262 Vth International Brick Masonry Conference

IV -6. A Model for the In-Plane Behaviour of Masonry and a Sensitivity Analysis of its CriticaI Parameters

A.W. Page Lecturer, Department of Civil Engineering, The University of Newcastle, N.S. W., Australia.

ABSTRACT

A finite element model for the in-plane behaviour of elay masonry is presented. The model reproduces the nonlinear characteristics of masonry caused Iry material nonlinearity and progressive joint failure. M asonry is considered as an assemblage of elastic brick continuum elements acting in conjunction with linkage elements simulating the mortar joints. These joint elements are assumed to have nonlinear deformation characteristics with limited shear and tensile capacity. The model is aPPlied to a mason'ry deep beam, and analytical and experimental results are presented.

Sensitivity analyses of the model parameters are carried out with reference to the deep beam testo Accurate definition of joint failure is shown to be the most important characteristic in a model of this nature, with joint and brick deformation characteristics being less significant.

INTRODUCTION

The analysis of day masonry walls subjected to in-plane loading is a problem commonly encountered in the design of masonry structures. Masonry is a two-phase material which typica\ly consists of elastic, brittle bricks set in inelastic mortar joints. These joints impart directional properties to the masonry assemblage, and act as planes of weakness due to their low bond strength. Depending upon the degree of compression present, failure can occur in the joints alone, or as a combined brick-joint failure, The mechanism of failure is not fu\ly understood, and no completed failure criterion has yet been developed,4 although significant results for concrete masonry have been recently presented by Hegemier et a\.5

Most previous analyses have considered masonry to be an assemblage of bricks and mortar with average prop­erties, Isotropic linear elastic behaviour has usua\ly been ass umed, Models of this nature may be satisfactory at low stress leveis, but will be inadequate at higher stress lev~l s

when extensive stress redistribution will occur. This redis­tribution is caused by nonlinear material behaviour and progressive joint failure.

The emphasis in this investigation is on the determi­nation of stress distributions in masonry, rather than the accurate prediction of final failure which will occur after progressive failure and/or slip in a number of joints. In modeling masonry, an analogy has been drawn with the behaviour of jointed rock as modeled by Goodman, Taylor and Brekke using the finite element technique.' Masonry is considered as an elastic continuum of brick elements, with a regu lar array of joint (linkage) elements embeeldeel between them. These joint elements have low tensile capacity, high compression capacity, and a shear capacity which is a funct ion of the bond strength and superim­posed compression. The nonlinear characteristics of the modeled masonry thus result from the nonlinear defor­mation characteristics of the joints under shear anel compression , and the local failure anel slip that occurs in the joints. The detailed derivation of this moelel has been previously elescribed by the author.~·".

This papel' describes a typical application of a model of this type. A test on a masonry eleep beam is used as the basis of comparison between predicted and observed per­formance. Sensitivity analyses of the criticaI parameters of the model are also carrieel out to determine their relative importance.

FINITE ELEMENT MODEL

A typical finite element subdivision is shown in Figure 2(b). Bricks are moeleled using conventional eight param­eter rectangular plane stress elements with four internaI degrees of freedom and isotropic elastic properties. The joint element concept developed by Goodman et aI' has been adap ted to simulate the mortar joints. Due to its elongated nature the joint element is assigned different characteristics to the typical continuum element. Its defor­mation is limited to normal and parallel to the joint direc­tion, so that only normal and shear stresses can be trans­mitteel across the joint. Failure occurs when a shear or tensile bond criterion is exceedeel, with residual shear capacity remaining in some cases. The material character­istics requireel to define the behaviour of the joint element are therefore the deformation characteristics of the mortar joints and a suitable failure criterion. These charactel'istics are summarised in Figure 1. The values shown are for the tests on the sample of masonry described later in this paper.

joint failure occurs when the criterion shown in Figure I (c) is violateel. If the combination of shear and tensile stress lies outside Region I, a tensile bond failure is sim­ulated by assuming zero joint stiffness. If failure occurs under a combination of shear anel compressive stress, (out­side Regions 2 or 3), a shear bond failure is simulated. In this case , some resielual capacity will remain in the joint. The stiffness of the joint in the normal direction is assumeel to remain unchanged, anel a reduced shear stiff­ness is a llocated to simu late the frictional resistance of the joint aher an initial shear bonel fai lu re.

The above element types have been incorporated into an incrementaI finite element computer program which simulates masonry behaviour at various leveis of applied

Session IV, Paper 6, A Modelfo)' lhe In-Plane Behaviou)' of Masomy and a Sensitivity Analyses 263

load. Two major ite l'ations are involved at each load leveI. T he first iteration reproduces the nonlinear response of the mortar, and the second allows for the changing struc­ture stiffness as the joints progressively fail in tension or shear.

EXPERIMENTAL EV ALU A TION OF MASONRY PROPERTIES

The parameters required to define the analytical model are the e lastic properties of the bricks, the deformation characte ristics of the morta r joints, and a criterion of fail­ure for the joints. Since the experimental evaluation of these properties has been previously desuibed in detail ,2 only the main features will be outlined. Medium strength, half scale masonry was used for ali tests. The mean value of brick elastic modulus and Poisson's ratio were 6740 MPa and 0. 17 I·espectively.

Mortar properties were obtained indirectly from tests o n panels of masonry constructed from bricks with known average e lastic properties. I f brick defol"mations are sub­tracted f ro m total measured masonry deformations, the remainder can be attributed to the mortar. A series of uniaxial compl'ession tests were ca rried out on panels of masonry with varying bed joint angles to the applied load , producing varying ratios of shear stress ("T) to normal stress (On) o n the joints. Fwm these tests, average mortar stress­strain curves were derived. 2 A failure criterion for the mo rta r joints in terms of th e ultimate shear stress ("Tu) and ultimate normal stress (Onu) can also be obtained. These curves are shown in Figure 1. The strength of the joint under pure tension was obtained from a separate senes of uniax ia l tension tests on masonry couplets.

DEEP BEAM VERIFICATION TEST

To check lhe adequacy of the analylical model, tests were performed on a half scale masonry dee p beam with restrained supports subjected to a centralload . The lesting arrangement is shown schematica lly in Figure 2(a). These lests have a lso been described in more detail elsehwere.:'

To all ow compal'ison with the analytical model, the ver­tical stress distribulion along A-A was measured by means of e lectrica l resistance stra in gauges auached to previously ca librated bricks. The beam was load ed to fai lure. The ultimate load was 11 0 kN .

The finite e lemenl model simulating the behaviour of the beam is shown in Figure 2(b) . The load was applied in 10 kN increments. For purposes of comparison a con­ventional finite element analysis was also carried out assuming masonry to be an isotropic linear elastic contin­uum.

An elastic modulus of 4700 MPa, (the initial tangent va lue to the masonry stress-su'ain curve of Fig. I ), and a Poisson 's ratio of 0.2 were assumed in this analysis.

The calcuIated and obse rved vertical stress disLI"ibutions a long the fi fth course of bricks, (plane A-A, Figure 2), are compareci in Figu re :~ for various leveis of applied load. 1I can be seen thal good agreemenl is exh ibited between experiment and analytical model throughout the load range, particu larly in view of the inherent variabli ty of lhe material. The increasing di fTel'ence between the isotwpic

elastic analysis and observed values as the applied loacl increases, illustrates the d egree of stress redistribution tak­ing place.

The pattern of progressive j oint failure as predicted by the computer simulation is shown in Figure 4. j oims fail­ing in tension or slipping in shear a re indicated. The large number of joints tha t have progressive!y failed illustrates the significant influence of joint properties on masonry behaviour. Progressive cracking patterns cluring the test were difficult to observe due to small crack size. Final col­lapse of the panel involvecl both brick and joint failure. For accurate prediction of ultimate load, a critel"ion for brick failure would have to be included in the model.

SENSITIVITY ANALYSIS OF THE MODEL PARAMETERS

Brick Elastic Properties

Va riations in brick stiffness were obtained by holding the joint properties consta nt and va rying the e lastic prop­e rties of the bricks. Values of 0.5, 3 a nd 6 times th e ori g­inai brick elastic modulus were used . The analyses IVere performed with one 10 kN increment of load being applied in each case. At this low loacl levei, no j oin t fa ilures occurred, ensuring that brick stiffness was the only vari­able.

It was found from these analyses that variations in stresses throughout the pane! were quite small even for large va riation in brick elastic modulus. T ypical values a re summarized in Table I.

h can be concluded there fore , that brick stiffness is not a sensitive parameter for this mod el, and the experimental determination of its values need not be particularly pre­CIse.

Joiot Deformatioo Characteristics

Two sensitivity analyses were ca rried out. The first was to detennine lhe influence of mortar stress-strain char­acteristics on the nonlinear behaviour of masonry. The second was to test the sensitivity of the assumed mortar su"ess-strain curves used in the original model.

Mortar Deformatíon Propertíes

Comparison was made between the original ana lys is incorporating nonlinear morta r properties and an analysis assuming linear elastic properties for the mortar (in iti al tangent va lues to the mortar curves shown in Figure I (a) and I(b» . In both cases, the ca paci ty for pwgressivejoint failure was maintained . Ten load incre ments of 10 kN were ap plied.

The vertical stress clistributions a long the fifth brick course for a load levei of 100 kN , (when nonlinear effects will be greatesl), are shown in Figure 5. The stress distri­bution for the isotropic e!astic case, with no provision for joint failure, is a lso included.

It can be seen that there is a sign ificant diffe rence between the isotl"opic e lastic ana lysis and the two analyses performed using the proposecl model. However, the dif­ference between the analyses with e lastic and inelastic joint deformation characteristics is not great, indicating th at the

264

predominate cause of masonry non linearity is progressive join l failure rather than inelastic mortal' characte ristics.

Sensitivity 01 Mortar Stress-Strain Characteristics

Since the mortal' streSs-strain curves are no nlinear, their infl uence on masonry behaviour will depend on local stress leveI. To determine the significance of this influ­ence, two additional analyses of the deep beam test were carried out. The first assumed the elastic and shear moduli of the mortal' joints to be twice the original value at ali stress leveis, and the second assumed them to be half the original value. In both cases, the joint l'ailure cri terion and deformation characteristics arter initial fai lure were left the same as in the original model. Ten load increments of 10 kN \Vere again applied.

From the analyses it was found that changing joint stiff­ness did not significantly influence the solution unless the joint flexibility increased to such an extent that premature joint fa ilures occurred with accompanying redistribution of stress.

The two ~'oregoing sensitivity analyses indicate that lin­ear elastic properties co uld be assigned to the mortar joints witho ut unduly affecting the performance of the model. T hi s would result in significant savings in computer time, since one of the two major iterations at each load levei would be eliminated. Cenain ly, small inaccuracies in esti­mates of j oint stiffness caused by the simplifying assump­tions mad e in the derivation of joint p roperties should nOl be signi fi cant.

Joint Failure Criterion

To test the sensitivity of the j oint failure criterion , four alternatives were considered as shown in Figure 6. Empha­sis was placed upon the lower regions of the Tu - Onu ran ge (Region A), since the stress state in the joints will lie in th is region for most 01' th e load range.

Cri teria # I and #2 correspond to joints constmcted from the same basic materiais but with di ffering standa rds of wo rkma nship achieving varying bond strengths. An extreme case was also investigated (C riterion #3) , corre­sponding to zero tensile and shear bond strength. The d eep beam problem was analysed for each case with ali other properties being held conslant. Ten load incre men ts of 10 kN were appl ied.

T he analyses revealed that with the exception o f Crite­ri on #4 , .ioint failure patte rns were significantly influenced by varyingjoint failure criterion. For the case of zero shear and te nsi le bond strengths, solution failed to converge even at the 10 kN load levei due to excessive joint crackin g anel slip , indicating that some nominal bond strength is essential. For the other cases, cracking beca me less exces­sive as the joint strength was increased , with failures fol­lowing a similar partern but a t different load leveis . As woulel be expected , the amou n t of stress reelistribu tion tak ing place correlated with the degree of cracking. A sig­nificant improvement in perfo rmance, (compared to Cri­terion # ,1), was achi eveel by supplying some nominal shear and tensile bonel capacily (as for Cri te rion #2). In this case , despite substantia l joint fail ure, lhe beam was capable

Vth lnternational Brick Masonry Conference

of carrying the load fo r the full loael ra nge in marked contl'ast to Criterion #3.

It can be concluded therefore, that bond strength is extremely important, anel the in-plane behavior of masonry is quite sensitive to its value. The fai lure cri terion used in the model should be evaluateel wilh care, particularl y for th e lower regions of the criterion wh ich elefine tensil e and shear bonel failure .

CONCLUSION

This paper has describeel a llIethou for lhe simulation of the in-plane behaviour of masonry using the fini te e lement technique. Masonry is considereel as a continuum of iso­tropic, elastic bricks acti ng in conjunctio n with inelastic morta l' joints with limiteel shear anel tensile bonel capacity . A moelel of this nature is capable of reproelucing the non­linea r response of masonry for most of the loael range. Final failure can on ly be preel icteel if a crite rion for bl-ick failure is inclueled in the analysis. A crite rion of lhis nature is elifficult to eletermine due to the complex triaxial stress state proeluceel by morta r-brick interaction , the inherenl variability of the material anel the local e /Tects of boneling partern .

The use or th e joint element concept enables the non­linear characteristics 01' masonry to be rep roeluceel from properties elefineel from rela ti vely sim pie un iaxia l tests proelucing various combinations of shear anel normal stress on the mortal' joints. The neeel for biaxia l testi ng, with its associateel elifficulties, is therefore e liminateel.

From a sensitivity anal ysis of lhe parametel's ele finin g the moelel, it appears that of lhe three cha racte ristics that musl be el etennineel , the accurate elefinitio n of joinl fail­ure is the most in1portan t. Some inaccuracies in the exper­imentai evaluation o f the brick a nel joint elefo rmation characteristics can be lolerateel. It has also been shown that lhe major cause of masonry non linearity is p rogres­sive joint failure rather than inelastic mortal' ele formation characteristics.

The accuracy of the anal ysis will be intluenceel by the degree o f variabil ity inherent in lhe material prope rties. Stress history , creep anel wo rkma nship e ffects will also exe rl some innuence in actual wall pane ls.

ACKNOWLEDGEMENTS

The work described in this papel' was ca rrieel ou t in the Department of Civil Engineering at the University of New­castle. The assistance of P. W. Kleeman , Sen ior Lectu reI' anel the staff or the Civi l Engi neering Laboralory is grate­fullyacknowleelgeel.

REFERENCES

I. Goodman, R.E. , Taylor , R.L. , Brekke, T. L. , "A Model fo r lhe Mechanics oI' joinled Rock" , joum. oI" the Soil Mechan ics anel FOllnelation Division , ASCE, Vol. 94, No. SM:\, Pwc. Papel' 59:\7, May 1968, pp. 6:~7- 659:

2. Page, A.W., " Finile Elemenl Moelel 101' Masonr)''', j ourn. oI" lhe Slrllct.ural Div. , ASCE, Vo l. 104, No. STH, Pwc. Pa pel' 1:\957, AuguM 1978,pp. 1267- 1285.

Session IV, Papel' 6, A Model for lhe In-Plane BehaviouT of Masonry and a Sensitivity Analyses 265

3. Page, A.W. , "A Model for the In-P lane Deformation and Fai l­ure of Brickwork ", Engineering Bu lletin CE8, The University of Newcast le, Austra lia, March 1978. -I. Yokel, F.Y. and Faltai, S.C., "Fai lure Hypothesis for Masonry Shear \"'a lls", journal of the Structural Division, ASCE, Vol. 102 ,

No. ST3, Proc. Paper 11 992 , March 1976, pp . 5 15-532. 5. Hegemier, C.A. , Nun n , R.O., Arya, S.K., "Behaviour of Con­crete Mason ry under Biax ial Stress", Proceed ings of the Nonh American Masonry Conference, Bou lder , CoIOl'ado, August 1978, Papel' No. I.

TABLE 1-Variation of Stress Within Masonry Deep Beam For Varying Brick Stiffness

T ypes o f Stress

Vertical Stress on 5th Brick Course* (Lal'gest Variation)

j oint Sheal' Stress Adjacent to 5th Brick Cou rse (Largest Variation)

Maximum Brick Principal Stress* (Adjacent to Support)

'A li Slresses compressive

2·0 I I I / I I ~ , .

:li I

8 · 0

~ :li

Stress (MPa)

0.5E E 6.0E

0. 187 0. 19 1 0.199

0.048 0.045 0.035

0.65 1 0.670 0.80-1

./

~~ ./ 1 · 5 I -

6 -0

./ .. I I t .. ,1 ./ ~

V>

L

~ ~ V>

/ l -O V>

o E L o z

0 -5

o -O 0----1 ... 0-0-0--2-0 ..... 0-0---30-'-0-0-

4 ' 0

/ 2 ' 0

800 1200 1600

AvrroQr 8riek ... -. -... : Av~ro9r 8rickwork ........-. : Mortor

Norma l Sfra in x 10~ la) Ib)

L

" .. .c:

'" .. "

3 '0

: 10 ::>

Rr Q,on I

1·0

MORT AR STRE SS - ST RAIN CURVES

• R~glon 2 RrQ,on 3

2·0 4·0 6·0

eom p. UlI ,molr Nor mol Slr ess - MPo

le) JOIN T FAILURE CRITERION

Figure 1. Materia l Characteristics to Define Joint Element

266

I Lood Sprcoder

I

: : Sfrom Cougc Posit ions •

(o) SCHEMATlC TESTINC ARRANCEMENT

(~om Dimcnsions 710mm" 484mm J( 54mm)

Figure 2. Deep Beam Test

Figure 3. Vertical StI-ess Distribution Along A-A Deep Beam T es t

o Q.

~

-< .o: c o

'" '" " L-... <f)

o u ... L-o. >

Vth lnternational Brick Masonry Conference

, I 8com { . / I

r"

r"

II

(b) FlriiTE ELEMENT SUBDIVISION

..... .,j~ A •• ~

3 ·6

2 ·e

2·0

1 ·2

0-4 2-8

2·4

1·6

o·e

O-O 1 -6

O-S

0·0 1-2

o-s

I !

; I

I ~~ ! ~

-i ~-~- --- I

I - r~ I f - ---- --1 - ' L ____ - -~;~-om~uter- Model ~ , , =t~ . ü,,'im<o' ~ " 0/ t P= Applied Lood f;f ~ __ .~ nn'Li"" fi"", Thro, ,

! i ~ I I

t I

--I

-I 7 ~ ~ - - --,--~ :-..... I

~ - ~, bJ - ---- -~---- -- --

I

L I

-----r---- I

----- - ---- ---------

p= 60 KN I

bJ -- .. t- --~ -~=----~ ~ ~ f--

f9 1---------.y

-- ------ ----- -

P= 40 KJ< I

-- --- -- - - : p~-- - I'----- - --

p= 20 K~ 0.01

Bcom Edge

1 - Stroin Gouge Locotions _

Session IV, PajJer 6, A Model for lhe In-Plane Behaviour of Masomy and a Sensilivily Analyses 267

~ ::I

.. .. OI "-~

'"

o u ~

"-OI >

P - 20 KN P - 40 KN

I j I I I

I I I I I

I ·J-rY . I • : . , •

I • I . .

i . •

_ : Sheor Bond Foilur.

---- : Tensi le Bond Foilur.

p. Totol Applied lood

~

1 1 1 :1 J j

J I 1 I

1 --.l I I •

• • · • . • . •

I I j j · i 1 .! .... . " .....

p- 100 KN

I I -.l I I I

f--L~.---L-1--""", Figure -I. j o int Crackin g Patterns, Deep Beam Test

p- 60 KN P-SO KN

4·0

~

~ / '/ ,

,V/" ...,- - ---~, f\-- -, ,

3,0

:; \ tI' i\.

V - --- ,

í I

2·0

I • • Inelostie Joints --- Elostie Joints

0- - --o Elostie Anolysis (bosed -

J on average br i ek .... ork

I properties)

- . .. - I -------- _.- - l 1·0

, 0·0 1·0

eam B Edge - (2 Beam W Idth -

Beam ~

Crt fer ion Reqlon A Req io n B

Ortqinol Tu - 0·85<r;,u . O· 24 Tu=O 1I000u. 1'92

I Tu = 0·850;,u + OA8 As Above

2 Tu = 0·85<)"u + 0·12 As Above

3 Tu = 0·850;,u As Above

4 Tu = 0'850"nu + 0'24 Tu= 0·13o;;.t 1·56

t-?

Rtq lon 'A' Req ion ' S'

T 7''''' ~ :::f 2'0

~

~ v,

~ .. .c:: Vl

.. 1·0 o E

=>

1'0 2·0 3'0 4 '0

UI1 1mote Normol S1ress - MPo I O"nu)

Figllre 5. Vertical Stress Distributio ns Along Fifrh Brick Cou rse for Varying j oint Propenies

Figurr 6. j o int Failure Cri teria Used iLl Sensitivity Ana lys is.