Calculation Method Of Inherent Characteristics Of Beam Element Considering Initial State And Its Application In Bridge Engineering

The natural frequencies and natural modes of the bridges are of great significance in the dynamic analysis of large span bridges.In the actual bridge engineering,the inherent characteristics of the bridge change due to initial deflections caused by the structural weight,the concentrated mass of the bridge,the load of the motor vehicle and so on.Based on the energy conservation equation and variational principle,the vibration governing equations and the analytical solution of natural frequency under the initial deflection are deduced.The variation law of the natural frequency under initial deflection and the influence of the high-span ratio and the static load size are determined.The paper analyzes the change of the inherent characteristics of the long span bridge under the action of static load,and reveals its application prospect in the large span bridge engineering field.The influence of the initial state on the inherent characteristics of the beam element is studied theoretically.According to energy conservation equation and Hamilton principle,the vibration governing equation of the beam considering the initial deflection is obtained.The boundary conditions of the simple supported beam and the fixed beam are introduced,and the analytical solution of the natural frequency and natural formation of the beam under uniform and concentrated static load are obtained by using the separation variable method and Galerkin method.The influence of span and static load on the natural frequency of beam is analyzed.According to the results of theoretical research,the natural frequencies and the natural formation of fixed beams with 5mm and 3mm thickness under initial concentrated static load are obtained.The model test of the inherent characteristics of fixed beam under concentrated static load is designed and carried out.The inherent mode of the fixed beam under concentrated static load is tested by using the LMS vibration test system.When the other parameters of the beam are constant,the initial deflection of the beam increases with the increase of the static load on the beam.The change of the geometrical configuration leads to the increase of the stiffness of the beam and the shift of corresponding vibration modes of the beams.When the high-span ratio decreases,the beam is more likely to produce large deflection,and the influence of the initial static load on the natural frequency of the beam becomes larger.Besides,the initial deflection caused by the initial static load will have certain influence on the response frequency of the fixed beam under the dynamic load excitation.With the increase of the initial deflection,the response peak of the beam will drift slightly to the high frequency.The inherent characteristics of the Minpu Bridge and the Fengpu Bridge are tested by the pulsation detection technique.The first two natural frequencies and the natural vibration modes of the Minpu Bridge and the Fengpu Bridge are obtained.The measuring points of Minpu Bridge and Fengpu Bridge are determined according to the analysis of low-order vibration modes of the bridge.The FFT method is used to analyze the data.The spectrum curve of the whole bridge is obtained from the micro tremor response signal of each point,and the inherent characteristic information is obtained.The natural frequencies of the first and second order of Minpu Bridge are 0.17 Hz and 0.48 Hz respectively,and the formation curves corresponding to the natural frequency of each stage are in good agreement with that of the theoretical analysis results,which indicates that the test results are good.The finite element analysis and numerical calculation method are used to study the natural frequency of Minpu Bridge with considering and not considering the static load.Two-dimensional finite element model of Minpu Bridge is established by using beam element and truss element system.The natural frequencies of the first and second order of Minpu Bridge are 0.12 Hz and 0.33 Hz,respectively,without considering the static load.The Minpu Bridge is simplified as a fixed beam with uniform static load and concentrated static load on the beam.Compared with the natural frequency of Minpu Bridge field test,which considers the static load,the structural rigidity of the Minpu Bridge is increased due to the static load on the beam,the first and second natural frequencies of the Minpu Bridge are increased by 42% and 45% respectively.It can be concluded that the initial deflection produced by the static load has a significant effect on the natural frequency of the long-span bridge.In the analysis of dynamic characteristics of long-span bridges,the influence of initial state on natural frequencies and natural modes should be fully considered.