箱型梁剪力滞效应及其可靠度随机有限元分析
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摘要
箱型梁是一种具有箱型截面的薄壁杆件,在桥梁、建筑等结构中得到广泛应用,并取得了良好的效果。箱型梁在横向外荷载作用下产生弯曲变形时存在剪力滞效应,使得正应力沿翼缘横向呈曲线分布。研究表明,剪力滞效应能够降低箱型梁的整体弯曲刚度,带来结构失稳和局部破坏的安全隐患。另外,在实际工程中,结构的材料性能、构件尺寸以及外部荷载等物理量大多都是随机的。随着结构计算理论的进步以及计算机信息处理能力的强大,需要运用不确定性的计算力学模型处理实际工程问题,并在此基础上对结构的可靠度进行计算。因此,对箱型梁的剪力滞效应问题、随机分析和可靠度分析进行全面深入的研究,具有重要的学术意义和工程应用价值。本论文的主要研究工作有：
1.提出了箱型梁附加挠度的概念,利用附加挠度代替剪力滞函数建立了新的翼板纵向位移表达式,利用外荷载代替内力建立箱梁总势能泛函,导出了箱梁附加挠度的四阶微分方程、边界条件和附加挠度的解析解,建立了基于挠度的箱型梁剪力滞效应分析的能量变分解析方法,从而解决了传统剪力滞理论存在的剪力滞函数物理意义不明确、不能正确处理自由端边界条件、难以求解固定-简支箱型梁的响应量等问题。进而,基于挠度和附加挠度定义剪力滞系数,分析了箱型梁支撑条件、截面几何尺寸等参数对剪力滞效应的影响。与传统的基于应力的剪力滞系数相比,基于挠度的剪力滞系数能够更加准确反映箱型梁截面的剪力滞效应及其分布规律；为减少剪力滞效应的影响,应适当控制箱型梁的宽高比和宽跨比。
2.以挠度、附加挠度及其一阶导数作为箱梁单元的结点位移参数,研究建立了基于挠度的箱型梁剪力滞效应分析的一维离散有限元法,分析了不同边界条件下箱梁截面的剪力滞效应及其变化规律。研究结果表明：本文建立的一维离散有限元分析方法可以保证箱梁挠度和附加挠度的有限元位移场之间的协调性,易于处理边界条件,计算格式简单,所得到的箱梁挠度和翼缘截面上的应力都能够与解析解良好吻合,有较高的计算精度。
3.研究建立了适用于随机参数大变异且能够同时处理弹性模量和外荷载具有随机性的结构响应随机分析的层递随机有限元法。利用层递展开方法建立了一组可以递归计算的基向量,根据变分原理建立了层递随机有限元法控制方程,利用混沌多项式的性质导出了层递刚度元素和荷载列阵元素的简化计算公式,给出了结点位移基于不同阶基向量展开的表达式,推导了结点位移向量的均值和协方差的计算公式。根据箱型梁的结构特点进一步建立了箱型梁随机分析的层递随机有限元方法,同时研究了剪力滞效应下箱型梁随机分析的蒙特卡洛有限元法和摄动随机有限元法。用三种随机有限元法对箱型梁进行了不确定性分析,计算了箱梁结构响应的数字特征值,对比分析了三种方法的计算结果。研究结果表明：在随机参数大变异的情况下,相较于摄动随机有限元法,层递随机有限元法的计算结果与蒙特卡洛有限元法的计算结果吻合较好,计算效率较高。
4.研究建立了基于层递随机有限元法的可靠指标求解方法：基于验算点的嵌入式迭代算法和基于均值点的分离式迭代算法。同时,研究了蒙特卡洛有限元法直接通过抽样分析和统计分析求得结构可靠指标的方法、摄动随机有限元法和结构可靠度分析的优化迭代算法相结合计算可靠指标的方法。将基于这三种随机有限元法的可靠度分析方法运用于箱型梁的可靠度分析中。研究结果表明：与基于蒙特卡洛有限元法的可靠度算法相比,基于均值点的层递随机有限元分离式迭代算法和基于验算点的摄动随机有限元法的可靠度算法都具有良好的计算精度。随机参数变异系数较大时,基于层递随机有限元的结构可靠度分析方法具有更好的适用性和计算精度。
Box-girders is a thin-walled beam with box section, which is widely used in bridge and building. The classical Euler beam theory does not hold for bending box-girders under external load because of the shear lag effect, the stress is non-uniformly distributed along flange cross section. Investigations show that shear lag effect can reduce the bending stiffness of box-girders, result in cracks or serious accident such as local collapse. Moreover, in practical engineering, the material porperties, member size and external load are random physical quantities. With the progressive of computation theory and powerful of information processing ability of computer, it is need to handle practical engineering problems using uncertain computational mechanics model, based on which reliability is calculated. Therefore, it is necessary to study shear lag effect, stochastic analysis and reliability analysis comprehensively and deeply. The main contents of the thesis include:
1. The additional deflection of box-girder was defined for characterizing the nonuniformity of normal stress along flange section. The potential energy functional is developed using external loads instead of inner forces. A set of4th order ordinary differential equations on additional deflection is derived as well as the boundary conditions, based on which the closed form solutions of the additional deflection were achieved. Furthermore, the shear lag coefficients based on deflection was presented to investigate the variation of shear lag effect on box girders, and examined the effects of the boundary conditions and the section's geometric dimensions on the shear lag.
2. The deflection, additional deflection and their first derivatives are chosen as nodal displacement parameters of the box-girder element, so that a one-dimensional finite element is developed for shear-lag effect on box girders, analyzed the shear lag effect and its variation law under different boundary conditions. Studies show that the shear lag coefficient based on the additional deflection can take place of the conventional one based on the stress; the one-dimensional finite element method has high precision.
3. Hierarchical Stochastic Finite Element Method was built for structure with large randomness in elastic modulus and external loads. This method established a set of base vector using hierarchical expand method, and the control equations of Hierarchical Stochastic Finite Element Method were built based on variational principle. The simplified formulae of stiffness elements and load array elements were derived on the basis of chaos polynomial, and the expression of node displacements were given under different order base vector, then derived the formulae for the mean and the covariance of node displacement were derived. Stochastic element methods were built for box-girders, including Monte Carlo Stochastic Analysis, Perturbation Stochastic Analysis and Hierarchical Stochastic Analysis in this paper.The statistical characteristics of structure response were analyzed using these three stochastic finite methods, and the calculate results were compared. Researches indicate that the results calculate by Hierarchical Stochastic Finite Element Method tallied with the Monte Carlo Simulation Method under stochastic parameters with large variations, and had high computational efficiency.
4. Calculation methods for reliability index based on Hierarchical Stochastic Finite Element Method were established, including embedded iterative algorithm based on cheking point and separate iterative algorithm based on mean value point. Studied using Monte Carlo simulation method to calculate reliability index by sampling and statistics, combined Perturbation Stochastic Finite Element Method and iterative algorithm for structural reliability analysis to calculate reliability index. Reliability of box-girder was analyzed using reliability analysis methods based on these three methods and calculations were compared. Studies manifest that accuracy of separate iterative algorithms based on mean value point of Hierarchical Stochastic Finite Element Method and of perturbation stochastic finite element method based on design point, are higher comparing with the algorithm accuracy of Monte Carlo stochastic finite element method. What is more, as the variation of stochastic parameter is lager, the accuracy of separate iterative algorithm based on mean value point is higher than that of perturbation stochastic finite element method based on design point.
1.提出了箱型梁附加挠度的概念,利用附加挠度代替剪力滞函数建立了新的翼板纵向位移表达式,利用外荷载代替内力建立箱梁总势能泛函,导出了箱梁附加挠度的四阶微分方程、边界条件和附加挠度的解析解,建立了基于挠度的箱型梁剪力滞效应分析的能量变分解析方法,从而解决了传统剪力滞理论存在的剪力滞函数物理意义不明确、不能正确处理自由端边界条件、难以求解固定-简支箱型梁的响应量等问题。进而,基于挠度和附加挠度定义剪力滞系数,分析了箱型梁支撑条件、截面几何尺寸等参数对剪力滞效应的影响。与传统的基于应力的剪力滞系数相比,基于挠度的剪力滞系数能够更加准确反映箱型梁截面的剪力滞效应及其分布规律；为减少剪力滞效应的影响,应适当控制箱型梁的宽高比和宽跨比。
2.以挠度、附加挠度及其一阶导数作为箱梁单元的结点位移参数,研究建立了基于挠度的箱型梁剪力滞效应分析的一维离散有限元法,分析了不同边界条件下箱梁截面的剪力滞效应及其变化规律。研究结果表明：本文建立的一维离散有限元分析方法可以保证箱梁挠度和附加挠度的有限元位移场之间的协调性,易于处理边界条件,计算格式简单,所得到的箱梁挠度和翼缘截面上的应力都能够与解析解良好吻合,有较高的计算精度。
3.研究建立了适用于随机参数大变异且能够同时处理弹性模量和外荷载具有随机性的结构响应随机分析的层递随机有限元法。利用层递展开方法建立了一组可以递归计算的基向量,根据变分原理建立了层递随机有限元法控制方程,利用混沌多项式的性质导出了层递刚度元素和荷载列阵元素的简化计算公式,给出了结点位移基于不同阶基向量展开的表达式,推导了结点位移向量的均值和协方差的计算公式。根据箱型梁的结构特点进一步建立了箱型梁随机分析的层递随机有限元方法,同时研究了剪力滞效应下箱型梁随机分析的蒙特卡洛有限元法和摄动随机有限元法。用三种随机有限元法对箱型梁进行了不确定性分析,计算了箱梁结构响应的数字特征值,对比分析了三种方法的计算结果。研究结果表明：在随机参数大变异的情况下,相较于摄动随机有限元法,层递随机有限元法的计算结果与蒙特卡洛有限元法的计算结果吻合较好,计算效率较高。
4.研究建立了基于层递随机有限元法的可靠指标求解方法：基于验算点的嵌入式迭代算法和基于均值点的分离式迭代算法。同时,研究了蒙特卡洛有限元法直接通过抽样分析和统计分析求得结构可靠指标的方法、摄动随机有限元法和结构可靠度分析的优化迭代算法相结合计算可靠指标的方法。将基于这三种随机有限元法的可靠度分析方法运用于箱型梁的可靠度分析中。研究结果表明：与基于蒙特卡洛有限元法的可靠度算法相比,基于均值点的层递随机有限元分离式迭代算法和基于验算点的摄动随机有限元法的可靠度算法都具有良好的计算精度。随机参数变异系数较大时,基于层递随机有限元的结构可靠度分析方法具有更好的适用性和计算精度。
Box-girders is a thin-walled beam with box section, which is widely used in bridge and building. The classical Euler beam theory does not hold for bending box-girders under external load because of the shear lag effect, the stress is non-uniformly distributed along flange cross section. Investigations show that shear lag effect can reduce the bending stiffness of box-girders, result in cracks or serious accident such as local collapse. Moreover, in practical engineering, the material porperties, member size and external load are random physical quantities. With the progressive of computation theory and powerful of information processing ability of computer, it is need to handle practical engineering problems using uncertain computational mechanics model, based on which reliability is calculated. Therefore, it is necessary to study shear lag effect, stochastic analysis and reliability analysis comprehensively and deeply. The main contents of the thesis include:
1. The additional deflection of box-girder was defined for characterizing the nonuniformity of normal stress along flange section. The potential energy functional is developed using external loads instead of inner forces. A set of4th order ordinary differential equations on additional deflection is derived as well as the boundary conditions, based on which the closed form solutions of the additional deflection were achieved. Furthermore, the shear lag coefficients based on deflection was presented to investigate the variation of shear lag effect on box girders, and examined the effects of the boundary conditions and the section's geometric dimensions on the shear lag.
2. The deflection, additional deflection and their first derivatives are chosen as nodal displacement parameters of the box-girder element, so that a one-dimensional finite element is developed for shear-lag effect on box girders, analyzed the shear lag effect and its variation law under different boundary conditions. Studies show that the shear lag coefficient based on the additional deflection can take place of the conventional one based on the stress; the one-dimensional finite element method has high precision.
3. Hierarchical Stochastic Finite Element Method was built for structure with large randomness in elastic modulus and external loads. This method established a set of base vector using hierarchical expand method, and the control equations of Hierarchical Stochastic Finite Element Method were built based on variational principle. The simplified formulae of stiffness elements and load array elements were derived on the basis of chaos polynomial, and the expression of node displacements were given under different order base vector, then derived the formulae for the mean and the covariance of node displacement were derived. Stochastic element methods were built for box-girders, including Monte Carlo Stochastic Analysis, Perturbation Stochastic Analysis and Hierarchical Stochastic Analysis in this paper.The statistical characteristics of structure response were analyzed using these three stochastic finite methods, and the calculate results were compared. Researches indicate that the results calculate by Hierarchical Stochastic Finite Element Method tallied with the Monte Carlo Simulation Method under stochastic parameters with large variations, and had high computational efficiency.
4. Calculation methods for reliability index based on Hierarchical Stochastic Finite Element Method were established, including embedded iterative algorithm based on cheking point and separate iterative algorithm based on mean value point. Studied using Monte Carlo simulation method to calculate reliability index by sampling and statistics, combined Perturbation Stochastic Finite Element Method and iterative algorithm for structural reliability analysis to calculate reliability index. Reliability of box-girder was analyzed using reliability analysis methods based on these three methods and calculations were compared. Studies manifest that accuracy of separate iterative algorithms based on mean value point of Hierarchical Stochastic Finite Element Method and of perturbation stochastic finite element method based on design point, are higher comparing with the algorithm accuracy of Monte Carlo stochastic finite element method. What is more, as the variation of stochastic parameter is lager, the accuracy of separate iterative algorithm based on mean value point is higher than that of perturbation stochastic finite element method based on design point.
引文
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