# An Innovative Construction Technique for Curved Structures
## 3 Design Procedure  
This section aims to report a design-oriented model for the geometry of the HLA. In the absence of external force, a suspended body reaches a stable equilibrium depending on the gravity force. The rotation can be defined by the GRA, which strongly correlates with the aspect ratio (length over width) of the body itself. For this reason, the relationship between GRA and \(l_{m}\)/s was investigated, and results are reported in Figure 4, where \(l_{m}\) is the length of the middle line of each block, and s is the width of the block. Demonstrably, when an annular section segment is lifted up, as illustrated in Figure 4, the GRA is the same as the angle reached by an Equivalent Rectangle (ER). The segment connecting the CIR and the intercept with the horizontal (passing through G)--is the first side of the ER. The segment connecting the intercept of the horizontal axis with the normal line to the free face of the block, passing through CIR--is the second side of the ER.  
Figure 4: Correlation between the Gravity Rotation Angle (GRA) and the aspect ratio of the block.  
Figure 3: A block of the HLA: base supported (_left_) and suspended with respect to the Center of Instant Rotation (CIR), in stable equilibrium (_right_).  
Given the nomenclature illustrated in Figure 5, the design procedure in Figure 6 can be implemented. The goal is to define the optimal aspect ratio so that all the blocks of the HLA are in contact when the arch is completely lifted. In this way, potential defects due to the positioning of the HLA are prevented. In fact, in the case of the wrong aspect ratio of the blocks, especially the terminal ones, unwanted movement of the HLA can be expected during the positioning of the support-devoted structure (e.g., buttresses).  
Figure 5: Nomenclature of the Hinged Lifting Arch (HLA).  
Figure 6: Flowchart of the proposed analytical model for the design of the HLA.  
The steps for application of the proposed model are as follows:  
1. Definition of the inputs. In particular, the geometry of the arched element, the mechanical characteristics of the substrate, and the mechanical and geometric characteristics of the FRP material.
2. Determination of the effective bond length, \(l_{\text{ed}}\), for detachment or peeling. The weight of the half arch has to be considered as a debonding force.
3. Calculation of the geometric parameters, such as: 1. Impose \(l_{\text{ed}}\) according to \(l_{D}\geq l_{\text{ed}}\); 2. \(\beta=\frac{180l_{D}}{\pi R_{e}}\); 3. \(l_{m}=\frac{\pi R_{m}\beta}{180}\); 4. \(GRA=\text{arctg}\left(\frac{l}{b}\right)\) according to Figure 4.
4. Verification of the angles. If GRA\(\leq\)\(\alpha\), the verification is satisfied, and then the input geometry is valid for the purposes of the construction technique. If not, one of the following solutions should be chosen: 1. Impose \(GRA_{new}\leq\alpha\) and determine \(l_{m,new}=tg(GRA)\) 2. Calculate a new aspect ratio by \(s_{new}=\frac{l_{m}}{tg(GRA)}\) 3. Impose \(\alpha_{new}\geq\max(\beta;GRA)\) 5. Calculation of the geometric characteristics related to the final configuration: 1. Number of blocks, \(n=\frac{\pi R_{m}-\frac{R_{m}\beta}{10}}{l_{m,new}}\) (odd and integer number); 2. \(l_{m}=\frac{\pi R_{m}}{n}\); 3. \(\beta_{d}=\frac{180l_{D}}{\pi R_{e}}\); 4. \(l_{D}=\frac{\pi R_{e}\beta_{d}}{180}\geq l_{\text{ed}}\). In other words:
* Solution \(a\) consists of decreasing the number of blocks.
* Solution \(b\) consists of reducing the arch thickness.
* Solution \(c\) consists of changing the global shape of the arch. Combined solutions are also valid. Alternatively, the use of advanced numerical methods can be considered for the assessment of such structures, e.g., in [23; 24; 25; 26].