# ANN-Based Fatigue Strength of Concrete under Compression

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## Abstract

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## 1. Introduction

_{max}, a value between 0 and 1. In a classic fatigue test of a concrete specimen (most often a cylinder) under compression, the load is applied as a sine wave between fixed lower and upper values. These loads induce stresses in the concrete specimen that fluctuate with S

_{min}f

_{c}and S

_{max}f

_{c}. In some experiments, other sequences of loading have been used, including loading with rest periods and using variable amplitude fatigue load testing [43,44,45]. The focus of this work is only on constant amplitude loading. When S

_{min}and S

_{max}are chosen as the input values for an experiment, the outcome of the experiment then is the number of cycles to failure, N.

_{max}) and the logarithm of the number of cycles to failure N is given, and it is called the Wöhler curve. Such curves can be derived for different values of S

_{min}. For the design of a new structure, we usually know the number of cycles the structure needs to withstand (for example, one million cycles) and carry out the design or the assessment based on the reduced strength associated with this number of cycles. Therefore, in this work, we selected the number of cycles, N, as one of the input variables and the reduced strength ratio, S

_{max}, as the output value.

_{min}and S

_{max}. Mix properties, such as amount of cement, entrained air, water–cement ratio, curing conditions, and age at testing, were found not to be of significant influence on the number of cycles to failure for a given value of S

_{max}[1]. The influence of testing frequency f on the fatigue life is a topic of discussion; some authors observed that, for high values of S

_{max}, there is a decrease in fatigue life for a decrease in frequency [1]. For high strength concrete, Hsu [13] came to the opposite conclusion, whereas for ultra-high performance concrete (UHPC), the same observation was made [46]. The influence of the concrete compressive strength is important on the fatigue life. Experimental work [12,47,48] indicated that the fatigue life is reduced for high strength concrete, but no consensus exists on this topic. To remain on the conservative side, older codes prescribe a lower fatigue life for high strength concrete. Fibers were not found to influence the fatigue strength [46,49].

## 2. Materials and Methods

#### 2.1. Data Gathering

_{max}as the output value, since this approach was in line with the design procedure of finding the compressive capacity under fatigue. In experiments, the output value was the number of cycles, N, but for design, this value was an input based on the required service life of the structure. The number of cycles from S

_{min}to S

_{max}and back (see Figure 3) that can be completed in one second is called the frequency, f, which has units [Hz]. The most commonly used frequency is 1 Hz, but the database includes experiments with frequencies ranging from 0.0625 Hz [81] to 65 Hz [49]. We did not select the frequency as an input variable for our model, since the frequency is a property of experiments and not a value used for design or assessment of concrete structures.

_{c}. We used the reported average measured values here. As can be seen in Table 2, the dataset encompasses low to very high strength concrete samples. In terms of the loading conditions, the input values were the lower limit of the stress range, S

_{min}, and the number of cycles to failure, N. The ranges of parameters in Table 2 show that the dataset included a wide range of values for S

_{min}and that the dataset included low- and high-cycle fatigue tests.

_{max}. Again, the range of values in Table 2 shows the wide range of stresses covered.

#### 2.2. Characteristics of Artificial Neural Networks in This Study

## 3. Results

#### 3.1. Proposed ANN-Based Model

^{−5}s. Figure 4 depicts a simplified scheme of some of the network key features. The max error was 5.1%, performance of all data was 1.2%, and the percentage of errors larger than 3% was 10.3% based on the original input and output values (before normalization and dimensional analysis). The properties of the microprocessor used in this work were OS: Win10Home 64 bits; RAM: 48 GB; Local Disk Memory: 1 TB; CPU: Intel® Core™ i7 8700 K @ 3.70–4.70 GHz.

_{1}. The input vector Y

_{1}contained f

_{c,cyl}, S

_{min}, and N, as shown in Table 2. After input normalization, the new input dataset ${\left\{{Y}_{1}\right\}}_{n}^{after}$ was defined as:

_{1}}

_{n}

^{after}(3 × 1 matrix), the next step was to present it to the proposed ANN to obtain the predicted output dataset Y

_{5}(single value, S

_{max}as shown in Table 2).

_{5}, thus eliminating all rumors that ANNs are “black boxes”.

#### 3.2. Performance Indicators of Results

_{max,test}the experimental value of S

_{max}and S

_{max,ANN}the predicted value based on the neural network. Figure 5 shows the tested versus the predicted values for the ANN-based model for each datapoint as well as the R-value, which was 0.99238 for this case.

#### 3.3. Comparison between ANN-Based and Existing Methods

_{max}with the code formulas, we used average values for the concrete compressive strength instead of design or characteristic values for the calculation of f’

_{b,rep,v}, f

_{cd,fat}and f

_{ck,fat}but not for the correction terms of f

_{ck}/250 MPa and f

_{ck}/400 MPa in the Eurocode and the Model Code expressions, respectively. Here, f

_{ck}was determined as f

_{c,avg}—8 MPa [51]. Figure 6 shows the comparison between the tested and the predicted values. Note that the values of NEN-EN 1992-1-1+C2:2011 [51] were not included, since these are only for one million cycles and were thus not applicable to most of the datapoints in our input dataset. We can also see that a few datapoints gave a negative value for the predicted S

_{max}, which was, of course, physically not possible. In total, seven datapoints calculated with NEN-EN 1992-2+C1:2011 [52] gave a negative value, and one point with NEN 6723:2009 [50] resulted in a negative value. The expression from NEN-EN 1992-2+C1:2011 [52] was not developed for high strength concrete. As such, for datapoints with a high compressive strength and/or a high value of S

_{min}, we could not calculate a value of S

_{max}that was larger than S

_{min}

_{,}and the expressions resulted in a negative solution, which was physically not possible. This effect was more pronounced for high strength concrete because of Equation (5), where the reduction term f

_{ck}/250 increased for increasing concrete compressive strengths. From a computational point of view, we note that taking the log of both sides of Equation (12) resulted in more stable results for the outcome of S

_{max}when using the MathCad 15.0 [89] solver to find S

_{max}for given input values of f

_{c,avg}, S

_{min}, and N. The one datapoint that resulted in negative values with NEN-EN 1992-2+C1:2011 [52] and NEN 6723:2009 [50] had f

_{c,avg}= 41 MPa (normal strength concrete) and S

_{min}= 0.836. Since the value of S

_{min}was very high, the expressions could not find a solution for S

_{max}larger than S

_{min}

^{,}giving a negative value instead. We should remark as well that the expressions of fib model code 2010 [26] always resulted in a value for S

_{max}but that, for cases where S

_{min}was large, the calculated value for S

_{max}could be smaller than S

_{min}, which was also physically not possible.

_{max}or a value of S

_{max}smaller than S

_{min}. From this analysis, we can see that the expressions of the fib model code [53] led to the best results of the expressions in Table 1 with an average tested to predicted value of 1.37 and the coefficient of variation equal to 20.5%. We can also observe that our proposed model led to a better prediction with an average tested to predicted value of 1.00 and a coefficient of variation of 1.7%.

## 4. Discussion

_{cd}instead of f

_{c,avg}as an input value and the use of S

_{min}resulting from the serviceability limit state load combination. The reason for the latter recommendation is that the experimental results indicate that the fatigue life increased as S

_{min}increased, thus it would be a more conservative approach to use the load combination that results in the lowest value for S

_{min}.

_{min}up to 0.836. As seen in Figure 6, the existing code formulas could not predict the value for S

_{max}when S

_{min}was larger. While such cases are uncommon in practice, it is an advantage of our proposed ANN-based model that this model can address cases with a large value for S

_{min}as input.

_{max}for high strength concrete, as physically impossible values for S

_{max}were found.

_{max}did not decrease further in the Wöhler curve. The fatigue limit in concrete is, however, subject to discussion. While the dataset had a maximum value of N close to 64 million cycles, there are no experimental results available that cover the range of 250 million cycles and higher. The reason why such experimental results are not available is that the amount of time needed becomes very large. For example, say we want to test 500 million cycles with a loading frequency of 1 Hz, as is commonly used in fatigue testing. Such an experiment would take 500 million seconds, or close to 16 years, to complete. Then, given the large scatter inherent in fatigue testing, we would need to repeat the experiment a number of times, which would only add to the time required for obtaining these test data. It would, however, be interesting to have such data available—not only for studying the fatigue limit and the number of cycles used for design and assessment but also to study the code formulas of the fib Model Code 2010 [53], which defines a change in Wöhler curve for 100 million cycles. Given the considerations in the previous paragraphs, the input database and the resulting proposed model form an improvement with respect to the existing code formulas.

_{max}. The first parameter analyzed was S

_{min,}see Figure 7. We can see in this figure that the ANN-based model performed consistently over the full range of values of S

_{min}. We remarked earlier that we could not find a solution for NEN 6723:2009 [50] for the datapoint with the largest value of S

_{min}and that various datapoints with a large value of S

_{min}did not lead to a physically possible solution with the expressions from NEN-EN 1992-2+C1:2011 [52]. The values of the tested to predicted S

_{max}seemed to slightly decrease as S

_{min}increased for the predictions with the fib Model Code 2010 [53].

_{max}. We can observe from this plot that our proposed model performed equally well over the full range of concrete compressive strengths. We can see that the expressions from NEN-EN 1992-2+C1:2011 [52] were overly conservative. However, we need to keep in mind that C90/105 is the highest strength concrete class in NEN-EN 1992-1-1:2005 [93], thus some high strength concrete specimens in our dataset were outside the scope of the Eurocodes. In particular, the term f

_{ck}/250 MPa in Equation (5) was overly conservative for high strength concrete. We can see in Figure 8 that the fib Model Code term of f

_{ck}/400 MPa from Equation (13) led to better results from high strength concrete. Figure 8 also shows that the predictions for S

_{max}were still more conservative for high strength concrete than for normal strength concrete. In that regard, the expressions from NEN 6723:2009 [50] seemed to have a more uniform performance over the full range of concrete compressive strengths.

_{max}are shown as a function of logN. We can see that the code equations were less conservative for low-cycle fatigue than for high-cycle fatigue. This observation was stronger for NEN 6723:2009 [50] than for the other codes. From experiments [48], we know that the Wöhler curve starts to be linear after 100 cycles. As such, it was expected that the datapoints for logN ≤ 2 would be more difficult to predict. Again, our proposed model performed well over the full range of cycles in the input dataset.

_{max}itself. Figure 10 shows the ratio of tested to predicted ratios of S

_{max}as a function of S

_{max}. We can see from this plot that the code predictions tended to become more conservative as S

_{max}increased, whereas our proposed model performed well and consistently over the full range of values of S

_{max}in the input dataset.

^{−5}s per datapoint. Since we provided all expressions for the readers in this work and the W and the b arrays in the public domain, direct implementation of our proposed model is easy. The reader can set up a spreadsheet with the equations from Section 3.1 and from then on can use our proposed model quickly and easily. This observation again underlines the improvement of our proposed model with respect to existing models.

## 5. Summary and Conclusions

- We derived an input dataset with 203 datapoints obtained from experiments reported in the literature. Each datapoint was unique. Where necessary, the geometric average of number of cycles to failure of repeat tests was determined;
- We selected three input parameters for the input dataset (concrete compressive strength, lower bound of the stress ratio, and number of cycles to failure) and one output parameter (upper bound of the stress ratio) in line with the parameters used in the currently used code expressions;
- We used different methods for 14 features of the ANN models to find the most suitable features. We looked at 219 combinations of features and selected the neural net with the best performance.

- Of the studied methods in the current codes, we found that the expression from the fib model code performed best when compared to the experimental results gathered in the dataset. The average value of tested to predicted upper bound of the stress ratio was 1.37 with a coefficient of variation of 20.5%;
- We can see that our proposed model outperformed the code equations for the prediction of the upper bound of the stress ratio, since the average value of tested to predicted upper bound of the stress ratio was 1.00 with a coefficient of variation of 1.7%;
- The tested to predicted values obtained with our proposed model did not show any dependence on any of the input or the output parameters, i.e., our model performed consistently well over the full range of the parameters. In contrast, plotting the tested to predicted ratios obtained with the code equations showed that these depended on the input and the output parameters. In particular, the predicted values for the upper bound of the stress ratio with the code equations became overly conservative as the concrete compressive strength increased;
- The computational time of our proposed neural net is small (0.07 milliseconds per datapoint).

- Experiments on high-cycle fatigue are necessary. Given the required time for such experiments, however, it is unlikely that such experiments can be carried out. Perhaps, numerical analyses can be used to generate datapoints for N > 64 million cycles;
- The proposed method was derived from test results and aims at average values. Further studies are necessary to define the safety factors for design and assessment based on our proposed method;
- The proposed ANN-based model is only valid for the parameter ranges in the input dataset. However, these ranges cover most practical cases, except, as mentioned earlier, N > 64 million cycles. In practice, we need 250 or 500 million cycles for design and assessment;
- The mechanics of the problem and the reasons for the large scatter in fatigue tests were not addressed in this study. The parameter studies presented here, however, give insight in the governing parameters and can be used in the future for comparison to expressions that are derived based on mechanics.
- Regardless the high quality of the predictions yielded by the proposed model for the used data, the reader should not blindly accept that model as accurate for any other instances falling inside the input domain of the design dataset. Any analytical approximation model must undergo extensive validation before it can be taken as reliable (the more inputs, the larger the validation process). Models proposed until that stage are part of a learning process towards excellence.

## Author Contributions

## Funding

## Conflicts of Interest

## Preprint

## Notations

b | bias of neuron |

e | relative error |

f | frequency |

f’_{b,rep,k} | characteristic value of the uniaxial short term concrete compressive strength |

f’_{b,rep,v} | characteristic value of the concrete compressive strength in the limit state of fatigue |

f’_{b,v} | fatigue reference strength |

f_{c} | concrete compressive strength |

f_{c,cyl,mean} | average measured value of the concrete cylinder compressive strength |

f_{cd} | design concrete compressive strength |

f_{cd,fat} | design fatigue strength |

f_{ck} | characteristic concrete compressive strength |

f_{ck,fat} | characteristic concrete compressive strength for the limit state of fatigue |

f_{c,mean,max} | maximum measured concrete compressive strength |

k_{1} | a factor from the Eurocode expression for fatigue |

m | the number of cycles of constant amplitude |

n_{i} | the number of cycles with a constant amplitude at interval i |

s | a factor that depends on the strength class of the cement |

t | the concrete age in days |

t_{0} | the time of the start of the cyclic loading on the concrete |

t_{T} | the concrete age in days, corrected for temperature |

AVG | average value |

CHAR | characteristic value based on a normal distribution |

COV | coefficient of variation |

E_{cd,max,equ} | the maximum compressive stress level |

E_{cd,min,equ} | the minimum compressive stress level |

E_{cd,max,i} | the maximum compressive stress level for the considered interval |

E_{cd,min,i} | the minimum compressive stress level for the considered interval |

N | number of load cycles |

N_{1} | first part of the S-N curve in the fib Model Code |

N_{2} | second part of the S-N curve in the fib Model Code |

N_{i} | the number of cycles to failure with a constant amplitude at interval i |

R | stress ratio |

R_{equ} | stress ratio |

R_{i} | stress ratio of the considered interval |

R_{i}* | the stress ratio for S_{max,EC} |

S | fraction of compressive strength applied in load cycle |

S_{c,max} | maximum fraction of compressive strength applied in load cycles |

S_{c,min} | minimum fraction of compressive strength applied in load cycles |

S_{max} | maximum fraction of strength applied in load cycles |

S_{max,ANN} | the predicted value based on the neural network. |

S_{max,EC} | the value of S_{max} associated with 10^{6} cycles |

S_{max,pred} | the value of S_{max} from the proposed methods for a given number of cycles N |

S_{max,test} | the value for S_{max} in the experiments for a given number of cycles N |

S_{min} | minimum fraction of strength applied in load cycles |

STD | standard deviation |

T(Δt_{i}) | temperature during time period Δt_{i} |

W | weight of connection between neurons |

Y | expression used in the fatigue formula of the fib Model Code |

β_{cc}(t_{0}) | coefficient for the concrete strength at first load application |

β_{cc}(t) | describes the strength development with time |

β_{c,sus}(t,t_{0}) | factor for sustained loading |

γ_{m} | partial factor for the material |

γ_{c} | partial factor for concrete |

γ_{c,fat} | partial factor for concrete in the limit state of fatigue |

σ’_{b,d,max} | design value of the maximum compressive stress in the concrete |

σ’_{b,d,min} | design value of the minimum compressive stress in the concrete |

σ_{cd,max,equ} | upper stress of the ultimate amplitude for N cycles |

σ_{cd,min,equ} | lower stress of the ultimate amplitude for N cycles |

σ_{c,max} | upper compressive stress of the ultimate amplitude for N cycles |

σ_{c,min,} | lower compressive stress of the ultimate amplitude for N cycles |

Δt_{i} | number of days with temperature T |

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**Figure 3.**Input (green) and output variables (red), shown on an example of a loading scheme in an experiment.

**Figure 5.**Regression plot for the proposed ANN for the output variable, S

_{max}. The expression for the blue line is: S

_{max,ANN}= 0.98 S

_{max,test}+ 0.012 and R = 0.99238.

**Figure 8.**Tested to predicted values for all considered methods as a function of the average concrete compressive strength.

**Figure 9.**Tested to predicted values for all considered methods as a function of the number of cycles to failure, N.

**Figure 11.**Comparison between Wöhler curve resulting from experimental data and from ANN-based predictions.

Code | Ref | Equations | Nr |
---|---|---|---|

NEN 6723:2009 | [50] | ${f}_{b,v}^{\prime}=\frac{{f}_{b,rep,v}^{\prime}}{{\gamma}_{m}}$ with γ_{m} = 1.2 | (1) |

${f}_{b,rep,v}^{\prime}=0.5\left({f}_{b,rep,k}^{\prime}-0.85\times 30\right)+0.85\times 30\text{}\mathrm{in}\text{}\left[\mathrm{MPa}\right]$ | (2) | ||

$Log\left(N\right)=\frac{10}{\sqrt{1-R}}\left(1-\frac{\sigma {\prime}_{b,d,max}}{f{\prime}_{b,v}}\right)\text{}\mathrm{for}\text{}\frac{\sigma {\prime}_{b,d,max}}{f{\prime}_{b,v}}0.25$ | (3) | ||

$R=\frac{{\sigma}_{b,d,min}^{\prime}}{{\sigma}_{b,d,max}^{\prime}}=\frac{{S}_{min}}{{S}_{max}}$ | (4) | ||

NEN-EN 1992-1-1+C2:2011 | [51] | ${f}_{cd,fat}={k}_{1}{\beta}_{cc}({t}_{0}){f}_{cd}\left(1-\frac{{f}_{ck}}{250}\right)$ with f_{ck} in MPa and k_{1} = 0.85 | (5) |

${\beta}_{cc}({t}_{0})=\mathrm{exp}\left\{s\left[1-{\left(\frac{28}{{t}_{0}}\right)}^{0.5}\right]\right\}$ | (6) | ||

${E}_{cd,max,equ}+0.43\sqrt{1-{R}_{equ}}\le 1$ | (7) | ||

${R}_{equ}=\frac{{E}_{cd,min,equ}}{{E}_{cd,max,equ}}$ | (8) | ||

${E}_{cd,max,equ}=\frac{{\sigma}_{cd,max,equ}}{{f}_{cd,fat}}$ | (9) | ||

${E}_{cd,min,equ}=\frac{{\sigma}_{cd,min,equ}}{{f}_{cd,fat}}$ | (10) | ||

NEN-EN 1992-2+C1:2011 | [52] | $\sum _{i=1}^{m}\frac{{n}_{i}}{{N}_{i}}\le 1$ | (11) |

${N}_{i}={10}^{\left(14\frac{1-{E}_{cd,max,i}}{\sqrt{1-{R}_{i}}}\right)}$ | (12) | ||

fib model code 2010 | [53] | ${f}_{ck,fat}={\beta}_{cc}(t){\beta}_{c,sus}(t,{t}_{0}){f}_{ck}\left(1-\frac{{f}_{ck}}{400}\right)$ with f_{ck} in MPa, β_{c,sus}(t, t_{0}) = 0.85 and s = 0.25 for cement class 42.5 N | (13) |

${\beta}_{cc}(t)=\mathrm{exp}\left\{s\left[1-{\left(\frac{28}{t}\right)}^{0.5}\right]\right\}$ | (14) | ||

${t}_{T}={\displaystyle \sum _{i=1}^{n}\Delta {t}_{i}\mathrm{exp}\left(13.65-\frac{4000}{273+T(\Delta {t}_{i})}\right)}$ | (15) | ||

$\mathrm{log}{N}_{1}=\frac{8}{Y-1}\left({S}_{c,max}-1\right)$ | (16) | ||

$\mathrm{log}{N}_{2}=8+\frac{8\mathrm{ln}(10)}{Y-1}\left(Y-{S}_{c,min}\right)\mathrm{log}\left(\frac{{S}_{c,max}-{S}_{c,min}}{Y-{S}_{c,min}}\right)$ | (17) | ||

$\mathrm{log}N=\{\begin{array}{l}\mathrm{log}{N}_{1}\text{}\mathrm{if}\text{}\mathrm{log}{N}_{1}\le 8\\ \mathrm{log}{N}_{2}\text{}\mathrm{if}\text{}\mathrm{log}{N}_{1}8\end{array}$ | (18) | ||

$Y=\frac{0.45+1.8{S}_{c,min}}{1+1.8{S}_{c,min}-0.3{S}_{c,min}^{2}}$ | (19) | ||

${S}_{c,max}=\frac{\left|{\sigma}_{c,max}\right|}{{f}_{ck,fat}}$ | (20) | ||

${S}_{c,min}=\frac{\left|{\sigma}_{c,min}\right|}{{f}_{ck,fat}}$ | (21) | ||

$\Delta {S}_{c}=\left|{S}_{c,max}\right|-\left|{S}_{c,min}\right|$ | (22) |

Input Parameters | Input Number | Min | Max | ||
---|---|---|---|---|---|

Concrete properties | f_{c,cyl} (MPa) | average concrete compressive strength | 1 | 24 | 170 |

Loading | S_{min} (-) | lower limit of stress range | 2 | 0 | 0.836 |

N (-) | number of cycles to failure | 3 | 3 | 63,841,046 | |

Output | S_{max} (-) | upper limit of stress range | 1 | 0.465 | 0.960 |

**Table 3.**Implemented artificial neural network (ANN) features (F) 1–7. The highlighted cells show the features that were used to derive the final neural net.

F1 | F2 | F3 | F4 | F5 | F6 | F7 |
---|---|---|---|---|---|---|

Qualitative Var Represent | Dimensional Analysis | Input Dimensionality Reduction | % Train-Valid-Test | Input Normalization | Output Transfer | Output Normalization |

Boolean Vectors | Yes | Linear Correlation | 80-10-10 | Linear Max Abs | Logistic | Lin [a, b] = 0.7[φ_{min}, φ_{max}] |

Eq Spaced in ]0,1] | No | Auto-Encoder | 70-15-15 | Linear [0, 1] | - | Lin [a, b] = 0.6[φ_{min}, φ_{max}] |

- | - | - | 60-20-20 | Linear [−1, 1] | Hyperbolic Tang | Lin [a, b] = 0.5[φ_{min}, φ_{max}] |

- | - | Ortho Rand Proj | 50-25-25 | Nonlinear | - | Linear Mean Std |

- | - | Sparse Rand Proj | - | Lin Mean Std | Bilinear | No |

- | - | No | - | No | Compet | - |

Identity |

**Table 4.**Implemented ANN features (F) 8–14. The highlighted cells show the features that were used to derive the final neural net.

F8 | F9 | F10 | F11 | F12 | F13 | F14 |
---|---|---|---|---|---|---|

Net Architectue | Hidden Layers | Connectivity | Hidden Transfer | Parameter Initialization | Learning Algorithm | Training Mode |

MLPN | 1 HL | Adjacent Layers | Logistic | Midpoint (W) + Rands (b) | BP | Batch |

RBFN | 2 HL | Adj Layers + In-Out | Identity-Logistic | Rands | BPA | Mini-Batch |

- | 3 HL | Fully-Connected | Hyperbolic Tang | Randnc (W) + Rands (b) | LM | Online |

- | - | - | Bipolar | Randnr (W) + Rands (b) | ELM | - |

- | - | - | Bilinear | Randsmall | mb ELM | - |

- | - | - | Positive Sat Linear | Rand [−Δ, Δ] | I-ELM | - |

- | - | - | Sinusoid | SVD | CI-ELM | - |

Thin-Plate Spline | MB SVD | - | ||||

Gaussian | - | - | ||||

Multiquadratic | - | - | ||||

Radbas | - | - | ||||

Thin-Plate Spline | MB SVD | - |

**Table 5.**Statistical properties of V

_{utot}/V

_{pred}for all datapoints with AVG = average of V

_{utot}/V

_{pred}, STD = standard deviation on V

_{utot}/V

_{pred}, and COV = coefficient of variation of V

_{utot}/V

_{pred}.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Abambres, M.; Lantsoght, E.O.L. ANN-Based Fatigue Strength of Concrete under Compression. *Materials* **2019**, *12*, 3787.
https://doi.org/10.3390/ma12223787

**AMA Style**

Abambres M, Lantsoght EOL. ANN-Based Fatigue Strength of Concrete under Compression. *Materials*. 2019; 12(22):3787.
https://doi.org/10.3390/ma12223787

**Chicago/Turabian Style**

Abambres, Miguel, and Eva O.L. Lantsoght. 2019. "ANN-Based Fatigue Strength of Concrete under Compression" *Materials* 12, no. 22: 3787.
https://doi.org/10.3390/ma12223787