Abstract: Bending-torsion-liquid coupling free vibration equat- ions of a semi-circule curved pipe conveying fluid is derived, based on D. Alembert principle. For this class formula, it is difficult to uncouple and to get bending and torsion analytic solutions separately. An approching method--direct method is presented. First, test functions selected to satisfy natural boundary conditions, but not to assure meeting equations, are regarded as the approximate solutions of equations and let weighted-integrating of errors in whole space domain equal to zero. It is explained that the sum of virtual work done by generalized forces equal to zero (the weak integration of equilibrium eqation). Then approximate analytic formula of system frequency are obtained. Meanwhile, approximate analytic formulation of limit velocity is obtained. An example shows the influence of flowing velocity on the natural frequency of the curved pipe conveying fluid. More precise effect is presented.

Key words: Bending-torsion-liquid coupling, Direct method, Limit velocity, Natural boundary conditions, Semi-circule curved pipe conveying fluid