# Semianalytical Lower-Bound Limit Analysis of Domes and Vaults
## 6 Conclusions  
This work presented a comparison between the semianalytical method developed in [18] with experiments reported in [14] and numerical calculations in [13; 20]. The SAM, whose formulation was developed for spherical domes of a general shape under a generic load pattern, was particularized to slender caps to compare the strategy with the experimental results and theoretical methods on the basis of fracture lines and the kinematic approach. The constitutive law and the corresponding limit domain of the structural material were derived considering the not-tensile-resistant material model of the unreinforced concrete, such as that used for the reference concrete caps in [13; 14; 20]. The collapse multiplier was determined by evaluating the existence of a statically admissible state of stress under loads of prescribed intensity. The statically admissible stress state consisted of the sum of the semianalytical solution of the balance equation of spherical domes and the simple linearly elastic solution under the prescribed load, obtained through a finite-element linear solution. The comparison between the experimental results and the simplest closed-form solution of domes under radial pressure shows that the method could accurately predict the collapse load. The method can be applied to spherical domes under axial-symmetrical loads; moreover, it can be applied to randomly variable loads to obtain the shakedown limit of the structures without any complications. The method requires the knowledge of a linearly elastic solution under the prescribed loads; this solution can be derived from a closed-form one if it exists or from any numerical solution obtained through well-established methods. However, the required elastic solution is obtained by only using linearly elastic methods and does not require any iterative ultraelastic calculation. When the collapse multiplier for a prescribed load path must be numerically calculated, one must apply an iterative strategy and nonlinear plasticity constitutive laws, with the latter consisting not only of a limit domain, but also of the flow rule that influences the calculation effort and the computational time [24]. Lastly, the proposed procedure also calculates the structure's Melan residuals and the compatible stress distribution at the incoming collapse that can be used to calculate the upper bounds of the plastic dissipation at the collapse [19]. In [17; 18], the procedure was applied to domes of revolution of different shapes and under several load conditions. In particular, parabolic and conical domes were analyzed with the geometry setup inspired by Santa Maria del Fiore and Saint Paul domes, respectively. The domes were loaded with uniform radial pressure, radial outward load, radial inward load, and vertical load on a generic parallel. The obtained outcomes were compared to FE-based commercial code ones. In the present work, the results were compared to experimental data, confirming the proposed procedure's feasibility. In conclusion, the results obtained in the present work were first compared with the analytical solution proposed by [20], and then with the numerical experiment proposed by [14]. The proposed method demonstrates its ability to obtain the load multiplier with no assumptions about the collapse mechanism. Independence from collapse-mechanism knowledge is another relevant advantage of such a static approach that allows for formulating the limit equilibrium of the structures with no a priori intuition of kinematics and collapse shape. The proposed strategy will be applied to a broader set of geometry and load conditions, paving the way to a standard procedure for collapse-load multiplier evaluation. Conceptualization, R.Z. and V.M. ; methodology, V.M. ; software, V.M. and R.Z. ; validation, R.Z., V.M. and L.E. ; formal analysis, V.M. ; investigation, R.Z., S.P., L.E., E.T. and V.M. ; resources, V.M. ; data curation, R.Z., S.P. and L.E. ; writing--original draft preparation, R.Z. ; writing--review and editing, R.Z. and V.M. ; visualization, R.Z., V.M.