# Fatigue-Life Prediction of Mechanical Element by Using the Weibull Distribution

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Generalities of Fatigue Analysis

#### 2.1. Static Stress Analysis

_{1}value as:

## 3. Proposed Method

#### 3.1. Generalities of the Weibull Analysis

_{y}is the mean of the used Y vector, determined based on the median rank approach. Notice that if $n=21$ elements are tested, then ${\mu}_{y}=-0.545624$; and ${\eta}_{s}$ is determined as:

_{1}and σ

_{2}values in Equations (8)–(10), the Weibull stress parameters are both completely determined, and that because ${\mu}_{y}=-0.545624$ is constant, then the efficiency of ${\beta}_{s}$ and ${\eta}_{s}$ only depends on the efficiency on which the ${\sigma}_{1}$ and ${\sigma}_{2}$ values were determined. Based on the addressed stress $({\beta}_{s},{\eta}_{s}$) parameters, let us present the method to determine the random behavior of the addressed ${\sigma}_{1}$ and ${\sigma}_{2}$ values, based on which the random behavior of the expected cycles to failure is formulated.

#### 3.2. Weibull Stress Family Estimation and Its Random Behavior Analysis

**Note 1.**Note that from Equations (11) and (12), we only know whether the design is safe or not, but at this point it is not possible to determine either the designed reliability or the corresponding expected failure times.

_{1}and σ

_{2}is determined as follows:

_{i}) elements as:

_{i}elements as shown in Equation (15). Then, from the Y

_{i}elements, determine its corresponding mean value, obtained by using Equation (16).

_{1i}value that corresponds to each one of the ${Y}_{i}$ elements is given as:

_{1}value, the ${t}_{01}$ element that corresponds to the ${\sigma}_{1}$ and ${\sigma}_{2}$ values, is calculated.

**Note 2**. It is important to mention that the R(t) index obtained in Equation (22) corresponds to an element which has a strength equivalent to the ${\sigma}_{1}$ value. Thus, if ${S}_{y}\ne {\sigma}_{1}$, the reliability of the designed element is found by replacing the ${S}_{y}$ value with ${\sigma}_{1}$ in Equation (20).

#### 3.3. Weibull Cycle Family Estimation and Its Random Behavior

#### 3.3.1. Estimation of the N Value that Corresponds to the Equivalent Stress Value

_{a}= surface condition modification factor, K

_{b}= size modification factor, K

_{c}= load modification factor, K

_{d}= temperature modification factor, K

_{e}= reliability factor and K

_{f}= miscellaneous-effects modification factor. After that, using Equations (25) and (26) and the values for principal stresses ${\sigma}_{1}$ and ${\sigma}_{2}$ from Section 2.1, the corresponding mid-range stress ${\sigma}_{m}$ and the alternating stress ${\sigma}_{a}$ values are set.

#### 3.3.2. Determination of the Weibull Cycle Parameters

**Note 3.**The ${\beta}_{s}$ value is used on the assumption that the failure mode remains constant.

## 4. Mechanical Application with Weibull/Finite Element Analysis (FEA)

_{SR}= Allowable stress range, ${C}_{f}$ = Fatigue constant load, $N$ = Number of cycles of stress range.

#### 4.1. Weibull FEA Stress Data Analysis

_{1}and σ

_{2}stress values of the FEA simulation given in Table 1 is compared to the static stress analysis. Both results are given in Table 2, where it can be seen that no significant difference exists.

#### 4.1.1. Validation of the Applied Principal Stresses σ_{1} and σ_{2} Values

#### 4.1.2. Weibull Stress Family Determination

_{i}elements are generated, with a mean of ${\mu}_{y}=-0.545624$ and a standard deviation of ${\sigma}_{y}=1.175117$. Finally, from Equation (8), and by using the ${\mu}_{y}$, σ

_{1}and σ

_{2}FEA values, the Weibull ${\beta}_{s}$ value is:

#### 4.1.3. Stress Random Behavior

_{2}stress value, in this section the random behavior of the stresses ${\sigma}_{1}$ and ${\sigma}_{2}$ values are determined by Equations (18) and (19) (see fifth and sixth columns in Table 4). First, by using the ${\beta}_{s}$ and the ${Y}_{i}$ values in Equation (17), the basic Weibull elements ${t}_{0i}$ defined for each one of the ${Y}_{i}$ elements are determined, as shown in Table 4. Additionally, by using the ${\eta}_{s}$ and the σ

_{1}values in Equation (20), the expected Weibull ${t}_{01}$ value, from which the ${\sigma}_{1}$ and ${\sigma}_{2}$ stress values are both reproduced, is determined as ${t}_{01}=301.455469/491.75=0.617339$, and it is included in Table 4 as well.

**Note 4**. Observe that R(t) = 0.713156 is not the reliability of the design component; it only represents the reliability of the ${t}_{0max}$ element in the Weibull analysis. The R(t) of the element is that determined by using the ${Y}_{1}$ value that corresponds to the S

_{y}value in Equation (22). It is located between rows 2 and 3 in Table 4.

_{y}value as ${\sigma}_{1}$ is as follows:

_{y}value is ${t}_{0Sy}=301.455469/827=0.364517$ and from Equation (21) the corresponding ${Y}_{Sy}$ value is ${Y}_{Sy}=ln\left(0.364517\right)\ast \left(2.248519\right)=-2.269166$. Thus, from Equation (22), the reliability index for the ${Y}_{Sy}$ value is $R\left(t\right)=exp\left\{-exp\left\{-2.269166\right\}\right\}=0.901768$. From the above it can be concluded that the reliability of the design element is $R\left(t\right)=0.901768$.

_{oi}elements are used to determine the corresponding random behavior of the expected cycles to failure. The analysis is as follows.

#### 4.2. Weibull Cycle to Failure (N) Analysis

#### 4.2.1. Weibull Cycle Parameters

#### 4.2.2. Cycle Random Behavior

_{oi}elements in Equation (34), the corresponding expected cycle to failure values for each one of the ${Y}_{i}$ elements are determined (see the seventh column in Table 5). Additionally, by using ${t}_{0sy}=0.364517$ in Equation (34) the ${N}_{sy}$ value that corresponds to the ${S}_{y}$ value is ${N}_{sy}=453,250,454$

#### 4.2.3. Weibull Stress Analytical Static Family

^{7}. The joint S–N curve for the welded joint category B is shown in Figure 7.

_{SR}and the allowable cycle number $N$ is given by Equation (36). Therefore, by applying Equation (36) in Equation (37) with ${C}_{f}$ = 120 × 10

^{8}, the allowable cycles N are calculated. Table 7 shows a summary of the comparison between the static method and the FEA simulation fatigue life welded joint results.

## 5. Conclusions

- Because the input’s method are the ${\sigma}_{1}$ and ${\sigma}_{2}$ values, then the proposed method can be applied in any mechanical analysis where ${\sigma}_{1}$ and ${\sigma}_{2}$ are known.
- The efficiency of the proposed method is that the Weibull $\beta $ (see Equations (8)) and $\eta $ (see Equation (9)) only depends on the ${\sigma}_{1}$ and ${\sigma}_{2}$ values. Consequently, by performing the Weibull analysis we can always reproduce them (see row between rows 5 and 6 Table 4).
- In the proposed method the mechanical element reliability can be determined by either the stress ${\sigma}_{21}$ values (see column ${\sigma}_{2i}$ in Table 5) or by the corresponding cycle to failure values.
- The random behavior of both ${\sigma}_{2i}$ and ${N}_{i}$ values is determined based on the ${\sigma}_{eq}$ and on the its corresponding ${N}_{eq}$ value, here determined by Basquin’s equation, but if they are determined in any other form the proposed method will be also efficient at determining the random behavior around them.
- As demonstrated in [14], the standard fatigue methodologies based on the stress-strain analysis and on the mechanical fracture both converge to the same solution; it might then be possible to extend the proposed method to be used in the strain and mechanical fracture analysis by related the equivalent stress ${\sigma}_{eq}$ to the total deformation given by the elastic (Basquin’s equation) and plastic (Coffin–Mason equation) areas, although further research should be undertaken.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 5.**Stress distribution picture of the flat spring under the different applied load (

**a**) 4.63 N; (

**b**) 1.74 N.

**Figure 6.**Deformation picture of the flat spring under the different applied forces (

**a**) 4.63 N; (

**b**) 1.74 N.

Maximum Principal Stress (MPa) Force F _{1} = 4.63 N | Minimum Principal Stress (MPa) Force F _{2} = 1.74 N | Deformation (mm) Force F _{1} = 4.63 N | Deformation (mm) Force F _{2} = 1.74 N |
---|---|---|---|

491.75 | 184.8 | 8.02 | 2.99 |

**Table 2.**Comparison of the stresses and deformation results of the static stress analysis and FEA simulation.

Application | Maximum Principal Stress (MPa), Force F _{1} = 4.63 N | Minimum Principal Stress (MPa), Force F _{2} = 1.74 N | Deformation (mm) Force F _{1} = 4.63 N | Deformation (mm) Force F _{2} = 1.74 N |
---|---|---|---|---|

Static | 470 | 176 | 8.00 | 3.00 |

FEA | 491.75 | 184.8 | 8.02 | 2.99 |

Parameter | Condition to Be Met for Safe Design | Status | |
---|---|---|---|

Maximum Shear Stress | $\left({\sigma}_{1}\right)<Sy/SF$ | 491.8 < 501.8 | Ok |

Distortion Energy | $\left({\sigma}_{1}^2-{\sigma}_{1}\ast {\sigma}_{2}+{\sigma}_{2}^2\right)^0.5{S}_{y}/SF$ | 430.2 < 501.2 | Ok |

**Table 4.**Statistics of Weibull FEA stress analysis for data in Section 4.

n Equation (13) | Y_{i} Equation (15) | t_{oi} Equation (17) | R(t_{oi}) Equation (22) | σ_{2i} Equation (18) | σ_{1i} Equation (19) | σ_{med} Equation (25) | σ_{alt} Equation (26) | σ_{eq} (ASME) Equation (29) |
---|---|---|---|---|---|---|---|---|

1 | −3.4034833 | 0.220104 | 0.96729 | 66.35155 | 1308.124 | 687.2377 | 620.8861 | 1116.157138 |

−2.9701952 | 0.26688 | 0.95 | 80.45246 | 1078.849 | 579.6507 | 499.1982 | 699.8903536 | |

2 | −2.491662 | 0.330174 | 0.920561 | 99.53286 | 872.034 | 485.7834 | 386.2506 | 477.2687796 |

−2.2691664 | 0.364517 | 0.901768 | 109.8856 | 827 | 468.4428 | 358.5572 | 435.0865005 | |

3 | −2.0034632 | 0.410239 | 0.873832 | 123.6689 | 701.842 | 412.7555 | 289.0865 | 333.6083957 |

4 | −1.6616459 | 0.477593 | 0.827103 | 143.9731 | 602.8627 | 373.4179 | 229.4448 | 257.1517697 |

5 | −1.3943983 | 0.537869 | 0.780374 | 162.1435 | 535.3039 | 348.7237 | 186.5802 | 205.7685332 |

−1.0845459 | 0.617339 | 0.713156 | 184.8 | 491.75 | 338.275 | 153.475 | 168.1886402 | |

6 | −1.1720537 | 0.593774 | 0.733645 | 178.9966 | 484.9034 | 331.95 | 152.9534 | 166.9966579 |

7 | −0.9793812 | 0.646898 | 0.686916 | 195.0109 | 445.0831 | 320.047 | 125.0361 | 135.6021558 |

8 | −0.8074473 | 0.698303 | 0.640187 | 210.5073 | 412.3183 | 311.4128 | 100.9055 | 108.9229489 |

9 | −0.6504921 | 0.748789 | 0.593458 | 225.7266 | 384.5185 | 305.1225 | 79.39597 | 85.4226406 |

10 | −0.5045088 | 0.799016 | 0.546729 | 240.8679 | 360.3471 | 300.6075 | 59.73961 | 64.12598946 |

11 | −0.3665129 | 0.84959 | 0.5 | 256.1134 | 338.8969 | 297.5051 | 41.39172 | 44.36161829 |

12 | −0.2341223 | 0.901115 | 0.453271 | 271.6459 | 319.519 | 295.5825 | 23.93655 | 25.62949661 |

13 | −0.1052851 | 0.954255 | 0.406542 | 287.6654 | 301.7256 | 294.6955 | 7.030101 | 7.524013293 |

0 | 1 | 0.367879 | 301.4555 | 301.4555 | 301.4555 | 0 | 0 | |

14 | 0.0219284 | 1.0098 | 0.359813 | 304.4098 | 285.129 | 294.7694 | 9.640397 | 10.31807533 |

15 | 0.14952577 | 1.068761 | 0.313084 | 322.1837 | 269.3992 | 295.7915 | 26.39226 | 28.26181481 |

16 | 0.279845 | 1.132534 | 0.266355 | 341.4085 | 254.2293 | 297.8189 | 43.58962 | 46.7245514 |

17 | 0.4159621 | 1.203211 | 0.219626 | 362.7145 | 239.2957 | 301.0051 | 61.70939 | 66.25374946 |

18 | 0.56250196 | 1.284238 | 0.172897 | 387.1405 | 224.1978 | 305.6691 | 81.47139 | 87.68037713 |

19 | 0.72761583 | 1.382091 | 0.126168 | 416.639 | 208.3243 | 312.4817 | 104.1573 | 112.4970518 |

20 | 0.92931067 | 1.511797 | 0.079439 | 455.7394 | 190.451 | 323.0952 | 132.6442 | 144.096238 |

21 | 1.22965981 | 1.727846 | 0.03271 | 520.8685 | 166.6371 | 343.7528 | 177.1157 | 194.7355493 |

n Equation (13) | Y_{i} Equation (15) | t_{oi} Equation (17) | R(t_{oi}) Equation (22) | σ_{2i} Equation (18) | σ_{1i} Equation (19) | N (Cycles) Equation (33) |
---|---|---|---|---|---|---|

1 | −3.4034833 | 0.220104 | 0.96729 | 66.35155 | 1308.124 | 273,683,422.9 |

−2.9701952 | 0.26688 | 0.95 | 80.45246 | 1078.849 | 331,846,104.7 | |

2 | −2.491662 | 0.330174 | 0.920561 | 99.53286 | 872.034 | 410,547,979.5 |

−2.2691664 | 0.364517 | 0.901768 | 109.8856 | 827 | 453,250,454.5 | |

3 | −2.0034632 | 0.410239 | 0.873832 | 123.6689 | 701.842 | 510,103,112.7 |

4 | −1.6616459 | 0.477593 | 0.827103 | 143.9731 | 602.8627 | 593,852,935.8 |

5 | −1.3943983 | 0.537869 | 0.780374 | 162.1435 | 535.3039 | 668,801,050.2 |

−1.0845459 | 0.617339 | 0.713156 | 184.8 | 491.75 | 767,615,910 | |

6 | −1.1720537 | 0.593774 | 0.733645 | 178.9966 | 484.9034 | 738,315,719.9 |

7 | −0.9793812 | 0.646898 | 0.686916 | 195.0109 | 445.0831 | 804,370,622.7 |

8 | −0.8074473 | 0.698303 | 0.640187 | 210.5073 | 412.3183 | 868,289,756.6 |

9 | −0.6504921 | 0.748789 | 0.593458 | 225.7266 | 384.5185 | 931,065,175.7 |

10 | −0.5045088 | 0.799016 | 0.546729 | 240.8679 | 360.3471 | 993,519,280.8 |

11 | −0.3665129 | 0.84959 | 0.5 | 256.1134 | 338.8969 | 1,056,403,357 |

12 | −0.2341223 | 0.901115 | 0.453271 | 271.6459 | 319.519 | 1,120,470,957 |

13 | −0.1052851 | 0.954255 | 0.406542 | 287.6654 | 301.7256 | 1,186,547,445 |

0 | 1 | 0.367879 | 301.4555 | 301.4555 | 1,243,427,849 | |

14 | 0.0219284 | 1.0098 | 0.359813 | 304.4098 | 285.129 | 1,255,613,543 |

15 | 0.14952577 | 1.068761 | 0.313084 | 322.1837 | 269.3992 | 1,328,926,684 |

16 | 0.279845 | 1.132534 | 0.266355 | 341.4085 | 254.2293 | 1,408,224,098 |

17 | 0.4159621 | 1.203211 | 0.219626 | 362.7145 | 239.2957 | 1,496,105,985 |

18 | 0.56250196 | 1.284238 | 0.172897 | 387.1405 | 224.1978 | 1,596,857,176 |

19 | 0.72761583 | 1.382091 | 0.126168 | 416.639 | 208.3243 | 1,718,530,710 |

20 | 0.92931067 | 1.511797 | 0.079439 | 455.7394 | 190.451 | 1,879,810,288 |

21 | 1.22965981 | 1.727846 | 0.03271 | 520.8685 | 166.6371 | 2,148,451,296 |

Application | Maximum Principal Stress (MPa), Force F _{1} = 4.63 N | Minimum Principal Stress (MPa), Force F _{2} = 1.74 N | Weibull β Parameter | Cycles to Failure (N) | Reliability R(t) |
---|---|---|---|---|---|

Static | 470 | 176 | 2.240388 | 691,910,584 | 0.91 |

FEA | 491.75 | 184.8 | 2.248519 | 453,250,454 | 0.90 |

Application | Allowable Stress Range F_{SR} _{(MPa)} Equation (37) | Cycles to Failure (N) Equation (36) | Reliability R(t) |
---|---|---|---|

Static | 294 | 155,358 | 0.91 |

FEA | 307 | 136,446 | 0.90 |

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**MDPI and ACS Style**

Barraza-Contreras, J.M.; Piña-Monarrez, M.R.; Molina, A.
Fatigue-Life Prediction of Mechanical Element by Using the Weibull Distribution. *Appl. Sci.* **2020**, *10*, 6384.
https://doi.org/10.3390/app10186384

**AMA Style**

Barraza-Contreras JM, Piña-Monarrez MR, Molina A.
Fatigue-Life Prediction of Mechanical Element by Using the Weibull Distribution. *Applied Sciences*. 2020; 10(18):6384.
https://doi.org/10.3390/app10186384

**Chicago/Turabian Style**

Barraza-Contreras, Jesús M., Manuel R. Piña-Monarrez, and Alejandro Molina.
2020. "Fatigue-Life Prediction of Mechanical Element by Using the Weibull Distribution" *Applied Sciences* 10, no. 18: 6384.
https://doi.org/10.3390/app10186384