Lower-bound Analysis of Masonry Vaults

This paper applies Thrust-Network Analysis, a three-dimensional computational method for obtaining lower-bound solutions of masonry vaults with complex geometries. The method extends thrust-line analysis to three-dimensional problems by finding equilibrium force networks within the vault’s geometry, representing possible paths of the compression forces. Through two case studies, this paper demonstrates the potential of the method as a powerful tool for understanding, visualizing and exploring the equilibrium of compression-only structures. First, an analysis of a series of groin and quadripartite rib vaults investigates the interrelationship between different parameters and the range of possible equilibrium solutions of these vaults. A second case study analyzes the fan vaults of King’s College Chapel in Cambridge, England. 2 EQUILIBRIUM ANALYSIS FOR VAULTS IN UNREINFORCED MASONRY 2.1 Pseudo-3D equilibrium analysis methods In order to analyze three-dimensional structures using the same intuitive methods discussed above, the analyst typically must slice the structure, reducing it to a combination of two-dimensional problems. In this way, structural behavior is reduced to a combination of arch actions. This process obviously does not capture the full threedimensional behavior of the structure and must rely heavily on the chosen discretization. These limitations have been the main reasons why thrustline analysis has not been used extensively for the assessment of complex 3D structures. Wolfe (1921) demonstrated how by slicing up a structure a global pseudo-3D analysis can be done by combining local thrust lines (Fig. 1). This methodology was entirely manual and quickly becomes tedious (Boothby 2001). Block et al. (2006b) proposed a method to produce models which contain the graphical construction but which are parametric and interactive, reducing the tedious iterative nature of traditional graphic statics. The models demonstrate the range of possible solutions and can be used to investigate the effects on the global stability of the vault of varying different parameters such as buttress thickness, arch thickness, level of fill, etc. Smars developed in his PhD thesis (2000) computational tools to perform such a pseudo-3D analysis in an automated fashion, starting from the actual measured geometry of the vaults. 2.2 Fully 3D equilibrium analysis methods The main problem with three-dimensional equilibrium analysis for masonry vaults is that they are highly indeterminate structures. Antoni Gaudí’s physical form-finding process for the church of the Colonia Guell can be used to explain this. First, before starting to construct a hanging string model, Gaudi had to decide on a suitable force pattern topology to represent the structural action of the vaults. Then, after choosing the structural logic, it is still challenging to control or even predict the final shape, since the equilibrium of each string influences the equilibrium of the entire network. It is a tedious, iterative process of adjusting and refining. New form-finding programs which explore hanging models in the virtual world, based on dynamic relaxation, such as Kilian’s CADenary tool have to deal with the same issues (Kilian and Ochsendorf 2005). It is very hard to control and predict how the final shape of the compression network will look like if local changes are being made or a string model is being assembled and hung under gravity. This is true for both physical and virtual string models as for graphical methods. In order to analyze a three-dimensional indeterminate system these unknowns need to be understood and controlled. This can be achieved by describing the problem as an optimization problem. The equilibrium requirements then are formulated as a set of constraints which have to be satisfied while optimizing a certain objective function. O’Dwyer (1999) implemented optimization methods to investigate masonry vaults by finding possible compression-only force networks which are entirely contained within the boundaries of the vault. Recently, the problem of controlling a virtual hanging strings network has been elegantly and efficiently been implemented within a similar optimization framework by Andreu et al. (2007). A hanging string network is found which fits within the inverted geometry of the vault to be analyzed. The following section briefly summarizes ThrustNetwork Analysis, a new fully three-dimensional extension to thrust-line analysis based on projective geometry, duality theory and linear optimization (Block and Ochsendorf 2007). Examples in Sections 4 and 5 will demonstrate its value for the analysis of vaulted structures in unreinforced masonry. 3 THRUST NETWORK ANALYSIS