不同梁理论之间简支梁特征值的解析关系

ANALYTICAL RELATIONSHIPS OF SIMPLY-SUPPORTED BEAM’S EIGENVALUES USING DIFFERENT BEAM THEORIES

利用Euler-Bernoulli梁理论(EBT)、Timoshenko梁理论(一阶理论,TBT)和Reddy三阶梁理论(RBT)之间,梁的特征值问题在数学上的相似性,研究了不同梁理论之间特征值的关系。将特征值问题的求解转化为一个代数方程的求解,并导出了不同梁理论之间梁的特征值之间的精确解析关系。因此,只要已知梁的经典结果(临界载荷和固有频率),便很容易从这些关系中获得一阶和三阶梁理论下的相应结果。另外,从这些关系中获得的含有剪切变形影响的结果,可以用于检验一阶和三阶梁理论下梁数值结果的有效性、收敛性以及精确性等问题。

Based on the mathematical similarity in the eigenvalue problem of Euler-Bemoulli beam theory, Timoshenko beam theory and Reddy＇s third-order beam theory, relationships of the eigenvalues of the three theories for simply-supported beams are investigated. Solving of the eigenvalue problem is converted into an algebra equation to be solved and the analytical relationships of the three theories are expressed explicitly. These relationships enable the conversion of the classical （Euler-Bernoulli） beam solutions to their shear deformable counterparts using the Timoshenko beam theory and Reddy＇s third-order beam theory. The shear deformable results obtained from these relationships may be used to check the validity, convergence and accuracy of numerical results of the Timoshenko beam theory and Reddy＇s third-order beam theory.