The minimum friction angle ϕmin\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi _{\mathrm{min}}$$\end{document} [∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^{\circ }$$\end{document}] as a function of the eccentricities cith contours, black line: the optimum of the minimum friction angle for discrete β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}-s. (Color figure online)

The minimum friction angle ϕmin\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi _{\mathrm{min}}$$\end{document} [∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^{\circ }$$\end{document}] as a function of the eccentricities cith contours, black line: the optimum of the minimum friction angle for discrete β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}-s. (Color figure online)

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The paper investigates how the shape affects the Couplet–Heyman minimum thickness of the masonry pointed arch. The minimum thickness is such a structural thickness, at which a vault made of rigid voussoirs is stable for self-weight. It is expressed as a function of the pointed generator curve’s deviation from the semicircle. The arch is analysed in...

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... Moreover, Galassi et al. [12] studied the vulnerability of masonry pointed arches subjected to support displacements through a novel numerical approach that uses combinatorial laws and adopts Heyman's assumptions. Further studies on pointed arches have been summarized and presented by Lengyel [13]. ...
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Pointed arches are important architectural elements of both western and eastern historical built heritage. In this paper, the effects that different geometrical (slenderness and sharpness) and mechanical (friction and cohesion) parameters have on the in-plane structural response of masonry pointed arches are investigated through the implementation of an upper bound limit analysis approach capable of representing sliding between rigid blocks. Results, in terms of collapse multipliers are presented and quantitatively analyzed following a systematic statistical approach, whereas collapse mechanisms are qualitatively explored and three different outcomes are found; pure rotation, pure sliding and mixed collapse mechanisms. It is concluded that the capacity of the numerical approach implemented to reproduce sliding between blocks plays a major role in the better understanding of masonry pointed arches structural response.
... Within a previous mainstream of work by the present authors [7][8][9][10][11][12][13][14][15], such a classical optimization problem in the Mechanics (statics) of symmetric circular masonry arches has been revisited, and framed within the relevant, now updated, literature [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34], by providing new, explicit analytical derivations and representations of the solution characteristics. Specifically, classical Heyman solution [3] has been shown to constitute a sort of approximation of the true solution (here labelled as "CCR" [7]), in Heyman assumption of uniform selfweight distribution (location of the centres of gravity along the geometrical centreline of the circular arch), while Milankovitch solution [35][36][37] may as well be consistently derived, in the consideration of the true self-weight distribution, though at the price of an increasing complexity in the explicit analytical handling of the governing equations (here newly resolved to a very end). ...
... As per the treatment adopted in the paper, namely a full analytical one, the present study is framed between similar related investigations that have enquired the problem by analytical or semi-analytical approaches, between which specific important developments that shall be mentioned in sharing the vision and perspectives here pursued are those concisely outlined as follows: pioneering and fundamental Heyman contributions [1][2][3][4][5][6]; attempts by the present authors [7][8][9][10][11][12][13][14][15], specifically: analytical [7], analytical-numerical, accompanied by a Discrete Element Method (DEM) investigation (Discontinuous Deformation Analysis, (DDA) [8,9,15], and including reducing friction effects and resulting mixed collapse modes [10,11]; analytical-numerical, by a Complementarity Problem/Mathematical Programming formulation truly accounting for finite friction implications [12][13][14][15]; other studies on mainstream arch derivation and masonry arch mechanics after Heyman, specifically considering the issue of least thickness [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34]; Milankovitch formulation, as a cornerstone of thrust-line analysis [35][36][37]; modern and recent thrust-line-like analyses, particularly in view of form optimization [38][39][40][41][42][43][44]. ...
... Least-thickness symmetric circular masonry arch A further wider literature framing within the vast subject of the analysis of masonry arches has already been provided in [7], and therein quoted references, with the integration of the above-quoted updates. Thus, the here-mentioned literature setting does not seek an intent of completeness, just that of focusing on selfcoherent formulations that may consider further related, though different and complementary features, such as: different geometrical shapes of the masonry arch [22,25,[30][31][32]34,39,41], issues of "stereotomy" (i.e., the shape of the cut of the masonry blocks; here, a continuous circular arch with just potential rupture radial joints is considered) [6,28,33], dedicated use of "thrust-line analysis" or graphical-analytical methods, and so on, though all sharing the common characteristics of seeking and setting an analytical method of analysis, or being linked to the optimization target of acquiring the least-thickness condition. Numerical methods, which are massively employed in the analysis of masonry structures, including for masonry arches (vaults and domes), shall also be pursued and mentioned (see, for instance, about the adoption of the Discrete Element Method for the evaluation of the least-thickness condition [8,9,15,30,31], and therein quoted references), but these are here intentionally taken out from an explicit discussion. ...
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... The results obtained in Section 5.1 corroborate the qualitative findings in [30], where a 2D analytical analysis using thrust lines showed the same 7-hinge mechanism for the optimal radius over length encountered. In [30], for β = 0 • , the optimal r/l 0 = 0.94 was obtained, corresponding to a minimum thickness equal to 3.7% of the span. ...
... The results obtained in Section 5.1 corroborate the qualitative findings in [30], where a 2D analytical analysis using thrust lines showed the same 7-hinge mechanism for the optimal radius over length encountered. In [30], for β = 0 • , the optimal r/l 0 = 0.94 was obtained, corresponding to a minimum thickness equal to 3.7% of the span. In this section, the last arch of the fan-system is analysed separately, i.e., the arch on the unsupported boundary edge. ...
... This analysis yielded the same hinge pattern and minimum thickness as the one obtained for the three-dimensional problem. It, however, differs slightly to the values from [30], due to the different load distribution considered. The solution of the 3D, and 2D problem, for β = 20 • are presented in Figure 7, showing the 7-hinge pattern. ...
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... If the no sliding assumption is relaxed, the arch can also fail due to a mixed (with rotating and sliding joints) and a pure-sliding failure mode (see also Fig. 12). We refer to e.g., Bagi (2014), Aita Lengyel (2018) for an extended description of on non-Heymanian collapse modes. In the present paper, we only consider symmetrical failure modes (due to the symmetry of the problem). ...
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Friction is much needed for the equilibrium of masonry arches as it transfers load between the voussoirs. In this paper, applying an analytical formulation of the problem, the angle of friction as a geometric constraint on the stereotomy (bricklaying pattern) is investigated to find the possible range of minimum thickness values of circular and elliptical masonry arches under static loads based on the lower bound theorem of limit state analysis. The Heymanian assumptions regarding material qualities are adopted; however, limited capacity in friction is accounted for. It has been shown earlier that considering stereotomies a-priori unknown, a considerably wide range of minimum thickness values is obtained for fixed loading and global geometry conditions. It is found that a stereotomy constrained by an angle of friction, characteristic of masonry, renders the effect of stereotomy on the minimum thickness value negligible because the range of minimum thickness values is significantly reduced in this case. Hence, the present study ultimately justifies the intuitive assumption of radial stereotomy, widely used in the literature, whenever the safety of masonry arches is studied.
... As is well known, such arches are structures of great architectonical interest, widely studied also in the recent literature on the mechanical behaviour of masonry structures. In particular, without pretending to be complete, we recall that the search for the minimum thickness of these types of arches has been tackled -trough different approaches -by Heyman (1966Heyman ( , 1969, Sinopoli et al. (1997), and more recently, Romano and Ochsendorf (2010), Alexakis and Makris (2015), Cavalagli et al. (2016), Nikolić (2017), Lengylel (2018), Aita et al. (2019) and Cocchetti and Rizzi (2020). In the cited contributions, the stability of the arch is studied by considering an infinite or a finite value of the friction coefficient, by assuming as known the direction of the joints (radial or perpendicular to the intrados line). ...
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This research is inspired by an issue strictly linked to the art of stereotomy and rarely tackled in the contributions on the statics of arches and vaults, i.e., the search of the inclination to be assigned to each joint able to ensure the respect of the equilibrium conditions when the friction between the voussoirs is absent, by assuming that the intrados and extrados curves are known. After presenting some brief notes on the state-of-the-art on this subject, both a numerical and an analytical approach, based on the maxima and minima Coulomb method revisited through a re-edition of Durand-Clayes method, is developed in order to determine the inclination of the joints as well as the minimum archs thickness compatible with equilibrium. The analysis is performed for frictionless pointed and circular arches for different values of the embrace angle (i.e., the complement of springing angle).
... A least-thickness self-standing evaluation of the masonry arch may be elaborated by using Discrete Element Method (DEM) quasi-static simulations of discrete voussoir arches [36][37][38]. To provide an independent numerical validation of the achieved analytical results, an available Discontinuous Deformation Analysis (DDA) tool was adopted in [8,9], to deliver the estimates of the critical thickness and the appearance of the corresponding collapse mode (notice that the five-hinge purely rotational collapse mode is assumed from scratch, in the analytical analysis, while in such a case is numerically evaluated, out of the analysis). ...
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... Fraddosio et al. [23] proposed a computational equilibrium analysis approach using the optimization method and applied it to the three-dimensional masonry vaults. Lengyel [24] investigated the effect of shape on the Couplet-Heyman minimum thickness of a masonry pointed arch using the optimization method, and compared it with the results by the discrete element method. Ricci et al. [25] proposed a numerical method for determining the admissible thrust curves of masonry arches under arbitrary loads. ...
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... The semicircular arch with its respective types, see Figure 1A, such as the stilted arch, Figure 1B, the segmental arch, Figure 1C, or the arch which is attached to the barrel vault of a nave to reinforce it and divides it into sections, were also characteristic of Romanesque architecture [4]. However, it lost importance in the Gothic, which gave way to the three-pointed arch, Figure 1E, with its different variants, as an acute arch, Figure 1G, or the depressed arch, Figure 1H [5]. ...
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The arches were a great advance in construction with respect to the rigid Greek linteled architecture. Its development came from the hand of the great Roman constructions, especially with the semicircular arch. In successive historical periods, different types of arches have been emerging, which in addition to their structural function was taking aesthetic characteristics that are used today to define the architectural style. When, in the construction of a bow, the rise is less than half the springing line, the semicircular arch is no longer used and the segmental arch is used, and then on to another more efficient and aesthetic arch, the basket-handle arch. This study examines the classic geometry of the basket-handle arch also called the three-centered arch. A solution is proposed from a constructive and aesthetic point of view, and this is approached both geometrically and analytically, where the relationship between the radius of the central arch and the radius of the lateral arch is minimized. The solution achieved allows the maximum springing line or clear span to be saved with the minimum rise that preserves the aesthetic point of view, since the horizontal thrust of a bow is greater than the relationship between the springing line of the arch and the rise. This solution has been programmed and the resulting software has made it possible to analyse existing arches in historic buildings or constructions to check if their solutions were close or not from both points of view. Thus, it has been possible to verify that in most of the existing arches analyzed, the proposed solution is reached.
... Within such a framework, classical problems as that of finding the critical condition of least thickness for a circular masonry arch under self-weight (Couplet-Heyman problem) have found further consistent solutions by different analytical and numerical methodologies. For instance, worthwhile to be mentioned shall be contributions (Como, 1992;Blasi and Foraboschi, 1994;Boothby, 1994;Lucchesi et al., 1997;Foce, 2007;Ochsendorf, 2002;Block et al., 2006;Ochsendorf, 2006;Romano and Ochsendorf, 2010;Gago, 2004;Gago et al., 2011;Rizzi et al., 2010;Cocchetti et al., 2012;Rizzi et al., 2012Rizzi et al., , 2014Rizzi, 2018, 2019;Aita et al., 2012Aita et al., , 2019Makris and Alexakis, 2013;Makris, 2013, 2015;Cavalagli et al., 2016;Nikolić, 2017;Angelillo, 2019;Gáspár et al., 2018;Bagi, 2014;Lengyel, 2018;Auciello, 2019;Sinopoli et al., 1997;Casapulla and Lauro, 2000;Gilbert et al., 2006;Casapulla and D'Ayala, 2001;D'Ayala and Casapulla, 2001;D'Ayala and Tomasoni, 2011;Smars, 2000Smars, , 2008Smars, , 2010Olivito et al., 2016;Portioli et al., 2014;Cascini et al., 2016;Milani and Tralli, 2012;Sacco, 2012;Baggio and Trovalusci, 2000;Beatini et al., 2017Beatini et al., , 2019Angelillo et al., 2018;Tempesta and Galassi, 2019;Trentadue and Quaranta, 2013). Specifically, Como (1992), Blasi and Foraboschi (1994), Boothby (1994), Lucchesi et al. (1997), Foce (2007), Ochsendorf (2002), Block et al. (2006), Ochsendorf (2006), Romano and Ochsendorf (2010), Gago (2004), Gago et al. (2011), Rizzi et al. (2010), Cocchetti et al. (2012), Rizzi et al. (2012Rizzi et al. ( , 2014, Rizzi (2018, 2019), Aita et al. (2012Aita et al. ( , 2019, Makris and Alexakis (2013), Makris (2013, 2015), Cavalagli et al. (2016), Nikolić (2017), Angelillo (2019), Gáspár et al. (2018), Bagi (2014), Lengyel (2018) and Auciello (2019) concern the general analysis of masonry arches, possibly in conjunction with the determination of the critical least-thickness condition and considering different issues, such as various arch shapes (e.g., pointed, oval, elliptical, etc.) and possible variable stereotomy of the arch blocks (differing from that of classical radial joints, as here analysed). ...
... For instance, worthwhile to be mentioned shall be contributions (Como, 1992;Blasi and Foraboschi, 1994;Boothby, 1994;Lucchesi et al., 1997;Foce, 2007;Ochsendorf, 2002;Block et al., 2006;Ochsendorf, 2006;Romano and Ochsendorf, 2010;Gago, 2004;Gago et al., 2011;Rizzi et al., 2010;Cocchetti et al., 2012;Rizzi et al., 2012Rizzi et al., , 2014Rizzi, 2018, 2019;Aita et al., 2012Aita et al., , 2019Makris and Alexakis, 2013;Makris, 2013, 2015;Cavalagli et al., 2016;Nikolić, 2017;Angelillo, 2019;Gáspár et al., 2018;Bagi, 2014;Lengyel, 2018;Auciello, 2019;Sinopoli et al., 1997;Casapulla and Lauro, 2000;Gilbert et al., 2006;Casapulla and D'Ayala, 2001;D'Ayala and Casapulla, 2001;D'Ayala and Tomasoni, 2011;Smars, 2000Smars, , 2008Smars, , 2010Olivito et al., 2016;Portioli et al., 2014;Cascini et al., 2016;Milani and Tralli, 2012;Sacco, 2012;Baggio and Trovalusci, 2000;Beatini et al., 2017Beatini et al., , 2019Angelillo et al., 2018;Tempesta and Galassi, 2019;Trentadue and Quaranta, 2013). Specifically, Como (1992), Blasi and Foraboschi (1994), Boothby (1994), Lucchesi et al. (1997), Foce (2007), Ochsendorf (2002), Block et al. (2006), Ochsendorf (2006), Romano and Ochsendorf (2010), Gago (2004), Gago et al. (2011), Rizzi et al. (2010), Cocchetti et al. (2012), Rizzi et al. (2012Rizzi et al. ( , 2014, Rizzi (2018, 2019), Aita et al. (2012Aita et al. ( , 2019, Makris and Alexakis (2013), Makris (2013, 2015), Cavalagli et al. (2016), Nikolić (2017), Angelillo (2019), Gáspár et al. (2018), Bagi (2014), Lengyel (2018) and Auciello (2019) concern the general analysis of masonry arches, possibly in conjunction with the determination of the critical least-thickness condition and considering different issues, such as various arch shapes (e.g., pointed, oval, elliptical, etc.) and possible variable stereotomy of the arch blocks (differing from that of classical radial joints, as here analysed). Moreover, Sinopoli et al. (1997), Casapulla and Lauro (2000), Gilbert et al. (2006), Casapulla and D'Ayala (2001), D'Ayala, and , D'Ayala and Tomasoni (2011), Smars (2000Smars ( , 2008Smars ( , 2010 and Olivito et al. (2016) particularly deal with the issue of finite friction, and its implications in the statics of masonry arches and the possible failure modes. ...
... Within such a framework, classical problems as that of finding the critical condition of least thickness for a circular masonry arch under self-weight (Couplet-Heyman problem) have found further consistent solutions by different analytical and numerical methodologies. For instance, worthwhile to be mentioned shall be contributions (Como, 1992;Blasi and Foraboschi, 1994;Boothby, 1994;Lucchesi et al., 1997;Foce, 2007;Ochsendorf, 2002;Block et al., 2006;Ochsendorf, 2006;Romano and Ochsendorf, 2010;Gago, 2004;Gago et al., 2011;Rizzi et al., 2010;Cocchetti et al., 2012;Rizzi et al., 2012Rizzi et al., , 2014Rizzi, 2018, 2019;Aita et al., 2012Aita et al., , 2019Makris and Alexakis, 2013;Makris, 2013, 2015;Cavalagli et al., 2016;Nikolić, 2017;Angelillo, 2019;Gáspár et al., 2018;Bagi, 2014;Lengyel, 2018;Auciello, 2019;Sinopoli et al., 1997;Casapulla and Lauro, 2000;Gilbert et al., 2006;Casapulla and D'Ayala, 2001;D'Ayala and Casapulla, 2001;D'Ayala and Tomasoni, 2011;Smars, 2000Smars, , 2008Smars, , 2010Olivito et al., 2016;Portioli et al., 2014;Cascini et al., 2016;Milani and Tralli, 2012;Sacco, 2012;Baggio and Trovalusci, 2000;Beatini et al., 2017Beatini et al., , 2019Angelillo et al., 2018;Tempesta and Galassi, 2019;Trentadue and Quaranta, 2013). Specifically, Como (1992), Blasi and Foraboschi (1994), Boothby (1994), Lucchesi et al. (1997), Foce (2007), Ochsendorf (2002), Block et al. (2006), Ochsendorf (2006), Romano and Ochsendorf (2010), Gago (2004), Gago et al. (2011), Rizzi et al. (2010), Cocchetti et al. (2012), Rizzi et al. (2012Rizzi et al. ( , 2014, Rizzi (2018, 2019), Aita et al. (2012Aita et al. ( , 2019, Makris and Alexakis (2013), Makris (2013, 2015), Cavalagli et al. (2016), Nikolić (2017), Angelillo (2019), Gáspár et al. (2018), Bagi (2014), Lengyel (2018) and Auciello (2019) concern the general analysis of masonry arches, possibly in conjunction with the determination of the critical least-thickness condition and considering different issues, such as various arch shapes (e.g., pointed, oval, elliptical, etc.) and possible variable stereotomy of the arch blocks (differing from that of classical radial joints, as here analysed). ...
... For instance, worthwhile to be mentioned shall be contributions (Como, 1992;Blasi and Foraboschi, 1994;Boothby, 1994;Lucchesi et al., 1997;Foce, 2007;Ochsendorf, 2002;Block et al., 2006;Ochsendorf, 2006;Romano and Ochsendorf, 2010;Gago, 2004;Gago et al., 2011;Rizzi et al., 2010;Cocchetti et al., 2012;Rizzi et al., 2012Rizzi et al., , 2014Rizzi, 2018, 2019;Aita et al., 2012Aita et al., , 2019Makris and Alexakis, 2013;Makris, 2013, 2015;Cavalagli et al., 2016;Nikolić, 2017;Angelillo, 2019;Gáspár et al., 2018;Bagi, 2014;Lengyel, 2018;Auciello, 2019;Sinopoli et al., 1997;Casapulla and Lauro, 2000;Gilbert et al., 2006;Casapulla and D'Ayala, 2001;D'Ayala and Casapulla, 2001;D'Ayala and Tomasoni, 2011;Smars, 2000Smars, , 2008Smars, , 2010Olivito et al., 2016;Portioli et al., 2014;Cascini et al., 2016;Milani and Tralli, 2012;Sacco, 2012;Baggio and Trovalusci, 2000;Beatini et al., 2017Beatini et al., , 2019Angelillo et al., 2018;Tempesta and Galassi, 2019;Trentadue and Quaranta, 2013). Specifically, Como (1992), Blasi and Foraboschi (1994), Boothby (1994), Lucchesi et al. (1997), Foce (2007), Ochsendorf (2002), Block et al. (2006), Ochsendorf (2006), Romano and Ochsendorf (2010), Gago (2004), Gago et al. (2011), Rizzi et al. (2010), Cocchetti et al. (2012), Rizzi et al. (2012Rizzi et al. ( , 2014, Rizzi (2018, 2019), Aita et al. (2012Aita et al. ( , 2019, Makris and Alexakis (2013), Makris (2013, 2015), Cavalagli et al. (2016), Nikolić (2017), Angelillo (2019), Gáspár et al. (2018), Bagi (2014), Lengyel (2018) and Auciello (2019) concern the general analysis of masonry arches, possibly in conjunction with the determination of the critical least-thickness condition and considering different issues, such as various arch shapes (e.g., pointed, oval, elliptical, etc.) and possible variable stereotomy of the arch blocks (differing from that of classical radial joints, as here analysed). Moreover, Sinopoli et al. (1997), Casapulla and Lauro (2000), Gilbert et al. (2006), Casapulla and D'Ayala (2001), D'Ayala, and , D'Ayala and Tomasoni (2011), Smars (2000Smars ( , 2008Smars ( , 2010 and Olivito et al. (2016) particularly deal with the issue of finite friction, and its implications in the statics of masonry arches and the possible failure modes. ...