Reverberation-Ray Analysis Of Inhomogeneous Elastodynamic Equations And Nondestructive Inspection Of Structures

Dynamics of structures composed of a large number of structural members connected by joints has been analyzed traditionally by modeling all structural members as a continuous system of rods and beams or by a discrete system of many mass particles with weightless elastic connectors.Due to recent development of large span bridges,high-rise framed buildings and lightweight large span aerospace structures subject to variety of dynamic loadings,there is revival of interest in rational analysis of a structure as continuum.Based on one-dimensional elastic theory,the reverberation-ray matrix analysis(RRM) was developed in 1998,and has been successfully applied to analyze transient wave in planar trusses,layered media and three-dimensional framed structures.The newly developed method could be an alternative to finite element method(FEM) or other discrete system analysis,and is particularly effective to determine precisely early-time transient responses.So far all papers on this method are limited to steady-state and transient responses of concentrated loads applied at joints or at one or several points of a structural member.In this thesis,we extend the method of RRM to investigate the dynamics of a structure subject to distributive loads or moving forces on surface-chord members of the structure, and conduct detailed evaluations and experiments of transient wave responses in a multi-bay structure subject to both types of loading.Axial and flexural wave motions of the surface-chord members of multi-bay framed structures or a multi-span continuous beam are governed respectively by standard wave equation and Timoshenko beam equation in coupled bending and shear wave modes with inhomogeneous term of loading functions.The complete solution for each equation is obtained by superposing complementary solution for the homogeneous part and particular solution for the inhomogeneous part of the equation.The latter includes a source function representing distributive load or moving force.A concentrated load is treated as a distributive load with a delta function in space and moving force is represented by a delta function with travelling phase in space x and time t.The double Fourier transform in x and t is applied to obtain the particular solution in transformed domains,which are then combined with complementary solutions to satisfy the boundary conditions at both joints.The method of RRM reduces a two points boundary value problem in single coordinate axis for each member to a one point boundary value problem in dual coordinate (two opposite axes),which simplifies greatly the determination of unknown coefficients in the general solution from complicated boundary conditions.The transient solution is then obtained by double inverse transform with respect to the wave numberλand the frequency parameterω.Carrying out the inverse transform inλfirst,we obtain the steady-state wave solution of the structural member excited by the distributive loads oscillating at a single frequency.This solution contains amplitude factor in x andω,which is a convolution of distributive loading function shifted by a spatial parameter s.For the case of stationary distributive loading,the convolution integral in s is carried out first and the inverse Fourier transform inωis performed by numerical integration as reported in previous papers.For moving distributive loading,the order of integrations in s andωmust be revised so as to avoid the artificially introduced singularities.Such a modification circumvents the difficulty of having singularities at critical moving load speeds in transient responses obtained by expansion in normal modes.In this thesis,we have applied afore mentioned matrix analysis and proper order of inverse Fourier transform to determine the transient response of a cracked structure under the action of stationary and moving concentrated load.The crack is modeled by discontinuity of rotational angle of the defective cross-section,characterized by several geometric parameters and a rotational spring constant.The precisely determined transient responses recorded at a point is then analyzed by the newly developed Hilbert-Huang transform(HHT).From the time spectrum of HHT,we can then identify the location of the crack,which could be developed into a nondestructive testing technique for detecting a crack locally.It is known that the natural frequencies of a continuous beam could be determined from the Fourier frequency spectrum of a transient wave record generated by a moving force.We have also calculated the transient response of a continuous beam or framed structure subject to a moving force,and then determine the location of a crack from the time spectrum of HHT.It could be developed into another global nondestructive testing technique.In the meantime,we have conducted experiments on a simply supported beam with and without the defect.Transient wave generated by a concentrated impact loading are recorded with strain gauges and broadband electronic instruments.The recorded signals and its time spectrum of HHT are compared with theoretical results with very good agreements.The experimental investigation substantiates the proper modeling of a crack in a beam and the effectiveness of time spectrum analysis of HHT for detecting the location of the defect.