Structural Analysis of Historic Construction – D’Ayala & Fodde (eds)© 2008 Taylor & Francis Group, London, ISBN 978-0-415-46872-5

Some considerations on out-of-plane collapse modes of masonry walls

G. Brandonisio, E. Mele & A. De LucaDepartment of Structural Engineer (DIST), University of Naples “Federico II”, Naples, Italy

ABSTRACT: In this paper the problem of out-of-plane collapse of masonry walls under vertical and horizontalloads is addressed through the review of ancient building rules and the study of the European (Eurocode 6 and 8)and new Italian seismic codes on unreinforced masonry buildings.

The limitations of the analysed seismic codes and the empirical geometrical rules of ancient constructors arecompared with the results of some structural analyses carried out on rectangular masonry panels with differentslenderness h/s, aspect ratio h/b (height/width) and boundary conditions. In particular, the analyses have beencarried out both with the limit analyses and non linear FEM analyses with Abaqus computer code.

The results of these parametric analyses show that the respect of the seismic codes limitations on the geomet-rical slenderness h/s allows to implicitly have a minimum resistance to out-of-plane collapses of the masonrywalls.

A proposal is made to overcome the unjustified provisions of some codes which do not allow to have distanceamong orthogonal walls larger then 7 m.

1 INTRODUCTION

Traditionally, the study of the behaviour of masonrypanels focuses the attention on the in-plane failuresand resistant mechanisms. Nevertheless, the out-of-plane collapses of the walls are often activated bothunder vertical loads and horizontal forces. These fail-ure modes depend on many aspects, as quality of bricksand mortar, bond pattern, geometrical characteristicsand boundary conditions of the wall.

In this paper the influence of the last two aboveaspects (geometrical characteristics and restraint con-ditions) are investigated. Particularly, the target of thepaper is to underline the relationships among: the h/sratio (height to thickness ratio) of the wall; the aspectratio h/b (height to width ratio); the edge restraint con-ditions; and the out-of-plane strength of masonry wallsubjected to vertical loads and horizontal forces.

In this aim, in the first part of the paper the problemof the buckling of a masonry wall under axial load isaddressed in order to find a relationship between thewall slenderness λ (as defined in the stability theoryof columns or plates) and the load-carrying capacity.

Then, the behaviour of masonry wall under horizon-tal actions is analyzed through both the evaluation ofthe collapse multipliers and the study of geometricallimitations, recommended both by the ancient rules ofbuilding and modern European seismic codes (EC8’03and NTC’07). The results are presented in a homoge-nous approach in order to explicit the relationship

between the recommended h/s ratio and the aspectratio h/b. Then, for several edge restraints conditionsof walls, designed according to required h/s ratio,the relationships between the h/b ratio and the col-lapse multipliers are illustrated and compared with theresults of a parametric nonlinear F.E.M. analysis.

The results herein obtained provide useful indica-tions on the seismic capacity of the masonry walls.

2 WALL UNDER VERTICAL LOADS

As in to the case of a steel column under axial force,the resistance of a masonry wall under pure axial loaddepends on the slenderness λ, defined by the followingrelationship:

where: ho is the buckling length of the wall in thebuckling plane considered; i is the radius of gyrationof the cross section of the wall; β is a coefficient thatdepends on the type of restraint; h and s are the heightand the thickness of the wall, respectively.

From the Eq. (1) it is possible to note that the slen-derness λ is proportional to h/s ratio, which can bedefined as geometrical slenderness of the masonrypanel.

The Figure 1 shows the relationship between theaxial resistance Nf of the wall and the slenderness λ

565

1.0

Nf/Npl

λ

N

Elasticbuckling

Transitionregion

Crushing

N

sb

h

Figure 1. Effect of slenderness on resistance of a masonrywall under axial load.

(in the chart Nf is normalized to the squash load Nplof the wall). It can be observed that the diagram canbe divided in three fields, characterized by differentbehaviour of the wall. In fact, if the slenderness λ islow (stocky wall), the failure would occur for compres-sion of the material; whereas the slenderness λ is high(slender wall), the failure would occur for elastic buck-ling of the wall; in the case of intermediate values ofλ, the failure would occur for interaction between thecrushing and the elastic buckling of the wall (Hendryet al. 1997).

The European (EC6’03) and Italian (NTC’07) pro-visions for masonry structures do not refer to theslenderness λ, but they introduce the conventionalslenderness λc defined as:

where: hef is the effective height of the wall; sef isthe effective thickness of the wall, that is usuallytaken equal to the actual thickness s (for cavity wallsthe EC6’03 suggests the following formula: sef =(t31 + t32)1/3 where t1 and t2 are the actual thickness ofthe leaves; the Italian code NTC’07, instead, in everycase assumes sef = s)1; ρ is a reduction factor depend-ing on the edge restraint or stiffening of the wall.

In the examined codes (EC6’03 and NTC’07) thesuggested values of the coefficient of restraint ρ arerelated to the aspect ratio h/b and to the degree of theedge restraints. Particularly, with reference to the wallsof Figure 2, the EC6’03 makes the difference amongthe following three cases:

– wall 2L (Fig. 2b): the value of ρ (appointed as ρ2)is depending on the degree of restraint (ρ2 = 0.75in presence of r. c. floors at the top and bottom ofthe wall; ρ2 = 1.0 for walls restrained at the top andbottom by timber floors);

1 In the following part of this paper, we will assume sef = s.

1L

(a)

2L

(b)

3L

(c)

4L

(d)

Figure 2. Edge restraints of the wall: (a) wall restrained atthe bottom (wall 1L); (b) wall restrained at the top and bottom(wall 2L); (c) wall restrained at the top, at the bottom and onone vertical edge, with the other vertical edge free (wall 3L);(d) wall with 4 edges restrained (wall 4L).

0

20

40

60

80

100

0 1 2 3 4 5 6

h/s

4L

3L

(7): NTC'07(2): EC6'03 (ρ2=1)

(1): EC6'03 (ρ2=0.75)

(8):

NTC

'07

(4): EC6'03 (ρ2=1)

(3): EC6'03 (ρ2=0.75)

(6):

EC6'

03(ρ

2=1)

(5):

EC6'

03(ρ

2=0.

75)

2L

h/b

Figure 3. Comparison among the limitations of EC6’03 andNTC’07 on h/s ratio.

– wall 3L (Fig. 2c): the value of ρ (appointed as ρ3)is depending on the degree of restraint ρ2 at the topand bottom and on aspect ratio h/b:

– wall 4L (Fig. 2d): the value of ρ (appointed as ρ4)is again depending on the degree of restraint ρ2 atthe top and bottom and on aspect ratio h/b:

In the EC6 the slenderness λc should not be greaterthan 27 when the wall is subjected mainly to verticalloading. This limitation on λc, together with the sug-gested limitations on ρ, may be used to establish themaximum h/s ratio allowed by the EC6.

In Figure 3 the relationships between h/s and h/bare shown in graphical form. In particular, the curves(1)–(2) are associated to the limits of the wall 2L,while the curves (3)–(4) and curves (5)–(6) are asso-ciated to the limitations (3) and (4), for wall 3L and4L, respectively.

The comparison among the curves (1) ÷ (6) ofFigure 3 underlines the effect of the lateral edgerestraints on the required h/s ratio. For example, inthe case of wall with aspect ratio h/b = 1, the maxi-mum values of geometrical slenderness h/s is equal to36 (ρ2 = 0.75) or 27 (ρ2 = 1.0), for the wall 2L; it isequal to 38 (ρ2 = 0.75) or 30 (ρ2 = 1.0), for the wall3L; it is equal to 56 (ρ2 = 0.75) or 54 (ρ2 = 1.0), forthe wall 4L.

Instead, the new Italian code NTC’07 does not con-sider the case of wall 3L. In fact, in NTC’07 are

566

recommended reduction factors ρ only in the case ofwall 2L (ρ2 = 1.0) and wall 4L, for which ρ dependsonly by the aspect ratio: ρ4 = f(h/b).

The maximum slenderness λc permitted in theNTC’07 is 20; this limit, together with the recom-mended values of ρ, allows to plot in Figure 3 thecurves (7) and (8) associated to the walls 2L and 4L,respectively.

The comparison among the curves (1)÷(6) associ-ated to the limits of EC6’03 and to the curves (7) and(8) derived by the NTC’07, shows that the Italian codeis more conservative than the EC6 both for wall 2L andwall 4L (Fig. 3). For the case of wall 4L, in Figure 4 it isproposed the comparison between the coefficients ofrestraint β and ρ, that define the relationships betweenthe slendernesses λ and λc and the h/s ratio (Eqs. (1)and (2), respectively).Therefore, the diagram allows tocompare the differences between the theoretical slen-derness λ (Eq. 1) and the conventional slenderness λc(Eq. 2).

To define the reduction factors β of the wall 4L, itis utilized the elastic theory of the buckling of the rect-angular plates under compressive loads. If the wall’sedges may be considered pinned, the critical stress σcrof the wall 4L becomes:

where: m is the number of half waves in witch theplate buckles in the vertical direction (m = 1, 2, . . ., n);D = E·s3

12·(1−ν2) ; E is the modulus of elasticity; ν is thePoisson’s ratio (in Figure 4 it is utilized E = 1100 MPaand ν = 0.1).

The condition:

allows to define the slenderness λ in the case of wall4L with all pinned edges:

where the reduction factor β assumes the expression:

The trends of the curve (1)÷(3) of Figure 4 showthat the factors β and ρ are equal only in the case ofEC6’03 in the hypothesis ρ2 = 1; the associated curve(2), indeed, is situated on the bisecting line. On thecontrary, in the cases of EC6 with ρ2 = 0.75, and of

1.20 0.2 1.00.80.60.40

0.2

0.4

0.6

0.8

1.0

1.2

ρ=λc/(h/s)

12·β=λ/(h/s)

4L

(3): NTC'07

(2): EC6'03 (ρ2=1)

(1): EC6'03 (ρ2=0.75)

Figure 4. Comparison between the coefficients of restraintβ and ρ.

NTC’07, only for small values of reduction factors(lower than 0.5) there is coincidence between β and ρ;for higher values (greater than 0.5), in EC6 (ρ2 = 0.75)the factor β is larger than ρ, while in the NTC’07 β islower than ρ.

By observing the curves plotted in Figure 4, it ispossible to found an approximate relationship betweenλ and λc. In fact the curves (1), (2) and (3) suggest thefollowing identities:

The relationships between capacity reduction fac-tor Nf /Npl and conventional slenderness λc, proposedin EC6’03 and NTC’07, are plotted in Figure 5aand b, respectively. In the diagrams are reportedseveral curves corresponding to different values ofeccentricity e of axial load N.

The application of Eq. (9) allows to individuatethe influence of slenderness λ on the reduction fac-tor Nf /Npl. The Figure 5, indeed, suggests that Nf /Nplis almost equal to 1.0 (almost no reduction) whenλ is lower than about 20 (stocky wall). When λ isgreater than about 50÷60, instead, the reduction ofresistance is significant (slender wall). Finally, whenλ is comprised between 20 and 50÷60, the wall showsan intermediate behaviour, with interaction betweencrushing and elastic buckling phenomena.

3 WALL UNDER HORIZONTAL LOADS

Also the out-of plane resistance of a wall subjected tohorizontal loads is influenced by the h/s ratio. Indeed,since the ancient rules of the art of building masonrystructures, the h/s ratio has been properly limited in

567

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30

Nf/Npl

cλ

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30

Nf/Npl

m=6e/s=0

m=6e/s=0.5

m=6e/s=1m=6e/s=1.5m=6e/s=2

cλ

(a)

(b)

e/s=0.05

e/s=0.10

e/s=0.15

e/s=0.20

e/s=0.25

e/s=0.30

e/s=0.33

0 34 51 68 85 102

λ

0 20 40 60 80 100 120

λ

17 ~~~~~~

~ ~ ~ ~ ~ ~

EC6'03

NTC'07

c,maxλ

c,maxλ

27

108

Figure 5. Reduction of resistance vs. slendernesses λc andλ in EC6’03 (a) and NTC’07 (b).

order to reduce the vulnerability of the walls to theout-of-plane collapses.

In this paragraph, the out-of-plane collapse mech-anisms of a wall with different restraint conditions atthe edges are initially examined and the collapse mul-tipliers are evaluated; then, the ancient rules of theart and the European (EC8’03) and Italian (NTC’07)seismic provisions for the unreinforced masonry build-ings, which mainly consist of geometrical limitations,are analyzed; finally the resistance of masonry wallsto horizontal loads, designed according to the abovegeometrical limitations (ancient rules of the art andseismic codes) is evaluated.

3.1 Collapse multipliers of horizontal loads

In the case of uniform distribution of horizontal loads,the application of the principle of virtual works to thewall 1L of Figure 2a, allows to explicit the relationshipbetween the geometrical h/s ratio and the out-of-plane collapse multiplier α1 (defined as maximumhorizontal load to self weight load ratio):

h/s=10

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 0.5 1 1.5 2 2.5 3

h/b

α4L

2L

3L

1L

(4) (3)

(2)

(1)α1=0.1

α2=0.8α3=1.2

α4=2.4

Figure 6. Comparison among the multipliers α of the walls1L, 2L, 3L and 4L.

When the wall is sufficiently retrained at the topand bottom by the floor diaphragms (wall 2L) themultiplier α becomes (Hendry et al. 1997):

Instead, if the wall is stiffened on one or on twovertical edges (wall 3L and 4L, respectively), in thehypothesis that the yield lines are slanted at 45◦, thecollapse multipliers α are, respectively:

The Eqs. (12) and (13) show that the multipliers α3and α4 depend on the geometrical slenderness h/s andon the aspect ratio h/b of the wall.

In the case of h/s = 10, the curves (1)÷(4) associ-ated to Eqs. (10)÷(13) are plotted in Figure 6. It can beobserved the influence of both edge restraints condi-tions and aspect ratio h/b on the out-of-plane resistanceof the wall. In fact, the diagram of Figure 6 showsthat: (i) the curves (1) and (2) (corresponding to wall1L and 2L, respectively) do not depend on the aspectratio h/b; (ii) in presence of floor diaphragms at thetop of the wall (wall 2L), the multiplier α increasesof 8 times with respect to the case of wall 1L; (iii) inpresence of wall 3L and 4L with 1 or 2 stiffened ver-tical edges, the collapse multiplier α shows a furtherincrease when the h/b ratio increases. For example, inFigure 6 a comparison among the multipliers α1, α2,α3 and α4 is provided when h/b = 1.

3.2 Ancient rules of art

Regarding the stability of unreinforced masonry wallsunder horizontal loads, J. B. Rondelet (1802), in his

568

0 1 20

10

20

h/s

3h/b

30

(b) h/s=10(a) h/s=8

(c) h/s=12

(d) h/s=12 1+(h/b)2

4L

1Lh/s=17

h/s=21

1.5

Figure 7. Comparison among the rules of J. B. Rondelet.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 0.5 1 1.5 2 2.5 3

h/b

α4L

1L(c): h/s=12, α1=0.083(b): h/s=10, α1=0.10(a): h/s=8, α1=0.125

(d): α4=4

3· 1+(h/b) · (1-2/3·h/b)2

Figure 8. Collapse multipliers for the rules of Rondelet.

historical treatise asserts that a wall will have strongstability if h/s=8, medium stability if h/s = 10, lowstability if h/s = 12.

From the well-known study on the collapse mecha-nisms for unreinforced masonry panels characterizedby different restraint conditions, Rondelet realizesthat, for the same degree of stability, the thickness s canbe reduced when the distance b between the transver-sal wall decreases. Particularly, in the case of a wall ofone floor and unique room building, when the beamsof the roof are well fixed to the wall and there are noroof thrust, the minimum value of the thickness s whichguarantees an adequate degree of stability for the wall,can be calculated using the following empirical rule byRondelet:

In term of h/s ratio, the Eq. (14) can be written asfollow:

The geometrical Rondelet’s rule (15) is plotted inFigure 7 (curve (d)) together with the limits that Ron-delet gives for wall 1L: h/s = 8, strong stability (curve

(a)); h/s = 10, medium stability (curve (b)); h/s = 12,low stability (curve (c)).

Particularly, the curve (d) shows that the requiredh/s ratio to guarantee the stability of the wall increaseswith the aspect ratio h/b of the masonry panel; forexample, if h = 3.5 m, for h/b = 1 it is necessary tohave a minimum value of h/s = 17, i.e. a wall thick-ness s = 21 cm; instead, for h/b = 1.5 it is necessary tohave a minimum value of h/s = 21, i.e. a wall thicknesss = 17 cm.

The above Rondelet’s rules, in terms of collapsemultiplier α (Eqs. (10) and (13)), lead to the curves(a)÷(d) reported in Figure 8. The comparison amongthese curves shows the great increase of resistance toout-of-plane loads of wall 4L (curve (d)) with respectto wall 1L (curves (a)÷(c)). Further, the trend of curve(d) shows that the multiplier α4 increases with h/bratio; indeed, the curve (d) is asymptotic to the verticalline h/b = 1.5. In such meaning, Rondelet writes thatthe collapse of wall is impossible when the distance bbetween the transversal stiffening walls is low.

3.3 Seismic codes for unreinforced masonrybuildings

In order to reduce the attitude to out-of-plane col-lapse of unreinforced masonry panels, also the modernEuropean seismic codes (EC8’03 and NTC’07 (seis-mic part)) limit the h/s ratio through the conventionalslenderness λc (Eq. (2)). Particularly, in presence ofmasonry wall with natural stone units, the Eurocode 8prescribes the following limit:

while, the new Italian code NTC’07 recommends thefollowing limits:

The above limitations on conventional slendernessλc, and consequently on h/s ratio, are more restrictivewith respect to the case of buildings in non-seismiczone. In Figure 9 the comparison among the analyzedseismic limitations are provided in term of h/s ratio.In the cases of wall 1L and 4L, it can be observeda substantial coincidence between the limitations ofEC8’03 and NTC’07, especially when h/b < 1 (for wall4L); instead, for h/b > 1 the limitations prescribed byEC8’03 are more conservative than the Italian provi-sions ones. Finally, also for this diagram it is possibleto note the effect of edge restraints on the h/s limits.

569

0 1 2 3h/b0

10

20

30 h/s

(1): EC8'03 (ρ2=0.75);NTC (seismic zones 3 and 4)(2): EC8'03 (ρ2=1)

2L(7):NTC (seismic zones 1 and 2)

(4): EC8'03 (ρ2=1)

(3): EC8'03 (ρ2=0.75)

(9): NTC'07 (seismic zones 3 and 4)

(6):

EC8'03

(ρ2=

1)

(5):

EC

8'03

(ρ2

=0.

75)

(8): NTC'07 (seismic zones 1 and 2)

Figure 9. Comparison among the limitations of EC8’03 andNTC’07 (in seismic zones) on h/s ratio.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 0.5 1 1.5 2 2.5 3

h/b

α

(f) (ρ2=1)(e) (ρ2=0.75)

(l) (ρ2=1)(i) (ρ2=0.75) (h) (ρ2=1)

(g) (ρ2=0.75)

3L

4L

2L

Figure 10. Collapse multipliers for EC8’03.

In terms of collapse multiplier α (Eqs. (11)÷(12)),the condition (16) recommended in EC8’03 allows toobtain the curves plotted in the diagram h/b-α of Fig-ure 10. Particularly in figure are reported: the curves(e) and (f) for the wall 2L; the curves (g) and (h) for thewall 3L; the curves (i) and (l) for the wall 4L. Each cou-ple of curves is associated to the condition ρ2 = 0.75or ρ2 = 1 that the EC6’03 considers in the definitionof hef .

The limitations (17) and (18) of Italian codeNTC’07 (seismic part) are plotted in the h/s-α dia-grams of Figure 11 for wall 2L (curves (m) and (n))and for wall 4L (curves (p) and (q)). The diagramshows that in high seismicity zones (curves (m) and(p)), the Italian code recommends a minimum value ofcollapse multiplier α higher than the case of low seis-micity (curves (n) and (q)). Moreover, the curves (p)and (q) associated to wall 4L show a constant trend inthe range h/b = 0.5÷1, i.e. in the most common rangeof aspect ratio h/b for the masonry walls. The pres-ence of the horizontal plateau in the curves (p) and (q)depends on the fact that the Italian code uses the sameexpressions of the collapse mechanisms as the onesexamined in §3.1 to define the reduction factor ρ.

In Figure 12 the comparison among the limitationsof Rondelet, EC8’03 and NTC’07 (in seismic zones)on collapse multiplier α, is provided. In the cases of

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 0.5 1 1.5 2 2.5 3

h/b

4L (q): seismic zones 3 and 4 (λc=12)(p): seismic zones 1 and 2 (λc=10)

(n): seismic zones 3 and 4 (λc=12)(m): seismic zones 1 and 2 (λc=10)

2L

α

Figure 11. Collapse multipliers for NTC’07.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 0.5 1 1.5 2 2.5 3

h/b

α

(n)(e)=(m)

(q)(p)

(f)

(g)(h)

(i)(l)

2L

4L3L

1L(a)

(c)(b)

(d)

Figure 12. Comparison among the limitations of Ron-delet, EC8’03 and NTC’07 (in seismic zones) on collapsemultiplier α.

walls 2L and 4L it is possible to note a certain coin-cidence among the plotted curves; particularly, it canbe observed that the Rondelet’s curve (d) is overlaid tothe curve (q) (associated to NTC’07) and to the curve(g) of EC8’03 (wall 3L, ρ2 = 0.75) when h/b > 1 andh/b < 0.8, respectively.

4 F.E.M. PARAMETRIC ANALYSIS

In this paragraph, non linear analyses on rectangularwalls are carried out through the F.E.M. computer codeAbaqus 6.7-1 [Simulia, 2007]. On the basis of theseanalysis results, some considerations about the col-lapse multiplier α of walls 1L, 2L, 3L and 4L subjectedto horizontal loads, are derived.

4.1 Modelling of the masonry walls

In Table 1 are reported the characteristics of the 42examined models. The specimens have been labelledaccording to the following criteria: h-b-s Ln, where theh, b and s are the geometrical dimensions of the wall(in cm) and Ln (n = 1÷4) defines the edge restraintof the wall as schematized in Figure 2. Regarding the

570

Table 1. Examined models.

Specimen h/b h/s

350-350-45 L1 1 8350-350-45 L2 1 8350-1400-45 L3 0.25 8350-700-45 L3 0.5 8350-470-45 L3 0.75 8350-350-45 L3 1 8350-280-45 L3 1.25 8350-233-45 L3 1.5 8350-1400-45 L4 0.25 8350-700-45 L4 0.5 8350-470-45 L4 0.75 8350-350-45 L4 1 8350-280-45 L4 1.25 8350-233-45 L4 1.5 8

350-350-35 L1 1 10350-350-35 L2 1 10350-1400-35 L3 0.25 10350-700-35 L3 0.5 10350-470-35 L3 0.75 10350-350-35 L3 1 10350-280-35 L3 1.25 10350-233-35 L3 1.5 10350-1400-35 L4 0.25 10350-700-35 L4 0.5 10350-470-35 L4 0.75 10350-350-35 L4 1 10350-280-35 L4 1.25 10350-233-35 L4 1.5 10

350-350-30 L1 1 12350-350-30 L2 1 12350-1400-30 L3 0.25 12350-700-30 L3 0.5 12350-470-30 L3 0.75 12350-350-30 L3 1 12350-280-30 L3 1.25 12350-233-30 L3 1.5 12350-1400-30 L4 0.25 12350-700-30 L4 0.5 12350-470-30 L4 0.75 12350-350-30 L4 1 12350-280-30 L4 1.25 12350-233-30 L4 1.5 12

geometry, all the walls have height h = 350 cm; thewidth b varies between 233 and 1400 cm in order tohave walls with aspect ratio between 1.5 and 0.25. Thethickness s is set equal to 30, 35, 45 cm, in order to havewalls with h/s ratio equal to 12, 10 and 8, respectively.

Four-nodes shell elements (S4R5 elements) areused to model the masonry tuff walls; reduced inte-gration is used for the shell elements; the numberof integration points through the thickness of shellelement is equal to five.

All the examined walls have been subjected to nonlinear analyses using a smeared cracking approach asimplemented in the computer code Abaqus.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 5 10 15 20 25

α

d [mm]

4L

3L

2L1L

Specimen 350-350-35h/b=1; h/s=10

α1=0.1

α2=0.8α3=1.2

α4=2.4F.E.M. analysisLimit analysis

(a)

(b)

(c)

Figure 13. Specimens 350-350-35: (a) comparisonamong the push-over curves and the collapse multipliers;(b) deformed shapes with stress tensor vectors of the F.E.M.models; (c) collapse mechanisms considered for limitanalyses.

In order to correctly calibrate the model parameters,reference has been made to the curve fitting procedurewas made by Giordano A. (2002), which utilizes theresults of the experimental tests on masonry tuff walls.

The applied loads are the self weight and the hor-izontal load, which increases with monotonic low upto the end of the analysis.

4.2 Results and comparisons

In Figure 13a, a summary of the non linear analysesfor the walls 350-350-35 is reported; particularly, thepushover curves are depicted. In this diagram, the col-lapse multipliers calculated through the limits analysesand already provided in Figure 6 are also reported.The comparison among the nonlinear static and limitanalyses shows that the F.E.M. results provide collapsemultipliers higher than limit analyses, because the ten-sile strength of masonry is considered in the Abaqusmodels.

Indeed, in Figure 13b and c the visualization ofthe F.E.M. deformed shapes, with stress tensor vec-tors, and the hypothesized collapse mechanisms arereported. The comparison shows a good agreementbetween the Abaqus and limit analyses.

In Figure 14 the comparison among the collapsemultipliers computed by means of F.E.M. analysis andof limit analyses (Eqs. (10)÷(13)) is provided. Par-ticularly, the diagrams refers to walls characterizedby h/s ratio equal to 10. In almost all analyzed spec-imens (Table 1) the diagrams confirm the previous

571

h/s=10

0

1

2

3

4

5

0 0.5 1 1.5 2 2.5 3h/b

α4L

2L

3L

Abaqus 3L

Abaqus 4L

(2)

(4) (3)6

Figure 14. Comparison among the collapse multipliersof walls with h/s = 10 evaluated through F.E.M. and limitanalyses.

(a)0

1

2

3

4

5

6

0 0.5 1 1.5 2 2.5 3h/b

α4L

Abaqus 3L

Abaqus 4L

3L(l) (ρ2=1) (h) (ρ2=1)

(b)0

1

2

3

4

5

6

0 0.5 1 1.5 2 2.5 3h/b

α4L

Abaqus 4L

(q): seismic zones3 and 4 (λc=12)

(p): seismic zones1 and 2 (λc=10)

Figure 15. Comparison among the collapse multipliersevaluated through nonlinear static and limit analyses onwalls designed according to the geometrical requirements ofEC8’03 (a) and NTC’07 (in seismic zones) (b).

observation regarding the multipliers α; in fact, thevalues of α computed through pushover analyses aregenerally higher than the values calculated with thelimit analyses. Finally, also the curves associated to theresults of nonlinear static analyses carried out on walls3L and 4L show an increasing trend with the aspectratio h/b; this confirms the beneficial effect of the ver-tical edge restraints on the resistance of the walls toout-of-plane collapses.

The Figure 15 shows the comparison among theF.E.M. results and the curves relative to collapse mul-tipliers α of the walls 3L and 4L associated to EC8’03

(Fig. 15a) and NTC’07 (Fig. 15b). The thicknesses s ofthe analyzed specimens have been obtained by Figure9 curves (4) (wall 3L) and (6) (wall 4L) for EC8’03,and by Figure 9 curves (8) and (9) for NTC’07. In thecase of EC8’03, the results of F.E.M. analyses showan increasing trend of the curves (Fig. 15b), while inthe case NTC’07 it possible to note a sub-horizontaltrend of the F.E.M. curves.

Moreover, when h/b ratio is larger than one, theNTC’07 curves show a slight decrease of collapse mul-tipliers; this seems to underline that the limitations ofNTC’07 give very large values of maximum h/s ratiowhen the wall is characterized by high values of aspectratio h/b.

Finally, the comparison among the F.E.M. resultsreported in the curves (a) of Figure 15 shows that theEC8’03 is on the safe side with respect to NTC’07; infact, it can be observed that the curve “Abaqus 4L” ofFigure 15a is characterized by higher values of collapsemultipliers α than the curves “Abaqus 4L” plotted inFigure 15b.

5 CONCLUSIONS

The study of the out-of-plane collapses of masonrywalls subjected to vertical loads has suggested therelationship between the slendernesses λ and λc or,equivalently, between the coefficients of restraint βand ρ. In term of slenderness λ, it has been observedthat a wall characterized by λ < 20 can be defined asstocky wall, having a low reduction of axial strengthNf with respect to the squash resistance Npl of the wall;when λ > 50÷60, instead, the wall can be consideredas slender wall. Finally, if the slenderness λ is com-prised between 20 and 50÷60, the wall is characterizedby an intermediate behaviour, with interaction betweenthe crushing and the elastic buckling of the wall.

In the second part of this paper, the issue ofout-of-plane collapses of masonry walls subjected tohorizontal loads has been studied through (i) the anal-yses of the collapse mechanisms; (ii) the review ofancient rules of the art; (iii) the study of the European(EC8’03) and new Italian seismic codes (NTC’07) forunreinforced masonry buildings.

In the case of walls 3L and 4L, the comparisonamong the geometrical limitations underlines that therecommended h/s ratio becomes greater than the val-ues h/s = 8, 10 or 12, that Rondelet suggests for thewall 1L. For example, in the range h/b = 0.5÷1.5, therequired geometrical slenderness h/s for the wall 4Lis equal to or smaller than: 13÷21 according to Ron-delet’s rule (Eq. 15); 11÷27, according to EC8’03;10÷32, according to NTC’07.

These observations are also confirmed in the prac-tice for a wider range of aspect ratio h/b, as shownin Figure 16 for the 8 classes of macro-elements of 10masonry churches studied in Brandonisio et al. (2008).

572

0

10

20

30

40

50

0 1 2 3 4 5 6

h/b

h/s

h/s=8h/s=10h/s=12

Figure 16. h/s vs. h/b relationships for the masonry walls of10 Italian masonry churches analyzed in Brandonisio et al.(2008).

The study of the collapse mechanisms togetherwith the analyses of the geometrical requirementsderived from the ancient rules and the modern seis-mic provisions, show the influence of the h/s, h/b andedge restraints conditions on the out-of-plane collapsemultiplier α.

The h/b-α curves suggest a remark regarding the“Rules for simple masonry buildings” reported inthe EC8’03 and NTC’07. In fact, both EC8’03 andNTC’07 require a maximum spacing of 7 m betweentwo bracing walls; this geometrical requirement, forthe common values of storey height h = 3.5÷5 m, con-ducts to walls characterized by aspect ratios h/b greaterthan 0.5. In absence of holes, this leads to masonrypanels which resist to out-of-plane lateral loads equalto: 87% (when ρ2 = 0.75) or 106% (when ρ2 = 1) oftheir self weigh, in the case of walls designed accord-ing to EC8’03; 120% (in seismic areas 1 and 2) or100% (in seismic areas 3 and 4), in the case of Italianseismic code.

In the last part of the paper, the results of a paramet-ric nonlinear F.E.M. analysis carried out on rectangular

masonry walls has been presented. In the F.E.M. anal-yses both h/s and h/b ratios and the edge restraints havebeen varied. The results confirm the observation donethrough the application of the limit analyses, i.e. therespect of the seismic codes limitations on the geo-metrical slenderness h/s allows to implicitly have aminimum resistance to the out-of-plane collapses ofthe masonry walls subjected to horizontal loads.

In the light of the previous consideration, it couldbe suggested to overcome the indications concerningthe maximum spacing between the cross walls equalto 7 m often recurrent both in ancient treatises and inthe modern seismic codes. In particular, it seem thata more appropriate limitation on orthogonal elementsshould be given in non dimensional forms, i.e. in termsof aspect ratio of the masonry panel: h/b ≥ 0.5.

REFERENCES

Brandonisio G., Mele E., Santaniello R., De Luca A., 2008.Seismic safety of basilica churches: analysis of ten casestudies. VI SAHC’08, Bath, UK. 2–4 July.

Giordano A., 2000. Numerical Modelling of Masonry Struc-tures Using the Abaqus “Concrete”. Model ABAQUSUsers’ Conference. Newport, Rhode Island.

Hendry, A. W., Sinha B. P., Davies S. R., 1997. Desig ofMasonry Strucutures, E & FN Spon.Great Britain.

NTC 2007. Norme Tecniche per le costruzioni. Draft of 6thDecember 2007 (in Italian).

PrEN1996-1. Eurocode 6: Design of masonry structures –Part 1-1: Common rules for reinforced and unreinforcedmasonry structures. March 2003.

PrEN1998-1. Eurocode 8: Design of Structures for Earth-quake Resistance. January 2003.

Rondelet, J. B., 1802.Trattato teorico e pratico dell’arte di edi-ficare, Tomo IV - Teoria delle Costruzioni. Prima edizionenapolitana notabilmente migliorata e corretta da RaffaelePepe. Napoli, 1840 (in Italian).

Simulia 2007. ABAQUS Theory Manual. USA.

573