Regarding the funicular analysis of arcuated structures, the literature provides several methods based on equilibrium, i.e. exploiting the static theorem of limit analysis upon assumption of the Heyman’s hypotheses (masonry is a no-tension material, it has unlimited compressive strength and no friction between blocks is allowed). Among the others, O’Dwyer presented an approach that linearizes the equilibrium equations of the nodes of the funicular network in terms of a given distribution of thrusts. This approach adopts, iteratively, methods of mathematical programming to maximize the load multiplier of the applied vertical loads. Block and Lachauer proposed a method exploiting the concept of force density, i.e. the ratio of the force to the length of any element in the grid. They searched for the closest funicular network with respect to the middle surface of a vault subject to given vertical loads. Unfortunately, this method does not allow for a local control of the height of the nodes in the network. The approach implemented in this thesis exploits the Force Density Method (FDM) to control point by point the coordinates of the funicular polygon/network through a multi-constrained optimization problem. This is solved by means of methods of mathematical programming that were originally conceived to handle large scale structural optimization problems. The adopted method is based on the detection of an independent sets of branches in the case of networks with fixed plan geometry and allows deriving funicular polygons/networks with minimum or maximum norm of the horizontal thrusts. The dependent force densities are written as a function of the independent ones, by exploiting the horizontal equilibrium. The nodal heights of the restrained nodes are used along with the set of independent force densities as unknows of the problem. The vertical coordinate of each node of the funicular polygon/network is constrained, through local enforcements, to lie within the upper and lower bounds of the envelope of the arcuated structure. Additional constraints are imposed to enforce no-tension elements, according to the Heyman’s assumptions. The solution is sought by means of a sequential convex programming technique that solves a sequence of explicit subproblems which are convex approximations of the original one. The proposed method can be used to investigate structures subjected to both vertical loads (self-weight) and horizontal loads, such as earthquake ones. Moreover, it is possible to determine the collapse load multiplier of arcuated structures subjected to self-weight along with increasing vertical/horizontal loads by tackling a set of optimization problems. The same algorithm applies to 2D and 3D structures, converges for different geometries of the plane grid, and allows detecting load paths in arcuated structures with given geometry. All these features have been tested in 2D and 3D case studies in this thesis. Two arches of constant thickness (50 cm and 25 cm) have been studied, the Mosca Bridge in Turin (which is an arch with variable thickness), a radial wall over the dome of the Cathedral of Milan. Afterwards, hemispherical domes of different thickness (9 cm and 18 cm) have been investigated under the effect of self-weight only or the combined effect of self-weight and horizontal loads. The role of meridional and hoop forces has been especially investigated in the achievement of equilibrated solutions. Finally, octagonal domes under self-weight only have been analysed, assuming different geometries and degrees of refinement of the grids.
Nell’ambito dell’analisi funicolare di strutture arcuate in muratura (archi/volte/cupole), in letteratura sono disponibili diversi metodi che implementano il teorema statico dell’analisi limite basandosi sulle ipotesi di Heyman, ovvero assumendo che (i) la muratura non resiste a trazione, (ii) la resistenza a compressione è infinita, (iii) lo scorrimento di un blocco su un altro non è ammesso. Tra gli altri, O’Dwyer ha proposto un metodo che linearizza le equazioni di equilibrio dei nodi del reticolo funicolare in funzione di una distribuzione di spinte assegnata, applicando iterativamente metodi di programmazione matematica per massimizzare il moltiplicatore dei carichi verticali. Block e Lachauer hanno elaborato un metodo basato sull’utilizzo delle densità di forza, ovvero dei rapporti tra la forza e la lunghezza di ogni elemento del reticolo, implementando un approccio ai minimi quadrati per avvicinare il reticolo funicolare alla superficie media della volta sotto l’effetto di carichi verticali assegnati. Ciò non consente un controllo puntuale delle quote dei nodi del reticolo funicolare. L’approccio numerico presentato in questo lavoro di tesi utilizza il metodo delle densità di forze per controllare nodo per nodo le quote del poligono/reticolo funicolare attraverso un problema di ottimizzazione multi-vincolato. Esso è risolto attraverso metodi di programmazione matematica originariamente concepiti per la risoluzione di problemi di ottimizzazione strutturale di larga scala. Il metodo ricerca un insieme di rami indipendenti per griglie con geometria fissata nel piano e consente di ricavare poligoni/reticoli funicolari di minima/massima spinta per assegnate distribuzioni di carico. Le densità di forze dipendenti sono espresse in funzione di quelle indipendenti sfruttando l’equilibrio in direzione orizzontale. Le quote dei nodi che sono sede di reazioni vincolari sono utilizzate come incognite del problema, in aggiunta alle densità di forza indipendenti. La coordinata verticale di ogni punto del poligono/reticolo funicolare è vincolata, attraverso vincoli di tipo locale, a rimanere tra l’intradosso (limite inferiore) e l’estradosso (limite superiore) della struttura arcuata. Ulteriori vincoli locali sono imposti per garantire elementi non resistenti a trazione, in accordo alle ipotesi di Heyman. La soluzione è ricercata mediante un approccio sequenziale di programmazione matematica che utilizza particolari approssimazioni di tipo convesso (sequential convex programming), ovvero risolve una sequenza di sotto-problemi espliciti che sono approssimazioni convesse del problema originale. Il metodo consente di risolvere strutture soggette a carichi verticali (peso proprio), anche in combinazione con carichi orizzontali, come richiesto ad esempio nel caso di azioni sismiche. È inoltre possibile determinare il moltiplicatore di collasso di strutture arcuate soggette all’azione combinata di peso proprio e forze verticali/orizzontali di intensità crescente, attraverso la risoluzione di più problemi di ottimizzazione. Lo stesso algoritmo tratta problemi bi- e tri- dimensionali, converge per diverse topologie di griglia nel piano, consente di determinare i percorsi di carico su archi e cupole di varia geometria e strutture arcuate in genere. Le caratteristiche enunciate sono state testate all’interno di questo lavoro di tesi su strutture bidimensionali (archi) e su strutture tridimensionali (cupole). In particolare, sono stati studiati due archi a spessore costante (pari a 50 cm e a 25 cm), il Ponte Mosca di Torino (arco a spessore variabile), il muro radiale che sovrasta la cupola del Duomo di Milano. Sono state quindi analizzate cupole sferiche di spessori differenti (9 cm e 18 cm) con diverse griglie di partenza, sotto l’azione del solo peso proprio o del peso proprio in combinazione con carichi orizzontali, indagando in particolare il ruolo di forze meridiane e parallele ai fini della determinazione di soluzioni equilibrate. Infine, sono state analizzate cupole ottagonali di varia geometria sottoposte al solo peso proprio, utilizzando diverse tipologie di griglie.
Un metodo numerico per l'analisi funicolare di archi e cupole
Simioni, Anna
2019/2020
Abstract
Regarding the funicular analysis of arcuated structures, the literature provides several methods based on equilibrium, i.e. exploiting the static theorem of limit analysis upon assumption of the Heyman’s hypotheses (masonry is a no-tension material, it has unlimited compressive strength and no friction between blocks is allowed). Among the others, O’Dwyer presented an approach that linearizes the equilibrium equations of the nodes of the funicular network in terms of a given distribution of thrusts. This approach adopts, iteratively, methods of mathematical programming to maximize the load multiplier of the applied vertical loads. Block and Lachauer proposed a method exploiting the concept of force density, i.e. the ratio of the force to the length of any element in the grid. They searched for the closest funicular network with respect to the middle surface of a vault subject to given vertical loads. Unfortunately, this method does not allow for a local control of the height of the nodes in the network. The approach implemented in this thesis exploits the Force Density Method (FDM) to control point by point the coordinates of the funicular polygon/network through a multi-constrained optimization problem. This is solved by means of methods of mathematical programming that were originally conceived to handle large scale structural optimization problems. The adopted method is based on the detection of an independent sets of branches in the case of networks with fixed plan geometry and allows deriving funicular polygons/networks with minimum or maximum norm of the horizontal thrusts. The dependent force densities are written as a function of the independent ones, by exploiting the horizontal equilibrium. The nodal heights of the restrained nodes are used along with the set of independent force densities as unknows of the problem. The vertical coordinate of each node of the funicular polygon/network is constrained, through local enforcements, to lie within the upper and lower bounds of the envelope of the arcuated structure. Additional constraints are imposed to enforce no-tension elements, according to the Heyman’s assumptions. The solution is sought by means of a sequential convex programming technique that solves a sequence of explicit subproblems which are convex approximations of the original one. The proposed method can be used to investigate structures subjected to both vertical loads (self-weight) and horizontal loads, such as earthquake ones. Moreover, it is possible to determine the collapse load multiplier of arcuated structures subjected to self-weight along with increasing vertical/horizontal loads by tackling a set of optimization problems. The same algorithm applies to 2D and 3D structures, converges for different geometries of the plane grid, and allows detecting load paths in arcuated structures with given geometry. All these features have been tested in 2D and 3D case studies in this thesis. Two arches of constant thickness (50 cm and 25 cm) have been studied, the Mosca Bridge in Turin (which is an arch with variable thickness), a radial wall over the dome of the Cathedral of Milan. Afterwards, hemispherical domes of different thickness (9 cm and 18 cm) have been investigated under the effect of self-weight only or the combined effect of self-weight and horizontal loads. The role of meridional and hoop forces has been especially investigated in the achievement of equilibrated solutions. Finally, octagonal domes under self-weight only have been analysed, assuming different geometries and degrees of refinement of the grids.| File | Dimensione | Formato | |
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https://hdl.handle.net/10589/164568