曲线加筋Kirchhoff-Mindlin板自由振动分析

相比传统加筋板,曲线加筋板能够更充分地发挥材料力学性能.在加筋板力学分析中,厚板通常采用Reissner-Mindlin理论,然而当板厚较薄时易出现剪切自锁,离散的Kirchhoff-Mindlin理论采用假设剪切应变场可避免该问题.针对曲线加筋Kirchhoff-Mindlin板自由振动分析,采用离散的Kirchhoff-Mindlin三角形单元和Timoshenko曲梁单元分别模拟板和加强筋,根据板的位移插值函数及筋板交界面的位移协调条件,建立基于板单元位移自由度的有限元方程.为了验证方法的有效性和准确性,采用直线加筋薄板、曲线加筋薄板和厚板3种模型进行算例研究,通过收敛性和精度分析来选择合理的有限元网格密度.直线加筋薄板前20阶固有频率均与文献结果吻合良好;曲线加筋板算例中,本文方法满足收敛条件的板单元数目为2469,Nastran模型板单元数目为6243;本文所得曲线加筋板固有频率与Nastran计算结果最大误差为3.4%.研究结果表明,本文方法无需筋板单元共节点,可使用较少的有限元网格数量,并能够保证计算精度;在离散Kirchhoff-Mindlin三角形板单元基础上构造Timoshenko梁单元可同时适用于曲线加筋薄板与厚板自由振动分析.

FREE VIBRATION ANALYSIS OF CURVILINEARLY STIFFENED KIRCHHOFF-MINDLIN PLATES

Compared with traditional stiffened plates, curvilinearly stiffened plates can deliver the mechanical properties of materials more adequately. In mechanical analysis of stiffened thick plates, Reissner-Mindlin theory is usually adopted. However, diffculties are encountered in connection with shear locking when the plate thickness approaches zero. In order to avoid the above problem, the discrete Kirchhoff-Mindlin theory was investigated by employing the assumption of shear strain field. An efficient finite element approach for free vibration analysis of curvlinearly stiffened Kirchhoff-Mindlin plates is presented in this paper. The discrete Kirchhoff-Mindlin triangular (DKMT) element and the Timoshenko curved beam element are employed for modeling the plate and the stiffeners, respectively. The finite element equation is established through the displacement interpolation function of plate and the displacement compatibility conditions at the plate-stiffener interfaces. In order to verify the efficiency and accuracy of the present method, linearly stiffened thin plate and curvilinearly stiffened thin and thick plates are used as numerical examples. The reasonable finite element mesh density is selected by convergence and accuracy analysis. The first 20 natural frequencies of the linearly stiffened plate are in good agreement with the literature. In the examples of the curvilinearly stiffened plate, the number of plate elements satisfying the convergence condition is 2469, while the number in Nastran model is 6243. The maximum error of the natural frequency between the present method and Nastran is 3.4%. Results show that present approach can guarantee the accuracy of calculation with less number of elements. The present method can be applied to the free vibration analysis of both stiffened thin and thick plates.