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采用变分原理,给出了地基弹性模量呈线性变化的Gibson地基上梁稳定性的解析方法。将非均匀地基以改进Vlasov双参数模型模拟,并对地基-梁系统的最小势能取变分,得到梁稳定性计算的控制微分方程,然后建立了特征参数γ与失稳模态的相关关系。梁的失稳荷载可通过求解特征值得到,采用简单的迭代程序可求得失稳荷载与失稳模态。结果表明:地基土表层特性对地基模型的竖向弹性和连续特性影响较大,地基下部的影响有限。地基非均匀性将影响梁的失稳荷载和失稳模态,上硬下软地基上梁失稳将需要更大的失稳荷载。
By using the principle of variational principle, an analytical method for the stability of a beam on a Gibson foundation with a linear change in the elastic modulus of the foundation is given. The nonhomogeneous foundation is improved to simulate the Vlasov two-parameter model, and the minimum potential energy of the foundation-beam system is taken as a variable to obtain the governing differential equation of the beam stability calculation. Then, the correlation between the characteristic parameter γ and the instability mode is established. Unstable load of beam can be obtained by solving eigenvalues. The instability load and instability mode can be obtained by a simple iterative procedure. The results show that the surface soil properties of foundation soil have a great influence on the vertical elastic and continuous characteristics of the foundation model, and the influence of the lower part of the foundation is limited. The nonuniformity of the foundation will affect the instability load and instability mode of the beam, and the instability of the upper beam on the soft foundation will require greater instability load.