Progressive failure of masonry shear walls a distinct element approach *

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1 Joural of Appled Mathematc ad Phyc, 2016, *, *-* ISSN Ole: ISSN Prt: Progreve falure of maory hear wall a dtct elemet approach * (Afflato): School of Cvl Egeerg ad Surveyg, Uverty of Souther Queelad, Brbae, Autrala Emal: ya.zhuge@uq.edu.au How to cte th paper: Author 1, Author 2 ad Author 3 (2016) Paper Ttle. ******, *, *- *. Receved: **** **, *** Accepted: **** **, *** Publhed: **** **, *** Copyrght 2016 by author() ad Scetfc Reearch Publhg Ic. Th work lceed uder the Creatve Commo Attrbuto Iteratoal Lcee (CC BY 4.0). Ope Acce Abtract I th paper a umercal model ha bee developed to tudy the behavor of maory wall uder plae hear. Maory ot a mple materal, the fluece of mortar jot a a plae of weake a gfcat feature ad th make the umercal modellg of maory very dffcult epecally whe dyamc (emc) aaly volved. Although fte elemet method a very powerful umercal method for the aaly of tructure ad t ha bee ued to model maory, mcro modellg ature ha made the aaly complex. It wa foud that a dtct elemet method (DEM) could be ued to mulate the behavor of dcotute. I DEM, a old repreeted a a aembly of dcrete block. Jot are modelled a terface betwee dtct bode. It a dyamc proce ad could be appled to mulate the dyamc repoe of maory. I the paper, the umercal oluto obtaed from DEM are valdated by comparg the reult wth thoe obtaed from extg expermet ad fte elemet modellg. Keyword Dtct elemet method (DEM), Ureforced maory (URM), Shear wall, Progreve falure, Numercal modellg 1. Itroducto Fte elemet method a very powerful umercal method for the aaly of tructure ad t ha bee ued to model the behavour of ureforced maory (URM) for more tha 20 year. For tatc aaly, ether a oe-phae (macro modellg of compote) materal model or a two-phae (mcro modellg of brck ad mortar) materal model ha bee ued. Maory ot a mple materal, t compoed of two materal a geometrc array - a aemblage of brck et a mortar matrx. The fluece of mortar jot ad bod a a plae of weake a gfcat feature whch ot preet cocrete ad th make the umercal modellg of maory very dffcult epecally whe emc aaly volved. To mplfy the problem, a homogeeou oe-phae fte elemet model whch ** Specal decrpto of the ttle. (dpeable) DOI: /***.2016.***** **** **, 2016

2 corporate the olear materal behavour ha bee adopted by may vetgator, epecally for dyamc aaly [1], [2]. I the model, ome vetgator adopted a homogeato procedure to corporate the effect of mortar jot [3]. However, the weak jot behavour, whch domate the local falure of maory may cae, could ot be mulated well by a cotuum model. Due to the ature of the problem ad mproved computatoal trategy, a more complcated two-phae fte elemet model ha bee ued by may reearcher recet year [4], [5]. I the model, the brck are modelled a cotuum elemet ad the jot are modelled by le terface elemet. However, uch model udoubtedly made aaly more complex ad t early mpoble to apply thee model to the dyamc/emc aaly of maory. There are other umercal model avalable whch are maly ued for Rock Mechac, amely the Boudary Elemet method (BEM) ad the Dtct Elemet Method (DEM), although they are ot commoly ued for tructural aaly. DEM wa frt troduced by [6] ad ha bee progrevely developed over the pat two decade. The method wa mplemeted the computer program [7] ad t major applcato ha bee the feld of rock mechac ad mg egeerg. However, t teretg to otce the mlarte the geeral materal behavour of rock ad maory, a both could be clafed a o-homogeeou, dcotuou materal. Therefore, t poble to apply the ewly developed aalytcal techque for rock egeerg to mcro modellg of maory ad DEM foud t uefule here. The frt applcato of the DEM to model maory, foud the lterature, wa by [8], where DEM wa appled to vetgate the tablty of freetadg clacal colum uder earthquake exctato. A mplfed 2D model wa adopted ad the deformablty of the block wa eglected. Ther tude proved that the DEM very effcet mulatg the progreve collape of blocky type tructure. More recetly, [9] appled DEM to tudy materal parameter detfcato for maory cottutve model. The tructural repoe of low bod tregth maory wall pael wa mulated by DEM. Materal parameter for DEM model were tued ug a optmzato proce order to acheve mlar repoe to thoe obtaed the laboratory. I th paper, a umercal model baed o the DEM ha bee appled to mulate the repoe of a ureforced maory hear wall pael wth ad wthout a opeg, where the expermetal tetg reult are avalable. The DEM formato troduced frt, followed by the materal model for brck ad jot a well a the falure crtera. The aaly performed ad the reult betwee the dtct elemet model ad expermet ad fte elemet model are compared ad dcued. 2. 2D dtct elemet modellg of URM The Dtct Elemet Method ha bee progrevely developed over the pat three decade. It wa frt troduced by [6] to mulate progreve movemet blocky rock ytem ad the model ha bee mplemeted to a computer program UDEC ce the. DEM mulate the repoe of dcotuou meda ubjected to ether tatc or dyamc loadg. 2

3 Author, Author I the DEM method, a old repreeted a a aembly of dcrete block. Jot are modelled a terface betwee dtct bode. The cotact force ad dplacemet at the terface of a treed aembly of block are foud through a ere of calculato, whch trace the movemet of the block [7]. At all the cotact, ether rgd or deformable block are coected by prg lke jot wth ormal ad hear tffe k ad k, repectvely (Fg 1), that repreet the force-dplacemet relatohp of the jot. Smlar to the Fte Elemet Method (FEM), the ukow the DEM are alo the odal dplacemet ad rotato of the block. However, ulke FEM, whch baed o a cotuum mechac formulato, DEM provde the capacty to repreet the moto of multple, terectg dcotute explctly. DEM ue a explct tme-marchg cheme to olve the equato of moto drectly. The calculato alterate betwee applcato of a force-dplacemet law at all jot ad Newto ecod law at all block. The cottutve law of jot ued to fd cotact force/tree from kow dplacemet. Newto ecod law the gve the moto of the block reultg from the kow force/tree actg o them. Fgure 1. Cotuum ad dcotuum elemet DEM Cottutve law ad falure crtero of jot I geeral, all tele ad hear damage have bee aumed to take place the relatvely weak jot whch ha bee proved to be true for mot maory hear wall pael. I the model, the mortar jot are repreeted umercally a a cotact urface formed betwee two block edge. The cottutve law appled to the cotact are: k u (1) k u (2) where k ad k are the ormal ad hear tffe of the cotact, ad are the effectve ormal ad hear tre cremet, ad u ad u are the ormal ad hear dplacemet cremet. Stree calculated at grd pot located alog cotact are ubmtted to the elected falure crtero. For the propoed model, there a lmtg tele tregth f t for the jot. If the tele tregth exceeded, the = 0. For hear, the Coulomb frcto formulated: C ta max (3) 3

4 where C the coheo ad the frcto agle. For the jot falure model, the dlato retrcted a follow: f max, The = 0 (4) f max ad c u u, the = 0 (5) where u c crtcal hear dplacemet Cottutve law ad falure crtero of block The ormal ad hear tree calculated Equato (1) ad (2) are the ued a the boudary tree to ether rgd or deformable block. If the block are deformable, they wll be terally dcreted to fte dfferece tragular elemet frt. The ue of tragular elemet elmate the problem of hourgla deformato that may occur wth cotat-tra fte dfferece quadrlateral. The vertce of the tragular elemet are grd pot (Fg 1). Newto ecod law appled at each grd pot a follow: ( t) t j jd F ( ) u g (6) t m where the urface eclog the ma m lumped at the grd pot, j the ut ormal (t) to, F the reultat of all exteral force appled to the grd pot ad g the gravtatoal accelerato. A et odal force vector, F (t) calculated at each grd pot. Th vector clude (g ) cotrbuto from appled load ad from body force due to gravty. Gravty force, F are computed from: ( g) F g m (7) where m g the lumped gravtatoal ma at the grd pot, defed a the um of oe-thrd of the mae of tragle coected to the grd pot. If the body at equlbrum, F (t) o the ode wll be zero; otherwe, the ode wll be accelerated accordg to the fte dfferece form of Newto ecod law of moto ad followg the cetral dfferece tegrato cheme: ( t) ( tt/ 2) ( tt/ 2) F u u t (8) m Where u the velocty, m the ma, ad t the tme. Durg each tme tep, tra ad rotato are related to odal dplacemet the uual form: 1 j ( u, j u j, ) 2 (9) 1 j ( u, j u j, ) 2 The the elected cottutve law for the block ued a cremetal form to determe tree at each grd pot. A olear model ca be ealy corporated a the oluto cheme ued by DEM the explct tme marchg cheme. To mply the problem, the Mohr-Coulomb falure crtero wth teo cut-off adopted the model. The ew poto of the block duce ew codto at block boudare ad thu ew cotact force. 3. Comparo of umercal ad expermetal reult g 4

5 Author, Author The umercal example preeted th paper were performed wth the umercal code UDEC [7]. A ere of URM hear wall were teted by [4] whch cluded reult for mcromodellg. Thee tetg reult are ued to valdate the umercal model. The hear wall pael ha a wdth/heght rato of 990 mm x1000mm, bult up wth 18 coure wth ad wthout a cetral opeg. The wall wa made of wre-cut old clay brck wth dmeo 210 mm x 52 mm x 100 mm ad 10 mm thck mortar. The wall wa ubjected to a vertcal compreve tre m = 0.3 MPa. Durg tetg, a horzotal dplacemet wa mootocally creaed utl falure. The expermetal crack patter ad FEM reult of the fal falure patter of the wall are how Fgure 2(a) ad 2(b) repectvely [4]. A the load creaed, a tepped dagoal crack through head ad bed jot wa formed, the crack progree the drecto of the upport ad fally a collape mecham formed (Fgure 2). The tele ad hear crack developmet ad the progreve falure of the wall modelled by DEM are dcated Fgure 3. It ca be ee that the behavour of the wall well captured by the propoed model. (a) (b) Fgure 2. (a). Expermetal crack at a dplacemet of 2 mm; (b). FEM model of the fal falure patter [4]. JOB TITLE : Maory w all; UDEC (Vero 3.10) LEGEND Apr-02 17:10 cycle block plot (a) (b) Fgure 3. DEM Model: (a). Crack at a dplacemet of 2 mm; (b). Fal falure patter. The tetg reult of the wall wth the ame geometry ad loadg codto, but wth a cetral opeg how Fgure 4(a). The cetral opeg of the wall produce two mall weak per. The collape mecham wth falure of the compreed toe, located at the bottom ad top of the wall ad at the bottom ad top of the mall per. The wall behave mlarly to four rgd coected by the hge how. The DEM reult how Fgure 5

6 4(b) clearly dcated the two compreve trut have bee formed ad the locato of the maxmum compreve tree agreed well wth tetg reult. (a) (b) Fgure 4. (a). Collape mecham wth four hged rgd block [4]; (b). DEM reult of mmum prcpal tree at peak load. 4. Cocluo Maory ot a mple materal, the fluece of mortar jot a a plae of weake a gfcat feature. Due to the ature of the problem, two-phae mcro-modellg of brck ad mortar requred. However, th make the umercal modellg of maory very complex. Th paper dcued a alteratve ad mple way of modellg maory, whch ug the Dtct Elemet Method. The prelmary reult obtaed the curret reearch have how the great potetal of the method, epecally for dyamc aaly. Referece [1] Vrataou, V. (1991) Determato of the behavour factor for brck maory pael ubjected to earthquake acto. Proc. of the Iteratoal Cof. o Sol Dyamc ad Earthquake Egeerg, Germay: [2] Zhuge, Y., Thambratam, D. P. ad Corderoy, J. (1998) Nolear dyamc aaly of ureforced maory. Joural of Structural Egeerg, ASCE, 124(3), [3] Gambarotta, L. ad Lagomaro, S. (1997) Damage model for the emc repoe of brck maory hear wall. Part II: the cotuum model ad t applcato. Earthquake Egeerg & Structural Dyamc, 26, [4] Loureco, P. B. (1996) Computatoal tratege for maory tructure, PhD The, Delft Uverty of Techology, The Netherlad. [5] Wag, C., Forth, J.P., Nkta, N. ad Sarho, V. (2016) Retrofttg of maory wall by ug a mortar jot techque: expermet ad umercal valdato. Egeerg Structure, 117, [6] Cudall, P. A. (1971) A Computer model for mulatg progreve large cale movemet blocky rock ytem. Proceedg of the Sym. of the Iteratoal Socety for Rock mechac, Nacy, Frac, Vol. 1, II-8: [7] ITASCA Coultg Group (2000) Uveral Dtct Elemet Code, ITASCA coultg Group, Ic., Meapol, Meota, USA. [8] Pychar, I., Papatamatou, D. ad Alexadr, A. (2000) Parametrc vetgato of the tablty of clacal colum uder harmoc ad earthquake exctato. Eart. Eg. Ad Struct Dyamc, 29, [9] Sarho, V. ad Sheg, Y. (2014). Idetfcato of materal parameter for low bod tregth maory. Egeerg Structure, 60,