Calculation Method For Dynamic Characteristics And Dynamic Response Of Nonuniform Timoshenko Beam Structures

Dynamic characteristics have been widely used in damage identification and state assessment of civil engineering structures,most of which are based on the differences between theoretical values and test values of dynamic characteristics.Due to the complexity of civil engineering structures,there are essential differences between analysis mechanism of static and dynamic performances and failure mode under static and dynamic loads.Traditional static analysis methods can not fully meet the requirements of complex engineering applications.Furthermore,civil engineering structures would inevitably be subjected to dynamic loads such as traffic vehicles,wind loads,earthquakes and so on.Therefore,it is of great significance to study the calculation methods for dynamic characteristics and dynamic response of structures.In the field of civil engineering,many structures or components can be regarded as beam structures,such as bridge,rail,immersed tunnel,etc.In engineering,beam structures are usually divided into Euler-Bernoulli beams and Timoshenko beams by the ratio of height to span(called slenderness ratio in material mechanics)and its essence is whether the influence of shear effect on mechanical characteristics of beam structures can be neglected.For example,in the field of bridge engineering,the main girders and cap beams of bridges with large height-span ratio belong to Timoshenko beams.However,with the deepening of understanding of the mechanical characteristics of complex beam structures,it is found that Timoshenko beam theory is more reasonable than Euler-Bernoulli beam at complex support conditions,even for beam structures with small height-span ratios.Even under simple support conditions,in order to improve the calculation accuracy of dynamic characteristics or dynamic response of beam structures with small height-span ratio by means of numerical methods such as finite element etc.,beam structures are usually refined into sub-beam elements with small length.Due to sub-beam element with small length,it could be regarded as Timoshenko beam.Therefore,Timoshenko beam theory has broad application prospects in the field of bridge engineering,even the civil engineering field.The mechanical properties of the nonuniform Timoshenko beam structure are complicated,which needs to be modified many times in the design process to achieve satisfactory results.How to quickly calculate the dynamic characteristics and dynamic responses in the design process is a technical problem that puzzles the engineering technicians.It is necessary to analyze the dynamic characteristics and dynamic response accurately and efficiently after the structure is put into use,in order to provide the reference value(theoretical calculation value)for the state evaluation method,which is also an urgent technical problem for engineering technicians.In order to solve dynamic characteristics and dynamic response of nonuniform Timoshenko beam structures,this paper was supported by National Natural Science Foundation of China(NSFC)“Research on natural frequency analysis method for the small and medium span beam bridges considering the coupling effect of vehicles and temperature”,and the specific works carried out are as follows:1.Aiming at nonuniform Timoshenko beam structures in the process of design and modification,a fast calculation method of dynamic characteristics has been developed by combining finite element method with matrix perturbation theory.The method considered that the nonuniform Timoshenko beam was based on uniform Timoshenko beam and modified many times.Firstly,the finite element method was used to calculate the dynamic characteristics of uniform Timoshenko beams.Secondly,the changes of structural mass matrix and stiffness matrix after each design and modification could be calculated.Finally,the dynamic characteristics of the structure after each modification were calculated by matrix perturbation theory based on the change of mass matrix and stiffness matrix.2.Aiming at nonuniform Timoshenko beam structures for use,the calculation method of dynamic characteristics was established based on transfer matrix method.Firstly,the free vibration equation of nonuniform Timoshenko beam represented by transfer matrix differential equation of undetermined coefficients was established based on the dynamic equilibrium equation of nonuniform Timoshenko beam.Secondly,according to the equilibrium and deformation continuity of shear and bending moments at the middle support and boundary conditions,the dynamic characteristic equation of multi-span Timoshenko beam supported by elastic supports was established.Finally,the Runge-Kutta method and iteration technique were used to solve the dynamic characteristic equation.3.Aiming at the characteristics of nonuniform Timoshenko beams,two key problems in the dynamic response solution by modal superposition method can be solved,i.e.,mode orthogonality and mode function continuity.Then the dynamic response analysis method of Timoshenko beams with elastic supports under spring-mass system was formed.Firstly,based on free vibration equation of nonuniform Timoshenko beams,the orthogonality of mode shapes of nonuniform Timoshenko beams was deduced.Secondly,the continuous mode functions were realized by cubic spline curve interpolation method.Finally,according to the interaction between spring-mass systems and Timoshenko beams,the forced vibration model of nonuniform Timoshenko beams under spring-mass system was established...