Abstract In this paper, the nonlinear bending analysis of Functionally Graded Porous (FGP) beams is investigated using an efficient numerical algorithm associating a meshless collocation technique uses the Multiquadric Radial Basis Function (MQRBF) approximation method and a higher-order Taylor series-based continuation procedure. Material properties of the FGP beams are described by adopting a modified power-law function taking into account the effect of porosities. Based on the First Order Shear Deformation Theory (FSDT) of beams with the von-Karman kinematic hypothesis, the strong form of nonlinear equations is established. For an efficient application of the proposed numerical approach, a quadratic matrix strong form of the problem is presented. The resulting nonlinear equations are solved numerically with the proposed algorithm which leaned on the following three steps: a higher-order Taylor series expansion to transform the quadratic nonlinear equations into a sequence of linear ones, a meshless collocation technique based on MQRBF approximation method to solve numerically the resulting linear equations and a continuation procedure to get the whole solution branch. To demonstrate the robustness of the developed algorithm, convergence and validation studies have been carried out. Furthermore, the effects of power-law index, porosity volume fraction, Young’s modulus ratio, loads and boundary conditions are investigated.

中文翻译：

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使用多二次径向基函数和基于泰勒级数的连续程序对功能梯度多孔梁进行非线性弯曲分析

摘要 在本文中，功能梯度多孔 (FGP) 梁的非线性弯曲分析使用一种有效的数值算法进行了研究，该算法将无网格配置技术使用多二次径向基函数 (MQRBF) 逼近方法和基于高阶泰勒级数的延拓程序。FGP 梁的材料特性通过考虑孔隙率的影响采用修正的幂律函数来描述。基于梁的一阶剪切变形理论(FSDT)，基于von-Karman运动学假设，建立了非线性方程的强形式。为了有效应用所提出的数值方法，提出了问题的二次矩阵强形式。得到的非线性方程用所提出的算法进行数值求解，该算法依赖于以下三个步骤：将二次非线性方程转换为线性方程序列的高阶泰勒级数展开，基于 MQRBF 近似方法的无网格配置技术求解数值计算所得的线性方程和延续过程以获得整个解分支。为了证明所开发算法的鲁棒性，已经进行了收敛和验证研究。此外，研究了幂律指数、孔隙体积分数、杨氏模量比、载荷和边界条件的影响。一种基于 MQRBF 近似方法的无网格配置技术，以数值方式求解所得线性方程，并采用连续过程来获得整个求解分支。为了证明所开发算法的鲁棒性，已经进行了收敛和验证研究。此外，研究了幂律指数、孔隙体积分数、杨氏模量比、载荷和边界条件的影响。一种基于 MQRBF 近似方法的无网格配置技术，以数值方式求解所得线性方程，并采用连续过程来获得整个求解分支。为了证明所开发算法的鲁棒性，已经进行了收敛和验证研究。此外，研究了幂律指数、孔隙体积分数、杨氏模量比、载荷和边界条件的影响。