## 3 Analytical Solution  
In this part of the research, an analytical solution based on the [PERSON]'s method was presented. To ensure the accuracy of the calculation, the analytical results were compared with the finite element results extracted from the FE software [34; 35]. Then, the analytical solutions were used to design the compartments and calculate different parameters including deflection, maximum tension, and thicknesses:  
\[p(x.y)=\sum\
olimits_{m-1}^{\infty}\sum\
olimits_{n-1}^{\infty}p_{mn}sin \frac{m\pi x}{a}sin\frac{n\pi y}{b}=p_{0} \tag{2}\]  
\[W(x.y)=\sum\
olimits_{m-1}^{\infty}\sum\
olimits_{n-1}^{\infty}a_{mn}sin \frac{m\pi x}{a}sin\frac{n\pi y}{b} \tag{3}\]  
where \(x\) and \(y\) are 2D rectangular coordinates. \(P(x,y)\) and \(P_{0}\) are the pressure function and constant pressure, respectively. \(a\) and \(b\) are the length and width, respectively, of a 2D rectangular shell plate on  
Figure 2: Final structures of reinforced compartments with T-shaped beams: (**a**) steel compartment, (**b**) aluminum compartment (depth of 20 m), and (**c**) aluminum compartment (depth of 60 m). any side of the compartment. Symptoms of \(m\) and \(n\) are counters. In Equation (3), \(W(x,y)\) represents the deflection function in terms of location, and the \(a_{mn}\) coefficient is defined as follows:  
\[a_{mn}=\frac{1}{\pi^{4}D}\frac{P_{mn}}{\left[\left(\frac{m}{a}\right)^{2}+\left( \frac{n}{b}\right)^{2}\right]^{2}} \tag{4}\]  
The parameter \(D\) expresses flexural rigidity of a plate or shell, and the 2D stress components can be calculated as follows:  
\[\sigma_{x}=-\frac{Ez}{1-\
u^{2}}\left(\frac{\partial^{2}W}{\partial x^{2}}+ \
u\frac{\partial^{2}W}{\partial y^{2}}\right) \tag{5}\]  
\[\sigma_{y}=-\frac{Ez}{1-\
u^{2}}\left(\frac{\partial^{2}W}{\partial y^{2}}+ \
u\frac{\partial^{2}W}{\partial x^{2}}\right) \tag{6}\]  
\[\tau_{xy}=-\frac{Ez}{1+v}\left(\frac{\partial^{2}W}{\partial x\partial y}\right) \tag{7}\]  
where \(\sigma_{x}\), \(\sigma_{y}\) are normal components of stress parallel to \(x\) and \(y\) axes, respectively. \(\tau_{xy}\) is the shearing stress component in the rectangular coordinate. Moreover, the Elastic modulus and Poisson's ratio are indicated by \(E\) and \(v\), respectively. In stress equations, z represents a distance from the middle surface. For the constant load of \(P_{0}\):  
\[ \begin{array}{l}P_{mn}=\frac{4}{ab}\int_{0}^{b}\int_{0}^{a}P_{0}sin\frac{m \tau_{xx}}{a}sin\frac{n\tau y}{b}dxdy\\ =\frac{4}{mm\pi^{2}}\left(\left[\left(-1\right)^{n}-1\right]\left[\left(-1 \right)^{m}-1\right]\right)\end{array}  \tag{8}\]  
Equation (8) can be simplified as follows:  
\[p_{mn}= \begin{cases}\frac{16p_{0}}{mm\pi^{2}}&\text{$m.n$ is not odd}\\ 0&\text{$other}\end{cases}  \tag{9}\]  
By replacing the \(p_{mn}\) in Equation (4):  
\[a_{mn}=\frac{16p_{0}}{mm\pi^{2}}\ast\frac{1}{mm\left[\left(\frac{m}{a}\right) ^{2}+\left(\frac{n}{b}\right)^{2}\right]^{2}} \tag{10}\]  
Thereupon, the parameter of \(a_{mn}\) is placed in the deflection equation:  
\[W(x.y)=\sum\sum\frac{16P_{0}}{\pi^{6}D}\frac{1}{mm\left[\left(\frac{m}{a} \right)^{2}+\left(\frac{n}{b}\right)^{2}\right]^{2}}sin\frac{m\pi x}{a}sin \frac{n\tau y}{b} \tag{11}\]  
Since \(m\) and \(n\) are odd, \(\tau_{xy}\) is always equal to zero. In the next step, the characteristics of the reinforced plates in the form of T-shaped reinforcements were obtained to minimize the total weight of the structure. In this regard, the required number of reinforcements was obtained as follows:  
\[\frac{1}{12}bh^{3}+bh(k-\frac{h}{2})^{2}+n\left[I_{XX}+S(e_{XX}+h-k)^{2}\right] =\frac{1}{12}bt^{3} \tag{12}\]  
and  
\[k=\frac{bh\frac{h}{2}+ns(e_{XX}+h)}{bh+ns} \tag{13}\]  
where \(b\), \(h\), and \(t\) represent the width, height, and thickness of sheet, respectively. The variable of \(n\) is number of reinforcements, \(e_{x}\) means the moment of inertia. s is the reinforcement cross-section,and \(e_{XX}=h-e_{X}\). The number of reinforcements was obtained based on the calculations done for the T-shaped reinforcements.