This paper presents a modified variational method for the dynamic analysis of functionally graded piezoelectric (FGPM) deep curved beams with porosity (FGPP beam) resting on an arbitrary number of discrete elastic foundations (EFs). The material properties including the electrical and porous property are assumed to continuously vary in the thickness direction. A modified principle is developed to impose the continuity constrains on the internal and boundaries of the beam in both mechanical and electric field, which allows any linearly independent, complete basis functions to achieve satisfactory solutions. The shear, inertial and curvature effects of the moderately thick deep curved beams are also introduced in the energy functionals. The superior convergence, accuracy and efficiency of the proposed method are validated by comparing the proposed results with those available in literature. Series of numerical examples are presented for the free and transient vibration of the FGPP beams with different material properties. The influences of the power-law index, porosity, applied electric voltage and the EFs on the natural frequencies and transient dynamic responses of the FGPP beams are examined. Especially, the combined effects of porosity properties with other material properties on the dynamic characteristics of the FGPP beams are firstly presented, which are of significance for optimal design of corresponding smart structures.

本文提出了一种改进的变分方法，用于对任意数量的离散弹性地基（EFs）上具有孔隙率的功能梯度压电（FGPM）深弯曲梁（FGPP梁）进行动力分析。假定包括电和多孔性质的材料性质在厚度方向上连续变化。开发了一种修改后的原理，以在机械和电场中将连续性约束强加于梁的内部和边界，这允许任何线性独立，完整的基函数来获得令人满意的解决方案。能量函数中还引入了中等厚度的深弯曲梁的剪切，惯性和曲率效应。卓越的融合，通过将提出的结果与文献中提供的结果进行比较，可以验证所提出方法的准确性和效率。给出了一系列数值示例，说明了具有不同材料特性的FGPP光束的自由振动和瞬态振动。研究了幂律指数，孔隙率，施加的电压和EF对FGPP光束的固有频率和瞬态动态响应的影响。特别是，首先提出了孔隙率特性与其他材料特性对FGPP梁动力特性的综合影响，这对于相应智能结构的优化设计具有重要意义。研究了幂律指数，孔隙率，施加的电压和EF对FGPP光束的固有频率和瞬态动态响应的影响。特别是，首先提出了孔隙率特性与其他材料特性对FGPP梁动力特性的综合影响，这对于相应智能结构的优化设计具有重要意义。研究了幂律指数，孔隙率，施加的电压和EF对FGPP光束的固有频率和瞬态动态响应的影响。特别是，首先提出了孔隙率特性与其他材料特性对FGPP梁动力特性的综合影响，这对于相应智能结构的优化设计具有重要意义。