曲线箱梁桥状态传递理论研究
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摘要
随着国民经济的发展、西部大开发和城市化进程的加速,各种曲线桥结构在我国已经被广泛采用。曲线梁桥线条流畅,意境生动,给人以美的享受；能够很好的适应地形、地物限制的要求,使交通线路的布置趋于合理和科学。无论从几何、美学,还是经济角度来看,曲线桥都有着重要的现实意义和良好的发展前景。但是,由于曲率的影响,曲线梁桥在平面内存在着纵横方向的耦合,在平面外存在着弯矩和扭矩的耦合,结构力学及动力特性十分复杂。随着科学技术的进步和施工机械化的发展,曲线桥的跨度越来越大。地震反应分析是评价大跨度桥梁抗震设计水平和结构安全性的一个重要方面。对大跨度桥梁进行地震反应分析时,需要考虑多点地震输入的影响已成为国内外学术界和工程界的共识。但是,由于其力学特性的复杂性和采用方法的不同以及出发点的不一致,关于曲线箱梁桥的动力特性、抗震理论和相关技术还不成熟,导致目前缺乏对曲线梁复杂动力性能全面、精确的分析,致使已建成的和正在建的曲线箱梁桥存在很多的未知隐患,有些甚至是致命性的。
因此,本文试图建立一种简洁、易懂、思路清晰、计算精度高且应用性较强的计算分析方法,为这类结构体系的分析研究提供理论依据和计算分析方法。从内容上分,本文可概括成以下几部分：
第一章详细地研究分析了曲线梁桥分析方法的研究进展和传递矩阵法的研究现状及其应用。总结了当前曲线梁分析方法及传递矩阵法的最新研究进展,并提出了传统传递矩阵法的不足之处。
第二章基于传递矩阵原理,结合曲线梁的变形微分方程的闭合解,对曲线梁桥的传递矩阵进行了认真、完善的推导。提供了曲线箱形梁场传递矩阵的具体表达式；并根据结构的振动原理,建立了曲线梁桥的振动传递矩阵理论；并研制了相应的计算机分析程序。结合结构抗震分析原理,将反应谱理论引入到传递矩阵法中,建立了曲线梁桥的抗震传递矩阵理论；并研制了相应的计算机分析程序。为传递矩阵法在曲线梁桥结构静力及动力分析中的应用提供了一条有效的途径。
第三章将指数矩阵的精细计算原理与传递矩阵法原理相结合,提出了曲线梁桥结构分析的精细传递矩阵法。根据结构的应力-应变关系、几何物理关系,对其进行了细致、完善的推导,建立了完整的精细传递矩阵理论；并研制了相应的计算机分析程序,对其进行了全面地数值分析和验证。
本文方法具有一般性,可用于任何结构的受力分析。对于复杂结构,本文建立的精细传递矩阵法的优势更为明显。
第四章基于曲线结构的温度变形规律,结合材料力学的相关内容,建立了求解曲线梁在温度荷载作用下的内力及变形的精细传递矩阵式；并研制了相应的计算机分析程序。该方法理论上可以求解任意温度场下曲线梁的内力及变形。将本文方法扩展运用与多跨曲线桥的求解,得出在变温和桥墩顶部摩擦力共同作用下多跨曲线梁的面内内力及位移的精细传递矩阵格式,并进行了相应的算例计算。
第五章将高精度的精细积分法和力学概念清晰的传递矩阵法结合起来,以微分方程和矩阵分析理论为基础,提出了一种新的精细传递矩阵形式,推导了该方法的计算公式,并用于分析曲线箱形梁桥的振动特性。与传统的传递矩阵法相比,无需对微分方程组进行求解,只需按照迭代公式进行计算,就可以得到所需要的传递矩阵。根据边界条件,运用频率搜索方法便可得到结构的自振频率；并研制了相应的计算机分析程序,进行了相应的算例计算。该法即适用于连续结构又适用于离散结构,而且可以应用于离散和连续相结合的混合结构的振动特性分析,为精确分析复杂结构的动力特性提供了一条便捷、有效途径。
第六章基于傅里叶变换和精细传递矩阵法,推导了曲线箱梁桥在水平方向和竖直方向的动力传递矩阵,建立了曲线箱梁桥在频域内地震反应分析的精细传递矩阵法。并研制了相应的计算机分析程序,进行了相应的算例分析。不仅丰富了曲线箱梁桥地震分析方法,而且将传递矩阵法的分析平台由时域扩展到了频域,这将使小波变换理论应用于传递矩阵理论成为可能。
第七章大跨度结构抗震分析的多点激励模型较地震波的均匀输入模型更符合实际,更为合理。笔者基于拟静力位移和傅里叶变换,提出了大跨曲线箱梁桥在多点输入下地震反应分析的频域精细传递矩阵法。并以曲率平面内的动力反应为例,给出了拟静力位移矩阵的求解思路；推导了多点输入下集中质量的频域状态传递矩阵。
第八章将大质量法与传递矩阵法原理相结合,提出了“底部大质量-传递矩阵法”：在基础加速度输入下结构地震反应分析的精细传递矩阵法。该法依据结构在地震作用下的实际受力机理和荷载的输入位置在频域内建立结构的总传递矩阵,对其进行动力反应分析。并基于以上理论,研制了相应的计算机分析程序。该方法更符合结构的实际受力状况,可以作为多点激励下结构动力反应分析的有效途径。
第九章对本论文的工作进行了一些总结,并对今后的工作进行了一些探讨。
With the process of western development and the building of high-grade highway, many kinds of curved bridge have been used, Curved bridge can not only is adapt to bridge site that is limited by the terrain and surface features, and also because the curve lines of the structure is smooth, crisp, lively mood, it can give people enjoyment. The design of the curved structure can enable the architectural beauty and the environment coherence,and coordinated with people's aesthetic demands. considering whether from the point of geometric, aesthetics nor economics,the curved bridge has important practical significance and good prospect.The curved bridge not only has the coupling of bending and torsion out plane,but also has the bending and axis deformation in plane.The characteristics mechanic and dynimic are very complex. Although the literature researching curved beam are too numerous to mention, but, because the complexity of mechanical and dynamic properties and the difference in the approach as well as the inconsistency of starting point,there is no comprehensive and accurate analysis on complex characteristics of curved bridges. With the progress of science and technology and the development of the mechanization construction, the span of curved bridge is increasing.seismic response analysis is an important method for studying seismic ability and guiding safe design of bridges. When do the seismic design of the long-span bridges, considering multi-support seismic impact has become the consensus of the engineering and academic at home and abroad.
Thus,The paper is attampt to establish a analysis calculation method which is simple,easily understand, clear, high precision and strongly applicable to provide a theoretical basis and computational methods for curesd structural analysis.
In this paper,some specific tasks have been done as follows:
In chapter 1,generalizes and summarizes the research progress of the analysis methods of curved bridges and the research status of transfer matrix method and its application,and find out the disadvantages of the transfer matrix method.
In chapter 2,based on transfer matrix theory, with the closed-form solution of the deformation differential equation of curved beams, the transfer matrix of curved bridges have been derivated seriously and perfectly. the specific expression of field transfer matrix of curved box girder has been provided, and in accordance with the vibration theory the curved girder bridge, the vibration transfer matrix theory is established, the corresponding calculation program has been worked out with Matlab.combining the seismic analysis principles of structure, sesponse spectrum is introduced to the transfer matrix method,the seismic transfer matrix theory of curved girder bridge is established. and the corresponding calculation program has been worked out with Matlab. provides an effective way for the application of the transfer matrix method in static and dynamic analysis of the curved girder bridge.
In chapter 3,the traditional transfer matrix method based on the precise answer of the structural differential equations.This method is simple and feasible to simple strut structure of. But to complex curved beam structure it is very difficult to solve,and its application is extremely inconvenient. Based on the precise algorithm of exponential matrix and transfer matrix theory, proposes a Precise Transfer Matrix Method (PTMM)for struture analysis of curved bridge.Perfectly derives the precise transfer matrix of curved bridges in detail,and establishse a complete theory of transfer matrix. Based on the numerical integration calculation program has been worked out with Matlab,and some comprehensive analysises and numerical validations and comparisons of PTMM are conducted.
This method is general, can be used for any structural mechanical analysis. For complex structures, the advantage of the transfer matrix method becomes more apparent.
In chapter 4,for the statically determinate structure, internal forces is not caused by the temperature change.Deformation and displacement result to free expansion and contraction of the material. But to statically indeterminate structure, the temperature change not only cause the deformation and displacement, but also caused structural internal force.The curved beams as one of the bridge structure are generally designed as statically indeterminate structure, under the action of variable temperature the mid-span transverse displacement of curved bridge is too large leading to destruction is a common failure mode. Based on the thermal expansion deformation of the curved structures,a precise transfer matrix format for solving internal forces and displacements of curved beams in-plane with pinned-pinned ends are derived explicitly. The corresponding calculation program has been worked out with Matlab, This method is theoretically possible to solve internal force and deformation of the beam under an arbitrary temperature field. It will expand the use of this method to the solution of multi-span curved bridge, and obtains the transfer matrix format for solving the internal force and displacement of the multi-span curved beams under the co-moderated action of the variable temperature and pier at the top surface.
In chapter 5,natural vibration characteristics of the structure is an important performance, which determines the seismic response behavior and seismic performance of the structure. combining the precise integration method which is high-precision and transfer matrix method which is clear mechanical concept, Based on the theory of differential equations and matrix analysis, a new precise transfer matrix form is provided by combining the high-precision precise integration and the clear concept transfer matrix method.The calculation formulas of the method are derivated.The natural frequencies of the curved box bridge are solved with the method. Compared with the traditional transfer matrix method, it no need to solve the differential equations, just follows the iterative formula, the necessary transfer matrixes are obtained. According to the boundary condition,the natural frequency of the structure can be get use by the search method of the frequency. The corresponding calculation program has been worked out with Matlab. The method applies to a continuous structure also applies to discrete structures, but also can be applied to analysis the vibration characteristics of hybrid structure which is combined with discrete and continuous structure, provides a convenient and effective way for precise analysis of the dynamic properties of complex structures.
In chapter 6,based on the FFT and FTMM,derives the power transfer matrix of curved in horizontal and vertical directions,and provides the FTMM for resolving the earthquake response of the curved box bridge in frequency domain.The corresponding calculation program has been worked out with Matlab,and some examples are analyzed, and the results indicate the accuracy and efficiency of this method. Not only enrich the seismic analysis of curved box girder bridges, but also the analysis platform of transfer matrix method extended from the time domain to the frequency domain, which would apply the possibility of the application of wavelet transform in the transfer matrix theory.
In chapter 7,the multi-point incentive analysis model of large-span structure is more realistic and more reasonable than uniform input model.Based on the FFT and pseudo-static displacements,proposes a precise transfer matrix method for long-span curved box bridges seismic response analysis under multi-support excitations in frequency domain.Takeing the dynamic response of curved box bridge in curvature plane as a example,gives the solving thinking of pseudo-static displacement matrix.Derives the transfer matrix of the point unit focused on quality under multi-support excitations in frequency domain.
In chapter 8,the dynamic analysis of structures under acceleration of bearing input is studied by the combination of precise transfer matrix method and great mass method. At first,according to the basic principle of great mass method,a great mass is attached to the bearing.Then,applying the precise transfer matrix method,the overall transfer matrix is set up in frequency domain.The equations are obtained and solved with the boundary conditions, and the dynamic reaction of the structure is obtained. And based on the above theory, the corresponding Matlab calculations program has been established.This method is more in line with the actual force situation of the structure,it can be regarded as an effective way to analysis dynamic response under multi-support excitations.
In chapter 9,the conclution and outlook are obtained,in which the major work and innovative points of dissertation are summarized,including some suggestions and prospect for the research in the future.
因此,本文试图建立一种简洁、易懂、思路清晰、计算精度高且应用性较强的计算分析方法,为这类结构体系的分析研究提供理论依据和计算分析方法。从内容上分,本文可概括成以下几部分：
第一章详细地研究分析了曲线梁桥分析方法的研究进展和传递矩阵法的研究现状及其应用。总结了当前曲线梁分析方法及传递矩阵法的最新研究进展,并提出了传统传递矩阵法的不足之处。
第二章基于传递矩阵原理,结合曲线梁的变形微分方程的闭合解,对曲线梁桥的传递矩阵进行了认真、完善的推导。提供了曲线箱形梁场传递矩阵的具体表达式；并根据结构的振动原理,建立了曲线梁桥的振动传递矩阵理论；并研制了相应的计算机分析程序。结合结构抗震分析原理,将反应谱理论引入到传递矩阵法中,建立了曲线梁桥的抗震传递矩阵理论；并研制了相应的计算机分析程序。为传递矩阵法在曲线梁桥结构静力及动力分析中的应用提供了一条有效的途径。
第三章将指数矩阵的精细计算原理与传递矩阵法原理相结合,提出了曲线梁桥结构分析的精细传递矩阵法。根据结构的应力-应变关系、几何物理关系,对其进行了细致、完善的推导,建立了完整的精细传递矩阵理论；并研制了相应的计算机分析程序,对其进行了全面地数值分析和验证。
本文方法具有一般性,可用于任何结构的受力分析。对于复杂结构,本文建立的精细传递矩阵法的优势更为明显。
第四章基于曲线结构的温度变形规律,结合材料力学的相关内容,建立了求解曲线梁在温度荷载作用下的内力及变形的精细传递矩阵式；并研制了相应的计算机分析程序。该方法理论上可以求解任意温度场下曲线梁的内力及变形。将本文方法扩展运用与多跨曲线桥的求解,得出在变温和桥墩顶部摩擦力共同作用下多跨曲线梁的面内内力及位移的精细传递矩阵格式,并进行了相应的算例计算。
第五章将高精度的精细积分法和力学概念清晰的传递矩阵法结合起来,以微分方程和矩阵分析理论为基础,提出了一种新的精细传递矩阵形式,推导了该方法的计算公式,并用于分析曲线箱形梁桥的振动特性。与传统的传递矩阵法相比,无需对微分方程组进行求解,只需按照迭代公式进行计算,就可以得到所需要的传递矩阵。根据边界条件,运用频率搜索方法便可得到结构的自振频率；并研制了相应的计算机分析程序,进行了相应的算例计算。该法即适用于连续结构又适用于离散结构,而且可以应用于离散和连续相结合的混合结构的振动特性分析,为精确分析复杂结构的动力特性提供了一条便捷、有效途径。
第六章基于傅里叶变换和精细传递矩阵法,推导了曲线箱梁桥在水平方向和竖直方向的动力传递矩阵,建立了曲线箱梁桥在频域内地震反应分析的精细传递矩阵法。并研制了相应的计算机分析程序,进行了相应的算例分析。不仅丰富了曲线箱梁桥地震分析方法,而且将传递矩阵法的分析平台由时域扩展到了频域,这将使小波变换理论应用于传递矩阵理论成为可能。
第七章大跨度结构抗震分析的多点激励模型较地震波的均匀输入模型更符合实际,更为合理。笔者基于拟静力位移和傅里叶变换,提出了大跨曲线箱梁桥在多点输入下地震反应分析的频域精细传递矩阵法。并以曲率平面内的动力反应为例,给出了拟静力位移矩阵的求解思路；推导了多点输入下集中质量的频域状态传递矩阵。
第八章将大质量法与传递矩阵法原理相结合,提出了“底部大质量-传递矩阵法”：在基础加速度输入下结构地震反应分析的精细传递矩阵法。该法依据结构在地震作用下的实际受力机理和荷载的输入位置在频域内建立结构的总传递矩阵,对其进行动力反应分析。并基于以上理论,研制了相应的计算机分析程序。该方法更符合结构的实际受力状况,可以作为多点激励下结构动力反应分析的有效途径。
第九章对本论文的工作进行了一些总结,并对今后的工作进行了一些探讨。
With the process of western development and the building of high-grade highway, many kinds of curved bridge have been used, Curved bridge can not only is adapt to bridge site that is limited by the terrain and surface features, and also because the curve lines of the structure is smooth, crisp, lively mood, it can give people enjoyment. The design of the curved structure can enable the architectural beauty and the environment coherence,and coordinated with people's aesthetic demands. considering whether from the point of geometric, aesthetics nor economics,the curved bridge has important practical significance and good prospect.The curved bridge not only has the coupling of bending and torsion out plane,but also has the bending and axis deformation in plane.The characteristics mechanic and dynimic are very complex. Although the literature researching curved beam are too numerous to mention, but, because the complexity of mechanical and dynamic properties and the difference in the approach as well as the inconsistency of starting point,there is no comprehensive and accurate analysis on complex characteristics of curved bridges. With the progress of science and technology and the development of the mechanization construction, the span of curved bridge is increasing.seismic response analysis is an important method for studying seismic ability and guiding safe design of bridges. When do the seismic design of the long-span bridges, considering multi-support seismic impact has become the consensus of the engineering and academic at home and abroad.
Thus,The paper is attampt to establish a analysis calculation method which is simple,easily understand, clear, high precision and strongly applicable to provide a theoretical basis and computational methods for curesd structural analysis.
In this paper,some specific tasks have been done as follows:
In chapter 1,generalizes and summarizes the research progress of the analysis methods of curved bridges and the research status of transfer matrix method and its application,and find out the disadvantages of the transfer matrix method.
In chapter 2,based on transfer matrix theory, with the closed-form solution of the deformation differential equation of curved beams, the transfer matrix of curved bridges have been derivated seriously and perfectly. the specific expression of field transfer matrix of curved box girder has been provided, and in accordance with the vibration theory the curved girder bridge, the vibration transfer matrix theory is established, the corresponding calculation program has been worked out with Matlab.combining the seismic analysis principles of structure, sesponse spectrum is introduced to the transfer matrix method,the seismic transfer matrix theory of curved girder bridge is established. and the corresponding calculation program has been worked out with Matlab. provides an effective way for the application of the transfer matrix method in static and dynamic analysis of the curved girder bridge.
In chapter 3,the traditional transfer matrix method based on the precise answer of the structural differential equations.This method is simple and feasible to simple strut structure of. But to complex curved beam structure it is very difficult to solve,and its application is extremely inconvenient. Based on the precise algorithm of exponential matrix and transfer matrix theory, proposes a Precise Transfer Matrix Method (PTMM)for struture analysis of curved bridge.Perfectly derives the precise transfer matrix of curved bridges in detail,and establishse a complete theory of transfer matrix. Based on the numerical integration calculation program has been worked out with Matlab,and some comprehensive analysises and numerical validations and comparisons of PTMM are conducted.
This method is general, can be used for any structural mechanical analysis. For complex structures, the advantage of the transfer matrix method becomes more apparent.
In chapter 4,for the statically determinate structure, internal forces is not caused by the temperature change.Deformation and displacement result to free expansion and contraction of the material. But to statically indeterminate structure, the temperature change not only cause the deformation and displacement, but also caused structural internal force.The curved beams as one of the bridge structure are generally designed as statically indeterminate structure, under the action of variable temperature the mid-span transverse displacement of curved bridge is too large leading to destruction is a common failure mode. Based on the thermal expansion deformation of the curved structures,a precise transfer matrix format for solving internal forces and displacements of curved beams in-plane with pinned-pinned ends are derived explicitly. The corresponding calculation program has been worked out with Matlab, This method is theoretically possible to solve internal force and deformation of the beam under an arbitrary temperature field. It will expand the use of this method to the solution of multi-span curved bridge, and obtains the transfer matrix format for solving the internal force and displacement of the multi-span curved beams under the co-moderated action of the variable temperature and pier at the top surface.
In chapter 5,natural vibration characteristics of the structure is an important performance, which determines the seismic response behavior and seismic performance of the structure. combining the precise integration method which is high-precision and transfer matrix method which is clear mechanical concept, Based on the theory of differential equations and matrix analysis, a new precise transfer matrix form is provided by combining the high-precision precise integration and the clear concept transfer matrix method.The calculation formulas of the method are derivated.The natural frequencies of the curved box bridge are solved with the method. Compared with the traditional transfer matrix method, it no need to solve the differential equations, just follows the iterative formula, the necessary transfer matrixes are obtained. According to the boundary condition,the natural frequency of the structure can be get use by the search method of the frequency. The corresponding calculation program has been worked out with Matlab. The method applies to a continuous structure also applies to discrete structures, but also can be applied to analysis the vibration characteristics of hybrid structure which is combined with discrete and continuous structure, provides a convenient and effective way for precise analysis of the dynamic properties of complex structures.
In chapter 6,based on the FFT and FTMM,derives the power transfer matrix of curved in horizontal and vertical directions,and provides the FTMM for resolving the earthquake response of the curved box bridge in frequency domain.The corresponding calculation program has been worked out with Matlab,and some examples are analyzed, and the results indicate the accuracy and efficiency of this method. Not only enrich the seismic analysis of curved box girder bridges, but also the analysis platform of transfer matrix method extended from the time domain to the frequency domain, which would apply the possibility of the application of wavelet transform in the transfer matrix theory.
In chapter 7,the multi-point incentive analysis model of large-span structure is more realistic and more reasonable than uniform input model.Based on the FFT and pseudo-static displacements,proposes a precise transfer matrix method for long-span curved box bridges seismic response analysis under multi-support excitations in frequency domain.Takeing the dynamic response of curved box bridge in curvature plane as a example,gives the solving thinking of pseudo-static displacement matrix.Derives the transfer matrix of the point unit focused on quality under multi-support excitations in frequency domain.
In chapter 8,the dynamic analysis of structures under acceleration of bearing input is studied by the combination of precise transfer matrix method and great mass method. At first,according to the basic principle of great mass method,a great mass is attached to the bearing.Then,applying the precise transfer matrix method,the overall transfer matrix is set up in frequency domain.The equations are obtained and solved with the boundary conditions, and the dynamic reaction of the structure is obtained. And based on the above theory, the corresponding Matlab calculations program has been established.This method is more in line with the actual force situation of the structure,it can be regarded as an effective way to analysis dynamic response under multi-support excitations.
In chapter 9,the conclution and outlook are obtained,in which the major work and innovative points of dissertation are summarized,including some suggestions and prospect for the research in the future.
引文
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