# Semianalytical Lower-Bound Limit Analysis of Domes and Vaults
## 2 Equilibrium and Self-Equilibrium of Domes  
This section briefly recalls a spherical shell equilibrium equation; the complete treatment can be found in [21]. The formulation used generalized stress as mechanical parameters; hence, the stress is described through internal forces N, T, and M that are the resultant components, axial, and shear forces, respectively, and the resultant bending moment of the stress acting on the section; see Figure 1. Only the axial force and bending moment were assumed to influence the structure's safety, since the shear effects were negligible due to the small thickness of the caps. The internal forces were statically equivalent to a single one of magnitude N acting on point C called the center of thrust of the section (Figure 2). The line connecting all the centers along the structure's axis is called the structure's thrust line, depending on the actual load condition. The procedure that follows was widely described in [16; 18]. The solution of the homogeneous form of the balance equations represents the domain of the eigenstress of the shell. The set of equilibrium equations of a spherical dome had the following expression, with the symbols summarized in Figure 3:  
\[ \begin{cases}\frac{d(N_{1}R_{1}sin\theta)}{d\theta}-N_{2}R_{1}cos\theta-T_{1 }R_{1}sin\theta&=-XR_{1}^{2}sin\theta\\ N_{1}R_{1}sin\theta+N_{2}R_{1}sin\theta+\frac{d(T_{1}R_{1}sin\theta)}{d\theta} &=ZR_{1}^{2}sin\theta\\ \frac{(M_{1}R_{1}sin\theta)}{d\theta}-M_{2}R_{1}cos\theta-T_{1}R_{1}^{2}sin \theta&=0\end{cases}  \tag{1}\]  
Due to the symmetry of the loads and shell, the stress depended only on colatitude angle \(\theta\). Hence, self-equilibrated generalized stress set \(\mathcal{C}=\left\{N_{1}^{0},M_{1}^{0},T_{1}^{0},N_{2}^{0},M_{2}^{0}\right\}\) satisfies the following equations:  
\[ \begin{cases}\frac{d(N_{1}^{0}r)}{d\theta}-N_{1}^{0}R_{1}cos\theta-T_{1}^{0}r =0\\ N_{1}^{0}r+N_{1}^{0}R_{1}sin\theta+\frac{d(T_{1}^{0}r)}{d\theta}=0\\ \frac{d(M_{1}^{0}r)}{d\theta}-M_{2}^{0}R_{1}cos\theta-T_{1}^{0}R_{1}r=0\end{cases}  \tag{2}\]  
Figure 1: Stress resultants on the dome infinitesimal element. Figure 3: (a) Geometric parameters; (b) infinitesimal element of the dome. Figure 2: Center of thrust at a cross-section. A set of polynomial shape functions of degree \(n\) are used to the numerical approximation of the solution of Equation (2). \[S_{n}(\theta)=\left[\theta^{0},\theta^{1},\ldots,\theta^{n}/n!\right] \tag{3}\]  
whose derivatives are:  
\[dS_{n}(\theta)=\left[0,1,\theta,\ldots,\frac{\theta^{n-1}}{(n-1)!}\right]=[0] \cup S_{n-1}(\theta) \tag{4}\]  
Five nodal parameters sets were used to approximate the five unknown self-equilibrated stresses. Using the same approximation for all stress functions results in the following expressions:  
\[ \begin{cases}N_{1}^{0}&=S_{n}(\theta)\mathbf{N_{1}}\\ N_{2}^{0}&=S_{n}(\theta)\mathbf{N_{2}}\\ M_{1}^{0}&=S_{n}(\theta)\mathbf{M_{1}}\end{cases}\begin{cases}\mathbf{N_{1}}&= [n_{10},n_{11},\ldots,n_{1n}]^{T}\\ \mathbf{N_{2}}&=[n_{20},n_{21},\ldots,n_{2n}]^{T}\\ \mathbf{M_{1}}&=[n_{10},m_{11},\ldots,m_{1n}]^{T}\\ \mathbf{M_{2}}&=[m_{20},m_{21},\ldots,m_{2n}]^{T}\\ \mathbf{T_{1}}&=[t_{10},t_{11},\ldots,t_{1n}]^{T}\\ \end{cases}\begin{cases}\frac{d\left(N_{1}^{0}r\right)}{d\theta}=dS_{n}( \theta)\mathbf{N_{1}}\\ \frac{d\left(N_{1}^{0}r\right)}{d\theta}=dS_{n}(\theta)\mathbf{N_{2}}\\ \frac{d\left(M_{1}^{0}r\right)}{d\theta}=dS_{n}(\theta)\mathbf{M_{1}}\\ \frac{d\left(M_{1}^{0}r\right)}{d\theta}=dS_{n}(\theta)\mathbf{M_{2}}\\ \frac{d\left(T_{1}^{0}r\right)}{d\theta}=dS_{n}(\theta)\mathbf{T_{1}}\\ \end{cases}  \tag{5}\]  
By substituting Equation (5) into Equation (2), and collecting the unknown parameters into a single vector \(\mathbf{x}=[\mathbf{N_{1}},\mathbf{N_{2}},\mathbf{M_{1}},\mathbf{M_{2}}, \mathbf{T_{1}}]\), the self-equilibrium equations can be rewritten in compact matrix form:  
\[\mathbf{A}\mathbf{x}=\mathbf{0} \tag{6}\]  
where \(\mathbf{A}\) is:  
\[\mathbf{A}(\theta)=\left[