The finite integral transform method is developed to explore the bending analysis of thin plates with the combination of simply supported, clamped, and free boundary conditions. Previous solutions mostly focused on simply supported and clamped boundary conditions, but the existence of free boundary conditions makes the solving process more complex, because it is difficult to find the exact solution which satisfies both deflection and internal force by conventional inverse/semi-inverse method or approximate method. Using this method, the plate high-order partial differential equation is simplified to a linear algebraic equation by the integral transformation. Then, through some mathematical manipulation, the analytical solution is elegantly achieved in a straightforward procedure. Compared with other methods, the present method is much simpler and general and does not need to pre-determine the deflection function, which makes it very attractive for calculating the mechanical responses of the plates. Comprehensive analytical results obtained in this paper illuminate the validity of the proposed method by comparison with the existing literature and finite-element method using (ABAQUS) software.

中文翻译：

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薄板的分析弯曲解决方案，其中两个相邻的边是自由的，而其他两个边则是通过有限积分变换法夹紧或简单支撑的

开发了有限积分变换方法，以通过简单支撑，夹紧和自由边界条件的组合来探索薄板的弯曲分析。先前的解决方案主要集中在简单支撑和约束的边界条件上，但是自由边界条件的存在使求解过程更加复杂，因为很难通过常规的逆/半逆方法找到同时满足挠度和内力的精确解。或近似方法。使用该方法，通过积分变换将板高阶偏微分方程简化为线性代数方程。然后，通过一些数学操作，可以在简单的过程中优雅地获得分析解决方案。与其他方法相比，本方法简单，通用，不需要预先确定挠度函数，这对于计算板的机械响应非常有吸引力。通过与现有文献和使用（ABAQUS）软件的有限元方法进行比较，本文获得的综合分析结果证明了该方法的有效性。