The FCSE controlling equation of pinned thinwalled curve box was derived and the indeterminate problem of continuous thin-walled curve box with diaphragm was solved based on flexibility theory. With Bayesian statistical theory, dynamic Bayesian error function of displacement parameters of indeterminate curve box was founded. The corresponding formulas of dynamic Bayesian expectation and variance were deduced. Combined with one-dimensional Fibonacci automatic search scheme of optimal step size, the Powell optimization theory was utilized to research the stochastic identification of displacement parameters of indeterminate thin-walled curve box. Then the identification steps were presented in detail and the corresponding calculation procedure was compiled. Through some classic examples, it is obtained that stochastic performances of systematic parameters and systematic responses are simultaneously deliberated in dynamic Bayesian error function. The one-dimensional optimization problem of the optimal step size is solved by adopting Fibonacci search method. And the Powell identification of displacement parameters of indeterminate thin-walled curve box has satisfied numerical stability and convergence, which demonstrates that the presented method and the compiled procedure are correct and reliable.During parameters' iterative processes, the Powell theory is irrelevant with the calculation of finite curve strip element (FCSE) partial differentiation, which proves high computation efficiency of the studied method.

A solution of probabilistic FEM for elastic-plastic materials is presented based on the incremental theory of plasticity and a modified initial stress method. The formulations are deduced through a direct differentiation scheme. Partial differentiation of displacement, stress and the performance function can be iteratively performed with the computation of the mean values of displacement and stress. The presented method enjoys the efficiency of both the perturbation method and the finite difference method, but avoids the approximation during the partial differentiation calculation. In order to improve the efficiency, the adjoint vector method is introduced to calculate the differentiation of stress and displacement with respect to random variables. In addition, a time-saving computational method for reliability index of elastic-plastic materials is suggested based upon the advanced First Order Second Moment (FOSM) and by the usage of Taylor expansion for displacement. The suggested method is also applicable to 3-D cases. Liu Ning et al.: Reliability of Elasto-Plastic Structure Using FEM 67 perturbation SFEM, but avoids the approximation in the partial differentiation calculation. In the late 80's, the Neumann expansion technique was introduced, associated with the Monte-Carlo simulation, to obtain the formulation of SFEM. One can refer, for example, to Shinozuka and Deodatis [6] and Yamazaki and Shinozuka [7]. In addition to the methods mentioned above, other sophisticated methods of stochastic finite elements have also been proposed, for example, the response surface method suggested by Faravelli Is], the weighted integral method suggested by Takada[ 9] and the matrix formulation of SFEM suggested by Kiureghian [1~ as well as the innovative methods suggested by Vanmarcke and Shinozuka [11], Hisada and Noguchi [12], Liu and Kiureghian [13], Haldar and Zhou [14], Papadrakakis and Papadopoulos [15], which, among many others, should be mentioned.Most of the studies concerning with SFEM are, however, within the scope of linear elastic problems. Materials such as soil, soft or joint rock, however, exhibit a nonlinear stress-strain relationship. It is obvious that the stochastic response of this kind of materials can not be handled by linear SFEM. The main difficulties are as follows: (1) both element and global stiffness matrices of elastic-plastic materials involve not only the elastic modulus, but also stress and strength parameters; (2) during each loading step, not only the mean value but also the gradients of stress and strain need to be obtained by some iterative algorithms; and (3) for the elastic problem, when using elastic SFEM to calculate the reliability index, only one iterative algorithm is necessary for searching of the design point (Madsen, Krenk and Lind[161). However, for the elastic-plastic problem, two kinds of iterative computations axe needed, it is hence difficult to obtain an efficient iterative algorithm for the two parts. Some attempts have been made to deal with materials whose properties are...

Based on the model of rigid-spring element suitable for homogeneous elastic problem, which was developed by Japanese professor Kawai, the interface stress element model (ISEM) for solving the problem of discontinuous media mechanics has been established. Compared with the traditional finite element method (FEM), the ISEM is more accurate and applicable. But the total number of freedom degree of ISEM in dealing with three-dimensional problems is higher than that of FEM, which often brings about the reduction on efficiency of calculation. Therefore, it is necessary to establish a mixed model by gathering the advantages of ISEM and FEM together. By making use of the good compatibility of ISEM and introducing the concept of transitional interface element, this paper combines ISEM and FEM and proposes a mixed model of ISEM-FEM which can solve, to a large extent, the contradictions between accuracy and efficiency of calculation. In addition, using natural coordinate, algorithm of ISEM for block elements of arbitrary shape has been performed. Numerical examples show that the method proposed in this paper is feasible and its accuracy is satisfactory.

The Bayesian method of statistical analysis has been applied to the parameter identification problem. A method is presented to identify parameters of dynamic models with the Bayes estimators of measurement frequencies. This is based on the solution of an inverse generalized eigenvalue problem. The stochastic nature of test data is considered and a norreal distribution is used for the measurement frequencies. An additional feature is that the engineer' s confidence in the measurement frequencies is quantified and incorporated into the identification procedure. A numerical example demonstrates the efficiency of the method.

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