# Prediction of Remaining Fatigue Life of In-Service Bridge Cranes
## 2 Fatigue Characteristic Analysis
### Stress-Time History Simulation of Fatigue Failure Point  
One working cycle of the bridge crane refers to the process of lifting and moving the weight from the ground to the designated position, then unloading, and then moving the weight to the next cycle. Due to the different actual working conditions of each bridge crane, the stress-time history of the bridge crane can only be counted according to the actual working conditions of one bridge crane. The maximum lifting capacity of the bridge crane used in this study is 40 t, and the working level is A3 (According to the crane design specification, the crane working level is classified according to the utilization level of the crane and the load status, and the A3 level is light.) and the span is 21.6 m. The established simulation model is a transient dynamic simulation with finite element analysis software to obtain the stress-time history of the fatigue failure point at the mid-span position of the box-type beam of the bridge crane. Figure 6 shows the stress-time history of a lifting cycle of the bridge crane, wherein sections 1-2 are the curves of the no-load trolley moving from the cross-end position of the main beam to the lifting object, sections 2-3-4 are the curves of the lifting object and the curve of the trolley moving from the lifting position to the unloading position, 5-6-7 are the curve of the unloading process, and 7-8 is the curve of the no-load trolley moving to the next cycle of lifting position. Latin hypercubic sampling [3; 4] is a method of approximate random sampling from a multivariate parameter distribution which belongs to the stratified sampling class of techniques and is commonly used in computer experiments or Monte Carlo numerical simulations. In this method, the sample space is divided into several equal sub-intervals (hypercube), and one sample point is randomly selected in each sub-interval. Such a sampling method can improve the randomness and representativeness of the sample point and avoid sampling error and variance. The LHS method is a sampling method that can improve the accuracy and reliability of the Monte Carlo numerical simulation method, and it is widely used in numerical analysis, prediction, and simulation in different fields. Therefore, to improve the reliability of the simulation, the LHS method is introduced in the Monte Carlo numerical simulation to simulate the load lifting and unloading position of each cycle, whether the trolley crosses the span, lifting load weight, and other parameters [3]. The specific method is shown as follows. In this paper, n random numbers \(x_{i}\)(i = 1, 2, 3,\(\ldots\), n) are generated in the interval from 0 to 1, and then the random number \(x_{i}\) is converted into a random number \(K_{i}\) in the i-th interval. \[K_{i}=[x_{i}+(i-1)]/n,i=1,2,3,\ldots,n \tag{1}\]  
Figure 6: Stress–time history curve of a lifting cycle. The upper boundary of the i-th interval is \(\rm i/n\), and the lower boundary is \(\rm(i-1)/n\), which can be known from Equation (1):  
\[\rm(i-1)/n<K_{i}<i/n \tag{2}\]  
Thus, only one constrained random number \(\rm K_{i}\) is generated in each interval, and the random realization of the random variable is obtained by solving for \(\rm K_{i}\) as:  
\[\rm R_{ji}=F_{x_{i}}^{-1}(K_{i}),j=1,2,3,\ldots,n \tag{3}\]  
where \(R_{ji}\) is the i-th random realization corresponding to the j-th random variable; \(\rm F_{x_{i}}^{-1}(K_{i})\) is the function of the cumulative distribution corresponding to the j-th random variable. If the cumulative distribution function corresponding to each characteristic parameter is known, the LHS method can be used to generate random values for each cyclic characteristic parameter.