# Numerical Analysis of the Bottom Thickness of Closed Rectangular Tanks Used as Pontoons
## 3 Results  
### Static Analysis of the Tanks  
By using the finite difference method in a variational approach, static calculations were made for five closed monolithic rectangular tanks with axial dimensions \(\text{l}_{\text{x}}\text{:}\text{l}_{\text{y}}\text{:}\text{l}_{\text{z}}=4\text {:}1\text{:}0.5\), assuming in each case thicker bottom plates compared to the thickness of tank walls. The bottom thickness of the pontoon was determined as \(\text{h}\), and the calculated bottom thickness values for individual tanks were \(1\) h, \(1.25\) h, \(1.5\) h, \(1.75\) h and \(2.0\) h. The division grid was adopted with square cell shape, with the mesh size equal to \(\text{l}_{\text{y}}/12\). With this division grid, by taking into account the plane of symmetry, a system of equations with 823 unknowns was obtained for each half of the tank. The following loads were assumed: hydrostatic load acting on tank walls, uniform load acting on the bottom (Figure 2a) and uniform load acting on the upper plate of the tank (Figure 2b). Figure 1: The schematic drawing of the pontoon under constant load (**a**) and the schematic drawing of the pontoon under constant and variable load (**b**). G, gravity centre of the pontoon; B, buoyancy centre of the pontoon at rest; K, keel; \(\text{h}_{\text{d}}\), immersion due to dead weight load; \(\text{h}_{\text{g}}\), location of gravity centre; C, centre of the visible area of the water surface; M, metacentre; \(\text{W}_{1}\), total weight of constant and variable loads; B’, buoyancy centre of the loaded pontoon; F, floating forces in water; \(\text{h}_{\text{mc}}\), metacentric height above gravity centre; \(\text{h}_{\text{mb}}\), metacentric height above buoyancy centre; \(\text{h}_{\text{1d}}\), height of buoyancy centre. Figure 2: Schematic drawing of the hydrostatic load acting on tank walls and the bottom (**a**) and uniform load acting on the upper plate (**b**) for which static calculations were made. Tables 1 and 2 summarise coefficients proportional to deflections and coefficients proportional to bending moments for selected characteristic points of the tanks shown in Figure 3. To calculate definite values of bending moments and deflections, the \(\beta\) and \(\beta\)1 coefficients provided in Tables 1 and 2 should be multiplied by \(10^{-2}\)q\({}_{1}\) ly\({}^{2}\) for bending moments and \(10^{-4}\)q\({}_{1}\) ly\({}^{4}\)/D for deflections, where D = Eh3/12 [32,33]. In order to analyse the effect of the bottom thickness on the change in deflections and bending moments, charts were produced that show the distribution of these values in the cross-sections running in the middle of the analysed tanks. The charts are shown in Figures 6-9. Figure 6 shows the distribution of deflections in the cross-section of the tank at half of its length from the acting hydrostatic load. This load causes the greatest bending moments in the bottom plate. As the thickness of the bottom plate increases, the value of the bending moment in the bottom in the middle of the tank width increases, while clamping moments decrease. The values of bending moments in the upper plate and tank walls also decrease. Figure 8 shows the distribution of deflections in the cross-section of the tank at half its length from the evenly distributed load acting on the upper plate. This load causes the greatest values of deflections in the upper plate. As the thickness of the bottom plate increases, the deflection in the bottom decreases, and the deflections in the upper plate and tank walls also decrease. Figure 9 shows the distribution of bending moments in the cross-section at half of its length from the evenly distributed load acting on the upper plate. This load causes the greatest bending moments in the upper plate. As the thickness of the bottom plate increases, the value of bending moment in the bottom, both in the middle of its width and the clamping moment, increases.