# Graphical and Analytical Quantitative Comparison
## 3 Problem Statement
### Materials and Methods
#### 3.3.2. Modified Thrust Line Method  
The MTLM is a graphic static tool developed by [PERSON] in 2006 [PERSON] (2006). Similarly to TLA, it allows for assessment of the state of axisymmetric masonry structures through the use of the thrust line. Indeed, according to the Safe Theorem, a balanced state is guaranteed if at least one thrust line lies entirely within the thickness of the analyzed structure. Differently from TLA, the lines of thrust obtained through MTLM also consider the contribution of the hoop forces. This implies that, in addition to the Heyman premises, this method requires further assumptions for the assessment of the equilibrium of the dome ([PERSON] and [PERSON], 2017)--namely, the load has to be axisymmetric, as well as the geometry of the structure. Under these additional constraints, the state of a dome is determined by studying the equilibrium of only one lune, representative of the whole dome. As shown in Figure 4, the lunes are obtained by dividing the structure into equal parts with vertical radial planes. Similarly to TLA, the lune is divided into blocks referred to with a index \(i\), running from the impost to the top of the vault, with mass \(m_{i}\) and centroid \(C_{i}\). For each block, one point \(A_{i}\) is determined as the intersection of the vertical line passing through \(C_{i}\) and the resultant of all the loads of the blocks from 1 to the current block \(i\). Therefore the thrust line is obtained as the set of all points \(A_{i}\). The determination of each point \(A_{i}\) follows the same logic as that adopted for TLA; it is affected by all forces and requires a solution to the equations of equilibrium of each block. \[\mathbf{P}_{i}-\mathbf{R}_{i}+\mathbf{w}_{i}+\mathbf{H}_{ri}-\mathbf{H}_{li}=0 \tag{2}\]  
For each block, \(i\). Considering a generic \(i\)-th block, Equation (2) expresses the equilibrium at the point \(\mathbf{A}_{i}\), and involves the thrust \(\mathbf{P}_{i}\) of the upper blocks, the reaction \(-\mathbf{R}_{i}\) from the lower blocks, the weight of the block itself \(\mathbf{w}_{i}\) and the two forces \(\mathbf{H}_{ri}\) and \(\mathbf{H}_{li}\) which represent the action of the hoop forces acting on the block. Due to static indetermination, Equation (2) cannot be solved: the thrust, reaction and the hoop forces are all unknown. For each block \(i\), the terms of Equation (2) are \(-\mathbf{R}_{i}\), \(\mathbf{H}_{ri}\), \(-\mathbf{H}_{li}\) and \(\mathbf{w}_{i}\); thanks to the symmetry of load and structure, the two hoop forces are equal and the unknown variables are two (the radial thrust and the hoop forces). Furthermore, for all blocks, the reaction which acts on the upper rigid body is the reverse of thrust applied on the lower block. Therefore, by estimating the hoop forces and starting from the uppermost block, the position of all points \(A_{i}\) is identified, and the thrust line is traced. To solve Equation (2), the hoop forces are estimated by an optimization algorithm [23] or by graphical manipulation [14]. In the present work, the MTLM has been implemented into the graphical editor algorithm Grasshopper [38] for Rhinoceros software [39], and the hoop forces are estimated by graphical manipulation [14]. According to the further assumptions (load and geometry are axisymmetric) and considering the compressive forces to be positive, the hoop forces are sought based on the guiding criterion that they act normally to the lateral block face, and their magnitude decreases from the crown to the base of the dome. With MTLM, it is possible to compute the geometrical safety factor with the same general definition given in Equation (1). The difference with respect to TLA is that, with MTLM, the thrust surface depends not only on two parameters, but also on the value of the hoop forces at each voussoir. Therefore, it is practically impossible to check all the possible equilibrium curves, and the search for the GSF of a structure must be done using well-established optimization tools.