This paper aims to analyze arbitrarily loaded isotropic rectangular thin plates with two opposite edges supported (simply supported or clamped), from which one or both are clamped, and the other edges have arbitrary support conditions. Rectangular thin plates with two opposite edges simply supported are routinely analyzed using for the deflection the simple trigonometric series of Lévy or the double trigonometric series of Navier, while the loads are expanded in Fourier series. In this paper the flexibility method (or force method) of the linear beam theory was applied whereby the unknowns were the bending moments along the opposite edges; in this regard, the primary system was the plate simply supported along the above-mentioned opposite edges and subjected to the external loads, system solved using the Lévy solution, and the redundant system was the plate simply supported along the opposite edges and subjected to bending moments along those edges. In the redundant system, a modified Lévy solution was introduced to account for the edge moments. The compatibility equations (vanishing of the slopes at selected positions of the opposite edges) were set to determine the unknowns and so the efforts in the plate. The results obtained were in good agreement with the exact results.

Although the Slicing Method (SM) is effective for calculating the volume of point cloud objects (PCOs), it is restricted in terms of applicability and practicability because of a certain contingency and directional defects. The Co-Opposite-Direction Slicing Method (CODSM) proposed in this paper is an improved method for calculating PCO volume by increasing parallel (co-opposite-direction) observation and considering the two-way mean as the result. This method takes full advantage of the mutual offsetting of random errors and the compensation of systematic directional errors, which can effectively overcome (or mitigate) the effect of random errors and reduce the effect of systematic errors in SM. In this paper, two typical objects, a cone model and a stone lion base, are the examples for calculating PCO volume using CODSM. The results show that CODSM has all the inherent advantages of SM and effectively weakens the volatility of random errors and the directionality of systematic errors from SM. Therefore, CODSM is a robust configuration upgrade of SM.