部分粘结组合梁的若干动静力问题
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摘要
钢筋混凝土组合粱是在钢材和混凝土的基础上发展出来的一种结构体系,由钢梁、钢筋混凝土翼板和剪切连接键三个主要部分构成。由于钢筋主要承受拉力,而钢筋混凝土翼板主要承受压力,从而这两种材料组合在一起受力作用时,钢筋和混凝土的性能能够得到最大程度地发挥,且要优于钢筋、混凝土单独使用时的受力性能。为了更好地发挥组合梁的优点,需要进一步对部分粘结作用组合梁在振动、屈曲、剪力滞和扭转等静动力特性方面进行深入的分析研究。
本文针对部分粘结Euler-Bernoulli和Timoshenko组合梁的振动问题,建立了以(广义)位移和(广义)力作为基本变量的状态空间列式,可方便处理边界和连续条件,还可避免方程中出现对荷载的导数等不合理现象。通过引进辛内积,建立了广义边界下部分粘结作用组合梁振动模态的两个正交关系,给出了固有频率和振动模态的求解过程,采用模态叠加法给出了非齐次状态方程的解析解,并详细推导了使用节点耦合矩阵(JCM)法分析单跨和多跨组合梁的具体步骤,可以对单跨和多跨组合梁的屈曲荷载、固有频率和模态进行分析,还可以进一步推广应用于分析单跨和多跨的非均匀组合梁。
本文在考虑上下不对称箱形截面纵向位移函数的基础上,基于辛体系力学方法推导了考虑剪力滞效应的部分粘结Timoshenko组合箱梁的Hamilton方程组,可退化为完全作用的组合箱梁情形。采用节点耦合矩阵(JCM)法可以很方便地分析单跨和多跨、均匀和非均匀组合箱梁的剪力滞效应。
本文分析了部分粘结组合梁的自由扭转和约束扭转,通过最小势能原理导出了相应的控制方程和边界条件,并进一步得到部分粘结矩形截面杆件自由扭转时翘曲函数的解析解,特意避免采用双曲函数,而采用适当的指数函数,有效地避免了计算时的数值不稳定问题。在最小势能原理的基础上建立的有限元列式,可以对任意截面形式组合梁的自由扭转进行分析。对于约束扭转可以采用迭代求解,并可退化到自由扭转情形。
Of the various types of structural system, steel-concrete composite beam developed from the steel and concrete mainly consists of three components including the steel girder, the steel-concrete slab and the shear connectors. Due to the individual advantage that steel mainly bears the tension force and the steel-concrete slab mainly bears the compression, the steel-concrete composite beam is superior to steel or concrete when they are used exclusively. To improve the performance of the composite beam, this paper aims at analyzing in depth the static and dynamic behavior of the composite beam with interlayer slip, including vibration, buckling, shear-lag and torsion.
This paper presents the state space formulation by adopting the generalized displacements and generalized forces as the basic variables for the vibration of the Euler-Bernoulli and Timoshenko partial-interaction composite beam. The formulation can deal with the boundary conditions and the continuity conditions conveniently and avoid the derivative of the load function in the equation as reported in literature. Based on the concept of symplectic inner product, two orthogonality relations for the free vibration modes of the partial-interaction composite beam are established. The procedure of solving the natural frequencies and free vibration modes, the analytical solution to the inhomogeneous state equation by virtue of mode superposition method and the joints coupling matrix method for single and multi-span composite beams are also presented. Then the buckling, natural frequency and modes are calculated and the analysis is further extended to nonuniform composite beams.
For unsymmetric box-girder composite beam, we suggest to use different longitudinal displacement functions for the top slab, web, bottom slab and flange. Based on the symplectic system mechanics, the Hamilton formulas of the partial-interaction Timoshenko composite beam with the shear-lag effect are established, which can be degenerated into the full-interaction situation. The shear-lag effect of the single or multi-span, uniform or nonuniform composite box-girder beams is then conveniently analyzed by virtue of joints coupling matrix.
The free torsion and constraint torsion of partial-interaction composite beams are also investigated. Based on the principle of minimum potential energy, the governing equation and the boundary conditions are derived. The analytical solution of warping function for the free torsion of bars with rectangular section is presented, in which we use the proper exponential functions instead of hyperbolic functions to avoid the numerical instability. The finite element formulation is also presented to analyze the free torsion of composite beams with arbitrary section. The solution of the constraint torsion which can be degenerated into the free torsion situation is obtained by iteration.
本文针对部分粘结Euler-Bernoulli和Timoshenko组合梁的振动问题,建立了以(广义)位移和(广义)力作为基本变量的状态空间列式,可方便处理边界和连续条件,还可避免方程中出现对荷载的导数等不合理现象。通过引进辛内积,建立了广义边界下部分粘结作用组合梁振动模态的两个正交关系,给出了固有频率和振动模态的求解过程,采用模态叠加法给出了非齐次状态方程的解析解,并详细推导了使用节点耦合矩阵(JCM)法分析单跨和多跨组合梁的具体步骤,可以对单跨和多跨组合梁的屈曲荷载、固有频率和模态进行分析,还可以进一步推广应用于分析单跨和多跨的非均匀组合梁。
本文在考虑上下不对称箱形截面纵向位移函数的基础上,基于辛体系力学方法推导了考虑剪力滞效应的部分粘结Timoshenko组合箱梁的Hamilton方程组,可退化为完全作用的组合箱梁情形。采用节点耦合矩阵(JCM)法可以很方便地分析单跨和多跨、均匀和非均匀组合箱梁的剪力滞效应。
本文分析了部分粘结组合梁的自由扭转和约束扭转,通过最小势能原理导出了相应的控制方程和边界条件,并进一步得到部分粘结矩形截面杆件自由扭转时翘曲函数的解析解,特意避免采用双曲函数,而采用适当的指数函数,有效地避免了计算时的数值不稳定问题。在最小势能原理的基础上建立的有限元列式,可以对任意截面形式组合梁的自由扭转进行分析。对于约束扭转可以采用迭代求解,并可退化到自由扭转情形。
Of the various types of structural system, steel-concrete composite beam developed from the steel and concrete mainly consists of three components including the steel girder, the steel-concrete slab and the shear connectors. Due to the individual advantage that steel mainly bears the tension force and the steel-concrete slab mainly bears the compression, the steel-concrete composite beam is superior to steel or concrete when they are used exclusively. To improve the performance of the composite beam, this paper aims at analyzing in depth the static and dynamic behavior of the composite beam with interlayer slip, including vibration, buckling, shear-lag and torsion.
This paper presents the state space formulation by adopting the generalized displacements and generalized forces as the basic variables for the vibration of the Euler-Bernoulli and Timoshenko partial-interaction composite beam. The formulation can deal with the boundary conditions and the continuity conditions conveniently and avoid the derivative of the load function in the equation as reported in literature. Based on the concept of symplectic inner product, two orthogonality relations for the free vibration modes of the partial-interaction composite beam are established. The procedure of solving the natural frequencies and free vibration modes, the analytical solution to the inhomogeneous state equation by virtue of mode superposition method and the joints coupling matrix method for single and multi-span composite beams are also presented. Then the buckling, natural frequency and modes are calculated and the analysis is further extended to nonuniform composite beams.
For unsymmetric box-girder composite beam, we suggest to use different longitudinal displacement functions for the top slab, web, bottom slab and flange. Based on the symplectic system mechanics, the Hamilton formulas of the partial-interaction Timoshenko composite beam with the shear-lag effect are established, which can be degenerated into the full-interaction situation. The shear-lag effect of the single or multi-span, uniform or nonuniform composite box-girder beams is then conveniently analyzed by virtue of joints coupling matrix.
The free torsion and constraint torsion of partial-interaction composite beams are also investigated. Based on the principle of minimum potential energy, the governing equation and the boundary conditions are derived. The analytical solution of warping function for the free torsion of bars with rectangular section is presented, in which we use the proper exponential functions instead of hyperbolic functions to avoid the numerical instability. The finite element formulation is also presented to analyze the free torsion of composite beams with arbitrary section. The solution of the constraint torsion which can be degenerated into the free torsion situation is obtained by iteration.
引文
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