# Numerical Analysis of the Bottom Thickness of Closed Rectangular Tanks Used as Pontoons
## 2 Materials and Methods  
### Calculation Method  
Monolithic rectangular tanks are complex plate structures characterised by spatial static work, so the load acting on one fragment of the structure causes displacement and creates cross-sectional forces throughout the system. Failure to consider it in the static calculations leads to incorrect determination of internal forces (bending moments and cutting forces), which in turn affects the value and distribution pattern of the adopted reinforcement and the correct design of the structure. One of the numerical methods that can be successfully used in calculating these structures is the finite difference method in a variational approach. It takes into account the spatial work of structures, the real geometric dimensions (E and \(\
u\)) corresponding to the material from which the tank will be made and any type of acting load, including temperature load. Apart from the finite element method, the finite difference method is one of the most popular numerical methods used for static calculations of building structures. The subject matter has been taken up in numerous outstanding and fundamental scientific works [13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26]. The method has been used in calculations regarding plate structures [3; 27; 28], tanks [2; 7] or surface girders. The finite difference method (FDM) is one of the methods used to solve differential equations that are difficult or impossible to solve analytically. It is based on the energy functional accumulated in the deformed system. The finite difference method adopted for the numerical analysis was validated in [2]. The verification process of this method consisted in calculating the deflections of the tank using the traditional approach, i.e., the solution of the matrix of the displacement equation system using the finite difference method and the calculation of the tank with a specialised computer program based on the finite element method and on measurements of actual deflections using a coordinate measuring arm with a contact head [2]. The results found the finite difference method was suitable for solving the given analytical problems. The solutions presented in the paper were obtained using the energy functional presented in the formula below (1) [23]. \[ \begin{array}{rcl}\mathrm{V}&=&\frac{\mathrm{D}}{2}\int\limits_{\mathrm{A}} \left\{\left(\frac{\partial^{2}w}{\partial x^{2}}\right)^{2}+2\left(\frac{ \partial^{2}w}{\partial x\partial y}\right)^{2}+\left(\frac{\partial^{2}w}{ \partial y^{2}}\right)^{2}+2v\left[\frac{\partial^{2}w}{\partial x^{2}}\frac{ \partial^{2}w}{\partial y^{2}}-\left(\frac{\partial^{2}w}{\partial x\partial y }\right)^{2}\right]\right.\\ &&\left.+2(1+v)\frac{\alpha_{\mathrm{A}}\Delta\Gamma}{\mathrm{h}}\left(\frac{ \partial^{2}w}{\partial x^{2}}+\frac{\partial^{2}w}{\partial y^{2}}+\frac{ \partial^{2}\Delta\Gamma}{\mathrm{h}}\right)\right]\mathrm{d}\mathrm{A}+ \frac{1}{2}\int\limits_{\mathrm{A}}\mathrm{Kw}^{2}\mathrm{d}\mathrm{A}\ -\int\limits_{\mathrm{A}}\mathrm{qwdA}\end{array}  \tag{1}\]  
where \(\mathrm{D}=\frac{\mathrm{Eh}^{3}}{12(1-v^{2})}\) is plate flexural rigidity, \(\mathrm{E}\) is elasticity modulus of material, \(\mathrm{v}\) is Poisson's ratio, \(\mathrm{h}\) is plate thickness, \(\mathrm{w}\) is plate deflection, \(\mathrm{q}\) is load perpendicular to the central surface of the plate, \(\mathrm{A}\) is plate area, \(\Delta\Gamma\) is difference in temperature between lower plate \(\mathrm{T_{d}}\) and upper plate \(\mathrm{T_{g}}\) determined by correlation: \(\Delta\Gamma=\mathrm{T_{d}}-\mathrm{T_{g}}\), \(\alpha_{\mathrm{t}}\) is coefficient of thermal expansion of the plate material and \(\mathrm{K}\) is subgrade stiffness reaction.