简支梁自振频率的预应力效应分析
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摘要
预应力桥梁在我国应用广泛，而竣工后桥梁中的有效预应力检测与识别一直是个难题。桥梁自振频率对预应力值具有一定的敏感性，利用桥梁的动力特性进行有效预应力识别是一个十分有价值的研究方向。由于预应力改变桥梁动力特性的机理还不十分明确，利用桥梁的动力特性进行有效预应力识别还未达到实用的程度。为此，本文针对预应力影响桥梁动力特性的机理这一问题开展研究。依托863国家高科技研究发展计划项目“季节冻土区大范围道路灾害参数监测与辨识预警系统研究”（2009AA11Z104）和高等学校博士点学科基金项目“预应力混凝土梁自振频率计算方法及有效预应力检测技术研究”项目，利用弹性动力学方法、非线性动力学理论、有限元方法，对两端锚固中间无接触预应力简支梁、无粘结预应力简梁和体外预应力简支梁等三种不形式预应力结构的动力特性进行了理论分析和试验研究。分析中考虑了预应力值大小、预应力筋布置形式、几何非线性效应、材料非线性效应等四种因素。对各因素影响预应力简支梁自振频率的机理进行探讨。
本文的主要创新在于将非线性动力学理论引入到预应力简支梁的动力分析当中，利用此理论推导了预应力简支梁的动力方程，并给出了自振频率的近似解，利用模型试验进行了验证；推导了不同布筋形式下的无粘结预应力简支梁的自振频率计算公式；在对无粘结预应力混凝土梁的动力分析中，考虑了初始静载作用下混凝土弹性模量改变对预应力混凝土梁自振频率的影响；提出了一种利用有限元方法分析体外预应力简支梁这种耦合振动体系动力特性的方法，介绍了利用频率法检测体外预应力筋有效预应力的方法。本文主要的研究工作及取得的成果如下：
（1）在考虑几何非线性情况下，利用非线性动力学方法对轴向受压简支梁进行了自由振动分析，推导并求解了自振频率的近似解。结果表明：考虑几何非线性效应的受轴向压力作用简支梁的横向自由振动方程是带有时变参数的非线性偏微分方程，在忽略梁的纵向惯性效应情况下，受轴向压力作用简支梁的横向自由振动方程为带三次项和五次项的自治系统。利用Galerkin法对方程进行求解，得出了考虑几何非线性效应的受轴向压力作用简支梁的1阶自振频率近似解。近似解的结构形式表明，梁的1阶自由振动频率将受到振动幅值的影响。通过对两端锚固的预应力简支钢梁进行动力试验，验证了考虑几何非线性效应的受轴向压力作用简支梁的1阶自振频率近似解有效性。利用HHT方法分析了试验梁的瞬时频率，结果表明，受轴向压力作用简支梁的固有频率是一个时变量，其值受到振动幅值和轴向作用力大小的影响。
（2）对轴心直线布筋、偏心直线布筋和曲线布筋三种无粘结预应力简支梁的自由振动方程进行了推导，对自振频率进行了求解。考虑几何非线性效应后，轴心直线布筋简支梁的简支梁的一阶自振频率会随着预应力值的增大而增大，增大程度受到位移振幅的影响。对于偏心直线布置的预应力简支梁，在同样的张拉力情况下，自振频率会随着偏心距的增大而增大。在同样偏心距的情况下，频率会随着预应力值的增大而增大，偏心距越大，增大趋势越明显。曲线布筋的无粘结预应力简支梁的频率会随着预应力值的增大而增大。
（3）参考国内外对受初始静载作用下混凝土弹性模量变化的研究成果，在预应力混凝土简支梁的动力分析中，考虑了预应力引起的混凝土弹性模量的改变对动力特性的影响。通过数值算例，分析了预应力引起弹性模量的改变对预应力简支梁自振频率的影响程度。得到三种无粘结预应力混凝土梁的修正后频率解析式。利用数值算例分析了1阶频率随预应力大小的改变规律。结果表明，在考虑材料非线性后，三种无粘结预应力混凝土梁的1阶自振频率均会随着预应力值的增大而增大，增大趋势比不考虑材料非线性的情况更加明显。
（4）利用耦合振动分析方法和有限元理论，分析了体外预应力简支梁的自振特性。分析结果表明，沿钢梁截面形心布置预应力筋情况下，钢梁的基频随预应力值的增大呈现非线性增大的关系；采用不同偏心距布置预应力筋情况，钢梁一阶频率均随预应力值的增大而增大；在预应力值相同的情况下，随着预应力筋布置偏心距的增大，钢梁一阶频率增大。以工程实际为例，对频率法在体外预应力桥梁有效预应力测试当的应用进行了分析。分析了体外预应力边界条件对其固有频率的影响。结果表明，固支边界条件下一阶频率比铰支边界大，二者的相对差随着预应力筋的张拉力的增大而减小；而在张拉力相同的情况下，二者的相对差则随着索长的减小而增大。在实际工程测试当中利用实测的预应力筋频率仅能得到有效预应力的上、下限值，实际有效预应力值介于二者之间。
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.
本文的主要创新在于将非线性动力学理论引入到预应力简支梁的动力分析当中，利用此理论推导了预应力简支梁的动力方程，并给出了自振频率的近似解，利用模型试验进行了验证；推导了不同布筋形式下的无粘结预应力简支梁的自振频率计算公式；在对无粘结预应力混凝土梁的动力分析中，考虑了初始静载作用下混凝土弹性模量改变对预应力混凝土梁自振频率的影响；提出了一种利用有限元方法分析体外预应力简支梁这种耦合振动体系动力特性的方法，介绍了利用频率法检测体外预应力筋有效预应力的方法。本文主要的研究工作及取得的成果如下：
（1）在考虑几何非线性情况下，利用非线性动力学方法对轴向受压简支梁进行了自由振动分析，推导并求解了自振频率的近似解。结果表明：考虑几何非线性效应的受轴向压力作用简支梁的横向自由振动方程是带有时变参数的非线性偏微分方程，在忽略梁的纵向惯性效应情况下，受轴向压力作用简支梁的横向自由振动方程为带三次项和五次项的自治系统。利用Galerkin法对方程进行求解，得出了考虑几何非线性效应的受轴向压力作用简支梁的1阶自振频率近似解。近似解的结构形式表明，梁的1阶自由振动频率将受到振动幅值的影响。通过对两端锚固的预应力简支钢梁进行动力试验，验证了考虑几何非线性效应的受轴向压力作用简支梁的1阶自振频率近似解有效性。利用HHT方法分析了试验梁的瞬时频率，结果表明，受轴向压力作用简支梁的固有频率是一个时变量，其值受到振动幅值和轴向作用力大小的影响。
（2）对轴心直线布筋、偏心直线布筋和曲线布筋三种无粘结预应力简支梁的自由振动方程进行了推导，对自振频率进行了求解。考虑几何非线性效应后，轴心直线布筋简支梁的简支梁的一阶自振频率会随着预应力值的增大而增大，增大程度受到位移振幅的影响。对于偏心直线布置的预应力简支梁，在同样的张拉力情况下，自振频率会随着偏心距的增大而增大。在同样偏心距的情况下，频率会随着预应力值的增大而增大，偏心距越大，增大趋势越明显。曲线布筋的无粘结预应力简支梁的频率会随着预应力值的增大而增大。
（3）参考国内外对受初始静载作用下混凝土弹性模量变化的研究成果，在预应力混凝土简支梁的动力分析中，考虑了预应力引起的混凝土弹性模量的改变对动力特性的影响。通过数值算例，分析了预应力引起弹性模量的改变对预应力简支梁自振频率的影响程度。得到三种无粘结预应力混凝土梁的修正后频率解析式。利用数值算例分析了1阶频率随预应力大小的改变规律。结果表明，在考虑材料非线性后，三种无粘结预应力混凝土梁的1阶自振频率均会随着预应力值的增大而增大，增大趋势比不考虑材料非线性的情况更加明显。
（4）利用耦合振动分析方法和有限元理论，分析了体外预应力简支梁的自振特性。分析结果表明，沿钢梁截面形心布置预应力筋情况下，钢梁的基频随预应力值的增大呈现非线性增大的关系；采用不同偏心距布置预应力筋情况，钢梁一阶频率均随预应力值的增大而增大；在预应力值相同的情况下，随着预应力筋布置偏心距的增大，钢梁一阶频率增大。以工程实际为例，对频率法在体外预应力桥梁有效预应力测试当的应用进行了分析。分析了体外预应力边界条件对其固有频率的影响。结果表明，固支边界条件下一阶频率比铰支边界大，二者的相对差随着预应力筋的张拉力的增大而减小；而在张拉力相同的情况下，二者的相对差则随着索长的减小而增大。在实际工程测试当中利用实测的预应力筋频率仅能得到有效预应力的上、下限值，实际有效预应力值介于二者之间。
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.
引文
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