Study On Calculation Methods For Seismic Second-order Effect Of Mdof System Under Horizontal Earthquake

The second order effect belong to the filed of the non-linear issues and the non-linear iteration should be undertaken if there is a require to calculate the second order effect of the structure exactly, which is not easy to carry out for the engineering application. In the view of the achievement of the second order effect at home and aboard, the research on the rule of the second order effect has been developed well, especially the simplified calculation method carried out by hands has been proposed on the static second order effect. However, there are few simplified methods for the computer programs, especially the application of current simplified methods on the dynamic second order effect needs the verification. Meanwhile, mode decomposition method and base shear method is the two fundamental methods for calculating the force of the earthquake and the way to the consider the dynamic second order effect of the structure in these two methods is also one of the work in this paper.The major work have been done based on the two points mentioned above as follow:①Summarizing several usual finite element methods managing the P ?Δeffect in theory and analyzing the advantages and disadvantages, practicability and accuracy.②For single dof and multi-dof systems, the results and the differences of virtual column method and exact method calculating the dynamic and static second order effect.③By the shift decoupling theory based on modal decomposition method, a method was proposed for defining the value of the stiffness reduction factor considering the second order effect and the accuracy was verified.The conclusions are listed as follow via the work above:①the finite method based on geometric stiffness takes no account of iterate, so its computational efficiency is higher though inaccurate. The finite method based on horizontal force or gravity load needs iteration, because of which it is inefficient while the result is more accuratel.②the exact method based on the information of the structure like geometric stiffness, internal force, deformation, the consideration on the variation of the initial location has been done when the equilibrium equations are established and it could be regarded as a calculation method which is fairly exact; However, the virtual column method considers the variation of the stiffness yielded by the axial force only and it is an approximate method.③For the static second order effect, the results of the virtual column method and the exact method are close and the distinction rises with the growth of the stable factor of the structure; However, for the dynamic second order effect, the error divergence of those two methods are notable and related to the stable factor and the seismic waves. The virtual column method doesn't work anymore when undertaking the exact analysis of the structure.④It is acceptable to calculate rigidity reduction coefficient according to structure first vibration mode rigidity while considering structure second-order effect, neglect rest of vibration mode 'effects.