基于一种广义梁理论的弹性地基FGM简支梁自由振动解析解

基于一种扩展的n阶广义剪切变形梁理论(n-GBT),应用Hamilton原理,建立了以轴向位移、横向位移及转角为未知函数的Winkler-Pasternak弹性地基功能梯度材料(FGM)梁的自由振动方程,采用Navier法获得了弹性地基FGM简支梁自由振动的精确解。与多种梁理论预测结果进行比较,讨论并给出了GBT阶次n的理想取值;分析了梯度指标、跨厚比及地基刚度对FGM梁频率的影响。结果表明:本文方法有效且适用范围广,若采用高阶剪切梁理论模型,宜取n≥3的奇数;FGM梁的自振频率随材料梯度指标的增大而减小;随跨厚比的增加而增大,但当跨厚比大于20,跨厚比增加对频率的影响很小;随地基刚度的增加而增大,地基刚度足够大时,频率趋于收敛。

Exact solutions for free vibration of FGM simply supported beams on elastic foundation based on a generalized beam theory

Based on an extension of the nth generalized shear beam theory(GBT),the free vibration equations offunctionally graded material(FGM)beams resting on Winkler-Pasternak elastic foundations are derived from the Hamilton’s principle,in which the unknown functions are axial displacement,deflection and rotation angle.Applying the Navier solution method,analytical solutions for the free vibration of FGM simply supported beams on elastic foundations are obtained.The availability and accuracy of the GBT are verified throughout numerical examples from the other various beam theories and herein proposed ideal value to n.The GBT can refine the beam theories and also be used to verify or modify other various shear beam theories.The exact solutions obtained herein can be used as benchmarks to verify other existed numerical methods.Finally,effects of the material graded index,slenderness ratios and elastic foundation stiffness on frequencies are analyzed.Results obtained show that frequencies of the structure will increase as both slenderness ratios and elastic foundation stiffness increase,but decrease while material graded index increase.Effect of slenderness ratios on frequencies is negligible for those thin and long beams when it is more than 20.Frequencies will become independent of elastic foundation stiffness and convergent when its value is high to a certain extent.