对称荷载作用下弹性地基梁的傅里叶级数解

Fourier Series Solution for Elastic Foundation Beams under Symmetric Loads

考虑地层位移荷载及梁与地基可能产生的脱空,针对长度在沉降槽内和长度延伸到沉降槽外的2种梁建立了弹性地基梁对称问题的数学模型.利用阶梯函数及脉冲函数,在所建数学模型基础上推导了求解弹性地基梁挠度的傅里叶级数系数的线性方程组,提出了计算方程组中脱空范围这一多余未知量的迭代步骤,利用有限元数值解对傅里叶级数解进行了验证.结果表明,傅里叶级数解精度高,可以作为带有脱空弹性地基梁问题的解析解,要达到相同的精度,傅里叶级数解的计算量远比有限元解的计算量小.此外,脱空范围的大小,不随级数项数的多寡而改变.傅里叶级数解法不但精度高,而且能够灵活处理不同形式的荷载,是求解复杂荷载条件下弹性地基梁问题的有效解析方法.

Taking into account the load due to ground displacement and the detachment between a beam and ground,a mathematical model was presented for two kinds of elastic foundation beams under symmetric loads.A kind of beams has a length larger than the width of settlement trough,and another with less length than the width of the trough.Heaviside step function and impulse function were introduced,and their Fourier series expansions were given.According to the established model and making use of the step function and impulse function,an equation set for solving the coefficients of the series that represents the deflection of elastic ground beam was derived.An iteration process to find detachment interval,which is a redundant unknown in the equation set,was put forward.Lastly,verification was done to the proposed series solution by comprising with finite element method.The results show that Fourier series solution is of high accuracy of computation,and can be used as an analytical solution of the elastic foundation beams.To achieve the same accuracy,series solution needs much less amount of calculation.The computed value of detachment interval in the series solution doesn't change with the number of employed series items.In a whole,Fourier series solution offers high accuracy and convenience in dealing with different kinds of loads.It provides an analytical way to effectively solve the problems for an elastic foundation beam under complicated loads.