# Optimum Design of Flexural Strength and Stiffness for Reinforced Concrete Beams Using Machine Learning

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Response Surface Model

_{1}, x

_{2}, …, x

_{k}). In general, such a relationship is unknown but can be approximated by a low-degree polynomial model in the form:

_{1}, x

_{2}, …, x

_{k}),

**f**(

**x**) is a vector function of

**p**elements that consists of powers and cross-products of powers of

**x**,

_{1}**x**, …,

_{2}**x**up to a certain degree, denoted by

_{k}**d**(≥ 1),

**β**is a vector of

**p**unknown constant coefficients, referred to as parameters, and

**ϵ**is a random experimental error assumed to have a zero mean. This is conditioned on the consideration that the model provides an adequate representation of the response. In this case, quantity $f\left(x\right)\beta $ represents the mean response, that is, the expected value of

**y**, and is denoted by

**μ(x)**. Two important models are commonly used in RSM. These are special cases of the model in Equation (1) and include the first-degree model (d = 1):

**d**= 2):

**D**, of order n × k, called the design matrix,

**u**th design setting of ${x}_{i}$ (

**i**= 1, 2,$\dots $, k;

**u**= 1, 2,$\dots $, n). Each row of

**D**represents a point, referred to as a design point, in a

**k**-dimensional Euclidean space. Let ${y}_{u}$ denote the response value obtained as a result of applying the

**u**th setting of

**x**, namely ${x}_{u}$= (${x}_{u1}$, ${x}_{u2}$, $\dots $, ${x}_{uk}$), (

**u**= 1, 2, $\dots $, n). From Equation (1), we then have:

**u**th experimental run. Equation (5) can be expressed in a matrix form as:

_{1}, $y$

_{2}, $\dots $, $y$

_{n}), $\mathit{X}$ is a matrix of order n × p of which the

**u**th row is ${f}^{\prime}\left({\mathit{x}}_{\mathit{u}}\right)$, and $\mathit{\u03f5}$ = ($\u03f5$

_{1}, $\u03f5$

_{2}, $\dots ,$ $\u03f5$

_{n}

**)**. Note that the first column of $\mathit{X}$ is the column of ones

**1**. Assuming that $\mathbf{\u03f5}$ has a zero mean, the so-called ordinary least-squares estimator of

_{n}**β**is [22]:

## 3. Dataset

## 4. Finite Element Models

## 5. Surrogate Models

^{2}is a tool used to identify the efficiency of surrogate models, where this parameter has a range value bounded from 0 to 1. This parameter makes use of a comparison between the experimental test results and the predicted results. When the value of this parameter is near 1, it means that the surrogate models are efficient and can be supported to predict the responses of any structural system.

#### 5.1. Flexural Stress

^{2}= 0.9566, which is an excellent value indicating the efficiency of the surrogate model to predict the flexural stress in reinforced concrete beam specimens.

#### 5.2. Maximum Deflection

^{2}= 0.9063, which is an excellent value, indicating the efficiency of the surrogate model to predict the maximum deflection in reinforced concrete beam specimens.

#### 5.3. Regression Analysis

^{2}is a tool used to identify the efficiency of the surrogate models, where this parameter has a range value starting from 0 to 1. This parameter makes use of a comparison between the experimental test results and the predicted results. When the value of the parameter is near 1, it means that the surrogate models are efficient and can be supported to predict the responses of any structural system.

#### 5.3.1. Flexural Stress

^{2}= 0.9566, which is an excellent value indicating the efficiency of the surrogate model to predict the flexural stress in reinforced concrete beam specimens (Figure 8). Only 4.34% of the system response was not predictable, which is very satisfactory.

#### 5.3.2. Maximum Deflection

^{2}= 0.9063, which is an excellent value indicating the efficiency of the surrogate model to predict the maximum deflection in reinforced concrete beam specimens (Figure 9). Only 9.37% of the system response was not predictable, which is satisfactory.

## 6. Optimization Results

#### 6.1. Flexural Stress

#### 6.2. Maximum Deflection

## 7. Conclusions

- The Box-Behnken design method manifested excellent strength in building the surrogate models to predict the responses of the structural system under loading and, as a result, a more efficient, safer, and lower-cost design.
- The numerical models generated in the ATENA program showed a good agreement with the experimental tests, where the results were very similar and very satisfactory. In addition, numerical models can be adopted to predict and optimize the design of the reinforced concrete beams for flexure and ductility.
- The optimization process produced minimum and maximum responses based on the factorial method of the predicted responses of the flexural strength and the stiffness from the surrogate models.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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Parameter | Symbol | Minimum Value | Middle Value | Maximum Value |
---|---|---|---|---|

Compression Area of Steel (mm^{2}) | ${X}_{1}$ | 157.00 | 235.50 | 314.00 |

Tension Area of Steel (mm^{2}) | ${X}_{2}$ | 157.00 | 235.50 | 314.00 |

Shear Area of Steel (mm^{2}) | ${X}_{3}$ | 169.56 | 197.82 | 226.08 |

Model | Area of Compression Steel Bars (mm^{2}) | Area of Tension Steel Bars (mm ^{2}) | Area of Shear Steel Bars (mm ^{2}) |
---|---|---|---|

1 | 157.00 | 157.00 | 197.82 |

2 | 157.00 | 314.00 | 197.82 |

3 | 314.00 | 157.00 | 197.82 |

4 | 314.00 | 314.00 | 197.82 |

5 | 157.00 | 235.50 | 169.56 |

6 | 157.00 | 235.50 | 226.08 |

7 | 314.00 | 235.50 | 169.56 |

8 | 314.00 | 235.50 | 226.08 |

9 | 235.50 | 157.00 | 169.56 |

10 | 235.50 | 157.00 | 226.08 |

11 | 235.50 | 314.00 | 169.56 |

12 | 235.50 | 314.00 | 226.08 |

13 | 235.50 | 235.50 | 197.82 |

14 | 235.50 | 235.50 | 197.82 |

15 | 235.50 | 235.50 | 197.82 |

Cylinder Specimen | Maximum Compression Load (kN) | Cylinder Specimen | Maximum Tensile Load (kN) |
---|---|---|---|

C1 | 101.74 | T1 | 85.90 |

C2 | 88.45 | T2 | 66.20 |

C3 | 99.08 | T3 | 100.19 |

Model | Maximum Flexure Load (kN) | Slope | Flexural Strain(mm/mm) | Young’s Modulus (N/mm^{2}) | Maximum Deflection (mm) | Flexural Stress (MPa) |
---|---|---|---|---|---|---|

1 | 77.240 | 0.7140 | 0.1639 | 104.860 | 4.430 | 17.190 |

2 | 94.680 | 0.8330 | 0.1722 | 122.340 | 5.68 | 21.070 |

3 | 79.320 | 0.7050 | 0.1704 | 103.540 | 5.62 | 17.650 |

4 | 97.040 | 0.7770 | 0.1892 | 114.120 | 6.24 | 21.590 |

5 | 79.260 | 0.7400 | 0.1623 | 108.680 | 5.35 | 17.640 |

6 | 97.900 | 0.8800 | 0.1685 | 129.240 | 5.56 | 21.780 |

7 | 91.900 | 0.8180 | 0.1702 | 120.140 | 5.61 | 20.450 |

8 | 93.560 | 0.7610 | 0.1862 | 111.770 | 6.10 | 20.820 |

9 | 78.750 | 0.9000 | 0.1326 | 132.180 | 4.37 | 17.520 |

10 | 75.240 | 0.4666 | 0.2442 | 68.530 | 8.06 | 16.740 |

11 | 97.380 | 0.4440 | 0.3323 | 65.210 | 10.96 | 21.670 |

12 | 107.700 | 0.6110 | 0.2670 | 89.740 | 8.81 | 23.960 |

13 | 93.930 | 0.6660 | 0.2137 | 97.810 | 7.05 | 20.900 |

Model | Flexural Stress ATENA (MPa) | Flexural Stress Experiment (MPa) | Error % |
---|---|---|---|

1 | 17.15611696 | 17.185900 | 0.17 |

2 | 21.09721253 | 21.066300 | 0.15 |

3 | 17.15664377 | 17.648700 | 2.79 |

4 | 21.59387950 | 21.591400 | 0.01 |

5 | 17.95554714 | 17.635350 | 1.82 |

6 | 21.44176454 | 21.782750 | 1.57 |

7 | 20.31767836 | 20.447750 | 0.64 |

8 | 20.70511776 | 20.817100 | 0.54 |

9 | 17.09409603 | 17.521875 | 2.44 |

10 | 16.33360424 | 16.740900 | 2.43 |

11 | 21.15750459 | 21.667050 | 2.35 |

12 | 24.50759912 | 23.963250 | 2.27 |

13 | 20.92028772 | 20.899425 | 0.10 |

Model | Max. Deflection ATENA (mm) | Max. Deflection Experiment (mm) | Error % |
---|---|---|---|

1 | 4.48 | 4.43 | 1.12 |

2 | 5.48 | 5.68 | 3.65 |

3 | 5.42 | 5.62 | 3.69 |

4 | 6.12 | 6.24 | 1.96 |

5 | 5.27 | 5.35 | 1.52 |

6 | 5.51 | 5.56 | 0.91 |

7 | 5.56 | 5.61 | 0.90 |

8 | 6.02 | 6.1 | 1.33 |

9 | 4.29 | 4.37 | 1.86 |

10 | 7.98 | 8.06 | 1.00 |

11 | 10.85 | 10.96 | 1.01 |

12 | 8.76 | 8.81 | 0.57 |

13 | 6.94 | 7.05 | 1.59 |

Model | Constant | X_{1} | X_{2} | X_{3} | X_{1}X_{1} | X_{2}X_{2} | X_{3}X_{3} |
---|---|---|---|---|---|---|---|

1 | 1.00 | 157.00 | 157.00 | 197.82 | 24,649.00 | 24,649.00 | 39,132.75 |

2 | 1.00 | 157.00 | 314.00 | 197.82 | 24,649.00 | 98,596.00 | 39,132.75 |

3 | 1.00 | 314.00 | 157.00 | 197.82 | 98,596.00 | 24,649.00 | 39,132.75 |

4 | 1.00 | 314.00 | 314.00 | 197.82 | 98,596.00 | 98,596.00 | 39,132.75 |

5 | 1.00 | 157.00 | 235.50 | 169.56 | 24,649.00 | 55,460.25 | 28,750.59 |

6 | 1.00 | 157.00 | 235.50 | 226.08 | 24,649.00 | 55,460.25 | 51,112.17 |

7 | 1.00 | 314.00 | 235.50 | 169.56 | 98,596.00 | 55,460.25 | 28,750.59 |

8 | 1.00 | 314.00 | 235.50 | 226.08 | 98,596.00 | 55,460.25 | 51,112.17 |

9 | 1.00 | 235.50 | 157.00 | 169.56 | 55,460.25 | 24,649.00 | 28,750.59 |

10 | 1.00 | 235.50 | 157.00 | 226.08 | 55,460.25 | 24,649.00 | 51,112.17 |

11 | 1.00 | 235.50 | 314.00 | 169.56 | 55,460.25 | 98,596.00 | 28,750.59 |

12 | 1.00 | 235.50 | 314.00 | 226.08 | 55,460.25 | 98,596.00 | 51,112.17 |

13 | 1.00 | 235.50 | 235.50 | 197.82 | 55,460.25 | 55,460.25 | 39,132.75 |

14 | 1.00 | 235.50 | 235.50 | 197.82 | 55,460.25 | 55,460.25 | 39,132.75 |

15 | 1.00 | 235.50 | 235.50 | 197.82 | 55,460.25 | 55,460.25 | 39,132.75 |

Model | X_{1}X_{2} | X_{1}X_{3} | X_{2}X_{3} |
---|---|---|---|

1 | 24,649.00 | 31,057.74 | 31,057.74 |

2 | 49,298.00 | 31,057.74 | 62,115.48 |

3 | 49,298.00 | 62,115.48 | 31,057.74 |

4 | 98,596.00 | 62,115.48 | 62,115.48 |

5 | 36,973.50 | 26,620.92 | 39,931.38 |

6 | 36,973.50 | 35,494.56 | 53,241.84 |

7 | 73,947.00 | 53,241.84 | 39,931.38 |

8 | 73,947.00 | 70,989.12 | 53,241.84 |

9 | 36,973.50 | 39,931.38 | 26,620.92 |

10 | 36,973.50 | 53,241.84 | 35,494.56 |

11 | 73,947.00 | 39,931.38 | 53,241.84 |

12 | 73,947.00 | 53,241.84 | 70,989.12 |

13 | 55,460.25 | 46,586.61 | 46,586.61 |

14 | 55,460.25 | 46,586.61 | 46,586.61 |

15 | 55,460.25 | 46,586.61 | 46,586.61 |

Coefficient | Value (10^{−6}) |
---|---|

${\mathsf{\beta}}_{0}$ | −13.075 |

${\mathsf{\beta}}_{1}$ | 138.61 × 10^{−3} |

${\mathsf{\beta}}_{2}$ | 27.47 × 10^{−3} |

${\mathsf{\beta}}_{3}$ | 77.41 × 10^{−3} |

${\mathsf{\beta}}_{11}$ | −0.11 × 10^{−3} |

${\mathsf{\beta}}_{22}$ | −0.14 × 10^{−3} |

${\mathsf{\beta}}_{33}$ | −81.39 × 10^{−6} |

${\mathsf{\beta}}_{12}$ | 2.43 × 10^{−6} |

${\mathsf{\beta}}_{13}$ | −0.42 × 10^{−3} |

${\mathsf{\beta}}_{23}$ | 0.35 × 10^{−3} |

Coefficient | Value |
---|---|

${\mathsf{\beta}}_{0}$ | −15.464 |

${\mathsf{\beta}}_{1}$ | 154.89 × 10^{−3} |

${\mathsf{\beta}}_{2}$ | 118.87 × 10^{−3} |

${\mathsf{\beta}}_{3}$ | −130.48 × 10^{−3} |

${\mathsf{\beta}}_{11}$ | −0.32 × 10^{−3} |

${\mathsf{\beta}}_{22}$ | 67.95 × 10^{−6} |

${\mathsf{\beta}}_{33}$ | 0.73 × 10^{−3} |

${\mathsf{\beta}}_{12}$ | −25.56 × 10^{−6} |

${\mathsf{\beta}}_{13}$ | 31.55 × 10^{−6} |

${\mathsf{\beta}}_{23}$ | −0. 66 × 10^{−3} |

Model | X_{1}(mm ^{2}) | X_{2}(mm ^{2}) | X_{3}(mm ^{2}) | ${\mathit{\sigma}}_{\mathit{f}}\phantom{\rule{0ex}{0ex}}\left(\mathbf{MPa}\right)$ |
---|---|---|---|---|

1 | 157.00 | 157.00 | 169.56 | 15.645 |

2 | 157.00 | 157.00 | 197.82 | 16.638 |

3 | 157.00 | 157.00 | 226.08 | 17.500 |

4 | 157.00 | 235.50 | 169.56 | 18.124 |

5 | 157.00 | 314.00 | 169.56 | 18.878 |

6 | 235.50 | 157.00 | 169.56 | 17.589 |

7 | 314.00 | 157.00 | 169.56 | 18.207 |

8 | 314.00 | 235.50 | 169.56 | 20.716 |

9 | 314.00 | 235.50 | 197.82 | 20.591 |

10 | 314.00 | 235.50 | 226.08 | 20.336 |

11 | 235.50 | 314.00 | 169.56 | 20.851 |

12 | 235.50 | 314.00 | 197.82 | 22.436 |

13 | 235.50 | 314.00 | 226.08 | 23.891 |

14 | 235.50 | 235.50 | 169.56 | 20.083 |

15 | 235.50 | 235.50 | 197.82 | 20.900 |

16 | 235.50 | 235.50 | 226.08 | 21.588 |

17 | 314.00 | 314.00 | 169.56 | 21.500 |

18 | 314.00 | 314.00 | 197.82 | 22.143 |

19 | 314.00 | 314.00 | 226.08 | 22.655 |

20 | 235.50 | 157.00 | 197.82 | 17.639 |

21 | 314.00 | 157.00 | 197.82 | 17.315 |

22 | 157.00 | 314.00 | 197.82 | 21.405 |

23 | 157.00 | 314.00 | 226.08 | 23.803 |

24 | 235.50 | 157.00 | 226.08 | 17.559 |

25 | 314.00 | 157.00 | 226.08 | 16.293 |

26 | 157.00 | 235.50 | 197.82 | 19.883 |

27 | 157.00 | 235.50 | 226.08 | 21.514 |

Model | X_{1}(mm ^{2}) | X_{2}(mm ^{2}) | X_{3}(mm ^{2}) | δ_{max}(mm) |
---|---|---|---|---|

1 | 157.00 | 157.00 | 169.56 | 2.776250 |

2 | 157.00 | 157.00 | 197.82 | 3.865000 |

3 | 157.00 | 157.00 | 226.08 | 6.116250 |

4 | 157.00 | 235.50 | 169.56 | 5.126250 |

5 | 157.00 | 314.00 | 169.56 | 8.313750 |

6 | 235.50 | 157.00 | 169.56 | 5.158750 |

7 | 314.00 | 157.00 | 169.56 | 3.588750 |

8 | 314.00 | 235.50 | 169.56 | 5.623750 |

9 | 314.00 | 235.50 | 197.82 | 5.392500 |

10 | 314.00 | 235.50 | 226.08 | 6.323750 |

11 | 235.50 | 314.00 | 169.56 | 10.381250 |

12 | 235.50 | 314.00 | 197.82 | 8.620000 |

13 | 235.50 | 314.00 | 226.08 | 8.021250 |

14 | 235.50 | 235.50 | 169.56 | 7.351250 |

15 | 235.50 | 235.50 | 197.82 | 7.050000 |

16 | 235.50 | 235.50 | 226.08 | 7.911250 |

17 | 314.00 | 314.00 | 169.56 | 8.496250 |

18 | 314.00 | 314.00 | 197.82 | 6.805000 |

19 | 314.00 | 314.00 | 226.08 | 6.276250 |

20 | 235.50 | 157.00 | 197.82 | 6.317500 |

21 | 314.00 | 157.00 | 197.82 | 4.817500 |

22 | 157.00 | 314.00 | 197.82 | 6.482500 |

23 | 157.00 | 314.00 | 226.08 | 5.813750 |

24 | 235.50 | 157.00 | 226.08 | 8.638750 |

25 | 314.00 | 157.00 | 226.08 | 7.208750 |

26 | 157.00 | 235.50 | 197.82 | 4.755000 |

27 | 157.00 | 235.50 | 226.08 | 5.546250 |

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**MDPI and ACS Style**

Nariman, N.A.; Hamdia, K.; Ramadan, A.M.; Sadaghian, H.
Optimum Design of Flexural Strength and Stiffness for Reinforced Concrete Beams Using Machine Learning. *Appl. Sci.* **2021**, *11*, 8762.
https://doi.org/10.3390/app11188762

**AMA Style**

Nariman NA, Hamdia K, Ramadan AM, Sadaghian H.
Optimum Design of Flexural Strength and Stiffness for Reinforced Concrete Beams Using Machine Learning. *Applied Sciences*. 2021; 11(18):8762.
https://doi.org/10.3390/app11188762

**Chicago/Turabian Style**

Nariman, Nazim Abdul, Khader Hamdia, Ayad Mohammad Ramadan, and Hamed Sadaghian.
2021. "Optimum Design of Flexural Strength and Stiffness for Reinforced Concrete Beams Using Machine Learning" *Applied Sciences* 11, no. 18: 8762.
https://doi.org/10.3390/app11188762