经典理论与一阶理论之间简支梁特征值的解析关系

利用Euler-Bernoulli梁理论(EBT)和Timoshenko梁理论(一阶理论,TBT)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梁理论下的相应结果。这些解析结果清楚地显示了横向剪切变形对经典结果影响的本质特点。另外,从这些关系中获得的含有剪切变形影响的结果,可以用于检验一阶理论下梁特征值数值结果的有效性、收敛性以及精确性等问题。

Analytical Relationships of Eigenvalue for a Simply Supported Beam Between EBT and TBT

Based on the mathematical similarity of the eigenvalue problem of the Euler-Bernoulli beam theory(EBT) and Timoshenko beam theory(TBT),relationships between the eigenvalues of the two theories for beams are investigated. Solving of the eigenvalue problem is converted into an algebra equation to be solved and the analytical relationships that are expressed explicitly between various theories are presented. These relationships enable the conversion of the classical (Euler-Bernoulli) beam solutions to their shear deformable counterparts using the Timoshenko 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.