The buckling load as well as the natural frequency under axial load for non-prismatic beam is a changeling problem. Determination of buckling load, natural frequency, and elastic deflection is very important in civil applications. The current paper used both perturbation method (PM), analytic method, and differential quadrature method (DQM), numerical method, to find buckling load and natural frequency with different end supports. The deflection of the beam resting on an elastic foundation under transverse distributed and axial loads is also obtained. Both PM and DQM are used for non-prismatic beams with rectangular and circular cross sections in the vibration analysis. The comparisons of results obtained from both PM and DQM showed perfect agreement with analytical solution for uniform beams with different end supports. The PM and DQM succeeded powerfully for investigating the buckling load as well as the natural frequency for non-prismatic beam. The percentage of relative error between DQM and PM doesn’t exceed than 5% if the gradient of rectangular section height and the gradient of circular section radius are less than 0.6. As the gradient of height and radius increase, the maximum deflection decreases and the location of maximum deflection displaced toward the smaller moment of inertia. The PM has not been used for solving the problem of non-prismatic beams resting on elastic foundations subjected to transverse distributed and axial loads. The current research proved the good ability of PM as an analytical solution for a complicated problem and defined its range of accuracy as compared to DQM. Also, it introduced accurate empirical formulae to find both natural frequency and buckling load of non-prismatic beams. These empirical formulae represent a good achievement in vibration analysis of non-prismatic beams.

中文翻译：

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弹性地基非棱柱梁变形解析与数值分析

非棱柱梁在轴向载荷下的屈曲载荷和固有频率是一个变化问题。屈曲载荷、固有频率和弹性挠度的确定在民用应用中非常重要。目前的论文同时使用了微扰法（PM）、解析法和微分正交法（DQM）、数值方法，来寻找不同端部支撑的屈曲载荷和固有频率。还获得了在横向分布和轴向载荷下放置在弹性基础上的梁的挠度。PM 和 DQM 都用于振动分析中具有矩形和圆形横截面的非棱柱形梁。从 PM 和 DQM 获得的结果的比较表明，与具有不同端部支撑的均匀梁的解析解完全一致。PM 和 DQM 成功地研究了非棱柱梁的屈曲载荷和固有频率。矩形截面高度梯度和圆形截面半径梯度小于0.6时，DQM与PM的相对误差百分比不超过5%。随着高度和半径梯度的增加，最大挠度减小并且最大挠度的位置向更小的转动惯量移动。PM 尚未用于解决非棱柱形梁在承受横向分布和轴向载荷的弹性基础上的问题。当前的研究证明了 PM 作为复杂问题的分析解决方案的良好能力，并定义了与 DQM 相比的精度范围。还，它引入了精确的经验公式来计算非棱柱梁的固有频率和屈曲载荷。这些经验公式代表了非棱柱梁振动分析的良好成果。