# Metaheuristic-Based Practical Tool for Optimal Design of Reinforced Concrete Isolated Footings: Development and Application for Parametric Investigation

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Modeling of the Design Problem

#### 2.1.1. Soil Pressure

#### 2.1.2. Depth of Footing

#### 2.1.3. Design for Shear

#### 2.1.4. Design for Flexure

#### 2.1.5. Spacing Criteria of Reinforcement

#### 2.1.6. Development Length Criteria

#### 2.2. Modeling of Optimization Problem

_{n}), as indicated in Equation (1).

_{j}(x) and g

_{k}(x), respectively, whereas Equation (4) gives upper (x

_{iU}) and lower (x

_{iL}) bounds on variables. In Equations (2)–(4), (p), (m), and (n) represent the total number of inequality constraints, equality constraints, and variables to be applied to the function respectively.

#### 2.2.1. Objective Function

#### 2.2.2. Variables

#### 2.2.3. Constraints

#### 2.3. Optimization Technique

#### 2.3.1. Evolutionary Algorithm (EA)

#### 2.3.2. Genetic Algorithm (GA)

#### 2.4. Spreadsheet Interface

## 3. Case Studies

## 4. Results and Discussion

^{3}, while the optimized quantity was 4.83 m

^{3}. The original quantity of steel was 208.48 kg while the optimized quantity was 165.47 kg. This resulted in the optimization of overall cost by 31.63%. By using the GA, one run of 54 s with total of 20,095 trials was made. This further reduced steel weight to 165.45 kg but without a change in concrete volume. This resulted in an optimization of overall cost by 31.65%.

^{3}was optimized to 8.22 m

^{3}and the total quantity of steel of 452.59 kg was optimized to 316.87 kg. This resulted in an optimization of overall cost by 43.01% by the two succeeding runs; run one of 31.797 s with 1160 sub-problems and run two of 32.434 s with 1292 sub-problems. By using the GA, one run of 56 s with total of 20,387 trials was made. This further reduced the concrete volume to 8.2 m

^{3}and steel weight to 306.8 kg. This resulted in an optimization of overall cost by 44.04%

^{3}was optimized to 8.19 m

^{3}and the original quantity of steel of 364.95 kg was optimized to 307.65 kg. This resulted in an optimization of overall cost by 18.4%. By using the GA, one run of 53 s with total of 20,001 trials was made which further reduced steel weight to 294.19 kg without a change in concrete volume resulting in an optimization of overall cost by 19.41%.

^{3}to 1.8 m

^{3}and the quantity of steel from 83.92 kg to 66.42 kg, which resulted in an optimization of overall cost by 26.45% by the two consecutive runs; run one of 57.157 s with 2874 sub-problems and run two of 31.625 s with 994 sub-problems. By using the GA, one run of 99 s with total of 32,222 trials was made. This further reduced steel weight to 53.91 kg but without a change in concrete volume. This resulted in an optimization of overall cost by 28.14%.

## 5. Parametric Investigation

#### 5.1. Effect of Geometric Parameters

#### 5.2. Effect of Reinforcement Parameters

#### 5.3. Effect of Material Strength

## 6. Conclusions

- The developed tool is not only a user-friendly and simple way of finding the most efficient designs with high computational efficiency compared to conventional methods, but it can also save up to 44% of the cost for the examples considered in this study. It does not get trapped in localized minimum solutions and can reach a global optimum solution for complex and non-linear problems.
- The cost analysis results show that optimization tends to increase the cheaper material, i.e., concrete, if required, and reduce the more costly material, i.e., steel, in such a way that all the constraints for design variables remain within the limits defined by the code, making this tool more efficient in terms of material usage.
- The parametric analysis suggested an optimal range of 0.9 to 1.1 for the aspect ratio of footings, 0.0025 to 0.003 for the reinforcement ratio, and the use of less compressive strength of concrete with high yielding strength of steel for a cost-effective design.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

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**Figure 1.**Reinforced concrete isolated footing sections and critical sections for moments, one way, punching and eccentric shear.

**Figure 2.**Pressure distribution diagram of eccentrically loaded footing subjected to bi-axial bending.

Symbol | Units | Variable | Lower Bound | Upper Bound |
---|---|---|---|---|

$h$ | mm | Footing thickness | 250 | 1000 |

$L$ | m | Length of the footing | 1 | 10 |

$B$ | m | Width of the footing | 1 | 10 |

$A{s}_{L}$ | mm^{2}/m | Area of steel along the length of footing | $A{s}_{min}$ | $A{s}_{max}$ |

$A{s}_{B}$ | mm^{2}/m | Area of steel along the width of footing | $A{s}_{min}$ | $A{s}_{max}$ |

${{f}_{c}}^{\prime}$ | MPa | Concrete compressive strength | 20 | 40 |

${f}_{y}$ | MPa | Steel yield strength | 275 | 500 |

Constraints | Formula | Description |
---|---|---|

${g}_{1}\left(x\right)$ | $V{u}^{\prime}+V{u}^{\u2033}\le \varphi Vc$ | $\mathrm{Concrete}\mathrm{shear}\mathrm{strength}(\varphi Vc$$)\mathrm{greater}\mathrm{than}\mathrm{maximum}\mathrm{punching}(V{u}^{\prime}$$)\mathrm{and}\mathrm{eccentric}\mathrm{shear}(V{u}^{\u2033}$) |

${g}_{2}\left(x\right)$ | $Vu\le \varphi Vc$ | $\mathrm{Concrete}\mathrm{shear}\mathrm{strength}(\varphi Vc$$)\mathrm{greater}\mathrm{than}\mathrm{maximum}\mathrm{one}-\mathrm{way}\mathrm{shear}(Vu$) |

${g}_{3}\left(x\right)$ | ${M}_{uL}\le {\varphi}_{b}Mn$ | $\mathrm{Flexural}\mathrm{strength}({\varphi}_{b}Mn$$)\mathrm{greater}\mathrm{than}\mathrm{maximum}\mathrm{flexure}\mathrm{moment}\mathrm{along}\mathrm{length}\mathrm{of}\mathrm{footing}({M}_{uL}$) |

${g}_{4}\left(x\right)$ | ${M}_{uB}\le {\varphi}_{b}Mn$ | $\mathrm{Flexural}\mathrm{strength}({\varphi}_{b}Mn$$)\mathrm{greater}\mathrm{than}\mathrm{maximum}\mathrm{flexure}\mathrm{moment}\mathrm{along}\mathrm{width}\mathrm{of}\mathrm{footing}({M}_{uB}$) |

${g}_{5}\left(x\right)$ | ${q}_{max}\le {q}_{net}$ | $\mathrm{Net}\mathrm{soil}\mathrm{pressure}\left({q}_{net}\right)$$\mathrm{greater}\mathrm{than}\mathrm{maximum}\mathrm{allowable}\mathrm{contact}\mathrm{pressure}\mathrm{under}\mathrm{service}\mathrm{load}\mathrm{incorporating}\mathrm{moments}({q}_{max})$ |

${g}_{6}\left(x\right)$ | ${A}_{required}\le {A}_{provided}$ | $\mathrm{Maximum}\mathrm{footing}\mathrm{area}\mathrm{provided}({A}_{provided})$$\mathrm{greater}\mathrm{than}\mathrm{required}\mathrm{area}\mathrm{of}\mathrm{the}\mathrm{footing}({A}_{required})$ |

${g}_{7}\left(x\right)$ | ${\mathrm{e}}_{L}\le L/6$ | $\mathrm{Eccentricity}\mathrm{along}\mathrm{length}({\mathrm{e}}_{L})$$\mathrm{less}\mathrm{than}\mathrm{one}\mathrm{sixth}\mathrm{of}\mathrm{the}\mathrm{length}\mathrm{of}\mathrm{footing}(L/6)$ |

${g}_{8}\left(x\right)$ | ${\mathrm{e}}_{B}\le B/6$ | $\mathrm{Eccentricity}\mathrm{along}\mathrm{width}({\mathrm{e}}_{B})$$\mathrm{less}\mathrm{than}\mathrm{one}\mathrm{sixth}\mathrm{of}\mathrm{the}\mathrm{width}\mathrm{of}\mathrm{footing}(B/6)$ |

${g}_{9}\left(x\right)$ | ${f}_{c\left(min\right)}\le {{f}_{c}}^{\prime}\le {f}_{c\left(max\right)}$ | $\mathrm{Upper}\mathrm{and}\mathrm{lower}\mathrm{bounds}\mathrm{on}\mathrm{compressive}\mathrm{strength}\mathrm{of}\mathrm{concrete}({{f}_{c}}^{\prime})$ |

${g}_{10}\left(x\right)$ | ${f}_{y\left(min\right)}\le {f}_{y}\le {f}_{y\left(max\right)}$ | $\mathrm{Upper}\mathrm{and}\mathrm{lower}\mathrm{bounds}\mathrm{on}\mathrm{bounds}\mathrm{on}\mathrm{yield}\mathrm{strength}\mathrm{of}\mathrm{the}\mathrm{steel}({f}_{y})$ |

${g}_{11}\left(x\right)$ | $A{s}_{min}\le A{s}_{B}$ | $\mathrm{Minimum}\mathrm{reinforcement}(A{s}_{min})$ along width |

${g}_{12}\left(x\right)$ | $A{s}_{B}\le A{s}_{max}$ | $\mathrm{Maximum}\mathrm{reinforcement}(A{s}_{max})$ along width |

${g}_{13}\left(x\right)$ | $A{s}_{min}\le A{s}_{L}$ | $\mathrm{Minimum}\mathrm{reinforcement}(A{s}_{min})$ along length |

${g}_{14}\left(x\right)$ | $A{s}_{L}\le A{s}_{max}$ | $\mathrm{Maximum}\mathrm{reinforcement}(A{s}_{max})$ along length |

${g}_{15}\left(x\right)$ | ${l}_{db}\le \frac{\mathrm{L}}{2}-{\mathrm{Cov}}_{f}-{\mathrm{d}}_{b}$ | $\mathrm{Development}\mathrm{length}\mathrm{along}\mathrm{length}({l}_{db})$ |

${g}_{16}\left(x\right)$ | ${l}_{db}\le \frac{\mathrm{B}}{2}-{\mathrm{Cov}}_{f}-{\mathrm{d}}_{b}$ | $\mathrm{Development}\mathrm{length}\mathrm{along}\mathrm{width}({l}_{db})$ |

${g}_{17}\left(x\right)$ | ${\mathrm{S}}_{min}\le {S}_{L}\le {\mathrm{S}}_{max}$ | $\mathrm{Spacing}\mathrm{of}\mathrm{steel}\mathrm{along}\mathrm{length}({S}_{L})$ |

${g}_{18}\left(x\right)$ | ${\mathrm{S}}_{min}\le {S}_{B}\le {\mathrm{S}}_{max}$ | $\mathrm{Spacing}\mathrm{of}\mathrm{steel}\mathrm{along}\mathrm{width}({S}_{B})$ |

Input Parameters | Example 1 (Wight and MacGregor 2011) [39] | Example 2 (Shah and Jain 2004) [40] | Example 3 (Kamara and Novak 2013) [41] | Example 4 (Mirza and Brant 2009) [42] | Unit |
---|---|---|---|---|---|

$\mathrm{Concrete}\mathrm{cover}({\mathrm{Cov}}_{f}$) | 63.5 | 75 | 63.5 | 72 | mm |

$\mathrm{Net}\mathrm{soil}\mathrm{pressure}({q}_{net}$) | 191.5 | 200 | 268 | 335 | kPa |

$\mathrm{Dead}\mathrm{load}({\mathrm{P}}_{D}$) | 800.68 | 2899 (P_{u}) | 2406.49 | 889.64 | kN |

$\mathrm{Live}\mathrm{load}({\mathrm{P}}_{L}$) | 533.79 | 862.95 | 444.82 | kN | |

$\mathrm{Dead}\mathrm{moment}\mathrm{about}\mathrm{axis}\mathrm{perp}.\mathrm{to}\mathrm{width}\mathrm{of}\mathrm{footing}\left({\mathrm{M}}_{DB}\right)$ | 0 | 26.4 (M_{u}) | 0 | 0 | kNm |

$\mathrm{Live}\mathrm{moment}\mathrm{about}\mathrm{axis}\mathrm{perp}.\mathrm{to}\mathrm{width}\mathrm{of}\mathrm{footing}\left({\mathrm{M}}_{LB}\right)$ | 0 | 0 | 0 | kNm | |

$\mathrm{Dead}\mathrm{moment}\mathrm{about}\mathrm{axis}\mathrm{perp}.\mathrm{to}\mathrm{length}\mathrm{of}\mathrm{footing}\left({\mathrm{M}}_{DL}\right)$ | 108.48 | 18.8 (M_{u}) | 0 | 0 | kNm |

$\mathrm{Live}\mathrm{moment}\mathrm{about}\mathrm{axis}\mathrm{perp}.\mathrm{to}\mathrm{length}\mathrm{of}\mathrm{footing}\left({\mathrm{M}}_{LL}\right)$ | 81.36 | 0 | 0 | kNm | |

$\mathrm{Length}\mathrm{of}\mathrm{the}\mathrm{column}({c}_{1}$) | 406 | 800 | 610 | 406 | mm |

$\mathrm{Width}\mathrm{of}\mathrm{the}\mathrm{column}({c}_{2}$) | 406 | 800 | 610 | 406 | mm |

$\mathrm{Diameter}\mathrm{of}\mathrm{bar}\mathrm{used}\mathrm{along}\mathrm{length}\mathrm{of}\mathrm{footing}({d}_{b}$) | 22 | 19 | 25 | 19 | mm |

$\mathrm{Diameter}\mathrm{of}\mathrm{bar}\mathrm{used}\mathrm{along}\mathrm{width}\mathrm{of}\mathrm{footing}({d}_{b}$) | 22 | 19 | 25 | 19 | mm |

$\mathrm{Reinforcement}\mathrm{location}\mathrm{factor}({\u1d2a}_{t}$) | 1 | 0.8 | 1 | 0.8 | - |

$\mathrm{Coating}\mathrm{factor}({\u1d2a}_{e}$) | 1 | 1 | 1 | 1 | - |

$\mathrm{Reinforcement}\mathrm{size}\mathrm{factor}({\u1d2a}_{s}$) | 1 | 1 | 1 | 1 | - |

$\mathrm{Lightweight}\mathrm{aggregate}\mathrm{concrete}\mathrm{factor}(\u028e$) | 1 | 1 | 1 | 1 | - |

Parameters | Conventional Design | GRG Optimized Design | EA Optimized Design | GA Optimized Design | Units |
---|---|---|---|---|---|

$\mathrm{Length}\mathrm{of}\mathrm{footing}(L$) | 3.70 | 3.20 | 3.10 | 3.10 | m |

$\mathrm{Width}\mathrm{of}\mathrm{footing}(B$) | 3.05 | 3.00 | 2.86 | 2.86 | m |

$\mathrm{Thickness}\mathrm{of}\mathrm{footing}(h$) | 660.00 | 560.00 | 543.75 | 543.69 | mm |

$\mathrm{Compressive}\mathrm{strength}\mathrm{of}\mathrm{concrete}({{f}_{c}}^{\prime}$) | 24 | 24 | 24 | 24 | MPa |

$\mathrm{Yield}\mathrm{strength}\mathrm{of}\mathrm{steel}({f}_{y}$) | 415 | 415 | 415 | 415 | MPa |

$\mathrm{Area}\mathrm{of}\mathrm{steel}(A{s}_{L}$) along the length of footing | 1188.00 | 1008.00 | 1306.63 | 1307.04 | mm^{2}/m |

$\mathrm{Area}\mathrm{of}\mathrm{steel}(A{s}_{B}$) along the width of footing | 1268.00 | 1548.00 | 1260.30 | 1260.30 | mm^{2}/m |

Total cost | 62,817.00 | 47,426.00 | 42,948.00 | 42,937.00 | INR |

% Optimization | 24.50 | 31.63 | 31.65 |

Parameters | Conventional Design | GRG Optimized Design | EA Optimized Design | GA Optimized Design | Units |
---|---|---|---|---|---|

$\mathrm{Length}\mathrm{of}\mathrm{footing}(L$) | 4.20 | 3.90 | 3.71 | 3.93 | m |

$\mathrm{Width}\mathrm{of}\mathrm{footing}(B$) | 4.20 | 3.90 | 4.01 | 3.78 | m |

$\mathrm{Thickness}\mathrm{of}\mathrm{footing}(h$) | 890.00 | 660.00 | 553.78 | 552.40 | mm |

$\mathrm{Compressive}\mathrm{strength}\mathrm{of}\mathrm{concrete}({{f}_{c}}^{\prime}$) | 20 | 20 | 20 | 20 | MPa |

$\mathrm{Yield}\mathrm{strength}\mathrm{of}\mathrm{steel}({f}_{y}$) | 415 | 415 | 415 | 415 | MPa |

$\mathrm{Area}\mathrm{of}\mathrm{steel}(A{s}_{L}$) along the length of footing | 1690.47 | 1529.20 | 1325.57 | 1459.86 | mm^{2}/m |

$\mathrm{Area}\mathrm{of}\mathrm{steel}(A{s}_{B}$) along the width of footing | 1690.47 | 1529.20 | 1554.26 | 1312.63 | mm^{2}/m |

Total cost | 132,554.00 | 89,159.00 | 75,539.00 | 74,173.00 | INR |

% Optimization | 32.74 | 43.01 | 44.04 |

Parameters | Conventional Design | GRG Optimized Design | EA Optimized Design | GA Optimized Design | Units |
---|---|---|---|---|---|

$\mathrm{Length}\mathrm{of}\mathrm{footing}(L$) | 3.66 | 3.60 | 3.47 | 3.49 | m |

$\mathrm{Width}\mathrm{of}\mathrm{footing}(B$) | 3.66 | 3.60 | 3.51 | 3.49 | m |

$\mathrm{Thickness}\mathrm{of}\mathrm{footing}(h$) | 760.00 | 760.00 | 671.47 | 671.45 | mm |

$\mathrm{Compressive}\mathrm{strength}\mathrm{of}\mathrm{concrete}({{f}_{c}}^{\prime}$) | 28 | 28 | 28 | 28 | MPa |

$\mathrm{Yield}\mathrm{strength}\mathrm{of}\mathrm{steel}({f}_{y}$) | 415 | 415 | 415 | 415 | MPa |

$\mathrm{Area}\mathrm{of}\mathrm{steel}(A{s}_{L}$) along the length of footing | 1811.47 | 1700.00 | 1671.85 | 1675.37 | mm^{2}/m |

$\mathrm{Area}\mathrm{of}\mathrm{steel}(A{s}_{B}$) along the width of footing | 1811.47 | 1700.00 | 1716.53 | 1676.57 | mm^{2}/m |

Total cost | 91,096.00 | 86,678.00 | 74,334.00 | 73,417.00 | INR |

% Optimization | 4.85 | 18.40 | 19.41 |

Parameters | Conventional Design | GRG Optimized Design | EA Optimized Design | GA Optimized Design | Units |
---|---|---|---|---|---|

$\mathrm{Length}\mathrm{of}\mathrm{footing}(L$) | 2.23 | 2.20 | 1.96 | 2.04 | m |

$\mathrm{Width}\mathrm{of}\mathrm{footing}(B$) | 2.23 | 2.20 | 2.04 | 1.96 | m |

$\mathrm{Thickness}\mathrm{of}\mathrm{footing}(h$) | 510.00 | 500.00 | 453.37 | 471.06 | mm |

$\mathrm{Compressive}\mathrm{strength}\mathrm{of}\mathrm{concrete}({{f}_{c}}^{\prime}$) | 28 | 28 | 28 | 28 | MPa |

$\mathrm{Yield}\mathrm{strength}\mathrm{of}\mathrm{steel}({f}_{y}$) | 415 | 415 | 415 | 415 | MPa |

$\mathrm{Area}\mathrm{of}\mathrm{steel}(A{s}_{L}$) along the length of footing | 1146.18 | 1161.81 | 1018.78 | 1043.99 | mm^{2}/m |

$\mathrm{Area}\mathrm{of}\mathrm{steel}(A{s}_{B}$) along the width of footing | 1146.18 | 1161.81 | 1307.31 | 943.47 | mm^{2}/m |

Total cost | 22,566.00 | 21,724.00 | 16,597.00 | 16,217.00 | INR |

% Optimization | 3.73 | 26.45 | 28.14 |

$\mathit{L}$$/\mathit{B}$ | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1 | 1.1 | 1.2 | 1.3 | 1.4 | 1.5 | Units |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Variables | ||||||||||||

$L$ | 2208.8 | 2391.8 | 2559.5 | 2715.2 | 2856.1 | 2994.8 | 3130.3 | 3244.2 | 3363.9 | 3479.4 | 3591.0 | mm |

$B$ | 4374.0 | 3953.3 | 3630.6 | 3373.3 | 3169.0 | 2990.1 | 2833.2 | 2713.2 | 2597.2 | 2494.1 | 2401.8 | mm |

$h$ | 570.3 | 545.0 | 559.6 | 544.3 | 544.1 | 543.9 | 543.6 | 543.5 | 554.5 | 572.7 | 543.0 | mm |

$A{s}_{L}$ | 1029.3 | 981.0 | 1041.5 | 979.8 | 1130.1 | 1218.9 | 1333.6 | 1487.9 | 1547.3 | 1611.7 | 1869.1 | mm^{2}/m |

$A{s}_{B}$ | 2953.6 | 2473.2 | 1952.1 | 1727.0 | 1475.0 | 1326.6 | 1155.3 | 1069.1 | 1085.6 | 1162.0 | 1030.8 | mm^{2}/m |

$\mathit{\rho}$ | 0.002 | 0.003 | 0.004 | 0.005 | 0.006 | 0.007 | 0.008 | Units |
---|---|---|---|---|---|---|---|---|

Variables | ||||||||

$L$ | 3031.4 | 3147.9 | 3043.7 | 3210.7 | 3210.7 | 3210.7 | 3210.7 | mm |

$B$ | 2946.1 | 2814.0 | 2931.6 | 2747.4 | 2747.4 | 2747.4 | 2747.4 | mm |

$h$ | 610.8 | 543.6 | 543.8 | 543.5 | 543.5 | 543.5 | 543.5 | mm |

$A{s}_{L}$ | 1099.4 | 1407.3 | 1877.1 | 2345.1 | 2814.1 | 3283.2 | 3752.2 | mm^{2}/m |

$A{s}_{B}$ | 1099.4 | 1407.3 | 1877.1 | 2345.1 | 2814.1 | 3283.2 | 3752.2 | mm^{2}/m |

Material | Strength (MPa) | Rate | Unit |
---|---|---|---|

Concrete | 20 | 7064 | Rs/m^{3} |

30 | 8053 | Rs/m^{3} | |

40 | 10,031 | Rs/m^{3} | |

Steel | 275 | 117 | Rs/kg |

420 | 120 | Rs/kg | |

500 | 122 | Rs/kg |

${{\mathit{f}}_{\mathit{c}}}^{\prime}$$\u2013{\mathit{f}}_{\mathit{y}}$ | 20–275 | 20–420 | 20–500 | 30–275 | 30–420 | 30–500 | 40–275 | 40–420 | 40–500 | Units |
---|---|---|---|---|---|---|---|---|---|---|

Variables | ||||||||||

$L$ | 3118.1 | 3065.0 | 3124.1 | 3126.9 | 3127.2 | 3168.5 | 3069.5 | 3061.2 | 3133.3 | mm |

$B$ | 2846.7 | 2906.7 | 2840.1 | 2836.9 | 2836.6 | 2791.8 | 2901.6 | 2911.2 | 2829.9 | mm |

$h$ | 569.0 | 569.1 | 569.0 | 514.0 | 514.0 | 513.9 | 481.6 | 478.0 | 477.9 | Mm |

$A{s}_{L}$ | 1915.7 | 1191.8 | 1047.6 | 2158.9 | 1404.3 | 1223.4 | 2198.8 | 1449.2 | 1282.9 | mm^{2}/m |

$A{s}_{B}$ | 1677.4 | 1126.7 | 1117.7 | 1824.7 | 1260.4 | 1013.8 | 2058.1 | 1385.8 | 1080.1 | mm^{2}/m |

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## Share and Cite

**MDPI and ACS Style**

Waheed, J.; Azam, R.; Riaz, M.R.; Shakeel, M.; Mohamed, A.; Ali, E.
Metaheuristic-Based Practical Tool for Optimal Design of Reinforced Concrete Isolated Footings: Development and Application for Parametric Investigation. *Buildings* **2022**, *12*, 471.
https://doi.org/10.3390/buildings12040471

**AMA Style**

Waheed J, Azam R, Riaz MR, Shakeel M, Mohamed A, Ali E.
Metaheuristic-Based Practical Tool for Optimal Design of Reinforced Concrete Isolated Footings: Development and Application for Parametric Investigation. *Buildings*. 2022; 12(4):471.
https://doi.org/10.3390/buildings12040471

**Chicago/Turabian Style**

Waheed, Junaid, Rizwan Azam, Muhammad Rizwan Riaz, Mansoor Shakeel, Abdullah Mohamed, and Elimam Ali.
2022. "Metaheuristic-Based Practical Tool for Optimal Design of Reinforced Concrete Isolated Footings: Development and Application for Parametric Investigation" *Buildings* 12, no. 4: 471.
https://doi.org/10.3390/buildings12040471