## 4 Fuzzy Fatigue Reliability Assessment of Welded A-Type Frame
### Fuzzy Fatigue Performance Function Considering Multi-Source Uncertainties
#### 4.5.2 Constructing Fuzzy Fatigue Reliability Performance Function  
In this paper, the Latin hypercube sampling method was used to obtain the sample data of random variables for constructing the fuzzy fatigue reliability performance function [26]. Once the random variables were determined, the maximum equivalent stress, as the output variable, could be calculated by the finite element analysis. All 103 samples are listed in Table 6. Due to the limited space, only six sample data is given here. Afterwards, based the second order response surface model [25], the sample data was fitted to gain the fuzzy fatigue reliability performance function, which was evaluated by:  
\[\begin{array}{l}S=5765.9829746838-0.046035259152328*-7620.518623468*+v+1.07724247031588*+ 911.320649940866*+-\\ 5.9753486639276*+t1-9.739473104683*+t2-0.81944395776015*t3-1.1987018215256*+ -400.00712008001152*-t1+0.00259170840474808*-f2-0.00409075701607125*t3-1.14514865650598*- -007.2+\(\,\)+8030.0070957048*+v^{2}-\\ 0.000684604009400945*+t2+1956172864825*+t2+0.0251740575496*+t2+0.0498468020095*+ t2+0.050318080892*t3+t3+0.0742673373931*t4+t4+5.53257484882952*-0.009 -t3+t2+7.3512127490948*-008-t2+1.959583299149086*-007*t3+\\ 4.8947001219944*-009*+\,\)/42 +0.04621026729009*+\,\)/42 +0.04621026729009*\(\,\)/4 +\(\,\)/2 +0.045610369986*-006*+\(\,\)Subsequently, the main effect chart between all random variables and the out variable is shown in Figure 15. The item \(x9\) had the maximum slope, which meant that the vertical force at front traction joint \(F_{1}\) had the most influence on the maximum equivalent stress \(S\). This conclusion is the same as the result from the Pareto chart. The item \(x5\) had the negative slope, which meant that the increase of thickness of tail plate \(t_{1}\) could lessen the maximum equivalent stress \(S\). In addition, the interaction effect between items \(x1\) and \(x10\), and the items \(x1\) and \(x12\), are also shown in Figure 16. It could be seen that there is no intersection between Young Modulus \(E\) and lateral force at right steering joint \(F_{2}\), which meant that they had no relevance. In contrast, there is relevance between Young Modulus \(E\) and lateral force at the front lateral stabilizer joint \(F_{4}\). Figure 14: Pareto chart of partial items in response surface model. Figure 15: Main effect between random variables and output variable. Another 10 samples of random variables were picked out, and their corresponding stress values were separately calculated through the approximation model and the finite element analysis, in order to verify the accuracy of the response surface model Equation (14). Afterwards, the simulated results from finite element analysis and the predicted results from the Equation (14) were compared, as shown in Figure 17. It could be seen that they are close to each other, which meant that the accuracy of the response surface model Equation (14) was pretty good. To observe the relationship between the maximum equivalent stress \(S\) and the random variables, the three-dimensional graph of response surface model is displayed in Figure 18. Figure 18a shows that there is an obvious nonlinear relationship between the maximum equivalent stress \(S\) and Young Modulus \(E\) and cyclic strength coefficient \(K^{\prime}\), while it could be seen in Figure 18b that there is linear relationship between maximum equivalent stress \(S\) and vertical force at front traction joint \(F_{1}\) and lateral force at front lateral stabilizer joint \(F_{4}\). Figure 18c shows that there is local linear relationship and nonlinear relationship between the maximum equivalent stress \(S\) and thickness of roof plate \(t_{2}\) and thickness of tail plate \(t_{1}\). Figure 16: Interaction effect between the items \(x1\) and \(x10\), and the items \(x1\) and \(x12\) ((a) \(x1\) and \(x10\); (b) \(x1\) and \(x12\)). Figure 17: Comparison of the simulated results and predicted ones.