Analysis Of Prestressing Effect On Natural Frequency For Simply Supported Beam

Prestressed bridges are widely used in China, however, the effective prestress detectionand identification for the bridges after completion has always been a problem. The bridgenatural frequencies have a certain sensitivity to the prestress values, so applying the bridgedynamic characteristics into the effective prestress identification is a valuable researchdirection. At present, the mechanism that prestress can change the bridge dynamiccharacteristics is not yet very clear. Accordingly, using the bridge characteristics to identifythe effective prestress has not reached the practical state. Considering the above situation,this paper conducts researches on the mechanism of the effects that prestress has on thebridge dynamic characteristics. Relying on the National High Technology Research andDevelopment Program of China (863Program)“Research on the System for Monitoring,Identification and Early-warning of Wide Range Road Hazards in Seasonal Frozen SoilRegion”(2009AA11Z104) and the Ph.D. Program Foundation of Ministry of Education ofChina “Research on the Calculation Method for Natural Frequencies of Prestressed ConcreteBeams and the Effective Prestress Detection Technique”, this paper uses the elastic dynamicmethod, nonlinear dynamic method and the finite element method to conduct researches onthe dynamic characteristics of three different types of prestressed structures, which areprestressed beams with pressure at both ends and no contact in the middle, simply supportedbonded prestressed beams and external prestressed simply supported beams. In analysis, theprestress values, the layout of prestressed steels, geometrical non-linearity and materialnonlinear behavior, all these four factors are considered. Then it discusses the mechanism ofthe effects of each factor on the natural frequencies of the simply supported prestressedbeams.The main innovation of this paper is to introduce the nonlinear dynamic theory into thedynamic analysis of the simply supported prestressed beam. And use this theory to deducethe dynamic equations of three different prestressed structures and calculate the approximatesolutions of natural frequencies which are testified by the model experiment; deduce thenatural frequency calculation formula of unbonded simply supported prestressed beam; inthe dynamic analysis of the unbonded prestressed concrete beams, consider the effects of the change in concrete elastic modulus on the natural frequencies of the prestressed concretebeam under the initial static loads; provide a method for analyzing the dynamiccharacteristics of the external prestressed simply supported beams by means of the finiteelement theory and detecting the effective prestress of the external prestressed tendons bymeans of frequency method. The main researches of this paper and the achieved results areas follows:(1) Considering the case of geometrical non-linearity, it analyzes the free vibration ofthe simply supported beams by nonlinear dynamic method, deduces and solves theapproximate solutions of the natural frequencies. The results show that, considering thegeometrical non-linearity, the free lateral vibration equation of the simply supported beamunder axial pressure is a nonlinear partial differential equation with a time-varying parameter.Regardless of the vertical inertial of the beam, the free lateral vibration equation of the beamunder axial pressure is an autonomous system with cubic term and quintic term. Solve theequation with Galerkin method and obtain the approximate solution of the first order naturalfrequency of the simply supported beam under axial pressure in the case of geometricalnon-linearity. The form of the approximate solutions shows that the first order free vibrationfrequencies of the beam are affected by the vibration amplitudes. The dynamic test of thesimply supported prestressed steel beam anchored at both ends testifies the practicability ofthe approximate solutions of the first order free vibration frequencies of the simplysupported beam under the axial pressure considering the geometrical non-linearity. Itanalyzes the instantaneous frequencies of the test beam by means of HHT method. And theresults show that, the natural frequencies of the simply supported beam under axial pressureare time-varying, which are affected by the vibration amplitude values and the axial forcevalues.(2) It deduces the free vibration equations of the three kinds of unbonded prestressedsimply supported beams, which respectively are with axial liner tendon distribution, witheccentric liner tendon distribution and with curved tendon distribution, and solves the freevibration frequencies. Analysis results show that, regardless of the geometrical non-linearity,the natural frequencies of the simply supported prestressed beam in linear tendon distributionhave nothing to do with the prestress values, but have something to do with the layout.However, to the simply supported prestressed beam in curved tendon distribution, the naturalfrequencies are not only affected by the prestress values, but also affected by the layoutshape and layout position. The first order natural frequencies increase with the eccentric moments of the prestressed tendons. Taking the geometrical non-linearity into consideration,the natural frequencies of the bonded prestressed simply supported beams are affected by theprestress values, the eccentric moments of the prestressed tendons, the prestressed tendonlayout shapes and the structural vibration amplitudes. And the natural frequencies increasewith the prestress values.(3) Referring to the research results of concrete elastic modulus changes under initialstatic loads at home and abroad, in the dynamic analysis of the simply supported prestressedconcrete beams, it considers the effects of the changes in the concrete elastic modulus causedby the prestress on the dynamic characteristics. Through a numerical example, it analyzesthe influence extent of the changes in elastic modulus caused by the prestress on the naturalfrequencies of the simply supported prestressed beams. Results show that, the concreteelastic modulus increases with the prestress, which increases the natural frequencies of thesimply supported prestressed concrete beam. Because elastic modulus increases with theprestress in nonlinearity, the natural frequency of the simply supported prestressed concretebeam increases in nonlinearity.(4) It analyzes the vibration characteristics of the external prestressed simplysupported beam by means of the finite element method. Results show that, in the conditionof arranging the prestress along the section gravity center of the steel beam, the fundamentalfrequency of the steel beam increases with the prestress in nonlinearity. Under differenteccentric moments of the prestressed tendons, the first order frequencies of the steel beamincrease with the prestress. With the same prestress, the first order frequencies of the steelbeam increase with eccentric moments of the prestressed tendons. Based on a practicalproject, it analyzes the application of the frequency method in the effective prestress test ofthe external prestressed bridges. And it analyzes the effects of the boundary condition of theexternal prestress on the natural frequencies. Results show that, the first order frequencyvalue in fixed boundary condition is higher than that in the simply supported boundarycondition, and the relative difference between them decreases with the tensions of theprestressed tendons; with the same tension, the difference between them increases with thedecrease of the cable length. In the practical project test, we can only obtain the maximumand the minimum values of the effective prestress based on the measured frequencies of theprestressed tendons, however, the actual effective frequency values are between them.