Investigated Lump Method For Wave Propagation Analysis Of Earthquake Response Of Structure

The essence of vibration of structure is the problem of wave propagation and reflection in the structure. Structural vibration is an exterior behavior of wave propagation in structure, and earthquake responses are the results caused by wave propagation and reflection in structure. Research on wave propagation and reflection in structure can not only provide the solution of wave equation for the earthquake response of structure, but also be helpful to show in detail the earthquake response in the structure. It is reasonable for obtaining the solution from the point of view of wave motion to the problem of structural instant response caused by earthquake, especially for the near-fault earthquake that contains high frequency content. In addition, the wave theory can explain well the duration phenomena of wave propagation in structure. Therefore, wave propagation method is an effective approach for studying structural earthquake response.Investigated lump methods are respectively presented for analyzing earthquake responses of high-rise structure, plane frame structure, and three-dimensional frame structure by simulating wave propagations in the structures. The investigated lump method is a kind of wave-based method. Bending, shear and axial deformations of the high-rise structure or the members in the frame structure are considered together in the proposed wave-based method. The main ideas of investigated lump method are:firstly the investigated lumps are constructed in the structure, and then the solutions of dynamic responses are obtained by using the dynamic equilibrium equations of every investigated lump. In this method, time histories of displacement obtained from seismic acceleration records by doing time integration are applied on the supports of structure, and the earthquake response of structure is obtained by doing recursive calculations in time domain rather than solving the conventional dynamic equations. The investigated lump method has characteristics of clear physical background and simplicity. For plane frame structure and three-dimensional frame structure, wave passage excitation can be implemented naturally, and there is no need to solve the classical governing equations for multiple support excitations. The major contributions are listed as follows:1. The investigated lump method is presented for analyzing earthquake responses of high-rise structures based on wave propagation approach. Firstly, the concept of investigated lump is introduced for the shear-bending MDOF mechanical model for high-rise structure. Secondly, the calculating formula of the median internal forces (axial force, shear force and bending moment) of discrete segment is derived by using the relations between the forces and displacements of the segment ends, and finally, the dynamic equilibrium equations for each investigated lump are established by considering the longitudinal eccentricity appearing between the discrete node and the centroid of investigated lump. The stability condition of the method has been derived and given. The algorithm is implemented by using in turn in time domain the dynamic equilibrium equations, the calculating formulae of the median internal forces of segment as well as the acceleration relationship between the centroid and discrete node of investigated lump.The validity of the investigated lump method is confirmed by comparing the bending wave velocity, the response of bending wave propagation in beam, and the internal force analysis of cantilever beam under static loading. It is also confirmed to be valid for the investigated lump method to consider the longitudinal eccentricity in structure by studying the dynamic responses of variable cross-section circle ring beam and the earthquake responses of equal cross-section chimney respectively. In addition, the numerical results also confirmed that the investigated lump method has compensation function. That is to say, the method can provide numerical results of high accuracy even in the case of non-uniform discretization in practical calculation.The earthquake responses of Nanjing TV tower show that the influence of vertical eccentricity in high-rise structure should be considered in earthquake response analysis because the stress values of the dangerous cross-section of the steel mast obtained by using the proposed method arc bigger than those given by the conventional finite element method without considering vertical eccentricity. The earthquake response of a24-storey frame structure shows that the investigated lump method can be used to study the elastoplastic earthquake response of high-rise structure.The dynamic equation of matrix form of the investigated lump method has also been given with the longitudinal eccentricities being considered in the dynamic equation for the shear-bending MDOF mechanical model. In fact, the conventional mass matrix and stiffness matrix have been modified when considering the longitudinal eccentricities.2. The investigated lump method is presented for analyzing earthquake responses of plane frame structures based on wave propagation approach. Firstly, the half segments connecting one of the spatial discrete nodes are used to construct one investigated lump in the plane frame structure, and the dynamic equilibrium equations of the investigated lump arc established about its centroid. Secondly, the calculating formulae of the median internal forces (axial force, shear force and bending moment) of one spatial discrete segment are derived in local coordinate system of two dimensions. Finally, the algorithm is implemented by using in turn in time domain the dynamic equilibrium equations of investigated lumps, the calculating formulae of median internal forces, transforming relations of the median internal forces and the displacements between the local and global coordinate systems of two dimensions, and the acceleration relation between the discrete node and the corresponding centroid of investigated lump.The validity of the investigated lump method for plane frame structures is confirmed by comparing the numerical results obtained by the investigated lump with the corresponding results of finite element method as well as the results of MIDAS software for a three-span6-storey RC frame structure under the uniform excitations and wave passage excitations respectively.The wave passage effects of four plane frame structures (including a three-span6-storey RC frame structure, a three-span3-storey cast-in-place slab RC frame structure, a three-span6-storey cast-in-place slab RC frame structure, and a two-span10-storey steel-concrete composite structure) are studied on site classes Ⅰ, Ⅱ, Ⅲ and Ⅳ respectively. The computing results demonstrated that the wave passage effect should not be overlooked even for short-span frame structure. The wave passage effects may cause the column shear forces and beam-end bending moments to increase remarkably, especially for the first-floor column and the beam-end facing the input wave. In addition, the horizontal deformation of frame structure increases and the deformations of the first-floor columns are non-uniform.3. The investigated lump method for analyzing the earthquake responses of three-dimensional frame structures is presented based on wave propagation approach. Firstly, the half segments that connect one of the spatial discrete nodes are used to construct one investigated lump in the three-dimensional frame structure, and the dynamic equilibrium equations of the investigated lump are established. Secondly, the calculating formulae of the median internal forces (axial force, shear force and bending moment) of one spatial discrete segment are derived in local coordinate system of three dimensions. Finally, the algorithm is implemented by using in turn in time domain the dynamic equilibrium equations of investigated lumps, the calculating formulae of median internal forces, transforming relations of the median intemal forces and the displacements between the local and global coordinate systems of three dimensions, and the acceleration relation between the discrete node and the corresponding centroid of investigated lump.The wave passage effect of a three-dimensional frame structure was studied. The numerical results showed that the wave passage effect may cause the column shear forces and beam-end bending moments to increase remarkably. It demonstrated again that the wave passage effect should not be overlooked even for short-span frame structure.