3D voxel homogenized limit analysis of single-leaf non-periodic masonry

Computers and Structures 229 (2020) 106186

Contents lists available at ScienceDirect

Computers and Structures journal homepage: www.elsevier.com/locate/compstruc

3D voxel homogenized limit analysis of single-leaf non-periodic masonry Simone Tiberti, Gabriele Milani ⇑ Department of Architecture, Built Environment and Construction Engineering, Technical University of Milan, Piazza Leonardo Da Vinci, 20133 Milan, Italy

a r t i c l e

i n f o

Article history: Received 15 October 2019 Accepted 12 December 2019

Keywords: Non-periodic masonry Voxel strategy Homogenisation Limit analysis Out-of-plane behaviour Single-leaf walls

a b s t r a c t This paper presents an extensive investigation on the out-of-plane collapse behavior of single-leaf nonperiodic masonry walls. This is achieved by deriving out-of-plane homogenized failure surfaces from testwindows that are extracted from various location within the same non-periodic masonry wall. The concept of ‘‘test-window” is inspired by that of Statistically Equivalent Periodic Unit Cell (SEPUC), which is an evolution of that of Representative Element of Volume (REV), and is needed for successfully applying homogenization to non-periodic masonry. An innovative feature introduced in this paper is the automatic generation of a suitable 3D finite element mesh directly from the sketch of the considered masonry testwindow, based on a so-called ‘‘voxel strategy” that converts each voxel (the 3D equivalent of a pixel) into a finite element. Moreover, this mesh generating procedure enables the correct representation of the transversal layout of the test-window, and it also contains a coarsening strategy that allows a reduction of the overall number of finite elements in the 3D mesh. For the derivation of the out-of-plane homogenized failure surfaces a linear programming problem is solved, which is based on the upper bound theorem of limit analysis coupled with a homogenization approach. The investigation is performed on six real case studies displaying different degrees of non-periodicity: for the four test-windows of each case study, two out-of-plane homogenized failure surfaces are extracted (flexural and torsional) as well as three relevant deformed modes at collapse. Eventually, the results are critically commented; comparisons are drawn among the case studies, highlighting the influence of each non-periodic masonry bond on the results. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction Starting from mid Nineties, the technique known as homogenization has been extensively dealt with in several works aiming at using it as a swift and effective tool for modelling masonry. Conceptually, this technique is based on the identification of a Representative Element of Volume (abbreviated with the acronym REV), which for a generic heterogeneous medium corresponds to the least-sized portion of material encompassing all the physical and geometrical properties needed for a full characterization of the investigated medium. Moreover, the translation of the REV (generally in a 2D space) must generate the considered bulk of material, which has then to display a periodic pattern. It is widely accepted that homogenization stands between the two classical approaches for representing heterogeneous materials, which are macromodelling [1–6] and micro-modelling [7–13]. In fact, homogenization requires the accurate modelling of the material at the ⇑ Corresponding author. E-mail address: [email protected] (G. Milani). https://doi.org/10.1016/j.compstruc.2019.106186 0045-7949/Ó 2019 Elsevier Ltd. All rights reserved.

microscale – as in the latter approach – but restricted only to the identified REV, thus greatly reducing the computational effort. On the other hand, homogenization aims at deriving macroscale properties for the considered material – as in the former approach – but starting from a limited portion of the material itself (i.e. the REV), so that no lengthy and expensive experimental campaigns are required. As previously mentioned, homogenization has been employed in various works seeking to give an insight into the behavior of masonry. Initially the aim has been focused on the derivation of the elastic properties of this material [14], later extended to include properties related to its post-peak behavior [15–17] and inelastic parameters linked to the formation of damage, degradation and cyclic behavior [18–20]. A different field of application is represented by the pairing of homogenization and the two theorems of limit analysis, with the goal of investigating the behavior of periodic masonry at collapse. Several works have focused on the in-plane behavior [21–25], with only a few venturing into the outof-plane behavior [26,27].

2

S. Tiberti, G. Milani / Computers and Structures 229 (2020) 106186

Some apparent complications arise when homogenization is applied to historical masonry. In fact, several old buildings consist of masonry in which the units are neither arranged in a periodic pattern, nor they display the same geometrical characteristics. This type of non-periodic masonry goes under the generic definition of ‘‘rubble masonry”. Seemingly, this undermines any possibility of effectively applying homogenization to such kind of masonry; actually, it is possible to override this issue by making use of statistical tools. In this way, it is still possible to identify a REV, provided that it is a statistically acceptable representation of the considered non-periodic masonry, Therefore, the concept of Representative Element of Volume must evolve into that of a Statistically Equivalent Periodic Unit Cell (abbreviated with the acronym SEPUC), whose origin are to be traced in an early work by Šejnoha and co-workers [28] and which is also successfully applied to other types of materials [29,30]. Actually, in these works the concept of SEPUC represents the actual microstructure derived by matching some statistical descriptor of the real and artificial geometry arriving at periodicity and exactly the same volume fractions of individual heterogeneities. Therefore, a similar concept named ‘‘testwindow” must be introduced, which represents a cell of finite size directly extracted from the considered non-periodic medium. Such concept is indeed employed by Tiberti and Milani [31] for the in-plane homogenized limit analysis of non-periodic masonry.

Conversely, Milani and Lourenço [32,33] use another kind of statistical REV that is modified through a long series of Monte Carlo simulations at every cycle so that a huge number of similar REVs are generated. These are then used for the in-plane [32] and out-ofplane [33] homogenized limit analysis of non-periodic masonry. It is rather evident from this short literature review that at the moment no studies have dealt with the out-of-plane behavior at collapse of real historical masonry structures, which is extremely useful in light of the disastrous consequences of several recent earthquakes occurred in Italy (L’Aquila 2009, Emilia-Romagna 2012, Central Italy 2016) where many historical masonry buildings partially or totally collapsed, often because of their poor response to the seismic-induced out-of-plane actions. This paper presents an extensive investigation on the out-ofplane collapse behavior of six case studies represented by historical masonry buildings that display different degrees of nonperiodicity. The out-of-plane collapse behavior is investigated through the extraction of out-of-plane homogenized failure surfaces according to the homogenized limit analysis modelling approach presented in [34]. This is integrated with the introduction of the so-called ‘‘voxel strategy” for the creation of a 3D finite element mesh directly from the rasterized sketch of a masonry test-window, which represents another innovation introduced by the present work.

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 1. (a) Sample image of a stone embedded in mortar; (b) mid-plane of the 3D FE mesh; (c) 3D FE mesh with extruded stone; (d) aerial view of the 3D FE mesh with ellipsoidal stone; (e) 3D FE mesh with ellipsoidal stone; (f) section of 3D FE mesh with ellipsoidal stone.

S. Tiberti, G. Milani / Computers and Structures 229 (2020) 106186

This paper is structured as follows: Section 2 offers a detailed description of the aforementioned ‘‘voxel strategy” aiming at creating the 3D finite element mesh. Section 3 presents a recap of the homogenized limit analysis problem used for extracting the outof-plane homogenized failure surfaces. Section 4 presents the results obtained for the considered six real case studies, which are critically compared and discussed in relation of the nonperiodic masonry type displayed by each case.

2. Voxel strategy for converting a picture into a 3D finite element mesh A swift, simple procedure for the creation of a 3D finite element mesh directly from the sketch of a real masonry test-window is described in this section. The approach used here goes under the name ‘‘voxel strategy” because it automatically generates finite elements from voxels, entities that are the 3D equivalent of 2D pixels. The procedure for creating the 3D mesh starts from obtaining the rasterized sketch of the considered masonry test-window, for instance by using the Image Processing Toolbox functions available in MATLAB [35]. This sketch represents the source image for the procedure and must be either greyscale or black-and-white so that units and mortar are uniquely identified by distinct colors. The source image is then imported into the MATLAB function purposefully written for the creation of the 3D mesh; the user must also input the real dimensions of the considered test-window, the number of finite elements desired in the transversal direction, and the transversal configuration of the considered test-window. The latter feature enables the user to choose between a simple transversal extrusion of the in-plane configuration, and a more complex transversal configuration where the masonry units are provided with an ellipsoidal shape. In this case, the in-plane configuration represents the mid-plane of the 3D mesh; the ellipsoidal shape is obtained by conveniently reducing the mid-plane surface of the units so that their 3D configuration resembles either a full ellipsoid or a truncated one.

3

Fig. 1 shows an example of the aforementioned feature: specifically, Fig. 1a shows the greyscale rasterized sketch of a stone embedded in mortar; Fig. 1b shows the mid-plane of the 3D finite element mesh obtained for the stone; Fig. 1c shows the 3D finite element mesh obtained considering the stone extruded along the transversal direction; Fig. 1d and e show the 3D finite element mesh obtained considering an ellipsoidal stone; finally, Fig. 1f shows a section of this version of the 3D finite element mesh, where inner layers of FEs pertaining to the ellipsoidal stone are denoted with different colors. The MATLAB function converts the source image into an M
N
3 array, where M and N are the number of pixels along the vertical and horizontal directions of the image, and the 3 layers each contain one entry of the pixel’s RGB triplet. Namely, the Red, Green, and Blue values of the triplet are listed in the first, second, and third layer, respectively. Then, a simple M
N matrix containing only the Red values of the triplet is extracted from the bigger array. Afterwards, an M
N
O array is constructed, where O is the number of transversal finite elements; each M
N matrix

Fig. 3. (a) Sample source image of masonry test-window; (b) 3D finite element mesh resulting from the voxel strategy.

Fig. 2. Voxel strategy for creating the 3D finite element mesh.

4

S. Tiberti, G. Milani / Computers and Structures 229 (2020) 106186

represents the configuration related to that specific finite element layer, which depends on the chosen transversal configuration. Each element of the M
N
O array corresponds to a voxel, which is treated as the centroid of a single finite element and is provided with XYZ coordinates that are determined from the input global dimensions (as shown in Fig. 2). These coordinates are evaluated according to a reference system whose origin is located at the centroid of the considered test-window, where X and Y represent the horizontal and vertical axes, respectively, while Z is the transversal

direction. This procedure enables the generation of solid brickshaped finite elements. The XYZ coordinates of each element’s eight nodes are then calculated starting from those of its centroid. A subscript is included aiming at the identification of the masonry units of the considered test-window, which represent ‘‘macro elements” needed in a later stage of the homogenized limit analysis problem: through functions made available in the Image Processing Toolbox library, each masonry unit is given an ID number, and the XYZ coordinates of its centroid are determined as well.

Fig. 4. Coarsening strategy for reducing the number of finite elements.

Fig. 5. (a) Black-and-white rasterized sketch of the sample masonry test-window; (b) mid-plane configuration with a 2
2 coarsening strategy; (c) mid-plane configuration with a 3
3 coarsening strategy; (d) mid-plane configuration with a 4
4 coarsening strategy; (e) mid-plane configuration with a 5
5 coarsening strategy.

S. Tiberti, G. Milani / Computers and Structures 229 (2020) 106186

Three distinct matrices are then created: the first is named ‘‘node matrix” and, for each node of the mesh, contains its XYZ coordinates and ID number – ordered in a top-to-bottom fashion that starts from the front-top-left corner and ends in the rear-bottomright corner. The second is named ‘‘matrix element” and, for each finite element of the mesh, contains its ID number, the ID number of its eight nodes (listed in a counterclockwise sense starting from the front-top-left node), the XYZ coordinates of its centroid, its ‘‘material flag” that indicates whether it pertains to mortar or to a masonry unit (depending on the Red value of the original pixel’s RGB triplet), and the ID number of its related masonry unit (in case of mortar elements, this ID number is set to zero). The third and final matrix is named ‘‘macro element matrix” and, for each masonry unit of the test-window, lists its ID number and its centroid’s XYZ coordinates. An example of the 3D finite element mesh resulting from this procedure – visualized through the patch function in MATLAB – is presented in Fig. 3b for the sample source image shown in Fig. 3a. Another subscript is included that sets a procedure to decrease the number of finite elements according to a ‘‘coarsening strategy” whose ultimate goal is to reduce the computational effort needed for the subsequent numerical analyses. In the original M
N matrix representing the in-plane configuration, this subscript condensates a square consisting of n
n entries (n = 2–5) into a single entry of a new Mr
Nr matrix (see Fig. 4), where Mr and Nr are the reduced number of pixels (which have increased dimensions) along the vertical and horizontal directions, respectively.

5

The physical nature of these new pixels is determined by a threshold representing the overall number of mortar pixels in the original configuration: if the number is lower than the selected threshold, the new pixel is treated as a unit pixel, otherwise it becomes a mortar pixel. An example of the result in terms of increase of the mesh size is shown in Fig. 5 applied to the same sample source image of Fig. 3a for the four considered coarse cases; in this case, the coarser meshes represent the mid-plane configuration of the overall 3D finite element mesh. It can be easily noted that, as the mesh becomes coarser, its accuracy in representing the original geometrical layout decreases. 3. Homogenized limit analysis problem formulation This section offers a recap of the formulation of the coupled homogenization-limit analysis problem that enables the creation of the out-of-plane homogenized failure surfaces. A complete treatment of the mathematical formulation is presented in [34], where the reader is referred to for further information. The out-of-plane homogenized failure surfaces are the final result of a linear programming problem expressed in a standard form and written into a MATLAB script. This problem is based on both homogenization and the upper bound theorem of limit analysis, so it is actually a minimization problem whose objective function is the dissipated internal power, and which is also subjected to four sets of equality constraints. Before listing them, it must be remarked that the chosen problem formulation requires an input

Fig. 6. (a) Velocity jumps for a generic interface normal to axis X; (b) Mohr-Coulomb failure criterion; (c) periodicity boundary conditions.

6

S. Tiberti, G. Milani / Computers and Structures 229 (2020) 106186

finite elements mesh consisting of regular parallelepiped 3D elements, thus making the 3D mesh resulting from the voxel strategy particularly suitable for use. The finite elements are rigid and bereft of rotation rate, so that their kinematics is fully determined
 by only the displacement rate field of their centroid u_ x ; u_ y ; u_ z ; the kinematics chosen for this problem is compatible with a KirchhoffLove plate model that is here employed for representing the outof-plane behavior, aiming at simplifying the computational effort. The three components of the centroid’s displacement rate field are evaluated according to the classic formulation of the kinematics in the framework of homogenization theory [21], and are the following:

u_ x ¼ u_ x;per þ E_ xx xG þ E_ xy yG þ v_ xx zG xG þ 0:5v_ xy zG yG

ð1Þ

u_ y ¼ u_ y;per þ E_ xy xG þ E_ yy yG þ v_ yy zG yG þ 0:5v_ xy zG xG

ð2Þ

u_ z ¼ u_ z;per  0:5v_ xx xG 2  0:5v_ yy yG 2  0:5v_ xy xG yG

ð3Þ

Here, fxG ; yG ; zG g are the coordinates of the element’s centroid with respect to a reference system located at the center of the
 masonry panel, u_ x;per ; u_ y;per ; u_ z;per are the periodic velocities of n o the element, E_ xx ; E_ xy ; E_ yy the components of the average strain

rate tensor (with E_ yx equal to E_ xy for symmetry), and n o v_ xx ; v_ xy ; v_ yy the components of the average curvature rate tensor (again, with v_ yx equal to v_ xy for symmetry). The four sets of equality constraints are the following:  Equality constraints coming from velocity jumps: since the elements are rigid, plastic dissipation only occurs across the interfaces of two adjoining elements. The chosen problem formulation assumes three accessory hypotheses: no dissipation can occur across interfaces orthogonal to axis Z (the transversal direction), no shear dissipation can occur along axis Z of the other interfaces, and no conditions are enforced in terms of the elements’ velocities along direction Z. For a generic interface normal to axis X (Fig. 6a), the normal and tangential velocity jumps are:

    j i Du_ n  Du_ x ¼ u_ x;per  u_ x;per þ E_ xx xjG  xiG þ v_ xx zjG xjG  ziG xiG

ð4Þ

    j i Du_ t  Du_ y ¼ u_ y;per  u_ y;per þ E_ xy xjG  xiG þ 0:5v_ xy zjG xjG  ziG xiG ð5Þ The plastic dissipations that are allowed across adjoining interfaces are governed by a user-defined failure criterion that causes velocity jumps once its bounding surfaces are attained. The failure criterion must always be expressed in the form Aqn rn þ Aqt s  C qI ¼ 0. For instance, in case a simple Mohr-Coulomb failure criterion is used (Fig. 6b), the normal and tangential velocity jumps become:

F ð s; rn Þ ¼

Du_ n ¼

Du_ t ¼





s þ rn tan /  c ¼0 s þ rn tan /  c

2 X

2 X q @F q q 1 2 k_ I k_ I Aqn ¼ k_ I tan / þ k_ I tan / ¼ @ r n q¼1 q¼1

2 X

q @F q k_ I ¼ @s q¼1

2 X

q 1 2 k_ I Aqt ¼ k_ I  k_ I

ð6Þ

ð7Þ

ð8Þ

q¼1

Eventually, Eqs. (4) and (5) are combined with Eqs. (7) and (8):

Fig. 7. Graphical representation of a masonry unit (M) and a unit finite element (S).

j i u_ x;per  u_ x;per 

2 X

    q k_ I Aqn þ E_ xx xjG  xiG þ v_ xx zjG xjG  ziG xiG ¼ 0

q¼1

Fig. 8. Moments applied to the test-window for extracting (a) the first and (b) the second out-of-plane homogenized failure surfaces.

ð9Þ

S. Tiberti, G. Milani / Computers and Structures 229 (2020) 106186

7

Fig. 9. The six case studies investigated in the present paper: (a) Case 1; (b) Case 2; (c) Case 3; (d) Case 4; (e) Case 5; (f) Case 6.

j i u_ y;per  u_ y;per 

2 X

    q k_ I Aqt þ E_ xy xjG  xiG þ 0:5v_ xy zjG xjG  ziG xiG ¼ 0

q¼1

ð10Þ The final equality constraints coming from velocity jumps and plastic dissipation is expressed with the following equation in matrix form: eq _ eq _ _ Aeq 11 uper þ A13 kI;ass þ A14 D ¼ 0

ð11Þ

Table 1 Mechanical properties for the material employed in all case studies. Cohesion [MPa]

Friction angle [°]

Tensile strength [MPa]

Compressive strength [MPa]

0.15

30

0.1

1.5

A&B

Vector u_ per contains the components of the periodic velocity

field of all the finite elements, vector k_ I;ass contains the plastic multiplier rates of the active interfaces, and vector D_ contains the components of the average strain and curvature rate tensors. The first two variables are defined at a structural level, whereas the third is defined for the considered test-window.  Equality constraints coming from the master-slave relations for unit finite elements: the kinematics of finite elements pertaining to a masonry unit is governed by master-slave relations that link the kinematics of a single finite element (‘‘slave element”, superscript S) to that of the masonry unit to which it belongs (‘‘master element”, superscript M), as shown in Fig. 7. The master-slave relations also entail an enriched kinematics that enables each master element to rotate about the global horizontal or vertical axis, albeit in a ‘‘pixeled” way:

C&D

Fig. 10. The four considered test-windows for Case 1.

8

S. Tiberti, G. Milani / Computers and Structures 229 (2020) 106186

Fig. 11. Out-of-plane homogenized failure surfaces for Case 1.

M S M u_ x ¼ u_ x þ h_ yy zSG  zM G

ð12Þ

M S M u_ y ¼ u_ y  h_ xx zSG  zM G

ð13Þ

 M S M M _M S u_ z ¼ u_ z þ h_ xx ySG  yM G  hyy xG  xG

ð14Þ

M M The quantities h_ xx and h_ yy represent the rotations about the X and Y axes of the master masonry unit, respectively. Eventually, the matrix form of the equality constraints coming from the master–slave relations for unit finite elements is:

eq _ eq _ _ Aeq 21 uper þ A22 R þ A24 D ¼ 0

ð15Þ

S. Tiberti, G. Milani / Computers and Structures 229 (2020) 106186

Fig. 12. Failure modes under M xx , M yy , and M xy for the four test-windows of Case 1.

9

10

S. Tiberti, G. Milani / Computers and Structures 229 (2020) 106186

A

B

C

D Fig. 13. The four considered test-windows for Case 2.

where vector R_ contains the periodic velocity fields and macroscopic rotations of all the masonry units of the considered testwindow. This variable is also defined at a structural level.  Equality constraints coming from the homogenization approach: the periodic components of the displacement rate fields of elements lying on opposite sides of the considered masonry test-window must be equal (elements pairs a-b and c-d, Fig. 6c). The matrix form of the equality constraints coming from the homogenization approach is:

_ Aeq 31 uper ¼ 0

ð16Þ

 Equality constraint coming from the normalization of the dissipated external power: this is required to enforce a restriction identifying a single collapse mechanism out of the countless ones associated to the collapse load (which is instead unique). In particular, the selected collapse mechanism is the one satisfying the normalization condition, expressed by Eq. (17). Moreover, a direct consequence of this condition is that the objective function of the minimization problem coincides with the dissipated internal power. The matrix form of the equality constraint coming from the normalization of the dissipated external power is expressed by Eq. (18).

Pext ¼ Rxx E_ xx þ Ryy E_ yy þ Rxy E_ xy þ M xx v_ xx þ M yy v_ yy þ M xy v_ xy ¼ 1

ð17Þ

_ Aeq 44 D ¼ 1

ð18Þ

The expression for dissipated internal power over a single interface whose area is A can be written as:

Z Pint ¼ A

ðrn Du_ n þ sDu_ t ÞdA

ð19Þ

After substituting Eqs. (7) and (8), and after performing some simple assemblage operations, the matrix form for the dissipated internal power is:

Pint ¼ C TI;ass k_ I;ass

ð20Þ

The homogenized limit analysis problem is eventually formulated as a linear programming problem. The objective function to minimize is the dissipated internal power of Eq. (20), whereas the equality constraints are those resulting Eqs. (11), (15), (16), and (18). For a smoother solution, this computational problem is formulated in a standard form where each unknown variable is greater or equal to zero. Namely, the elements’ periodic velocity field u_ per , the masonry units’ periodic velocity field and macro_ and the components of average scopic rotations contained in R, _ are here all strain and curvature rate tensors contained in D expressed as the difference of two nonnegative quantities to satisfy the requirements of the standard form. For instance, R_ must be

split into two nonnegative parts, labelled R_ þ and R_  . The standard form for the linear programming problem here presented is:

11

S. Tiberti, G. Milani / Computers and Structures 229 (2020) 106186

Fig. 14. Out-of-plane homogenized failure surfaces for Case 2.

T

Minimize C X

ð21Þ

Subject to AX ¼ B

ð22Þ

X0

ð23Þ

where

2

Aeq 11

6 Aeq 6 A ¼ 6 21 4 Aeq 31 0

Aeq 11

0

0

eq eq Aeq 21 A22 A22 eq 0 A31 0

0

0

0

Aeq 13 0 0

Aeq 14 Aeq 24 0

0 Aeq 44

Aeq 14 Aeq 24 0 Aeq 44

3 7 7 7 5

ð24Þ

12

S. Tiberti, G. Milani / Computers and Structures 229 (2020) 106186

Fig. 15. Failure modes under M xx , Myy , and M xy for the four test-windows of Case 2.

S. Tiberti, G. Milani / Computers and Structures 229 (2020) 106186

A

B

C

D

13

Fig. 16. The four considered test-windows for Case 3.

3 þ u_ per 6  7 6 u_ per 7 7 6 6 _þ 7 6 R 7 7 6 7 X¼6 6 R_  7 7 6 _ 6 kI;ass 7 7 6 6 _þ 7 4 D 5 2

ð25Þ

D_ 

2

0

3

607 6 7 B¼6 7 405

ð26Þ

1 2

0 6 0 6 6 6 0 6 C¼6 6 0 6 C I;ass 6 6 4 0

investigates the response under the combination of moments M xx and M xy , where the former is the one just introduced above and the latter is the torsional moment (Fig. 8b). After a single pair of macroscopic moments ½ M xx M yy  or ½ M xx M xy  is selected, the linear programming problem is solved and the kinematic limit multiplier v (coinciding with the dissipated internal power) is calculated. Then, this is multiplied to the initial macroscopic moments, leading to the calculations of the collapse bending (or torsional) moments. 41 pairs of collapse moments vM xx and vM yy (or vM xy ) are investigated, and each represents a point in the M xx -M yy or M xx -M xy plane; the resulting flexural or torsional outof-plane homogenized failure surface consists of segments linking every point, and is then piecewise linear. The post-processing phase also allows the extraction of the failure mode for a specific out-of-plane load condition through the MATLAB command patch.

3 7 7 7 7 7 7 7 7 7 7 5

4. Case studies

ð27Þ

0 Two different out-of-plane homogenized failure surfaces are extracted for a single test-window: one is named ‘‘flexural outof-plane homogenized failure surface” and investigates the outof-plane response under the combination of bending moments M xx and M yy , where the former is the vertical bending moment and the latter is the horizontal bending moment (Fig. 8a); the other is named ‘‘torsional out-of-plane homogenized failure surface” and

Six case studies are investigated in terms of homogenized failure surfaces: all of them are real old buildings consisting of nonperiodic masonry and are the same as those addressed in [31]. The masonry panels considered for each case study are pictured in Fig. 9, and for every panel four test-windows are extracted; both the flexural and torsional out-of-plane homogenized failure surfaces are derived for each test-window, which are then critically compared, as well as the three failure modes that come from the single application of M xx , M yy , and M xy to the considered testwindow. Eventually, the average out-of-plane homogenized failure surfaces are calculated for each case study. The six case studies are the following:

14

S. Tiberti, G. Milani / Computers and Structures 229 (2020) 106186

Fig. 17. Out-of-plane homogenized failure surfaces for Case 3.

 Case study 1, a rubble masonry building in Casola in Lunigiana, Tuscany.  Case study 2, a quasi-periodic masonry ruin in Codiponte, Tuscany.

 Case study 3, a quasi-periodic masonry parish church in Filattiera, Tuscany.  Case study 4, quasi-periodic masonry tower ruins in Mulazzo, Tuscany.

S. Tiberti, G. Milani / Computers and Structures 229 (2020) 106186

15

Fig. 18. Failure modes under M xx , M yy , and M xy for the four test-windows of Case 3.

 Case study 5, a quasi-regular masonry parish church in San Secondo Parmense, Emilia-Romagna.  Case study 6, a quasi-regular masonry grand corridor in Sabbioneta, Lombardy. It must be noted that ‘‘quasi-periodic masonry” is characterized by a clearly visible presence of bed joints, whereas the head joints

are not periodic at all; moreover, the height of the brick/stones is not uniform in this type of masonry. Conversely, ‘‘quasi-regular masonry” is characterized by bricks/stones of uniform height, but they have different length and are not assembled with a periodic arrangement in the various layers; nonetheless, this kind of masonry is very similar to the stretcher bond masonry type, which is periodic.

16

S. Tiberti, G. Milani / Computers and Structures 229 (2020) 106186

Since no tests whatsoever have been performed on the considered masonry type, there are no experimental data available for their mechanical properties. Hence, mechanical properties available in literature are here used [36,37]. As noted in Section 3, the linear programming problem requires the introduction of a microscopic failure criterion for addressing the velocity jumps between adjacent interfaces. For all the six case studies, a Mohr-Coulomb failure criterion with cutoffs in tension and compression is selected with the same set of mechanical parameters (cohesion, friction angle, tensile strength, and compressive strength) that are listed in Table 1. 4.1. Case study 1: rubble masonry building in Casola in Lunigiana, Tuscany The first case study is represented by a three-story residential masonry building located in Casola in Lunigiana, in the Province of Massa and Carrara, Tuscany, Italy. This case is characterized by a rubble masonry type: river pebbles of different geometry are randomly arranged in the walls, which also display the presence of tapered blocks and stone chips. The four test-windows of this case study are depicted in Fig. 10 in their original location within the considered rubble masonry wall. Their dimensions are 130
130
40 cm3, and the transversal configuration of their masonry units is ellipsoidal, since it is consistent with that of river pebbles (which are present in this case), and employs 16 finite elements over the thickness. The flexural and torsional out-of-plane homogenized failure surfaces for the four test-windows are depicted in Fig. 11, along with their envelope and mean: all the moment values are normalized by 0:5  t 2  f t , where t is the thickness of the considered single-

leaf wall and f t is the tensile strength of the interfaces. It can be noted that test-windows A and D display smaller out-of-plane failure surfaces with respect to the other two: this is because they contain some clearly visible bed joints (actually, pseudo bed joints), despite consisting of rubble masonry. In fact, testwindows B and C show a normalized value of M yy that is greater than 1: again, this is due to the more marked randomness in the arrangement of their masonry units. Nonetheless, the shape of both out-of-plane failure surfaces for the four test-windows reflects the high rate of randomness of this case study. Fig. 12 shows the deformed shapes at collapse (‘‘failure modes”) of the four test-windows of this case study coming from the single application of M xx , M yy , and M xy . All these failure modes are consistent with what is expected to occur under the application of the aforementioned moments. In particular, it can be noted how the application of M xx causes widespread cracks in all four test-windows; moreover, the application of Myy to test-window C does not induce a single horizontal crack across one of the pseudo bed joints but again it generates widespread cracking patterns. This confirms the resulting flexural out-of-plane homogenized failure surface for this test-window, which is the largest of the four and shows the biggest normalized collapse value of M yy (which is greater than 1, also). 4.2. Case study 2: quasi-periodic masonry ruin in Codiponte, Tuscany The second case study is represented by the ruin of a derelict two-story masonry structure located in the small hamlet of Codiponte, which is part of the municipality of Casola in Lunigiana. This case is characterized by a quasi-periodic masonry type: although the wall consists of roughly-cut ashlars, a few tapered blocks and

A

B

C

D Fig. 19. The four considered test-windows for Case 4.

S. Tiberti, G. Milani / Computers and Structures 229 (2020) 106186

17

Fig. 20. Out-of-plane homogenized failure surfaces for Case 4.

some occasional stone chips, it is easy to identify the presence of proper bed joints – even though there are a few masonry units that spread over two layers. The four test-windows of this case study are depicted in Fig. 13 in their original location within the considered quasi-periodic masonry wall. Their dimensions are 160
160
40 cm3, and the transversal configuration of their

masonry units is ellipsoidal to ensure consistency with the inplane shape of the ashlars in this case, and employs 16 finite elements over the thickness. The flexural and torsional out-of-plane homogenized failure surfaces for the four test-windows are depicted in Fig. 14, along with their envelope and mean: all the moment values are normal-

18

S. Tiberti, G. Milani / Computers and Structures 229 (2020) 106186

Fig. 21. Failure modes under M xx , Myy , and M xy for the four test-windows of Case 4.

ized by 0:5  t 2  f t . Unlike the previous case, the shape of the flexural out-of-plane homogenized failure surfaces is very similar for all the four test-windows, with no differences whatsoever in the nor-

malized collapse value of M yy . Conversely, more marked differences among the four test-windows can be observed in the torsional out-of-plane homogenized failure surfaces. Fig. 15 shows

S. Tiberti, G. Milani / Computers and Structures 229 (2020) 106186

19

chips are also present. Again, the bed joints are clearly visible in the wall. The four test-windows of this case study are depicted in Fig. 19 in their original location within the considered quasiperiodic masonry wall. Their dimensions are 160
160
40 cm3, and the transversal configuration of their masonry units is again ellipsoidal, and employs 16 finite elements over the thickness. The flexural and torsional out-of-plane homogenized failure surfaces for the four test-windows are depicted in Fig. 20, along with their envelope and mean: all the moment values are normalized by 0:5  t2  f t . In this case, both the flexural and torsional outof-plane homogenized failure surfaces almost coincide for three of the four test-windows: only test-window A displays larger homogenized failure surfaces, probably due to the presence of stone chips within some of its bed joints. Fig. 21 shows the failure modes of the four test-windows of this case study: once again, all are consistent with what is expected to occur under the application of M xx , M yy , and Mxy . 4.5. Case study 5: quasi-regular masonry parish church in San Secondo Parmense, Emilia Romagna

A, B, C & D Fig. 22. The four considered test-windows for Case 5.

the failure modes of the four test-windows of this case study: all are consistent with what is expected to occur under the application of M xx , M yy , and M xy . 4.3. Case study 3: quasi-periodic masonry parish church in Filattiera, Tuscany The third case study is represented by a Romanesque masonry parish church located in the small hamlet of Sorano, which is part of the municipality of Filattiera, in the Province of Massa and Carrara, Tuscany, Italy. This case is also characterized by a quasiperiodic masonry type: the wall consists of river pebbles and displays thick mortar joints, and the bed joints are even more visible with respect to the previous case. Moreover, there are no masonry units that spread over two layers. The four test-windows of this case study are depicted in Fig. 16 in their original location within the considered quasi-periodic masonry wall. Their dimensions are 160
160
40 cm3, and the transversal configuration of their masonry units is again ellipsoidal, and employs 16 finite elements over the thickness. The flexural and torsional out-of-plane homogenized failure surfaces for the four test-windows are depicted in Fig. 17, along with their envelope and mean: all the moment values are normalized by 0:5  t2  f t . As in the previous case, the shape of the flexural out-of-plane homogenized failure surfaces is very similar for all the four test-windows, whereas the torsional out-of-plane homogenized failure surfaces display noticeable differences. Fig. 18 shows the failure modes of the four test-windows of this case study: once again, all are consistent with what is expected to occur under the application of M xx , M yy , and M xy . 4.4. Case study 4: quasi-periodic masonry tower ruins in Mulazzo, Tuscany The fourth case study is represented by the ruins of a hexagonal masonry tower located in Mulazzo, in the Province of Massa and Carrara, Tuscany, Italy. This case is also characterized by a quasi-periodic masonry type: the wall consists of roughly-cut ashlars whose height is rather uniform, and some occasional stone

The fifth case study is represented by a Romanesque masonry parish church located in San Secondo Parmense, in the Province of Parma, Emilia Romagna, Italy. This case is characterized by a quasi-regular masonry type: the wall consists of brick-like units of different lengths. Moreover, the wall presents a single continuous head joint that spreads over the whole height. The four testwindows of this case study are depicted in Fig. 22 in their original location within the considered quasi-regular masonry wall. Their dimensions are 100
100
15 cm3; in this case the transversal configuration of the masonry units is simply the extrusion of their brick-like in-plane configuration, and employs 10 finite elements over the thickness. The flexural and torsional out-of-plane homogenized failure surfaces for the four test-windows are depicted in Fig. 23, along with their envelope and mean: all the moment values are normalized by 0:5  t 2  f t . This case is particularly interesting due to the peculiar behavior of test-window D: it is the only test-window containing part of the continuous head joint, which greatly affects its out-of-plane behavior. In fact, both its out-of-plane homogenized failure surfaces are considerably different than those of the other three test-windows and display a sensibly small normalized collapse value of Mxx , which is due to the presence of the continuous head joint. Fig. 24 shows the failure modes of the four testwindows of this case study: they are all consistent with what is expected to occur under the application of M xx , M yy , and M xy . In particular, it can be observed how the presence of the continuous head joint affects the failure modes of test-window D for Mxx and Mxy : in both cases the test-window displays a continuous vertical crack in correspondence of the aforementioned head joint, which is extremely marked for M xx where the test-window actually splits. 4.6. Case study 6: quasi-regular masonry grand corridor in Sabbioneta, Lombardy The sixth and final case study is represented by a masonry grand corridor located in Sabbioneta, in the Province of Mantua, Lombardy, Italy; it is part of a larger building that used to be the residence of the Duke of Sabbioneta. This case is also characterized by a quasi-regular masonry type: the masonry units are bricks that share the same height while displaying different lengths. Some brick heads are also present within the wall, and the overall arrangement of the units is almost regular. The four testwindows of this case study are depicted in Fig. 25 in their original

20

S. Tiberti, G. Milani / Computers and Structures 229 (2020) 106186

Fig. 23. Out-of-plane homogenized failure surfaces for Case 5.

location within the considered quasi-regular masonry wall. Their dimensions are 130
130
15 cm3, and the transversal configuration of their masonry units is again the mere extrusion of their in-plane configuration, and employs 10 finite elements over the thickness.

The flexural and torsional out-of-plane homogenized failure surfaces for the four test-windows are depicted in Fig. 26, along with their envelope and mean: all the moment values are normalized by 0:5  t2  f t . Both the flexural and torsional out-of-plane homogenized failure surfaces for the four test-windows present

S. Tiberti, G. Milani / Computers and Structures 229 (2020) 106186

21

Fig. 24. Failure modes under M xx , M yy , and M xy for the four test-windows of Case 5.

an elongated shape that is usually associated to a stretcher bond masonry, which means that the out-of-plane behavior of this case is the closest to that of periodic masonry among the six cases. The out-of-plane homogenized failure surfaces of test-window B are

actually slightly smaller than those of the other three testwindows, and this is due to the presence of a mortar spot in its top-left corner. Fig. 27 shows the failure modes of the four testwindows of this case study: once again, all are consistent with

22

S. Tiberti, G. Milani / Computers and Structures 229 (2020) 106186

A

B

C

D Fig. 25. The four considered test-windows for Case 6.

what is expected to occur under the application of M xx , M yy , and M xy . 4.7. Comparison of the out-of-plane homogenized failure surfaces for the six case studies Fig. 28 contains the comparison of the out-of-plane mean homogenized failure surfaces for the test-windows of the considered six case studies; all the moment values are normalized by 0:5  t2  f t . In both the flexural and torsional out-of-plane homogenized failure surfaces, it is possible to observe how the quasiregular sixth case study offers the highest normalized collapse value for Mxx and M xy . As previously observed, the shape of both failure surfaces closely resembles that usually obtained for a stretcher bond masonry (see, for instance, [34]). Although the fifth case study is itself quasi-regular, its out-of-plane mean homogenized failure surfaces are severely affected by the presence of the continuous head joint, so that they are actually reduced with respect to the expectations. This leads to an interesting observation: when considering historical masonry, the test-windows must be randomly extracted from the wall under investigation in order to encompass all the possible geometrical features. As clearly shown by the fifth case study, even though the units’ arrangement may be almost regular, the overall out-of-plane behavior is definitely affected by the presence of a feature such as the continuous head joint, and this must be always taken into account. As far as the other four cases are concerned, the quasi-periodic fourth case study offers the highest resistance to M xx and M xy with respect to the other three. Conversely, the rubble first case study offers the lowest resistance to M xx and M xy , but this is partially compensated by its high resistance to M yy , which is usually the out-of-plane load condition mostly relevant when assessing the vulnerability of masonry walls to out-of-plane actions, for instance induced by a

seismic event. It can then be remarked how a rubble masonry wall offers a greater resistance to such actions with respect to more periodic bonds, and this helps explaining the fact that historical rubble masonry buildings seem to be spared from collapse more frequently than other masonry structures. 5. Conclusions and future developments In this paper, an extensive study on the out-of-plane collapse behavior of non-periodic masonry has been presented. Such study is based on a MATLAB script enabling the derivation of out-ofplane homogenized failure surfaces: these are the ultimate results of a linear programming problem that blends the upper bound theorem of limit analysis and the technique known as homogenization. While the application of the latter approach is usually limited to periodic masonry, the concept of Representative Element of Volume (REV) – on which homogenization is based – is here evolved into that of test-window (itself inspired by that of Statistically Equivalent Periodic Unit Cell, or SEPUC), thus enabling the application of homogenization to non-periodic masonry as well. In the linear programming problem, the materials employ a frictional failure criterion provided with limited tensile and compressive strength as well as an associated flow rule. The main innovations introduced in this paper are:  The direct creation of a 3D finite element mesh from the sketch of the considered masonry test-window through a procedure implemented in a MATLAB script. This is based on the socalled ‘‘voxel strategy”, where each voxel is transformed into a finite element; the former is the 3D equivalent of the classic 2D pixel. The voxel strategy here presented enables the creation of a 3D FE mesh that is respectful of the actual transversal layout of the considered masonry test-window, giving the user the

S. Tiberti, G. Milani / Computers and Structures 229 (2020) 106186

23

Fig. 26. Out-of-plane homogenized failure surfaces for Case 6.

opportunity to choose between two typical 3D layouts (masonry units obtained either through the extrusion of the in-plane layout, or as ellipsoidal stone-like blocks). A coarsening strategy is also devised, aiming at reducing the total number of finite elements in the mesh for ensuring the swiftness of the subsequent numerical simulations. Moreover, the creation of solid brick-like finite elements whose axes are already oriented

along the local reference system of the considered masonry test-window makes the resulting 3D finite element mesh suitable for use in the broader MATLAB script containing the linear programming problem.  For the first time, a complete investigation on the out-of-plane collapse behavior of non-periodic masonry is carried out. In particular, six real case studies – each displaying a different type of

24

S. Tiberti, G. Milani / Computers and Structures 229 (2020) 106186

Fig. 27. Failure modes under M xx , Myy , and M xy for the four test-windows of Case 6.

non-periodicity – are here examined: their out-of-plane collapse behavior is assessed through the derivation of out-ofplane homogenized failure surfaces. For each case study, four different test-windows are considered: for a single test-

window, two out-of-plane homogenized failure surfaces are extracted (one investigating the flexural behavior, the other focused on the torsional one) as well as three relevant deformed shapes at collapse. The results show the influence of the actual

S. Tiberti, G. Milani / Computers and Structures 229 (2020) 106186

25

Fig. 28. Comparison of the flexural and torsional out-of-plane mean homogenized failure surfaces among the six case studies.

bond on the out-of-plane collapse behavior of each different masonry type, including the presence of possible structural features such as continuous head joints that reduce the out-ofplane strength of the masonry wall. It must be noted that the out-of-plane mean homogenized failure surfaces obtained for each case study help in giving an insight on the out-of-plane global mechanical behavior of the greater non-periodic walls.

The failure surfaces may also be employed as macroscopic failure criteria in the seismic assessment of masonry structural elements, such as church façades. An immediate development of this work will focus on the extraction of out-of-plane homogenized failure surfaces for multi-leaf non-periodic masonry walls. These have been rarely

26

S. Tiberti, G. Milani / Computers and Structures 229 (2020) 106186

dealt with in literature, but actually are of paramount interest since they are a common structural feature in many historical masonry buildings, especially in Italy. In fact, such kind of walls often shows a poor resistance against seismic-induced out-ofplane actions, and unfortunately their presence in masonry buildings is only revealed after the latter’s partial or total collapse due to an earthquake. For these reasons, a study of their out-of-plane collapse behavior is certainly necessary and interesting for the scientific community. Finally, the 3D mesh resulting from the voxel strategy show a great potential for other future developments in assessing the behavior of masonry, as corroborated by the results in terms of out-of-plane collapse behavior for non-periodic masonry. For instance, one of these possible developments is represented by employing this method to obtain a good discretization of masonry with the use of Discrete Element Method (DEM) and the Non-Smooth Contact Dynamics Method (NSCD), which has proved to be a reliable tool to model the seismic behavior of masonry (as shown in [6,38–41]).

[17]

[18]

[19]

[20]

[21]

[22]

[23]

[24]

Declaration of Competing Interest The authors declared that there is no conflict of interest. References [1] Ghiassi B, Soltani M, Tasnimi AA. A simplified model for analysis of unreinforced masonry shear walls under combined axial, shear and flexural loading. Eng Struct 2012;42:396–409. https://doi.org/10.1016/j. engstruct.2012.05.002. [2] Sarhosis V, Milani G, Formisano A, Fabbrocino F. Evaluation of different approaches for the estimation of the seismic vulnerability of masonry towers. Bull Earthquake Eng 2018;16(3):1511–45. https://doi.org/10.1007/s10518017-0258-8. [3] Saloustros S, Cervera M, Pelà L. Tracking multi-directional intersecting cracks in numerical modelling of masonry shear walls under cyclic loading. Meccanica 2018;53:1757–76. https://doi.org/10.1007/s11012-017-0712-3. [4] Chieffo N, Clementi F, Formisano A, Lenci S. Comparative fragility methods for seismic assessment of masonry buildings located in Muccia (Italy). J Build Eng 2019;25:100813. https://doi.org/10.1016/j.jobe.2019.100813. [5] Miccoli L, Garofano A, Fontana P, Müller U. Experimental testing and finite element modelling of earth block masonry. Eng Struct 2015;104:80–94. https://doi.org/10.1016/j.engstruct.2015.09.020. [6] Clementi F, Ferrante A, Giordano E, Dubois F, Lenci S. Damage assessment of ancient masonry churches stroked by the Central Italy earthquakes of 2016 by the non-smooth contact dynamics method. Bull Earthquake Eng 2019:1–32. https://doi.org/10.1007/s10518-019-00613-4. [7] Sarhosis V, Lemos JV. A detailed micro-modelling approach for the structural analysis of masonry assemblages. Comput Struct 2018;206:66–81. https://doi. org/10.1016/j.compstruc.2018.06.003. [8] Baraldi D, Reccia E, Cecchi A. In plane loaded masonry walls: DEM and FEM/ DEM models. A critical review. Meccanica 2018;53(7):1613–28. https://doi. org/10.1007/s11012-017-0704-3. [9] Milani G. 3D FE limit analysis model for multi-layer masonry structures reinforced with FRP strips. Int J Mech Sci 2010;52(6):784–803. https://doi.org/ 10.1016/j.ijmecsci.2010.01.004. [10] Nodargi NA, Intrigila C, Bisegna P. A variational-based fixed-point algorithm for the limit analysis of dry-masonry block structures with non-associative Coulomb friction. Int J Mech Sci 2019:161–2. https://doi.org/10.1016/j. ijmecsci.2019.105078. art. 105078. [11] Trentadue F, Quaranta G. Limit analysis of frictional block assemblies by means of fictitious associative-type contact interface laws. Int J Mech Sci 2013;70:140–5. https://doi.org/10.1016/j.ijmecsci.2013.02.012. [12] Petracca M, Pelà L, Rossi R, Zaghi S, Camata G, Spacone E. Micro-scale continuous and discrete numerical models for nonlinear analysis of masonry shear walls. Constr Build Mater 2017;149:296–314. https://doi.org/10.1016/ j.conbuildmat.2017.05.130. [13] Macorini L, Izzuddin BA. A non-linear interface element for 3D mesoscale analysis of brick-masonry structures. Int J Numer Meth Eng 2011;85 (12):1584–608. https://doi.org/10.1002/nme.3046. [14] Drougkas A, Roca P, Molins C. Analytical micro-modeling of masonry periodic unit cells – elastic properties. Int J Solids Struct 2015;69–70:169–88. https:// doi.org/10.1016/j.ijsolstr.2015.04.039. [15] Milani G, Bertolesi E. Quasi-analytical homogenization approach for the nonlinear analysis of in-plane loaded masonry panels. Constr Build Mater 2017;146:723–43. https://doi.org/10.1016/j.conbuildmat.2017.04.008. [16] Reccia E, Leonetti L, Trovalusci P, Cecchi A. A multiscale/multidomain model for the failure analysis of masonry walls: a validation with a combined FEM/

[25]

[26]

[27]

[28]

[29]

[30]

[31]

[32]

[33]

[34] [35] [36]

[37]

[38]

[39]

[40]

[41]

DEM approach. Int J Multiscale Com 2018;16(4):325–43. https://doi.org/ 10.1615/IntJMultCompEng. 2018026988. Milani G, Bruggi M. Simple homogenization-topology optimization approach for the pushover analysis of masonry walls. Int J Archit Herit 2018;12 (3):395–408. https://doi.org/10.1080/15583058.2017.1323248. Zucchini A, Lourenço PB. Mechanics of masonry in compression: Results from a homogenisation approach. Comput Struct 2007;3–4:193–204. https://doi.org/ 10.1016/j.compstruc.2006.08.054. Rekik A, Lebon F. Homogenization methods for interface modelling in damaged masonry. Adv Eng Softw 2012;46(1):35–42. https://doi.org/ 10.1016/j.advengsoft.2010.09.009. Berke PZ, Peerlings RHJ, Massart TJ, Geers MGD. A homogenization-based quasi-discrete method for the fracture of heterogeneous materials. Comput Mech 2014;53(5):909–23. https://doi.org/10.1007/s00466-013-0939-3. Milani G, Lourenço PB, Tralli A. Homogenised limit analysis of masonry walls. Part I: Failure surfaces. Comput Struct 2006;84(3–4):166–80. https://doi.org/ 10.1016/j.compstruc.2005.09.005. Milani G, Taliercio A. In-plane failure surfaces for masonry with joints of finite thickness estimated by a method of cells-type approach. Comput Struct 2015;150:34–51. https://doi.org/10.1016/j.compstruc.2014.12.007. Stefanou I, Sab K, Heck J-V. Three dimensional homogenization of masonry structures with building blocks of finite strength: a closed form strength domain. Int J Solids Struct 2015;54:258–70. https://doi.org/10.1016/j. ijsolstr.2014.10.007. Godio M, Stefanou I, Sab K, Sulem J, Sakji S. A limit analysis approach based on Cosserat continuum for the evaluation of the in-plane strength of discrete media: application to masonry. Eur J Mech A-Solid 2017;66:168–92. https:// doi.org/10.1016/j.euromechsol.2017.06.011. Milani G, Lourenço PB. A discontinuous quasi-upper bound limit analysis approach with sequential linear programming mesh adaptation. Int J Mech Sci 2009;51(1):89–104. https://doi.org/10.1016/j.ijmecsci.2008.10.010. Cecchi A, Milani G, Tralli A. A Reissner-Mindlin limit analysis model for out-ofplane loaded running bond masonry walls. Int J Solids Struct 2007;44 (5):1438–60. https://doi.org/10.1016/j.ijsolstr.2006.06.033. Milani G, Taliercio A. Limit analysis of transversally loaded masonry walls using an innovative macroscopic strength criterion. Int J Solids Struct 2016;81:274–93. https://doi.org/10.1016/j.ijsolstr.2015.12.004. Šejnoha J, Šejnoha M, Zeman J, Sy´kora J, Vorel J. Mesoscopic study on historic masonry. Struct Eng Mech 2008;30(1):99–117. https://doi.org/10.12989/ sem.2008.30.1.099. Šejnoha M, Zeman J, Valenta R. Macroscopic constitutive law for mastic asphalt mixtures from multiscale modeling. Int J Multiscale Com 2010;8 (1):131–49. https://doi.org/10.1615/IntJMultCompEng.v8.i1.100. Vorel J, Zeman J, Šejnoha M. Homogenization of plain weave composites with imperfect microstructure. Part II. Analysis of real-world materials. Int J Multiscale Com 2013;11(5):443–62. https://doi.org/10.1615/ IntJMultCompEng. 2013004866. Tiberti S, Milani G. 2D pixel homogenized limit analysis of non-periodic masonry walls. Comput Struct 2019;219:16–57. https://doi.org/10.1016/ j.compstruc.2019.04.002. Milani G, Lourenço PB. Monte Carlo homogenized limit analysis model for randomly assembled blocks in-plane loaded. Comput Mech 2010;46 (6):827–49. https://doi.org/10.1007/s00466-010-0514-0. Milani G, Lourenço PB. A simplified homogenized limit analysis model for randomly assembled blocks out-of-plane loaded. Comput Struct 2010;88(11– 12):690–717. https://doi.org/10.1016/j.compstruc.2010.02.009. Tiberti S, Milani G. Fast brick-based homogenized limit analysis for in- and out-of-plane loaded periodic masonry panels. Comput Struct (under review) MATLAB Release 2018b. Natick (Massachusetts, United States): The MathWorks, Inc. van der Pluijm R, Rutten HS, Ceelen M. Shear behaviour of bed joints. In: 12th IB2MaC: Proceedings of the 12th international brick/block masonry conference; 2000 Jun 25–28; Madrid, Spain. p. 1849–62. van der Pluijm. Material properties of masonry and its components under tension and shear. In: Proceedings of the 6th Canadian masonry symposium; 1992 Jun 15–17; Saskatoon, Canada. p. 675–86. Ferrante A, Clementi F, Milani G. Dynamic behavior of an inclined existing masonry tower in Italy. Front Built Environ 2019;5. https://doi.org/10.3389/ fbuil.2019.00033. art. 33. Ferrante A, Ribilotta E, Giordano E, Clementi F, Lenci S. Advanced seismic analyses of ‘‘Apennine churches” stroked by the Central Italy earthquakes of 2016 by the non-smooth contact dynamics method. Key engineering materials, vol. 817. Trans Tech Publications Ltd.; 2016. p. 309–16. https:// doi.org/10.4028/www.scientific.net/KEM.817.309. Gazzani V, Poiani M, Clementi F, Milani G, Lenci S. Modal parameters identification with environmental tests and advanced numerical analyses for masonry bell towers: a meaningful case study. Procedia Struct Integr 2018;11:306–13. https://doi.org/10.1016/j.prostr.2018.11.040. Poiani M, Gazzani V, Clementi F, Milani G, Valente M, Lenci S. Iconic crumbling of the clock tower in Amatrice after 2016 central Italy seismic sequence: advanced numerical insight. Procedia Struct Integr 2018;11:314–21. https:// doi.org/10.1016/j.prostr.2018.11.041.