EXPERIMENTAL DETERMINATION OF THE

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a homogenous continuum material made up of brick units, mortar joints and any unit- ... experimental testing of brick masonry prism material according to ASTM specification ... 16. 0. 5. 10. 15. 20. 25 immersion time (mins). A b s o rp tio n. (b y w e ig h t) % .... where the height to width ratio of the prisms is greater than 2.5.
Canadian Journal on Environmental, Construction and Civil Engineering Vol. 3, No. 3, March 2012

EXPERIMENTAL DETERMINATION OF THE MECHANICAL PROPERTIES OF CLAY BRICK MASONRY

T.C. Nwofor Department of Civil and Environmental Engineering University of Port Harcourt, Rivers State, Nigeria. P.M.B 5323, Port Harcourt [email protected] Abstract: The structural behaviour of masonry is influenced by the mechanical properties of the constituent materials. Therefore a full mechanical characterization is required for proper non-linear analysis of masonry structures. Hence uniaxial compressive test is carried out on unreinforced masonry and its constituents (clay bricks and mortar). From this study, compressive stress-strain relationships at different confining stress levels have been defined. In this paper the experimental results obtained is used to formulate simple analytical models for the purpose of estimating the modulus of elasticity of unreinforced masonry, utilizing the control points established in this work. The proposed material model can be employed in the non-linear finite element analysis of masonry structures.

Keywords: Brick-mortar masonry, compressive stress, strain and modulus of elasticity.

1.

Introduction

The properties of brickwork are influenced by variables of bricks, type of mortar, physical properties of the sand and lime used for the mortar, state of bricks before casting, curing workmanship and many others. Hence it can be deduced that in the experimental determination of mechanical properties of brickwork, a large number of variables can be considered. We should note that the analysis and design of buildings require the material properties of masonry, for example, the modulus of elasticity of masonry is require for the non-linear static analysis. Stress-strain curves of masonry are required for more detailed non-linear analysis of masonry structures. Limited research

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has been carried out by researches to obtain a realistic material property for masonry [1][4]. Also laboratory test have also been carried out on masonry, adopting the approach of a homogenous continuum material made up of brick units, mortar joints and any unitjoint interfaces [5]. The wide variation of test value of the mechanical property of masonry and its units especially when under the influence of compressive load [6]-[9] has not aided the formulation of realistic stress-strain curves for non-linear modeling of masonry and infilled frame structures. Hence, in this present study, extensive experimental testing of brick masonry prism material according to ASTM specification [10] would be performed to obtain stress-strain curves. Also experimental relationships would be obtained between the modulus of elasticity of masonry units of its compressive strength. Furthermore, simple analytical equations are developed using the experimental data to estimate the mechanical properties and plot the stress-strain curves for masonry. However, to maintain the scope of this research, the materials used have been kept constant. The bricks, cement, sand and lime used are described below. 2.

Materials and Experimental Procedure

The bricks employed are solid burnt clay bricks of average size of 224 x 106 x 72mm. Typical physical properties are shown in Table 1. The absorption rate of bricks immense in water at room temperature is also shown in Figure 1. Table 1: Physical properties of bricks No. of specimens Crushing strength

*

Length (mm) Breadth (mm) Depth (mm)

ψ * * *

20 102 20 20 20

Compressive Strength (x 106kN/m2) Mean Range 23.30 18.69-27.25 22.00 224 223.1-224.5 106 106.2-106.8 72 71.1-72.9

Water Absorption % by wt. after 25 hrs, immersion

12.49

% by wt. after 5 hrs, boiling

12.91

128

Std Dev 2.21 2.32 0.391 0.183 0.194

Canadian Journal on Environmental, Construction and Civil Engineering Vol. 3, No. 3, March 2012

(*)

Values obtained by author

(ψ) Values obtained by British Ceramic Research Association The mortal mixes will consist of cement, lime and sand, while ordinary Portland cement would be used for this investigation so as to use the 28th day strength. 16

Absorption (by weight) %

14 12 10 8 6 4 2 0 0

5

10

15

20

25

immersion time (mins)

Figure 1: Absorption of Bricks Also, graded sand classified in zone 2 was generally used for this investigation to prepare the mortar. The particles distribution analysis is shown below for a sample of 300grams. Sieve analysis was carried out on the river sand sample and the results shown in Table 2. The result revealed the sand sample was well graded falling into zone 2 near border of zone 1, which is very appropriate for concrete work in accordance with BS 882, part 2 1992 [11]. The fineness modulus of the sand was found to be 3.15, which makes the sand sample a rather coarse one.

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Table 2: Particle size analysis on fine aggregate sample Sieve 5.00mm 2.36mm 1.18mm 600.00µm 300.00µm 150.00µm Tray +

No.7 14 25 52 100

Mass on (g)

% on sieve

% retained

% passing

Zone 2 Limits

Max. on sieve permitted (g)

12 54 60 66 63+ 39 6

4 18 20 22 21 13 2

4 22 42 64 85 98 315

96 78 58 36 15 2 -

90-100 75-100 55-90 35-59 8-30 0-10 -

200 100 75 50 40

Needed dividing as it exceeded the maximum permitted on sieve.

Zone = zone 2, near border of zone 1. Fineness modulus (FM) 315 ÷ 100 = 3.15 (rather coarse) A mortar mix proportioning of 1:¼:3, 1:0.5:4.5 and 1:0:6 corresponding to a volume proportioning of cement, lime, and sand respectively are generally used, while maintaining reasonable workability for mortar is achieved by varying the water-cement ratio ( W C ) to produce suitable workability.

2.1

Water Cement Ratio ( W C ) of Mortar In concrete work the term water/cement ratio, if not qualified, could refer either to

effective water/cement ratio or total water/cement ratio.

In this investigation, the

water/cement ratio of fresh mortar is an important factor in determining the properties of the hardened mortar (as a joint) in brickwork. It is therefore necessary to clarify the definition of this term as used in this work. Assuming only fine aggregates (i.e. sand in the case of this investigation), the effective water/cement ratio for strength is defined as: Effective ( W C ) = total ( W C ) – as( S C )

(1)

where total W C = total amount of water added to the dry mix aS = 30mins absorption capacity of the sand (in the air-dry state). S/c = sand/cement ratio by weight. For the oven-dry sand used throughout this investigation as (in this case corresponding to maximum absorption capacity) was found to be 0.024.

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Hence from equation 1 Eff. W/C = total W/C – 0.024s/c

(2)

Table 3 gives a comparison of the effective W/C with the total W/C for two typical mixes used.

Table 3: Comparison of the effective W/C with the total W/C Mortar

Total W/C (by wt.)

Effective W/C (by wt.)

1:¼:3

1.0

0.93

0.8

0.73

1.0 2.6

0.88 2.48

1:05:45

It should be noted that the term W/C used throughout this work refers to the total W/C as defined above.

2.2

Actual W C of Mortar Joint Owing to absorption of water by bricks, the W/C of the mortar before placing is

different from that after placing between bricks. Therefore, while any control specimens (such as cubes or prisms) would be useful for indicating the relative qualities of different batches of mortar mixes, it should be appreciated that properties such as crushing strength, Young’s modulus, Poisson’s ratio, determined from such specimens may not bear much resemblance (except by correlation) to those of the same mortar in the hardened state (as a joint) between the bricks. A theoretical relation between the W/C’s of the control specimens and the mortar joint was derived by considering the amount of water absorbed from a known weight of fresh mortar placed between two bricks. The relationship is given by Wa C = W/C – k(C + L + S + W)/C

(3)

where Wa C = actual water/cement ratio of mortar between bricks (or of mortar joint)

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W C = water/cement ratio of the fresh mortar (or control specimen) K=

wt. of water absorbed by bricks wt. of original mortar

The value of k was determined experimentally. It is assumed that loss of water to the atmosphere from the bricks- mortar couplet and from the control specimen is the same. For 1:¼:3 mortar with W/C = 1.0 Wa C = 1.0 – 5.25k

(3a)

Corresponding expressions for other values of W C could be obtained similarly. These relationships are plotted in Figure 2. The conditions for saturated bricks and for bricks soaked for one minute before casting are indicated by vertical lines AB and CD respectively obtained from measured values of k. The intersections of these lines with the straight line represented by equation 3a give the values of actual water/cement ratio (Wa/C) for ‘saturated’ bricks and for bricks soaked for one minute.

1.4 1.2 1 W a /C

W/C =1.2

0.8

W/C =1.0

0.6

W/C =0.8

0.4

LINE AB

0.2

LINE C D

0 0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

K

Figure 2: Relationship between Wa C and K The respective values being 0.94 and 0.5. The indication is therefore that for saturated bricks, while the W/C for the control specimen is 1.0, the actual water/cement ratio ( Wa C ) of the corresponding mortar joint would be of the order of 0.94. For bricks 132

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soaked for one minute (or with a moisture content of about 6%) the corresponding values are W/C = 1.0, Wa C = 0.5. The deductions drawn from these results are as follows:i.

To determine properties of mortar joint, such as tensile or shear strength, realistic values may be directly obtained from test on brick-mortar couplets.

ii.

For elastic analysis of brick-mortar couplet or brick-work as a non-homogenous material, meaningful values of elastic modulus and Poisson ratio for the mortar joints may be obtained from tests on mortar specimens with W C ratio value equal to the value of the actual water/cement ratio ( Wa C ) of the mortar joint.

3.

Mechanical Properties of Bricks and Mortar

3.1

Compressive/Crushing Strength of Brick

The maximum compressive stress of a brick-mortar prism is determined by applying a compressive load in the direction parallel and perpendicular to the bedding planes, hence, the fabricated bricks are built into a prism of different layers with constant mortar joint thickness. It is important to note that while these test give reasonably good indication of the crushing strength of brick-mortar prism for control purposes it is doubtful if they give the compressive strength of brick work as a basic physical property. Several research work on concrete specimens which can also be related to brickwork has shown that some important factors influences the compressive strength of specimen such as size of specimen, surface condition, state of stress induced in the specimen, thickness of mortar joint and also the testing technique. Hence the most important factor arrived at, is that the direct measure of the uniaxial compressive strength of concrete which gave a realistic value of the value for the basic property of concrete in uniaxial compression is a case where the height to width ratio of the prisms is greater than 2.5. From the foregoing, compression tests were carried out on bricks with varying height to width ratios in the compression machine. For the case of loading parallel single bricks were tested, Figure 3a while for the case of loading perpendicular to the bedding planes of the brick the respective H/L ratios are obtained by varying the number of bricks in the prisms as is shown in Figure 3b.

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The crushing strength of the bricks were generally found to decrease with increase H/L ratio to a limiting value of 23.28N/mm2. This reflects the value shown in table 3.1. This value shows the average of the result obtain from both loading perpendicular and parallel to the bedding plane. The variation of brick crushing strength with H/L ratios is shown in Figure 4.

Hence it can be observed that more realistic results would be

obtained by loading single bricks parallel to these bedding plane.

Table 4: Summary of crushing strength of bricks Number of tests

H/L

10 10 10 10 10

0.32 0.90 1.51 2.15 2.12

Crushing mean strength (fb) × 103kN/m3 38.36 25.19 29.71 24.15 23.16

Range

description

33.92-44.33 22.59-28.08 19.26-29.82 22.48-25.89 18.69-27.25

Type (b) 1 brick Type (b) 3 brick Type (b) 5 brick Type (b) 7 brick Type (a) 1 brick

(a)

(b)

Figure 3: Compressive loading on brick specimens (a) Loading parallel to bedding planes (b) loading perpendicular to bedding plane

3.2

Crushing Strength of Mortar

Compressive strength test were carried out on 20mm mortar cubes and a graphical representation of the results shown in Figure 3.4 for different ages of a 1:¼:3 nominal mix of mortar by varying the water-cement ration of the mix.

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20

W/c = 0.8 W/c = 1.0

Crushing strength (x 103kN/m2)

18

W/c = 1.2

16 14 12 10 8 6 4 2 0 0

5

10

15

20

25

30

Age (days)

Figure 4: Crushing strength of 1:¼:3 grade of mortar mix.

3.3

Modulus of Elasticity and Poissons Ratio for Brick The static method was used to obtain elastic properties of brick. In order to

produce uniaxial compression in the middle portion, single bricks were loaded in compression in a direction perpendicular to the bedding planes. Longitudinal compressive strain

(ε ) and the corresponding lateral tensile strain (ε ) were measured y

x

in the middle region by using suitable electrical strain gauges. By measuring the compression load and the strains

αy

εy

and

αy

ε y and ε x as shown on table 5, the stress-strain curves

ε x can be obtained. Young’s modulus (E) and Poissons ratio (v) were

obtained from the slopes of the linear portions of the stress-strain curves as follows.

E=

αy εx

(4)

αy

v =

εy εx = εy αy εx

(5)

Table 5 shows values of Modulus of elasticity (E) and Poissons ratio (v) thus obtained. E1 and V1 denote values obtained from strain measurements on face 1, E2 and V2 correspond to values for face 2. Two sizes of bricks were tested. Type (a) is the

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normal single brick, and type (b) is the single brick made to produce a square section (72mm x 72mm) perpendicular to the loading direction.The values shown in Table 5 were obtained from test on twenty (20) specimen of bricks and the stress-strain curves were obtained from an average of about six sets of readings obtained by loading and unloading the specimen six times. Table 5: Summary of test result for brick units Description

No. of specimens

(x 103kN / m2) Fb

(x 106kN/m2) E1 E2 Av.E

10

23.10

8.83

8.14

10

23.18

8.76

8.89

V1

V2

Failure strain

8.49

0.09

0.08

0.0045

8.83

.06

.08

0.0048

Type (a)

2

1

Normal bricks

Type b

2

1

Brick with square cross-section

The stress-strain curve as a result of the two cases of loading for bricks is shown in Figure.5. The average value for crushing strength Fb and failure strain is also shown in Table 6. The variation of modulus of elasticity Eb with the crushing strength Fb is shown in Figure 6 and it is seen with the best line of fit drawn, that the average relationship in equation (6) can be obtained between Eb and Fb with a coefficient of correlation Cr = 0.60

Eb = 348.51Fb

(6)

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35

C om prees ive S tres s ( σ ) *10 3 K N/m 2

30

25

20

15 Type A loading 10

Type B loading

5

0 0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

S tra in(εε)

Figure.5: Stress-strain curves for brick

12000

E b = 348.51F b C r=0.5934

E las tic Modulus (Eb ) *103 K N/m 2

10000

8000

6000 Type A L oading 4000

Type B L oading

2000

0 0

5

10

15

20

25

30

C om pre ssive S tre ng ht (F b ) *103 K N/m 2

Figure 6: Variation of modulus of elasticity with crushing strength of bricks (40No. bricks)

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3.4

Modulus of Elasticity and Poissons Ratio for Mortar

It was seen earlier that due to the absorption of water by the bricks from the fresh mortar, the value of the actual water/cement ratio of the mortar joint is smaller than its original value. In order to obtain realistic values of the Modulus of elasticity and Poisson ratio for the mortar joint, a 50mm x 50mm x 125mm mortar Prisms with a particular water/cement cured for 21 days, were tested. Both longitudinal strain

(ε ) and lateral strain (ε ) were y

x

measured by electrical resistance strain gauges, and stress-strain curves can be plotted for εy and εx. Modulus (E) and Poissons ratio (v) were calculated by using Equations 4 and 5. Table 6: Summary of average of test result conducted on different mix proportions of mortar prisms Mortar grade 1:¼:3 1:0.5:4.5 1:0:6

No. of specimen 4 4 4

Er x 10 kN/m2 3.90 3.45 0.75

Average fr x 103kN/m2 22.03 13.10 4.2

6

V 0.18 0.18 0.17

Failure strain 0.019 0.0220 0.0090

The compressive stress-strain curves for the mortar mix for the adopted grades of mortar is shown in Figure 7 while the variation of the modulus of elasticity of mortar (Er) with the corresponding compressive strength (fr) is shown in Figure 8. 25

mortal mix 1:1/4:3

mortal mix 1:0:6 20

3

C omprees ive S tres s (σ) *10 K N/m

2

mortal mix 1:0.5:4.5

15

10

5

0 0

0.005

0.01

0.015

0.02

0.025

0.03

S tra in(εε)

Figure 7: Stress-strain curve for different grade of mortar prisms

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E r= 231.11F r C r = 0.9

5000

3

E la s tic M o d u lu s (E r ) *10 K N /m

2

6000

4000

3000

Mortal Mix 1:1/4:3 Mortal Mix 1:0.5:4.5 Mortar Mix 1:0:6

2000

1000

0 0

5

10

15

20 3

25

30

2

C ompres s iv e S treg ht (F r) *10 K N/m

Figure 8: Variation of modulus of elasticity to compressive strength for mortar

The relationship between the stress and strain was linear while observing the initial portion of the stress-strain curve up to about 35 percent of the mortar strength and then is followed by a non-linear curve extending beyond the strain limits. The stress reading beyond the strain limits of the mortar prisms with of grade 1:0:6 was difficult to read because of the relatively low strength of these samples with a mean value of the failure strain equal to 0.0090.

An observation of Figure 8 shows that an average

relationship can be obtained in equation 7 by drawing a line of best fit. Er = 231.11Fr

(7)

A good correlation coefficient of 0.9 was obtained between the values of the experiment test results. Also it would appear that more realistic values for E for the mortar joint in a brickwork may be obtained by substituting the compressive strength value (Fr) obtained from the test cubes of mortar with water/cement ratio value equal to the value of the actual water/cement ratio ( Wa C ) for the mortar joint given by the expression in equation (3). An average value of 0.177 for poisons ratio (v) can also be deduced from Table 6.

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4.

Mechanical Properties for Brickwork

Elastic modulus and poisons ratio values for brickwork were obtained from compression test on brick-mortar prisms by maintaining a length-height ratio of about 2. The longitudinal and lateral strains would be obtained for cases of loading parallel and perpendicular to the bedding plane as to obtain the modulus of elasticity in the two orthogonal directions. The longitudinal and lateral strains will be measured by use of electrical resistance strain gauges, and similarly the modulus of elasticity (E) and the poisons ratio (v) values would be obtained through equation 4 and 5. Table 7 shows summary of values obtained from the test carried out on a number of brick-mortar prisms specimens. Values are obtained for compressive strength, failure strains, modulus of elasticity and the poission’s ratio of masonry. Table 7: Summary of test results on brick-mortar prisms Description of loading

Mortar mix

Number of specimen

Perpendicular to bedding plane Parallel to bedding plane Note * Vyx

1:¼:3 1:0:4.5 1:0:6 1:¼:3 1:0:4.5 1:0:6  Ey  =   Vxy  Ex 

8 8 8 8 8 8

Max. strain

Compressive strength ( × 103kN/m2) Fm 13.46 11.54 10.58 8.5 7.2 5.1

0.0019 0.0020 0.0047 0.0057 0.0092 0.0086

Modulus of elasticity ( × 106kN/m2) E 8.41 7.21 6.61 5.32 4.67 3.21

(b)

(a)

Figure 9: Set up for loading of brick-mortar prisms (a) perpendicular to bedding plane (b) parallel to bedding plane. 140

Poisson’s ratio V

0.29 0.33 0.36 0.18 0.21 0.28

Canadian Journal on Environmental, Construction and Civil Engineering Vol. 3, No. 3, March 2012

The stress-strain curves are plotted using the average of values obtained from test on 8 specimens of masonry prisms made with a specific mortar mix. The height of the prisms was 400mm maintaining 10mm mortar joint thickness.

A general setup for

loading in the two major directions is shown is Figure 8. The stress-strain curves for the masonry prisms with mortar mix of 1:¼:3 is shown in Figure 10. Similar stress-strain curves of masonry prisms made with other grades of mortar mix can also be obtained. We should note here that in most cases, failure was as a result of vertical splitting cracks along the depth of the prisms, when loading is perpendicular to the bedding plane while a relatively diagonal splitting failure is notice when loading is parallel to the bedding plane. It was generally noticed that the strength properties of masonry reduced as weaker mortar is used. The stress-strain curve was seen to be linear to an extent after which a nonlinearity pattern is noticed. The variation of modulus of elasticity Em with compressive strength fm is shown in Figure 11. An average relationship can be obtained for Em and fm for perpendicular and parallel loading in Figure 8 and 9 respectively, with a coefficient of correlation of 0.9 between the experimental value obtained from test on the masonry. Em1 = 634.66Fm1

(8)

Em2 = 640.00Fm2

(9)

It would also be observed that the worst performance was seen in the prisms with weak mortar joint as the compressive strength (Fm) was low with a corresponding high value for strain.

C o m p re e s i v e S tre s s (s ) *1 0 3 K N /m 2

14 12 10 8 6 P erpendicular Loading 4

P arallel Loading

2 0 0

0.001

0.002

0.003

0.004

0.005

0.006

S train(εε )

Figure 10:

Stress-strain curve for brickwork

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A study of the stress-strain curve, especially for the case of parallel loading shows that certain salient points are easily observed on the stress-strain prisms pattern. The strain values used to determine these points of interest varying with the grade of mortar used in the prisms, as great difficulty is observed with deriving strain values when a weak grade of mortar is used, especially after the near linear range; due to brick-mortar bond failure and inivitable sudden collapse of the test specimens.

12000

E m1= 634.65F m1 C r = 0.9

Mortar Mix 1:0.5:4.5

Mortar Mix 1:1/4:3

10000

3

E las tic Modulus (E m 1 ) *10 K N/m

2

Mortar Mix 1:0:6

8000

6000

4000

2000

0 0

2

4

6

8

10

12 3

14

16

2

C ompre s s iv e S tre ng ht (F m1 ) *10 K N/m

(a) Mortar Mix 1:1/4:3

E m 2 = 640E m 2 C r = 0.9

Mortar Mix 1:0:4.5 7000 6000

3

E las tic Modulus (E m 2 ) *10 K N/m

2

Mortar Mix 1:0:6

5000 4000 3000 2000 1000 0 0

1

2

3

4

5

6

7 3

8

9

10

2

C ompre s s iv e S tre ng ht (F m2 ) *10 K N/m

(b) Figure 11: Variation of modulus of elasticity for brickwork with corresponding compressive strength (a) case of perpendicular loading (b) case of parallel loading

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Hence the following salient points are easily observed from this work and compares favourably with [11]. (a)

point 0.40Fm corresponding to the limit of the region at which the stress-strain curve is near linear as much as possible, after which regional cracks starts developing suggesting non-linearity.

(b)

Point 0.75Fm corresponding to the particular stress at which vertical splitting cracks are seen, but the masonry specimen still remains relatively stable.

(c)

0.95Fm corresponds to the stress level at which the splitting cracks have reached a very advanced level and failure is ready to occur.

(d)

Fm is the ultimate stress level in which the masonry is in a collapse state with a corresponding rapid increase in strain reaching an observable failure strain in the masonry.

5.

Analytical Model for prediction of Stress-strain Curves of Masonry Acknowledging that there exist a reasonable mathematical relationship between

the compressive strength of masonry and the modulus of elasticity of masonry, hence analytically modeling to obtain fm is necessary as it is not always very feasible to conduct test on masonry prisms. On the other hand, the compressive strength of bricks and mortar (Fb and Fr) can readily to obtained through tests. The compressive strengths of bricks, mortar and masonry can be properly related as proposed by Eurocode [12] by equation 10 f m = Kf bα f rβ

(10)

where k, α and β are all constants for effective relationship. Observing experimental stress-strain curves, fm depends on the brick strength more than the mortar strength, hence α must be higher than β. Conducting an unconstrained regression analysis of equation 10 using the data obtained from our experimental study the values of 0.61, 0.51 and 0.36 have been obtained for k, α and β respectively using the least – square fit method and the following equation proposed.

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Fm = 0.61 f b0.51 f r0.36

(11)

The effectiveness of this relationship can be tested by the parameter λ, which represents the standard error of estimate. A value of λ in equation 12 close to minimum reflects low scatter of the actual data from the value obtained by the regression analysis [13].

λ =

∑( f

i

− f Ri )

2

(12)

n−3 ^

where fi and f Ri = ith experimentally obtained and regression estimated prism strength, respectively. n

=

total number of data points.

From the experimental data a value of and 0.46 x 103kN/m2 is obtained for λ.

6.

Conclusion

From the foregoing the basic mechanical properties of masonry has been obtained by tests carried out on specimens. These mechanical properties are basic input parameters for the numerical modeling of masonry and infilled frame structure, noting that masonry is a composite material made up of brick units binded by mortar. Thus non-linear stressstrain curves have been obtain for masonry with salient points identified on the stressstrain curves with stress level of 0.40Fm corresponding to the limit of the near linear region. Also simple analytical model has been proposed for prediction of the modulus of elasticity of masonry, to aid the numerical analysis of masonry structures. Finally, compressive test result obtained from test on brick units and mortar is enough to predict the elastic properly of masonry, as simple relationships have been obtained for obtaining the modulus of elasticity of bricks, mortar and masonry from their corresponding compressive strengths.

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2

Lourenço, P.B., “Computational strategies for masonry structure,’’ PhD-Thesis, Delft University of Technology, 1996, Delft University Press: Delft. 144

Canadian Journal on Environmental, Construction and Civil Engineering Vol. 3, No. 3, March 2012

3.

Augenti, N., ‘’Il Calcolo Sismico Degli Edifici In Muratura’’, UTET: Turin (in Italian), 2004.

4.

Augenti, N. and Parisi, F., “Stress-strain relationships for yellow tuff masonry in direct shear,’’ 8th International Masonry Conference Dresden, 2010.

5

Lourenço, P.B., “Experimental and numerical issues in the modelling of the mechanical behaviour of masonry,’’ In: Proc. of the 2nd Conf. on Structural Analysis of Historical. Constructions, eds. P. Roca, J.L. González, E. Oñate & P.B. Lourenço, CIMNE: Barcelona, 57-91, 1998.

6.

Knutson H. H., “The Stress-Strain Relationship for Masonry,’’ Masonry International, Vol. 7, No. 1, pp. 31-33, 1993.

7.

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8.

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9.

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10.

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11.

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12

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13

Wesolowsky, G.O., “Multiple regression and analysis of variance”, Wiley, New York. 1976.

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