^2}{2}}},$$

(39)

where r = concrete cover.

Figure 6 shows A_sx, A_sy, s_x, s_y, and L_e.

The volume of steel “V_s” is shown below:

$$V_s=a_s\left(L_y+L_x+L_c\right).$$

(40)

The volume of concrete “V_c” is presented below:

$$V_c=\pi ab\left(d+r\right)-V_s.$$

(41)

Substituting Equations (40) and (41) into Equation (31) gives the following:

$$C_{min}=\left[\pi ab\left(d+r\right)-V_s\right]C_c+V_s{\gamma }_s{C}_s.$$

(42)

Now, substituting α = γ_sC_s/C_c → γ_sC_s = αC_c into Equation (42) gives the following:

$$C_{min}=\left[\pi ab\left(d+r\right)+\left(\alpha -1\right)V_s\right]C_c.$$

(43)

Substituting Equation (40) with their respective values of “L_y”, “L_x”, and “L_e” into Equation (43) yields the following:

$$\begin{array}{c}{C}_{min}=[a_s\left(\alpha -1\right)\left(2a+4{\displaystyle \sum _{i=1}^{\left(n_x-3\right)/2}}a\sqrt{1-{\displaystyle \frac{{\left(i{s}_x\right)}^2}{b^2}}}+2b+4{\displaystyle \sum _{j=1}^{\left(n_y-3\right)/2}}b\sqrt{1-{\displaystyle \frac{{\left(j{s}_y\right)}^2}{a^2}}}+2\pi \sqrt{{\displaystyle \frac{{\left(a-r\right)}^2{+\left(b-r\right)}^2}{2}}}\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\pi ab\left(d+r\right)]C_c.\hfill \end{array}$$

(44)

2.2.5. Constraint Functions

This subsection shows the equations provided by the code [51].

For the moments, the following is expressed [51]:

$$\left|M_{ua}\right|,\left|M_{ub}\right|,\left|M_{uc}\right|,\left|M_{ue}\right|\le {Ø}_f{f}_{y}d{A}_s\left(1-{\displaystyle \frac{A_s{f}_y}{1.7b_{w}d{f^{\prime }}_c}}\right),$$

(45)

where f_y = strength of the steel reinforcement in MPa, f’_c = compressive strength of the concrete after 28 days in MPa, A_sx = steel reinforcement in the X direction, A_sy = steel reinforcement in the Y direction, and Ø_f (bending strength reduction factor) = 0.90.

Note: For M_ua, A_s = A_sxa and b_w = b_wya; for M_ub, A_s = A_sxb and b_w = b_wyb; for M_uc, A_s = A_syc and b_w = b_wxc; for M_ue, A_s = A_sye and b_w = b_wxe.

For the shears, the following is expressed [51]:

$$\left|V_{uf}\right|,\left|V_{ug}\right|,\left|V_{uh}\right|,\left|V_{ui}\right|\le 0.17{\varnothing }_v\sqrt{{f^{\prime }}_c}{b}_{ws}d,$$

(46)

where Ø_v (shear strength reduction factor) = 0.85.

Note: For V_uf, b_ws = b_wsyf; for V_ug, b_ws = b_wsyg; for V_uh, b_ws = b_wsxh; for V_ui, b_ws = b_wsxi. Here, b_wsyf, b_wsygb, b_wsxh, and b_wsxi are determined as follows:

$$b_{wsyf}=2b\sqrt{1-{\displaystyle \frac{{\left(c_x+2d\right)}^2}{{4a}^2}}};b_{wsyg}=2b\sqrt{1-{\displaystyle \frac{{\left(c_x+2d\right)}^2}{{4a}^2}}};b_{wsxh}=2a\sqrt{1-{\displaystyle \frac{{\left(c_y+2d\right)}^2}{4b^2}}};b_{wsxi}=2a\sqrt{1-{\displaystyle \frac{{\left(c_y+2d\right)}^2}{4b^2}}}.$$

(47)

For the punching, the following is expressed [51]:

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

(48)

where β_c = long side of column/short side of column; b₀ = perimeter of the critical punching shear section in m; α_s = 20 for the corner column; α_s = 30 for the edge column; and α_s = 40 for the interior column.

For the percentages of steel, the following is expressed [51]:

$${\rho }_x,{\rho }_y\le 0.75\left[{\displaystyle \frac{0.85{\beta }_1{f^{\prime }}_c}{f_y}}\left({\displaystyle \frac{600}{600+f_y}}\right)\right],$$

(49)

$${\rho }_x,{\rho }_y\ge \left\{\begin{array}{c}{\displaystyle \frac{0.25\sqrt{{f^{\prime }}_c}}{f_y}}\\ {\displaystyle \frac{1.4}{f_y}}\end{array}\right.,$$

(50)

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

(51)

where β₁ = factor relating the depth of the equivalent rectangular compressive stress block to the depth of the neutral axis.

For the steel areas, the following is expressed [51]:

$$A_{sx}\ge \left\{\begin{array}{c}{A}_{sxa}={\rho }_x{b}_{wya}d\\ A_{sxb}={\rho }_x{b}_{wyb}d\end{array}\right.,$$

(52)

$$A_{sy}\ge \left\{\begin{array}{c}{A}_{syc}={\rho }_x{b}_{wxc}d\\ A_{sye}={\rho }_x{b}_{wxe}d\end{array}\right..$$

(53)

For the number of rods, the following is expressed [51]:

$$n_x={\displaystyle \frac{A_{sx}}{a_s}},$$

(54)

$$n_y={\displaystyle \frac{A_{sy}}{a_s}}.$$

(55)

For the separations, the following is expressed [51]:

$$s_x={\displaystyle \frac{b_{wy}{a}_s}{A_{sx}}},$$

(56)

$$s_y={\displaystyle \frac{b_{wx}{a}_s}{A_{sy}}},$$

(57)

The new model is developed in two parts. The first part is used to estimate the minimum area, and the second part is used to determine the minimum cost. For the minimum area, the independent parameters are σ_aasp, P, M_x, and M_y, and the design variables are A_min, a, b, σ_max, and σ_min. For the minimum cost, the independent parameters are a, b, P_u, M_xu, and M_yu, and the design variables are C_min, d, A_sx, A_sy, ρ_x, and ρ_y.

The flowchart algorithm for the design procedure of an elliptical footing under the minimum cost criterion appears in Figure A1 (see Appendix A).

The flowchart for using Maple software (v.15) to design an elliptical footing under the minimum cost criterion appears in Figure A2 (see Appendix A).

3. Numerical Examples

Two numerical examples are presented, and each example shows five variants to determine the minimum cost of elliptical footings subjected to biaxial bending. The general data for the two examples are c_x = c_y = 40 cm, r = 7.5 cm, as = 5.07 cm² (1Ø1”), σ_aasp = 200 kN/m², f’_c = 21 MPa, f_y = 420 MPa, and α = 90.

The input data (P, M_x, M_y; these loads and moments should not be factored) for example 1 appear in

Table 1. Input data for example 1.

Example	PD (kN)	PL (kN)	P (kN)	MxD (kN-m)	MxL (kN-m)	Mx (kN-m)	MyD (kN-m)	MyL (kN-m)	My (kN-m)
A.1	600	500	1100	300	200	500	500	400	900
B.1	600	500	1100	300	200	500	400	300	700
C.1	600	500	1100	300	200	500	300	200	500
D.1	600	500	1100	300	200	500	200	100	300
E.1	600	500	1100	300	200	500	100	50	150

. The output data (P_u, M_xu, M_yu; these loads and moments must be factored) for example 1 are shown in

Table 2. Output data for example 1.

Problem	a (m)	b (m)	σmax (kN/m2)	σmin (kN/m2)	Amin (m2)	Pu (kN)	Mxu (kN-m)	Myu (kN-m)	Asx (cm2)	Asy (cm2)	sx (cm)	sy (cm)	d (cm)	ρx	ρy	Cmin (USD)
A.1	4.60	2.60	58.48	0.07	37.57	1520	680	1240	148.05	114.82	17.77	40.49	37.50	0.00761	0.00333	31.11Cc
B.1	3.60	2.60	74.61	0.21	29.41	1520	680	960	109.52	89.66	24.03	40.60	37.50	0.00563	0.00333	22.39Cc
C.1	2.60	2.60	103.02	0.57	21.24	1520	680	680	73.07	73.07	35.94	35.94	37.50	0.00376	0.00376	14.88Cc
D.1	1.55	2.60	173.09	0.68	12.66	1520	680	400	73.03	71.37	35.82	21.95	42.50	0.00333	0.00535	10.58Cc
E.1	1.25	2.60	196.54	18.93	10.21	1520	680	200	81.14	58.55	32.05	21.56	47.50	0.00333	0.00495	9.34Cc

Note: C_min is shown as a function of C_c.

.

The input data (P, M_x, M_y; these loads and moments should not be factored) for example 2 appear in

Table 3. Input data for example 2.

Example	PD (kN)	PL (kN)	P (kN)	MxD (kN-m)	MxL (kN-m)	Mx (kN-m)	MyD (kN-m)	MyL (kN-m)	My (kN-m)
A.2	800	700	1500	300	200	500	200	100	300
B.2	700	600	1300	300	200	500	200	100	300
C.2	600	500	1100	300	200	500	200	100	300
D.2	500	400	900	300	200	500	200	100	300
E.2	400	300	700	300	200	500	200	100	300

. The output data (P_u, M_xu, M_yu; these loads and moments must be factored) for example 2 are shown in

Table 4. Output data for example 2.

Problem	a (m)	b (m)	σmax (kN/m2)	σmin (kN/m2)	Amin (m2)	Pu (kN)	Mxu (kN-m)	Myu (kN-m)	Asx (cm2)	Asy (cm2)	sx (cm)	sy (cm)	d (cm)	ρx	ρy	Cmin (USD)
A.2	1.60	2.60	196.98	32.57	13.07	2080	680	400	81.62	73.58	32.05	14.51	47.50	0.00333	0.00486	11.74Cc
B.2	1.55	2.50	198.24	15.34	12.17	1800	680	400	70.24	71.73	35.80	21.84	42.50	0.00333	0.00546	10.22Cc
C.2	1.55	2.60	173.09	0.68	12.66	1520	680	400	73.03	71.37	35.82	21.95	42.50	0.00333	0.00535	10.58Cc
D.2	1.90	3.15	95.50	0.24	18.80	1240	680	400	67.89	90.29	46.75	21.28	32.50	0.00333	0.00733	13.58Cc
E.2	2.45	4.05	44.77	0.14	31.17	960	680	400	87.81	91.11	46.60	27.21	32.50	0.00335	0.00573	20.99Cc

Note: C_min is shown as a function of C_c.

.

4. Results

The proposed model (elliptical footing) compared with that of Luévanos-Rojas et al. [47] (circular footing) is verified as follows.

Note: The axes parallel to the X axis are the c and e axes for moments and the h and i axes for bending shears, and the axes parallel to the Y axis are the a and b axes for moments and the g and f axes for bending shears (proposed model). The axes parallel to the X axis are the a and b axes for moments and the f and g axes for bending shears, and the axes parallel to the Y axis are the c and e axes for moments and the h and i axes for bending shears (proposed model).

For factored pressure:

• Substituting a = D/2, b = D/2 into Equation (12) gives σ_u(x, y), and substituting e_x = 0, e_y = 0 into Equation (8) of the reference [47] gives σ_u(x, y); both are equal.

For moments:

• Substituting a = D/2, b = D/2 into Equation (14) gives M_ua, and substituting e_x = 0, e_y = 0 into Equation (14) of the reference [47] gives M_uc, M_ua = M_uc (both are parallel to the Y axis).
• Substituting a = D/2, b = D/2 into Equation (16) gives M_ub, and substituting e_x = 0, e_y = 0 into Equation (16) of the reference [47] gives M_ue, M_ub = M_ue (both are parallel to the Y axis).
• Substituting a = D/2, b = D/2 into Equation (18) gives M_uc, and substituting e_x = 0, e_y = 0 into Equation (10) of the reference [47] gives M_ua, M_uc = M_ua (both are parallel to the X axis).
• Substituting a = D/2, b = D/2 into Equation (20) gives M_ue, and substituting e_x = 0, e_y = 0 into Equation (12) of the reference [47] gives M_ub, M_ue = M_ub (both are parallel to the X axis).

For bending shears:

• Substituting a = D/2, b = D/2 into Equation (21) gives Vuf, and substituting ex = 0, ey = 0 into Equation (22) of the reference [47] gives Vubh, Vuf = Vubh (both are parallel to the Y axis).
• Substituting a = D/2, b = D/2 into Equation (23) gives Vug, and substituting ex = 0, ey = 0 into Equation (24) of the reference [47] gives Vubi, Vug = Vubi (both are parallel to the Y axis).
• Substituting a = D/2, b = D/2 into Equation (25) gives Vuh, and substituting ex = 0, ey = 0 into Equation (18) of the reference [47] gives Vubf, Vuh = Vubf (both are parallel to the X axis).
• Substituting a = D/2, b = D/2 into Equation (27) gives Vui, and substituting ex = 0, ey = 0 into Equation (20) of the reference [47] gives Vubg, Vui = Vubg (both are parallel to the X axis).

For punching shear:

• Substituting a = D/2, b = D/2 into Equation (29) gives V_up, and substituting e_x = 0, e_y = 0, x₁ = c_x/2 + d/2, x₂ = −c_x/2 − d/2, y₁ = c_y/2 + d/2, y₂ = −c_y/2 − d/2 into Equation (26) of the reference [47] gives V_up, both are equal.

Also, the proposed model can be verified by continuity as follows.

For moments:

• Integrating Equation (13) from zero to “a” and obtaining moments at the free end gives M_ua = 0.
• Integrating Equation (15) from “−a” to “a” and obtaining moments at the free end gives M_ub = 0.
• Integrating Equation (13) from zero to “a” and obtaining moments at the center of footing gives M_ua = −2P_ua/3π − M_yu/2. Integrating Equation (15) from “−a” to zero and obtaining moments at the center of footing gives M_ub = 2P_ua/3π + M_yu/2. Therefore, M_ua and M_ub are the same, but in the opposite sense. Thus, continuity is guaranteed.
• Integrating Equation (17) from zero to “b” and obtaining moments at the free end gives M_uc = 0.
• Integrating Equation (19) from “−b” to “b” and obtaining moments at the free end gives M_ue = 0.
• Integrating Equation (17) from zero to “b” and obtaining moments at the center of footing gives M_uc = −2P_ub/3π − M_xu/2. Integrating Equation (19) from “−b” to zero and obtaining moments at the center of footing gives M_ue = 2P_ub/3π + M_xu/2. Therefore, M_uc and M_ue are the same, but in the opposite sense. Thus, continuity is guaranteed.

For bending shears:

• Integrating Equation (21) from zero to “a” and obtaining the bending shear at the free end gives V_uf = 0.
• Integrating Equation (23) from “−a” to “a” and obtaining the bending shear at the free end gives V_ug = 0.
• Integrating Equation (21) from zero to “a” and obtaining the bending shear at the center of footing gives V_uf = −P_u/2 − 8M_yu/3aπ. Integrating Equation (23) from “−a” to zero and obtaining the bending shear at the center of footing gives V_ug = P_u/2 + 8M_yu/3aπ. Therefore, V_uf and V_ug are the same, but in the opposite sense. Thus, continuity is guaranteed.
• Integrating Equation (25) from zero to “b” and obtaining the bending shear at the free end gives V_uh = 0.
• Integrating Equation (27) from “−b” to “b” and obtaining the bending shear at the free end gives V_ui = 0.
• Integrating Equation (25) from zero to “b” and obtaining the bending shear at the center of footing gives V_uh = −P_u/2 − 8M_xu/3bπ. Integrating Equation (27) from “−b” to zero and obtaining the bending shear at the center of footing gives V_ui = P_u/2 + 8M_xu/3bπ. Therefore, V_uh and V_ui are the same, but in the opposite sense. Thus, continuity is guaranteed.

For punching shear:

• Integrating Equation (29) from zero to “a” in the X direction and from zero to “b” in the Y direction to obtain the punching shear at the free ends of the footing results in zero.

Table 2 and Table 4 show the following. For examples A.1, B.1, C.1, D.1, C.2, D.2, and E.2 the minimum pressure governs, and for examples E.1, A.2, and B.2 the maximum pressure governs.

Table 2 presents the following. When M_yu is reduced, a, A_min, and C_min decrease; b is the same for all examples; A_sx, A_sy, and ρ_x decrease up to M_yu = 400 kN-m and subsequently increase (ρ_x is the same for examples D.1 and E.1); ρ_y is the same for examples A.1 and B.1, then increases up to M_yu = 400 kN-m, and subsequently decreases; σ_max, σ_min increase; s_x increases up to M_yu = 680 kN-m and then decreases; s_y increases up to M_yu = 960 kN-m and then decreases; d tends to increase. Example C.1 shows a circular footing.

Table 4 shows the following. When P_u is reduced, a, b, A_min, C_min decrease up to P_u = 1800 kN and subsequently increase; A_sx decreases up to P_u = 1800 kN, then increases up to P_u = 1520 kN, then decreases up to P_u = 1240 kN, and then increases; A_sy increases up to P_u = 1520 kN, then decreases up to P_u = 1240 kN, and then increases; σ_max increases up to P_u = 1800 kN and then decreases; σ_min decreases for all examples; ρ_y increases up to P_u = 1800 kN, then decreases up to P_u = 1520 kN, then increases up to P_u = 1240 kN, and then decreases; ρ_x is the same for examples A.2, B.2, C.2, and D.2 and then increases; s_x increases up to P_u = 1240 kN and then decreases; s_y increases up to P_u = 1520 kN, then decreases up to P_u = 1240 kN, and then increases; and d tends to decrease.

According to the results, this model can be applied to circular footings with axial load located at the center of gravity of the footing as shown in problem C.1 of Table 2.

Table 5. Output data of example 2 for circular footings.

Problem	D (m)	σmax (kN/m2)	σmin (kN/m2)	Amin (m2)	Pu (kN)	Mxu (kN-m)	Myu (kN-m)	Asx (cm2)	Asy (cm2)	sx (cm)	sy (cm)	d (cm)	ρx	ρy	Cmin (USD)
A.2	4.10	199.79	27.44	13.20	2080	680	400	64.54	64.54	32.21	32.21	47.50	0.00333	0.00333	10.85Cc
B.2	4.00	196.25	10.65	12.57	1800	680	400	56.33	63.52	35.82	31.77	42.50	0.00333	0.00376	9.73Cc
C.2	4.25	154.91	0.17	14.19	1520	680	400	59.87	63.86	35.82	33.58	42.50	0.00333	0.00355	10.79Cc
D.2	5.20	84.62	0.14	21.24	1240	680	400	64.69	64.69	40.60	40.60	37.50	0.00333	0.00333	14.12Cc
E.2	6.70	39.60	0.11	35.26	960	680	400	83.56	83.55	40.59	40.60	37.50	0.00333	0.00333	23.74Cc

D = diameter of the footing. Note: C_min is shown as a function of C_c.

shows the results of problem 2 using the equations for circular footings proposed by Luévanos-Rojas et al. [47]; this example is solved using e_x (eccentricity in the X direction) = 0, e_y (eccentricity in the Y direction) = 0, and the same input data as example 2 (see Table 3).

Table 5 presents the following. When P_u is reduced, D, A_sx, A_sy, A_min, and C_min decrease up to P_u = 1800 kN and subsequently increase; σ_max and σ_min decrease for all examples; ρ_x is the same for all examples; ρ_y increases up to P_u = 1800 kN, then decreases up to P_u = 1240 kN, and then it remains constant; s_x tends to increases; s_y decreases up to P_u = 1800 kN and then increases; d tends to decrease. For examples A.2, B.2 the maximum pressure prevails. For examples, C.2, D.2, and E.2 the minimum pressure prevails.

Figure 7 shows the comparison of the minimum area in contact with the ground and minimum design cost between EF (elliptical footings) and CF (circular footings) of example 2 (see Table 4 and Table 5).

Figure 7a presents the following. The minimum area is smaller for elliptical footings than for circular footings in all problems. The greatest savings occur in E2 with a saving of 11.60% using the elliptical footings model.

Figure 7b shows the following. The lowest minimum cost appears in examples A.2; B.2 for circular footings; and in examples C.2, D.2, and E.2 for elliptical footings. The greatest savings occur in A2 with a saving of 7.58% using the circular footings model. The greatest savings occur in E2 with a saving of 11.58% using the elliptical footings model.

5. Conclusions

This research shows the mathematical calculation for the optimal cost design of elliptical footings subjected to biaxial bending, assuming that the footing rests on elastic soils, the footing is rigid, and the pressure diagram is linear.

The new model for elliptical footings is shown in two parts. The first part is used to determine the minimum area, and the second part is used to obtain the minimum cost. For the minimum area, the independent parameters are σ_aasp, P, M_x, and M_y, and the design variables are A_min, a, b, σ_max, and σ_min. For the minimum cost, the independent parameters are a, b, P_u, M_xu, and M_yu, and the design variables are C_min, d, A_sx, A_sy, ρ_x, and ρ_y.

The contributions of this paper are as follows:

(1)

There is no literature on the subject of elliptical footings since the closest are circular footings; therefore, it is an innovative topic.

(2)

The moments, bending shears, and punching shear are verified by equilibrium (see Section 4).

(3)

The new model for elliptical footings can be applied to other building standards, taking into account the moments, bending shears, and punching shear that must be resisted.

(4)

When the factored moment about the Y axis “M_yu” is reduced, the minimum area “A_min” and the minimum cost “C_min” are reduced (see Table 2).

(5)

When the factored axial load “P_u” is reduced, the minimum area “A_min” and the minimum cost “C_min” increase from B2 both models (elliptical footings in Table 4 and circular footings in Table 5).

(6)

The minimum area is smaller for elliptical footings than for circular footings in all problems (see Figure 7a).

(7)

When the dimensions of the footing are governed by the maximum pressure, the minimum cost appears in circular footings (see Table 4 and Table 5).

(8)

When the dimensions of the footing are governed by the minimum pressure, the minimum cost appears in elliptical footings (see Table 4 and Table 5).

(9)

The elliptical footing model presents a saving of 11.60% in the minimum area and 11.58% in the minimum cost for E2 compared to the circular footing model (see Figure 7).

This paper presents, in detail, the mathematical equations of moments, bending shear, and punching shear, as well as the optimization algorithms, which is its main advantage over other works.

Suggestions for future research may include mathematical calculations for the cost-optimal design of elliptical footings working with the resulting moment only and locating the principal axes where the resulting moment appears, that is, rotating the principal axes; another investigation could be considering that the surface in contact with the ground works partially in compression.