Structural analysis of masonry vaulted staircases through rigid block limit analysis

Structures 23 (2020) 180–190

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Structural analysis of masonry vaulted staircases through rigid block limit analysis

T

M. Rossia, , C. Calderinia, B. Di Napolia, L. Cascinib, F. Portiolib ⁎

a b

Department of Civil, Chemical and Environmental Engineering, University of Genoa, Via Montallegro 1, 16145 Genoa, Italy Department of Structures for Engineering and Architecture, University of Naples Federico II, Via Forno Vecchio 36, 80134 Naples, Italy

ARTICLE INFO

ABSTRACT

Keywords: Masonry vaults Staircases Limit analysis Rigid block model Crushing failure Finite element model

This paper investigates the structural behaviour of masonry vaulted staircases and the efficacy of different modelling strategies for the assessment of their structural safety. In particular, the present study focuses on the structural analysis of the masonry vaulted stairs of the medieval Santa Maria delle Vigne bell tower in Genoa (Italy), dating back from XVI to XVII Century. Both a limit analysis tool and a static incremental analysis using a finite element model are adopted. The limit analysis problem is formulated for a rigid block model using an iterative solution procedure to take into account crushing failure. Results indicate that crushing failure is crucial in capturing the response of this structural typology.

1. Introduction

the flying buttresses or the brick lintels (Fig. 1). For these structures, interpenetration of material occurs for any hinge position, which is inconsistent with the hypothesis of infinite compressive strength. Their internal state of stress is always compressive and collapse may happen only in the case of moving supports, or the occurrence of compressive failure. The effects of the infill material could also exacerbate the role of crushing failure on the global behaviour of arched structures [22–24]. The infill, on the one hand, significantly increases the compressive stress state condition; on the other hand, it provides a physical constraint inhibiting hinge formation. The hypothesis of infinite compressive strength, indeed, appears to be unsuitable as compression stresses can be high enough to cause crushing in masonry. Crushing failure also affects the response of reinforced arches, as proved by Foraboschi [25] and Caporale et al. [26], which investigated the behaviour of single and multi-span arches with FRP reinforcement. Some researchers investigated the behaviour of masonry staircases, but referring to masonry helical and cantilever stairs. In these cases, the use of conventional limit analysis approaches is inadequate. Refined methods based on non-linear continuum modelling approaches [27–30], and discrete element modelling approaches [31,32] usually adopted for modelling masonry walls can be nowadays used for analysing curved structures taking into account also the compressive resistance. However, they all require high computational efforts. Recently, a rigid block model was developed which is based on limit

The structural safety of masonry vaults usually relies on their geometry and their collapse is determined by loss of equilibrium. For this reason, limit analysis (or equilibrium) methods are among the most used to investigate vault’s structural response [1]. These approaches are based on the classical hypotheses of Heyman [2], stating that the material has infinite strength in compression and zero strength in tension and that historic masonry structures can be analysed as assemblies of blocks with enough friction to avoid sliding. Whether in the form of graphic tools [3–6], or in the form of analytical formulations [7–9] or implemented in numerical codes [10–15], equilibrium methods allow for determining both the collapse mechanisms and ultimate collapse load. Although, in general, the reliability of these methods is largely proved, it should be recognized that in some cases they are unable to predict the collapse of masonry vaults since not always masonry vaults collapse for loss of equilibrium. This is the case of masonry staircases built on flying vaults, named a collo d’oca or zoppe in Italian [16,17]. Although several researchers focused their attention on investigating the structural behaviour of other types of masonry staircases, such as helical and cantilever stairs [18–20], no studies have been found about staircases on flying vaults. This type of masonry vaulted staircases can be classified within the class of “no-mechanism” structures as they are defined by Como [21], when an infinite compressive strength is assumed. Other examples are

⁎

Corresponding author. E-mail address: [email protected] (M. Rossi).

https://doi.org/10.1016/j.istruc.2019.10.015 Received 9 August 2019; Received in revised form 9 October 2019; Accepted 22 October 2019 2352-0124/ © 2019 Institution of Structural Engineers. Published by Elsevier Ltd. All rights reserved.

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Fig. 1. Examples of flat arch systems (Como, 2013): a) a flying buttresses, and b) a masonry lintel.

analysis and takes into account the limited compressive strength of masonry as well as the sliding failure condition. The model was conveniently applied to the case of masonry wall panels with regular bond pattern [33]. In this paper, the rigid block model is extended to the case of arched structures and is applied to the analyses of a set of vaulted staircases. The results obtained by using this new approach are compared with those obtained with traditional limit analysis approaches and with those obtained by a non-linear continuum constitutive model. The manuscript is organized as follows. In Section 2, a description is reported of the rigid block model and of the crushing failure conditions included in the limit analysis formulation. The results of a validation study against analytical and non-linear finite element modelling are illustrated in Section 3. In Section 4, an application of the developed model is presented to the safety assessment of masonry vaulted stairs of the medieval Santa Maria delle Vigne bell tower in Genoa.

include the shear force components t1k and t2k and the normal force nk at each contact point, which act along the tangent and normal direction to contact interface (Fig. 2) and are collected in vector ck. The normal force is assumed to be positive in the case of compression. It should be noted that this simple modelling approach for contact interactions, with concentrated forces at contact points representing the distributed stresses along the contact surface, is consistent with the failure conditions assumed for sliding and crushing failure. Those are expressed in terms of equivalent stress resultants and are formulated using a proper distribution for internal stresses in the case of crushing, as detailed in the following. External loads fi are applied at the centroid of rigid block i and are expressed as the sum of dead and live loads fDi + fLi . The collapse load multiplier of live loads represents the additional unknown of the implemented optimization problem. Considering that in the present formulation external loads cannot be applied to the faces at the extrados of the arches, equivalent external forces applied to the blocks centroids, also including corresponding bending moments, should be considered when the load given by the infill is taken into account. Another way to consider the infill loads or external forces applied to the extrados using the present formulation is to use additional rigid blocks, as reported in [34] for the case of a masonry arch subjected to eccentric load. Although the application of the load given by infill to the extrados of the arch could affect its load-bearing capacity, in this study live loads were considered as applied to the block centroid. The constraints of the optimization problem are the equilibrium and failure conditions at contact points. Equilibrium conditions for block i and contact point k can be expressed in matrix form as follows:

2. The modelling of crushing failure in rigid block limit analysis using mathematical programming The rigid block model for limit equilibrium analysis of masonry vaulted staircases was implemented in a software tool named LiABlock_3D [34]. The software is a MATLAB based tool with a simple interactive graphic user interface for handling the input data, the analysis type (live loads or moving supports) and the visualization of results (mechanism of collapse and collapse multiplier). LiABlock_3D is able to analyse 3D structures made of polyhedral blocks in contact by means of quadrilateral interfaces. The limit analysis problem is posed as a force-based (i.e. static) optimization problem, assuming as primary static variables the contact forces acting at the contact points k, k + 1, … k + 3, which are assumed to be coincident with the vertices of contact interface j (Fig. 2). Those

Ai, k c k = fDi + fLi ,

(1)

where Ai, k is the equilibrium matrix associated to the six degrees of freedom of block i and to the three force components at point k, with

Fig. 2. Rigid block model and block i (a); internal forces at contact point k on interface j and external loads at the block centroid expressed as a function of dead, live loads and collapse multiplier (b). 181

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Fig. 3. Sliding failure conditions at contact point k (a); Rocking, non-linear crushing and linearized failure conditions at interface j (b).

entries determined on the basis of positions of block centroid and contact point k [33]. Failure conditions include sliding failure and crushing failure. Sliding failure conditions at a contact point k are expressed by the following Coulomb friction cone (Fig. 3a):

Ck = {ck

3:

µnk

t12k + t22k , nk

0},

conditions also reveals that the results are not affected by the initial distribution of normal forces. 3. Validation of the rigid block model Two simple case studies are illustrated in the following to show the accuracy of the implemented numerical procedure for crushing failure. The case studies comprise a two-blocks flat arch and a vaulted staircase. Comparisons against closed-form solutions and results from finite element modelling were carried out to validate the rigid block model.

(2)

where µ is the friction coefficient. For crushing failure at contact interface j, a quadratic failure domain for bending moment and normal force resultants (Fig. 3b) was considered. The domain is associated to the assumption of uniform normal stress distribution under crushing failure for eccentric normal stress resultant. The domain was linearized with tangent hyperplanes expressed by the following relationship:

YTj cj

rj

3.1. Two-block flat arch The two-block flat arch, with dimensions l = 2000 mm, b = 500 mm and h = 460 mm, is loaded by two vertical external forces f, which are applied at 500 mm and 1500 mm along the x-axis (Fig. 4a). The arch is supported at the ends by two frictional surfaces, which are inclined with an angle α of 70° from the ground. The compressive strength fm is equal to 1.0 MPa and the friction coefficient is taken as 0.6. The failure mechanism obtained from the numerical model and the corresponding internal forces at contact points are shown in Fig. 4b,c. The arch fails with a three hinges failure mechanism, with hinges approximately located at the half depth of the arch, at a distance d of 219 mm from the top (Fig. 4d). It is worth noting that the failure mechanism involves interpenetration at interfaces when crushing occurs. This aspect of the formulation, which is related to the flow rule of displacement rates at contact interfaces, has been also discussed in [33]. The value of the external forces f associated to the activation of the collapse mechanism is equal to 54.4 kN. This value is in a good agreement with the failure load of 56.4 kN, which can be derived from the application of the virtual work principle to the failure mechanism depicted in Fig. 4d [21], where a uniform distribution of compressive strength corresponding to the calculated contact forces is considered.

(3)

where YTj is the matrix collecting coefficients proportional to the direction cosines of the outward normals to hyperplanes, cj is the vector of contact forces associated to interface j (i.e to nodes k, k + 1, … k + 3) and r j is the vector collecting the distances of hyperplanes from the origin. For the whole masonry block assemblage, the optimization problem corresponding to the static formulation of the limit analysis theorem can be posed as follows:

max s.t. Ac = fD + fL Y Tc r c C

(4)

The variables associated to displacement rates at contact points are recovered from the dual optimization problem to obtain a plot of the failure mode corresponding to the collapse load multiplier. A simple iterative solution procedure was implemented for the linearization of the quadratic crushing failure conditions, which can be summarized as follows. To start the iterative solution procedure, a tentative distribution of normal forces at contact points k is used for the linearization of the quadratic crushing failure conditions, rather than using the failure conditions corresponding to rocking failure (i.e. to infinite compressive strength, Fig. 3b), as it was presented in [33]. The load factor iter and a new distribution of normal forces are obtained from the solution of the corresponding optimization problem. Failure conditions are updated on the basis of the new distribution of normal forces and a new optimization problem is solved. The iterative solution procedure ends when the difference between the values of the collapse load multipliers is lower than the prescribed tolerance. Considering that the proposed modelling approach relies on plastic limit analysis, it should be noted that the obtained results are independent from loading history. Sensitivity analysis to starting

3.2. Vaulted staircase The importance of considering the compressive strength in the case of flat arches is also investigated by analysing a typical masonry vaulted staircase, as that shown in Fig. 5. In this case, the structural behaviour was studied according to two load cases (LC_1 and LC_2): in the first, a concentrated load is applied on the third step (F1, in Fig. 5); in the second, a concentrated load is applied on the sixth step (F2, in Fig. 5). In these cases, both the limit equilibrium method and an incremental static finite element analysis were adopted, in order to compare the results. The limit equilibrium analysis was performed with LiABlock by considering both the infinite and the limited compressive strength of masonry. The vault was modelled as an assembly of discrete blocks, 182

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Fig. 4. Geometry of the two-block flat arch (a); collapse mechanism obtained from LiABlock_3D (b); failure load and internal forces at contact points computed from the numerical solution expressed in kN (c); failure mechanism and compressive stresses distribution considered for the application of the virtual work principle (d).

minimum value of the range proposed by the Italian Technical Code [35] for clay brick masonry with poor mortar joints (Table C8.5.I, [36]) decreased by a confidence factor of 1.35. However, the analyses were conducted by considering a further reduced compressive strength in order to take account the actual softening behaviour of the material. This effective compressive strength fm,ef was calculated using the following expression, firstly used for the limit analysis of reinforced concrete by Nielsen [37], and then adopted for masonry by Orduña and Lourenço [38]:

fm, ef =

e fm ,

(5)

with e

= 0.7

fm 200

(6)

where νe is the effectiveness reducing factor. Thus, the effective compressive stress fm,ef adopted for the analysis was 1.23 MPa. The FEM analysis was carried out by the commercial software ANSYS, in which the non-linear constitutive laws proposed by Calderini and Lagomarsino [28] for the evaluation of the in-plane behaviour of brick masonry were implemented. The micromechanical model allows for describing damage to mortar joints because of decohesion and sliding, and failure of bricks caused by both tensile and shear stresses. The brick masonry was modelled using non-linear orthotropic shell elements (Fig. 6b). The masonry pattern was simulated by rotating each element’s reference system according to the actual orientation of mortar joints. The model was fixed at the vault’s springing. The mechanical parameters adopted for the material’s modelling are illustrated in Table 1. The analyses were performed in two steps: in the first one, the gravity loads were applied, while in the second one, a vertical displacement (d1 in LC_1 and d2 in LC_2) was applied to the nodes of the model corresponding to the loaded bricks of Fig. 6b. The sum of the

Fig. 5. Geometry of the brick vaulted staircase used for the benchmark.

whose height is equal to the vault’s thickness and width represents the geometry of two bricks plus the mortar joint, as shown in Fig. 6a. The vault was clamped at both ends and the loads F1 and F2 were applied as concentrated forces to the blocks corresponding to the third and the sixth steps, respectively, shown in dark grey in Fig. 6a. The masonry’s properties were the weight density pM (equal to 18 kN/m3) and the coefficient of friction µ (equal to 0.6). When the limit to compressive stress was included, the compressive strength of the masonry fm was assumed equal to 1.78 MPa, which corresponds to the 183

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Fig. 6. Vault modelling and applied loads in LiABlock (a), with the geometry of each discrete rigid block in red and in FEM (b) with shell elements. Table 1 Mechanical linear and nonlinear parameters adopted in the FEM analysis. Masonry

ρM EM GM σM

1800 1400 500 1.78

[kg/m3] [MPa] [MPa] [MPa]

Mass density Young modulus Shear modulus Compressive strength

Mortar joints

µ σm τm Gxy/Ĝxy

0.6 0.02 0.12 2

[–] [MPa] [MPa] [–]

βm

0.2

[–]

Friction coefficient Tensile strength Cohesion Ratio between the elastic and inelastic shear strain at failure in mortar joints Softening coefficient

σb Ey/Êy

0.59 2

[MPa] [–]

βb

0.2

[–]

Blocks

Tensile strength Ratio between elastic and inelastic strain in the masonry in compression at failure Softening coefficient

Fig. 7. Force-displacement curves obtained from the incremental static FEM analysis.

reaction forces in the considered nodes was measured during the analysis to evaluate F1 or F2. Table 2 shows the analyses results in terms of collapse loads obtained from LiABlock (considering or not the crushing failure conditions) and FEM analyses, respectively. In Fig. 7, the force-displacement curves are also illustrated. It can be observed that by adopting the conventional limit analysis, i.e. considering the infinite compressive resistance of masonry, no solution is found. On the contrary, by considering the compressive strength, the results in terms of collapse loads were in good agreement with those obtained from the incremental FEM analysis. In particular, it is worth noting that, in both the load cases, the FEM collapse loads are slightly higher than those obtained with the equilibrium analysis. This result can be explained by the fact that the FEM constitutive laws take into account the contribution of the mortar joints tensile strength, which is completely neglected in the limit analysis. The results in terms of collapse mechanisms and damage plots are illustrated in Table 3. The comparison between the three-hinge failure

mechanisms obtained from LiABlock_3D software (Table 3, on the left) and the tensile and compressive damage plots obtained from the FEM analysis (Table 3, on the right, where the crack formation corresponds to damage variable > 1) shows a good agreement. The position of each extrados (intrados) hinge predicted by LiABlock_3D conforms to a pronounced tensile stress damage at the intrados (extrados) and a compressive failure at the extrados (intrados) obtained from the FEM analysis. In the load case LC_1, the FEM damage in correspondence of the first two hinges starting from the bottom is largely developed (damage variable > 20). The third hinge, on the contrary, is not still fully formed having a damage in the range of 0.75 ÷ 1. In the load case LC_2, the three hinges are well identifiable and developed. 4. Application on the vaulted staircases of the Santa Maria delle Vigne bell tower The structural behaviour of eight brick masonry vaulted staircases of the Santa Maria delle Vigne bell tower (Fig. 8a) in Genoa (Italy) was analysed. The geometry and constructive techniques of the structures were fully described in [39].

Table 2 Collapse loads obtained from limit analysis with LiABlock and FEM analysis. LiABlock_3D

FEM

No crushing

Crushing failure

Load case

Collapse load [kN]

Collapse load [kN]

Collapse load [kN]

LC_1 LC_2

no solution no solution

62.9 52.9

68.5 63.7

4.1. Rigid block models and material properties The geometry of each vaulted staircase was deducted from a laser scan survey [38]. Fig. 8b shows an axonometric view of the bell tower and its eleven staircase systems, named from S1 to S11. The staircases S1, S9 and S10 were not object of the following analysis, as the first system is not a self-supporting body but it is entirely supported by the 184

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Table 3 Comparison between the collapse mechanisms obtained from LiABlock analyses with the crushing failure conditions and the damage plots from FEM analyses.

Load case

LiABlock_3D

FEM

Crushing failure

Tensile damage variable of the mortar joints (crack if > 1)

Compressive damage variable of the masonry (crack if > 1)

LC_1

LC_2

Fig. 8. Santa Maria delle Vigne bell tower: northern view (a) and axonometric view of the vaulted staircase systems (b). 185

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stone barrel vault of the first level, while the other two systems were made of wood. Two main categories of staircases can be distinguished based on their clear span and extrados profile curvature. The first set of vaults includes the vaulted staircases in the lower levels of the tower. Their clear span is in a range between 1.5 ÷ 2.7 m and their extrados can be approximated to an arch (S2, S3 and S5) or at least a curve with two centres of curvature (S4 and S6). The second set is made by the vaulted stairs that connect the higher levels of the bell tower. They connect double height spaces and their clear span length is around 4.8 m, corresponding to the full internal width of the tower. Their extrados can be approximated to a curve with multiple centres of curvature (S7, S8 and S11). Although it was not always possible to inspect in detail the constructive techniques of each vault, they were all assumed 0.24 m thick as made by two layers of bricks of size 0.12 m × 0.24 m × 0.037 m arranged with their longer side parallel to the directrix of the vault and assembled with mortar joints 6 mm thick. The representative masonry unit was defined as a trapezoidal prism, having a height equivalent to the vault’s thickness and a width corresponding to the size of two bricks plus the mortar joint, as already illustrated in Fig. 6a. For each vaulted staircase, four different models were investigated with LiABlock_3D, depending on two the following variables: masonry compressive strength (unlimited/limited); infill role (regarded/disregarded). The choice of regarding/disregarding the infill role depended on the lack of information on this building element. Actually, the physical nature of the space between the vaults and the overlying stairs is still unknown. In some cases, it may be expected that it is entirely filled with bulk material (see staircases S2, S3, S5 and S11); in other cases, the geometry suggests the presence of secondary substructures, such as masonry arches or walls or timber structures (staircases S4, S6, S7, S8). This uncertainty and the fact that it is quite hard to predict whether the infill increases or reduces the strength of the vault, led to consider two limit conditions. In the first one (infill role regarded), the volume between the vault and the stairs was considered as entirely filled with a rough material whose weight density pi is 14 kN/m3; in the second one (infill role disregarded), the volume was considered as empty. When the presence of the infill was taken into account, it was modelled as a dead load not giving any structural contribution in terms of strength and stiffness. The vaults were assumed to be clamped at both ends. However, further investigations should be conducted to better understand the validity of this assumption. For each staircase, n analyses were performed by applying concentrated forces to each of the n steps. Since it was not possible to determine the masonry mechanical properties through detailed investigations, they were estimated according to the values proposed by the Italian Technical Code [35] for clay brick masonry with poor mortar joints (Table C8.5.I, [36]), as already described in 2.2. Thus, the considered masonry’s properties were: the weight density pM = 18 kN/m3, the coefficient of friction µ = 0.6 and the compressive strength fm,ef = 1.23 MPa. The compressive strength fm,ef was calculated according to the expression already shown in Section 3.2. This value was reduced to 70% for the staircases S8 and S11 because of a widespread state of material decay detected during surveys, mainly caused by moisture and lack of maintenance.

was adopted (NC) solution was not always found. For the staircases S3 and S5, indeed, no solutions were found using the NC models, while for the staircase S2 only the analyses conducted loading the last two steps lead to reliable results. In all the other cases, the analyses carried out using the limited compressive strength (C) lead to lower values of the ultimate capacity, proving that the collapse of these structures is generally caused by crushing more than loss of equilibrium. The ratio between C and NC analysis results is particularly low when considering the models with the infill. With the infill, the C collapse loads can, indeed, be up to 10% of the NC collapse loads (see, for instance, the analysis of S4 loading the step 7, and the analysis of S7 loading the last seven steps). The presence of the infill increases the capacity of the structures, with the exception of the staircases S2, S3 and S5, where the collapse loads, especially those obtained by loading the first steps, are not significantly affected by the presence of the infill. It is worth noting that, however, these cases are characterized by a low amount of infill compared with the other vaulted staircases, in particular those characterized by a longer clear span (S7, S8, and S11). In some cases (see for instance the analysis of S2 at step 7, S4 at steps 11 and 12), the results of the C analysis coincide with those obtained from the NC analysis. This situation occur when the collapse is caused by the formation of the four hinges mechanism also when the crushing failure condition is considered. It can be noted that in several cases (in particular, S4 from step 3 to step 5, S6 steps 3 and 4, S7 from step 1 to step 4 and from step 10 to step 14, S8 from step 2 to step 14) when considering the models with the infill, the results obtained from the C analysis differ from those obtained from the NC analysis. However, in the analysis without the infill, there are no appreciable differences. This can be explained by the fact that the presence of the infill causes a higher initial state of compression. Thus, when the crushing failure is considered, the collapse is generally caused by the achievement of the ultimate compressive strength before than the loss of stability. On the contrary, in the models without the infill, the lower level of compressive stresses ensures that the main cause of collapse is the formation of the four-hinge mechanism even when considering the crushing failure conditions. The structural safety of staircases S7, S8 and S11 is particularly compromised, being their collapse loads lower than 20 kN in most of the analyses, with a minor peak of approximately 2 kN in the analyses of S7 loading the steps from 8 to 11. However, it is worth noting that, no significant damage was detected during the surveys on the real structures. The results of the analysis C with the infill of the staircases S2, S4, and S8 expressed in terms of collapse mechanisms, together with the collapse multipliers and maximum loads values, are shown in Fig. 9, Fig. 10 and Fig. 11, respectively. The mechanism of damage is in all the cases characterized by the formation of four hinges. However, the collapse occurs because stresses locally exceed the compressive strength of the material. The achievement of the compressive strength limit shows up as an interpenetration between blocks, as can be seen in the collapse mechanisms figures. 5. Conclusions This paper investigated the structural behaviour of masonry vaulted staircases using a novel rigid block model based on limit analysis, which takes into account the limited compressive strength of the material. A simple iterative solution procedure was implemented to take into account crushing failure in masonry flat arches using rigid block modelling and mathematical programming to solve the underlying limit analysis problems. The accuracy and computational efficiency of the model were evaluated against analytical and finite element modelling strategies, showing that accurate predictions of both collapse load and failure mechanism can be obtained with significant reduced computational costs. Finally, the application to eight different masonry vaulted

4.2. Results of rigid block limit analysis The Limit analysis results are shown in Table 4. The bar charts summarize the results in terms of collapse loads for each of the four analysed models: the black bars represent the models with the limited compressive strength condition (C), while the grey bars represent the models without the crushing conditions (NC). The solid fill indicates the models with the infill, while the diagonal stripes pattern fill indicates the models without the infill. In general, it can be observed that when the classic limit analysis 186

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Table 4 Collapse loads values for each vaulted staircase. S2

6

5

4

3

2

1

2.50 m

7

1.54 m 9

204

8

7

6

5

4

3

2

1

9

8

1.63 m

12 11 10

C_no infill

140

NC_infill

120

NC_no infill

100 80 60 40 20 0

13

C_infill

160

2.42 m

S4

199

180

2.13 m

10

Collapse load [kN]

S3

7

6

5

4

3

1

2

3

4

5

6

7 8 Step

9 10

21

2.66 m

S5

(continued on next page) 187

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Table 4 (continued)

8

7

290 180

6

5

4

3

2

1

2.27 m

13 12 11

C_infill C_no infill NC_infill

120

NC_no infill

100 80 60 40 20 1

2

3

4

5

6

7 8 Step

311 180

10

9

8

7

6

5

4

3

2

1

9

C_infill

160 Collapse load [kN]

14

228

140

0 S7

278

160

1.37 m

9

Collapse load [kN]

S6

C_no infill

140

NC_infill

120

NC_no infill

100 80 60

40

2.33 m

20

0

1

2

3

4

5

6

7 8 Step

9 10 11 12 13 14

4.81 m

13 12 11 12 10

180

C_infill

160

9 8 9 7

6 5 6 4 3 4 2 3 2

1

C_no infill

140

NC_infill

120

NC_no infill

100 80 60 40 20 0

3.44 m

14

Collapse load [kN]

S8

1

2

3

4

5

6

7 8 Step

9 10 11 12 13 14

4.82 m (continued on next page)

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Table 4 (continued) S11

8

7

3.56 m

6

5

4

3

2

1

4.82 m

Fig. 9. Collapse mechanisms, collapse multipliers, and collapse loads of the staircase S2 loading the steps 2, 4, and 6.

Fig. 10. Collapse mechanisms, collapse multipliers, and collapse loads of the staircase S4 loading the steps 2, 6, and 10.

Fig. 11. Collapse mechanisms, collapse multipliers, and collapse loads of the staircase S7 loading the steps 2, 7, and 12. 189

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staircases of a medieval bell tower were analysed. The main conclusions than can be drawn from the results are the following:

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• the use of the classic assumptions of limit analysis is not always possible to analyse the behaviour of this type of structures; • crushing failure involves a dramatic reduction of the safety factor • •

for this type of structures, when compared to the results associated to classic assumptions of limit analysis; the presence of the overlying infill material tends to increase the ultimate load capacity of the structures; the staircases with a larger clear span are the most vulnerable.

Further studies would be needed to investigate the characteristics of the infill volume, analysing the physical and mechanical properties of the material and, on the other hand, assessing the possible presence of secondary substructures (e.g., arches and walls) that support the steps and might change the loading conditions of the primary structure of the vaults. Finally, the effect of different boundary conditions may be also taken into account. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. References [1] Huerta S. Mechanics of masonry vaults: the equilibrium approach. In: Lourenco PB, Roca P, editors. International Seminar in Historical Constructions. Portugal: Guimarães; 2001. p. 47–69. [2] Heyman J. The stone skeleton. Int J Solids Struct 1966;2(2):249–79. [3] O’Dwyer D. Funicular analysis of masonry vaults. Comp Struct 1999;73:187–97. https://doi.org/10.1016/S0045-7949(98)00279-X. [4] Block P, Ciblac T, Ochsendorf J. Real-time limit analysis of vaulted masonry buildings. Comp Struct 2006;84(29–30):1841–52. https://doi.org/10.1016/j. compstruc.2006.08.002. [5] Fraternali F. A thrust network approach to the equilibrium problem of unreinforced masonry vaults via polyhedral stress functions. Mech Res Commun 2010;37(2):198–204. https://doi.org/10.1016/j.mechrescom.2009.12.010. [6] Andreu A, Gil L, Roca P. Computational analysis of masonry structures with a funicular model. J Eng Mech 2007;133(4):473–80. https://doi.org/10.1061/(ASCE) 0733-9399(2007) 133:4(473). [7] Smars P. Kinematic stability of masonry arches. Adv Mat Res 2010;133–134:429–34. https://doi.org/10.4028/www.scientific.net/AMR.133134.429. [8] D’Ayala FD, Tomasoni E. Three-dimensional analysis of masonry vaults using limit state analysis with finite friction. Int J Arch Herit 2011;5(2):140–71. https://doi. org/10.1080/15583050903367595. [9] Coccia S, Como M. Minimum thrust of rounded cross vaults. Int J Arch Herit 2014;9(4):468–84. https://doi.org/10.1080/15583058.2013.804965. [10] Milani E, Milani G, Tralli A. Limit analysis of masonry vaults by means of curved shell finite elements and homogenization. Int J Solids Struct 2008;45(20):5258–88. https://doi.org/10.1016/j.ijsolstr.2008.05.019. [11] Milani G, Rossi M, Calderini C, Lagomarsino S. Tilting plane tests on a small-scale masonry cross vault: experimental results and numerical simulations through a heterogeneous approach. Eng Struct 2016;123:300–12. https://doi.org/10.1016/j. engstruct.2016.05.017. [12] Ricci E, Fraddosio A, Piccioni MD, Sacco E. A new numerical approach for determining optimal thrust curves of masonry arches. Eur J Mech, A-Solid 2019;75:426–42. https://doi.org/10.1016/j.euromechsol.2019.02.003. [13] Tempesta G, Galassi S. Safety evaluation of masonry arches. A numerical procedure based on the thrust line closest to the geometrical axis. Int J Mech Sci 2019;155:206–21. https://doi.org/10.1016/j.ijmecsci.2019.02.036.

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