超静定梁-柱的解析解研究

ANALYTICAL SOLUTION OF STATICALLY INDETERMINATE BEAM-COLUMN

本文采用渐进积分法研究了超静定梁-柱的弯曲问题.首先建立超静定梁-柱的四阶挠度微分方程,考虑到边界条件和连续光滑条件,采用连续分段独立一体化积分法求解得到了挠度的精确解析解.为了满足工程设计需要,构造了超静定梁-柱的四阶挠度微分迭代方程,选取无轴向力作用时超静定梁的挠曲线作为梁的初函数,将初函数代入梁的四阶挠度微分迭代方程进行积分,利用边界条件和连续光滑条件确定积分常数,得到下一次迭代挠度函数,依次进行迭代积分运算.计算出了最大挠度、最大转角和最大弯矩等用轴向力放大系数表示的多项式解析函数解.本文选取了两种边界条件下受分布力作用的超静定梁-柱进行分析,计算结果表明,当超静定梁-柱所受的轴向力小于欧拉临界力的1/2时,迭代六次误差就可以控制在1%以内;不仅梁-柱最大位移和最大内力的大小随轴向力的增大而增大,而且其位置也随轴向力的增大而发生迁移.本文的研究对揭示轴向力对超静定梁-柱变形和内力的影响有重要意义,为超静定梁-柱的实际设计提供了一定的理论基础.

In this paper, the problem of statically indeterminate beam-column bending is studied by using the asymptotic integral method. Firstly, the fourth order deflection differential equation of statically indeterminate beamcolumn is established. Considering the boundary condition and continuous smooth condition, the exact analytical solution of the deflection is obtained by using continuous piecewise independent integration method. In order to meet the requirements of engineering design, the statically indeterminate beam-column of the fourth order equation is constructed. The deflection line of statically indeterminate beam without axial force is selected as the initial function of beam. The initial function is substituted into the fourth-order deflection differential iterative equation of the beam for integration. The boundary condition and continuous smooth condition are used to determine the integral constant to get the next iteration deflection function. The iterative integral operation is carried out in turn. The polynomial analytical function solutions of the maximum deflection, the maximum angle and the maximum bending moment expressed by the axial force amplification coefficient are calculated. The statically indeterminate beam-column subjected to distributed forces under two boundary conditions are analyzed in this paper. The calculation results show that when the axial force of statically indeterminate beam-column is less than half of Euler critical force, the error can be controlled within 1%after six iterations. Not only the beam-column’s maximum displacement and shear force increase with the increase of axial force, but also the position of the maximum displacement and internal force migrate with the increase of axial force. The research in this paper is of great significance to reveal the influence of axial force on statically indeterminate beam-column deformation and internal force, and provides a certain theoretical basis for the practical design of statically indeterminate beam-column.