均布和静水压力作用下固支矩形薄板力学特性

Mechanical properties of clamped rectangular thin plate under uniform load and hydrostatic pressure

利用重三角级数构造了均布和静水压力两种载荷作用下四边固支弹性矩形薄板的挠度函数,并依据最小势能原理,求解了构造挠度函数的系数。由于选取了完全满足边界条件的重三角级数作为四边固支矩形薄板的挠度函数,不需要采用叠加法或简化其边界条件,使得四边固支矩形薄板弯曲问题的求解过程思路清晰、计算简便,简化了现有解析方法求解的繁琐过程。最后,采用弹性薄板理论,对两种载荷作用下矩形薄板的应力、内力等力学特性进行了对比分析。结果表明:1两种不同载荷作用下,固支矩形薄板最大拉应力、压应力、剪应力、横向剪力的位置明显不同;2均布载荷作用下固支矩形薄板在其两条长边中点的位置拉应力较大,中部位置压应力较大,而在(0.5,2 m)和(1.5,2 m)的位置横向剪力最大;3静水压力载荷作用下固支矩形薄板仅其一条长边中点的位置拉应力较大,中部偏下的位置压应力较大,而在(0.80,2 m)和(1.69,2 m)的位置横向剪力最大;4两种载荷作用下,固支矩形薄板长边中点的位置容易出现拉破坏而导致其失稳,剪应力较大的位置则容易出现剪切破坏。

Using the double Fourier series, the deflection function of a rectangular thin plate with four edges clamped under uniform load and hydrostatic pressure is proposed, and the coefficient of the proposed deflection function is solved based on the principle of minimum potential energy. Due to the complete satisfaction with the boundary conditions for the double Fourier series that selected as the deflection function during solving process, and do not need to adopt the superposition method or simplify the boundary conditions, the solving process is simple in form and calculation process. Compared with the analysis of existing methods, the computational complexity of the used method is reasonable, which can simplify the complex solving process on the bending problem of rectangular thin plate with four edges clamped. Finally, the mechanical properties of the clamped rectangular thin plate under these two kinds of loads are compared by using the elastic thin plate theory, including stress, internal force and so on. Results obtained indicate that the position of maximum tension stress, compressive stress, shear stress and transverse shear stress is obviously different for these two kinds of loads. Under the uniform load, the tensile tress in the long side midpoint of the four edges clamped rectangular thin plate is larger and the compressive stress in the middle of the clamped rectangular thin plate is larger, and themaximum transverse shear stress is in the position of(0.5, 2m) and(1.5, 2m). While under the hydrostatic pressure, the tensile tress appeared only in one of the long side midpoint of the four edges clamped rectangular thin plate and the compressive stress in the downward position of the middle part of the clamped rectangular thin plate is larger as well as the maximum transverse shear stress is in the position of(0.80, 2m) and(1.69, 2m). For these two kinds of loads, the position of the long side midpoint of the clamped rectangular thin plate will more easily fail due to tensile yield, and the position with large shear stress will also fail due to shear yield.