Materials and Structuresl Matériaux et Constructions, 1989,22,364-373

Fracture mechanics of brick masonry: size effects and snap-backanalysis

PIETRO BOCCAIstituto Universitario di Architettura di Venezia, 30125 Venezia, Italy

ALBERTO CARPINTERI, SILVIO VALENTEPolitecnico di Torino, Dipartimento di Ingegneria Strutturale, Corso Duca degli Abruzzi 24, 10129 Torino, ltaly

Fracture energy, GF, and the criticai value of stress-intensity factor, K1C, are determined forbrick masonry specimens tested in bending with different notch depths. The experimentalresults are compared with numerical simulations, obtained through a cohesive crack modeldeveloped originally for concrete. Theoretical and experimentalload-defiection curves presentvery similar softening branches. In some cases, a snap-back instability is predicted by the modeland confirmed by the experimental data. A size-scale transition appears evident from anultimate strength collapse at the ligament to a brittle fracture due to stress-intensification. Such atransition demonstrates that LEFM is a very suitable model for brick masonry structures at theusual size-scale. A non-dimensional brittleness number is introduced as a measure of theLEFM applicability.

NOTATION

aEt.CFKyclL,H,BPòvWo(J"

w

w[k]

Fc

r

Crack lengthYoung's modulusUltimate tensile strengthFracture energyStress-intensity factor (criticaI value)Support spanLength, depth, thickness of the specimenLoadLoading point displacementPoisson ratioArea under the P-Ò curveStress acting on the crack surfacesCrack opening displacementCriticaI value of the crack opening displace-mentVector of the crack opening displacementsMatrix of the coefficients of influence (nodalforces)Vector of the nodal forcesVector of the coefficients of influence (externalload)Vector of the crack opening displacements dueto the specimen weight

l. INTRODUCTION

Brick masonry failures have been extensively investi-gated through uniaxial and biaxial loading conditions[1-3], and the cracking evolution, from microcracking tomacrofracture, can be analysed by means of recentIyestablished experimental techniques. On the other

0025-5432/89 © RILEM

hand, when the dissipative phenomena play an impor-tant role, as, for example, in the case of masonry struc-tures in seismic zones, energy criteri a analyse the failuremechanism better and more realistically than the classic-al stress criteria. The application of fracture mechanics,whose concepts are aIready used for steel and concretesuccessfully [4, 5], is extended herein to brick masonry.It is demonstrated how the theoretical results fit with theexperimental ones satisfactorily. Further investigationsare encouraged with the following objects:

(i) to characterize the different brick masonry mate-rials;

(ii) to simulate numerically the mechanical and fai-lure behaviour of brick masonry structures evenof complex geometrical shape.

The problem of the Mode I (opening) fracturemechanism in brick masonry is faced. Structural ele-ments working in compression often undergo this modeof failure [6]. The fracture energy CF and the criticaIvalue Krc of stress-intensity factor [4] are evaluated forthree-point bending specimens, directly drilled fromhistorical bricks. Such experimental data, obtained fordifferent materials and different notch depths, are thenput in comparison with numerical results derived from acohesive crack finite-element simulation, originally de-veloped for concrete [4]. Theoretical and experimentalload-deflection curves present very similar softeningbranches. In some cases, a snap-back instability is pre-dicted by the model and confirmed by the experimentalinvestigation. A softening branch with positive slope isrevealed experimentally 3Y controlling the crack mouth

Materials and Structures 365

____ ~L~=20cm ~./opening displacement, and numerically by controllingthe crack length. Both such quantities are thus monoto-nically increasing with time. Otherwise, when the con-trolling parameter is the beam deftection, unstable andcatastrophic behaviour occurs with a brittle fracture.

An evident transition appears from an ultimatestrength collapse at the ligament, to a brittle crackpropagation due to stress-intensification, by increasingthe dimensionless size Hft/CF. In the present paper, thebrittleness number variation is produced by varying thematerial properties t.. CF, and not the size scale H, asdone in previous studies [7].

2. SPECIMENPREPARATION ANDEXPERIMENTALPROCEDURE

The specimens' features (Fig. 1) are listed in Table 1.The capital letters A, B, -C refer to three differentstructures, built respectively in the 19th (A, B) and 20th(C) century.

Three-point bending tests according to the RILEMrecommendation for concrete [8] were carried out,through an MTS machine with maximum load up to20 kN. Load-deftection (P-O) diagrams were plotted bycontrolling the crack mouth opening displacement. Thelatter was increased at a constant rate of2.5 x 10-7 m S-l (Fig. 2).

3. COHESIVE CRACK MODEL

The cohesive crack model, developed originally forconcrete, is based on the following assumptions [4, 5]:

1. The cohesive fracture zone (plastic or process zone)begins to develop when the maximum principal stressachieves the ultimate tensile strength t. (Fig. 3a).

2. The material in the process zone is partially dam-aged but still able to transfer stress. Such a stress isdependent on the crack opening displacement w. Alinear (T-W law is assumed as in Fig. 3b.

The real crack tip is defined as the point where thedistance between the crack surfaces is equa] to the

I •1= 16 cm

"I

Fig. 1 Geometry of the masonry specimens and three-pointbending scheme.

Fig. 2 Testing apparatus and crack mouth opening contro Isystem.

critical value of crack opening displacement Wc and thenormal stress vanishes (Fig. 4a). On the other hand, thefictitious crack tip is defined as the point where thenorma] stress attains the maximum value tI and the crackopening vanishes (Fig. 4a).

Table 1 Specimen details

Specimen Size Notch depth Notch(10-2 m) (10-2 m) thickness

(10-2 m)L H B a a/H

Al 20 4 2.0 0.4 0.100 0.1A2 20 4 2.2 1.1 0.275A3 20 4 2.0 1.9 0.475B1 20 4 1.9 0.4 0.100 0.1B2 20 4 1.9 1.1 0.275B3 20 4 2.3 1.9 0.475Cl 20 4 1.9 0.4 0.100 0.1C2 20 4 2.0 l.1 0.275Cl 20 4 2.0 1.9 0.475

Density(kg m-3)

Young's Tensilemodulus strength(MPa) (MPa)

1900 2.2

1880 1.5

4900 10.0

1718

1750

1809

366 Bocca et al.

strain, é

bti1ftIl....1ft

opening, w

Fig. 3(a) Stress-strain law of the material outside the damagezone and (b) stress-crack opening displacement law of thedamage zone.

(

I

Ii

tt I 't\<, /

'\ r7Wc

)

<a)

../<,/'<,./

tiP node<,/'<,

-, ./<,./<,./<,./<,./r-,./<,

Fig. 4(a) Stress acting across the cohesive crack and (b)equivalent nodal forces in the finite-element mesh.

The closing stresses acting on the crack surfaces (Fig.4a) can be replaced by nodal forces (Fig. 4b). Theintensity of these forces depends on the opening of thefictitious crack, w, according to the (T-W constitutive lawof the material (Fig. 3b). When the tensile strength f isachieved at the fictitious crack tip (Fig. 4b), the top nodeis opened and a cohesive force starts acting across thecrack, while the fictitious crack tip moves to the nextnode.

With reference to the three-point bending test(TPBT) geometry in Fig. 5, the nodes are distributedalong the potential fracture line. The coefficients ofinftuence in terms of node openings and deftection arecomputed by a finite-element analysis where the ficti-tious structure in Fig. 5 is subjected to (n + 1) differentloading conditions. Consider the TPBT in Fig. 6a withthe initial crack tip in the node k. The crack openingdisp1acement at the n fracture nodes may be expressedas follows:

w = IkIF+cP+ l'

)

(b)

On the other hand, the initial crack is stress-free andtherefore

i=I, ... ,(k-l) (2a)

while at the ligament there is no displacement discon-tinuity:

i = k, (k+ 1), ... , n (2b)

Equations 1 and 2 constitute a linear algebraic systems of2n equations in 2n unknowns, i.e. the elements ofvectors w and F. If load P and vector Fare known, it ispossible to compute the beam deftection, 8:

(3)

(1)

where Dp is the deftection for P = 1 and D'I is thedeftection due to the specimen weight.

After the first step , a cohesive zone forms in front ofthe rea1 crack tip (Fig. ob), say between nodesj and l.

Materials and Structures 367

ft----nDde n

node j

Fj-----t t<--- Fj

nodel

Fig. 5 Finite-element nodes along the potential crack line.

Ft: F~nodek (a)

/'

PO-k+l )

node j(b)

Fig. 6 Cohesive crack at (a) the first loading step and (b) the(1- k + l)th loading step.

Then Equations 2 are replaced by

F; = O i = 1,2, ... , (j-1)

F; = Ft(1- w/wc) i = j, (j + 1), ... ,1

(4a)

(4b)

(4c)i = 1, ... , n

where F, is the uItimate-strength nodal force:

Equations 1 and 4 constitute a Iinear algebraical sys-tem of (2n + 1) equations and (2n + 1) unknowns, i.e.the elements of vectors w and F and the externalload P.

At the first step, the cohesive zone is missing(l = j = k) and the load P1 producing the uItimate-strength nodal force F, at the initial crack tip (node k) iscomputed. Such a value Pç, together with the related

(5)

deflection 01 computed through Equation 3, gives thefirst point of the P-O curve. At the second step, thecohesive zone is between the nodes k and (k + 1), andthe load P2 producing the force F; at the second fictitiouscrack tip (node k + 1) is computed. Equation 3 thenprovides the deflection 02' At the third step, the fictitiouscrack tip is in the node (k + 2), and so ono The presentnumerical program simulates a loading process wherethe controlling parameter is the fictitious crack depth.On the other hand the real (stress-free) crack depth,external load and deflection are obtained at each stepafter an iterative procedure.

The program stops with the untying of node n and,consequently, with the determination of the last coupleof values F; and on- In this way, the complete load-deflection curve is automatically plotted by the com-puter.

368 Bocca et al.

'\

A,j -;7I I-'

•f(.4

VI A2----;:::7

(\ ~

1\

V ~\l' \\\ ~

r/ \Ì\\ A3:7

\\.~r.r---

Fig. 7 Finite-element mesh (n = 39).

p( kg)

26

24

22

20

18

16

14

12

10

8

6

4

2

o 0.1 0.2 0.4 os

Fig. 8 Load-deflection diagrams obtained experimentally onmasonry ofthe 19th century (material A).

In the case under consideration, the finite-element Isimulation was realized with the mesh sketched in Fig. 7(n = 39), using constant-strain triangles.

4. DISCUSSION

The fracture energy CF is obtained frorn the load-deflection curves (Figs 8 to lO), taking into account thearea under the curve divided by the ligament area. TheCF values are reported in Table 2. It is worth noting thatthey are varying with the material and turo out to bealmost independent of notch depth. Such values are ofthe same arder of magnitude as those of standard con-cretes [9].

0.6 0.7

d (mm)

o.a

Materials and Structures

Ip

( kg)

28

369

A

I B1

I~

J

I/ B2VV ~

"

/V\t 1\V/\ '\ B3

V1/~ 1\ ~

l \~<,r-,

24

22

20

18

18

14

12

10

8

6

4

2

o 0.1 0.3 0.4 0.50.2

Fig. 9 Load-deflection diagrams obtained experimenta\ly onmasonry ofthe 19th century (material B).

The criticai value of stress-intensity factor is derivedfrom the following well-known relation [9]:

Kic = (GFE)1I2 (6)

The K1C values, listed in Table 2, are slightly lower thanthose relating to concrete, due to the smaller Young'smodulus of bricks.

In Figs 8 to lO it is shown how the softening branchvaries with the initial notch depth in the brick specimen.Specimens with shallow notches hehave hrittler thanthose with deeper notches.

The nurnerical sirnulation , performed with the cohe-

0.6 OB0.7

d (mm)

Table 2 Experimental results

Specimen CF CF mean K1c KIC mean(N m-I) value (kN m-3/2) value

(N m-l) (kN m-3/2)

Al 52.40 52.20 315.5 313.9A2 41.30 280.1A3 63.10 346.2

B1 33.80 37.20 252.0 263.7B2 32.40 246.8B3 45.50 292.4

Cl 59.50 52.10 540.0 505.0C2 50.20 496.0C3 46.70 478.0

Portland lO to 100 500 to 2000

370 Bocca et al.

p( kg)

~

1/ c,VI

Iir~vC2

/'I

VI 1/

hcli !P

~v~- 'Il \~y \

65

60

55

50

45

40

35

30

25

20

15

10

5

o 0.1 0.3 0.4 0.50.2

Fig. lO Load-defìection diagrams obtained experimentally onmasonry ofthe 20th century (material C).

sive crack model previously described, provides theload-deflection diagrams plotted in Fig. 11. Theoreticaland experimental results present very similar softeningbranches. For the specimens with shallow initial notch(A 1, B l, Cl) a snap-back instability is predicted by themode!. Such an instability is revealed by the ex-perimental investigation. As a matter offact, the soften-ing branch presents a positive slope in these cases and,whilc load and dcfìcction both decrease , in the mean-rime thc crack mouth opcning displaccmcnt (which ist hc controlling pararnctcr ) and crack lcngth incrcasc. Aphcnorncnon which is cat astrophic and unstablc in na-

0.6 0.7

d(nlnl)

OB

ture is made stable in practice by controlling a monotoni-cally increasing function of time. It is therefore demons-trated that the cohesive crack model describes brickmasonry failure as accurately as in the case of concrete.

The ratio of the experimental peak load to the max-imum load derived from a linear elastic fracture mecha-nies (LEFM) analysis, Pcxp/PLEFM, is represented in Fig.12 as a function of the relative crack depth a/H. Such aratio is nearly constant for each material and alwayslower than unity. LEFM, i.e. the criticai conditionK, = K!C, is not a conservative criterion when the speci-rncn sizes arc cornparativcly srnall [<)-121. In addition to

a r eI

o I oo oZ N I z N

/,CL /, CL

/,o o /, o oCI: o CI: ~o o-.J -.J

oo 2 3 4 2 3 4

LORDING POINT DISPLRCEMENT <5 (mxlO-4) LORDING POINT DISPLRCEMENT <5 -4(mx i o )

b fo oo o

Z N Z N

CL CL

o o o oCI: o CI: oo o-.J -.J

2 3 4 2 3 4-4

(mxlo-4)LORDING POINT DISPLRCEMENT <5 (mxlO ) LORDING POINT DISPLRCEMENT o

C o9 o

(JJ

ooZ N Z ooCL CL e-

o o oCI: o CI: oo o o-.J -.J

N

2 3 4 2 3 4

LORDING POINT DISPLRCEMENT <5-4 LORDING POINT DISPLRCEMENT <5 (mxlo-4)(mx lO )

d h :s(JJ

ooZ N Z ooCL CL

.,.

o o oCI: o CI: oo o o-.J -.J N

2 3 4 2 3 4

LORDING POINT DISPLRCEMENT <5 -4 LORDING POINT DISPLRCEMENT <5 (mxlo-4)(mx ì n ) :

ao(JJ

Fig. Il Load-deflection diagrams: comparison between (---) z oexperimental and (-) numerical results for cohesive crack o

CL -e-

rnodel. Specimen (a) Al, (b) A2, (c) A3, (d) Bl, (e) B2, (f)B3, (g) Cl, (h) C2, (i) C3. o

CI: oo o-.J N

2 3 4

LORDING POINT DISPLRCEMENT o -4(mxlO )

372 Bocca et al.

a :L b :LU. U.W w o...I o ...I

a.. a..-, ...: <,a.. a..x x *w wa.. a..

* * *LI) LI)

:L o :L oU. u,w W...I ...I

a.. a..-, -,tIl tIlW WI Io ou ua.. o 0.5 1.0 a.. o 0.5 1.0

RELRTIVE CRRCK DEPTH a/H RELRTIVE CRRCK DEPTH a/H

C :LU.W...I Oa.. ...:-,a..xw

*a..*

*LI)

:L Ou,

Fig. 12 Ratio of the fictitious to the real fracture toughness:w...I

a..comparison between (*) experimental and (-) numerical -,

tIlresults. (a) MaterialA:f,/E = 0.00116, Hf,/Gf = 1685. (b) w

IMaterialB:f,/E = 0.00134, Hf,/GF = 2688. (c) Material C: O

uf,/ E = 0.00204, Hf,/GF = 5000. a.. O 0.5 1.0

:L b :La u, u,w W...I o ...I oa.. a..-, -,a.. a..x xw w

*a..*

a..

* *lJ1

:L ou,W...I

a..-,tIlWIOUa..

2000 4000 5000o 1000 3000

DIMENSIONLESS SIlE: H ft/Gf

c ~W...I oa..-,a..xwa..

Fig. 13 Size transition towards brittleness and LEFM:comparison between (*) experimental and (-) numericalresults. a/H = (a) 0.1, (b) 0.275, (c) 0.475. For material A,f,/E = 0.00116; forB, 0.00134; forC, 0.00204.

lJ1

:L ou,W...I

a..-,tIlWIOUa..

1000 2000 3000 4000 5000o

DIMENSIONLESS SIlE: H ft/Gf

*lJ1

:L ou,W...I

a..-,tIlWIoua..

O 1000 2000 3000 50004000DIMENSIONLESS SIlE: H ft/Gf

Materials and Structures 373

the experimental data, the ratio of the theoretical cohe-sive peak load to the maximum LEFM load, Pcohe/

PLEFM, is plotted in Fig. 12 as a function of the relativecrack depth a/H. The theoretical curves approximate theexperimental data very accurately. For initial notchdepths lower or higher than those selected for the ex-perimental investigation, a decrease of the ratio Pexp/

PLEFM is expected. The theoretical curve Pcohe/PLEFM

thus presents a bell-shape, achieving a maximum valuefor a/H=O.3 and tending to zero for a/H----'>O+ anda/H ----'> r. Such a result may also be predicted by ap-plying dimensional analysis and considering the com-petition between brittle fracture collapse due to stress-intensification and ultimate strength collapse at the liga-ment [9-12].

The two non-dimensional numbers Ct = NE and 11SE = HNGF are reported in the legend of Fig. 12. Theratio Pexp/PLEFM unequivocally increases by increasingsuch numbers. This is due to the high notch sensitivityand brittleness that the structural response shows formaterial C (Fig. 10). The peak load may be predicted byLEFM much more accurately for specimens C than forspecimens A. On the other hand, the bell-shaped curvesfor material A would rise if larger specimens wereutilized. The physical behaviour is in fact similar if thebrittleness number HI/GF is the same. We can expectthe behaviour of material A to be as brittle as that ofmaterial C when the size-scale of specimen A is aboutthree times that of the present investigation, i.e.H= 12cm.

The same ratios previously discussed, PexpfPLEFM andPcohe/PLEFM, are reported in Fig. 13 as functions of thedimensionless size Hlt/GF. One numerical curve foreach material, i.e. for each dimensionless number NE,and for each crack depth a/H, is plotted in Fig. 13. Thecurves related to the same crack depth are very dose, themaximum load provided by the cohesive crack mode!being not very sensitive to the ratio It/E. In each caseselected for the investigation, an evident transitionappears from an ultimate strength collapse at the liga-ment

Pl 6O"nom = 4 b(H _ a)2 = I,

to a brittle crack propagation due to stress-intensifica-tion

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REFERENCES

(7)

1. Page, A. W., 'An experimental investigation ofthe biaxialstrength of brick masonry', in Proceedings of 6thI.B.Ma.C., Rome, May 1982, pp. 3-19.

2. Bocca, P. and Levi, F., 'Indagini sistematiche su muraturain blocchi di argilla espansa', Quaderni Anpae Nr. 3(1983) 3-10.

3. Drysdale, R. G. and Hamid, A. A., 'Tension failurecriteri a for plain concrete masonry', J. Struet. EngngASCE 110 (2) (1984) 228-244.

4. Carpinteri, A., 'Interpretation of the Griffith instability asa bifurcation of the global equilibrium', in Proceedingson 'Application of Fracture Mechanics to CementitiousComposites', NATO Advanced Research Workshop,September 1984, Northwestern University, edited by S.P. Shah (Nijhoff, 1985) pp. 287-316.

5. Hillerborg, A., Modeer, M. and Petersson, P. E., 'Analy-sis of crack formation and crack growth in concrete bymeans offracture mechanics and finite elements', Cem.Concr. Res. (1976) 773--782.

6. Scott, M. W. and Abrams, D. P., 'Mechanics ofmasonryin compression', J. Struet. Engng ASCE 111 (4) (1985)857-870.

7. Carpinteri, A., Marega, C. and Savadori, A., 'Size effectsand ductile-brittle transition of polypropylene', J. Ma-ter. Sci., 21 (1986) 4173-4178.

8. RILEM Draft Recommendation 'Determination of thefracture energy of mortar and concrete by means ofthree-point bend tests on notched beams', Mater.Struet. 18 (106) (1985) 285-290.

9. Carpinteri, A., 'Static and energetic fracture parametersfor rocks and concretes', ibid. 14 (1981) 151-162.

10. Idem, 'Notch sensitivity in fracture testing of aggregativematerials', Engng Fraet. Meeh. 16 (1982) 467-481.

11. Idem, 'Application of fracture mechanics to concretestructures', J. Struet. Div. ASCE 108 (1982) 833-848.

12. Wecharatana, M. and Shah, S. P., 'Prediction of nonlinearfracture process zone in concrete', J. Engng Meeh.ASCE 109 (1983) 1231-1246.