an orthotropic continuum model for the analysis of masonry structures

Delft University **of** TechnologyFaculty **of** Civil EngineeringAN ORTHOTROPIC CONTINUUM MODEL FORTHE ANALYSIS OF MASONRY STRUCTURESAuthor : P. B. LOURENÇODate : June 1995TU-DELFT report no. 03-21-1-31-27TNO-BOUW report no. 95-NM-R0712TNO Building **an**d Construction ResearchComputational Mech**an**ics

SummaryA **continuum** **model** **for** **the** **an**alysis **of** **masonry** **structures** subjected to in-pl**an**e loading is proposed.The **model** combines **an**isotropic elastic behaviour with **an**isotropic plastic behaviour.The proposed composite yield surface includes a Hill type failure criterion **an**d a R**an**kine typefailure criterion. Different uniaxial strengths **an**d post-peak behaviour are predicted by **the** **model**along **the** material axes both in tension **an**d compression. The **for**mulation **of** **the** elastoplasticalgorithm is made in modern plasticity concepts, including implicit Euler backward return mappingschemes **an**d consistent t**an**gent operators **for** all regimes **of** **the** **model**. The problem **of** localizationis tackled, in **an** engineering way, by **the** introduction **of** **an** equivalent length h relatedto **the** element size.The per**for**m**an**ce **of** **the** implementation **an**d behaviour **of** **the** **model** is assessed by me**an**s **of** singleelement tests. A comparison between numerical results **an**d experimental results available in**the** literature shows good agreement both **for** ductile **an**d brittle failure modes, providing that **the**size **of** **the** structure is large enough to permit a macro-**model**ling strategy.AcknowledgementsThe author wishes to express his gratitude to Dr. ir. J.G. Rots **an**d Dr. ir. P. Feenstra **for** **the**ir supportduring **the** course **of** this work.The fin**an**cial support by **the** Ne**the**rl**an**ds Technology Foundation (STW) under gr**an**tDCT-33.3052 is gratefully acknowledged.The calculations have been carried out with **the** Finite Element Package DIANA **of** TNO Building**an**d Construction Research on a Silicon Graphics Indigo R4000 workstation **of** **the** DelftUniversity **of** Technology.

TNO-95-NM-R0712 1995 11. INTRODUCTIONAn accurate **an**alysis **of** **masonry** **structures** in a macro-**model**ling (or composite) perspectiverequires a material description **for** all stress states. The difÞculties are, however, quite strong. Thisis due, not only, to **the** fact that almost no comprehensive experimental results (including pre- **an**dpost-peak behaviour) are available, but also to intrinsic difÞculties in **the** **for**mulation **of** **orthotropic**inelastic behaviour. It is noted that a representation **of** **an** **orthotropic** yield surface in terms **of** principalstresses or stress invari**an**ts is not possible. For pl**an**e stress situations, which is **the** case **of** **the**present report, a graphical representation in terms **of** **the** full stress vector (σ x , σ y **an**d τ xy ) is necessary.The material axes are assumed to be deÞned by **the** bed joints direction (x direction) **an**d **the**head joints direction (y direction). In some **of** **the** pictures shown below, **the** yield surface **an**d **the**experimental results are plotted in terms **of** principal stresses **an**d a **an**gle θ . The **an**gle θ measures**the** rotation between **the** principal stress axes **an**d **the** material axes. Clearly, different principalstress diagrams are found according to different values **of** θ .Tw o different strategies **for** **the** macro-**model**ling **of** **masonry** c**an** be used, namely:¥ Extend **the** conventional **for**mulation **of** isotropic quasi brittle materials in order to describe**orthotropic** behaviour. Current approaches consider different inelastic criteria **for** compression**an**d tension. A possible extension **of** conventional **model**s is to use a Hill type yield criterion**for** compression **an**d a R**an**kine type à yield criterion **for** tension (see Fig. 1). Thisapproach will be discussed in this report;¥ Describe **the** material behaviour with a single yield criterion. The H**of**fm**an** yield criterion isquite ßexible **an**d attractive to use, see Schellekens **an**d de Borst (1990) **an**d Scarpas **an**dBlaauwendraad (1993), but yields a non-acceptable Þt **of** **the** **masonry** experimental values(see Fig. 2). A least squares Þt **of** **the** experimental results from Page (1981) with a H**of**fm**an**type yield criterion turns out to show no tensile strength in uniaxial behaviour **an**d a m**an**ualÞt through **the** different uniaxial strengths plus **the** compressive failure obtained upon loadingwith σ 1 = σ 2 **an**d θ = 0 giv es a very poor representation **of** **the** diagrams **for** **the** o**the**r θ values.In fact a single surface Þt **of** **the** experimental values would lead to **an** extremely complexyield surface with a mixed hardening/s**of**tening rule in order to describe properly **the**inelastic behaviour. The author believes that this strategy is practically non-feasible.( ) The word type is used here because **the** original authors, see Hill (1948) **an**d H**of**fm**an** (1967), assumed a3-dimensional **for**mulation. The inßuence **of** **the** out-**of**-pl**an**e direction is generally unknown **an**d will be notconsidered in **the** present report. The proposed yield surface **for** compression should in fact be considered as aparticular case **of** **the** complete quadratic **for**mulation from Tsai **an**d Wu (1971).(à)The word type is used here because **the** R**an**kine yield criterion represents **the** material strength along **the** maximumprincipal stress. For **an** **an**isotropic material such deÞnition is clearly not possible. The proposed yield surface **for**tension represents only a Þt **of** **the** experimental results.

2 1995 TNO-95-NM-R0712σ2σ2σ2σ 1σ 1σ 1σσ 2σ22θθσ1σ1σ 1θ =0° b) θ = 22.5° c) θ = 45.0°Fig. 1 - Comparison between a Hill type + R**an**kine type composite yield surface**an**d experimental results from Page (1981)Through axes + biaxial compression fitBest fit with least squaresσ2σ 1σ 2σ1Through axes + biaxial compression fitBest fit with least squaresσ2σ 1θσ2σ1Through axes + biaxial compression fitBest fit with least squaresσ2σ 1θσ 2σ 1θ =0° b) θ = 22.5° c) θ = 45.0°Fig. 2 - Comparison between a H**of**fm**an** type single yield surface**an**d experimental results from Page (1981)The composite yield surface to be presented features **an**isotropic behaviour in tension **an**d compressionas well as non-isotropic s**of**tening. The **for**mulation **of** **the** **model** is given in modern plasticityconcepts including fully implicit Euler backward return mapping, a local Newton-Raphson methodto solve **the** return mapping, proper h**an**dling **of** **the** corners **an**d t**an**gent operators consistent with**the** integration **of** **the** update equations **for** all modes **of** **the** **model**, including **the** apex **an**d **the** cornerregimes.The application **of** **the** **model** is limited, at present, to a pl**an**e stress conÞguration. The **model** isimplemented in **the** DIANA Þnite element package. For **the** purpose **of** compatibility with **the** currentcode **an**d **for** simpler future extension **of** **the** **model** to a 3-dimensional stress-state, **the** **for**mulationwill be given in 4 stress components (i.e. pl**an**e strain). The exp**an**sion/compression mech**an**ismfrom 3 stress components to 4 stress components is described in de Borst (1991) **an**d will not bereviewed here.Only a few authors tried to develop speciÞc macro-**model**s **for** **the** **an**alysis **of** **masonry** **structures**, inwhich **an**isotropic elasticity is combined with **an**isotropic inelastic behaviour. To **the** knowledge **of****the** author **of** this report only Dh**an**asaker et al. (1985,1986) **an**d Seim (1994) dealt with **the** implementation**of** a speciÞc numerical **model** **for** **masonry**. Both **of** **the**se authors fail to include rationallys**of**tening in **the** **model**: brittle s**of**tening was included **for** tension, which leads to mesh sensitiveresults **an**d numerical instabilities, **an**d compressive s**of**tening was ei**the**r absent or brittle.Moreover, **the** numerical **an**alyses **of** **masonry** walls carried out by **the**se authors included interfaceelements in **the** boundaries. This yields a weak assessment **of** **the** material macro-**model** because **the**interface elements were responsible **for** most **of** **the** inelastic behaviour. Finally, **the** complex yield

TNO-95-NM-R0712 1995 3surfaces suggested by **the** above authors almost preclude **the** use **of** modern plasticity concepts. The**model** proposed by Dh**an**asaker et al. (1985,1986) **for** solid units **masonry** consists **of** three ellipticalcones, see Fig. 3. The **model** is based on **the** experimental Þndings **of** Page (1981) **an**d Þts **the**data extremely well, see Fig. 4. However, it is difÞcult to h**an**dle this **model** in modern plasticityconcepts. Not only **the** composite yield surface contains several corners **an**d apexes but also **the**inclusion **of** ÒrealisticÓ inelastic behaviour is practically impossible - how to control **the** exp**an**sion/shrinkage**of** **the** yield surface?σyσxσy0.10.20.3σ xyσxFig. 3 - Yield surface **for** solid units **masonry** proposed by Dh**an**asekar (1986) with iso-shear stress linesContour spacing: 0.1 f mxσ2σ 1σ 2σ1σ2σ 1θσ2σ1σ2σ 1θσ 2σ 1θ =0° b) θ = 22.5° c) θ = 45.0°Fig. 4 - Comparison between Dh**an**asekarÕs (1986) **masonry** yield surface**an**d experimental results from Page (1981)The **model** proposed by Seim (1994) is based on **the** failure surface proposed by G**an**z (1989) **for**hollow **masonry** units, see Fig. 5. The failure surface was derived from **the** **the**orems **of** Limit Analysis,assuming rigid-perfectly plastic behaviour **for** **the** **masonry** components. The assumptions **for****the** material behaviour **of** **the** components include a Mohr-Coulomb yield surface **for** **the** units **an**d aCoulomb friction law **for** **the** joints. The head joints do not feature **an**y strength. This yield surfaceis even more difÞcult to h**an**dle numerically in a consistent way, specially when s**of**tening isincluded in **the** **model** with different s**of**tening parameters **for** all regions. Seim (1994) assumedideally plastic behaviour in compression **an**d brittle tension failure.

4 1995 TNO-95-NM-R07120.50.10.20.30.4σyσxσyσ xyσxFig. 5 - Yield surface **for** hollow units **masonry** proposed by G**an**z (1989) with iso-shear stress linesContour spacing: 0.1 f mxFinally, it should be realised that a **masonry** macro-**model** always includes some degree **of** approximation.The basic features **of** a two-material composite c**an**not be reproduced but only smeared outin **the** **continuum**. The Þeld **of** applications **of** **the**se **model**s are indeed large **structures** where **the**state **of** stress **an**d strain across a macro-length c**an** be assumed uni**for**m. It is noted that, due to **the**difÞculties **of** carrying out experiments in large **structures**, **the** examples used in **the** present report**for** **the** assessment **of** **the** **model** per**for**m**an**ce (extracted from available literature) are, in general,ÒsmallÓ **for** a macro-**model**ling strategy.The **model** proposed in **the** present report seems however capable **of** reproducing **the** globalbehaviour **of** **the** **an**alysed **structures**. Satisfactory predictions **of** collapse loads are also found providedthat **the** **structures** do not show a highly localized failure mode. In such cases **the** interactionbetween units **an**d mortar c**an** be **of** capital import**an**ce **an**d a micro-**model**, in which both **masonry**components are **model**led separately, should be used instead.

TNO-95-NM-R0712 1995 52. TENSION - A RANKINE TYPE ANISOTROPIC YIELD SURFACEIn this section a possible extension **of** **the** st**an**dard R**an**kine yield criterion to **an** **orthotropic** **for**mulationis given. It is clear that **the** yield surface obtained c**an**not be derived from **the** materialstrength in **the** maximum principal stress direction. The proposed yield surface has to be regardedas pure curve Þtting from existing experimental results. The yield surface is coined as R**an**kine typeas **the** derivation is based on **the** original R**an**kine yield surface.The difÞculties **of** **for**mulating **the** R**an**kine yield criterion in **the** principal stress state are addressed,**for** example, in Feenstra (1993). Consider a pl**an**e-stress situation in which **the** major principalstress σ 1 **an**d **the** minor principal stress σ 2 are deÞned by me**an**s **of** a MohrÕs circle asσ 1, 2 = σ x + σ y22⎛σ ±√⎺⎺⎺⎺⎺x − σ y ⎞⎝ 2 ⎠+ τ 2 xy . (1)The hardening behaviour is assumed to be described by two internal variables κ 1 **an**d κ 2 which govern**the** corresponding principal stresses. The yield functions are **the**n given by **the** principal stressσ j **an**d **an** equivalent stress σ j as a function **of** **an** internal variable κ j according to⎧⎪⎪⎪⎨⎪⎪⎪⎩f I = σ x + σ y2f II = σ x + σ y22⎛σ +√⎺⎺⎺⎺⎺x − σ y ⎞⎝ 2 ⎠2⎛σ +√⎺⎺⎺⎺⎺x − σ y ⎞⎝ 2 ⎠+ τ 2 xy − σ I (κ I )+ τ 2 xy − σ II (κ II )The yield function is depicted in Fig. 6 in **the** principal stress space. The problem which occurswith **the** two yield functions is that **the** tr**an**s**for**mation between **the** stress space **an**d **the** principalstress space has to be deÞned uniquely **for** two yield functions with different hardening **model**s..(2)σ 2σ 1Fig. 6 - R**an**kine yield surface in **the** principal stress spaceIn Feenstra (1993), **the** **for**mulation is given by a single function which is governed by **the** Þrst principalstress **an**d one equivalent stress which describes **the** s**of**tening behaviour **of** **the** material. Theassumption **of** isotropic s**of**tening is not completely valid **for** a material such as concrete or **masonry**which c**an** be loaded to **the** tensile strength even if in **the** perpendicular direction **the** strength has

6 1995 TNO-95-NM-R0712been reduced due to s**of**tening **of** **the** material. This problem is partially solved in Feenstra (1993)by using kinematic s**of**tening such that **the** yield surface is shifted in **the** direction **of** **the** Þrst principalstress. It is noted that **the** above **for**mulation **of** kinematic s**of**tening is also not quite realistic: letus assume that **the** material is loaded initially along a certain direction until s**of**tening is completed.If now **the** material is loaded in a direction orthogonal to **the** crack previously open, ideal plasticbehaviour is found. This is due to **the** fact that all **the** fracture energy has been consumed during **the**opening **of** **the** Þrst crack. An eleg**an**t solution is found if two independent s**of**tening parameterscontrol **the** shifting **of** **the** yield surface. Such a **for**mulation **for** **the** R**an**kine yield surface is givenin Louren•o et al. (1995) **an**d reproduces exactly **the** material feature in tension just described. It isshown by Louren•o et al. (1995) that **the** response **of** **the** **model** seems to lie between **the** Þxed **an**drotating crack **model**s **an**d, **the**re**for**e, comprises **the** beneÞt **of** a **model** with memory **an**d a ßexibleshear response. Un**for**tunately, **for** certain values **of** **the** trial stress **the** return mapping becomes illposed**an**d **an** almost singular Jacobi**an** is found close to **the** solution. This precludes **the** use **of** sucha yield surface in large scale computations due to **the** lack **of** robustness **an**d need to considerextremely small steps.Here a different approach is used aiming at **an** **orthotropic** R**an**kine type yield surface controlled byonly one scalar that measures **the** amount **of** s**of**tening simult**an**eously in **the** two material axes but,still, two corresponding different fracture energies are considered. This approach is less attractivefrom a physical point **of** view but leads to a more robust algorithm **an**d should be preferred in practice.The R**an**kine yield surface reads, cf. eqs (1,2),f 1 = σ x + σ y22⎛σ +√⎺⎺⎺⎺⎺x − σ y ⎞⎝ 2 ⎠This expression c**an** however be rewritten asf 1 = (σ x − σ t (κ t )) + (σ y − σ t (κ t ))2+ τ 2 xy − σ t (κ t ) . (3)2⎛ (σ +√⎺⎺⎺⎺⎺⎺⎺⎺⎺⎺x − σ t (κ t )) − (σ y − σ t (κ t )) ⎞⎝2⎠+ τ 2 xy , (4)where coupling exists between **the** stress components **an**d **the** equivalent stress. Setting **for**th aR**an**kine type yield surface **for** **an** **orthotropic** material, with different tensile strengths along **the** x, ydirections, see Þg. 7, is now straight**for**ward if eq. (4) is modiÞed t**of** 1 = (σ x − σ t1 (κ t )) + (σ y − σ t2 (κ t ))22⎛ (σ +√⎺⎺⎺⎺⎺⎺⎺⎺⎺⎺⎺x − σ t1 (κ t )) − (σ y − σ t2 (κ t )) ⎞⎝2⎠+ ατ 2 xy . (5)Note that a parameter α is introduced to calibrate **the** shear strength. The parameter α readsα = f tx f tyτ 2 u, (6)where f tx , f ty **an**d τ u are, respectively, **the** uniaxial tensile strengths in **the** x **an**d y directions **an**d **the**pure shear strength. Note that **the** material axes are now Þxed **an**d it shall be assumed that allstresses **an**d strains **for** **the** elastoplastic algorithm are given in **the** material reference axes.

TNO-95-NM-R0712 1995 7τxyτ uσxftxftyσyFig. 7 - Orthotropic R**an**kine type yield surface (plotted **for** τ xy ≥ 0)Eq. (5) c**an** be recast in a matrix **for**m asf 1 = ( 1 / 2 ξ T P t ξ ) 1 / 2+ 1 / 2 π T ξ , (7)where **the** projection matrix P t reads⎡1/ 2 − 1 / 2 0 0 ⎤⎢ −P t =1 / 21/ 2 0 0 ⎥⎢⎥ , (8)⎢0 0 0 0⎥⎣ 0 0 0 2α ⎦**the** projection vector π readsπ = { 1, 1, 0, 0 } T , (9)**the** reduced stress vector ξ readsξ = σ − η (10)**an**d **the** back stress vector η readsη = { σ t1 (κ t ), σ t2 (κ t ), 0, 0 } T . (11)Exponential tensile s**of**tening is considered **for** both equivalent stress-equivalent strain diagrams,with different fracture energies (G fx **an**d G fy ) **for** **the** yield values, **an**d readsσ t1 = f tx exp ⎛ ⎝ − h f tx ⎞κG tfx ⎠**an**dσ t2 = f ty exp ⎛ ⎝ − h f tyG fyκ t⎞⎠ . (12)Here, **the** scalar κ t controls **the** amount **of** s**of**tening **an**d **the** equivalent length h, see Baº**an**t **an**d Oh(1983), is assumed to be related to **the** area **of** **an** element A e by, see Feenstra (1993),⎛h = α h √⎺⎺A e = α h ⎜⎝n ξΣ n ηξ = 1 η = 11 /2⎞Σ det(J) w ξ w η ⎟ , (13)⎠in which w ξ **an**d w η are **the** weight factors **of** **the** Gaussi**an** integration rule as it is tacitly assumed

8 1995 TNO-95-NM-R0712that **the** elements are always integrated numerically. The local, isoparametric coordinates **of** **the**integration points are given by ξ **an**d η. The factor α h is a modiÞcation factor which is equal to one**for** quadratic elements **an**d equal to √⎺2 **for** linear elements, see Rots(1988). With this approach **the**results which are obtained in **the** **an**alyses are reasonably objective with regard to mesh reÞnement.It is however possible that **the** equivalent length **of** **an** element results in a snap-back at **the** constitutive**model** if **the** element size is large. Then, **the** concept **of** fracture energy which has beenassumed is no longer satisÞed. In such a case, **the** strength limit has to be reduced in order toobtain **an** objective fracture energy by a sudden stress drop, resulting at a certain stage in brittlefailure, see Rots (1988). The condition **of** maximum equivalent length is given byh ≤ G f Eft2 , (14)where E is **the** YoungÕs modulus in **the** respective material axis. If **the** condition is violated, **for** **an**y**of** **the** material axes, **the** tensile strength in **the** respective axis is reduced according t**of** t = ⎛ 1G f E/2⎞. (15)⎝ h ⎠It is noted that eq. (15) yields a reduction on **the** material strength without **an**y physical ground. Theidea is solely to obtain **an** energy release independent **of** **the** mesh size but this objective should beaccomplished by me**an**s **of** a mesh reÞnement **an**d not with a strength reduction.Finally, **for** **an** **orthotropic** material with different yield values along **the** material axes, it wouldseem only natural to assume two different equivalent lengths along **the** material axes (this is **of**course irrelev**an**t in **the** special case **of** a mesh with square elements). However, **the** equivalentlength depends on so m**an**y factors that more complex assumptions are disregarded.The ßow rule is written in a st**an**dard fashion (non-associated s**of**tening) asúεúε p = ú∂g 1 λ t∂σ , (16)where **the** plastic potential g readsg 1 = ( 1 / 2 ξ T P g ξ ) 1 / 2+ 1 / 2 π T ξ (17)**an**d **the** projection matrix P g is given by⎡1/ 2 − 1 / 2 0 0 ⎤⎢ −P g =1 / 21/ 2 0 0 ⎥⎥⎥⎦⎢. (18)⎢0 0 0 0⎣ 0 0 0 2α gThe parameter α g is taken equal to **the** unit value (R**an**kine plastic ßow), unless o**the**rwise stated.The inelastic behaviour is described by a modiÞed strain s**of**tening hypo**the**sis given byúκ t = úε p α = úε p x + úε p y2+ 1 / 2 (úε x√⎺⎺⎺⎺⎺⎺p − úε y) p 2 + 1 (úγ xy) p 2 . (19)α gIt is noted that **the** above expression represents a ÒmodiÞedÓ maximum principal plastic strain, inwhich **the** shear component has been averaged by **the** inverse **of** α g . The expression **for** **the** s**of**teningscalar rate úκ t c**an** be recast in a matrix **for**m **an**d reads

TNO-95-NM-R0712 1995 9whereúκ t = úε p α = ⎛ 1 /2⎝ 1 / 2 (úεúε p ) T Q úεúε p ⎞+⎠1 / 2 π T úεúε p , (20)⎡ 1/⎢ 2⎢ − 1 / 2Q = ⎢⎢ 0⎢ 0⎣− 1 / 21/ 20000000⎤⎥⎥⎥⎥⎥⎦0. (21)012α g2.1 Return mapping algorithm - Tension regimeThe integration **of** **the** constitutive equations given above is a problem **of** evolution that c**an** beregarded as follows. At a stage n **the** total strain **an**d plastic strain Þelds as well as **the** hardeningparameter (or equivalent plastic strain) are known:{ε n , ε p n, κ t, n } given data . (22)Note that **the** elastic strain **an**d stress Þelds are regarded as dependent variables which c**an** bealways be obtained from **the** basic variables through **the** relationsε e n = ε n − ε p n **an**d σ n = D ε e n . (23)There**for**e, **the** stress Þeld at a stage n+1is computed once **the** strain Þeld is known. The problemis strain driven in **the** sense that **the** total strain ε n+1 is trivially updated according to **the** exact **for**mulaε n+1 = ε n +∆ε n+1 . (24)It remains to update **the** plastic strains **an**d **the** hardening parameter. These qu**an**tities are determinedby integration **of** **the** ßow rule **an**d hardening law over **the** step n → n+1. In **the** frame **of** afully implicit Euler backward integration algorithm this problem is tr**an**s**for**med into a constrainedoptimization problem governed by discrete Kuhn-Tucker conditions as shown by Simo et al.(1988). It has been shown in different studies, e.g. Ortiz **an**d Popov (1985) **an**d Simo **an**d Taylor(1986), that **the** implicit Euler backward algorithm is unconditionally stable **an**d accurate. Thisalgorithm results in **the** following discrete set **of** equations:ε n+1 = ε n +∆ε n+1σ n+1 = σ trial −∆λ t, n+1 D ∂g ⎪1⎪∂σ⎪ n+1, (25)ε p n+1 = ε n p ∂g 1⎪+∆λ t, n+1 ⎪∂σ⎪ n+1κ t, n+1 = κ t, n +∆κ t, n+1in which ∆κ t, n+1 results from **the** integration **of** **the** rate equation, eq. (19), **an**d **the** elastic predictorstep returns **the** value **of** **the** elastic trial stress σ trial asσ trial = σ n + D ∆ε n+1 . (26)The above equations must be satisÞed **an**d simult**an**eously **the** yield criterion must be fulÞlled

10 1995 TNO-95-NM-R0712f 1 = ( 1 / 2 ξ T n+1P t ξ n+1 ) 1 / 2+ 1 / 2 π T ξ n+1 = 0 . (27)It is noted that **the** update **of** **the** s**of**tening scalar ∆κ t, n+1 reduces to **the** particularly simple expression∆κ t, n+1 =∆λ t, n+1 . (28)The above equations c**an** be reduced to **the** following set **of** Þve equations containing 5 unknowns(σ n+1 **an**d ∆κ t, n+1 =∆λ t, n+1 )⎧⎪D −1 (σ n+1 − σ trial ∂g) 1⎪+∆λ t, n+1 ⎪ = 0⎪∂σ⎪⎨n+1⎪f⎪ 1 = ( 1 / 2 ξ n+1P T t ξ n+1 ) 1 / 2+ 1 / 2 π T ξ n+1 = 0⎩Due to **the** coupling **of** **the** σ n+1 **an**d κ t, n+1 values it is not possible to obtain **an** explicit one variablenon-linear equation. The system **of** non-linear equations is **the**re**for**e solved with a regular Newton-Raphson method. The Jacobi**an** necessary **for** this procedure reads (note that **the** subscript n+1 isdropped in **the** derivatives **an**d matrices **for** convenience).(29)where⎡D⎢−1 ∂ 2 g 1+∆λ t, n+1∂σ 2J = ⎢ −−−−−−−−−−−−⎢T⎢ ⎛ ∂ f 1 ⎞⎣ ⎝ ∂σ ⎠|+|∂g 1∂σ +∆λ ∂ 2 g 1 ⎤t, n+1∂σ ∂κ t ⎥⎥⎥⎥⎦−−−−−−−−−− , (30)∂ f 1∂κ t∂ f 1∂σ = P t ξ n+12( 1 / 2 ξ T n+1P t ξ n+1 ) 1 / 2+ 1 / 2 π ;∂g 1∂σ = P g ξ n+12( 1 / 2 ξ T n+1P g ξ n+1 ) 1 / 2+ 1 / 2 π∂ 2 g 1∂σ 2 = P gP g ξ n+1 ξ n+1P T g2( 1 / 2 ξ n+1P T −g ξ n+1 ) 1 / 24( 1 / 2 ξ n+1P T .g ξ n+1 ) 3 / 2∂g 1∂κ t=− ⎛ ∂g 1 ⎞T ∂η;⎝ ∂σ ⎠ ∂κ t∂ 2 g 1=− ∂2 g 1∂σ ∂κ t ∂σ 2∂η∂κ t;∂η= ⎧ ∂σ t1⎨ , ∂σ t2,0,0 ⎫ T⎬∂κ t ⎩∂κ t ∂κ t ⎭(31)The numerical algorithm explained above is howev er not stable through all **the** stress domain. In **the**apex **of** **the** yield surface **the** gradient **of** **the** plastic potential, cf. eq. (31.1), is not deÞned. It is fur**the**rnoted that **the** proposed plastic potential c**an** be written in a quadratic **for**m asg 1 = 1 / 2 ξ T P g ξ + 1 / 4 ξ T π |ξ T π | . (32)However, this **for**mulation does not overcome **the** problem **of** a non-deÞned gradient in **the** apex.The new expression **for** **the** gradient reads∂g 1∂σ = P gξ + 1 / 2 π |ξ T π | , (33)which degenerates to a point in **the** apex ( ∂g 1= 0) due to **the** singularity **of** yield surface.∂σFor **the** apex regime, **the** stress update, cf. eq. (25.2), in case **of** pl**an**e stress conÞguration, is independent**of** **the** trial stress. It is simply a return mapping to **the** apex **an**d reads

TNO-95-NM-R0712 1995 11σ n+1 = η n+1 . (34)However, **the** present yield surface is implemented in **the** exp**an**sion/compression concept **for** pl**an**estrain (four stress components), see de Borst (1991). This concept yields several adv**an**tages **for**multi-purpose Þnite elements packages but **for** **the** present yield surface no adv**an**tage is found. On**the** contrary, eq. (34) c**an**not be used because **the** third normal stress component is not zero during**the** global/local iteration procedure. In this case (σ n+1 ) z must be calculated so that (∆ε p n+1 ) z = 0.This c**an** be easily done from**an**d results in(∆ε p n+1 ) z = ⎛ ⎝ D−1 (σ trial − σ n+1 ) ⎞ ⎠ z= 0 (35)(σ n+1 ) z = (σ trial ) z + d−1 31 ⎛d −133⎝ σ trial ⎞− σ n+1+ d−1 32⎠ xd −133⎛⎝ σ trial ⎞− σ n+1 . (36)⎠ yHere **the** values d −1ij are terms from **the** compli**an**ce matrix D −1 . The o**the**rs stress components aregiven from eq. (34) **an**d read(σ n+1 ) x = σ t1 (κ t ) , (σ n+1 ) y = σ t2 (κ t ) **an**d (τ n+1 ) xy = 0 . (37)The above expression c**an** be adv**an**tageously recast in a matrix **for**mat **an**d, after some m**an**ipulations,**the** stress update readsσ n+1 = A 1 η n+1 + A 2 σ trial , (38)where **the** auxiliary matrices A 1 **an**d A 2 are given by⎡⎢⎢A 1 = ⎢⎢⎢⎣10− d−1 31d −133001− d−1 32d −133000100 ⎤⎥0 ⎥⎥0 ⎥⎥1 ⎦**an**d A 2 =⎡⎢⎢⎢⎢⎢⎣00d −131d −133000d −132d −133000100 ⎤⎥0 ⎥⎥0 ⎥. (39)⎥0 ⎦The stress update given is sufÞcient to fulÞll f n+1 = 0. It remains to update **the** s**of**tening scalaraccording to eq. (20). For this purpose a non-linear equation in one variable c**an** be written asF = F(∆κ t, n+1 ) =∆κ t, n+1 −∆ε p α ,n+1 = ⎛ 1 /2⎝ 1 / 2 (∆ε p n+1 )T Q∆ε p ⎞n+1+⎠1 / 2 π T ∆ε p n+1 = 0 , (40)where **the** increment **of** **the** plastic strain vector ∆ε p n+1c**an** be calculated from∆ε p n+1 = D−1 (σ trial − σ n+1 ) (41)**an**d **the** update **of** **the** stress vector σ n+1 is given by eq. (38). The sec**an**t method is used to solve thisnon-linear equation instead **of** **the** regular Newton method. This has proven robust **an**d fast, seeLouren•o (1994) **an**d **the** Appendix A.2.2 Consistent t**an**gent operator - Tension regimeIn order to obtain quadratic convergence when making use **of** a Newton-Raphson iterative solvingprocedure at **the** structural level, a t**an**gent operator consistent with **the** integration algorithm mustbe used, see Simo **an**d Taylor (1985). For **the** st**an**dard part **of** **the** yield criterion, differentiation **of**

12 1995 TNO-95-NM-R0712**the** update equations **an**d **the** consistency condition (d f 1, n+1 = 0) results inJ ⎧ dσ n+1⎫⎨ ⎬⎭ = ⎧ dε n+1⎫⎨ ⎬ . (42)⎩dλ t, n+1 ⎩0⎭Then, **the** consistent t**an**gent operator is given byD ep = ∂σ ⎪⎪ = J −14×4 (43)∂ε⎪ n+1in which J −14x4 is **the** top-left 4×4 submatrix **of** **the** inverse **of** J. The consistent t**an**gent operator c**an**also be written in o**the**r fashion by me**an**s **of** a condensation **of** **the** matrix J. Let us deÞne **the** modi-Þed compli**an**ce matrix H t **an**d **the** modiÞed ßow direction vector γ t asH t = D −1 ∂ 2 g 1+∆λ t, n+1∂σ 2 **an**d γ t = ∂g 1∂σ +∆λ ∂ 2 g 1t, n+1 . (44)∂σ ∂κ tCondensation **of** **the** Jacobi**an** J **an**d **the** Sherm**an**-Morrison **for**mula yield, after algebraic m**an**ipulation,D ep = ∂σH⎪−1 ⎛ ∂ f 1 ⎞⎪ = H −1t γ t⎝ ∂σ ⎠t −∂εT⎪ n+1 ⎛ ∂ f 1 ⎞⎝ ∂σ ⎠TH −1tH −1t γ t − ∂ f 1∂κ t. (45)The consistent t**an**gent **for** **the** apex regime is obtained from differentiation **of** eqs. (38,40-41). Thisresults in **the** following system⎡I|⎢⎢ −−−−−−−−−−− +⎢T⎢ ⎛ ∂∆κ t ⎞⎣ ⎝ ∂ε p D −1 |⎠∂η−A⎤1∂κ t⎥−−−−−⎥⎥1 ⎥⎦⎧ dσ n+1⎫⎨ ⎬⎭ =dκ⎩ t, n+1⎧⎫⎪ A 2 D dε n+1 ⎪⎬⎪⎭⎨⎛∂∆κ t ⎞, (46)⎪Tdε⎩⎝ ∂ε p ⎠ n+1with∂∆κ t∂ε p = Q∆ε p n+12( 1 / 2 (∆ε p n+1 )T Q∆ε p n+1 )1 / 2+ 1 / 2 π . (47)Let us call **the** matrix above A. Then, **the** consistent t**an**gent operator is given byD ep = A −14×4A 2 D + A −14×1⎛⎝T∂∆κ t ⎞∂ε p ⎠, (48)where **the** A −14×4 **an**d A −14×1 are submatrices **of** **the** inverse **of** A. The consistent t**an**gent operator c**an**also be written in o**the**r fashion by me**an**s **of** a condensation **of** **the** matrix A **an**d readsD ep = ∂σ ⎪⎪ = ⎡ −1∂η ⎛ ∂∆κ⎢I + A t ⎞1∂ε⎪ n+1 ⎣∂κ t ⎝ ∂ε p D −1⎤ ⎡⎢⎣ ∂η ⎛ ∂∆κ t ⎞⎠⎥ A 1⎦∂κ t ⎝ ∂ε p ⎠TT+ A 2 D ⎤ ⎥⎦. (49)An investigation on **the** per**for**m**an**ce **of** **the** numerical implementation is given in Appendix A.

TNO-95-NM-R0712 1995 132.3 Features **of** **the** **model** - Tension regimeOne pl**an**e stress element with unit dimensions is loaded under different conditions in order to discuss**the** behaviour **of** **the** **model**. Be**for**e considering **orthotropic** material behaviour, a Þrst examplewith isotropic material behaviour is presented. This is extremely relev**an**t **for** a comparison with different**model**s **an**d **an** assessment **of** **the** assumptions made in **the** previous section.2.3.1 Isotropic material behaviour. Tension-shear **model** problemThe elementary problem proposed by Willam et al. (1987) introduces biaxial tension **an**d shearloading in **the** element. This causes a continuous rotation **of** **the** principal strain axes after cracking,as it is typical **of** crack propagation in smeared Þnite element **an**alysis. The element is subjected totensile straining in **the** x-direction accomp**an**ied by lateral Poisson contraction in **the** y-direction tosimulate uniaxial loading. Immediately after **the** tensile strength has been reached, **the** element isloaded in combined biaxial tension **an**d shear strain, see Fig. 8. The ratio between **the** differentstrain components is given by ∆ε x : ∆ε y : ∆γ xy = 0. 5 : 0. 75 : 1. The material properties are givenin Table 1.Table 1 - Material properties (isotropic - α = 1.0)Material propertiesE 10000 N/mm 2ν 0.2f t 1.0 N/mm 2G f 0.00015 N.mm/mm 2The behaviour **of** different crack **model**s **for** this problem c**an** be found in Rots (1988). A comparisonwith different smeared ÒcrackingÓ **for**mulations (total, t**an**gential **an**d R**an**kine plasticity) c**an** befound in Feenstra (1993). The **an**alyses from Rots (1988) **of** this problem with **the** multi-directionalcrack **model** showed that **the** shear response becomes s**of**ter with decreasing threshold **an**gle, resultingin **the** limiting case **of** **the** rotating crack **model** with zero threshold **an**gle as **the** most ßexibleresponse. The **an**alyses from Feenstra (1993) showed that **the** R**an**kine **model** with kinematic hardeningis in very good agreement with **the** rotating crack **model**. A comparison between **the** proposed**model** **an**d **the** commonly used smeared crack **model**s is relev**an**t to assess **the** adequacy **of****the** **model** to describe cracking behaviour. The results **for** **the** different stress **an**d strain componentsare depicted in Fig. 9-Fig. 11.ε yε y =-νε xε xγ xyyxγ xyε xa) Tension up to cracking b) Biaxial tension with shear after crackingFig. 8 - Tension-shear **model** problem

14 1995 TNO-95-NM-R0712Initial shear modulusProposed **model**xy[MPa]τ0.1Fixed crack **model**( = 0.05) β0.0Rotating crack **model**-0.10.0 1.0 2.0-4γxy[10 ]Fig. 9 - Comparison **of** ÒcrackingÓ **model**s.τ xy - γ xy response1.0σ x [MPa]0.80.60.4Fixed crack **model**Rotating crack **model**Proposed **model**0.20.00.0 1.0 2.0 3.0-4εx[10 ]Fig. 10 - Comparison **of** ÒcrackingÓ **model**s.σ x - ε x response1.0Fixed crack **model**y [MPa]σ0.80.60.40.2Rotating crack **model**Proposed **model**0.0-1.0 0.0 1.0 2.0 3.0-4εy[10 ]Fig. 11 - Comparison **of** ÒcrackingÓ **model**s.σ y - ε y responseThe main conclusions from **the** above results are:Ñ The shear stress-strain behaviour gives a good impression about **the** **model** because a ßexibleresponse is obtained. This is clearly in opposition with **the** Þxed crack **model** with const**an**tshear retention factor but is close to **the** rotating crack **model**;

TNO-95-NM-R0712 1995 15Ñ The normal stress-strain response in **the** x-direction shows **an** implicit coupling between normalstress **an**d shear stress. This is also in opposition with **the** Þxed crack **model**, where nocoupling is found, but also characterizes **the** rotating crack **model**, though to a less extent;Ñ The normal stress-strain response in **the** y-direction shows **the** implicit coupling between normalstresses. This is also in opposition with **the** Þxed crack **model**, where no coupling isfound, but also characterizes **the** rotating crack **model**, though to a less extent. The largeramount **of** coupling found in **the** proposed **model** is due to **the** isotropic s**of**tening.2.3.2 Orthotropic material behaviourThe **orthotropic** behaviour **of** **the** **model** is now discussed in a single element test under pure uniaxialtension. The material properties given in Table 2 are assumed, in which **the** y-direction is penalizedby a factor 2. Two different fracture energies are considered **for** **the** y-direction: G fx /2 **an**d500 × G fx (almost ideally plastic behaviour).Table 2 - Material properties (**orthotropic** - α = 1.0)Material propertiesE x 10000 N/mm 2 E y 5000 N/mm 2ν xy 0.2 G xy 3000 N/mm 2f tx 1.0 N/mm 2 f ty 0.5 N/mm 2G fx 0.0002 N.mm/mm 2 G fyCase 1 Case 20.0001 N.mm/mm 2 0.1 N.mm/mm 2The values chosen **for** **the** material properties conÞrm **the** fact that completely different behaviouralong **the** two material axes c**an** be reproduced, see Fig. 12. In **the** Þrst example isotropic s**of**teningis considered. This me**an**s that **the** ratio **of** **the** material strength along **the** material axes is const**an**t,see Fig. 13a, during **an**y load history. It is import**an**t that this deÞnition is not confounded with **the**deÞnition **of** isotropy used in damaged **model**s. Isotropic s**of**tening is related to **the** current yieldstrength values **an**d, not necessarily, to all **the** components **the** current stress vector. When all **the**fracture energy is exhausted a no-tension material is recovered, see Fig. 14a. In **the** second exampleideally plastic behaviour in **the** y-direction is considered. This me**an**s that **the** ratio **of** **the** materialstrength along **the** material axes ( f tx / f ty) tends to zero, see Fig. 13b. The yield surface is onlyallowed to shrink along **the** x-axis, see Fig. 14b.

16 1995 TNO-95-NM-R0712[MPa]1.00.80.6x-directiony-direction[MPa]1.00.80.6x-directiony-directionσσ0.40.40.20.20.00.00.0 1.0 2.0 3.0 4.00.0 1.0 2.0 3.0 4.0-4-4ε [10 ]ε [10 ]a) Case 1 (isotropic s**of**tening) b) Case 2 (ideally plastic behaviour in y-direction)Fig. 12 - Stress-strain response in uniaxial tension along **the** two material axes/ f tyf tx2.52.01.51.00.5/ f tyf tx2.52.01.51.00.50.00.00.0 2.0 4.0 6.0 8.00.0 2.0 4.0 6.0 8.0-4-4κ [10 ]κ [10 ]a) Case 1 (isotropic s**of**tening) b) Case 2 (ideally plastic behaviour in y-direction)Fig. 13 - Equivalent plastic strain vs. strength ratio along **the** material directions(tension regime)σ yInitial yieldsurfaceσ xσ yInitial yieldsurfaceσ xResidual yieldsurfaceResidual yieldsurfacea) Case 1 (isotropic s**of**tening) b) Case 2 (ideally plastic behaviour in y-direction)Fig. 14 - Trace **of** **the** yield surface in **the** pl**an**e τ xy = 0 (tension regime)

TNO-95-NM-R0712 1995 173. COMPRESSION - A HILL TYPE ANISOTROPIC YIELD SURFACEIn this section, a possible Þt **of** **the** experimental results in **the** compression regime is given, see alsoFig. 1. The Hill type yield criterion here introduced is capable **of** reproducing different behaviouralong two orthogonal material axes. The yield surface is coined Hill type because **the** **for**mulation islimited to pl**an**e stress material properties. The properties in **the** out-**of**-pl**an**e direction are usuallyunknown **an**d are not included in **the** **model**, in opposition to **the** original **for**mulation from Hill(1948). The proposed yield surface should, in fact, be considered a particular **for**m **of** **the** completequadratic **for**mulation from Tsai **an**d Wu (1971).τxyτ uσxfmxfmyσyFig. 15 - The Hill type yield surface (plotted **for** τ xy ≥ 0)The simplest yield surface that features different compressive strength along **the** material axes is arotated centered ellipsoid in **the** full pl**an**e stress vector (σ x , σ y **an**d τ xy ), see Fig. 15. The expression**for** such a quadric isf 2 = Aσ 2 x + Bσ x σ y + Cσ 2 y + Dτ 2 xy − 1 = 0 , (50)where A, B, C **an**d D are four material parameters such that B 2 − 4AC < 0, in order to ensure convexity.For **the** numerical implementation **the** yield surface will be recast in a square root matrix**for**m **an**d **the** variables will be rewritten in a more amenable way. Thus, **the** proposed yield surfaceis given byf 2 = ( 1 / 2 σ T P c σ ) 1 / 2− σ c (κ c ) , (51)where **the** projection matrix P c readsP c =⎡ 2 σ c2(κ c )⎢ σ c1 (κ c )⎢⎢β⎢⎢⎢⎢ 0⎢ 0⎣β2 σ c1(κ c )σ c2 (κ c )00**the** yield value σ c is given by00000 ⎤⎥⎥⎥0⎥⎥⎥0 ⎥2γ ⎥⎦, (52)

18 1995 TNO-95-NM-R0712σ c = √⎺⎺⎺⎺⎺ σ c1 σ c2 (53)**an**d **the** single scalar κ c controls **the** amount **of** hardening **an**d s**of**tening. The current yield stressvalues along **the** materials axes (σ c1 (κ c )**an**d σ c2 (κ c )) follow **the** inelastic law giv en below as a function**of** **the** material strength along **the** material axes (respectively f mx **an**d f my as shown in Fig. 15).It is noted that **the** β **an**d γ values introduced in eq. (52) are additional material parameters thatdetermine **the** shape **of** **the** yield surface. The parameter β controls **the** coupling between **the** normalstress values **an**d must be obtained from one additional experimental test, e.g. biaxial compressionwith a unit ratio between principal stresses. If this test is used to obtain **the** parameter β , **the**collapse load under biaxial compression (σ x = σ y =− f 45° **an**d τ xy = 0) leads toβ = ⎡ ⎢⎣1f 2 45°− 1f 2 mx− 1f 2 my⎤⎥ f mx f my . (54)⎦The parameter γ , which controls **the** coupling between **the** normal stress values **an**d **the** shearstrength, c**an** be obtained fromγ = f mx f myτ 2 u, (55)where τ u is **the** material pure shear strength.Parabolic hardening followed by parabolic/exponential s**of**tening is considered **for** both equivalentstress-equivalent strain diagrams, with different compressive fracture energy (G fcx **an**d G fcy ) along**the** material axes. The problem **of** mesh objectivity **of** **the** **an**alyses **of** strain s**of**tening materials is awell debated issue, at least **for** tensile behaviour. Due to localization **of** de**for**mation in a single elementor row **of** elements **the** stress-strain diagram must be adjusted according to a characteristiclength h to provide **an** objective energy dissipation. Here, **the** same expression **for** h is used as **for****the** tension regime even if it is recognized that tensile fracture is a surface driven process **an**d compressivefailure is a volume driven process. The inelastic law shown in Fig. 16 features hardening,s**of**tening **an**d a residual plateau **of** ideally plastic behaviour. The compressive fracture energy G fc(shaded area in Fig. 16) is deÞned only as **the** non-local contribution **of** **the** stress-strain diagram.The basis **for** **the** present deÞnition is only numerical, so that objective **an**alyses with regard tomesh reÞnement are obtained, see also Section 3.4. Howev er some experimental evidence exists ona local **an**d non-local component **for** **the** total compressive fracture energy, see Vonk (1992). Thepeak strength value is assumed to be reached simult**an**eously on both materials axes, i.e. isotropichardening, followed by **an**isotropic s**of**tening as determined by **the** different fracture energies. Aresidual strength value is considered to avoid a cumbersome code (precluding **the** case when **the**compressive mode falls completely inside **the** tension mode) **an**d a more robust code (precludingdegeneration **of** **the** yield surface to a point).

TNO-95-NM-R0712 1995 19σ cσ pσ mσ iσ I (κ c )σ II (κ c )G fcσ III (κ c )σ I (κ c ) = σ i + (σ p − σ i)√⎺⎺⎺2κ cκ p− κ 2 cκ 2 pσ II (κ c ) = σ p + (σ m − σ p ) ⎛ 2κ c − κ p ⎞⎝κ m − κ p ⎠σ III (κ c ) = σ r + (σ m − σ r )exp ⎛ ⎝ m κ c − κ m ⎞σ m − σ r ⎠σ rκ p κ m κ cwith m = 2 σ m − σ pκ m − κ pFig. 16 - Hardening/s**of**tening law **for** compressionFor practical reasons, it is assumed that all **the** stress values **for** **the** inelastic law are determinedfrom **the** peak value σ p = f m as following: σ i = 1 / 3 f m , σ m = 1 / 2 f m **an**d σ r = 1 / 10 f m . The equivalentplastic strain corresponding to **the** peak compressive strength, κ p , is assumed to be **an** additionalmaterial parameter. In **the** case that no experimental results are available, this material parameterc**an** be calculated assuming a total peak stress equal to 2 × 10 −3 . Then, in order to obtain a meshindependent energy dissipation **the** parameter κ m is given byκ m = 7567G fch f m+ κ p . (56)To avoid a possible snap-back at constitutive lev el, **the** conditionκ m ≥ f mE + κ p (57)must be fulÞlled. O**the**rwise, **the** strength limit, in order to obtain **an** objective fracture energy, isreduced t**of** m = ⎛ 67⎝ 751G fc E/2⎞. (58)h ⎠The ßow rule is written in a st**an**dard fashion (associated s**of**tening) asúεúε p = ú∂ f 2 λ c∂σ . (59)The inelastic behaviour is described by a work hardening hypo**the**sis given byúκ c = 1 σ T úεúε p = ú λ c . (60)σ c3.1 Return mapping algorithm - Compression regimeThe return mapping algorithm in **the** frame **of** a implicit Euler backward integration scheme isgiven in Section 2.1 **an**d, **for** **the** present yield surface, results in **the** following set **of** Þve equationscontaining 5 unknowns (σ n+1 **an**d ∆κ c, n+1 =∆λ c, n+1 ), cf. eqs. (29),

20 1995 TNO-95-NM-R0712⎧⎪D −1 (σ n+1 − σ trial ∂ f) 2⎪+∆λ c, n+1 ⎪ = 0⎪∂σ⎪⎨n+1⎪f⎪ 2 = ( 1 / 2 σ n+1P T c, n+1 σ n+1 ) 1 / 2− σ c, n+1 = 0⎩This set **of** equations c**an** be reduced to one non-linear equation, namely f 2 (∆λ c, n+1 ) = 0, if **the**stress update is m**an**ipulated to obtain.σ n+1 = ⎡ ⎢⎣I + ∆λ c, n+12σ c, n+1DP c, n+1⎤⎥⎦−1σ trialn+1 . (62)This approach will, however, not be followed here so that a const**an**t framework is obtained **for** **the**several modes **of** **the** composite yield surface. Alternatively, **the** system **of** 5 non-linear equations issolved with a regular Newton-Raphson method. The Jacobi**an** necessary **for** this procedure reads(note that **the** subscript n+1 is dropped in **the** derivatives **an**d matrices **for** convenience)(61)where⎡D⎢−1 ∂ 2 f 2+∆λ c, n+1∂σ 2J = ⎢ −−−−−−−−−−−−⎢T⎢ ⎛ ∂ f 2 ⎞⎣ ⎝ ∂σ ⎠∂ f 2∂σ = P c σ n+12( 1 / 2 σ T n+1 P cσ n+1 ) 1 / 2;|+|∂ f 2∂σ +∆λ ∂ 2 f 2 ⎤c, n+1∂σ ∂κ c ⎥⎥⎥⎥⎦−−−−−−−−−− , (63)∂ f 2∂κ c∂ 2 f 2∂σ 2 = P c2( 1 / 2 σ T n+1 P cσ n+1 ) 1 / 2−P c σ n+1 σ T n+1P c4( 1 / 2 σ T n+1 P cσ n+1 ) 3 / 2∂ f 2∂κ c=σ n+1T ∂P cσ n+1∂κ c4( 1 / 2 σn+1 T P −cσ n+1 ) 1 / 2∂σ c1∂κ cσ c2, n+1 + ∂σ c2∂κ cσ c1, n+12σ c, n+1.∂ 2 f 2∂σ ∂κ c=∂P cσ n+1 (σ T ∂P cn+1 σ n+1 )P c σ n+1∂κ c ∂κ2( 1 / 2 σn+1 T P −ccσ n+1 ) 1 / 28( 1 / 2 σn+1 T P cσ n+1 ) 1 / 2(64)⎧ ⎡ ∂σ c2σ∂P c⎪ ⎢ c1, n+1 − ∂σ c1∂κ= diag⎨2c⎢∂κ c ⎪ ⎢σc1, 2 n+1⎩ ⎣⎤ ⎡σ c2, n+1∂κ c⎥ ⎢⎥,2⎢⎥ ⎢⎦ ⎣3.2 Consistent t**an**gent operator - Compression regime∂σ c1∂κ cσ c2, n+1 − ∂σ c2σ 2 c2, n+1⎤ ⎫σ c1, n+1∂κ c⎥ ⎪⎥,0,0⎬⎥ ⎪⎦ ⎭Differentiation **of** **the** update equations **an**d **the** consistency condition (d f 2, n+1 = 0) results, afteralgebraic m**an**ipulation, in

TNO-95-NM-R0712 1995 21[MPa]-10.0-8.0-6.0D ep = dσH⎪−1 ⎛ ∂ f 2 ⎞⎪ = H −1c γ c⎝ ∂σ ⎠c −dεT⎪ n+1 ⎛ ∂ f 2 ⎞⎝ ∂σ ⎠TH −1cH −1c γ c − ∂ f 2∂κ c, (65)where **the** modiÞed compli**an**ce matrix H c **an**d **the** modiÞed ßow direction vector γ c readH c = D −1 ∂ 2 f 2+∆λ c, n+1∂σ 2 **an**d γ c = ∂ f 2∂σ +∆λ ∂ 2 f 2c, n+1 . (66)∂σ ∂κ cAn investigation on **the** per**for**m**an**ce **of** **the** numerical implementation is given in Appendix A.3.3 Features **of** **the** **model** - Compression regimeOne pl**an**e stress element with unit dimensions is loaded under different conditions in order to discuss**the** behaviour **of** **the** **model**.The material properties given in Table 3 are assumed, in which **the** material strength **an**d YoungÕsmodulus in **the** y-direction are penalized by a factor 2. Three different fracture energies are considered**for** **the** y-direction: 0. 3G fcx ,G fcx / 2 (isotropic s**of**tening) **an**d 500 × G fcx (almost ideally plasticbehaviour).Table 3 - Material properties (β = -1.0, γ = 3.0 **an**d κ p = 0.0005)Material propertiesE x 10000 N/mm 2 E y 5000 N/mm 2ν xy 0.2 G xy 3000 N/mm 2f mx 10.0 N/mm 2 f ty 5.0 N/mm 2G fcx 0.05 N.mm/mm 2 G fcyCase 1 Case 2 Case 30.015 N.mm/mm 2 0.025 N.mm/mm 2 7.5 N.mm/mm 2The **orthotropic** behaviour **of** **the** **model** is now discussed in a single element test under pure uniaxialcompression. The uniaxial stress-strain responses **for** **the** different cases considered are illustratedin Fig. 17. As shown in this picture, **the** **model** is capable **of** reproducing different behaviouralong **the** two material axes.x-directiony-direction[MPa]-10.0-8.0-6.0x-directiony-directionσ-4.0-2.0σ-4.0-2.00.00.00.0 -2.5 -5.0 -7.5 -10.0 0.0 -2.5 -5.0 -7.5 -10.0-3-3ε [10 ]ε [10 ]a) Case 1 b) Case 2 (isotropic s**of**tening)Fig. 17 - Stress-strain response in uniaxial compression along **the** two material axes (cont.)

22 1995 TNO-95-NM-R0712-10.0[MPa]-8.0-6.0x-directiony-directionσ-4.0-2.00.00.0 -2.5 -5.0 -7.5 -10.0-3ε [10 ]c) Case 3 (ideally plastic behaviour in y-direction)Fig. 17 - Stress-strain response in uniaxial compression along **the** two material axes (contd.)Fur**the**r insight on **the** behaviour **of** **the** **model** c**an** be obtained from Fig. 18 **an**d Fig. 19. Fig. 18gives **the** ratio **of** **the** material strength along **the** material axes **an**d Fig. 19 shows **the** trace **of** **the**yield surface in **the** pl**an**e τ xy = 0. In **the** Þrst case, **the** material strength in y-direction degradesfaster th**an** **the** material strength in x-direction. In **the** second case, degradation **of** **the** materialstrength in both directions occurs with **the** same rate **an**d isotropic s**of**tening is obtained. Finally, in**the** third case, degradation only occurs in **the** material strength in x-direction.Note that, in **the** case **of** isotropic s**of**tening, **the** post-peak stress-strain diagrams under uniaxialloading conditions along **the** two material axes, see Fig. 17b, is not scaled by a factor 2. This issolely due to **the** deÞnition **of** **the** s**of**tening scalar **an**d **the** fact that **the** yield value σ c is not equal to**the** uniaxial strength along each material axis. This also me**an**s that **the** deÞnition **of** **the** Òcompressivefracture energyÓ c**an** be argued because a perfect equivalence to **the** stress-strain diagram is notobtained. This limitation **of** **the** **model** c**an** be solved by e.g. if a unit norm is used **for** **the** plasticßow vector **an**d a strain s**of**tening hypo**the**sis is used **for** **the** s**of**tening scalar. The additional difÞcultyintroduced in **the** **for**mulation ( úκ c ≠ ú λ c ) is not particularly difÞcult to solve but, with **the** notoriouslack **of** experimental results on **the** material behaviour, **the** initial assumptions are kept inorder to simplify **the** implementation.8.08.0/ f myf mx6.04.0/ f myf mx6.04.02.02.00.00.00.0 5.0 10.0 15.0 20.0 0.0 5.0 10.0 15.0 20.0-3-3κ c [10 ]κ c [10 ]a) Case 1 b) Case 2 (isotropic s**of**tening)Fig. 18 - Equivalent plastic strain vs. strength ratio along **the** material directions(compression regime) (cont.)

TNO-95-NM-R0712 1995 238.0/ f myf mx6.04.02.00.00.0 5.0 10.0 15.0 20.0-3κ c [10 ]c) Case 3 (ideally plastic behaviour in y-direction)Fig. 18 - Equivalent plastic strain vs. strength ratio along **the** material directions(compression regime) (contd.)Residual yield surfaceσyResidual yield surfaceσyPeak yield surfacePeak yield surfaceσxσxa) Case 1 b) Case 2 (isotropic s**of**tening)Residual yield surfaceσyPeak yield surfaceσxc) Case 3 (ideally plastic behaviour in y-direction)Fig. 19 - Trace **of** **the** yield surface in **the** pl**an**e τ xy = 0 (compression regime)3.4 About **the** definition **of** a mesh independent energy releaseFor strain-s**of**tening materials, **the** need to introduce **an** equivalent length h in **the** stress-strain diagramto obtain **an**alyses which are objective with respect to mesh reÞnement is a well debated issue

24 1995 TNO-95-NM-R0712since **the** original work **of** Baº**an**t **an**d Oh (1983). As stated in Section 2., in **the** present work h isassumed to be related to **the** area **of** **an** element, cf. eq. (13). However, this approach is generallyused in engineering practice only **for** **model**ling tensile behaviour with linear elastic pre-peakbehaviour followed by inelastic s**of**tening until total degradation **of** strength.The constitutive relation shown in Fig. 16 features pre-peak hardening **an**d a residual plateau.Clearly, **the** hardening br**an**ch **of** **the** constitutive relation is stable **an**d should not be adjusted as afunction h but also **the** residual plateau is const**an**t **an**d independent **of** **the** h value. To demonstrate**the** veracity **of** **the** deÞnition **of** a mesh independent release **of** energy upon mesh reÞnement **an**example **of** a simple bar loaded in uniaxial is given. The problem is similar to **the** well-known problem**of** a simple bar loaded in tension proposed by CrisÞeld (1982).Consider **the** bar shown in Fig. 20 which is divided in n elements with n = 10, 20 **an**d 40 elements.The length **of** **the** bar is 50 mm **an**d **the** tr**an**versal section **of** **the** bar has unit dimensions(1. 0 mm 2 × 1. 0 mm 2 ). The compressive fracture energies are assumed to equal G fcx = 10. 0 N/mm**an**d G fcy = 5. 0 N/mm. For **the** rest **of** **the** material properties **the** values used in **the** previous sectionare assumed. One element is slightly imperfect (10%) to trigger **the** localization: f cx = 9. 0 N/mm 2 ,f cy = 4. 5 N/mm 2 ,G fcx = 9. 0 N/mm **an**d G fcy = 4. 5 N/mm. The o**the**r material parameters remain**the** same.Imperfect elementFig. 20 - Simple bar with imperfect element loaded in compressionThe load-displacement response **of** **the** bar is depicted in Fig. 21a **for** **the** energy-based regularizationmethod (note that ÒdisplacementÓ is understood as **the** relative displacement between **the** ends**of** **the** bar). It c**an** be observed that **the** response is totally independent from **the** number **of** elements.The response **of** **the** bar with a constitutive **model** which has not been modiÞed by **the** size **of** **the**Þnite element mesh, see Fig. 21b, shows a dramatic mesh-dependent behaviour in **the** post-peakresponse. The brittleness **of** **the** response increases with **an** increasing number **of** elements.10.010.0Load [MN]8.06.04.0n = 10, 20 **an**d 40Load [MN]8.06.04.0n = 20n = 102.02.0n = 400.00.0 10.0 20.0 30.0 40.0 50.0-3Displacement [10 mm]0.00.0 10.0 20.0 30.0 40.0 50.0-3Displacement [10 mm]a) Energy-based regularization b) No regularizationFig. 21 - Load-displacement diagram **for** simple bar with imperfect element

TNO-95-NM-R0712 1995 254. A COMPOSITE YIELD SURFACE FOR MASONRYThe two yield surfaces detailed in **the** previous section are now combined in a composite yield surfaceas illustrated in Fig. 22.σyσxσy0.1 0.2 0.3σ xyσxFig. 22 - Proposed composite yield surface with iso-shear stress linesf tx = 1. 0, f ty = 0. 5, f mx = 10. 0, f my = 5. 0 [N/mm 2 ] − α = 1. 0, β =−1. 0, γ = 3. 0Contour spacing: 0.1 f mxA full description **of** **the** compressive **an**d tensile parts **of** **the** composite yield surface are given in**the** previous sections **an**d will not be repeated here. Only **the** aspects relative to **the** corner regimewill be addressed in **the** present section.One **of** **the** most import**an**t issues **of** multi-surface plasticity is to deÞne **the** number **of** active yieldsurfaces. Simo et al. (1988) have proposed **an** algorithm in which **the** assumption is made that **the**number **of** active yield surfaces in **the** Þnal stress state is less th**an** or equal to **the** number **of** activeyield surfaces in **the** trial stress state. This implies that it is not possible that a yield surface, whichis inactive in **the** trial state, becomes active during **the** return mapping. This is not valid **for** **the** proposedyield surface. Due to **the** small number **of** yield surfaces (two), a trial **an**d error iterative procedureis used, in which **the** return mapping process is restarted if a non-admissible solution isfound, see Louren•o (1994) **for** a complete description. From **the** experience **of** **the** author, thisleads to a robust **an**d always convergent algorithm. The disadv**an**tage is that **the** return mappingalgorithm might have to be restarted be**for**e **the** correct solution is obtained.4.1 Return mapping algorithm - Corner regimeThe return mapping algorithm in **the** frame **of** a implicit Euler backward integration scheme isgiven in Section 2. **for** single surface plasticity. For multisurface plasticity, **the** most import**an**tassumption is KoiterÕs (1953) generalisation **of** **the** plastic strain rate asúεúε p = ú∂g 1 λ t∂σ+ ú∂ f 2 λ c∂σ . (67)Note that no coupling is assumed between **the** compressive **an**d tensile regimes.Upon algebraic m**an**ipulation, **the** return mapping algorithm **for** **the** corner regime results in **the** followingset **of** six equations containing 6 unknowns (σ n+1 , ∆κ t, n+1 =∆λ t, n+1 **an**d ∆κ c, n+1 =∆λ c, n+1 ),cf. eqs. (29,61),

26 1995 TNO-95-NM-R0712⎧⎪D −1 (σ⎪ n+1 − σ trial ∂g) 1⎪∂ f 2⎪+∆λ t, n+1 ⎪ +∆λ c, n+1 ⎪ = 0∂σ∂σ⎪⎪ n+1 ⎪ n+1⎨ f 1 = ( 1 / 2 ξ n+1P T t ξ n+1 ) 1 / 2+ 1 / 2 π T ξ n+1 = 0 .⎪⎪ f⎪ 2 = ( 1 / 2 σ n+1P T c, n+1 σ n+1 ) 1 / 2− σ c, n+1 = 0⎩This system **of** non-linear equations is solved with a regular Newton-Raphson method **an**d **the** Jacobi**an**necessary **for** this procedure reads (note that **the** subscript n+1 is dropped in **the** derivatives**an**d matrices **for** convenience)(68)⎡⎢D −1 ∂ 2 g 1+∆λ t, n+1∂σ⎢2 +∆λ ∂ 2 f 2c, n+1∂σ 2−−−−−−−−−−−−−−−−⎢TJ = ⎢⎛ ∂ f 1 ⎞⎢⎝ ∂σ ⎠⎢ −−−−−−−−−−−−−−−−⎢T⎢⎛ ∂ f 2 ⎞⎣⎝ ∂σ ⎠|+|+|∂g 1∂σ +∆λ ∂ 2 g 1t, n+1∂σ ∂κ t−−−−−−−−−−∂ f 1∂κ t−−−−−−−−−−0|+|+|∂ f 2∂σ +∆λ ∂ 2 f 2c, n+1∂σ ∂κ c−−−−−−−−−−⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦0 . (69)−−−−−−−−−−∂ f 2∂κ c4.2 Consistent t**an**gent operator - Corner regimeDifferentiation **of** **the** update equations **an**d **the** consistency conditions (d f 1, n+1 = 0 **an**d(d f 2, n+1 = 0) results inJ⎧dσ⎫ ⎧⎪ n+1 dε⎫⎪⎬⎪⎭ ⎪ n+1⎪⎨dλ t, n+1 = ⎨ 0 ⎬⎪dλ c, n+1⎪ 0 ⎪⎩⎩ ⎭. (70)Then, **the** consistent t**an**gent operator is given byD ep = ∂σ ⎪⎪ = (J −1 ) 4×4∂ε⎪ n+1, (71)in which (J −1 ) 4x4 is **the** top-left 4×4 submatrix **of** **the** inverse **of** J. The consistent t**an**gent operatorc**an** also be written in o**the**r fashion by me**an**s **of** a condensation **of** **the** matrix J. The derivation **of****an** expression equivalent to eqs. (45,65) is given in Louren•o (1994).An investigation on **the** per**for**m**an**ce **of** **the** numerical implementation is given in Appendix A.

TNO-95-NM-R0712 1995 275. EXAMPLESIn **the** present section **the** **masonry** **model** proposed in this report is used to **an**alyse different **structures**found in **the** literature.5.1 TU Eindhoven tests on shear wallsSeveral tests on **masonry** shear walls were carried out within **the** scope **of** **the** STRUCTURALMASONRY program from **the** Ne**the**rl**an**ds, see Raijmakers **an**d Vermeltfoort (1992) **an**d Vermeltfoort**an**d Raijmakers (1993). In Louren•o (1994) some **of** **the** specimens are **an**alysed with a micro**model****an**d very good agreement is found with **the** experimental results. The parameters necessaryto characterize **the** micro-material **model** are available from micro-experiments but almost noexperimental data are available on **the** behaviour **of** **the** composite.The purpose **of** this report is not a sharp reproduction **of** **the** experimental results **an**d **the** main concern**of** **the** author is to reproduce **the** behaviour observed in **the** experiments. For this purpose **the**exact knowledge **of** **the** macro-material parameters is not capital (**an**d, in **an**y case, it is not known**for** this particular structure).The Þrst shear wall **an**alysed by Louren•o (1994) is used **for** comparison between **the** micro- **an**dmacro-**model**. The specimen consists **of** a pier with a width/height ratio **of** one (990 × 1000 mm 2 ),built up with 18 layers (16 layers unrestrained) **of** Joosten solid clay bricks (dimensions 204 x 98 x50 mm 3 ) **an**d 10 mm thick mortar (1:2:9, cement:lime:s**an**d by volume). The pier was subjected toa vertical precompression **for**ce P = 30 kN be**for**e a horizontal load was monotonically increasedunder top displacement control d until failure (see Fig. 23). The cracking patterns are depicted inFig. 24 **for** **the** two tests carried out.Pda) Phase 1 - Vertical loading b) Phase 2 - Horizontal loadingFig. 23 - Loads **for** Vermeltfoort shear wall

28 1995 TNO-95-NM-R0712Fig. 24 - Experimental failure patterns **for** Vermeltfoort shear wallsFor **the** numerical **an**alysis linear pl**an**e stress **continuum** elements (4-noded) with full Gauss integrationare utilised. A regular mesh **of** 15 × 15 elastoplastic elements is used. It is noted that **the** top**an**d bottom layers **of** bricks are almost completely restrained **an**d, **the**re**for**e, two additional rows **of**linear elastic elements are also included in **the** **model**. The same assumption is used **for** micro**model**ling**an**d yield good results, see Louren•o (1994). The elastic properties **of** **the** composite areobtained from homogenisation, see Louren•o (1995), **of** **the** elastic properties **of** brick(E b = 16700 N/mm 2 **an**d ν b = 0. 15) **an**d mortar (E m = 780 N/mm 2 **an**d ν m = 0. 125) **an**d read:E x = 7520 N/mm 2 , E y = 3960 N/mm 2 , ν xy = 0. 090 **an**d G xy = 1460 N/mm 2 . The only inelasticparameters available from experimental results are **the** vertical tensile strength f ty , **the** vertical stackbond compressive strength f my **an**d **the** vertical fracture energy G fy . The rest **of** **the** material parametersare estimated **an**d **the** full parameters list reads: f tx = 0. 35 N/mm 2 , f ty = 0. 25 N/mm 2 ,α = α g = 1. 0, G fx = 0. 05 N/mm, G fy = 0. 018 N/mm, f mx = 10. 0 N/mm 2 , f my = 8. 8 N/mm 2 ,β =−1. 0, γ = 2. 5, G fcx = 20. 0 N/mm, G fcy = 15. 0 N/mm **an**d κ p = 0. 0012.The results **of** **the** numerical **an**alysis are given in Fig. 25 to Fig. 29. The comparison betweenexperimental **an**d numerical results, in terms **of** load-displacement diagrams, is given in Fig. 25.Fig. 25a shows **the** comparison between macro-, micro- **an**d experimental results **an**d Fig. 25bshows **the** agreement between **the** collapse load values. Clearly, **the** macro-**model** yields **an** enormousover-prediction **of** strength. The difference found c**an**not be explained solely by a possiblywrong estimation **of** **the** material parameters but must be due to some deÞciency **of** **the** **model** or**model**ling strategy. A physical reason **for** this phenomena does exist **an**d is given fur**the**r below in**the** text.The behaviour **of** **the** wall is depicted in Fig. 26 to 29 in terms **of** total de**for**med meshes, incrementalde**for**med meshes, cracked **an**d crushed Gauss points **an**d principal stresses. Note that **the** cracksare plotted normal to **the** tensile principal plastic strain directions with a thickness proportional to**the** tensile equivalent plastic strain (**the** lowest 5% values are discarded in order to obtain legiblepictures). Masonry crushing is represented by dotted tri**an**gles with a size proportional to **the** compressiveequivalent plastic strain (only **the** points in **the** post-peak regime are plotted, i.e. criticalpoints in **the** s**of**tening regime). Finally, note that **the** principal stresses in **the** elastic bricks are notshown. In **the** elastoplastic elements, **the** negative principal values are represented by a dotted line**an**d **the** positive principal values by a solid line.These pictures demonstrate that **the** behaviour **of** **the** structure is quite well captured by **the** **model**.Initially, two horizontal cracks develop at **the** top **an**d bottom **of** **the** wall, see Fig. 26. Upon

TNO-95-NM-R0712 1995 29Horizontal **for**ce F [kN]100.075.050.025.0ExperimentalMicro-**model**, Lourenco (1994)Macro-**model**0.00.0 1.0 2.0 3.0 4.0 5.0Horizontal displacement d [mm]Horizontal **for**ce F [kN]200.0150.0100.050.0Experimental collapse loadMacro-**model**0.00.0 10.0 20.0 30.0Horizontal displacement d [mm]a) Vs. experimental result **an**d micro-**model** prediction b) Full responseFig. 25 - Load-displacement diagram **for** Vermeltfoort shear wallincreasing de**for**mation, a diagonal crack arises, see Fig. 27. The diagonal crack progresses in **the**direction **of** **the** compressed corners **of** **the** wall, accomp**an**ied by crushing **of** **the** toes, see Fig. 28,until total degradation **of** strength in **the** compressive strut, see Fig. 29. This behaviour agrees wellwith Fig. 24 **an**d **the** micro-**an**alysis **of** Louren•o (1994).a) Total de**for**med mesh b) Incremental de**for**med meshc) Cracks **an**d crushing d) Principal stresses σ min = -1.85 N/mm 2Fig. 26 - Numerical results **for** a horizontal displacement d = 0.5 mm. 15 × 15 elements

30 1995 TNO-95-NM-R0712a) Total de**for**med mesh b) Incremental de**for**med meshc) Cracks **an**d crushing d) Principal stresses σ min = -3.52 N/mm 2Fig. 27 - Numerical results **for** a horizontal displacement d = 1.0 mm. 15 × 15 elementsa) Total de**for**med mesh b) Incremental de**for**med meshc) Cracks **an**d crushing d) Principal stresses σ min = -10.1 N/mm 2Fig. 28 - Numerical results **for** a horizontal displacement d = 6.9 mm (peak). 15 × 15 elements

TNO-95-NM-R0712 1995 31a) Total de**for**med mesh b) Incremental de**for**med meshc) Cracks **an**d crushing d) Principal stresses σ min = -9.65 N/mm 2Fig. 29 - Numerical results **for** a horizontal displacement d = 30.0 mm (ultimate). 15 × 15 elementsTw o o**the**r points **of** **the** per**for**m**an**ce **of** **the** **model** deserve special attention. A very good impressionabout **the** robustness **of** **the** **model** is obtained from **the** load-displacement diagram because it ispossible to follow **the** complete load path until total degradation **of** strength. Ano**the**r import**an**tissue is **the** dependency **of** **the** **model** upon mesh reÞnement. Fig. 30 shows **the** comparison between**the** initial mesh **an**d a 2 × Þner mesh. The results are, practically, mesh insensitive **an**d **the**behaviour encountered **for** **the** Þner mesh, see Fig. 31 **an**d Fig. 32, shows no difference from **the**behaviour **of** **the** coarser mesh.200.0Horizontal **for**ce F [kN]150.0100.050.030 x 30 elements15 x 15 elements0.00.0 10.0 20.0 30.0Horizontal displacement d [mm]Fig. 30 - Mesh dependency **of** **the** solution

32 1995 TNO-95-NM-R0712a) Total de**for**med mesh b) Incremental de**for**med meshc) Cracks **an**d crushing d) Principal stresses σ min = -10.2 N/mm 2Fig. 31 - Numerical results **for** a horizontal displacement d = 6.9 mm (peak). 30 × 30 elementsa) Total de**for**med mesh b) Incremental de**for**med meshc) Cracks **an**d crushing d) Principal stresses σ min = -10.5 N/mm 2Fig. 32 - Numerical results **for** a horizontal displacement d = 30.0 mm (ultimate). 30 × 30 elements

TNO-95-NM-R0712 1995 33The expl**an**ation **for** **the** over prediction **of** **the** failure load will now be detailed. Fig. 25a shows that**the** experimental load-displacement diagram is well reproduced until a horizontal displacement dequal to 1.0 mm, which coincides with **the** opening **of** **the** diagonal shear crack. After this point, itseems that **the** behaviour **of** **the** structure is not captured by **the** **model**. The crucial question iswhe**the**r **the** **masonry** macro-**model** here proposed is not acceptable or **the** **model**ling strategy isinadequate. The author believes **the** latter to be true.Fig. 24 clearly shows **an** extremely localized diagonal failure type in which all inelastic phenomenaconcentrate, basically, in a single stepped crack. A complete underst**an**ding **of** **the** behaviour **of** **the**structure is obtained by **the** distribution **of** internal **for**ces calculated by **the** micro-**model**, see Fig.33. Fig. 33a shows that, initially, **the** direction **of** **the** compressive strut is determined by **the** geometry**of** **the** bricks **an**d deviates from **the** strut direction found in **the** macro-**an**alysis. The discrep**an**cybetween **the** micro- **an**d macro-**model** gets only larger under increasing de**for**mation. Fig. 33bshows that in reality two independent struts occur once **the** diagonal crack is fully open **an**d a welldeÞned b**an**d completely unloads, with approximately **the** width **of** a brick at **the** center **of** **the** specimen.This phenomenon c**an**not be captured by a (smeared) macro-**model** unless some internallength scale, that reßects **the** **masonry** micro-structure, is incorporated in **the** **model**. This problemis outside **the** scope **of** **the** present report **an**d, presently, no simple solution is envisaged. It is notedthat **the** task is quite complex because it is not clear yet if **the** above length scale is a geometricalproperty dependent on **the** unit size or a structural property that depends on **the** micro-structure,geometry **of** **the** structure **an**d loading conditions.a) d = 1.0 mm d) d = 4.0 mmFig. 33 - Minimum principal stresses **for** a horizontal displacement d obtained by a micro-**model**Louren•o (1994)For **the** sake **of** completeness a new **an**alysis is carried out with **the** macro-**model** **for**mulated insuch a way that degradation **of** tensile strength induces degradation **of** compressive strength, i.e. acoupled **model**. Coupling is introduced via **an** equation **of** **the** following type:

34 1995 TNO-95-NM-R0712(σ peak ) c = f mf tσ t , (72)Horizontal **for**ce F [kN]where **the** compressive yield value is made dependent **of** **the** current tensile yield value. This **for**mulationleads to additional terms in **the** derivations **of** **the** numerical algorithm given in **the** previoussections but details about **the** coupled **for**mulation will not be given here. By making use **of** Sections2. to 4., it is relatively straight**for**ward to obtain **the** new algorithm. The results obtained with**the** coupled **model** are given in Fig. 34 **an**d Fig. 35. A much better agreement is found between **the**new results **an**d **the** experimental values. Fig. 34a shows that **the** collapse load is well predicted by**the** **model** (note that **the** Þrst peak in **the** load-displacement diagram is obtained when **the** top **an**dbottom horizontal bending cracks occur) **an**d Fig. 35a shows that **the** expected two independentstruts c**an** be reproduced. There**for**e, **for** this particular structure, with a good estimate **of** **the** width**of** **the** inactive shear b**an**d, it seems possible to reproduce **the** experimental results, at least to someextent. In fact, **the** coupled **model** introduces **an** internal length that is related to **the** element sizebecause **the** release **of** compressive fracture energy is coupled with tensile s**of**tening. This leads to aresponse totally inobjective with regard to **the** mesh size, see Fig. 34b **an**d Fig. 35b. Upon meshreÞnement **the** solution in terms **of** load-displacement diagram converges to **the** uncoupled **model****an**d, simult**an**eously, **the** width **of** **the** inactive shear b**an**d tends to zero.75.050.025.0ExperimentalIsotropic **model** - 8x8 elements0.00.0 1.0 2.0 3.0 4.0 5.0Horizontal displacement d [mm]Horizontal **for**ce F [kN]75.050.025.030 x 30 elements15 x 15 elements8 x 8 elementsExperimentalCoupled **model**Uncoupled **model**0.00.0 1.0 2.0 3.0 4.0 5.0Horizontal displacement d [mm]a) Vs. experimental results b) Mesh dependency **of** **the** solutionFig. 34 - Load-displacement diagram **for** Vermeltfoort shear wall (isotropic **model**)a) 8 × 8 elements b) 30 × 30 elementsFig. 35 - Principal stresses at peak **for** isotropic **model**

TNO-95-NM-R0712 1995 355.2 ETH Zurich tests on shear wallsLarge shear wall tests made with hollow clay bricks were carried out at ETH Zurich, Switzerl**an**dby G**an**z **an**d ThŸrlim**an**n (1984). These experiments are very well suited **for** validation **of** **the****model** presented here not only because **the**y are large **an**d feature well distributed cracking patternsbut also because experimental tests on **the** strength **of** **the** composite material are available, seeG**an**z **an**d ThŸrlim**an**n (1982). Fig. 36 shows **the** geometry **of** **the** specimens, which consist **of** a**masonry** p**an**el **of** 3600 × 2000 × 150 mm 3 **an**d two ß**an**ges **of** 150 × 2000 × 600 mm 3 . Additionalboundary conditions are given by two concrete slabs placed in **the** top **an**d bottom **of** **the** specimen.Initially, **the** wall is subjected to a vertical load p uni**for**mly distributed over **the** length **of** **the** wall.This is followed by **the** application **of** a **for**ce F on **the** top slab along a horizontal displacement d.pd1501602000200 3600 2001806009001400Fig. 36 - Geometry **for** G**an**z specimensFor **the** Þrst specimen **an**alysed here (Wall W1) **the** result**an**t **of** **the** conÞning pressure p equals415 kN **an**d **for** **the** second specimen (Wall W2) **the** result**an**t **of** **the** conÞning pressure p equals1287 kN. The Þrst specimen shows a very ductile response with tensile **an**d shear failure along **the**diagonal stepped cracks, see Fig. 37a,b, **an**d **the** second specimen shows a brittle failure with explosivepost-peak behaviour due to crushing **of** **the** compressed toes, see Fig. 37c,d. Note that **the**sepictures present a back view **of** **the** structure, i.e. **the** horizontal load shown is applied from left toright.

36 1995 TNO-95-NM-R0712a) Wall W1 - Peak b) Wall W1 - Ultimatec) Wall W2 - Peak d) Wall W2 - UltimateFig. 37 - Experimental failure patterns **for** G**an**z walls(pictures rotated 90°)

TNO-95-NM-R0712 1995 37For **the** numerical **an**alysis linear pl**an**e stress **continuum** elements (4-noded) **an**d const**an**t strain tri**an**glesin a cross diagonal patch with full Gauss integration are utilised. A regular mesh **of** 24 × 154-noded elements is used **for** **the** p**an**el **an**d 2 × 15 cross diagonal patches **of** 3-noded tri**an**gles areused **for** each ß**an**ge. The properties **of** **the** composite material are obtained from G**an**z **an**dThŸrlim**an**n (1982). The elastic properties read E x = 2460 N/mm 2 ,E y = 5460 N/mm 2 , ν xy = 0. 18**an**d G xy = 1130 N/mm 2 . The inelastic parameters **for** **the** p**an**el read f tx = 0. 05 N/mm 2 ,f ty = 0. 25 N/mm 2 , α = 1. 66, α g = 1. 0, G fx = 0. 02 N/mm, G fy = 0. 02 N/mm, f mx = 1. 87 N/mm 2 ,f my = 7. 61 N/mm 2 , β =−1. 05, γ = 1. 2, G fcx = 5. 0 N/mm, G fcy = 10. 0 N/mm **an**d κ p = 0. 0008.The inelastic parameters that control **the** shape **of** **the** yield surface are obtained from a least squaresÞt from **the** experimental results but no data are available **for** **the** post-peak r**an**ge. It is noted that, in**the** ß**an**ges, a stack bond **masonry** is used **an**d different material properties must be considered. Theinelastic parameters **for** **the** ß**an**ges read f tx = 0. 68 N/mm 2 , f ty = 0. 25 N/mm 2 , α = α g = 1. 0,G fx = 0. 05 N/mm, G fy = 0. 02 N/mm, f mx = 9. 5 N/mm 2 , f my = 7. 61 N/mm 2 , β =−1. 05,γ = 3. 0, G fcx = 10. 0 N/mm, G fcy = 10. 0 N/mm **an**d κ p = 0. 0008. Finally, note that **the** selfweight**of** wall **an**d top slab is also considered in **the** **an**alysis.5.2.1 Wall W1The results **of** **the** numerical **an**alysis are given in Fig. 38 to Fig. 44. The comparison betweenexperimental **an**d numerical results, in terms **of** load-displacement diagrams, is given in Fig. 38. Inopposition to **the** Þrst example shown in **the** present report, good agreement is found between bothresults. This is due to **the** distributed nature **of** **the** process prior to collapse. The behaviour **of** **the**wall is depicted in Fig. 39 to 44 in terms **of** total de**for**med mesh, incremental de**for**med mesh,cracked Gauss points **an**d minimum principal stresses contour. Note that **the** center node **of** **the**crossed diagonal patch is not shown in **the** meshes to obtain a more legible picture. For **the** samereason **the** contour **of** minimal principal stresses is shown instead **of** **the** representation at eachGauss point. The comparison between experimental **an**d numerical behaviour results is more difÞcultbut reasonable agreement seems to be found. Immediately after starting loading **the** structure,extensive diagonal cracking **of** **the** p**an**el is found, see Fig. 39. Upon increasing de**for**mation crackingtends to concentrate in a large shear b**an**d going from one corner **of** **the** specimen to **the** o**the**r,see Fig. 40 **an**d Fig. 41. This is accomp**an**ied by ßexural cracking **of** **the** right ß**an**ge **an**d, at a laterstage, also **the** left ß**an**ge, see Fig. 42 **an**d Fig. 43. At ultimate stage, see Fig. 44, a well deÞned failuremech**an**ism is **for**med with a Þnal shear b**an**d going from one corner **of** **the** specimen to **the**o**the**r **an**d intersecting **the** ß**an**ges. This me**an**s that cracks rotate signiÞc**an**tly since initiation governedby MohrÕs circle to failure in a sort **of** shear b**an**d, which agrees extremely well with **the**experiments (see Fig. 37a,b).300.0Horizontal **for**ce F [kN]200.0100.0ExperimentalNumerical0.00.0 4.0 8.0 12.0Horizontal displacement d [mm]Fig. 38 - Load-displacement diagram **for** wall W1 (low conÞning pressure)

38 1995 TNO-95-NM-R0712a) Total de**for**med mesh b) Incremental de**for**med mesh-2.40-2.05-1.70-1.36-1.01-0.67-0.320.03[MPa]c) Cracks d) Minimum principal stressesFig. 39 - Numerical results **for** a horizontal displacement d = 1.0 mm. Wall W1a) Total de**for**med mesh b) Incremental de**for**med mesh-3.94-3.37-2.79-2.22-1.64-1.06-0.490.09[MPa]c) Cracks d) Minimum principal stressesFig. 40 - Numerical results **for** a horizontal displacement d = 2.0 mm. Wall W1

TNO-95-NM-R0712 1995 39a) Total de**for**med mesh b) Incremental de**for**med mesh-5.43-4.64-3.85-3.06-2.27-1.49-0.700.09[MPa]c) Cracks d) Minimum principal stressesFig. 41 - Numerical results **for** a horizontal displacement d = 4.0 mm. Wall W1a) Total de**for**med mesh b) Incremental de**for**med mesh-6.48-5.53-4.58-3.64-2.69-1.75-0.800.14[MPa]c) Cracks d) Minimum principal stressesFig. 42 - Numerical results **for** a horizontal displacement d = 6.0 mm. Wall W1

40 1995 TNO-95-NM-R0712a) Total de**for**med mesh b) Incremental de**for**med mesh-6.86-5.86-4.86-3.87-2.87-1.87-0.870.12[MPa]c) Cracks d) Minimum principal stressesFig. 43 - Numerical results **for** a horizontal displacement d = 8.0 mm. Wall W1a) Total de**for**med mesh b) Incremental de**for**med mesh-7.00-5.98-4.95-3.93-2.90-1.87-0.850.18[MPa]c) Cracks d) Minimum principal stressesFig. 44 - Numerical results **for** a horizontal displacement d = 11.5 mm (ultimate). Wall W1

TNO-95-NM-R0712 1995 415.2.2 Wall W2The results **of** **the** numerical **an**alysis are given in Fig. 45 to Fig. 51. Fig. 45 shows **the** comparisonbetween experimental **an**d numerical results, in terms **of** load-displacement diagrams. Again, goodagreement is found between both results. The collapse load value found in **the** numerical **an**alysis is20% higher th**an** **the** experimental value but apart from this difference **the** same tendency in foundin both diagrams (note that **the** sharp reproduction **of** **the** collapse load value is not **the** main issuehere). As in **the** experiment, a brittle collapse is obtained shortly after **the** peak load due to compressivefailure **of** **the** specimen.The behaviour **of** **the** wall is depicted in Fig. 46 to 51 in terms **of** total de**for**med mesh, incrementalde**for**med mesh, cracked Gauss points **an**d minimum principal stresses contour. Again, reasonableagreement seems to be found between **the** experimental **an**d numerical behaviour. Initially, extensivecracking leads to some separation between **the** left ß**an**ge **an**d **the** p**an**el while **the** rest **of** **the**structure remains elastic, see Fig. 46. Shortly afterwards, three diagonal cracks occur, see Fig. 47.Upon mode I cracking **of** **the** right ß**an**ge, **the** previous diagonal cracks in **the** p**an**el concentrate intwo well deÞned diagonal crack b**an**ds starting from each corner **of** **the** p**an**el **an**d progressing to **the**opposite side **of** **the** p**an**el, see Fig. 48 **an**d Fig. 49. This agrees well with **the** experimental crackingpatterns **of** Fig. 37c. Finally, crushing **of** **the** compressed bottom toe leads to **the** explosive type **of**failure shown in Fig. 50 **an**d Fig. 51.600.0Horizontal **for**ce F [kN]400.0200.0ExperimentalNumerical0.00.0 2.0 4.0 6.0 8.0 10.0Horizontal displacement d [mm]Fig. 45 - Load-displacement diagram **for** wall W1 (high conÞning pressure)

42 1995 TNO-95-NM-R0712c) Cracks d) Minimum principal stressesFig. 46 - Numerical results **for** a horizontal displacement d = 1.0 mm. Wall W2a) Total de**for**med mesh b) Incremental de**for**med mesha) Total de**for**med mesh b) Incremental de**for**med meshc) Cracks d) Minimum principal stressesFig. 47 - Numerical results **for** a horizontal displacement d = 2.0 mm. Wall W2

TNO-95-NM-R0712 1995 43c) Cracks d) Minimum principal stressesFig. 48 - Numerical results **for** a horizontal displacement d = 4.0 mm. Wall W2a) Total de**for**med mesh b) Incremental de**for**med mesha) Total de**for**med mesh b) Incremental de**for**med meshc) Cracks d) Minimum principal stressesFig. 49 - Numerical results **for** a horizontal displacement d = 6.0 mm (peak). Wall W2

44 1995 TNO-95-NM-R0712c) Cracks d) Minimum principal stressesFig. 50 - Numerical results **for** a horizontal displacement d = 8.0 mm (be**for**e unloading). Wall W2a) Total de**for**med mesh b) Incremental de**for**med mesha) Total de**for**med mesh b) Incremental de**for**med mesh-10.7-9.19-7.64-6.09-4.55-3.00-1.450.10[MPa]c) Cracks d) Minimum principal stressesFig. 51 - Numerical results **for** a horizontal displacement d = 8.0 mm (ultimate). Wall W2

TNO-95-NM-R0712 1995 455.2.3 Comparison **of** **the** responses from specimens W1 **an**d W2A comparison between **the** behaviour **of** both walls in terms **of** load-displacement diagrams isshown in Fig. 52. Wall W1, with a low conÞning pressure, exhibits a extremly ductile behaviourwhereas Wall W2, with a high conÞning pressure, exhibits a relatively small plateau followed bybrittle failure, which agree well with **the** experiments. This gives a good impression **of** **the** **model**because both types **of** failure c**an** be captured.600.0Wall W2Horizontal **for**ce F [kN]400.0200.0ExperimentalNumericalWall W10.00.0 5.0 10.0 15.0Horizontal displacement d [mm]Fig. 52 - Comparison **of** load-displacement diagrams **for** walls W1 **an**d W2

46 1995 TNO-95-NM-R07126. CONCLUSIONSAn **an**isotropic pl**an**e stress **continuum** **model** **for** **the** **an**alysis **of** **masonry** **structures** has been presented.The **model** combines **orthotropic** elasticity with **orthotropic** plasticity. A composite yieldsurface that includes a Hill type yield criterion **for** compression **an**d a R**an**kine type yield criterion**for** tension has been developed. The **model** is novel in **the** sense that a relatively simple compositeyield surface is proposed **an**d hardening **an**d s**of**tening are included in a rational way. Modern plasticityconcepts as unconditionally stable implicit Euler backward return mappings, local Newton-Raphson iterative procedures **an**d consistent t**an**gent operators are used **for** all **the** modes **of** **the****model**. The per**for**m**an**ce **of** **the** implementation is assessed by me**an**s **of** single element tests. It isshown that **the** numerical algorithm is robust **an**d numerically efÞcient.The **model** predicts different yield strengths **an**d fracture energies along **the** material axes, both intension **an**d compression. These features deÞne **an** extremely ßexible **model** capable **of** accommodating**the** behaviour obtained from experiments. The behaviour **of** **the** **model** is detailed by me**an**s**of** single element tests, in which **the** response is evaluated upon different choices **of** **the** materialparameters. Un**for**tunately **the** number **of** test results **of** **the** composite material is relatively scarce**an**d almost non-existent in **the** post-peak regime. The setup **of** experimental programs coordinatedby experimentalists **an**d **an**alysts seem **the**re**for**e crucial.A comparison between numerical results **an**d experimental results available is also included. Previousattempts to use macro-**model**s, speciÞcally developed **for** **masonry** **structures**, included interfaceelements at **the** boundaries. The interface elements were responsible **for** most **of** **the** non-linear phenomenaobserved **an**d, thus, only a poor validation **of** previously proposed macro-**model**s exists. Itis shown that **the** **model** proposed in **the** present report is able to predict well **the** behaviour **of****masonry** **structures**, with both ductile **an**d brittle failure modes, as well as sufÞciently accurate collapseload values. However, this statement is only true if **the** structure is sufÞciently large that amacro-**model**ling strategy c**an** be applied.

TNO-95-NM-R0712 1995 477. REFERENCESde Borst, R., The zero-normal stress condition in pl**an**e-stress **an**d shell elasto-plasticity, Comm. in.Appl. Numer. Methods, Vol. 7, pp. 29-33 (1991)de Borst, R. **an**d Feenstra, P.H., Studies in **an**isotropic plasticity with reference to **the** Hill criterion,Int. J. Numer. Meth. Engrg., Vol.29, pp. 315-336 (1990)Baº**an**t, Z.P. **an**d Oh, B.H., Crack b**an**d **the**ory **for** fracture **of** concrete, Materials **an**d Structures,RILEM, Vol. 93(16), pp. 155-177 (1983)CrisÞeld, M.A., Accelerated solution techniques **an**d concrete cracking, Comp. Meth. Appl. Mech.Engrg., Vol. 33, pp. 585-607 (1982)Dh**an**asekar, M., Kleem**an**, P.W. **an**d Page, A.W., Biaxial stress-strain relations **for** brick **masonry**, J.Struc. Engrg., Vol. 111(5), pp. 1085-1100 (1985)Dh**an**asekar, M. **an**d Page, A.W., The inßuence **of** brick **masonry** inÞll properties on **the** behaviour**of** inÞlled frames, Proc. Intsn. Civ. Engrs., Part 2, Vol.81, pp. 593-605 (1986)Feenstra, P.H., Computational aspects **of** biaxial stress in plain **an**d rein**for**ced concrete, Dissertation,Delft University **of** Technology, Delft, The Ne**the**rl**an**ds (1993)G**an**z, H.-R., Failure criteria **for** **masonry**, 5 th C**an**adi**an** Masonry Symposium, V**an**couver, C**an**ada,pp. 65-77 (1989)G**an**z, H.-R. **an**d ThŸrlim**an**n, B., Experiments about **the** strength **of** biaxially loaded **masonry** p**an**els,Report 7502-3, Institute **of** Structural Engineering, ETH Zurich, Switzerl**an**d (in Germ**an**)(1982)G**an**z, H.-R. **an**d ThŸrlim**an**n, B., Experiments **of** **masonry** walls under normal **an**d shear loading,Report 7502-4, Institute **of** Structural Engineering, ETH Zurich, Switzerl**an**d (in Germ**an**) (1984)Hill, R., A **the**ory **of** **the** yielding **an**d plastic ßow **of** **an**isotropic metals, Proc. Roy. Soc. (London)A, Vol. 193, pp. 281-288 (1948)H**of**fm**an**, O., The brittle strength **of** **orthotropic** materials, J. Comp. Mat., Vol. 1, pp. 200-206(1967)Koiter, W.T., Stress-strain relations, uniqueness **an**d variational problems **for** elastic-plastic materialswith a singular yield surface, Q. Appl. Math., Vol. 11, pp. 350-354 (1953)Louren•o, P.B., Analysis **of** **masonry** **structures** with interface elements: Theory **an**d applications,Delft University **of** Technology, Report 03-21-22-0-01, Delft, The Ne**the**rl**an**ds (1994)Louren•o, P.B., The elastoplastic homogenisation **of** **masonry** **structures**: With **an** extension to**masonry** **structures**, Delft University **of** Technology, Report 03-21-1-31-02, Delft, The Ne**the**rl**an**ds(1995)Louren•o, P.B., Rots, J.G. **an**d Feenstra, P.H., A ÕtensileÕ R**an**kine type **orthotropic** **model** **for****masonry**, Proc. 3 rd Int. Symp. on Computer Methods in Structural Masonry, Lisbon, Portugal(1995)Ortiz, M. **an**d Popov, E.P., Accuracy **an**d stability **of** integration algorithms **for** elastoplastic constitutiverelations, Int. J. Numer. Methods Engrg., Vol. 21, pp. 1561-1576 (1985)Page, A.W., The biaxial compressive strength **of** brick **masonry**, Proc. Intsn. Civ. Engrs., Part 2,Vol. 71, pp. 893-906 (1981)

48 1995 TNO-95-NM-R0712Raijmakers, T.M.J. **an**d Vermeltfoort, A.Th., De**for**mation controlled meso shear tests on **masonry**piers, Report B-92-1156, TNO-BOUW/TU Eindhoven, Building **an**d Construction Research, TheNe**the**rl**an**ds (in Dutch) (1992)Rots, J.G., Computational **model**ling **of** concrete fracture, Dissertation, Delft University **of** Technology,Delft, The Ne**the**rl**an**ds (1988)Scarpas, A. **an**d Blaauwendraad, J., Non-local plasticity s**of**tening **model** **for** brittle materials, in:Fr acture **an**d Damage **of** Concrete **an**d Rock - FDCR-2, Eds. H.P. Rossm**an**ith, E & FN. Spon.,London, U.K., pp. 44-53 (1993)Schellekens, J.C.J. **an**d de Borst, R., The use **of** **the** H**of**fm**an** yield criterion in Þnite element **an**alysis**of** **an**isotropic composites, Comp. Struct., Vol. 37(6), pp. 1087-1096 (1990)Seim, W., Isotropic or **an**isotropic? Simulation **of** in-pl**an**e loaded **masonry** **structures** close to reality,10 th Int. Brick/Block Masonry Conf., Calgary, C**an**ada, pp. 77-86 (1994)Simo, J.C. **an**d Taylor, R.L., Consistent t**an**gent operators **for** rate-independent elastoplasticity,Comp. Meth. Appl. Mech. Engrg., Vol. 48, pp. 101-118 (1985)Simo, J.C. **an**d Taylor, R.L., A return mapping **for** pl**an**e stress elastoplasticity, Int. J. Numer. MethodsEngrg., Vol. 22, pp. 649-670 (1986)Simo, J.C., Kennedy, J.G. **an**d Govindjee, S., Non-smooth multisurface plasticity **an**d viscoplasticity.Loading/unloading conditions **an**d numerical algorithms, Int. J. Numer. Methods Engrg., Vol.26, pp. 2161-2185 (1988)Tsai, S.W. **an**d Wu, E.M., A general **the**ory **of** strength **of** **an**isotropic materials, Comp. Materials,Vol. 5, pp. 58-80 (1971)Vermeltfoort, A.Th. **an**d Raijmakers, T.M.J., De**for**mation controlled meso shear tests on **masonry**piers, Part 2, Draft report, TU Eindhoven, The Ne**the**rl**an**ds (in Dutch) (1993)Vonk, R.A., S**of**tening **of** concrete loaded in compression, Dissertation, Eindhoven University **of**Technology, Eindhoven, The Ne**the**rl**an**ds (1992)Willam, K.J., Pramono, E. **an**d Sture, S., Fundamental issues **of** smeared cracked **model**s, Proc.SEM/RILEM Int. Conf. Fracture **of** Concrete **an**d Rock, eds. S.P.Shah **an**d S.E. Swartz,Springler-Verlag, New-York, U.S.A., pp. 142-157 (1987)

TNO-95-NM-R0712 1995 49APPENDIX APERFORMANCE OF THE NUMERICAL IMPLEMENTATIONIn this appendix **the** per**for**m**an**ce **of** **the** numerical implementation is investigated.One pl**an**e stress element with unit dimensions is loaded under different conditions. The load isintroduced in **the** element under direct displacement control **an**d a number **of** equally spaced plasticincrements **of** **the** displacement follow one elastic increment. If **the** direction **of** **the** plastic ßow isconst**an**t, no dependence is found on **the** step size, i.e, **the** integration is exact because **the** algorithmis strain driven. O**the**rwise **an** extremely small (< 1%) dependence on **the** mesh size was found butno discussion will be given here about **the** accuracy **of** **the** return mapping algorithm. For this purposesee e.g. Ortiz **an**d Popov (1985), de Borst **an**d Feenstra (1990) **an**d Schellekens **an**d de Borst(1990). The toler**an**ce **for** **the** local return mapping algorithm equals **an** absolute value **of** 1 × 10 −9**for** all residuals **an**d **the** toler**an**ce **for** **the** **for**ce norm **of** **the** structural response equals 1 × 10 −8 .Aregular Newton-Raphson is used **for** **the** global iterative procedure.A.1 TENSION REGIMEThe material properties given in Table A.1 are assumed.A.1.1 Uniaxial tensionTable A.1 - Material properties (isotropic - α = 1.0)Material propertiesE 10000 N/mm 2ν 0.2f t 1.0 N/mm 2G f 0.00015 N.mm/mm 2The element is subjected to uniaxial tension along **the** x axis as shown in Fig. A.1. The results areillustrated in Fig. A.2.1.00.8yxdx [MPa]σ0.60.40.2Fig. A.1 - Uniaxial tension test0.00.0 1.0 2.0 3.0 4.0-4εx[10 ]Fig. A.2 - Stress-strain response in uniaxial tension

50 1995 TNO-95-NM-R0712The per**for**m**an**ce **of** **the** **model** is given in Table A.2 to Table A.4. Quadratic convergence is indeedfound at local **an**d global level.Table A.2 - Number **of** local iterations per global iteration **for** uniaxial tensionPlastic steps Av erage Maximum100 3.24 410 4.24 5A.1.2 General pathTable A.3 - Number **of** global iterations **for** uniaxial tensionPlastic steps Av erage Maximum100 2.08 310 2.40 3Table A.4 - Convergence **for** selected steps (uniaxial tension)100 plastic steps10 plastic stepsPlastic step1Force norm0. 453 × 10 00. 370 × 10 −30. 270 × 10 −8Plastic step1Force norm0. 371 × 10 00. 310 × 10 −30. 876 × 10 −8501000. 930 × 10 −40. 564 × 10 −110. 219 × 10 −40. 537 × 10 −135100. 406 × 10 −30. 703 × 10 −80. 853 × 10 −40. 886 × 10 −10The general path shown in Fig. A.3 is now considered. The response in terms **of** a **for**cedisplacementdiagram is given in Fig. A.4.0.51F, d1.5yxσ y = -1.0MPaF [MN]1.00.5Fig. A.3 - General test0.00.0 2.0 4.0 6.0-4d [10 mm]Fig. A.4 - Force-displacement response **for** general testThe per**for**m**an**ce **of** **the** **model** until peak (thicker line in Fig. A.4) is given in Table A.5 to TableA.7. After this point two integration points reach **the** apex **an**d **the** apex algorithm is used. This caseis **an**alysed independently in **the** next section. Note that quadratic convergence is indeed found.

TNO-95-NM-R0712 1995 51Plastic step150100A.1.3 ApexTable A.5 - Number **of** local iterations per global iteration **for** general path100 plastic stepsPlastic steps Av erage Maximum100 3.15 410 4.08 5Table A.6 - Number **of** global iterations **for** general pathPlastic steps Av erage Maximum100 3.01 410 3.80 5Table A.7 - Convergence **for** selected steps (general path)Force norm0. 212 × 10 20. 712 × 10 −10. 190 × 10 −40. 344 × 10 −120. 113 × 10 −20. 956 × 10 −70. 000 × 10 −∞0. 233 × 10 −30. 172 × 10 −60. 126 × 10 −12Plastic step151010 plastic stepsForce norm0. 816 × 10 1 0. 419 × 10 −50. 776 × 10 0 0. 407 × 10 −120. 134 × 10 −10. 103 × 10 10. 113 × 10 −30. 952 × 10 −80. 498 × 10 −20. 213 × 10 −30. 284 × 10 −60. 512 × 10 −12The apex algorithm per**for**m**an**ce is now evaluated. The test illustrated in Fig. A.5 is considered.The obtained **for**ce-displacement diagram is given in Fig. A.6.2.0F, d11F [MN]1.51.0yx0.5Fig. A.5 - Apex test0.00.0 2.0 4.0 6.0-4d [10 mm]Fig. A.6 - Force-displacement response **for** apex testThe per**for**m**an**ce **of** **the** **model** is given in Table A.8 to Table A.10. Quadratic convergence at globallevel is indeed found. At local level quadratic convergence is not found but supra-linear convergenceis found. As **the** apex is a well-deÞned area **of** **the** yield surface **the** number **of** local iterations

52 1995 TNO-95-NM-R0712is approximately **the** same as in **the** **for**mer cases. Note that **the** inconvenience **of** using **the** exp**an**sion/compressionmech**an**ism, de Borst(1991), **for** a basically pl**an**e stress yield criterion is againevident. In a pl**an**e stress algorithm global convergence would be found in one step because all **the**displacements are prescribed whereas, with **the** exp**an**sion/compression mech**an**ism, a non-zero out**of**-pl**an**enormal stress component must be reduced to zero.Table A.8 - Number **of** local iterations per global iteration **for** **the** apex algorithmPlastic steps Av erage Maximum100 3.40 410 4.32 5Table A.9 - Number **of** global iterations **for** **the** apex algorithmPlastic steps Av erage Maximum100 2.01 310 2.10 3Table A.10 - Convergence **for** selected steps (apex algorithm)100 plastic steps10 plastic stepsPlastic step1Force norm0. 515 × 10 00. 480 × 10 −50. 210 × 10 −11Plastic step1Force norm0. 516 × 10 00. 491 × 10 −40. 557 × 10 −1250100A.2 COMPRESSION REGIME0. 529 × 10 −50. 168 × 10 −120. 101 × 10 −50. 139 × 10 −135100. 693 × 10 −30. 500 × 10 −110. 120 × 10 −30. 112 × 10 −13The material properties given in Table A.11 are assumed, in which **the** material strength **an**dYoungÕs modulus in **the**y-direction are penalized by a factor 2. Three different fracture energiesare considered **for** **the** y-direction: 0. 3G fcx ,G fcx / 2 (isotropic s**of**tening) **an**d 500 × G fcx (almostideally plastic behaviour).Table A.11 - Material properties (β = -1.0, γ = 3.0 **an**d κ p = 0.0005)Material propertiesE x 10000 N/mm 2 E y 5000 N/mm 2ν xy 0.2 G xy 3000 N/mm 2f mx 10.0 N/mm 2 f ty 5.0 N/mm 2G fcx 0.05 N.mm/mm 2 G fcy 7.5 N.mm/mm 2A.2.1 Uniaxial compressionThe element is subjected to uniaxial compression along **the** x axis as shown in Fig. A.7. The resultsare illustrated in Fig. A.8.

TNO-95-NM-R0712 1995 53-10.0-8.0yxdx [MPa]σ-6.0-4.0-2.0Fig. A.7 - Uniaxial compression test0.00.0 -2.0 -4.0 -6.0 -8.0 -10.0-3εx[10 ]Fig. A.8 - Stress-strain response in uniaxial compressionThe per**for**m**an**ce **of** **the** **model** is given in Table A.12 to Table A.14. Note that, in **the** calculation **of****the** maximum value **of** local iterations per global iteration, **the** Þrst plastic step is ignored (valuegiven inside brackets). The author believes that this value is misleading because **the** problem is illdeÞned**for** **the** Þrst plastic step, i.e. **the** derivative **of** **the** equivalent stress-equivalent strain diagramequals +∞ **for** κ c = 0. Note also that quadratic convergence is indeed found at local **an**d globallevel.Table A.12 - Number **of** local iterations per global iteration **for** uniaxial compressionPlastic steps Av erage Maximum200 4.82 6 (10 )20 7.68 13First stepTable 13 - Number **of** global iterations **for** uniaxial compressionPlastic stepsAv e