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FINITE ELEMENT MODEL FOR BRICK MASONRY

Dr. James Co1vi11e Prof. and Chairman Department of Civil Engineering University of Mary1and Co1lege Park, MO 20742 USA

by

ABSTRACT

Dr. W. Samarasinghe Former1y Fu1bright Fellow Department of Civil Engineering University of Maryland College Park, MO 20742 USA

An approach to the ana1ysis of masonry that accounts for material non­linearity and cracking, and the inf1uences of the Poisson's ratio of mortar and the resulting brick-mortar interaction, is presented. Detai1s of the development of the stiffness contributions of each component of masonry construction along with the associated fai1ure cri teria are given. A numerical solution of a deep masonry beam s tructure is compared to existing experimental results. This comparison includes both stress profi1es at various loadings and failure load values.

INTRODUCTION

The finite element method of analysis has been applied to masonry construction in recent years. Although the complex nature of masonry, and, in particular, the presence of mortar joints acting as planes of weakness, has been well understood and reported (1,2,3,4), early applications of the finite element method to masonry structures have considered the masonry as an assemblage of units and mortar with average properties and the influence of weak mortar joints has been ignored (5,6). More recent1y, the effects of non-linear material properties along with separate modeling of the masonry units and mortar joints have been included. Some of these researchers (1,2) have modeled brick masonry as an assemblage of elastic brick continuum elements acting in conjunction with linkage elements that simulate the mortar j oints. In these studies, mortar j oints were limited to deform normal and parallel to the joint direction, so that only normal and shearing stresses could be transmitted across the joint. An advantage of this approach is that aspect ratio problems in modeling narrow mortar joints have been avoided. Crack propagation along the j oints can be predicted but the influence of the Poisson' s ratio of mortar and consideration of brick-mortar interaction are neglected .

A method that accounts for the non-1inear behavior of brick masonry whi1e considering the interaction between brick and mortar is presented in this paper . The masonry is considered as a two-phase material consisting of elastic bricks set in an inelastic mortar. An e1ement stiffness matrix which assumes a uniform deformation along the element boundaries has been used to represent the mortar joints. Brick units are modeled using conventional eight parameter rectangular plane stress elements . These brick and mortar elements have been incorporated into an incrementaI finite element program which models the non-linear joint properties and allows progressive failure in the mortar joints and bricks. The details of

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the failure cri teria used are described and the effectiveness of the model is demonstrated by comparison with existing experimental results (2).

ANISOTROPIC NATURE OF HASONRY

Masonry exhibits distinct directional properties due to the influence of the mortar joints . Depending upon the orientation of the joints with respect to the applied loads, failure can occur in the joints alone, or in some form of combined mechanism involving both the mortar joints and the masonry units.

Researchers have long been aware that the deformation characteristics of brick and mortar are different; for example, the tensile splitting of masom;v assemblages under uniaxial compression 1s solely caused by this effect. Henc~ ; in theoretical modeling , the brick and mortar should be considered separately, allowing the interaction between unit and mortar to be simulated .

MODELING

The basic concepts of formulating an element stiffness matrix are well documented (7 ) and will not be repeated herein .

Each brick is modeled up to failure of the unit using a conventional eight parameter rectangular plane stress element wi th isotropic elastic properties . After failure , orthogonal properties which depend on the failure mode and the direction of cracking are assigned to the brick.

In developing a stiffness matrix for horizontal mortar joints, it is assumed that uniform deformations exist along the vertical and horizontal boundaries under ax ial stresses in each respec tive direction . Similarly , during horizontal shearing de formations horizontal surfaces will remain horizontal while the other surfaces rema in parallel to each other .

The resulting stiffness matrix is given below :

pNfJ + q pNII/4 -q _ pnN -pNfJ pNv O pNII Z- 4'" --r 4'" 4'"

p/ 2fJ pNII/ 4 - p/ 2fJ -pNII/ 4 O -pNII O ---z;-pNB pNII O pNII _ pNfJ _pNII Z- + q 4'" 4'" Z- 4'"

[K] - t p/ 2fJ pNII /4 O pNv/ 4 O

pNfJ + _ pNII pnN "'2 4'" -q

4'"

s y mmetri c a l p/2 fJ _ pNII -p/2fJ 4'"

pBfJ + q pNII "'2 4'"

p/2fJ

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Note : E~ - modulus of elasticity for load parallel to bed joint; E~ - modulus of elasticity for load normal to the bed joint;

u Poisson's ratio for load normal to the bed joint. p - element aspect ratio; q - G/2P; N - E~E~; P - E~(1-Nu2).

AI though the vertical j oints can be modeled in the same way as the horizontal j oints, the effect of the mortar Poisson' s ratio does not typically have a significant effect on the stress distribution in the brick. Hence, the vertical j oints are considered to be linkage elements which are capable of transmitting only horizontal and shearing stresses. It is ass\PDed that a uni.form deformation prevails along the joint under the action of horizontal stress.

Each brick unit is subdivided horizontally into two rectangular elements. The length of the vertical joint equals the height of the brick while the length of a typical horizontal joint is equal to the half length of the brick unit plus half the vertical mortar joint thickness.

The properties of brick and mortar used in the model were taken from Ref. (2). Bricks are assumed to be isotropic and elastic up to failure. Average values of E~ - 5920N/mm2 and Eoo - 7550 N/mm2 are used for the brick elastic moduli. The compressive strength of the masonry is 36.25 N/mm2 and Poisson's ratio, u - 0.167. The reported deformation characteristics of mortar joints are shown graphically in Figure 1 (a) and 1 (b).

FAILURE CRITERIA

In brick masonry panels subjected to in-plane loads, the joints will fail most often by separation of the brick and mortar either in tension or shear. Brick failure results from a criticaI combination of biaxial stresses (ignoring the three-dimensional effect).

Test data is available to predict the bond failure at the brick-mortar interface. These results have been presented in terms of average shear stress and normal stress at the interface without any regard to the stress paraI leI to the joint a p. Also, a broad scatter of results is quite evident. In fact, the failure criterion can be substantially affected (3) by the influence of stress parallel to the direction of the joint. However, the criterion adopted in this model also ignored the effect of ap and used the criterion proposed by Page (2) since other material properties required for the model were aIs o taken from these experimental resul ts. The criterion used for the j oint failure is diagramically shown in Figure 2.

Since brick is considered to be a brittle material, a failure criterion similar to concrete was adopted for the brick. The tensile strength of brick has not been reported in the test data obtained for this model and hence it is assumed to be equal to 2.0 N/mm2.

Stress HPa

Stress HPa

8.0

4.0

2.0

1.5

1.0

0 .5

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Brick / I

I ... Masonry

I

I I

I /

/ I

I '" I ./

/

/

400

" " ."

." . -.;.::." Mo r t a r

800 1200 1600

Stress- Strain Curves

Fig. 1 (a)

Brick /

I

I ~Masonry

I . "-I

..-..-I ..- ·<-Mortar

I ..- ' I ...-I ".

'"

1000 2000 3000

-5 St r ain x 10

-5 Shear Strain x 10

Shear Stress-Strain Curves

Fig . l (b)

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3.0

Shear 2. 0 St r ess ~lPa

1. 0

2 . 0 4 .0 6. 0 8 .0

Compres sive Stre s s -MPa

Fi g . 2 J o i nt Failure Envelope

PROGRAM STRUCTURE

A plane stress finite element program was developed in which the loads were applied incrementally up to the final failure of the structure.

Prior to failure, the brick elements were assumed to be linearly elastic and isotropic. If a brick element failed under biaxial tension-tension or tension­compression, the major principal stress was set to zero by introducing a very low elastic modulus perpendicular to the direction of crack. In such a case, shear stress can only be transmitted by aggregate interlocking action . This action was simulated by assuming G equal to 10% of the original shear stiffness value . The elastic modulus parallel to the crack was unchanged. In a crushing type of failure (biax ial compression- compression failure) very low elastic moduli parallel to the directions of principal stresses were introduced and the shear resistance capacity of the element was assumed to be negligible .

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If the failure criterion was violated for a joint element, the element properties were changed (2). In a tensile bond failure, no residual capacity was ·assumed. If a shear-bond failure occurs, the stiffness of the joint perpendicular to its direction is assumed to remain unchanged, and the shear stiffness is reduced depending on the leveI of precompression applied normal to the joint. The shear modulus value varies from 25.0 N/mm2 (when 0n - 2.3N/mm2) to zero (when 0n - O). When 0n > 2.3N/mm2, G is assumed equal to 25.0 N/mm2.

For each load leveI applied, two iterative procedures were involved. At a particular load leveI, stresses were computed and checked for failure. If none of the elements were cracked, pseudo loads were calculated to simulate the effects of non-linearly elastic mortar joints and displacements and stresses recalculated. (If the normal stress on a joint is tensile, the material is assumed to be linearly elastic-brittle) . This procedure was repeated until a convergent displacement solution was obtained. . If any element failed within this iterative cycle the structure stiffness was modified to reflect this failure and the iterative scheme continued.

The load was incremented until failure occurred . Final failure is indicated by lack of convergence of the displacements or by lack of symmetry in the displacements in a symmetrical problem. This latter condition was considered an indication of impending instability of the structured modelo

VERIFICATION OF THE FINITE ELEMENT MODEL

Experimental results reported by Page (2) on a deep masonry beam were used to compare the effectiveness of the theoretical model, since the material data required for the model analysis was well documented in Ref. (2). A schematic diagram of the structure analyzed is shown in Figure 3.

!!! !l~~ I I I I I

I I I I I I I I I I I

I I I 1 I I I I I I I

I I I I I I I I I I I

I I I I I I I I I I I

I I I I I I I I I I I

Rigid Supports

Fig. 3 Deep Beam Structure

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The wall was constructed with half scale bricks (112 mm long x 54 mm wide x 37 mm high) and 5 mm thick mortar joints. The ultimate load at failure was reported as 109 . 2 kN .

Although the supports were restrained to prevent separation of the base while t es ting, the exact nature of the restraint imposed is not clear from Ref. (2). In the ana lytical model used herein alI nodal points at the base were fixed.

The vertical stress distribution across line A-A (see Figure 3) computed from mea sured vertical strains (2) are compared to the computer model results for s evera l values of applied loading in Figure 4.

3 . 45

2 . 7b

2 . 07

Stress NPa

. 69

I I

I

I

I

/

/

/

I i

/

I i

/

I

/ /

-- ..... -- -".-... ' ...... ..... '

--=--,-

';-- . , , \\

- --. ..

, \ \\ ,\

\ \

\ \ 80KN \

\ \

.;::- .":.,,,,- ...... 20KN

Ha lf-Panel Width

Fig . 4 Vertical Stress Distr i bution

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It can be seen that the analytical model reproduces the actual stress distribution reasonably well . The peak stresses are in good agreement, although the location of the peak stress differs slightly. This may be due to formation of cracks in the wall at some joints which have not been simulated in the theoretical modelo

Predicted crack patterns at different load leveIs are shown in Figure 5 . (Experimental results are not available for comparison).

- Shear

Initial cracks at 20 KN _ Tension

Fig. 5 Cr acking Pattern at 100 KN.

Splitting of bricks occurred at P - 120 kN and displacements in the lower central region of the panel were not exactly symmetrical, indicating some instability of the structure. This imminent instability was confirmed in the next load step wi th many more bricks cracked and the development of a non-positive definite structure stiffness matrix . Hence the failure load was considered to be 120.0 kN compared to the 109.2 kN experimental resulto

However, the lack of knowledge of the exact boundary conditions at the base of the test wall complicates the comparison of theory and experiment o For example, when the wall was completely supported on rollers, the numerical analysis indicated a failure load of only 20 kN . When one support was completely restrained and the other support was on rollers, the ultimate load capacity of the wall was 90 kN.

SUMMARY AND CONCLUSIONS

An approach to the analysis of masonry that includes material non-linearity and cracking is presented. The analytical model for masonry presented herein differs from previously used models in that the Poisson' s effect of mortar is considered and, thereby, the interaction between brick and mortar is modeled effectively while the aspect ratio problem of thin mortar joints is eliminated.

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Experimental results reported in Ref. (2) are within reasonable agreement with the theoretical results.

This model can be used to study any in-plane loading problem up to failure provided the necessary material properties are derived.

REFERENCES

1. Hegemier, G.A., Arya, S.K., Krishnamoorthy, G., Nachbar, W., and Furgerson, R., "Qn the Behavior of Joints in Concrete Masonry," proceedings of the North American Conference, Boulder, Colorado, 1978, pp. 4.1-4.21.

2. Page, A.W., "A Finite Element Model for Masonry," Journal of the Structural Division, ASCE., Vo1. 104, No. ST8, August 1978, pp. 1267-1285.

3. Samarasinghe, W., "The In-Plane Failure of Brickwork, " Ph.D. Thesis, University of Edinburgh, U.K., 1980.

4. Stafford-Smith, B., and Carter, C., "Distribution of Stresses in Masonry Walls Subjected to Vertical Loading," Proceedings of the 2nd International Conference on Brick Masonry. Stoke-0n-Trent, 1970, pp. 119-124.

5 Saw, C. B., "Linear Elastic Finite Element Analysis of Masonry Walls on Beams," Building S~ience, Vol. 9, 1974, pp. 299-307.

6. Yettram, A.L., and Hirst, M.T.S., "An Elastic Analysis for the Composite Action of Walls Supported on Simple Beams, " Building Science, Vol. 6, 1971, pp. 151-159.

7. Zienkiewicz, O.C., "The Finite Element Method, " Third Edition, McGraw-Hill Book Co. (UK) Ltd., 1977.