# A Statistical Model Updating Method of Beam Structures with Random Parameters under Static Load
## 3 Numerical Examples
### A Two-Span Continuous Beam  
Consider a two-span continuous beam as shown in Figure 4. The mechanical parameters of the continuous beam are the same as those of the simply supported beam in Section 3.1. The concentrated load \(P=200\) kN is applied in the middle of the left span of the continuous beam. Here, the 10% reduction of the elastic modulus of the 3 rd and 4 th element is assumed. Additionally, it is assumed that the coefficients of variation of the measured displacements at all nodes are 0.01, and they are of Beta distributions.  
Figure 4: A two-span continuous beam.  
Figure 3: The statistics of deflections of the simply supported beam based on the updated finite element model. (**a**) Means of deflections; (**b**) Standard deviations of deflections; (**c**) Probability density functions of the deflection at the 4 th node.  
Figure 2: The statistical results of the element updated factors of the simply supported beam from the proposed high-order perturbation statistical model updating method (HP-SMUM) and the Monte Carlo simulation method (MC). (**a**) Means of element updated factors; (**b**) Standard deviations of element updated factors; (**c**) Probability density functions of the 4 th element updated factor.  
By the simulated measurement data, the proposed HP-SMUM and the Monte Carlo simulation method with 100,000 samples are used to obtain the statistics of the element updated factors, and the results are plotted in Figure 5. From Figure 5, it is found that the means and standard deviations of the element updated factors from the proposed HP-SMUM are in good agreement with those of the Monte Carlo simulation method. Furthermore, the means of the 3 rd and 4 th element shown in Figure 5a testify that a 10% reduction of the element stiffness has happened.  
However, the probability density function of the 4 th element selected from the proposed method is little different from the results of the Monte Carlo simulation method. This may be attributed to the non-symmetry of the continuous beam. Meanwhile, using the obtained element updated factors to solve Equation (10), the statistics of deflections of the updated continuous beam can be determined and shown in Figure 6 which shows that the first two moments of deflections by the updated finite element model match with the assumed measurement values very well. For the mean of deflection, actual calculated relative errors at all nodes are less than 0.54%. On the other hand, for the standard deviation of deflection, the relative errors are less than 7%. In addition, the pdf of the deflection in the mid-span by the updated model hardly deviates from the measured result.  
Figure 5: The statistical results of the element updated factors of the continuous beam from the proposed HP-SMUM and the Monte Carlo simulation method (MC). (**a**) Means of element updated factors; (**b**) Standard deviations of element updated factors; (**c**) Probability density functions of the 3 rd element updated factor; (**d**) Probability density functions of the 4 th element updated factor.