# The Influence of the Load Term of the Strain Energy Sensitivity in the Shell Shape Optimization
## 4 Effect of the Load Term on Strain Energy Sensitivity  
The load vector \(\mathbf{F}\) in the global coordinate system can be divided into components of the external load and the self-weight load, which can be expressed by formula (11). \[\mathbf{F}=\mathbf{F}^{e}+\mathbf{F}^{m} \tag{11}\]  
\(\mathbf{F}^{e}\) and \(\mathbf{F}^{m}\) respectively represent the load vector in the global coordinate system formed by external load and self-weight weight. The load vector \(\mathbf{F}^{e}\) of the vertical distribution load is the function of the element area and the load vector \(\mathbf{F}^{m}\) of the self-weight is a linear function of the thickness t and the element area. For \(\mathbf{F}=\sum\mathbf{T}^{T}\mathbf{F}^{L}\), Equation (12), the partial derivatives of the load vector in the global coordinate system with respect to zj which is the coordinate of node j, can be obtained as follows. \[\frac{\partial\mathbf{F}}{\partial z_{j}}=\sum\left(\frac{\partial\mathbf{T}^{T}}{ \partial z_{j}}\mathbf{F}^{L}+\mathbf{T}^{T}\frac{\partial\mathbf{F}^{L}}{\partial z_{j}}\right) \tag{12}\]  
where \(\mathbf{T}\) represents the coordinate transformation matrix of the element, and \(\mathbf{F}^{L}\) represents the load vector in the local coordinate system of the element. \(\sum\) means the sum of all nodes. Correspondingly, the load vector for each element is composed of two parts, as shown in Equation (13). \[\mathbf{F}^{L}=\mathbf{F}^{Le}+\mathbf{F}^{Lm} \tag{13}\]  
\(\mathbf{F}^{Le}\) is the element load vector produced by the external load of the element; \(\mathbf{F}^{Lm}\) is the element load vector produced by the element weight. Thus, the Equation (12) can be rewritten as Equation (14),  
\[\frac{\partial\mathbf{F}}{\partial z_{j}}=\sum\left(\frac{\partial\mathbf{T}^{T}}{ \partial z_{j}}\mathbf{F}^{Le}+\mathbf{T}^{T}\frac{\partial\mathbf{F}^{Le}}{\partial z_{j }}\right)+\sum\left(\mathbf{T}^{T}\frac{\partial\mathbf{F}^{Lm}}{\partial z_{j}}+ \frac{\partial\mathbf{T}^{T}}{\partial z_{j}}\mathbf{F}^{Lm}\right) \tag{14}\]  
In the above Equation, the first term is the external load term, and the second term is the structural weight term. When the external load is vertically uniformly distributed \(q_{z}\) and the external load remains unchanged in the evolution process, the vertical load vector produced by the element in the global coordinate system is \(\left[ \begin{array}{ccc}0&0&q_{z}\overline{\Delta}\end{array} \right]\), in which \(\overline{\Delta}\) is the area of the triangular element enclosed by the horizontal projection coordinate of the three nodes. Correspondingly, the self-weight of the element is \(\left[ \begin{array}{ccc}0&0&q_{z}\\ \end{array} \right]=\left[ \begin{array}{ccc}0&0&\Delta\end{array} \right]\mathbf{ t}\rho\). Here, \(\Delta\) is the area of the triangle element. When the thickness of the shell structure is constant, the role of self-weight and the vertically uniformly distributed \(q_{z}\) will play the same role because the relationship between them is fixed. When the external load forms the load vector, the load of a node is determined according to 1/3 of the sum of the area of the element containing the node. Then, since the curved surface node only changes in z coordinate, the horizontal projection area \(\overline{\Delta}\) remains unchanged. Although the element size changes, the external load keeps constant, namely, \(\frac{\partial\mathbf{F}^{Le}}{\partial z_{j}}=0\). If the external load is a point load, the change of node position does not change the size of the load and the corresponding load vector. So, \(\frac{\partial\mathbf{F}^{\mathrm{Lm}}}{\partial z_{j}}=0\). Equation (14) can be written as Equation (15).