Finite Element Modelling of Anisotropic Elasto-plastic Timber Composite Beams With Openings

8/12/2019 Finite Element Modelling of Anisotropic Elasto-plastic Timber Composite Beams With Openings

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Engineering Structures 31 (2009) 394403

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Engineering Structures

journal homepage:www.elsevier.com/locate/engstruct

Finite element modelling of anisotropic elasto-plastic timber composite beamswith openings

Z.W. Guan a,, E.C. Zhu ba Department of Engineering (Civil), University of Liverpool, Brownlow Street, Liverpool L69 3GQ, UKb School of Civil Engineering, Harbin Institute of Technology, 202 Haihe Road, Harbin, PR China

a r t i c l e i n f o

Article history:

Received 7 May 2008

Received in revised form

18 June 2008

Accepted 1 September 2008

Available online 1 October 2008

Keywords:

Anisotropic

Composite timber beam

Finite element

Interaction

Orthotropic

OSB

User subroutine

Web opening

a b s t r a c t

In this paper, constitutive equations to model anisotropic elasto-plastic timber composite beams withopenings were formulated and implemented into the finite element (FE) package ABAQUS, via a user-defined subroutine. The TsaiHill criterion was applied to judge failure of Oriented Strand Board (OSB)

and timber in tension. Both OSB and timber in tension were modelled as linear orthotropic elasticmaterials,and in compressionas orthotropicelasto-plastic materials.Good correlationhas been obtained

between the experimental results and the FE simulations. The user subroutine was used to check andremove critical elements, through which crack growth was simulated. In addition, interactions between

two openings were modelled, which gave the corresponding critical distance.

2008 Elsevier Ltd. All rights reserved.

1. Introduction

OSB webbed timber I-beams have been widely used inconstruction industry in Europe and North America, due to anumber of advantages, such as engineered features, materialsavings, low handling costs and its environmental friendlynature [14]. Openings in webs, usually square or circularshaped, are often needed to allow services to pass through.Openings made through the webs will affect the structuralperformance of beams to different extents, depending upon theopening location, size and beam depth[5,6]. Stress distributionsaround an opening are complicated, varying between tension and

compression, depending on loading conditions and deformationmodes. OSB and timber can be either treated as anisotropic ororthotropic materials, dependent upon whether the structuralbehaviour through the panel thickness plays a more importantrole in comparison with in-plane behaviour. Both materials behavedifferently in tension and compression. Experiments show thatOSB in tension behaves almost linearly up to failure, whilst incompression it exhibits obvious plasticity [7]. Therefore, it isnecessary to develop appropriate constitutive models that can

Corresponding author. Tel.: +44 151 7945210; fax: +44 151 7945218.E-mail address:[email protected](Z.W. Guan).

deal with the constituent materials under different stress statesautomatically, by implementing them into computer models.

Research on modelling of crack initiation and crack growth intimber structures has been limited so far [810]. Premrov andDobrila[11]developed a semi-analytical modelling for simulatingsmall cracks on gypsum plasterboards (GPB) of prefabricatedtimber-framed walls, using the modified -method. The mostchallenging part, is to simulate crack growth in timber, sincesuch growth is fast and the growth path cannot be accuratelydetermined beforehand.More recently, Smith et al. [12] undertooka thorough review on discrete, continuum and hybrid finiteelement approaches to address failure mechanisms in wood-based

materials. The discrete element approaches [1315]have shownthat discrete element models are very effective at simulating thefailure mechanisms and failure capacities of components of woodproducts. The continuum element approaches are largely based on2-D linear elastic fracture mechanics to predict the failure of loadsof notched wood components[16,17], to model fracture of wood,using the bridging crack model [14,18] and to simulate failuremechanisms of single bolt specimen [19]. Using the continuumelement approach, studies carried out by the authors [20] haveshown some promising developments in the first stage of usingmanual control in the modeling of crack growth initiated from anopening in OSB webbed timber I-beam.

In this paper, timber composite I-beams with openingswere modelled, using a user defined subroutine, which was

0141-0296/$ see front matter 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2008.09.007
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Z.W. Guan, E.C. Zhu / Engineering Structures 31 (2009) 394403 395

implemented into the FE package ABAQUS. The TsaiHill criterion

was applied to judge failure of OSB in tension. The user-defined

subroutine is capable of distinguishing and tracing tension and

compression zones in the flange and the web, and modelling

those zones with different constitutive models accordingly. Both

OSB and timber in tension were modelled as linear orthotropic

elastic materials, and in compression as orthotropic elasto-plastic

materials [21,22]. Good correlation has been obtained betweenthe experimental results and the FE simulations. Crack initiation

and growth were also simulated by element removal techniques

controlled by the user-defined subroutine. In addition, interactions

between two openings were modelled, which produced the

corresponding critical distances between two circular openings,

two square openings, and a circular opening and a square opening.

2. Description of problems

For a beam without any openings, subjected to bending, its

tension and compression zones, which should be treated with

different constitutive models, are clearly separated by the neutral

axis. Its behaviour can be simulated with little difficulty. However,for a beam with openings subjected to similar loading conditions,

the situation is quite different, since both tensile and compressive

zones around the openings are variable, dependent upon the

applied loading types and levels. Therefore, an algorithm, which

can detect tension and compression zones automatically, and treat

them with properconstitutivemodels accordingly,must be sought.

Automatic checking has to be undertaken in every iteration.

Both timber and OSB are treated as orthotropic materials in

this paper, which behave differently in tension and compression.

In tension, both materials follow linear orthotropic elasticity;

however, in compression, they demonstrate certain amount of

plasticity [7]. Therefore, orthotropic elasto-plasticity is employed

in the compression zone, whilst linear orthotropic elasticity inthe tension zone. When deal with stresses around an opening,

shear stresses act with both tensile and compressive stress which

add complexity of stress state there. In the modelling, the OSB

web is assumed to be fully bonded with the timber flange, which

is generally true, based on observations of the experimental

work [23].

Failure can occur either in the OSB web or the timber flange.

For a beam with openings, failure is dependent upon the shape,

the size and the location of an opening. The lower bound failure is

based on the initial crack, and the upper bound failure is based on

the ultimate failure. The corresponding failure stresses, which are

obtained from material tests, are required to be implemented into

the failure criteria.

3. Finite element formulation

Before implementing the user-defined constitutive models into

an ABAQUS programme, finite element formulation of anisotropic

elastic behaviour and plastic behaviour need to be developed.

3.1. Orthotropic elasticity

{}=[D]orth{} (1)

where{}is strain tensor, {} is stress tensor and [D]orth isorthotropic elastic matrix(66), i.e.

{} = {L RTRTLTLR}T (1a){} = {LRTRTLTLR}T (1b)

[D]orth=

DLLLL DLLRR DLLTT 0 0 0DRRRR DRRTT 0 0 0

DTTTT 0 0 0

Symmetric DLRLR 0 0D

LTLT 0DRTRT

. (2)

All components in Eq.(2)are defined below[28]:

DLLLL=EL(1TRRT) (2a)DRRRR=ER(1LTTL) (2b)DTTTT= ET(1LRRL) (2c)DLLRR=EL(RL+TLRT) (2d)DLLTT= EL(TL+RLTR) (2e)DRRTT= ER(TR+LRTL) (2f)DLRLR=ER(LR+TRLT) (2g)

DLTLT= ET(LT+LRRT) (2h)DRTRT= ET(RT+RLLT) (2i)= 1/(1LTTLTRRTRLLR2TLRTLR). (2j) is an effective factor based on Poissons ratios, which is used to

give simplified expressions of Eqs. from(2a)to(2i).

The above values also satisfy the stability requirements[28]:

DLLLL, DRRRR, DTTTT, DLRLR,DLTLT, DRTRT >0 (3a)

DLLRR < (DLLLLDRRRR)0.5 (3b)

DLLTT < (DLLLLDTTTT)0.5 (3c)

DRRTT < (DRRRRDTTTT)0.5 (3d)

det([D]orth) >0 (3e)i.e.

DLLLLDRRRRDTTTT+2DLLRRDLLTTDRRTTDRRRRD2LLTTDLLLLD2RRTTDTTTTD2LLRR > 0. (3f)

Substituting material parameters shown inTable 1,all compo-

nents of the orthotropic elastic matrix in Eq.(2)can be calculated.

3.2. Anisotropic plasticity

Hills yield criterion (stress potentials) [24] has been adopted

to simulate anisotropic plastic behaviour of the OSB web and thetimber flange. The Hill stress potentials are dependent only upon

the deviatoric stress, so that the plastic part of the response is

incompressible. This implies that, in cases where the plastic flow

dominates the response, except for plane stress problems, the

finite elements should be chosen to accommodate incompressible

flow. Ultimate load calculations for OSB-webbed timber I-beams

are such cases. Hills stress potentials, in terms of rectangular

Cartesian stress components, are

f(ij)=

F11(2233)2 +F22(3311)2

+ F33(1122)2 +2N12212+2N23223+2N31212

12 (4)

where Fii(i = 1, 2, 3) and Nij(i = j = 1, 2, 3) are constantsobtained by tests of the material in different orientations, and are


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396 Z.W. Guan, E.C. Zhu / Engineering Structures 31 (2009) 394403

Table 1

Material properties of the OSB and the timber

Component EL ER ET LT TL LR RL TR RT

OSB 3 708 2660 130 0.364 0.013 0.184 0.144 0.019 0.312

Timber 10 500 900 500 0.470 0.020 0.370 0.029 0.250 0.430

N/mm2 for all modulus, for OSB L1,R2,T3.

defined as follows:

Fii=

02

2

1

2jj+ 1

2kk 1

2ii

= 12

1

R2jj+ 1

R2kk 1

R2ii

i=1, 2, 3

j=2, 3, 1k=3, 2, 1

(4a)

Nij=3

2

0

ij

2= 3

2R2ij(i= j=1, 2, 3). (4b)

Rij are yield ratios which relate the yield level for stress

componentijto the reference yield stress0 of the material. The

yield ratios are defined as follows:

Rij=

ij

o, ifi=j

ij

o, ifi=j

(5)

0=0

3. (6)

For the orthotropic material plasticity, the associated flow ruleused is given by:

d{}pl=d

f

{}

= d

f{} (7)

where d is a proportionality constant termed the plasticmultiplier. Before yielding, there is f(ij, Fii, Nij) < 0. Therefored=0. From Eqs.(4)and(7),there is

f

ij

= 1

f[] (8)

where

{} =

F22(3311)+F33(1122)F11(2233)F33(1122)

F11(2233)+F22(3311)2N12122N31312N2323

. (9)

Furthermore, based on Eq. (8), the second order of partialdifferential of the yield function can be expressed as

2f

ijij

= 1

f

[]ij

1f2

[][]

(10)

where

[]ij

=

F22+F33 F33 F22 0 0 0F33 F11+F33 F22 0 0 0F22 F11 F11+F22 0 0 0

0 0 0 2N12 0 0

0 0 0 0 2N31 0

0 0 0 0 0 2N23

. (10a)

At TEM (transmission electron microscopy) solution level, timberhas a cellular structure that is made up of lumens and cell

walls[25]. Reiterer and Stanzl-Tschegg [26] showed that due tothis structure, timber cells buckle and collapse under compressive

loads. In the longitudinal direction, they reported plastic softening

(post-peak) in their specimens. Perpendicular to the grain, theyreported extended yielding plateaux followed by significant

hardening after the cell walls had collapsed into the lumens andthe wood densified. This type of hardening is not feasible in the

timber flange. Therefore, no material hardening is assumed for

modelling the timber in the current research. Also, as tougheningis not included, the current models likely produce lower bound

predictions.

4. Finite element implementation

Finite element code ABAQUS [27,28] was utilised to obtainnumerical simulations. First order iso-parametric solid elementsandthe related interfaceelementswere used forthe 3-dimensional

problems. The purpose of using interface elements, is to model the

possible interaction between the flange and the web.

An attempt was made to simulate the cracking behaviour ofbeams with openings, by removing the damaged elements along

the fracture line [5]. Great effort was needed to manually read

the stress output and to make judgements of material failure.The efficiency of modelling beams in this way was low. To

improve the situation a user-defined constitutive subroutine was

developed, which automatically assesses whether a particularelement in a beam, especially around an opening, is in tension

or in compression, and whether a failure criterion is satisfied.

The appropriate material constitutive model could thus be appliedand the propagation of cracks traced along progressively damaged

elements.

In addition to the above special purposes, thecoreof this devel-opment is to update current stresses, strains and material proper-

ties, which are dependent upon current solutions, for computation

of the following load increments. Using the subroutine, materialnonlinearity, fracturing of OSB, crack propagation and failure of a

beam can be dealt with.

In the subroutine, computation procedures follow two material

routes OSB andtimber. Foreach of the twoorthotropic materials,there are two sub-routes to follow tension and compression.

For OSB in tension, stresses are checked at every iteration.Cracking load is reached when the TsaiHill criterion is first

satisfied. The material properties in cracked zones are then set tobe zero, and the elements there lose stiffness in the subsequentiterations. With an increase of load, more damaged elements on

the OSB web will occur to lead crack growth. Crack propagationcan thus be traced till the cracked zones extend to the flanges, and

the ultimate load is reached. The above numerical processes can

stand alone without the element removal, which will be able topredict the initial cracking load and the ultimate load, but without

showing actual cracks.

For OSB in compression, the initial yield stress is set to be

60% of the ultimate stress [7]. After initial yield the orthotropicelasto-plastic constitutive relationship is applied. Failure of OSB in

compression is defined as when the equivalent strain reaches the

statistical ultimate value of 0.008 [7].

If thetensile stress in a flangereaches the strength of thetimber,although this is less likely to happen to beams with openings, the


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Fig. 1. Flowchart for a load increment in the subroutine.

beam will collapse. For timber in compression, the initial yield

stress isset to be 75%of the ultimate strength,and the failure strain

is 0.0085, based on the statistics of the compression test data [7].

The flowchart for an iteration in the subroutine is shown in

Fig. 1, where the treatment of OSB is illustrated. Timber was

similarly treated.

The Model Change Option in ABAQUS [28], in conjunction

with the user-defined subroutine, is used to simulate the damage

behavior of the web with openings, i.e. to remove the fractured

elements along a path where stresses are judged to have reached

a critical level, determined by tests. Once a critical element is

removed, stress redistributions occur in its surrounding areas.

Elements located in the crack growth path will bear moreredistributed stresses than other surrounding elements. Since this

is a static problem, a loading rate does not numerically apply here.Stresses in the crucial areas are checked in every iteration. Whenthe web fractures fully and de-bonding between the flange andthe web takes place, the stresses in flanges become critical. If theirvalues reach the maximum strength of a wood, beam failure islikely to occur.

The procedures of the element removal are presented asfollows:

a. Check principal stresses or strains in elements in high stressareas against the critical stress or strain

max(pri)e cri or max(pri )e cri (11)

wheree is an element number, which links to the elements tobe checked, and cri and cri are the critical stress and strain


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Fig. 2. Mesh generations with boundary and loading conditions, forexamplebeams (a)A beam without opening,(b) A beam with circular openings, (c)A beam with square

openings.

obtained from uniaxial tension tests, which are 13 MPa and4000 micro-strain respectively, obtained from averaging thecorresponding longitudinal and transverse values.

b. Once an element checked satisfies conditions in Eq. (11),it isthen removed from the beam by setting its material propertiesas zero. There will be a redundancy in Jacobian matrix, whichwill affect the tangential stiffness matrix[K]T, i.e. assumeelementiireaches the critical condition there is

[K]T=

Ke11. . . Ke1i. . . . . . K

e1n

... . . .

Kei1. . . Keii= 0 . . . Kein

...

Ken1. . . Keni. . . . . . K

enn

(12)

wherenis the total number of elements.c. Start a new step based on a redundant structural tangential

stiffness and repeat procedures a and b.d. Terminate the program when the load starts to go horizontally,

i.e. there is no load increase in subsequent increment.

It is worth pointing out, that using Mode Change or ElementRemoval a possible de-bonding failure between the flange andthe web can be dealt by removing the critical elements in the

de-bonding region, provided the de-bonding stress is known. Inthe current study, all critical elements were evaluated against

the corresponding critical principal stress or strain, which may

overestimate the de-bonding resistance of the beam. However, de-

bonding only occurs in few situations, such as flaws in the flange

in the critical region and the extreme loading conditions.

Interaction between two openings (see Fig. 7) was also

modeled using the user-defined constitutive subroutine. Whentwo openings approach each other, the interactive stress state is

changed with the inter-distance between the two openings, i.e. the

corresponding tension zone and compression zone are constantly

varied. The only approach to deal with such variable stress zone, is

to use the user-defined subroutine so that appropriate constitutive

models and material properties can be applied to those zonesaccordingly.

5. Examples

Three examples that do not involve the element removal are

shown in the first, one for a beam without any opening, one for abeam with circular openings, and another with square openings.

All beams are spanned in 4800 mm, with overall depth of 450 mm.

The opening is 1000 mm away from the symmetrical section of the

beam, with the square opening sized as 180 mm180 mm andthe circular opening in diameter of 180 mm. Fig. 2 shows the mesh

generations, boundary, loading and geometrical conditions.Material properties are listed as follows.


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Fig. 3. Loaddeflection relationships for a beam without opening.

Timber (Sitka Spruce of strength class C24)[23,29]:

EL=1.05104 N/mm2, ER=9.0102 N/mm2,ET= 5.0102 N/mm2,

GLR

=7.5

102 N/mm2, GLT

=7.2

102 N/mm2,

GRT= 39 N/mm2,LR=0.37, LT= 0.47, RT= 0.43, TR=0.25,

RL=0.029, TL=0.020.OSB[23]:

0=11=14.1 N/mm2, 22=12.62 N/mm2,33

=6.31 N/mm2,

0=14.10/3=8.14 N/mm2, 12=7.5 N/mm2,23=1.5 N/mm2, 13=1.5 N/mm2.

Substituting the strengths into the expressions of the con-stants [27,28], gives the following values.

F11=20

2

1

222

+ 1233

111

2

= 14.12

2

1

12.622+ 1

6.222 1

14.12

=2.69

F22=20

2

1

233

+ 1211

122

2

Fig. 4. Numerical simulations for a beam with circular openings at an ultimate load of 24 kN. (a) Loaddeflections curves, (b) Shear stress(S12, N/mm2

)in the web atultimate load (24.0 kN), (c) Tensile and compressive principal stresses at failure SP3(N/mm2)and SP1(N/mm2).


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Fig. 5. Numerical simulations for a beam with square openings at an ultimate load of 20.2 kN (a) Loaddeflections curves, (b) Shear stress(S12, N/mm2)in the web at

ultimate load (20.2 kN), (c) Tensile and compressive principal stresses at failure, SP3(N/mm2 )and SP1(N/mm2).

= 14.12

2

1

6.222+ 1

14.12 1

12.622

=2.45

F33=20

2

1

2

11

+ 12

22

133

2

= 14.12

2

1

14.12+ 1

12.622 1

6.222

= 1.45

N12=3

2

0

12

2= 3

2

8.14

7.5

2= 1.77

N13=3

2

0

13

2= 3

2

8.14

1.5

2= 44.17

N23=3

2

0

23

2= 3

2

8.14

1.5

2= 44.17.

Fig. 3shows the loaddeflection curves for the beam without

opening, obtained from the FE simulation and the experimental

work. Reasonably good correlation has been obtained. The initialdamaging load was 27.0 kN and the ultimate load was 41.7 kN,

compared with testload of 37.7 kN. Discrepancies in the later stagemay be caused by local instabilities such as wrinkling or warpingin the loading points and supports.

Fig. 4a shows the loaddeflection curves for the beam withcircular openings. It can be seen that the overall behaviour of

the beam was well simulated. Fig. 4b shows shear stress (S12)distributions in the web at the ultimate load. There are shearstress concentrations around the opening. Fig. 4c shows the tensile(SP3) and compressive (SP1) principalstresses at the ultimate load.The damaged areas expanded in the tension zones and reachedthe bottom flange, while yielded areas expanded in compressionzones with more plasticity being developed in the upper rightcorner. However, at a predicted cracking load of 16.5 kN, tensileand compressive stress concentrations appeared at the diagonallyopposite corners of the opening, with the maximum tensile stressat the lower right corner, and the maximum compressive stressat the upper right corner. This indicates that cracking would takeplace initially from the lower right corner and OSB would yieldfrom the upper right corner.

Fig. 5shows the modelling results for the beam with squareopenings. It first cracked from thelower right cornerof the opening


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Fig. 6. Comparison of experimentally failed modes and numerically simulated failed modes (ratio of the opening size to the web depth is 0.75). (a) A beam with circularopenings. (b) A beam with square openings,

Fig. 7. Principal tensile stress(SP3, N/mm2)in beam webs at ultimate load. (a) Openings 500 mm apart, (b) openings 250 mm apart.

at a load of 9.0 kN (14.0 kN in test), collapsed at 20.2 kN (21.3 kN

in test). The lower predictions were likely caused by precluding

toughening in the models. The maximum tensile and compressive

stresses in the flanges were 19 and26 N/mm2 respectively.Crack initiation and growth were then modelled, using the

elementremoval controlled by the stress checking. Fig.6 shows the

failure modes obtained from FE simulations forbeamswith circular

and square openings, respectively. Experimentally failed beamsare

also shown in the same figure for better comparison. Compared

with beam with circular openings (see Fig.6a),the behaviour of thetwo beams are similar, both in tests and FE simulations. They both

cracked from the lower right corners of the opening, and collapsed

when cracks reached the flanges. However, the cracking load and

the ultimate load of the latter are significantly lower than those of

the former, proving the fact that a square opening imposes more

damage to a beam than the same sized circular opening does.

Both tests and FE analysis have shown that when two openings

are sufficiently far away from each other, the interaction effect is

not significant. As they become closer to each other, interactions

become more severe, and the load carrying capability of a beam is

further reduced. The distance between two openings is defined asthe critical distance at which interactions become obvious. This is


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Fig. 8. Initial cracking loadversus distance between openings(ratio of the opening

size to the web depth is 0.5).

symbolised in FE analysis as the point at which the initial cracking

load is first reduced as a result of the interaction.Interactions, in terms of the critical distance, between two

circular openings, a circular opening and a square opening, andtwo square openings were simulated using the developed models.

Here,the maximum principal stressdistributions from interactionsbetween a circular opening and a square opening are shown in

Fig. 7. Fig. 7a shows the stress distributions around the squareopening and the circular opening 500 mm apart. There is littleinteraction effect, since the openings are sufficiently far away fromeach other.Fig. 7b shows that interactions between the openings250 mm apart become more severe. There are moderate effects

on both tensile and compressive stress between the right handcorners of the square opening and the left hand side corners ofthe circular opening. As a result, the tension and the compressionzones at the far corners of the openings are much more severethan those in the beam shown in Fig. 7a. In fact, there are high

tensile stress regions between the two openings almost merged,which could contribute a damage linking both openings. It should

be noted that interaction between the openings would be alsodependent upon the size of the opening, which is not included inthe current study.

To investigate the critical distance, parametric studies werecarried out, in which one opening was kept at a distance of1000 mm from the mid-span, just at theposition where the squareopening was in the tests. The other opening was moved graduallytowards the fixed position opening at 50 mm intervals, starting

from 750 mm away. Fig. 8 shows the relationships between theinitial cracking load and the distance between two openings. Theinitial cracking loads shown in Fig. 8 indicate that the criticaldistance between two circular openings and between a square anda circular opening is 500 mm, and between two square openingsis 550 mm. If openings of these sizes are spaced at distances

greater than the critical distances found, interactions will havelittle influence on bending behaviour of a beam. Otherwise, initialcracking and ultimate loads of a beam will be further reduced andthe failure mode maychange. It is interesting to compare the abovecritical distance, which is about 2.2 times the opening width, with

industrialrecommendations that areusually 22.5 times the widthof the largest opening. However, the calculated critical distanceswere based on the initial cracking load, which is usually 50%60%of the ultimate load. Therefore, the predictions are well into thesafe side.

The developed user subroutine can also be used to simulatede-bonding between two bonded sections, provided critical de-bonding stresses are obtained from further experimental work.Tracing of de-bonding paths can be carried out by checking

selected key elements where de-bonding is likely to occur througheach iteration. However, if there is no clue where the possible

de-bonding will be, the number of elements to be checked must

be increased. This will inevitably increase computing costs. In

practical modelling, mesh sizes in possible critical regions need to

be small enough, so that removal of de-bonded elements will not

cause unrealistic loss of load carrying capacity in a structure.

6. Conclusions

A user-defined constitutive subroutine has been successfully

implemented into a commercial FE code to simulate 3-dimensional

structural behaviourof composite timber beams, with and without

openings. The model developed can deal with various constituent

materials, such as OSB and timber, under variable stress states,

by selecting appropriate constitutive relationships and the corre-

sponding material properties automatically. Reasonably good cor-

relation between the experimental results and the FE simulations

has been obtained. The model can identify the location of the ini-

tial cracking around an opening and the related load. In conjunc-

tion with the element removal, the model is capable of simulating

crack growth in an efficient way. It can also predict the ultimate

load, by assessing stress states in key elements. Using validated

models, interactions between two openings were also studied anddiscussed, by which the critical distances between openings with

various combinations were produced.

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Documents

Finite Element Modelling of Anisotropic Elasto-plastic Timber Composite Beams With Openings