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Investigation of Fracture in Reinforced Concrete Masonry Walls

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Abstract— The study of fracture of materials under compression is a comparatively new field. So there are quite more to discover in this sphere. The basic concept of fracture of materials under compression such as concrete, masonry, rock etc. has been developed though there are many disputes regarding various topics yet. In masonry it is still unknown what should be the relation between the compressive strength of the material and the tensile stress that causes the first cracking. This paper made an attempt to find out this relation by examining various test results of reinforced concrete masonry walls. The tensile strength while the first cracking is determined assuming two types of flaws in the masonry: spheroidal and cylindrical. Study is also made to find the types of flaws that might actually exist in masonry. It was found that the ratio of compressive to tensile strength of masonry might range from 3-10 as obtained from the results reported herein. Again spheroidal assumption seem to give close result to the actual tensile strength.

Index Term— Compression Fracture, Cohesive Strength, Cracking Stress, Flaw, Grouting.

I. INTRODUCTION

Masonry is one of the most important construction materials in the history of mankind. It has been used as the basic construction material for public and residential buildings in the past several thousand years. But still there is a lot to understand about this material. The behaviour of masonry is quite complex as it is a composite of block, grout, reinforcement etc. Among the features that are yet to unravel, the fracture of masonry is one of the prime factor that need to be understood clearly in order to evaluate various property related to fracture such as the initiation and propagation path of fracture. Cracks usually initiate from flaws present in the material. There are different types of flaws assumed to exist in materials. Among them the most common shapes are the cylindrical and the spheroidal type of flaws. But it cannot be said confidently the type of flaw present in the masonry. Again crack initiation in masonry also involves tensile strength that exceeds the tensile capacity of the material. So in this paper the tensile capacity while the initiation of cracks will be determined for reinforced concrete masonry walls assuming two types of flaws mentioned earlier and then it will be discussed which types of flaws might be an exact

Nusrat Hoque is with the Department of Civil Engineering, Chittagong University of Engineering & Technology, Chittagong, Bangladesh Nusrat Hoque, Bangladesh. (First author is the corresponding author,

phone: +880-1712173544; e-mail: nusrat_hoque@yahoo.com).

assumption for masonry. Again the relationship between the compressive strength of masonry and the tensile capacity will also be discussed.

II. LITERATURE REVIEW

In uniaxial compressive testing of brittle specimens, the failure is accompanied by numerous cracks parallel to the compression direction. The phenomena of multiple cracking and crack direction are in direct contrast with the tensile fracture. From the two basic criteria of fracture it is obvious that there must be tension to break the inter-atomic bonds and the release of a sufficient amount of energy should occur. There is a factor that is common to both tensile as well as compression fracture and that is the presence of a void or flaw. If a spherical particle in an infinite medium is an air void, tension develops parallel to the surface at the top of the void, which is perpendicular to the applied compression. That is the tension that is required for the crack to develop. Similar stress concentrating abilities can be observed for different shaped voids. Many voids seem to have similar effects in compression. As a result, cracks can be initiated at a number of voids rather than a particular one and that results in multiple cracking. It is generally recognized that flaws such as cracks, pores, voids and fissures exist in materials even before the application of load [1]. These flaws act as stress raisers and the crack can initiate in the compressive stress fields.

Many structural elements, such as shear walls, are subjected to multi-axial stress states, rather than purely uniaxial. Therefore, the strength of concrete, masonry and stone, when subjected to stress states other than uniaxial, is of importance. In biaxial stress states, loads are applied in a single plane. The application of shear loads reduces the principal stresses which results in a pair of principal stresses and the principal stresses can be determined from Mohr’s circle. The principal stress in the third principal direction becomes zero. The crack patterns for brittle materials under varying biaxial stress shown in Fig.1 and Fig. 2 in a general form where cracking is perpendicular to the principal tensile stress though this pattern can be changed due to the presence of a particular weak plane.

Cylindrical and spheroidal shaped voids are two types of flaw that might be present in masonry such as concrete. It remains unknown whether the flaws present in masonry are cylindrical or spheroidal in shape. The following paragraphs discuss the stress due to these two types of flaw and the computation of cracking stress will be carried out for both before comparing the results to see which one better fits the assumption.

Investigation of Fracture in Reinforced

Concrete Masonry Walls

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A. Effect of Circular Holes on the Stress Distribution in a Plate

If a plate with a circular hole in the middle of it is subjected to uniform tension, S, as shown in Fig.3, the stress distribution in the neighborhood of the hole increases though it remains unchanged at a distance away which is large compared to the radius of the circle.

The stress at the edge of the hole can be found from the solution given by Kirsch’s equation as mentioned by

Fig. 2. a) Biaxial Tension-Cracking is Perpendicular to the Tension and Parallel to the Compression b) When the Second Stress is Small (Compared With the First) the Cracking is Parallel to the Main Compression. c) When Two Compressions are Applied the Cracking is Parallel to Both.

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Timoshenko and Goodier [2]   2 2 2 4 2 2 2 cos ) 4 3 1 ( 2 ) 1 ( 2 r a r a s r a s

r      (1)

  (1 3 )cos2

2 ) 1 ( 2 4 4 2 2 r a s r a

s

 (2)

  (1 3 2 )sin2

2 2 2 4 4 r a r a s

r    (3)

Where ‘a’ is the radius of the hole. Therefore,

σθ=3S at the boundary at m and n and σθ = -S at the boundary

at p and q.

Where S=Stress Applied on Plate.

The maximum tensile stress at points m and n is three times the stress applied S, while at points p and q the stress, –S, is a compressive stress of the same magnitude as that developed around the hole in the direction parallel to the applied stress. So if a compressive stress of value S1 is applied in the

direction perpendicular to the tensile stress then a tensile stress of magnitude S1 would develop at points ‘m’ and ‘n’. So the

maximum tensile stress in the plate would then be equal to S1

+3S at points m and n. The circular flaw in a solid body becomes cylindrical in three dimensional spaces.

B. Effect of Spheroidal Holes on Stress Distribution

If a spheroidal flaw is present in a solid body as shown in Fig.4, under the externally applied stresses S1, S2 and S3, the

resulting stress at the surface of the flaw due to the presence of the flaw can be estimated by the formula given by Goodier [3]. The stress σ22 at point A is then given in (4).

σ22(In direction of

B)=

( 3 15 ) (13 5 ) ( 3 15 )

10 14 1 3 2 1

2 S    S   S   

S

(4)

Where S1, S2 and S3= Stress applied in three orthogonal

direction of a solid body.

So the stress is found to be dependent on Poisson’s ratio but not on the modulus of elasticity where Poisson’s ratio, υ=0.15-0.2 for masonry.

III. TENSILE STRENGTH WHILE CRACKING

The modified version of Griffith’s theory is that fracture starts when the tensile stress induced at or near the tip of an existing crack exceeds the molecular cohesive strength of the material. Since the molecular cohesive strength is difficult to determine by direct measurement the fracture criterion is in terms of the uniaxial tensile strength of the material [4]. So the initiation of diagonal shear cracks in masonry walls requires the presence of tensile forces as known from the concept of cohesive strength as mentioned above. However, the tensile stress that initiates cracks in masonry is an issue that needs more study.

In this section, the force or stress causing first cracking and final cracking (while failure), as reported by various researchers, and according to the test results of walls obtained from various sources, are examined and an attempt is made to determine the tensile stresses of masonry using the concept of cylindrical and spheroidal cracks in masonry .The walls that are discussed here in this paper are all built of concrete blocks and the length and height remains in the range of 700 mm-1800 mm. All were tested under in-plane cyclic loading under the vertically applied stress to assess the effect of various parameters in walls in resisting in-plane shear. The first cracking horizontal load, the horizontal load causing failure of the wall and masonry compressive strength are obtained from the test results. The determination of tensile strength at the time of cracking requires the computation of the principal stresses applying the procedure described in literature review. The cracking tensile strength is dependent on the vertical stress as well.

A. Analysis of Results for First Cracking Stress

For the purposes of this analysis, data of Schultz [5] and Yancey [6], Sveinsson et al. [7] are selected where the first cracking loads were reported. These data are then grouped into partially and fully grouted. All the data that are used in the analysis were tested under fixed-fixed support conditions at the top and bottom. The tensile strengths computed are plotted

Fig. 3. Effect of Circular Hole in a Plate (Timoshenko & Goodier, 1951)

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in Fig.5. As shown in the figure, in all cases the spheroidal equation gives a lower estimation of tensile stress at cracking than the cylindrical assumption, irrespective of the type of grouting and for the partially grouted walls the calculated cracking stresses are much lower than for the fully grouted walls. The data are also analyzed to identify a relationship between compressive strength of masonry and the tensile strength at first crack as shown in Table I.

From the table it is obvious that fully grouted walls showed greater coefficient of variation than partially grouted walls. Although a spheroidal assumption of the initiating flaw gives lower values than for a cylindrical flaw type, it also gives slightly higher coefficient of variation. Although the COVs are quite high, this could be due to the high COV of the masonry compressive strength itself. The walls that are tested under high vertical stresses give high cracking stress which in turn gives high principal stress and as a result, high fracture tensile stress. For this reason some high peaks can be observed in Fig.5.

It is does not appear that any particular ratio of masonry compressive strength to tensile cracking strength can be identified for masonry unlike the value obtained for concrete which varies between 5 and 10. This ratio remains low for fully grouted walls than that of partially grouted walls. However, it may be that with further research, a range could be proposed for masonry as well.

B. Analysis of Results for Final Cracking Stress

In this section data from different researchers have been analyzed to find the ratio of masonry compressive strength to that of tensile strength while the failure that means the tensile strength related to final load before failure. For this purpose test data obtained from Voon [8], Matsumura [9], Shing [10], Tomazevic & Lutman [11], Sveinsson [7] have been chosen. These tests vary in end conditions in testing and there are differences in changes of parameter too. The ratio of compressive strength to tensile strength is presented in Table 2 with standard deviation as well as the coefficient of variance. As can be seen from the table normalization of the masonry compressive stresses with respect to the tensile strength deteriorates the COV. So it can be said that the concept of cohesive strength may be an effective approach for determining the cracking stress, however, masonry compressive strength depends on the conditions in the compressive strength test (specimen size, capping, etc.) Therefore normalizing with compressive strength may not be the best approach to estimating cracking strength. The increase in variability when using the cohesive strength concept for the stresses at a flaw could be due to several factors – alone or in combination. For example, the compressive strength is determined on a specimen that is confined between steel platens and involves not just crack initiation, but also crack propagation before failure. Thus the “strength” reflects the boundary conditions on the specimen, the amount of

Fig. 5. Plot of Tensile Stress for Different Types of Flaw and for Different Grout Conditions

TABLE I

COMPRESSIVE STRENGTH TO TENSILE STRENGTH RATIO FOR FULL AND PARTIAL GROUTING

Grouting

type Flaw type

Ratio of Masonry Compressive strength to tensile strength

Standard

Deviation COV

COV of Masonry Compressive Strength

(%) Partial Cylindrical 5.7 1.2 21.5 28.9

Spheroidal 9.5 2.1 22.4

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confinement, the material used at the specimen steel interface etc. as well as crack initiation. Fig.6 presents the masonry tensile strength of the walls at failure tested by different researchers and from the figure it can be said that the cylindrical assumption gives higher estimation of tensile strength than the spheroidal assumption in all cases. The results are then organized according to the type of support

condition while testing and the type of grouting. The estimated tensile strength is higher for walls tested under fixed-fixed support condition than that of walls tested under cantilever type of support and this may due to the restraint provided to walls by both fixed end that increases the tensile strength capacity. The higher tensile capacity is also obtained for fully grouted walls than that of partially grouted walls.

TABLE II

COMPRESSIVE STRENGTH TO TENSILE STRENGTH RATIO FOR FULL AND PARTIAL GROUTING Name of

The Researcher

End

Conditions Grouting

type Flaw type

Ratio of Masonry Compressive strength to tensile strength

Standard

Deviation COV

COV of Masonry Compressive Strength (%) Voon Cantilever Full/Parti

al

Cylindrical 4.9 0.90 17.4

14.3 Spheroidal 8.0 1.40 17.5

Shing Cantilever Full Cylindrical 3.4 0.45 13.4 14.3 Spheroidal 5.6 0.80 13.7

Matsumura Fixed

Full Cylindrical 3.4 1.6 29.8 13.4 Spheroidal 5.6 2.6 28.6

Partial Cylindrical 5.4 1.6 29.9 32.0 Spheroidal 8.9 2.6 28.6

Sveinsson Fixed Full Cylindrical 2.7 1.4 52.4 25.2 Spheroidal 4.6 2.3 49.9

Tomazevic

and Lutman Cantilever Partial

Cylindrical 6.2 1.2 18.9

7.9 Spheroidal 11.0 2.3 20.4

Schultz Fixed Partial Cylindrical 5.5 1.0 17.8 - Spheroidal 9.3 1.7 18.7

Yancey and

Scribner Fixed Full

Cylindrical 12.1 3.3 27.7

11.4 Spheroidal 19.3 5.4 27.7

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(b)

Fig. 6. Plot of Tensile Stress at Failure for Different Types of Flaw and for a) Different support Condition b) Different Grout Conditions

This also may be the result of the hollow part of the walls that lead to less tensile capacity of partially grouted walls. From the analysis of results presented in Table 2 it can be said the table it can also be said that the coefficient of variation of the ratio of compressive to tensile strength is very insignificant both using the cylindrical and spheroidal assumption. The coefficient of variation is quite high for walls tested under fixed-fixed conditions and fully grouted walls. On the other hand it is moderate for partially grouted walls and quite less for cantilever walls. But in no case it does not fall below 10%. On the other-hand when the ratio of masonry compressive strength to tensile strength is checked it could be concluded that the ratio varies from 2-12 for cylindrical assumption and from 4-20 for spheroidal assumption. So may be the formula obtained from cylindrical and spheroidal assumption can be used without much reliance and further study is required to get the most accurate estimation of masonry tensile strength.

IV. CONCLUSION

The performance of reinforced concrete masonry shear walls under in-plane cyclic loading is discussed from the point of view of fracture. The main focus was the tensile strength that initiates the fracture in the walls and the tensile stress at is required to initiate the fracture while failure. The tensile cracking stress was computed for all walls applying the basic theory of fracture and cohesive strength applying two types of flaw assumption. The study reveals that the application of cylindrical and spheroidal assumption can give an idea of the masonry tensile strength but without much reliability. No definite relationship can be established of the ratio between the masonry compressive strength and the tensile strength. The approach used in this paper is pretty simple that gives a general overview of the fracture phenomena. The study of fracture of masonry requires more comprehensive investigation.

ACKNOWLEDGMENT

The authors wish to acknowledge the financial support provided by NSERC (Natural Science and Engineering Research Council), Canadian masonry design center (CMDC).

REFERENCES

[1] X. Xiao, "Fracture Mechanics of Masonry," PhD Dissertation, Department of Civil Engineering, University of Calgary, 2009. [2] S. Timoshenko and J. Goodier, "Theory of Elasticity", 2nd ed.,

Mc-GRAW HILL BOOK Company Inc. York, Pennsylvania, 1951. [3] N. Shrive, "Compression Testing and Cracking of Plain Concrete,"

Magazine of Concrete Research, vol. 35, no. 20, pp. 27-39, March 1983. [4] E. Hoek and Z. Bieniawski, "Brittle Rock Fracture Propagation in Rock

Under Compression," Internation Journal of Fracture Mechanics , vol. 1, no. 3, pp. 137-155, 1965.

[5] A. E. Schultz, " Seismic Resistance of Partially Grouted Masonry Shear Walls," in Worldwide advances in structural concrete and masonry, proceedings of CCMS symposium ASCE., Acapulco, 1996.

[6] C. Yancey and C. Scribner, "Influence of Horizontal Reinforcement on Shear Resistance of Concrete Masonry Walls," Gaithersburg, 1989. [7] B. I. Sveinsson, R. L. Mayes and H. D. McNiven, "Cyclic Loading of

Masonry Single Piers," University of California, Barkeley, Volume 4 report no UCB/EERC-85/15, Berkeley, 1985.

[8] K. Voon . “In-plane seismic design of concrete masonry structures”.

Phd thesis, Department of Civil and Environmental Engineering, University of Auckland, 2011.

[9] A. Matsumura . "Shear Strength of Reinforced Hollow Unit Masonry Walls". Proceedings of Ninth World Conference On Earthquake Engineering, (pp. 50-1-50-16.). Los Angeles, 1987.

[10] P. Shing, M. Schuller, & V. Hoskere,. “In-Plane-resistance of reinforced masonry shear walls”. Journal of structural Engineering, Volume 116, No.3, paper no 24411, 1990.

Figure

Fig. 1. Pure Shear Results in a Diagonal Tension Crack Perpendicular to the Equivalent Principal Tension
Fig.4. Spheroidal Hole in a Solid Body.
Fig. 5. Plot of Tensile Stress for Different Types of Flaw and for Different Grout Conditions
TABLE II COMPRESSIVE STRENGTH TO TENSILE STRENGTH RATIO FOR FULL AND PARTIAL GROUTING
+2

References

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