# Study on Prediction and Application of Initial Chord Elastic Modulus with Resonance Frequency Test of ASTM C 215

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Materials and Preparation of Specimens

#### 2.2. Destructive Tests for Elastic Modulus and Compressive Strength

#### 2.3. Dynamic Elastic Modulus Measurements with Resonance Frequency Tests

#### 2.4. Initial Chord Elastic Modulus Measurement

## 3. Machine Learning Methods

#### 3.1. Ensemble Method

#### 3.2. Artificial Neural Network (ANN)

## 4. Results and Discussion

#### 4.1. Experimental Consistency of Static and Dynamic Tests

#### 4.2. Relationship among Static and Initial Chord and Dynamic Elastic Modulus

^{3}) [12].

^{3}of concrete [13].

#### 4.3. Prediction of Initial Chord Elastic Modulus with Multiple Linear Regression

#### 4.4. Prediction of Initial Chord Elastic Modulus with Ensemble Method

#### 4.5. Relationship between Initial Chord and Static Elastic Modulus

#### 4.6. Relationship between Initial Chord and Compressive Strength

## 5. Conclusions

- The Ed values calculated by three theoretical equations (ASTM, Rayleigh Ritz) of the resonance frequency test were in the order of f1,f2.LT > ASTM.LT > ASTM.TR, and had nearly the same values. The size of the elastic modulus as measured by static and dynamic tests was Ed > Ei > Ec. In addition, it is determined that it is desirable to utilize the Ei, as the correlation with Ec is analyzed as Ei > Ed.
- The Popovis equation for the relationship between Ec and Ed gives results similar to the Eds of the ASTM, and the Lydon and Balendran equations are similar to Ei values. BS8110 Part 2 is not suitable, as because it has a large error from the Ed and Ei in the resonance frequency test.
- As a result of comparing an E-modulus based on ASTM.LT, f1,f2.LT, and ASTM.TR had a clear linear relationship, and they were close in the line of equality. They were identified as having a nonlinear relationship with the Ei and Ec. As the theoretical equations assumed the concrete as a perfectly elastic body for microscopic stress, it was difficult to overcome the nonlinear behavior of the actual Ei and Ec, owing to challenges in considering inhomogeneity and inelasticity of concrete. Thus, it is more appropriate to accurately predict and utilize the Ei, which has a similar nonlinear behavior with the Ec.
- As a result of applying ASTM.LT, f1,f2.LT, and ASTM.TR to the correction factors, the MAPE in the Ei could be lowered to 6.40%, 6.50%, and 6.43%, respectively. In addition, the Ed in the three equations and Ei of the MAPE decreased in order in days 4, 7, 14, and 28. The theoretical equation is suitable for concrete after 28 days but is considered difficult to use to accurately predict lower ages.
- In the relationship between Ei and frequency, the correlation of Ei-f1 was the largest, the nonlinearity increased as the mode appeared later, and the density and consistency of the data gradually decreased. In addition, the Ei and Ec values and first frequency of the resonance frequency test tended to be similar to an exponential function, indicating that prediction of the Ei and Ec based on frequency was possible.
- As a result of predicting the Ei using only frequencies through the ensemble and ANN methods, the MAPE decreased by 3.90% in the case of using only f1, and by 3.51% in the case of using f1-f4. Accordingly, the nonlinear behavior could be overcome by using ML.
- As a result of analyzing the contributions of variables in predicting the Ei, f1 and f2 were dominant, the RI of the size factor was 0, and 0.3% of the day variables contributed to the Ei prediction. Therefore, it is possible to predict a sufficiently accurate Ei using only the frequencies, i.e., without other variables.
- As a result of predicting the Ec by applying a correction factor of 0.89 to the predicted Ei in four ways, the MAPE ranged from 4.6% to 6.57%, and the correlation between the predicted Ec and fc was high. Therefore, far more accurate Ei values can be predicted by the ASTM method in the future, and more accurate design, construction, and maintenance will be possible if this approach is used for calculating the Ec and fc.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

ANN | artificial neural network |

ASTM.LT | dynamic elastic modulus measured by the first frequency in longitudinal modes |

ASTM.TR | dynamic elastic modulus measured by the first frequency in transverse modes |

COV | coefficient of variation |

f1,f2.LT | dynamic elastic modulus measured by the first and second frequency in longitudinal modes |

GBFS | granulated blast furnace slag |

LSBoost | least squares boosting |

LT | longitudinal |

MAPE | mean absolute percentage error |

ML | machine learning |

MLP | multilayer perceptron |

MLR | Multiple linear regression |

MSE | mean squared error |

RI | relative importance |

RMSE | root mean square error |

SCMs | supplementary cementitious materials |

SVM | support vector machine |

TR | transverse |

Ec | static elastic modulus |

Ed | dynamic elastic modulus |

Ei | initial chord elastic modulus |

fc | compressive strength |

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**Figure 3.**Resonance frequency by curing ages according to ASTM resonance test. (

**a**) frequencies of longitudinal mode, (

**b**) frequencies of transverse mode.

**Figure 8.**Comparison of Ed and Ei by ages and mixes according to ASTM.LT. (

**a**) Mix 1 (20 MPa), (

**b**) Mix 2 (40 MPa).

**Figure 10.**Relationship between the frequencies and Ei of the resonance frequency test. (

**a**) LT frequencies-Ei, (

**b**) TR frequencies-Ei.

**Figure 11.**Comparison of prediction results with two frequencies. (

**a**) Predicted Ei with multiple linear regression (MLR), (

**b**) Predicted Ed with Equation (4).

**Figure 14.**Comparison of Ei predicted with machine learning (ML) and actual Ei. (

**a**) Ei predicted by frequency of LT mode, (

**b**) Ei predicted by frequency of TR mode.

ID | Cement Type | W/B | S/A | W | C | S | G | Unit Quantity (kg/m^{3}) Mineral Admixture | Chemical Admixture | ||
---|---|---|---|---|---|---|---|---|---|---|---|

FA | GBFS | AE (Binder%) | SP (Binder%) | ||||||||

Mix1 (20 MPa) | Type I | 0.45 | 0.46 | 259 | 121 | 777 | 934 | 58 | 69 | 0.9 | - |

Mix2 (40 MPa) | 0.35 | 0.47 | 308 | 166 | 761 | 886 | 81 | 85 | - | 1 |

^{1}SCMs: Supplementary cementitious materials, W: water, C: cement, S: sand, G: crushed cobblestone, FA: fly ash, SC: slag cement, AE: air-entraining agent, SP: superplasticizer, GBFS: granulated blast furnace slag.

w/c | The Number of Specimens | Weight (kg) | Dimension | Density (kg/m3) | Age (Day) | fc (MPa) | Ec (MPa) | |
---|---|---|---|---|---|---|---|---|

Diameter (mm) | Length (mm) | |||||||

0.45 (Mix1) | 91 | 10.66 ~12.12 (11.19) | 150 ~150 (150) | 290.95 ~298.10 (293.97) | 2051.58 ~2340.38 (2153.14) | 4 | 7.33~8.56 (7.91) | 9202~15,929 (10,368) |

7 | 9.31~10.85 (10.07) | 10,930~13,256 (11,784) | ||||||

14 | 12.67~14.63 (13.87) | 12,723~16,883 (14,735) | ||||||

28 | 17.65~20.68 (19.26) | 14,552~20,915 (17,041) | ||||||

0.35 (Mix2) | 194 | 11.10 ~12.08 (11.63) | 150 ~150 (150) | 293.30 ~299.70 (297.50) | 2125.88 ~2290.24 (2212.76) | 4 | 20.96~26.19 (24.12) | 14,566~23,879 (16,796) |

7 | 27.21~32.56 (29.62) | 15,431~19,807 (18,079) | ||||||

14 | 32.82~42.38 (38.17) | 15,523~24,009 (20,904) | ||||||

28 | 40.21~47.34 (43.99) | 18,686~26,530 (22,860) |

Days/Variable | ASTM.LT [MPa] | f1f2.LT [MPa] | ASTM.TR [MPa] | Ei [MPa] | Ec [MPa] | fc [MPa] | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Mix 1 | Mix 2 | Mix 1 | Mix 2 | Mix 1 | Mix 2 | Mix 1 | Mix 2 | Mix 1 | Mix 2 | Mix 1 | Mix 2 | ||

Day 4 | N | 24 | 49 | 24 | 49 | 24 | 49 | 24 | 49 | 24 | 49 | 24 | 49 |

μ | 15,378 | 24,348 | 15,643 | 24,813 | 14,850 | 23,914 | 10,745 | 18,650 | 10,368 | 16,796 | 7.91 | 24.12 | |

COV | 4.36% | 3.81% | 4.71% | 4.00% | 4.56% | 4.01% | 7.89% | 7.36% | 14.31% | 7.84% | 4.29% | 5.10% | |

Day 7 | N | 19 | 49 | 19 | 49 | 19 | 49 | 19 | 49 | 19 | 49 | 19 | 49 |

μ | 17,643 | 26,166 | 18,003 | 26,561 | 17,151 | 25,294 | 13,301 | 20,467 | 11,784 | 18,079 | 10.07 | 29.62 | |

COV | 4.30% | 2.39% | 4.36% | 2.57% | 5.16% | 2.71% | 4.36% | 4.31% | 4.90% | 4.60% | 4.57% | 3.03% | |

Day 14 | N | 25 | 50 | 25 | 50 | 25 | 50 | 25 | 50 | 25 | 50 | 25 | 50 |

μ | 20,696 | 28,543 | 21,057 | 28,973 | 20,074 | 27,530 | 16,804 | 24,003 | 14,735 | 20,904 | 13.87 | 38.17 | |

COV | 3.72% | 2.47% | 3.80% | 2.64% | 3.88% | 3.01% | 5.35% | 5.81% | 6.10% | 6.96% | 3.35% | 5.47% | |

Day 28 | N | 23 | 46 | 23 | 46 | 23 | 46 | 23 | 46 | 23 | 46 | 23 | 46 |

μ | 23,814 | 30,348 | 24,239 | 30,805 | 22,959 | 29,652 | 19,648 | 25,853 | 17,041 | 22,860 | 19.26 | 43.99 | |

COV | 3.94% | 2.14% | 4.35% | 2.35% | 3.92% | 2.73% | 6.25% | 4.90% | 6.88% | 6.21% | 4.09% | 4.32% |

Correlation | ASTM.LT-Ec | f1,f2.LT-Ec | ASTM.TR-Ec | Initial Chord Elastic Modulus (Ei)-Ec |
---|---|---|---|---|

Value | 0.9376 | 0.9362 | 0.9364 | 0.9645 |

Type of Errors | ASTM.LT | f1,f2.LT | ASTM.TR |
---|---|---|---|

Mean square error (MSE) | 2.53 × 10^{7} | 2.95 × 10^{7} | 1.88 × 10^{7} |

Root MSE (RMSE) | 5031 | 5427 | 4338 |

Mean absolute percentage error (MAPE) | 26.32% | 28.39% | 22.59% |

R | 0.9572 | 0.9556 | 0.9551 |

ID | Mix | Day 4 | Day 7 | Day 14 | Day 28 | Average |
---|---|---|---|---|---|---|

ASTM.LT | Mix 1 | 0.70 (6.38%) | 0.75 (2.81%) | 0.81 (3.12%) | 0.83 (3.43%) | 0.77 (7.47%) |

Mix 2 | 0.77 (4.69%) | 0.78 (3.42%) | 0.84 (4.69%) | 0.85 (3.19%) | 0.81 (5.94%) | |

f1,f2.LT | Mix 1 | 0.69 (7.02%) | 0.74 (3.35%) | 0.80 (3.51%) | 0.81 (3.91%) | 0.76 (7.63%) |

Mix 2 | 0.75 (5.54%) | 0.77 (4.14%) | 0.83 (4.81%) | 0.84 (3.24%) | 0.80 (6.12%) | |

ASTM.TR | Mix 1 | 0.72 (6.25%) | 0.78 (4.17%) | 0.84 (4.29%) | 0.86 (4.79%) | 0.80 (7.32%) |

Mix 2 | 0.78 (4.81%) | 0.81 (3.82%) | 0.87 (5.28%) | 0.87 (4.13%) | 0.83 (5.98%) |

Type of Mode | The Number of Specimens | Variable | f1 (Hz) | f2 (Hz) | f3 (Hz) | f4 (Hz) | Weight (kg) | Diameter (mm) | Length (m) | Density (kg/m3) |
---|---|---|---|---|---|---|---|---|---|---|

LT.f1 | 285 | Range | 4450 ~6400 | 10.66 ~12.12 | 150 | 290.95 ~299.70 | 2051.58 ~2340.38 | |||

Average | 5641 | 11.49 | 150 | 296.38 | 2193.72 | |||||

LT.f2 | 283 | Range | 8500 ~12,100 | 10.66 ~12.12 | 150 | 290.95 ~299.70 | 2051.58 ~2340.38 | |||

Average | 10,757 | 11.49 | 150 | 296.39 | 2193.79 | |||||

LT.f3 | 275 | Range | 10,550 ~15,500 | 10.66 ~12.12 | 150 | 295.00 ~299.40 | 2051.58 ~2340.38 | |||

Average | 13,677 | 11.50 | 150 | 297.41 | 2194.86 | |||||

LT.f4 | 230 | Range | 12,400 ~20,250 | 10.66 ~12.12 | 150 | 295.00 ~299.60 | 2051.58 ~2340.38 | |||

Average | 15,838 | 11.48 | 150 | 297.68 | 2191.51 | |||||

LT.f1,f2 | 283 | Range | 4450 ~6400 | 8500 ~12,100 | 10.66 ~12.12 | 150 | 290.95 ~299.70 | 2051.58 ~2340.38 | ||

Average | 5640 | 10,757 | 11.49 | 150 | 296.39 | 2193.79 | ||||

LT.f1,f2,f3 | 275 | Range | 4450 ~6400 | 8500 ~12,100 | 10,550 ~15,500 | 10.66 ~12.12 | 150 | 295.00 ~299.40 | 2051.58 ~2340.38 |

Type of Mode | The Number of Specimens | Variable | f1 (Hz) | f2 (Hz) | f3 (Hz) | Weight (kg) | Diameter (mm) | Length (m) | Density (kg/m^{3}) |
---|---|---|---|---|---|---|---|---|---|

TR.f1 | 285 | Range | 2750 ~3900 | 10.66 ~12.12 | 150 | 290.95 ~299.70 | 2051.58 ~2340.38 | ||

Average | 3441 | 11.49 | 150 | 296.38 | 2193.72 | ||||

TR.f2 | 105 | Range | 5150 ~7700 | 10.71 ~12.12 | 150 | 290.95 ~299.50 | 2069.88 ~2340.38 | ||

Average | 6276 | 11.42 | 150 | 295.99 | 2182.91 | ||||

TR.f3 | 105 | Range | 7750 ~12,250 | 10.71 ~12.12 | 150 | 290.95 ~299.50 | 2069.88 ~2340.38 | ||

Average | 9919 | 11.42 | 150 | 295.99 | 2182.91 | ||||

TR.f1,f2 | 105 | Range | 2750 ~3850 | 5150 ~7700 | 10.71 ~12.12 | 150 | 290.95 ~299.50 | 2069.88 ~2340.38 | |

Average | 3351 | 6276 | 11.42 | 150 | 295.99 | 2182.91 | |||

TR.f1,f2,f3 | 105 | Range | 2750 ~3850 | 5150 ~7700 | 7550 ~12,250 | 10.71 ~12.12 | 150 | 290.95 ~299.50 | 2069.88 ~2340.38 |

Average | 3351 | 6276 | 9919 | 11.42 | 150 | 295.99 | 2182.91 |

Correlation | LT.f1-Ei | LT.f2-Ei | LT.f3-Ei | LT.f4-Ei | TR.f1-Ei | TR.f2-Ei | TR.f3-Ei |
---|---|---|---|---|---|---|---|

Values | 0.9420 | 0.9397 | 0.8629 | 0.7071 | 0.9413 | 0.8992 | 0.7915 |

LT Mode | f1 | f2 | f3 | f4 | f1~2 | f1~3 | f1~4 |
---|---|---|---|---|---|---|---|

MSE | 1.18 × 10^{6} | 1.21 × 10^{6} | 2.75 × 10^{6} | 4.73 × 10^{6} | 1.08 × 10^{6} | 9.74 × 10^{5} | 9.06 × 10^{5} |

RMSE | 1080 | 1100 | 1660 | 2170 | 1040 | 987 | 952 |

MAPE | 3.90% | 3.95% | 5.69% | 8.00% | 3.67% | 3.52% | 3.51% |

R | 0.9716 | 0.9705 | 0.9332 | 0.8900 | 0.9738 | 0.9768 | 0.9798 |

TR Mode | f1 | f2 | f3 | f1~2 | f1~3 |
---|---|---|---|---|---|

MSE | 1.36 × 10^{6} | 1.64 × 10^{6} | 1.83 × 10^{6} | 1.13 × 10^{6} | 7.74 × 10^{5} |

RMSE | 1160 | 1280 | 1350 | 1060 | 879 |

MAPE | 4.31% | 4.44% | 5.79% | 3.87% | 3.40% |

R | 0.9668 | 0.9620 | 0.9577 | 0.9740 | 0.9822 |

Type | Day 4 | Day 7 | Day 14 | Day 28 | Average |
---|---|---|---|---|---|

Mix 1 | 0.95 (5.51%) | 0.89 (2.97%) | 0.87 (3.94%) | 0.91 (6.31%) | 0.91 (5.27%) |

Mix 2 | 0.90 (4.15%) | 0.88 (2.97%) | 0.88 (4.78%) | 0.89 (4.88%) | 0.89 (4.14%) |

Lithological Type of Coarse Aggregate | k1 | Type of Addition | k2 |
---|---|---|---|

Crushed limestone, calcined bauxite | 1.20 | Silica fume, ground-granulated blast-furnace slag, fly ash fume | 0.95 |

Crushed quartzitic aggregate, crushed andesite, crushed basalt, crushed clay slate, crushed cobblestone | 0.95 | Fly ash | 1.10 |

Coarse aggregate, other than above | 1.00 | Addition other than above | 1.00 |

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## Share and Cite

**MDPI and ACS Style**

Yoon, Y.G.; Choi, H.; Oh, T.K.
Study on Prediction and Application of Initial Chord Elastic Modulus with Resonance Frequency Test of ASTM C 215. *Appl. Sci.* **2020**, *10*, 5464.
https://doi.org/10.3390/app10165464

**AMA Style**

Yoon YG, Choi H, Oh TK.
Study on Prediction and Application of Initial Chord Elastic Modulus with Resonance Frequency Test of ASTM C 215. *Applied Sciences*. 2020; 10(16):5464.
https://doi.org/10.3390/app10165464

**Chicago/Turabian Style**

Yoon, Young Geun, HaJin Choi, and Tae Keun Oh.
2020. "Study on Prediction and Application of Initial Chord Elastic Modulus with Resonance Frequency Test of ASTM C 215" *Applied Sciences* 10, no. 16: 5464.
https://doi.org/10.3390/app10165464