Reverberation-Ray Matrix Method For 3D Framed Structures And Its Application

The vast majority of investigations on dynamic response for structures and structural members focus on vibration behavior and long time response on the basis of modal analysis.In the effort for accurate prediction of transient response,there is a need in analyzing the features of wave motions in structures.In this dissertation,a reverberation-ray matrix method(RRMM),as a new method in structure dynamics,is developed for 3D framed structures,and has been applied to analysis of static and dynamic responses of such structures.Based on exact equations of wave motions including longitudinal wave, transverse wave and torsional wave,corresponding generalized mathematical forms for scattering matrix,phase matrix,permutation matrix and reverberation-ray matrix are formulated.In such a wave model,two basic vectors,i.e.,arriving wave vector and departing wave vector,are introduced to calculate the displacement and internal forces in members,and they can be determined by evaluating the reverberation-ray matrix.The reverberation-ray matrix reflects entire information induced from the scattering and transmitting waves in the structures.Using the wave model,initial transient response can be obtained accurately.Especially,for a static case,the scattering matrix is replaced by a carry-over and distribution matrix,and both the arriving and departing wave vectors represent the displacements and rotational angles at two ends in the members,respectively.In application of RRMM,a main task is to determine the arriving and departing wave vectors prior to calculating the internal forces at any joint on the basis of the exact wave solution forms.Meanwhile,the displacement and internal forces at any cross-section in a member can be determined accurately using the wave solution form instead of other existing interpolation algorithms.A specialized FORTRAN code for 3D framed structures is compiled to determine stead-state or transient response values for internal forces and moments at any joints as well as at any cross-section in a member,including one axial force,two transverse shear forces,one twisting moment and two transverse bending moments. As an application,a two-story framed structure under a step time loading is chosen as a computational model.The structure is composed of 16 beam members with lengths of 7l₀,4l₀ and 2l₀ in three orthogonal directions respectively and with the same circular cross-section having a diameter of l₀/2.The RRMM shows the accuracy in some aspects,e.g.,the first arrival time for the stress waves,long time satisfaction of force and moment balances at joints,and dynamic response,in general,gradually approaching to the static results,etc.The numerical solutions of dynamic analysis show that the initial transient response due to impact loading can be calculated accurately,which is very difficult to be obtained by applying other approaches.On the other hand,numerical results show that,a remarkable change in twisting moment probably occurs for a dynamic case.