大跨度桥梁钢箱梁加劲板的动力行为研究
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摘要
钢箱梁具有高度低、自重轻、极限承载力大、易于加工制造且结构连续等特点，从长远看比较经济，因此在大跨度桥梁中得到普遍应用。本文在总结前人研究成果和各国设计规范的基础上，针对钢箱梁加劲板的结构特点，通过理论推导与数值分析相结合的方法，较为系统地研究了钢箱梁加劲板的动力特性，主要完成以下工作：
(1)运用能量法分析钢箱梁加劲板的线性振动，分析中包括三点假设：第一，将钢箱梁加劲板视为双向加劲板处理，纵向加劲肋与横隔板均按质量与刚度进行等效；加劲肋视为梁单元，同时考虑梁的扭转对横向振动的影响；母板按经典薄板理论计算，不计扭转的影响。第二，考虑加劲肋的偏心，同时考虑母板的膜应变能。第三，板的振型函数用两个独立的梁的振型函数的乘积表示。
(2)运用组合板梁单元法分析钢箱梁加劲板的线性振动，针对梯形肋加劲板的结构特点，构建了组合板梁单元这一特殊的单元，其中顶板作为结构的基本部分，按平板壳单元分析，而梯形加劲肋作为结构的附属部分，各个板件作为板梁子单元进行分析。通过能量变分原理分别推导出组合板梁单元的刚度矩阵、一致质量矩阵与一致荷载列阵，通过编制出相应的有限元计算程序，求解梯形肋加劲板的局部振动。
(3)针对四边简支加劲板，提出了动态屈曲临界荷载的求解方法，运用Hamilton原理建立加劲板动态屈曲特征方程。分析中考虑初始几何缺陷的影响，并讨论了初始几何缺陷、加劲肋的数量及其刚度的变化对动态屈曲临界荷载的影响。
(4)基于能量原理确定母板与肋的应变能与动能，然后运用Lagrange方程推导出加劲板的非线性振动微分方程。运用单模态方法研究了四边简支、四边固定与移动边界三种情况下加劲板的非线性振动。对于自由振动，通过对非线性微分方程进行积分，并利用初始条件直接求得单模态自由振动的解析解。通过数值算例分析了四边简支与四边固定的加劲板自由振动前几阶模态的非线性特征，并通过给定不同的加劲肋布置情况分别分析了振幅与非线性自振频率的关系。对于四边简支与四边固定加劲板的受迫振动，运用多尺度法求得单模态系统非共振与主共振的一次近似解。并通过数值算例讨论了两个方向设置不同数量加劲肋的情况下加劲板的非共振稳态响应与主共振的幅频响应。对于具有动边界加劲板的非线性振动，将动边界条件的效应转化为等效激励，板的阻尼视为粘弹性类型。通过参数算例分析，讨论了两个方向设置不同数量加劲肋的情况下加劲板的非共振稳态响应与主参数共振-主共振的幅频响应。
(5)研究了四边简支与四边固定加劲板在主共振激励作用下的双模态非线性动力响应。将板的阻尼视为粘弹性类型，加劲板的运动方程通过Lagrange方程建立，通过设定振型函数将运动方程简化为双自由度。考虑3:1内共振情况，利用多尺度法分析加劲板的双模态运动，同时结合数值算例讨论了两个方向设置不同数量加劲肋的情况下加劲板的主共振稳态响应与幅频关系。
(6)研究了四边简支与四边固定加劲板在面内周期激励作用下的动力稳定性，运用Hamilton原理建立加劲板的参数振动控制方程，选取第(1,1)阶模态进行分析，将控制方程转换成Mathieu-Hill方程，并利用傅立叶级数进行求解。最后，给定参数讨论了加劲肋的数量与刚度变化对加劲板动力不稳定区域的影响。
Due to the low structural height, light self-weight, great load capacity, and easymanufacture of the steel box girder, it is used widely in long-span bridges. On the basis ofprevious research and design code of some countries and combining the construction featuresof stiffened plates of steel box girder, the dynamic behavior of stiffened plates of steel boxgirder is investigated through both theoretical derivation and numerical analysis. The mainresearch work in this thesis is as follows:
(1) Energy principle is used to investigate the linear vibration of the stiffened plate ofsteel box girder. In this work, the basic assumption are as follows: first, it is considered to be astiffened plate in double directions; both longitudinal stiffeners and transverse diaphragms areconsidered to be beam elements according to the equivalent principle of mass and rigidity, andthe torsional effect is taken into account; the plate is computed according to the classical thinplate theory without the torsional effect taken into account. Second, the eccentricity ofstiffeners is taken into consideration, as well as the membrane strain energy of the plate. Third,the mode shape function of the plate is expressed by the product of two independent beamfunctions.
(2) A combined plate beam element method is presented to investigate the localvibration of the steel bridge deck with trapezoidal stiffeners. The top plate is taken as shellelement. The trapezoidal stiffener is taken as plate beam element, and its displacement modelis built according to the deformation coordinate relationship between plate and stiffener. Thestiffness matrix, consistent mass matrix and consistent load matrix of the combined platebeam element are obtained based on energy-variation principle. The local vibration of thesteel bridge deck with trapezoidal stiffeners can be analyzed through finite element program.
(3) An approach is presented to study dynamical buckling of stiffened plates with fouredges simply supported. The Hamilton principle and modal superposition method are used toderive the eigenvalue equations of the stiffened plate according to energy of the system. Theinitial geometrical imperfection is considered in the equations. Detailed discussion on how theinitial geometrical imperfection, the number and the flexural rigidity of stiffeners influencethe critical load is carried out.
(4)The strain and kinetic energy of both the plate and stiffeners are established, and thenLagrange equation is used derive the governing equation of motion. Single-modal method ispresented to investigate the nonlinear vibration of stiffened plates with four edges simplysupported, four edges clamped and moving boundary conditions. For the free vibration, the exact single-mode solution can be obtained according to the integral of nonlinear differentialequations and the initial conditions. For the stiffened plates with four edges simply supportedand four edges clamped, the relationship between nonlinear natural frequency and itsamplitude is discussed with the number of stiffeners in the two directions varying. For thenonlinear forced vibration of stiffened plates with four edges simply supported and four edgesclamped, the first approximation solutions of the non-resonance and the primary resonance ofthe single-mode system are obtained by means of the method of multiple scales. Numericalexamples for different stiffened plates are presented to discuss the steady response of thenon-resonance and the amplitude-frequency response of the primary resonance. For thenonlinear vibration of stiffened plates with moving boundary conditions, the effect caused byboundary movement is transformed into equivalent excitations and the damping of the plate istaken into account as viscoelastic damping. Numerical examples for different stiffened platesare presented to discuss the steady response of the non-resonance and theamplitude-frequency response of the primary parametric resonance and primary resonance.
(5) Double-modal nonlinear dynamic response of stiffened plates is investigated withfour edges simply supported and four edges clamped. The damping of the plate is taken intoaccount as viscoelastic damping.The governing equations of motion, which are derived byusing the Lagrange equation, are reduced to a two-degree-of-freedom nonlinear system byassuming mode shapes. Three-to-one internal resonance is taken into consideration, and themethod of multiple scales is used to investigate the double-modal motion. Finally, numericalexamples for different stiffened plates are presented to discuss the steady response andamplitude-frequency response of the primary resonance.
(6) The dynamic stability of stiffened plates under in-plane periodic excitation isinvestigated with four edges simply supported and four edges clamped. The governingequation of parametric vibration is derived through Hamilton principle. Selecting the (1,1)thmode, the governing equation is converted into Mathieu-Hill equation, which can be sovledthrough Fourier series. Finally, numerical examples for different stiffened plates are presentedto discuss how the number and rigidity of stiffeners influence the unstable region.
(1)运用能量法分析钢箱梁加劲板的线性振动，分析中包括三点假设：第一，将钢箱梁加劲板视为双向加劲板处理，纵向加劲肋与横隔板均按质量与刚度进行等效；加劲肋视为梁单元，同时考虑梁的扭转对横向振动的影响；母板按经典薄板理论计算，不计扭转的影响。第二，考虑加劲肋的偏心，同时考虑母板的膜应变能。第三，板的振型函数用两个独立的梁的振型函数的乘积表示。
(2)运用组合板梁单元法分析钢箱梁加劲板的线性振动，针对梯形肋加劲板的结构特点，构建了组合板梁单元这一特殊的单元，其中顶板作为结构的基本部分，按平板壳单元分析，而梯形加劲肋作为结构的附属部分，各个板件作为板梁子单元进行分析。通过能量变分原理分别推导出组合板梁单元的刚度矩阵、一致质量矩阵与一致荷载列阵，通过编制出相应的有限元计算程序，求解梯形肋加劲板的局部振动。
(3)针对四边简支加劲板，提出了动态屈曲临界荷载的求解方法，运用Hamilton原理建立加劲板动态屈曲特征方程。分析中考虑初始几何缺陷的影响，并讨论了初始几何缺陷、加劲肋的数量及其刚度的变化对动态屈曲临界荷载的影响。
(4)基于能量原理确定母板与肋的应变能与动能，然后运用Lagrange方程推导出加劲板的非线性振动微分方程。运用单模态方法研究了四边简支、四边固定与移动边界三种情况下加劲板的非线性振动。对于自由振动，通过对非线性微分方程进行积分，并利用初始条件直接求得单模态自由振动的解析解。通过数值算例分析了四边简支与四边固定的加劲板自由振动前几阶模态的非线性特征，并通过给定不同的加劲肋布置情况分别分析了振幅与非线性自振频率的关系。对于四边简支与四边固定加劲板的受迫振动，运用多尺度法求得单模态系统非共振与主共振的一次近似解。并通过数值算例讨论了两个方向设置不同数量加劲肋的情况下加劲板的非共振稳态响应与主共振的幅频响应。对于具有动边界加劲板的非线性振动，将动边界条件的效应转化为等效激励，板的阻尼视为粘弹性类型。通过参数算例分析，讨论了两个方向设置不同数量加劲肋的情况下加劲板的非共振稳态响应与主参数共振-主共振的幅频响应。
(5)研究了四边简支与四边固定加劲板在主共振激励作用下的双模态非线性动力响应。将板的阻尼视为粘弹性类型，加劲板的运动方程通过Lagrange方程建立，通过设定振型函数将运动方程简化为双自由度。考虑3:1内共振情况，利用多尺度法分析加劲板的双模态运动，同时结合数值算例讨论了两个方向设置不同数量加劲肋的情况下加劲板的主共振稳态响应与幅频关系。
(6)研究了四边简支与四边固定加劲板在面内周期激励作用下的动力稳定性，运用Hamilton原理建立加劲板的参数振动控制方程，选取第(1,1)阶模态进行分析，将控制方程转换成Mathieu-Hill方程，并利用傅立叶级数进行求解。最后，给定参数讨论了加劲肋的数量与刚度变化对加劲板动力不稳定区域的影响。
Due to the low structural height, light self-weight, great load capacity, and easymanufacture of the steel box girder, it is used widely in long-span bridges. On the basis ofprevious research and design code of some countries and combining the construction featuresof stiffened plates of steel box girder, the dynamic behavior of stiffened plates of steel boxgirder is investigated through both theoretical derivation and numerical analysis. The mainresearch work in this thesis is as follows:
(1) Energy principle is used to investigate the linear vibration of the stiffened plate ofsteel box girder. In this work, the basic assumption are as follows: first, it is considered to be astiffened plate in double directions; both longitudinal stiffeners and transverse diaphragms areconsidered to be beam elements according to the equivalent principle of mass and rigidity, andthe torsional effect is taken into account; the plate is computed according to the classical thinplate theory without the torsional effect taken into account. Second, the eccentricity ofstiffeners is taken into consideration, as well as the membrane strain energy of the plate. Third,the mode shape function of the plate is expressed by the product of two independent beamfunctions.
(2) A combined plate beam element method is presented to investigate the localvibration of the steel bridge deck with trapezoidal stiffeners. The top plate is taken as shellelement. The trapezoidal stiffener is taken as plate beam element, and its displacement modelis built according to the deformation coordinate relationship between plate and stiffener. Thestiffness matrix, consistent mass matrix and consistent load matrix of the combined platebeam element are obtained based on energy-variation principle. The local vibration of thesteel bridge deck with trapezoidal stiffeners can be analyzed through finite element program.
(3) An approach is presented to study dynamical buckling of stiffened plates with fouredges simply supported. The Hamilton principle and modal superposition method are used toderive the eigenvalue equations of the stiffened plate according to energy of the system. Theinitial geometrical imperfection is considered in the equations. Detailed discussion on how theinitial geometrical imperfection, the number and the flexural rigidity of stiffeners influencethe critical load is carried out.
(4)The strain and kinetic energy of both the plate and stiffeners are established, and thenLagrange equation is used derive the governing equation of motion. Single-modal method ispresented to investigate the nonlinear vibration of stiffened plates with four edges simplysupported, four edges clamped and moving boundary conditions. For the free vibration, the exact single-mode solution can be obtained according to the integral of nonlinear differentialequations and the initial conditions. For the stiffened plates with four edges simply supportedand four edges clamped, the relationship between nonlinear natural frequency and itsamplitude is discussed with the number of stiffeners in the two directions varying. For thenonlinear forced vibration of stiffened plates with four edges simply supported and four edgesclamped, the first approximation solutions of the non-resonance and the primary resonance ofthe single-mode system are obtained by means of the method of multiple scales. Numericalexamples for different stiffened plates are presented to discuss the steady response of thenon-resonance and the amplitude-frequency response of the primary resonance. For thenonlinear vibration of stiffened plates with moving boundary conditions, the effect caused byboundary movement is transformed into equivalent excitations and the damping of the plate istaken into account as viscoelastic damping. Numerical examples for different stiffened platesare presented to discuss the steady response of the non-resonance and theamplitude-frequency response of the primary parametric resonance and primary resonance.
(5) Double-modal nonlinear dynamic response of stiffened plates is investigated withfour edges simply supported and four edges clamped. The damping of the plate is taken intoaccount as viscoelastic damping.The governing equations of motion, which are derived byusing the Lagrange equation, are reduced to a two-degree-of-freedom nonlinear system byassuming mode shapes. Three-to-one internal resonance is taken into consideration, and themethod of multiple scales is used to investigate the double-modal motion. Finally, numericalexamples for different stiffened plates are presented to discuss the steady response andamplitude-frequency response of the primary resonance.
(6) The dynamic stability of stiffened plates under in-plane periodic excitation isinvestigated with four edges simply supported and four edges clamped. The governingequation of parametric vibration is derived through Hamilton principle. Selecting the (1,1)thmode, the governing equation is converted into Mathieu-Hill equation, which can be sovledthrough Fourier series. Finally, numerical examples for different stiffened plates are presentedto discuss how the number and rigidity of stiffeners influence the unstable region.
引文
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