Analytical approximate solutions for large post-bucklingresponse of a hygrothermal beam
Analytical approximate solutions for large post-bucklingresponse of a hygrothermal beam
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DOI：
10.12989/sem.2012.43.2.211
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发表时间：
2012-07
影响因子：
2.2
通讯作者：
Yongping Yu;Youhong Sun
中科院分区：
文献类型：
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作者：
Yongping Yu;Youhong Sun
本文研究了具有轴向不可移动销钉端部的线弹性湿热梁的大变形后屈曲，并通过替代方法使膨胀显着增加。提出了几何非线性问题的解析近似解。还获得了字符串极限情况的解。通过著名的麦克劳林级数展开式与正交切比雪夫多项式的耦合，可以将具有正弦非线性的控制微分方程简化为三次非线性方程，并且具有余弦非线性的补充条件也可以简化为多项式积分方程。通过将牛顿法与谐波平衡法相结合，建立了所产生的边界条件问题的解析近似。针对梁端小转角和大转角，建立了沿轴载荷、自由湿热膨胀潜在应变和周期解的两个近似公式。选择示例并与射击法获得的“参考”解进行比较，以证实近似解析方法的准确性和正确性。
This paper deals with large deformation post-buckling of a linear-elastic and hygrothermal beam with axially nonmovable pinned-pinned ends and subjected to a significant increase in swelling by an alternative method. Analytical approximate solutions for the geometrically nonlinear problem are presented. The solution for the limiting case of a string is also obtained. By coupling of the well-known Maclaurin series expansion and orthogonal Chebyshev polynomials, the governing differential equation with sinusoidal nonlinearity can be reduced to form a cubic-nonlinear equation, and supplementary condition with cosinoidal nonlinearity can also be simplified to be a polynomial integral equation. Analytical approximations to the resulting boundary condition problem are established by combining the Newton`s method with the method of harmonic balance. Two approximate formulae for load along axis, potential strain for free hygrothermal expansion and periodic solution are established for small as well as large angle of rotation at the end of the beam. Illustrative examples are selected and compared to "reference" solution obtained by the shooting method to substantiate the accuracy and correctness of the approximate analytical approach.