In this article, structural probabilistic and non-probabilistic reliability have been evaluated and compared under big data condition. Firstly, the big data is collected via structural monitoring and analysis. Big data is classified into different types according to the regularities of the distribution of data. The different stresses which have been subjected by the structure are used in this paper. Secondly, the structural interval reliability and probabilistic prediction models are established by using the stress-strength interference theory under big data of random loads after the stresses and structural strength are comprehensively considered. Structural reliability is computed by using various stress types, and the minimum reliability is determined as structural reliability. Finally, the advantage and disadvantage of the interval reliability method and probability reliability method are shown by using three examples. It has been shown that the proposed methods are feasible and effective.

The reliability problem has existed since the beginning of human social activities, but the real cause of human attention should have started during the Second World War. At that time, the United States military’s communications and electronic equipment frequently failed during the service period, and some electronic equipment had not even been put into use, and a large number of them had failed.

Since the end of the 1940s, many industrially developed countries led by the United States have successively carried out research on reliability. Reliability research in the field of structures and systems can be traced back to the late 1940s and early 1950s. In 1947, the famous stress-strength interference model was established, which formed the basis of structural reliability design. Since 1954, the normal-normal model of the stress-strength interference model was established, scholars from the Soviet Union, the United States, Japan, Canada and other countries have made outstanding contributions to the study of mechanical and structural reliability. So that structural reliability has been gradually developed into an independent engineering subject.

In modern manufacturing companies, big data is being captured by using lasers, sensors, wireless networks, etc. These big data include the processing of manufacturing processes, the effects of temperature, vibration, reliability data, etc. Big data is processed in computers by using different algorithms for improving the products and to increase the competitiveness of organizations in the market. Big data is widely used to discover and develop new technologies, methodologies and decisions. Big data has been paid attention to and researched in modern society. The 5 V (volume, velocity, variety, veracity, value) of big data have been defined and they are believed as important as the meaning of new technology [

For most engineering structures, massive data would be generated during their service life such as bridges, wind towers, pipelines and automobile chassis [

The traditional structural probability reliability model has been developed based on the known probability distribution function of structural strength and imposed load. The structural reliability can be directly obtained by using the integration method or the first-second moment method, and the prediction result is sensitive to the accuracy of the probability distribution parameters [

Interval data with both heavy censoring and batch effects are considered, simulation studies demonstrate the superiority of the proposed method over two other commonly-used bootstrap methods [

With the massive data advantage, big data processing technology has been applied to the reliability research of large-capacity power and electronic systems [

In this paper, based on the research of structural probability reliability and interval reliability, big data collected from structural monitoring will be classified according to different types of data. The stress response of structure will be obtained by analyzing the big data as shown in

During the structural service lifetime, massive structural health monitoring data of a structure can be produced because it is constantly subjected to imposed loads. The loads can be classified according to their characteristics, they are classified as A_{1}, A_{2},…,A_{i},…,A_{n}, respectively. The number of the loads is denoted as _{i} in each A_{i}, where

In each A_{i}, a structural load is denoted as _{i}, the maximum load is determined and it is denoted as _{i} load of _{i} load of _{i} times of the maximum load _{i} times of _{i}.

The function of the interval non-probabilistic reliability can be calculated as follows by using the strength-stress interference theory:

The structural interval non-probabilistic reliability index can be obtained as follows:

If

In each big data load classification A_{i}, the maximum load is

The mean and dispersion of ^{I}, respectively.

The mean and dispersion of ^{c} and ^{r}, respectively. The structural interval non-probabilistic reliability index can be obtained by using _{i} as follows:

A structure is considered similar to a series system that is comprised of _{i}.

Stress-strength interference is referred to the basic idea of interference analysis of probabilistic variables in reliability estimation. The stress and strength are considered as two random variables to compare their relative magnitudes in the same coordinate system. The distribution function curves of two random variables will be drawn in one coordinate system, they will generally be intersected, that is, there is an interference area, indicating the possibility that the stress is greater than the strength.

The probabilistic density function of the stress is noted as

The concepts of stress and strength are broad. It can be considered that stress is any physical factor that can lead to failure of the structure, such as load, temperature, corrosion, radiation, etc. The strength of structure and system components is their performance index against corresponding stress. According to the traditional view and the assumption of independent failure of components, the reliability of the system can be directly calculated based on the reliability of the system components.

The reliability model of a series system can be calculated as follows:

The reliability model of a parallel system can be calculated as follows:_{sys} is the reliability of the system,

The limit state function of a structure can be established by using the stress-strength interference theory as follows:

The reliability of the structure that the structure is in a safe state can be computed by using the first-order second-moment method._{s} is the mean of the structural stress, and _{s} is the standard deviation of the structural stress.

Suppose the probability distribution function and the probability density function of _{i} times random loads can be written as follows:

The probability density function

Suppose the probability density function _{i} type of loads with the big data condition as follows:

Based on the _{1}, A_{2},…,A_{i},…,A_{n} of structural loads, a formulation of structural reliability can be calculated by using

The minimum value of

A chassis of a tractor is the basic component for the tractor, the chassis of a tractor is shown in

The chassis was made of 45# steel, its interval of the stress is given as

The probability reliability of the chassis can be computed by using

The reliability of the chassis is computed by using Monte Carlo (MC) method with 10^{6} samples. The reliability is 0.9999 which is considered as a practical value in structural reliability engineering. The reliability index is 6.025. It can be converted to the interval reliability index of 1.0346 [

The probability reliability index of the chassis is more than 4.5. Based on the literature review, in comparison to the probability reliability index and interval reliability index, the probability reliability index may be smaller. This is because for the probability reliability index, if less data is used for random distribution parameters, the probability distribution of the stress and strength of the chassis will not be modelled sufficiently, which also reflects the advantages of the interval reliability method.

The structural health monitoring data was collected for a large spanning suspension bridge in China with 3 GB data per day that is 1 TB of big data every year [

The welding joint of a U stiffener and roof is an important part of the large spanning suspension bridge. The welding joint of a U stiffener and roof is shown in

The passing vehicles of the whole year are divided into six types according to the number of axles of the vehicle as follows:

Type I, 2 axles 2 axles,

Type II, 3 axles 2 axles,

Type III, 4 axles 3 axles,

Type IV, 5 axles 3 axles,

Type V, 6 axles 3 axles,

Type VI, 6 axles 4 axles.

The total weight of six types of vehicles is shown in

Vehicle type | I | II | III | IV | V | VI |
---|---|---|---|---|---|---|

Total weight (T) | 1.1 | 11.5 | 20.8 | 26.2 | 31.0 | 31.4 |

Vehicle type | I | II | III | IV | V | VI |
---|---|---|---|---|---|---|

Ratio | 95.60 | 1.171 | 1.15 | 0.43 | 0.744 | 0.37 |

Vehicle type | Mean (MPa) | Standard deviation (MPa) |
---|---|---|

I | 22.37 | 1.961 |

II | 41.25 | 3.602 |

III | 58.80 | 5.135 |

IV | 75.64 | 6.606 |

V | 80.53 | 7.032 |

VI | 90.75 | 7.925 |

The strength of the welding joint of the U stiffener and roof is also assumed to obey the normal distribution with the mean of 345 MPa and the standard deviation of 15 MPa. In this example, the theoretical lifetime of the part is 204 years, whereas the actual observed lifetime is 165 years [_{1}…,_{i},…,_{6} is computed by using Monte Carlo (MC) method with 10^{6} samples, respectively, and the reliability results are shown in

Vehicle type | I | II | III | IV | V | VI |
---|---|---|---|---|---|---|

_{i} |
5.736 | 0.1026 | 0.069 | 0.0258 | 0.0444 | 0.0222 |

Vehicle type | I | II | III | IV | V | VI |
---|---|---|---|---|---|---|

Reliability _{m} |
0.9999 | 0.9999 | 0.9999 | 0.9999 | 0.9999 | 0.9999 |

The reliability _{1},…,_{i},…,_{6} is computed by using

Vehicle type | I | II | III | IV | V | VI |
---|---|---|---|---|---|---|

Reliability _{i} |
0.9999 | 0.9999 | 0.9999 | 0.9999 | 0.9999 | 0.9999 |

Vehicle type | I | II | III | IV | V | VI |
---|---|---|---|---|---|---|

Interval reliability index _{i} |
2.841 | 2.280 | 1.887 | 1.591 | 1.516 | 1.374 |

Based on

Hollow cylindrical cantilever beams (HCCB) have been widely used in engineering practice as shown in

Variable | Distribution | Parameter 1 | Parameter 2 |
---|---|---|---|

Normal | 5 | 0.1 | |

Normal | 42 | 0.5 | |

_{2}/mm |
Uniform | 59.75 | 60.25 |

_{1}/N |
Gumbel | 3000 | 300 |

_{2}/ |
Gumbel | 3000 | 300 |

Normal | 12000 | 1200 | |

_{1} |
Interval | 0 | 10 |

_{2} |
Interval | 5 | 15 |

Normal | 300 | 30 | |

Normal | 90 | 9 | |

_{1}/ |
Interval | 119.75 | 120.25 |

Normal | 200 | 20 | |

Interval | 0 | 1.9 |

The reliabilities of the strength, stiffness, shear force, torsional angle and deflection of the HCCB are considered in this example. The interval reliability indices of the strength, stiffness, shear force, torsional angle and deflection of the HCCB are shown in ^{6} samples under big data conditions, and the reliability results are shown in

Type | Strength | Stiffness | Shear | Torsional angle | Deflection |
---|---|---|---|---|---|

Interval reliability index _{i} |
1.023 | 1.056 | 1.133 | 1.164 | 1.031 |

Type | Strength | Stiffness | Shear | Torsional angle | Deflection |
---|---|---|---|---|---|

Probabilistic reliability index _{H} |
6.136 | 6.334 | 6.796 | 6.982 | 6.183 |

Type | Strength | Stiffness | Shear | Torsional angle | Deflection |
---|---|---|---|---|---|

Reliability | 0.9952 | 0.9999 | 0.9999 | 0.9999 | 0.9989 |

Type | Strength | Stiffness | Shear | Torsional angle | Deflection |
---|---|---|---|---|---|

Reliability | 0.9952 | 0.9999 | 0.9999 | 0.9999 | 0.9989 |

If only considering one of these five reliabilities, it will be a parallel system. However, by considering all these five reliabilities, it will be a series system. If three of them do not fail, then the cantilever beam does not fail. The cantilever beam will be a 3/5 voting system. The reliabilities of these systems are shown in

System type | Series system | Parallel system | 3/5 system |
---|---|---|---|

Reliability | 1 | 0.9940 | 0.9991 |

Based on

The collected big data has been analyzed according to its source and distribution. The different types of stress responses of the structure have been classified.

The structural interval and probabilistic reliability computational models have been proposed using the stress-strength interference theory according to different types of structural stress responses and structural strength. Different structural reliabilities were computed, and the minimum of these reliabilities is the reliability of the structure.

It has been shown that the proposed methods are feasible and effective. The probabilistic reliability method and interval reliability method have their advantages and disadvantages which have been shown by using the three examples under the condition of big data. The structural reliability can be obtained by using the probabilistic reliability method when the probability distribution parameters can be determined by using big data. Alternatively, the interval reliability method can be used.

It should be pointed out that both the structural interval reliability method and probabilistic reliability method are no better or simpler than each other under the condition of big data, and each has different applicable conditions. They are not developed to replace each other, and the interval reliability method is a useful complement to the probabilistic reliability method under the condition of big data.

The repairable system and other complex systems with random and interval parameters under big data will be studied in future work.

The original intention of the interval algorithm was used to calculate the reliability of the structure, but people soon discovered that the interval algorithm can have a wider range of applications. In interval mathematics, all variables can be considered as an interval quantity and the uncertain factors of the variables are converted into intervals. Due to the set properties of the interval itself, the set operations between the intervals can be easily implemented.

Interval analysis is a set of operation rules defined on the interval. Its main feature is that it can process uncertain data, automatically record the truncation and rounding errors generated in computer floating-point operations, and efficiently and reliably estimate the value range of a function in a certain independent variable area, so it is widely used in all areas of science. Interval analysis emerged in the 1960s and has been widely used in computer graphics and computer-aided design since the early 1980s.

There are two real numbers

For two intervals ^{I} = ^{I}.

The

If ^{I} would be degenerated to the ordinary real numbers, also known as point interval numbers.

The mean of the interval number is given as follows:

The dispersion of the interval number is given as follows:

The interval

These are the rules for the addition, subtraction, multiplication, and division of two intervals.

Let

If

On the other hand, if

If ^{I}) can be written as follows:_{min}(_{max}(

The four arithmetic operations of the interval can be satisfied with some algebraic properties of the four arithmetic operations on real numbers, such as the associative and commutative laws of addition and multiplication. However, the law of distribution does not always hold. In general, the following equation does not hold, that is as follows:

However, the following sub-distributive law expression is true, that is as follows:^{I}⊆ ^{I}, ^{l} ≤ ^{l} ≤ ^{u} ≤ ^{u}, the interval ^{I} is contained to the interval ^{I}.

If there are some common parts in the interval ^{I}, ^{I} ∈

If

Based on the above, it is obtained that in the interval calculation process, the arithmetic operation of the interval is closed, and the set operation of the interval is also closed. However, their algebraic properties are different from the algebraic properties of real numbers. When performing interval operations, the following expressions should be given special attention to, otherwise, serious errors may occur.