Masonry structures made of monolithic blocks with an application to spiral stairs

The broad topic of the present work is Statics and Kinematics of masonry structures, made of monolithic blocks, modelled a la Heyman, that is rigid bodies loaded by external forces, submitted to unilateral constraints, and undergoing small displacements, under the simplifying assumption that sliding on rough interfaces is prevented. Specifically, in this work, we study the effect, in terms of internal forces, of specified loads, by using given settlements/eigenstrains to trigger special regimes of the internal forces. Although our scope here is the analysis of masonry structures composed by monolithic pieces, and whose blocks are not likely to break at their inside, the method we propose can also be applied to generic masonry structures, such as those made of bricks or small stones. Heyman’s assumptions translate, for unilateral continua, into a normality assumption which allows to employ the two theorems of Limit Analysis. These continuous structures may actually fracture everywhere at their inside, forming rigid blocks in relative displacement among each other. Such piecewise rigid-body displacements in masonry are physiological, and rather than the result of over-loading, are most likely the direct product of small changes of the displacement type boundary conditions. However, when in a part of the structure a specific piecewise rigid-body displacement nucleates, that part of the structure exhibits a one degree of freedom mechanism, and becomes statically determined. Therefore, the internal forces can be computed, despite the original uncracked structure being abundantly overdetermined, and then admitting infinite many statically admissible stress regimes. With these assumptions, in the present paper we study the equilibrium and the effect of settlements in a masonry structure made of monolithic blocks. In particular, the triple helical stair of the convent of San Domingos de Bonaval, located in the Bonaval district of Santiago de Compostela, is considered as case study.

[1]  Philippe Block,et al.  Real-time limit analysis of vaulted masonry buildings , 2006 .

[2]  Yin Zhang,et al.  Solving large-scale linear programs by interior-point methods under the Matlab ∗ Environment † , 1998 .

[3]  Antonio Gesualdo,et al.  Constitutive behaviour of quasi-brittle materials with anisotropic friction , 2015 .

[4]  M. Angelillo,et al.  Limit analysis for unilateral masonry-like structures , 2016 .

[5]  G. Dantzig,et al.  Notes on Linear Programming: Part 1. The Generalized Simplex Method for Minimizing a Linear Form under Linear Inequality Restraints , 1954 .

[6]  Antonio Gesualdo,et al.  Equilibrium formulation of masonry helical stairs , 2017 .

[7]  Antonio Fortunato,et al.  Equilibrium of Masonry Vaults , 2004 .

[8]  Fernando Fraternali,et al.  The Lumped Stress Method and the Discrete-Continuum Approximation. , 2002 .

[9]  M. Lippiello,et al.  Analytic solutions for the stress field in static sandpiles , 2016 .

[10]  C. Casapulla,et al.  Formulating the Torsion Strength of Dry-Stacked Stone Blocks by Comparing Convex and Concave Contact Formulations and Experimental Results , 2016 .

[11]  Antonio Gesualdo,et al.  On the bounding of limit multipliers for combined loading , 2010, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[12]  Fernando Fraternali,et al.  Limit analysis of masonry structures with free discontinuities , 2018 .

[13]  Matthew Gilbert,et al.  Limit analysis of masonry block structures with non-associative frictional joints using linear programming , 2006 .

[14]  R. Livesley.  Limit analysis of structures formed from rigid blocks , 1978 .

[15]  Antonio Gesualdo,et al.  Modelling the cracks produced by settlements in masonry structures , 2018 .

[16]  M. Angelillo,et al.  Structural capacity of masonry walls under horizontal loads , 2014 .

[17]  M. Angelillo,et al.  Singular stress fields in masonry structures: Derand was right , 2014 .

[18]  Mariateresa Guadagnuolo,et al.  The role of friction in the seismic risk mitigation of freestanding art objects , 2014, Natural Hazards.

[19]  M. Angelillo,et al.  A lumped stress method for plane elastic problems and the discrete-continuum approximation , 2002 .

[20]  Philippe Block,et al.  Three-dimensional funicular analysis of masonry vaults , 2014 .

[21]  Sanjay Mehrotra,et al.  On the Implementation of a Primal-Dual Interior Point Method , 1992, SIAM J. Optim..

[22]  Enrico Babilio,et al.  Numerical solutions for crack growth based on the variational theory of fracture , 2012 .

[23]  Jacques Heyman,et al.  The stone skeleton , 1995 .

[24]  Takeo Igarashi,et al.  Parametric self-supporting surfaces via direct computation of airy stress functions , 2015, ACM Trans. Graph..

[25]  Katalin Bagi,et al.  Discrete element analysis of stone cantilever stairs , 2018 .

[26]  Maurizio Angelillo,et al.  Structural failures due to anthropogenic sinkholes in the urban area of Naples and the effect of a FRP retrofitting , 2016 .

[27]  Dermot O'Dwyer,et al.  Funicular analysis of masonry vaults , 1999 .

[28]  C. Casapulla,et al.  The comparative role of friction in local out-of-plane mechanisms of masonry buildings. Pushover analysis and experimental investigation , 2016 .

[29]  Maurizio Angelillo,et al.  The Equilibrium of Helical Stairs Made of Monolithic Steps , 2016 .

[30]  Fernando Fraternali,et al.  A thrust network approach to the equilibrium problem of unreinforced masonry vaults via polyhedral stress functions , 2010 .

[31]  Pierre Alliez,et al.  On the equilibrium of simplicial masonry structures , 2013, ACM Trans. Graph..

[32]  Johannes Wallner,et al.  Design of self-supporting surfaces , 2012, ACM Trans. Graph..

[33]  Santiago Sánchez-Cervera Huerta,et al.  Galileo was Wrong: The Geometrical Design of Masonry Arches , 2006 .

[34]  Jacques Heyman,et al.  The Mechanics Of Masonry Stairs , 1970 .

[35]  Maurizio Angelillo,et al.  Static analysis of a Guastavino helical stair as a layered masonry shell , 2015 .