ON THE APPLICABILITY OF FRACTURE MECHANICS TO MASONRY

Transcription

1 1027 ON THE APPLICABILITY OF FRACTURE MECHANICS TO MASONRY PIETRO BOCCA Associate Professor Istituto Universitario di Architettura di Venezia Tolentini VENEZIA - ITALY ALBERTO CARPINTERI Full Professor Dipartimento di Ingegneria Strutturale politecnico di Torino C.so Duca degli Abruzzi TORINO SILVIO VALENTE Graduate Research Assistant Dipartimento di Ingegneria Strutturale politecnico di Torino C.so Duca degli Abruzzi TORINO ABSTRACT The article explores the possibility of applying to masonry a number of concepts formulated in the field of Fracture Mechanics. The problem of mode 1 failure (opening) is examined. Fracture energy G F and criticai value of the stress intensity factor, K IC ' are determined for masonry specimens tested in bending with different notch depths. The experimental results are compared with values, obtained through a cohesive crack numerical simulation developed originally for concrete. NOTATION Alig = ligament area a = crack length E = Young's modulus ft = ultimate tensile strength G F fracture energy K IC = stress-intensity factor criticai value 1 = support span L,H,B = length, depth, thickness of the specimen P = load o = loading point displacement V = Poisson ratio

2 Wo = o = w W c = {w} = [k] = {F} = {c} = P = {r} = 1028 area under the P-ô curve stress acting on the crack surfaces crack opening displacement critical value of the crack opening displacement vector of the crack opening displacements matrix of the coefficients of influence (nodal forces) vector of the nodal forces vector of the coefficients of influence (external load) external load vector of the crack opening displacements due specimen weight. to the INTROOUCTION After extensive experimental investigation [1-3], where masonry specimens have been tested in uniaxial and biaxial loading conditions, close to the real situations, it is clear today how failure occurs in brick masonry. Moreover, cracking evolution (from microcracking to macrofracture) and failure may be studied by means of recent1y established experimental techniques. The application of fracture mechanics, whose concepts are already used for steel and concrete structures successful1y [4,5], is extended herein to brick masonry, so that the theoretical results can fit with the real situations satisfactorily. As a matter of fact, when dissipative phenomena play an important role (as, for example, in the case of masonry structures in sismic zones), the energy criteria analyze failure better and more realistically than the stress criteria. In the present paper, the problem of failure in mo de 1 (opening) is faced. Structura1 elements working in compression, generally undergo this mode of fracture [6]. Fracture energy GF and critical value K IC of stress-intensity factor [4] are evaluated for bending specimens, directly drilled from historical bricks. Such experimental data, obtained for different notch depths, are then compared with numerica1 results coming from a cohesive crack finite element simulation, originally developed for concrete [4]. SPECIMEN PREPARATION ANO EXPERIMENTAL PROCEOURE The specimens' features (Fig.1) are listed in Tab.1. The capital letters A, B, C are referred to three different structures, built respectively in the 19th (A,B) and 20th (C) century. Three point bending tests, according to the RILEM Reccomendation for concrete [7], were carried out, through a MTS machine with maximum load up to 20 kn. Load-deflection (P-ô) diagrams were plotted by controlling the crack mouth opening displacement. The latter was increased at the constant rate m/s (Fig.2).

3 ,..." Mate= Sizes Notch Notch Density Young's ria1 depth thickness modu1us 10 2 [m] 1õl [m] 10 2 [m] [Kg/m 3 ] [MPa] L H B a a/h A A A B B B C C C Tab. 1. Specimens' features... L- 20 em.' B"2an// / H.4cmI At V ta HI f ~ ~,- I.. 16em -, Fig. 1. Geometry of the Historica1 Masonry Specimen and Testing Scheme

4 1030 Fig.2. Testing Scheme and Notch Mouth Opening Control System COHESIVE CRACK MODEL The cohesive crack model, developed originally for concrete, is based on the following assumptions [4] [S] : (1) The cohesive fracture zone (plastic or process zone) begins to develop when the maximum principal stress achieves the ultimate tensile strength ft (fig. 3-a.). (2) The material in the process zone is partially damaged but still able to transfer stress. Such a stress is dependent on the crack opening displacement w. A bilinear o-w law is assumed as in fig. 3-b. The real crack tip is defined as the point where the distance between lhe crack surfaces is equal to the criticai value of crack opening displacement W c and the normal stre~s vanishes (fig.4-a). On the other hand, the fictitious crack t~p is defined as the point where the normal stress attains the maximum value ft and the crack opening vanishes (fig.4-a). The closing stresses acting on the crack surfaces (fig. 4- a) can be replaced by nodal forces (fig.4-b). The intensity of these forces depends on the opening of the fictitious crack, W, according to the o-w constitutive law of the material (fig. 3- b). When the tensile strength ft is achieved at the fictiti~us crack tip (fig.4-b), the top node is opened and a cohes~ve force starts acting across the crack, while the fictitious crack tip moves to the next node. With reference to the three point bending test (TPBT) geometry in fig.s, the nodes are distributed along the potential fracture line. The coefficients of influence in terms of node openings and deflection ~ re computed by a finite

5 1031 element analysis where the fictitious structure in fig.s. is subjected to (n+1) different loading conditions. Consider the TPBT in fig.6-a with the initial crack tip in the node k. The crack opening displacement at the n fracture nodes may be expressed as follows: {w} = [k]{f} + {c}p + {r} (1 ) On the other hand, the initial crack is stress-free and therefore: Fi = 0, for i = 1,...,(k-1) (2-a) while at the ligament there is no displacement discontinuity: wi = 0, for i = k,(k+1),..,n ( 2-b) Eqs. (1) and (2) constitute a linear algebraic system of 2n equations in 2n unknowns, i.e. the elements of vectors {w} and {F}. rf load P and vector {F} are known, it is possible to compute the beam deflection, 5: 5 = {ClT {F} + Dp P + Dy (3) where Dp is the deflection for P 1 and Dyis the deflection due to the specimen weight. After the first step, a cohesive zone forms in front of the real crack tip (fig.6-b), say between nodes j and 1. Then eqs (2) are replaced by: Fi = Fi = F t (l-wi/wc) wi = where F t is the ultimate F t for i 1,2,...,(j-1) for i = j,(j+1),...,1 for i = 1,...,n strength nodal force: = ft H/(n+l) (4-a) ( 4-b) (4-c) Eqs (1) and (4) constitute a linear algebrical system of (2n+1) equations and (2n+1) unknowns, i.e. the elements of vectors {w} and {F} and the external load P. At the first step, the cohesive zone is missing (l=j=k) and the load P1 producing the ultimate strength nodal force F t at the initial crack tip (node k) is computed. Such a value P1' together with the related deflection 51 computed through Eq.(3), gives the first point of the P-5 curve. At the second step, the cohesive zone is between the nodes k and (k+1), and the load P2 producing the force F t at the second fictitious crack tip (node k+1) is computed. Eq. (3) then provides the deflection 52. At the third step, the fictitious crack tip is in the no de (k+2), and so on. The present numerical program simulates a loading process where the controlling paramet"er is the fictitious crack depth. On the other hand, the real (stress-free) crack depth, external load and deflection are

6 1032 obtained at each step after an iterative procedure. The program stops with the untieing of the no de n and, consequentely, with the determination of the last couple of values Fn and õ~. In this way, the complete load deflection curve is automat~cally plotted by the computer. ft b! ; b :.. i straln, 8 openlng, W Fig. 3. Stress-Strain law of the material outside the damage zone (a) and stress-op~ng displacement law of the damage zone (b) ( ft f. W c ) tlp node./..../..../... /'...,./..../ )... /'...,...,././...,./ I"'--. (a) (b) Fig. 4. Stresses acting across the fictitious crack (a) and equivalent nodal forces in the finite element mesh (b).

7 1033 node n node i Fi ---+ P'---Fi node1 Fig. 5. Finite element nodes along the fictitious crack line. ~ F t nodek (a) /' PU-k+1 ) ~ F t node I node j (b) Fig.6. Fictitious crack line at the first loading step (a) and (l-k+l)th loading step (b).

8 1034 n 38 Fig. 7. Finite element mesh (n=38) P ( kg) O 0.1 Q.2 03 OA o.s o.e 0.7 o.a d (mm) Fig. 8. Load vs Displacement experimental diagrams of the 19th centu~ masonry (A).

9 1035 In the case under consideration, the simulation was realized with the mesh sketched in Fig. 7 (n=38), using finite elements of the constant strain triangles type. Specimen properties are listed in Tab.1. P C kg) 2& A 20 1a 1& a li 4 2 I I B, ;7 ~ j B 2 7 /,..,/!/ J r\ B3 I / VI 1\ \ Y' It ~ \ \ ~I/ I~ ~ ~ ~ o 0.1 o.z Q3 OA & 0.7 o.a dcmm) Fig. 9. Load vs Displacement experimental diagrams of the 19th century masonry (B).

10 P 65 ( kg) &O o OA CUS o.a Q7 08 d (mm) Fig. 10. Load vs Displacement experimental diagrams of the 20th century masonry (e).

11 1037 Material A A A B E B C C C Portland I Gt:" Gfmean value KI C Klc mean value [N/m] [N/m] [N/m 3 1] [N/m 3 1] Tab. 2. Experimental Results. (a) Z o... N Q a: o -.J I / í / \ \ I ~ \ \ COHESIVE CRRCK MODEL EXPERIMENTRL VRLUES LORDING POINT DISPLRCEMENT:Ó (M/10.000)

12 (b) í ] z (\J!l. o a: o -.J / l,.' A-' EXPERIMENTRL " COHESIVE CRRCK MODEL VRLUES (c ) z (\J!l. 234 LORDING POINT DISPLRCEMENT:Ó (M/10.000) EJ COHESIVE CRRCK MODEL EXPERIMENTRL VRLUES o a: o -.J,-:; 234 LORDING POINT DISPLRCEMENT:Ó (M/ ) Fig. 11 (a) (b) (c). Comparison between experimental and theoretical results DISCUSSION The fracture energy G F is obtained from the load-deflection curves (Figs. 8-10), taking into account the are a under the curve divided by the 1igament area. The G F values are reported in Tab. 2. It is worth noting that they are varying with the material and result to be almost independent of specimen size. Such values are of the same order of magnitude as those of standard concretes [8]. The critical value of stress-intensity value is derived from the following well-known relation [8]: K 1C = JGF ' E'J

13 " 1039 The K IC values, listed in Tab.2, are slightly lower than those related to concrete, due to the smaller Young's modulus of bricks. In Figs.8-10 it is shown how the softening branch varies with the initial notch depth in the brick specimen. Specimens with shallow notches behave brittler than those with deeper notches. The numerical simulation, performed with the cohesive crack model previously described, provides the diagrams in Fig.ll, which are related only to bricks of type A. Theoretical and experimental results, summarized in Fig.11, provide very near softening branches. Therefore, it is proved that that the cohesive crack model describes masonry failure so accurately as in the case of concrete. REFERENCES [1] Page A.W. "An experimental Investigation of the Biaxial Strength of Brick Masonry" 6th I.B.Ma.C., Rome, May, 1982, pp [2] Bocca P., Levi F. "Indagini sistematiche su muratura in blocchi di argilla espansa" Quaderni Anpae, nr.3, 1983 pp [3] Drysdale R.G., Hamid A. A. "Tension Failure Criteria for Plain Concrete Masonry". Journal of Structural Engineering, ASCE, Vol. 110, Nr.2, 1984, pp [4] Carpinteri A., Interpretation of the Griffith instability as a bifurcation of the global equilibrium. Application of Fracture Mechanics to Cementitious Composites. NATO Advanced Research Workshop. September 4-7, 1984, Northwestern University, edited by S.P. Shah, Martinus Nijoff Publishers, (1985) pp [5] Hillerborg A., Modeer M., Petersson P.E., Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements, Cement and Concrete Research 6, (1976) pp [6] Scott Mc Nary W., Abrams Daniel P. - "Mechanics of Masonry in compression" - Journal of Structural Engineering, ASCE, Vol 111, Nr. 4, April 1985 pp [7] Rilem Draft Reccomendation "Determination of the fracture energy of mortar and concrete by means of threepoint bend tests on notched beams" Materials & Structures, Vol. 18, nr 106, 1985, pp [8] Carpinteri A., Static and energetic fracture parameters for rocks and concretes, Materials & Structures, R.I.L.E.M., Vol.14, 1981, pp