Parametric analysis of structures from flat vaults to reciprocal grids

International Journal of Solids and Structures 54 (2015) 50–65

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International Journal of Solids and Structures journal homepage: www.elsevier.com/locate/ijsolstr

Parametric analysis of structures from flat vaults to reciprocal grids Maurizio Brocato a,b,⇑, Lucia Mondardini a a b

University Paris-Est, Laboratoire GSA – Geométrie Structure Architecture (Ecole nationale supérieure d’architecture Paris-Malaquais), 14 rue Bonaparte, 75006 Paris, France University Paris-Est, Laboratoire Navier, 6 et 8 Av. Blaise Pascal, 77455 Champs-sur-Marne, France

a r t i c l e

i n f o

Article history: Received 12 May 2013 Received in revised form 15 May 2014 Available online 20 November 2014 Keywords: Stone structure Flat vault Stereotomy Reciprocal grid Parametric analysis

a b s t r a c t We propose a parametric analysis of a class of structural systems stemming from the 17th century invention of Joseph Abeille usually called flat vault and reaching the field of reciprocal grids. Our purpose is to understand the load descent paths taking place in the various specimens of this class and their relations with basic structural principles such as that of the inverted catenary, useful to deal with vaults, or that of the lever, more appropriate for grids. The analysis is performed on changes of the geometry and of the topology that preserve the logic of the bonding of stone blocks characterizing Abeille’s invention. Shown results concern the distribution of the elastic energy, the reactions, the chirality and the stress and displacement maps. Our findings support the idea that the structures belonging to this class of structures (and having reasonable proportions) are rather to be considered as deflected grids than as compressed vaults. Furthermore, a local bending interests the blocks, for because of the bonding, in a way that is typical of reciprocal structures. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction The starting point of this work is a stone structure, called a flat vault, invented by the French engineer Joseph Abeille in 1699, where the appropriate interweaving of stone blocks allows the covering of a rectangular space. In the structure designed by Abeille, the standard block has two orthogonal vertical sections in the shape of isosceles trapezia (each with one pair of sides horizontal, see Fig. 1). Some variations exist that present more complex shapes, but we will not study them here. In the past years many researchers—especially in the field of construction history—have analyzed the mechanical behavior of these objects to give a synthetic interpretation of their nature (a bibliographical survey is presented in Brocato and Mondardini (2011); see also Fleury (2009), Nichilo (2003), Rabasa-Dìaz (1998), Sakarovitch (2006) and Uva (2003)). These interpretations are based on two main positions: (V1) The structure behaves like a vault, where the stone elements are primarily subject to axial stress and transmit a sensible thrust on the confinement structures (catenary effect). ⇑ Corresponding author at: University Paris-Est, Laboratoire GSA – Geométrie Structure Architecture (Ecole nationale supérieure d’architecture Paris-Malaquais), 14 rue Bonaparte, 75006 Paris, France. E-mail addresses: [email protected] (M. Brocato), [email protected] (L. Mondardini). http://dx.doi.org/10.1016/j.ijsolstr.2014.11.007 0020-7683/Ó 2014 Elsevier Ltd. All rights reserved.

(V2) The structure can be seen as a kind of stacked timber grid, where elements are primarily subject to bending (‘levery’ effect, if one admits the neologism). If the rationale behind the first thesis is obvious, the second is supported by a kinship between Abeille’s system and a particular family of structures called nexorades or reciprocal frames (Baverel et al., 2000). In particular the structural assembly of timbers called Serlio floor (a proposal appearing in the literature since the middle-age; see Yeomans (1997)) seemingly works like Abeille’s: in both examples, each element supports and is supported at the same time by other equivalent components in order to cover a space with shorter pieces than the span. Our purpose is to understand at which rate, depending on its geometry, the structural system is represented by either model. This insight is useful in the practice of structural civil engineering, both because it helps deciding on the adequacy of the system to any particular application and because it helps assigning criteria for the design of the individual pieces, of the supports and of the abutments. In the last decade, following a proposal by Dyskin et al. (2001) that renewed the attention on an even older theme, some authors have dealt with the subject of the topological interlocking of blocks (Dyskin et al., 2003a; Dyskin et al., 2003b; Estrin et al., 2011; Khandelwal et al., 2012), showing interest on structural systems kin to those studied here as a basis to develop new materials. In

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these papers the investigation on the mechanical properties of the system focuses predominantly on the force-deflection behavior under large displacements and until ruin. Experiments made to gather information on this behavior show evidence of the setting of a catenary effect during failure, as confirmed by comparison with some theoretical results obtained modeling the structure as an arch (Khandelwal et al., 2012). This dominance of the catenary effect can be explained by the fact that the considered systems—unlike those examined here— are made of regular polyhedra (especially tetrahedra), i.e. elements that have no dimension prevailing on the others. Hence our investigation might contribute extending the picture put forward in the quoted papers. It must be noticed that Khandelwal et al. (2012) seems to deduce from the literature on Abeille’s bonds that these flat vaults do not withstand loads of inverted direction and work only under a stabilizing self-weight. This is actually not the case: Abeille’s flat vaults can withstand orthogonal and transversal loads in all directions and do not need be stabilized by their weight. An investigation on a similar system is presented in Brocato and Mondardini (2012), where spherical domes are considered and a method to define an optimal structure is proposed. Here the study of flat vaults is developed through the generation of three parametric families of topologically equal structures, plus one family of topological variants. When considering the two previously listed viewpoints (V1 and V2), clearly, the first can be expected to be more consistent of the second if the elements composing the structure are bulky, less if they are slender. A slenderness parameter, given by the ratio of the horizontal dimensions of the blocks, can be introduced to help

51

gauging this issue. Similarly, considering that the catenary effect is likely to be more efficient when the thickness of the structure grows with respect to the span, less in the opposite condition, a thickness parameter is introduced, given by the ratio between the thickness of the vault and its overall length. Also the inclination of the contact planes between blocks can be assumed to contribute, when they are closer to vertical planes, more to the behavior of a catenary system than to that of a ‘levery’ one and vice versa in the opposite condition. This wedge effect can be measured by what we name the ‘splice angle’, i.e., the inclination from the vertical of the legs of the trapezia describing the cross sections of blocks. The chiral geometry of the considered structures has a consequence on the nature of their thrust, which, under a symmetric load, is necessarily expressed by a chiral set of force vectors. Nevertheless this effect should diminish when the number of pieces composing the vault grows, as the characteristic length of the chirality becomes much smaller than the span. A parameter that can be introduced to represent the importance of chirality is the number of stone rows the structure is made of or, equivalently, the ratio of the distance between two such rows on the span. Hence, we have taken into account the four quoted parameters to generate several structures bearing equivalent loads with equivalent boundary conditions and study their variations. To evaluate the difference between members of these families, the definition of the shear, bending and membrane energies within the structure is adapted to the case and their quotas in terms of total elastic energy computed for a given structure. Then these quotas are compared for flat vaults having different slendernesses and splice angles, showing that the influence of the latter on the behavior of the vault is but slightly important. The same quotas are compared for vaults with different number of rows and for vaults with different thickness parameter. Maps of the bending part of the elastic energy are also proposed to show the difference between the structure considered here and a grid of continuous interlacing beams. In the next section the parametric families of considered structures, the model used for numerical simulations, and the different measures used to indicate the mechanical performances of the vault are presented. The numerical results are given, in Section 3, with a discussion on these performances, considering four different kind of analyses, which focus respectively on the distribution of the elastic energy, the boundary reactions, the geometric chirality and its mechanical effect, and a class of topological variations of the system.

2. Assumptions 2.1. Parametric geometry The geometry of blocks is presented in Fig. 2. Any cross section of the block orthogonal to the 2a and 2d long edges has the shape of an isosceles trapezium; the same happens for the cross sections orthogonal to the 2c and 2b edges. The splice angle u is the angle between the vertical and the legs of these trapezia, and it is the same for all cross sections in both x- and y-direction (see Fig. 2). The height of these cross sections, or structural thickness of the vault, is denoted by h. These six measures are related by the two conditions

c ¼ b þ h tan u;

Fig. 1. Examples of flat structures presented by Frézier, 1980. Abeille’s proposal appears in the red box. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

d ¼ a  h tan u;

ð1Þ

so that two of them, namely c and d, will in this paper be always considered as dependent variables defined by (1). The overall length L and the span S of the vault indicated in Fig. 3 are related to the even integer number 2N representing the number of rows spanning in the orthogonal direction by the equations:

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Fig. 2. Nomenclature of the lengths of edges and of the height of a typical block and sections considered for computations. Images on the left show also the different values of the splice angle considered in the analysis.

L ¼ 2Nða þ bÞ þ a  b;

S ¼ 2Nðd þ cÞ þ d  c;

ð2Þ

as we concentrate on square vaults this information is the same in the two directions. Clearly, L ¼ S þ 2h tan u and we do not need to take both L and S into account as this condition eliminates one of them. Only L will be considered henceforth. The shortest distance between axes of blocks spanning in the same direction on different lines is

p¼aþb

ð3Þ

The whole geometrical information is thus given by a set of seven parameters (a; b; h; u; N; L; p) with the two conditions given by the first expression of (2) and by (3). Our purpose is to evaluate the relation between some key geometrical parameters and the mechanical behavior of the structure; the parameters we take into account are: (P1) (P2) (P3) (P4)

the blocks’ slenderness ratio a=b (see Fig. 2), the splice angle u (see Fig. 2), the vault’s thickness ratio h=L (see Figs. 2 and 3), and half the number of rows N (see Fig. 3).

Notice also that the splice angle has not only a mechanical influence on the structure, but also an important effect on the fragility of the quoins and is usually to be limited in applications by considerations on the nature of the stone.

p

We will consider four different families of structures corresponding to changes of these parameters: (F1) A first family is generated by changing a preserving the thickness h ¼ 25 cm, the splice angle u ¼ p=6 (rad), the number of rows 2N ¼ 8, and the overall length L ¼ 5 m, b and p being computed with (2) and (3). Six structures where considered taking a from 47 cm to 52 cm with 1 cm steps, which gives slenderness ratios from 4 to 11. All parameters are listed in Table 1 (fixed values are not repeated, see Fig. 4): (F2) A second family of structures is built by changes of the splice angle. In the change, we keep the edges at the extrados constant and make those at the intrados change accordingly. Also the thickness and the number of rows are left unchanged. For this family, six structures were tested taking u ranging from p=9 to p=4 with step p=36, a ¼ 50 cm, b ¼ 10 cm, h ¼ 18 cm, and 2N ¼ 8; L and p being computed from (2) and (3). See Table 2. (F3) A third family is obtained by changing the thickness parameter h=L, all other measures being constant; we took a ¼ 40 cm, b ¼ 10 cm, h ranging from 9 cm to 30 cm with step 3 cm (8 specimens), u ¼ 5p=36, 2N ¼ 8; L and p computed from (2) and (3). See Table 3. (F4) The fourth and last family is obtained by changing the number of rows. We have taken into account three structures with fixed b ¼ 20 cm, u ¼ p=6; h ¼ 18 cm, and L ¼ 4 m, N equal to 1, 2 or 4, and a and p computed from (2) and (3). See Table 4. Notice that the topology of the vault changes with the number of rows, so that the transformation generating family F4 is not simply a geometrical one. Nevertheless it is useful introducing it to analyze the differences in the thrust of structures within this family, especially the modification of the effects of chirality, which are supposed to become smaller and smaller when the number of rows grows larger.

Table 1 Nomenclature of the studied specimens in family F1 and corresponding parameters. Unrepeated values are kept constant.

S L Fig. 3. Sketch of the vault with indication of the parameters p; S, and L.

Family and specimen

a [m]

b [m]

h [m]

u [rad]

N

L [m]

p [m]

1.1 1.2 1.3 1.4 1.5 1.6

0.47 0.48 0.49 0.50 0.51 0.52

0.11 0.10 0.08 0.07 0.06 0.05

0.25

p/6

4

5.0

0.58 0.58 0.57 0.57 0.57 0.57

M. Brocato, L. Mondardini / International Journal of Solids and Structures 54 (2015) 50–65 Table 2 Nomenclature of the studied specimens in family F2 and corresponding parameters. Unrepeated values are kept constant. Family and specimen

a [m]

b [m]

h [rad]

u [m]

N

L [m]

p [m]

2.1 2.2 2.3 2.4 2.5 2.6

0.50

0.10

0.18

p/9 5p/36 p/6 7p/36 2p/9 p/4

4

5.2

0.60

Table 3 Nomenclature of the studied specimens in family F3 and corresponding parameters. Unrepeated values are kept constant. Family and specimen

a [m]

b [m]

h [m]

u [rad]

N

L [m]

p [m]

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8

0.40

0.10

0.09 0.12 0.15 0.18 0.21 0.24 0.27 0.30

5p/36

4

4.3

0.50

53

constant whatever the change of the geometry. The dimensions and the shape of the loaded surface depend on the set of parameters defining the vault’s geometry, but the number of blocks interested by it is the same for all specimens of the family: the load is distributed on the upper faces of the sixteen central blocks of the vault. In Fig. 6 this surface is indicated in blue. When the number of rows changes as in family F4, it is not possible to preserve the topology of this surface. Hence, in this case, we compare structures that are charged by a uniform load on the whole upper face. Families F2 and F3 were loaded with 2 kN or with 50 kN. Though the former can be considered small relative to the weight of the structures, it will be shown that computations made with respect to the latter give sensibly equal results concerning the energy distribution. The boundary conditions are given by a rigid confinement structure supporting the ends of all blocks at the boundary of the vault, with a contact law that is the same as between blocks (a law explained in the next paragraph). For instance, with reference to Fig. 6, there are four blocks supported at their outmost end at each side of the vault; in Fig. 5, the blocks supported at the boundary are one, two or four per side in the structures shown from left to right. 2.3. Materials and mechanical model

Table 4 Nomenclature of the studied specimens in family F4 and corresponding parameters. Unrepeated values are kept constant. Family and specimen

a [m]

b [m]

h [m]

u [rad]

N

L [m]

p [m]

4.1 4.2 4.3

1.27 0.68 0.29

0.20

0.18

p/6

1 2 4

4

1.47 0.88 0.49

2.2. Loading and boundary conditions Loading and boundary conditions change in computations according to the parametric change of the geometry. In the families from F1 to F3, the central part of the vault is charged by a uniformly distributed load whose resultant is kept

In Brocato and Mondardini (2011) a detailed presentation is given of the models that are available to study structures consisting of solid blocks separated by thin layers of mortar, and a comparison of them is given which led the authors to the choice of the model that will also be used here: a finite element model where each block is a distinct solid with linear elastic behavior (defined by density, Young modulus and Poisson ratio). The interfaces are considered to have a unilateral Mohr–Coulomb behavior (defined by density, Young modulus, Poisson ratio, cohesion, friction angle with associated dilatancy and tensile resistance): the compression and shear stresses are limited by friction and are transmitted throughout the joints; the stones may separate or slide under the action of contact forces; the joint behaves elastically

Fig. 4. First and last samples from the parametric family F1 obtained by changing the slenderness ratio a=b from the lowest (on the left) to the highest considered value (on the right).

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M. Brocato, L. Mondardini / International Journal of Solids and Structures 54 (2015) 50–65

Fig. 5. The three different structures of family F4, considered to evaluate the chirality.

on the nodes if they tend to come closer to each others, while no force at all is applied on them when they tend to move apart. Step-by-step integration of the static equilibrium equation is done with an adaptive solver (modified Newton–Raphson). 2.4. Energetic indicators of the structural behavior

Fig. 6. Sketch of a vault with the loaded surface of families F1 to F3 in blue. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

under compressive forces, sliding apart. We consider that non-linearities are possible only in the interfaces. Ruptures in blocks are not taken into account. For the purpose of the sought analyses, an Abeille’s vaults made of 64 blocks at most has been considered (Fig. 6). The results shown in the next paragraph have been obtained starting from the following numerical data. The material properties are, for the stone: density 2000 kg/m3, Young modulus 27:375 GPa, Poisson ratio 0:27 (shear modulus 10:778 GPa), and for the joints: thickness 5 mm, density 2500 kg/m3, Young modulus 3:0 MPa, Poisson ratio 0:00 (shear modulus 1:5 GPa), cohesion 0:00, friction angle p=4 (rad). Numerical modeling was made using CAST3M, a finite elements code developed by the French Commission for the Atomic Energy (Commissariat à l’Energie Atomique, CEA), with routines for dimension stone buildings written by researchers of the European Commission’s Joint Research Centre (Pegon et al., 2001; Pegon, 1999). The blocks are meshed into 4 nodes tetrahedra so their surfaces are meshed into 3 nodes triangles. Surfaces facing each others are meshed in such a way that every node of one surface faces one and only one node of the other. These couples of nodes are used to model the contact condition as a relaxed Signorini one: a linear elastic repulsive force is applied

When an Abeille’s vault is ideally transformed by increasing its slenderness ratio or decreasing its thickness ratio, the deep beam model that can possibly describe the mechanics of a single block if bulky (Brocato and Mondardini, 2011) leaves way to that of a plain beam with compressed extrados, neutral middle plane and stretched intrados. Similarly to what have been proposed in the investigation on shell structures by Ramm and Wall (2004), we assume here that looking at the main load carrying phenomena would lead to a better understanding of the behavior of our structures. Membrane actions and bending are the predominant mechanism in a traditional vaulted system: the more the first dominates on the second, the more the structure works like a vault by the catenary effect, conversely it could be considered more like a stack of timbers carrying loads as levers. In order to discern the difference, we look at the definition of the elastic energy in a beam model: each block can be seen as a beam the longitudinal axis of which is parallel to the longer edges of the block (i.e. the x axis is parallel to the 2a long edge, see Fig. 2), the transversal axes being one in the plane of the vault and the other orthogonal to this plane (say, y and z in that order). The elastic energy of a block can be computed accepting the approximation given by Saint Venant’s solution of the beam problem, a solution that looks at the beam as a slender prism, disregarding local effects and assuming that the only existing external forces act on the end faces of the prism (in such a way as to be locally equilibrated by the stress vector). Further simplifications are necessary to cope with the case at issue according to the aim, and we will admit working within the limits of a simplified solution with respect to the shear forces and to torsion, neglecting their possible coupling. According to this solution, the elastic energy of the beam is the sum of the elastic energies associated with the individual internal forces and moments. For the kth block: ðMy Þ

ðNÞ

Ek ¼ Ek þ Ek ðNÞ

Ek ¼

1 2

Z Lk

ðM z Þ

þ Ek

N2k dx; EA

ðM y Þ

Ek

ðV y Þ

þ Ek ¼

1 2

ðV z Þ

þ Ek Z Lk

ðTÞ

þ Ek ;

M2ky dx; EIy

ðM z Þ

Ek

¼

1 2

Z Lk

M2kz dx; EIz

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M. Brocato, L. Mondardini / International Journal of Solids and Structures 54 (2015) 50–65

ðV y Þ

Ek

¼

Z

1 2

Lk

V 2ky dx; GAv y

ðV z Þ

Ek

¼

1 2

Z Lk

V 2kz dx; GAv z

ðTÞ

Ek ¼

1 2

Z Lk

qT 2k dx; GI0

where Nk is the axial force on the block, M ky the bending moment about y; Mkz the bending moment about z; V ky the shear force in the y direction, V kz the shear force in the z direction, T k the torque, A the area of the cross section of the block, Iy the moment of inertia of this section relative to its central y axis and Iz is the moment of inertia relative to the z axis, Av y is the area of the cross section effective with respect to the y shearing and Av z with respect to z shearing, q is the torsion factor, I0 ¼ Iy þ Iz the polar moment of inertia, and Lk is the interval along the x axis of the block to be taken into account. (E is the Young modulus of the material and G its shear modulus, already given in Section2.3). Notice that all geometric parameters do not depend on k, as all blocks are geometrically equal, but the introduced coordinates have a local meaning, even though for shortness we have not indicated their belonging to k by an index. Then, neglecting the elastic energy of the mortar layers, the elastic energy of the vault can be computed as the sum of the energies of all blocks:

Eb ¼

X Ek

Fig. 7. Nomenclature of the internal actions and marks of the ideal sections, numbered from 1 to 5, used to compute the integrals.

k

(where k runs on all blocks). We can distinguish the contributions to the total energy coming from the axial deformation, the shearing deformations, the bending deformations and the torsional deformation of all blocks, simply taking the sums of the corresponding terms. For instance the total energy coming from the axial deformation is ðNÞ

Eb ¼

X ðNÞ Ek ; etc: k

and the total energy of the vault: ðM y Þ

ðNÞ

Eb ¼ Eb þ Ek

ðMz Þ

þ Eb

ðV y Þ

ðV z Þ

þ Eb

þ Eb

Both the numerical approximation of this rule and the assumptions made to adapt to Saint Venant’s generate errors that can be estimated as a ‘residual’ energy:

Eres ¼ Ec  Eb where Ec is the total elastic energy computed as in a continuum with the numerical rule associated with the used finite elements. To compare results obtained on different structures, we conclude introducing the following energy quotas:

eðNÞ ¼

The internal forces and moments can be computed from the stresses according to the definitions:

Nk ðxÞ ¼

Z A

ðkÞ rxx dA; Mky ðxÞ ¼

Z

V ky ðxÞ ¼ rðkÞ V kz ðxÞ ¼ xy dA; Z A ðkÞ T k ðxÞ ¼ ðrðkÞ xz y  rxy zÞ dA;

Z A

Z A

ðkÞ rxx z dA; M kz ðxÞ ¼ 

Z A

rðkÞ xx y dA:

rxzðkÞ dA;

eðMy Þ ¼

Eb y ; Eb

ðV Þ

eðV y Þ ¼

eres ¼

where the components of the stress tensor depend in general on x; y; z and on k as indicated by the exponent. To evaluate integrals over any Lk numerically, each block is sliced into six parts by five inner sections on which the components of the elastic energy are computed (see Fig. 7). The total energy of the kth block is then obtained by the trapezoidal rule: ðiÞ if f ðkÞ ðxÞ is a function defined on Lk ; f k ; i ¼ 1; . . . 5, the value it takes on the ith section, and Dx the equal distance between sections, we take:

Lk

Eb ; Eb

Eb y ; Eb

ðM Þ

eðMz Þ ¼

Eb z ; Eb

ðV Þ

eðV z Þ ¼

Eb z ; Eb

ðTÞ

eðTÞ ¼

Eb ; Eb

and the dimensionless residual

A

Z

ðM Þ

ðNÞ

ðTÞ

þ Eb :

4   1X ðiÞ ðiþ1Þ f þ fk Dx 2 i¼1 k   1 ð1Þ 1 ð5Þ ð2Þ ð3Þ ð4Þ ¼ f k þ f k þ f k þ f k þ f k Dx 2 2

f k ðxÞ dx ¼

and the integral of f over the whole vault is then

 X1 ð1Þ 1 ð5Þ ð2Þ ð3Þ ð4Þ f k þ f k þ f k þ f k þ f k Dx: 2 2 k

ð4Þ

Eres : Eb

Values of eðMy Þ approaching 1 represent the aptitude of the structure to work according to the principle of the lever while values of eðNÞ approaching 1 suggest a vault-like behavior according to that of the inverted catenary. Hence these functions can be taken as indicators of the structural performance of the system, as a catenary or as a ‘levery’ one. The shear part eðV y Þ is presumably relatively higher here than in thinner structures and is thus computed to have a more complete examination (eðV y Þ =eðMy Þ would be larger in a deep beam than in a slender one). eðMz Þ ; eðV z Þ ; eðTÞ , representing in plane and torsional effects, are also evaluated to have a more detailed picture. The six terms of the beam model give all together but a first approximation of the phenomenon; eres is computed to take the deviation from the ideal beam model into account. This deviation is expected to be more relevant for catenary structures than for ‘levery’ ones because the former are supposed to be made of less slender elements than the latter and the beam model is more appropriate for a slender element than for a bulky one. eres (neglecting errors related to the approximation in computation) is probably related to three physical issues:

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 the lack of information in the Saint Venant’s model about the local effects, which are not necessarily negligible here, due to the different contacts taking place on a block;  the presence of traversal actions across the blocks, giving rise to ryy and ryz stresses;  the simplifying assumptions in the treatment of shearing and torsion, leading to disregard possible secondary torsion effects (associated with the deformation of the cross section in its plane, neglected by Saint Venant’s primary torsion), which might be effective when the slenderness ratio and the thickness ratio are small. A further consideration can be made on how the six quoted parts of the elastic energy are spatially distributed on these structures. For this purpose we also consider an ancillary linear elastic model of them, not taking the existence of joints into account, and compare the results. In both kind of models the ratio between the local value of a component of the elastic energy (either due to the axial deformation, to the shearing deformations, to the bending deformations, or to the torsional deformation) and the maximum value that this component takes on the whole structure is computed in order to display the spatial distribution of this energy in terms of a non dimensional field taking values in the ½0; 1 interval. Thus we will display piecewise constant fields like the following one:

gðNÞ k ¼

ekðNÞ n o ðNÞ maxk ek

ð5Þ

k running on all blocks and the apex indicating the different internal forces and moments. 2.5. Chirality and thrust indicators A vault withstands vertical loads discharging inclined forces at its buttresses, whose horizontal component we will refer to as thrust. This catenary mechanism is not significant for a flat grid undergoing small deflections, though it may take some extent if some elements of the grid are out of plane, as it is the case for a nexorade, or if the boundary conditions impose so. In the cases at issue, each block at the boundary discharges a force having, vertical component apart, a thrust component orthogonal to the ideal boundary line (i.e. along the x axis of the block as depicted in Fig. 7) and another horizontal component parallel to this line. As already mentioned in the Introduction, the chiral geometry of these structures must entail, under a uniform vertical load, a chiral set of boundary reactions. This effect appears more evidently when observing the local vectorial components of the thrust than with respect to the vertical reactions, so that it can more be attributed to catenary structures than to ‘levery’ ones. In any case a measure of the chirality can be adopted as an indicator of the structural performance of Abeille’s vaults. Due to its chiral geometry any structure of the type considered here is not superimposable on its mirror image. A geometric measure of the chirality can be given by the parameter p=L : the larger this number the larger the discrepancy between the structure and any image of it made by reflection about a line passing through its center and parallel to the bonding. p=L is also related to the distance between the center of the contact area at each side of the boundary and the axis of symmetry of the ideal square inscribing the vault. Clearly, it carries no information on the slenderness, splice angle, or thickness of the vault and thus indicate equally a catenary and a ‘levery’ structure from all the other points of view considered here so far.

On another side, the chirality can be measured as a mechanical effect, considering the number

MH ; FV L

ð6Þ

where MH is the moment with respect to the center of the vault of all thrusts at one of the four sides of the boundary, F V is the vertical resultant of all the external forces, and L (overall length of the vault) is introduced to make the result dimensionless. Hence, a measure of chirality that lets the load descent path emerge can be given by the ratio of the two quoted numbers:

v¼

M H =p ; FV

ð7Þ

and is expected to take larger values when the behavior of the system comes closer to that of a vault. 3. Results In this section the numerical results are illustrated focusing on the following issues. (A) The elastic energy, especially its partition into the six already quoted terms (bending moments, axial force, shear forces and torsion), (A1) for the family F1 (different values of the slenderness ratio a=b), (A2) for the family F2 (different values of the splice angle u), (A3) for the family F3 (different values of the thickness ratio h=L), ðNÞ (A4) in terms of spatial distribution (gk and alike results). (B) The reactions, especially the thrust at the boundary, (B1) for the family F2 (different values of the splice angle u), (B2) for the family F3 (different values of the thickness ratio h=L). (C) The chirality of the structure, for the family F4. (D) The stress distribution and the displacements. 3.1. Elastic energy The computed quotas of the elastic energy for structures changing the parameter a=b (A1 analysis) is shown in Fig. 8. It can be observed that the bending energy dominates in any case on all the others, which suggests that the structure works more as a deflected grid than as a compressed vault. As expected, the quota of the energy of the axial deformation, eN , decreases, if slightly, when the geometry changes from bulky to slender blocks (a=b going from 4 to 11), while the bending energy quota, eM , grows correspondingly. In this passage, the residual energy quota, eres , decreases too, probably due to the fact that it contains the effect of transversal compressions that are larger for structures made of less slender blocks and working more as vaults. This idea is confirmed by the analogous decrease of the torsional energy quota, eT , certainly associated with the compression forces transversal to the axes of the blocks. The energetic comparison of structures having different splice angles (A2 analysis) is shown in Fig. 9. Again, the bending energy dominates in any case, so that the conclusion drawn from A1 can be confirmed on the preeminence of the flexural response. A small dependence of the bending and axial force energies from the splice angle shows anyway that the latter are less negligible if the joints are closer to the vertical than in the other cases, so that structures with smaller splice angles can be seen as closer to catenary ones. This interpretation is confirmed by the fact that the axial and torsional energies display divergent tendencies, similar to those

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described above for F1—with the former decreasing and the latter increasing when the angle grows. Opposite to what observed in the previous analysis, the quota of the residual energy augments significantly when the structure moves away from the (slightly) more compressed states observed for smaller splice angles, possibly indicating an increasing importance of the energy associated with the distortion of the blocks’ cross section (bi-moments or secondary torsion) which is not computed with the Saint Venant’s model. Notice that the numerical precision of the trapezoidal rule (4) grows when the splice angle decreases, so that the difference in the computed energies must be attributed exclusively to some inaccuracy of the model, not of the computation. Comparing structures with different thickness ratios (A3 analysis) one still observes the prevailing of the bending energy quota on the others and a little sensibility of this result to changes within the family. Observing Fig. 10, the prevalence is smaller than in the previous cases; this is due to the fact that, in this family, the lengths a; h, and L are such as to generate structures that withstand vertical loads with smaller highest stresses (both a and L are smaller while h is larger). This remark is confirmed by the comparison of Fig. 22 with Fig. 20 and 21, which shows evidence of the reduced stress level about the middle section of blocks in the central area of the structures F3.1 and F3.8 with respect to the structures F1.1, F1.6, F2.1, and F2.6. Hence one can assert that the ‘levery’ mechanism, sustained by the bending energy, is paramount when the structure experiences large stresses, and its dominance reduced when the maximum stress level decreases.

Notice also that the residual energy quota grows to the largest registered value in all case when h=L tends to its inspected maximum (1=14, see Fig. 10), a symptom telling that blocks are in this case to be mostly seen as deep beams, whose behavior is poorly captured by Saint Venant’s model for cause of the non negligible role of the local effects of the contact forces. Dealing with the spatial distribution of the energy quotas (A4 analysis), we observe that the distribution of the energy of the axial deformation computed taking the joints into account is chiral like the geometry of the structure, and displays higher values in correspondence to the supports—as expected considering the catenary mechanism (see the left-hand side of Fig. 11). When the blocks are ideally bound to each others across their joints, this part of the elastic energy is much smaller (no sensible catenary effect existing) and both the chirality and the increase of the energy content next to the boundary can only be read because of the resolving power of the plot—made larger for a field with a small maximum value by the transformation of the field onto the ½0; 1 interval given by Eq. (5) (right-hand side of Fig. 11). The energy associated with the bending moment computed for the model with Mohr–Coulomb interfaces is higher in the middle of the vault than at its boundary, as it could be expected for a structure that is simply supported or close to be so (Fig. 12 on the left). The continuous structure, being on the contrary fixed at the supports, displays the largest bending energies at the boundary and a relative maximum at the center, passing through a neutral zone in between (same figure, on the right). As the plot of the von Mises stress level will show in a forthcoming paragraph, the difference between a continuous structure and one made of distinct blocks results in a higher concentration of this energy in the central part of the ashlars, which is not visible in Fig. 12, where only the average energy of each block is shown (compare with Fig. 21). In the non linear model, the shearing energies are, like the bending ones, higher in the middle of the vault. Though unusual for a structure supported at its boundary, this behavior appears in this case as a consequence of the ‘levery’ mechanism that rules the assembly of ashlars (see the left-hand side of Fig. 13). In fact, observing the right-hand side of Fig. 13, we see that the shearing energy field takes higher values at the boundary when the structure is not separate into distinct elements, and decreases approaching to the center. We may deduce from these observation that—at least schematically—the lack of continuity of Abeille’s vaults entails a sort of superposition of two bearing mechanisms: on the one hand, there is a global bending and membrane effect that equilibrates the

Fig. 9. Quotas of the total elastic energy for the parametric family F2 vs the splice angle (angles are measured in degrees from the vertical). The picture is obtained for a resultant of the external load of 2 kN and is equal to that obtained for 50 kN.

Fig. 10. Quotas of the total elastic energy for the parametric family F3 vs the thickness ratio.

Fig. 8. Quotas of the total elastic energy for the parametric family F1 vs the slenderness ratio of the blocks.

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Max

Min

Fig. 11. The ratio of the energy of the axial deformation on its maximum value, computed with the numerical model including non-linear joints (F2.1 specimen), is plotted on the left. On the right the same field is plotted, but computed using, with the same geometry, a linear elastic model that does not take the existence of joints into account. Notice that fields are normalized to 1 and thus the plots represent the unscaled fields with different resolutions.

Max

Min

Fig. 12. On the left: the ratio of the bending energy on its maximum value for the considered model (F2.1 specimen); on the right, the same field computed using a linear elastic model with no joints and the same geometry. Notice that the purple area (smallest values) in the result on the left is close to red (highest values) in that on the right and vice versa. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

loading like in a continuous system, on the other hand, there is a local bending that is summoned by the bonding to make each ashlar simply support two others at its center while being similarly hold up at its ends. This secondary, local and certainly ‘levery’, effect introduces in all studied cases such a large amount of bending and shearing energy as to dominate over the energies needed by the primary, global and possibly catenary, mechanism, and allow us to draw the conclusion that Abeille’s vaults are rather to be considered as deflected systems.

3.2. Reactions The (A) part of the analysis shown in the previous paragraph focuses on phenomena occurring in the bulk of the structure; the (B) part presented in this paragraph concentrates on what happens at the boundary. If, by the previous results, it appears that Abeille’s vaults mainly work as a deflected grid with a relatively small membrane effect, i.e. rather like the principle of the lever than by that of the inverted catenary, the analysis of the reactions should give some more information on their membrane behavior. A grid, in

Max

Min

Fig. 13. On the left: the ratio of the shearing energy on its maximum value for the considered model (F2.1 specimen); on the right, the same field computed using a linear elastic model with no joints and the same geometry. Notice that, as in Fig. 12, min and max areas occupy opposite places on the shown solutions.

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fact, exerts no horizontal actions at its boundary, at least within the limits of a first order theory and anyway when the constraints do not allow for such reactions. A membrane, on the contrary, do exert such actions necessarily and needs the appropriate constraints. A result that has been observed for the first time in the analysis of Abeille’s vaults by Brocato and Mondardini (2012), is that the thrust is not locally directed toward the radial direction: the chirality of the structure results into a chiral system of horizontal reactions that is invariant under rotations of p=2 about the vertical axis passing through the center of the vault. Couples are created by the horizontal reactions normal to the ideal boundary line (R1 to R4 in Fig. 14) that must be balanced by a set of non zero horizontal reactions tangential to the same line (T 1 to T 4 in Fig. 14) for the rotational equilibrium of the vault about the vertical axis. In the next paragraph we will concentrate on the importance of this effect in different structures (C analysis on family F4), here we rather look at the ratios of the two horizontal components of the reactions on the total vertical load to understand the part that the catenary effect (which these ratios represent) plays in the structural system. When the splice angle is large it can be expected that blocks bear each others with a smaller horizontal component of the contact force than when this angle is small (the ‘wedge’ effect being reduced accordingly). Hence in the family begot by the splice angle, the thrust indicating a catenary effect should decrease when the angle grows (B1 analysis made for the F2 family). This effect is confirmed by the calculations shown in Fig. 15. In the same family the tangential reactions tend to disappear when the angle grows (Fig. 16), so that vaults with large splice angle can be seen as having a negligible mechanical chirality and a behavior similar to that of a simply supported plate, with most of the load discharged in the middle of the boundary sides (see how the R2 and R3 components of the thrust in Fig. 15 dominate over the other two terms). In the same matter, notice that the corner of the structure are restrained by the joint action of two forces in each direction. E.g. looking at the left upper corner of the vault in Fig. 14: 90% by the R4 normal thrust and 10% by the T 1 tangential thrusts in one

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Fig. 15. Radial components of the reactions per unit weight of the vault in the parametric family F2; their decreasing with increasing u testifies of a larger and larger superiority of the ‘levery’ effect.

Fig. 16. Tangential components of the reactions per unit weight of the vault in the parametric family F1: they decrease faster than the radial components when the splice angle grows; the values of the three concordant forces tend to become equal in this passage.

Fig. 14. Sketch of a typical vault and nomenclature of the horizontal reactions of the confinement ring (not drawn).

direction and 80% by R1 and 20% by T 4 in the orthogonal direction. This means that the first and last row of the structure are held at the boundary partly by the friction they exert on the confinement parallel to them, partly by a discharge arch across their span. In other terms, there is a diagonal catenary effect observable especially from corner to corner of the vault. These observations are valid also for the outcome of the next (B2) analysis. In the case of an ideal inverted catenary for the given load conditions, the thrust is inversely proportional to the depth of the structure. To check if this effect can be observed in our case we refer to Fig. 17, where the reactions are plotted as functions of the structural height h (B2 analysis for the F3 family). It can be observed that the function thrust versus depth has a lesser pitch and concavity then it should have following the principle of the inverted catenary. In the ideal catenary case this function should be an hyperbole, but in the numerical result the difference between the first and the last value is less than half that between the corresponding points on an hyperbole. This confirms again the previous conclusions on the ‘levery’ nature of Abeille’s vaults. In any case, the tangential components of the thrust are relatively small compared with the previous case (B1), taking values which are one order of magnitude less than the radial components (while the two components have the same order of magnitude for

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Fig. 17. Radial components of the reactions per unit weight of the vault in the parametric family F3: the thrust per unit carried load should, in a catenary system, decreases at the rate of 1=h, which is not the case in this picture.

the F2 family). Increasing the structural depth decreases the tangential components of no more than 20% with respect to their larger value (taken for the smallest depth, i.e. when the structure is closer to a grid), while this sensibility goes up to 50% when the splice angle ranges from the smallest to the largest of the considered values (see Fig 18). The discussion about the reactions can be concluded by the assertions that the thrust is less sensible to the structural depth than it ought to be in a catenary system and that the structure mainly discharges loads in the middle of the four sides, with a smaller contribution of the corners and a reduced chirality. Furthermore, the presence of tangential thrusts (clearly associated with an in plane deflection of the blocks) is more important for structures behaving more like grids, less for structures behaving like vaults, as if the catenary effect tended to clear up this feature. 3.3. Chirality The outcome of the (C) analysis of the chirality is shown in Fig. 19. In this plot two curves are displayed, which represent two distinct features as functions, normalized to one, of the number of rows of the structure: the thrust at one side of the structure H (i counting all the blocks at one side of the vault)

H¼

X Ri ; i

Fig. 18. Tangential components of the reactions per unit weight of the vault in the parametric family F3.

Fig. 19. Plots of the normalized measures of the chirality functions of the p=L parameter of family F4.

v and of the thrust H as

and the chosen measure of the chirality v. The functions are

H ; Hmax

1

v : vmax

The plot indicates that changing the number of rows (or p=L) changes chiefly the mechanical behavior of the structure: less rows imply a smaller thrust and a larger chirality, i.e. a ‘levery’ system, vice versa more rows make a more catenary system. Clearly, all previous results still demonstrate that this catenary effect, though larger for larger numbers of blocks, is dominated by the ‘levery’ one. 3.4. Stress and displacement analyses Figs. 20 and 21 show the von Mises stress level in the vaults of family F1 and F2 respectively. Both fields are similar, with the central part of blocks interested by most of the deviatoric energy they represent—as it could be expected for simply supported beams carrying a concentrated load at their center. This feature confirms the previous observations of a prevailing ‘levery’ mechanism over the catenary one. Furthermore, notice that there is a small change in the stress level within family F2, i.e. changes of the splice angle only slightly affect the structural system. The distribution of vertical displacements is similar in all cases, but there is an important difference between the extreme specimens of each family: the maximum deflection is 17% larger for F1.1 than for F1.2, +82% for F2.1 with respect to F2.6, +50% between F3.1 and F3.8 (see Figs. 23–25). These results can be interpreted in the light of the fact that the catenary effect, generally, gives rise to stiffer systems than the ‘levery’ one does, because of a more even distribution of the elastic energy within the structure and under the condition of a reduced compliance of joints. At the same time, the stiffness of a deflected grid depends at least linearly on the moment of inertia of the beams (the bending-torsion interaction between orthogonal beams might enhance the stiffness with respect to that given by the sum of the flexural ones) and thus grows, more than linearly in family F2 as the splice angle grows, or as the cube of the thickness in family F3. Still in these families, the self weight grows with the quoted generating parameters, so that the deflections can be expected to decrease also as a consequence of the ‘levery’ mechanism. To check if the stiffening of joints favors the occurrence of a catenary effect, the displacements of an equivalent structure

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without joints were computed. The result shows that the latter are of one order of magnitude smaller than those computed with joints. It can be deduced from this result that the quoted change in the maximum deflection should be more attributed to the increase of the stiffness of the ‘levery’ shear in the structural behavior going from F1.6 to F1.1, from F2.1 to F2.6, and from F3.1 to F3.8. A last piece of information coming from computations concerns the behavior of joints, i.e. wether and where they open or not when the structure is loaded and where, approximately, a catenary representing a limit ‘line of pressure’ or ‘surface of pressure’ can be

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expected to pass. It is worth reminding that such line is defined in an arch by the set of tangents given by the resultant at all sections of the internal actions, and that the homonymous surface is the generalization of this concept to shells. Fig. 26 shows the chart of the parameter indicating the opening of joints, for the structure F2.1 and in the two cases of loading with 2 kN (left) or with 50 kN (right); only a quarter of the two structures is shown to increase the resolution of plots. Blue areas indicate parts that are not in contact, red ones those that are in contact. It can be noticed that there is only a slight difference between the two loading conditions.

Fig. 20. Comparison of the von Mises stress chart seen from above, in the first (F1.1 on the left) and last (F1.6 on the right) vault of the parametric family F1 (generated by changes of the slenderness ratio).

Fig. 21. Comparison of the von Mises stress chart seen from above, in the first (F2.1 on the left) and last (F2.6 on the right) vault of the parametric family F2 (generated by changes of the splice angle).

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Furthermore, it is possible to remark that the central area of the structures, with the exception of the very first row of stones next to the boundary, has joints compressed at the extrados and opened at the intrados, while the opposite condition appears at the boundary. Hence, following the direction of the thrust from the center to the supports on any row of stones, it is possible to imagine a line of pressure staying horizontal and close to the extrados across all stones but one and bending toward the intrados crossing the last stone before the support. This first image need be made more precise with the support of Fig. 27, showing the cross section of a structure and some idealized static fields. In a purely catenary system, the line of pressure is the

inverted catenary of the load and needs to fall inside all contact surfaces—if the assumption of no traction bonds is emitted—to be statically admissible. These conditions entail, in the limit of admissibility, a given ratio between the thrust and the vertical reactions at the boundary, a condition that is not respected by the systems studied here, as it was shown in Section 3.2. Hence, a false line of pressure with a larger arrow than admissible should be considered, failing to obey to the aforementioned contact law; we associate to this line a ‘primary’ bending moment (given by the product of the thrust with the distance from the line of pressure to the central line of the structure); both are sketched in red in the quoted figure. A ‘secondary’ bending is due to the bonding and can be

Fig. 22. Comparison of the von Mises stress chart seen from above, in the first (F3.1 on the left) and last (F3.8 on the right) vault of the parametric family F3 (generated by changes of the thickness ratio).

Fig. 23. Comparison of the displacement chart seen from above, in the first (F1.1 on the left) and last (F1.6 on the right) vault of the parametric family F1 (generated by changes of the slenderness ratio).

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imagined as a series of three points bending moments on the individual blocks (represented in green). The diagram of the true bending moment is then obtained superposing those of the primary and secondary bending and the line of pressure corresponding to it should be considered the actual one (in orange). It can be noticed that this line crosses all bonds consistently with the information on their opening and in a statically admissible manner. It is consistent with the computational observations and with the model above to affirm that the behavior expected for a catenary system takes place across the row that is directly supported, while one more typical of deflected systems reigns in the rest of the structure.

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4. Conclusions The study shows that Abeille’s vaults are mainly ‘levery’ systems, with most of the elastic energy captured by a bending mechanism and spatially distributed in the blocks as the sum of a ‘primary’ term, related to the global equilibrium of the plate, and a ‘secondary’ one, kin to what is observable in three points deflected beams. The catenary mechanism is limited to the stones that are in contact with the confining supports and thus characterized by a reduced sensibility to the thickness ratio of the system. Clearly the mechanism cannot disappear as some thrust is necessary to avoid the slipping of joints.

Fig. 24. Comparison of the displacement chart seen from above, in the first (F2.1 on the left) and last (F2.6 on the right) vault of the parametric family F2 (generated by changes of the splice angle).

Fig. 25. Comparison of the displacement stress chart seen from above, in the first (F3.1 on the left) and last (F3.8 on the right) vault of the parametric family F3 (generated by changes of the thickness ratio).

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Fig. 26. Comparison of the joints contact charts for F2.1 vault loaded with 2 kN or 50 kN. A quarter of the structures is shown, seen from above, with the center of the structure at the bottom right corner of the picture. Remember that the largest base of each trapezium stays at the intrados. Blue indicates surfaces that are no more in contact, red those that are still in contact, the non linear neutral line falling in the green zones. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 27. Qualitative analysis of the load descent path in the structure: the red line represents an ideal line of pressure corresponding to the given loads and the computed thrust; the red diagram ðaÞ is the corresponding primary bending moment. Such internal action is not statically admissible in the assumption of no traction bonds, as the pressure falls outside of the contact surfaces. The blue line represents the apparent line of pressure obtained interpolating the results about the opening of the contact surfaces given in Fig. 26. The superposition of a secondary bending moment ðbÞ, shown in green, on the primary one ðaÞ gives the true bending moment ðcÞ and the actual line of pressure, which are shown in orange. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

These structures discharge loads with a relatively small thrust (if compared to purely catenary ones) and display a chirality that has a reduced mechanical effect. The catenary mechanism becomes more important when the structure becomes thicker, but—contrary to expectations—it never prevails. A model of the structure with ideally continuous joints is not to be deemed predictive, being unable to capture the main feature related to the quoted distribution of the bending energy. Nevertheless, an equivalent homogeneous model of the structure can be sought for the analysis of materials like suggested by Dyskin et al. (2001) and subsequent literature, studying the limit of infinitely many blocks, but under the condition that the energy of ‘secondary’ bending deformation appearing in the blocks, which disappears in the limit, does not play a role at the scale one wants to study. Acknowledgments This work was made during Ms. Mondardini doctorate within the framework of an industrial agreement for training through research (CIFRE n.1506/2010) jointly financed by the company ‘‘Société Nouvelle le Bâtiment Régional’’ (SNBR), and the National Association for Research and Technology (ANRT) of France.

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