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a b s t r a c tIn this paper, a simple and efficient approach is presented to compute the eigenvalues of the fourth-order Sturm-Liouville equations with variable coefficients. By transforming the governing differential equation to a system of algebraic equation, we can get the corresponding polynomial characteristic equations for kinds of boundary conditions based on the polynomial expansion and integral technique. Moreover, the lower and higher-order eigenvalues can be determined simultaneously from the multi-roots. Several examples for estimating eigenvalues are given. The convergence and effectiveness of the method are confirmed by comparing numerical results with the exact and other existing numerical results.

SUMMARYA multilevel genetic algorithm (MLGA) is proposed in this paper for solving the kind of optimization problems which are multilevel structures in nature and have features of mixed-discrete design variables, multi-modal and non-continuous objective functions, etc. Firstly, the formulation of the mixed-discrete multilevel optimization problems is presented. Secondly, the architecture and implementation of MLGA are described. Thirdly, the algorithm is applied to two multilevel optimization problems. The ÿrst one is a three-level optimization problem in which the optimization of the number of actuators, the positions of actuators and the control parameters are considered in di erent levels. An actively controlled tall building subjected to strong wind action is considered to investigate the e ectiveness of the proposed algorithm. The second application considers a combinatorial optimization problem in which the number and conÿguration of actuators are optimized simultaneously, an actively controlled building under earthquake excitations is adopted for this case study. Finally, some results and discussions about the application of the proposed algorithm are presented.

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An analytical solution for the bearing capacity of a large‐diameter pile is proposed based on the load transfer function in this paper. The load–settlement relationship or the P–S curve of such a pile can be determined by the proposed method for practical application. Comparison is made between the results obtained by the proposed method and the test data, and close agreement has been found, thus verifying the accuracy of the proposed method. It is noted that the most reliable approach to the determination of the vertical load capacity of a pile is to use the ­P–S curve, which is usually obtained through on‐site load test. However, for a large‐diameter pile (e.g. a diameter over 3 m), this kind of test requires a thousand tons of test load, which sometimes is impossible to provide. It is expected that the proposed method can be used in this kind of situation. Copyright © 2002 John Wiley & Sons, Ltd.