摘要
对实际工程中运营的压杆进行精确的内力识别和临界力求解是准确判别压杆的稳定安全系数的重要方面。对于结构的内力测试与识别中,频率法作为一种方便且实用的方法被广泛地运用,但主要集中于受拉力的索或杆,将这种思想用于压杆的内力识别与稳定分析还较少。事实上,运用频率法不仅可以通过频率测试压杆的内力,而且由动力准则,可以通过频率为零时对应的内力来求得其临界力,对其稳定性进行分析。所以对压杆的振动频率法进行研究和完善,对于压杆的安全系数评估具有重要的意义。
本文结合振动频率法和有限元法的基本思想,对压杆进行内力识别与临界力求解。该方法对于任意边界条件的不均匀变截面杆均可以适用。
1)构造了一种能够综合考虑各种边界条件和变截面影响的高精度长压杆单元。在综合考虑压杆结构特点与振动特性的基础上,选择合适的位移模式,基于能量变分原理推导出压杆单元的刚度矩阵,形成压杆单元自由振动方程,进而得到频率方程,确定频率与内力之间的对应关系;运用动力准则,通过求得压杆频率为零时的内力值来求压杆稳定的临界力。
2)根据压杆振动频率法有限元理论,编制相应的MATLAB程序,包括单元特性程序(单元刚度矩阵与单元质量矩阵)、频率求解程序、内力识别程序、临界力求解程序,并通过算例验证了本文方法及程序的正确性。
3)对压杆临界力计算的欧拉公式中的计算长度系数针对不同的弹性边界进行精确求解,并制成表格,通过查阅表格并进行插值计算,可以方便地得到任意弹性边界条件下的计算长度系数并进行压杆临界力求解。
4)对东江大桥的长压杆进行稳定分析,并对边界条件、计算长度、孔洞率、内力等因素对压杆频率的影响大小进行了分析。结果表明,孔洞的产生对欧拉梁的临界承载力没有明显影响;边界条件、孔洞率、计算长度等对压杆的频率影响较大,而内力远小于临界力时,内力对压杆的频率影响较小。
Internal force identification and critical pressure are two most important aspects to determine the stability safety factor of a compressed bar in engineering project. Today frequency method has been widely used in force identification of structures such as cables and tie rods, but less used in compressed bar. Actually, based on frequency method, not only the pressure force can be identified, but also the critical pressure can be gained according to dynamic criterion. Doing research in vibration frequency method in safety coefficient evaluation of compressed bar has important meanings.
Combined with the vibration frequency method and finite element method, several research is done in this thesis about pressure force identification and critical pressure analysis. It is suitable for variable section roll and inhomogeneous slender pressure roll as well.
1) Use the appropriate function to be the displacement function, then base on the energy variation principle, the frequency equation is derived, which considers the pressure force and the boundary conditions of the compressed bar. This method can take the influence of arbitrary boundary and variable cross-section into account. As the corresponding relation between the pressure and frequencies is obtained, the critical force can get by ? ? 0 according to dynamic criterion.
2) According to the finite element method, the computer executable procedure is setted up. Programmed with MATLAB, several programmes are designed, including solving of stiffness matrix, mass matrix, frequencies, pressure force, critical pressure and so on. Then, an example is given to testify this method.
3) Effective length factor ? in Euler's Formula is calculated accurately according to different kinds of elastic boundary. The effective length factor can be looked up or interpolation calculation through the tables made. It is convenient to use ? in these tables to calculate the critical pressure of compressed bar with arbitrary elastic boundary.
4) With the programs, stability analysis is done to the slender pressured bar of Dongjiang river. What is more, the influence of boundary condition, effective length, hole ratio in section and internal force to frequencies are analyzed. The result shows that, hole ratio has no obvious influence on critical pressure. However, boundary condition, effective length, and hole ratio in section all have significant influence on frequency of compressed bar. The internal force has little influence on frequency when the it is much less than critical pressure.
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