# A Statistical Model Updating Method of Beam Structures with Random Parameters under Static Load
###### Abstract  
This paper presents a new statistical model updating method of beam structures with random parameters under static load. The new updating method considers structural parameters and measurement errors to be random. To reduce the unmeasured degrees of freedom in the finite element model, a static condensation technique is used in this method. A statistical model updating equation with respect to element updated factors is established afterwards. The element updated factors are expanded as random multivariate power series  
evaluate the unknown stiffness parameters of the structural components. [PERSON] et al. [8] proposed a superelement-based virtual distortion method to improve the efficiency of the finite element model updating of large-scaled bridges by using static information. [PERSON] et al. [9] presented a method for updating a finite-element model based on the measured static responses of structures with the aid of the response surface methodology. [PERSON] et al. [10] proposed a method based on the meta-model for updating the finite element model of bridges by using measured static and dynamic results. [PERSON] et al. [11] collected strain data during the load test to calibrate a detailed baseline finite element model in an effort to represent the 3D system behavior of a bridge. For the above research, the uncertainty of the measurement errors is seldom involved. Considering the measurement errors as random quantities, many statistical approaches have been developed to update the parameters of structures based on the measured data. These approaches include the stochastic perturbation methods [3; 12; 13; 14; 15; 16], the Monte Carlo simulation methods [17; 18], the Bayesian updating methods [19; 20; 21; 22; 23; 24; 25; 26] and so on. For example, [PERSON] et al. [12] proposed a random matrix approach to derive the closed-form expressions for the mean matrix and the covariance matrix of the updated stiffness matrix by the perturbation technique. [PERSON] et al. [13] considered the statistical properties of experimental data and updating parameters as random variables, and used the perturbation method to update the parameters. Combined with the sensitive method, [PERSON] et al. use a Monte Carlo simulation method to solve the updating parameters [18]. Although the results from the Monte Carlo simulation methods are traditionally regarded as the exact solution of random problems, these methods generally require much greater computational efforts to obtain an accurate solution, and are impractical in the case of large-scale problems. [PERSON] et al. [23] studied the behaviour of the Bayesian updating framework when both static and dynamic data are used to update the model. However, all these studies mainly focus on using dynamic data to update structural models. Although some researchers have tried to use different random solution methods based on the finite element model to conduct statistical model updating, few investigations involve the direct use of random finite element methods to perform this work with static data. In light of the evidence, it is clear that further research is needed in this area. In this paper, a new statistical model updating method of beam structures with random parameters under static load is proposed. This new model updating method considers structural parameters and measurement errors as random quantities. A static condensation technique is used to reduce the unmeasured degrees of freedom in the random finite element model of the beam. Then, a statistical model updating equation with respect to element updated factors is established. The element updated factors are expanded as random multivariate power series. Using the high-order perturbation technique, the statistical model updating equation can be solved to determine the coefficients of the power series expansions of the element updated factors.