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索杆张力结构需依靠预张力来获得初始刚度和维持稳定性,但实际工程中索长误差等因素所引起的预张力偏差不可忽视。分别以单元绝对和相对预张力偏差平方和作为结构预张力偏差的评价指标,建立了两类指标与索长误差间的二次型解析表达式。利用其中二次型矩阵Rayleigh商的极性解释了结构最不利预张力偏差的有界性。进一步对二次型矩阵进行谱分解,将结构最不利预张力偏差表示为以该矩阵特征值为权重、以索长误差在特征向量上的投影值为因子的加权平方和形式。发现二次型矩阵的特征值衰减迅速,故近似采用其一阶特征值和特征向量便能有效估计结构的最不利预张力偏差大小以及对应的索长误差分布。采用该文方法对一个实际索杆张力结构的最不利预张力偏差进行求解,并与两种常规优化算法的计算结果进行对比考察了该方法的计算精度和有效性。
Cable tension structure needs to rely on pre-tension to obtain the initial stiffness and maintain stability, but the actual project length error and other factors caused by pre-tension deviation can not be ignored. Based on the sum of unit absolute and relative prestress deviation sum of squares, respectively, the quadratic analytic expressions of the two types of indexes and the length of cable are established. The polarity of the quadratic matrix Rayleigh quotient is used to explain the boundedness of the most unfavorable pretension deviation of the structure. The spectral decomposition of the quadratic matrix is further carried out. The most unfavorable pretension deviation of the structure is expressed as the weighted square sum with the weight of the matrix eigenvalue and the projection value of the euphemism error on the eigenvector as a factor. It is found that the eigenvalues of the quadratic matrix decay rapidly. Therefore, the first-order eigenvalues and eigenvectors of the quadratic matrix can effectively estimate the most unfavorable pretension deviation and the corresponding error distribution of cable length. In this paper, the most unfavorable pretension deviation of a real cable tension structure is solved, and the calculation accuracy and effectiveness of the method are compared with those of two conventional optimization methods.