boundary elements modeling of plain concrete notched beams by means of crack propagation...

7/30/2019 BOUNDARY ELEMENTS MODELING OF PLAIN CONCRETE NOTCHED BEAMS BY MEANS OF CRACK PROPAGATION FOR

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BOUNDARY ELEMENTS MODELING OF PLAIN CONCRETE NOTCHED

BEAMS BY MEANS OF CRACK PROPAGATION FORMULA

Gospodin Gospodinov1 and Irina Kerelezova2

1. Introduction

Cracks are present to some degree in all plain and reinforced concrete structures. They exist either as

basic defects in materials or they may be induced in the construction during the time of the service. Cracksoften act as stress concentrators, so in many cases they are the main reason for catastrophic crack

propagation and structural failure under increasing load. Such failures being of a very great concern have

led to the evolution and development of the theory of fracture mechanics.

The present concrete structural design does not take into account the tensile carrying capacity of

concrete and is almost entirely based on the elasticity and plasticity theories. Still these structures aredesigned with no regard to the propagation of large cracking zones and the energy failure criterion is not

employed. As a result the classical failure theories, such as the limit state theory, which are based on

conventional strength criterion, are unable to adequately explain certain phenomena typical for concrete

structures. These are for example the strain softening associated with the post-cracking behaviour, the non-

simultaneous failure due to propagating cracks and especially the size effect phenomena. It was obvious inthe last few decades that a new failure theory is needed, so fracture mechanics provided such a general

failure theory, see references [1], [2] and [3].

Fracture mechanics is a theory that determines the material failure by energy criteria but it could be

used in conjunction with strength criteria. While this theory has been already generally accepted for the

analysis and failure of metal structures, its application in the field of concrete structures is relatively new.

There are strong arguments which support the idea of incorporating the theory of fracture mechanics in the

analysis of concrete structural elements [1], namely: (1) the proper structural analysis must capture the size

effect; (2) the results of numerical solution must be objective respecting different finite/boundary element

meshes; (3) the fracture energy, required for a unit crack propagation, must be known in advance and taken

into account in the theory.

Based on the theory of fracture mechanics, several models have been created and employed for the

simulation of tensile behaviour of plain concrete structures. In general, we classify these models as one, twoand three parameter models, depending on the number of the fracture mechanics constants to be known in

advance. These constants are material-related and are usually determined experimentally. The simplest,

one-parameter model is the well-known model of the linear fracture mechanics. Under certain conditions

this highly simplified and still sophisticated model can be applied to plain concrete. The two and three

parameter models are nonlinear and even approximate, some of them have the potential to describe the

whole force-displacement relationship. A short overview of the existing cracking models for plain concrete

is given in section 2, see paper [4] for more details and additional literature.

In this paper an original two-parameter fracture mechanics nonlinear model, proposed by Nielsen [5], is

developed by means of boundary element method. The crack propagation formula derived by Nielsen is

incorporated in a boundary element software, developed for analysis of two dimensional plane objects

having an existing crack of given initial length. The program is able to perform the required number of

linear solutions within one run in order to get the final results. Few numerical simulations of concrete beamswith arbitrary geometry and openings are carried out and comparisons with other numerical and

experimental solutions are made. Some examples and graphics explaining the size effect phenomena are

also given and some conclusions are made.

1 Associate professor, Department of Civil Engineering, University of Architecture Civil Engineering and Geodesy,

1 Hristo Smirnenski blv., 1046 Sofia, Bulgaria, E-mail address: [email protected] PhD student, Department of Civil Engineering, University of Architecture Civil Engineering and Geodesy,1 Hristo Smirnenski blv., 1046 Sofia, Bulgaria, E-mail address: [email protected]


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=w hc

u displacement

w

E w=hc

E

FF

band with "smeared" crackshc

E unction

deformation

E

line crack

smooth functions u,

F

u displacement

deformation

(a)

(b) and (c)

Figure 2 Kinematic description of the fracture process zone:

(a) strong discontinuity; (b) weak discontinuity;(c) no discontinuity

(c) "smeared" model(a) real crack in a body (b) discrete model

Figure 1 Discrete and smeared (band) modeling of fracture zone

2. An overview of the existing cracking models for concrete

The mechanical behaviourof quasibrittle materials, such as concrete, rock etc., is mainly characterized

by the localization of strain and damage in a relatively narrow zone (fracture process zone), where a

macroscopic stress-free crack is gradually developed, see Figure 1 (a). The mathematical modeling of such

a zone of highly concentrated microcracks is the key issue in the numerical simulation, therefore it is

usually used as a basis of classification of the cracking models.

Discrete and smeared types of modeling of the fracture process zone

Two different approaches are available for the proper simulation of the tensile cracking of concrete in

the fracture process zone. These are discrete andsmearedcrack models, see Figure 1 (b) and (c). In its

earliest applications , see reference

[6], the cracks in concrete were

modelled discretely. The discrete

crack is usually formed by

separation of previously defined

finite element edges, if the finite

element method is used in the

analysis. The nonlinear process islumped into a line and nonlinear

translational springs are usually

employed. The symmetric mode I is

assumed for simplicity, therefore the

crack path is known in advance. The discrete crack approach is very attractive from physical point of view

as it reflects the localised nature of the tensile cracking and associated displacement discontinuity. Some

drawbacks are however inherent for this approach, namely: the constraint that the crack trajectory must

follow the predefined element boundaries and the increasing cost of the computer time because of the

additional DOF.

That was the reason for researchers to search for another method and they introduced the smeared crack

approach, see references [7] and [8], where the nonlinear strain issmearedover a finite area or band with

given thickness. Before going further, it is worth considering the mentioned models from the point of viewof the kinematic description of the fracture process zone. Consider a simple homogeneous bar with linear,

one-dimensional behaviour, until the fracture process zone develops, Figure 2. Then, following references

[1] and [9], we distinguish three types of kinematic descriptions, depending on the regularity of the

displacement field u(x). The first one, see Figure 2 (a), incorporatesstrong discontinuity, i.e. jump indisplacement w and strain

field (x), which consists

of a regular part, obtained

by differentiation of the

displacement function

u(x), and a singular part,

which can be expressed by

Dirac delta function. It isvery natural therefore, torelate the kinematic

description of Figure 2 (a)

to discrete crack approach.

At this stage we do not

discuss the material model,

that is (w) or()

constitutive relationship,


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or mathematical difficulties associated with discrete cracks upon application of kinematic model with

strong discontinuity.

The mathematical problem can be regularized by employing another type of kinematic description,

given in Figure 2 (b), which corresponds to smeared crack model of Figure 1 (c). That is a continuum model

very similar to plasticity models, but when the crack propagation process is continuously developing, the

stress-strain relation exhibits softening, instead of hardening. From kinematic point of view, it represents

the region of localized deformation by a band of a small but finite thickness hc, separated from theremaining part of the body by two weak discontinuities. The displacement field remains continuous but the

strain components experience a jump, so if we assume that the nonlinear (due to cracks) strain is uniformly

distributed over the band, we find f=hcw, where w is easily calculated from the FE/BE solution. We shall

leave the question for the choice of band thickness hc open for further discussion.

In its pioneers paper [7], Rashid did not employ properly the constitutive (- w ) relation, so the

numerical solution was mesh dependent. The method was therefore reformulated in paper [8] by including

an energy criteria and fracture mechanics principles in it. Fixed and rotational smeared crack versions were

developed in [10] and [11], suitable for FE program implementation. Although after 1970s the smeared

crack approach was continuously implemented and used in the general-purpose programs like ANSYS,

ABAQUS, DIANA etc., it has not escaped criticism. The principle objection against was that it would tend

to spread the crack formation over the entire structure, so it was incapable to predict the real strain

localisation and local fracture. As a result a new return to the discrete cracks is noted in recent time usingdifferent numerical methods plus interactive-adaptive computer graphics techniques [12[, as an integral part

of the analysis process.

Finally, we can explore the most regular description - see Figure 2 (c) with smooth functions u(x) and

(x), where the displacement field is continuously differentiable and the strain field remains continuous.

These are the so-called non-local, gradient or "enhanced strain" models [1], [9], [13] and [14], but we shall

not make comments on these methods in the present paper.

One-parameter model of Linear Elastic Fracture Mechanics (LEFM)We begin with the introduction of the simplest model of the linear elastic fracture mechanics called

one-parameter model. Consider a specimen made of ideally brittle material (for example glass), having a

crack of length a. When a load is applied, the structure can supply a potential energy Uat the rate

dU/da=G, termed as an energy release rate. On the other hand, the crack propagation at the crack tip needsto consume some energy, which we denote as Wat the rate dW/da=R, termed as a fracture resistance.Consequently, the linear elastic fracture mechanics criterion for crack growth is defined as follows:

G=R , (1)

In the LEFM,R is a material constant and is denoted sometimes as Gc, whereas G is a function of the

structural geometry and applied loads. The latter can be easily obtained by performing a linear elastic

solution using a certain numerical method such as FEM or BEM. The energy-based criterion (1) can be also

written in terms of stress intensity factor (SIF)K, using the relation:

G=K2/E , (2)

whereE=Efor plane stress, andE=E/(1-2) for the plane strain, andEand are the elastic modulus and

the Poissons ratio of the material.

As well known SIFKrelates the intensity of the crack-tip stresses and deformations to the imposed

loading. Therefore, the criterion (1) can be alternatively written as:

K=Kc , (3)

where as known from LEFM theoryKc=(E.R)1/2 is termed as the critical stress intensity factor and is

considered to be a material constant, obtained through an experiment.


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Figure 3 Brittle to ductile transitional behaviour of TPB test made from

quasibrittle material due to change of dimensions

Deflection

Loadrittle behaviour

Ductile-brittle behaviour

Ductile behaviour

It should be also mentioned that according to LEFM principles, the supplied energy during fracture is

dissipated only at one point that is the crack tip. We classify this approach of LEFM as one-parameter

model, since the only new material constant related to fracture isR orKc and the only equation, postulating

the beginning of the crack propagation is equation (1) or its equivalent equation (3).

The main problem for the direct

application of the LEFM to the

concrete structures, having an initialcrack, is the fact that the concrete is a

typical quasibrittle material. Wespecify as quasibrittle those materials

that exhibit moderate strain

hardening prior to the attainment of

ultimate tensile strength and tension

softening thereafter. It is quite cleartoday that the principal reason for the

deviation of the fracture behavior of

concrete and other quasibrittle materials from LEFM is the existence of a fracture process zone ahead of the

crack tip, which is not small enough compared to the structure dimensions. That is the consequence of the

progressive softening of the material due to microcracking.The LEFM, even the one-parameter model, could be applied for concrete provided we have to analyse a

largesized concrete element. In the literature that is called structural size-effect phenomenon and is a

matter of great amount of scientific publications [1], [2]. The size of the concrete element (in most cases we

use the height of the beamD, as a characteristic size) is closely related to its behaviour and the mode of

fracture, see Figure 3, where three typical failure modes are shown, depending on the size of the concrete

beam.

An assessment is made in reference [15] on the influence of the dimensions of the concrete beam to the

type of behaviour and the mode of the structural failure. A special spring FE cohesive crack model is

employed, and the fracture mechanics data is specially adjusted in the springs constants, bearing in mind

the size of the FE mesh. A number of numerical solutions are performed on a three-point concrete bend

beam, loaded with a concentrated forceFat the middle, having an initial crack of length a0=D/2 and it is

proven that the model is able to successfully analyse the whole range of beams having different dimensions.It is shownwhen the methods of LEFM could be applied to concrete and how that relates the beam

dimensions and other FM parameters. In [15] the performance of the LEFM model is improved by using the

Irwins idea of small but non zero fracture process zone, taking into account first and second order

estimate of the nonlinear zone ahead of crack tip. In general, that is the simplest application of the idea of

the effective elastic crack approach involving an iterative procedure in solution and we call it an enhanced

Irwins approach.

Finally, it is worth to note that the only output result from a LEFM solution, using one-parameter

model, is the value of peak (maximum, critical) loadFcr.

Two-parameter model of Jenq and Shah

It belongs to the class of effective crack or two-parameter models, and can be classified as to Griffith-

Irwin model, see references [1], [2], [3]. Sometimes in the literature they are called equivalent elastic crack

methods and they belong to the class of approximate nonlinear fracture mechanics models. The principal

idea is that the actual crack is replaced by an effective elastic crack substitute, governed by LEFM criteria.

The equivalence between the actual and the corresponding effective crack is prescribed explicitly in the

model and it usually involves an element of nonlinear behaviour. Those type of models are able to predict,

in principle, peak loads only of pre-cracked specimens of any geometry and size, but they have no potential

(unlike cohesive models) to describe the fullforce-displacement relationship.

Jenq and Shah [16], proposed a two-parameter fracture model shown, in Figure 4. In their approach, theindependent material fracture properties are the critical stress intensity factorKsIcand the critical crack tip


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opening displacement (CTOD)c , which are defined in terms of the effective crack. The fracture criteria for

an unstable crack are:

KI=KsIc ,

CTOD=(CTOD)c , (4)

whereKI is the stress intensity factor, calculated for the given load at the tip of the assumed effective crack

of length aeff=a0+a, and (CTOD) is the crack tip opening displacement, calculated at the initial crack tip.

In this model, the effective crack exhibits compliance equal to the unloading compliance of the actual

structure, see the Figure 4 (b). It is measured from the test of a three point bending specimen, so the

nonlinear effect is included in the solution by experimentally obtained material parameters of the method.

The procedure for obtaining experimentally the parameters of the Jenq and Shah method is given in the

1990 RILEM recommendation. As shown in Figure 4, a three-point bend fracture specimen is tested under

the crack mouth opening displacement (CMOD). After the peak, within 95% of the peak loadFmaxthe

unloading compliance Cu is measured. Using Cu and the initial compliance C0 along with LEFM equation,

the corresponding critical effective crack length aeffcan be determined. Consequently, again using the given

LEFM relations,KsIc and (CTOD)cof the material can be obtained at the critical loadFmax [1], [3]. This

approach is proven to yield size-independent values for the two material constants and for any other

geometry as well, [3].

Another well-known two-parameter model is proposed by Karihaloo [2]. However, the effective crack

length aeffis calculated not from the unloading compliance, but from secant stiffness of the real concrete

specimen at a peak load. The suggested fracture criteria are very similar to equations (4) and the procedure

for obtaining the two material parameters is explained in details in reference [2].

The class of effective crack, two-parameter models (such as: the size effect model; the R curve

concept; elastic equivalence methods etc) take advantage of the well-documented pre-peak nonlinear

behaviour of concrete. They are able to give a good estimate of the critical (peak) load, but in order toobtain a complete description of the response past the peak load, using these models, additional assumptions

have to be invoked. The answer is either development of the class of three-parameter models or the

enhanced, two-parameter model of Nielsen, which will be given a special attention in this paper.

Fmax

F

Cu

C0

1

1

KI=KsIc

CTOP=(CTOD c

ac

a0 ac

a0

ac

ac

F

notch

CTOP

CTOPelCTOPcr

unloading

at peak load

Figure 4 Jenq and Shah two-parameter model: (a) Fracture criteria: KI=KsIc and

CTOP= (CTOD )c ; (b) Determination of KsIc and (CTOD) c from C0 and Cu

(a) C0 and Cu are the initial compliance and theunloading compliance at the peak load

(b)


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Three-parameter models of fracture mechanics (cohesive type models)

The first three-parameter nonlinear theory of fracture mechanics of concrete was proposed by

Hilleborg et al., [17], and it was named fictitious crack method. It originated from the idea of the cohesive

approach due to Dugdale-Barenblatt. Distributed cohesive forces, along the crack faces, which tend to close

the crack, simulate the toughening effect of the fracture process zone (FPZ). Therefore, there is no

singularity at the crack tip unlike in LEFM and stresses there have finite values.

Hilleborg assumes that the stress-strain constitutive relation is not a unique material property due to the

localization of the deformation in theFPZ. Instead, two separate constitutive relations, inside and outside

the fracture zone, are needed for describing the material response, see Figure 5 (a) for the two dissipation

mechanisms proposed. The thickness of the fracture zone (the fictitious crack) is zero and the constitutive

relation representing theFPZmust include, in addition to continuous displacements, the displacements due

to cracking. It is important also to point out that the distribution of the closing stresses (w), is a function

which depends on the opening of the crack faces, w. The area GFunder the curve in the Figure 5 (a) istermed fracture energy and that is the energy consumed in increasing the width of the crack from zero to wc.

The fracture energy GF is considered to be a material property and its value can be calculated from the

following integral:

)(,)( 5dwwfGcw

0

F =

where wc is the critical crack separation corresponding to (w)=0.We consider the cohesive approach of Hilleborg as a three-parameter model. Two of them are defined

as material constants, say tensile strengthftand critical separation wc, so the third parameter is usually

geometrical, e.g. the type off(w) function. In the numerical simulation we choose either linear or bilinear

form off(w) function, therefore in this case the fracture energy GF is "adjusted" to the model being

calculated from equation (5). Any other combination of parameters GF, ftand wcis possible, provided theform off(w) function is known (or accepted) in advance.

The idea of Hilleborg was first used in the context of the discrete crack approach having the kinematic

description of Figure 2 (a). As a result the mesh dependency, as one of the drawbacks of the analysis of

concrete structures, was removed. A similar approach, proposed by Baant and Oh [18], is the crack band

model, see Figure 5 (b). In this model, a crack band zone of thickness hc is introduced to represent theFPZ.

In contrast to Hilleborg's approach, the crack band is treated as a continuum described by a stress-strain

relation reflecting both: the localized deformation and the continuous deformation. Naturally Baant's

approach turns out to have a kinematic description, shown in Figure 2 (b). It is also considered to be a three-

F F FF

t

w

t

area

GF

wc

(w)= f(w)

area

GFhc

wc/hc

(w)t

(a) cohesive method of Hilleborg (a) "band" method of Baant

Figure 5 Relationship between constitutive curves of cohesive and smeared (band) methods

crackline

band of thickness hc

t


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n-1

FnF2

F1

1

Y

n 321

b2

interface line

two crack aces1

FnF2

F1Y

b

a

1

opening

(a) (b)

Figure 6 (a) Plane body with initial crack subjected to arbitrary loading; (b)Thestructure discretized with boundary elements divided into two sub-regions 1and2

parameter model, where the third parameter is of geometrical type and that is the band width hc. It is proven

in references [10] and [11], that the two models, given in Figure 5, are equivalent under certain conditions

and this relationship is also shown in the figure. In the same references another variant of the "band" model

is developed (fixed and rotational smeared crack models), which seems to be more convenient for a finite

element implementation.

3. The BEM applied to crack problems using multi-domain formulation approach

The boundary element method (BEM) is already well established and powerful numerical technique,

which is a good alternative of the finite element method. Its main advantages are the higher accuracy of the

solution and the fact that the discretization is only on the boundary of the investigated body. Therefore, the

number of the unknowns is very small when compared with FEM, which makes the method very efficient.

The application of the boundary element method in fracture mechanics and crack propagation analysis has

been given a growing attention recently, see references [15] and [19], but when compare with FEM fracture

mechanics applications, it is still in a premature state.

In this paper we develop a multi-domain variant of BEM, which means that the crack path is known in

advance. Decomposition is made of the plane body into sub-regions with a line boundary between,

containing the crack to be eventually developed. A short description of the theory follows.

Consider a two dimensional domain with an openingunder plane stress or plane strain condition. Itis subjected to distributed load b in the domain and concentrated forcesF1, F2,..Fn on its boundary , see

Figure 6 (a). The plane body

has an arbitrary shape of the

contour and contains an

initial crack of length a plus

an opening with contour1.

There are some supporting

links on the boundary or it

could be continuously

supported. Part of the

boundary can be subjected to

external tractions. It is wellknown that starting fromBetti's reciprocal theorem

the integral equation of the

problem may be derived,

from where the boundary integral equation may be formulated, [20]. Without going into details, using

tensor notation for simplicity and disregarding the effect of the concentrated forces for clarity of theequation, we write the following boundary integral equation:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

+=+ xdxbxuxdxpxuxdxuxpuc jijjijjijjij ,,,*** , (6)

where i, j =1,2 ; and x are the observation and source points, respectively; uj() is the displacement

at ; represents the boundary of the body and includes the boundary of the opening 1; represents thedomain of the body; bj(x) is the intensity of the body forces atx; uj (x) is the displacement atx;pj (x) is the

traction atx; and the function

=

.,

,,

)(nodesboundartysmoothfor

2

1c

ij

ij

ij

(7)

The well-known Kelvin fundamental solutions ),(),( ** xpandxu ijij are given by:


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typical corner point

with incompatible

elements

Figure7 Three types of linear boundary elements-compatible, right and left incompatible

x

the real boundary2

1 =1

= -1

compatible linear boundary element

y

incompatible right linear boundary element

incompatible left linear boundary element

a

[ ] ,))(()()(

,ln)()(

,,,,*

,,*

+

=

+

=

ijjijiijij

jiijij

nrnr21n

rrr221

r14

1p

rrr

143

G18

1u

(8)

where is the Poisson's ratio; G is the shear modulus; ris the distance between the point and the pointx;nj is the direction cosines of the outward normal to the boundary; ij is the Kronecker delta symbol.

In order to solve numerically the integral equation (6), we make discretization on the boundary

(including 1) dividing it to n linear segment and introducing four boundary discrete values - two

displacements and two tractions at each boundary node, see Figure 6 (b). Two of these are known and two

unknown, so they should be obtained from the numerical solution. If a loading in the domain b is given,

a discretization in the domain is also needed usingtwo dimensional elements but with no new unknowns

introduced. Note, that in order to consider properly the stresses around the crack tip we divide the body

into two sub-regions 1and

2. We choose appropriate

shape functions for

representing the boundary

displacements and tractions.In order to obtain convergence

in the solution the minimum

order of approximation is

linear for this type of

problems [20], so linear

boundary shape functions are

used for representing the

displacement and traction

functions within the linear

type boundary elements, see Figure 7.

In local coordinate (-1 to 1 variation within the element) we make the following approximation for

the vectors of boundary displacement and traction functions ue, pe:

ue()=Ne()ue, pe()=Ne()pe, (9)

where ue=(uxuy)Tis the vector of displacement functions for the element,pe=(pxpy)

Tis the vector of

traction functions for the element, ue=(u1xu1y u2xu2y)T is the vector of discrete values of displacements of

local points 1 and 2 for the element,pe=(p1xp1y p2xp2y)Tis the vector of discrete values of tractions of local

points 1 and 2 for the element, Ne() is matrix of shape functions for compatible element, which has the

form:

( ) ( )( ) ( )

.)(

+

+=

1010

0101

2

1Ne (10)

Dealing with corner points of the plane body is one of the most laborious problem to be resolved inBEM. In the present work we use the idea, developed in reference [20], where so called "incompatible" left

and right linear boundary elements are introduced, see figure 7. The corner point is not treated directly, but

the nodal point of the element is moved to a small distance a (in our case we take a= 1/3), therefore it is not

necessary to satisfy the integral equation at the corner point, because sometimes the boundary conditions are

different at both sides of the corner. The matrix of shape functions Ne() for a left incompatible element is:

( ) ( )( ) ( )

,)(

+

+=

230130

023013

5

1Ne

(11)


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and for the right incompatible element we have:

( ) ( )( ) ( )

.)(

+

+=

130320

013032

5

1Ne (12)

It is well known that the calculation of the stress intensity factors (SIF) of the LEFM requires very

accurate results for the displacement and stress fields around the crack tip. On the other hand the stresses atthe crack tip are singular, which makes the task very complex. We use the so-called "singular" displacement

boundary elements, derived for first time in reference [15], which put at the crack tip, will lead to

considerable improvement of the numerical results for a relatively coarse mesh. The corresponding

displacement shape functions are of such a type, that the first derivative with respect to local coordinate

has singularity of orderO(r1/2), which is in accordance with the required stress singularity.

In order to get the discretized analog of the boundary integral equation (6) we put equation (9) into (6)

and do the integration. Usually a numerical integration is performed (in this case we use 10 points Gaussion

quadratures) and this leads to the following system of matrix equation:

HU+B=GP, (13)

where Uis the vector of all nodal boundary displacements,Pis the vector of all nodal boundary tractions,

B is calculated vector, representing the effect of the external loads,Hand Gmatrices are calculated using

the numerical evaluation of the boundary integrals, which contain the fundamental solutions and shape

functions.

After the boundary conditions are satisfied in equations (13), we obtain and solve the final system of

linear algebraic equation and get the numerical set of boundary displacements and tractions. Once we have

the boundary data available, we go further and obtain the displacements and stresses at a given point from

the domain , using again equation (6) for displacements and the following integral equation for stresses:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

+= xdxbxuxdxuxpxdxpxu kijkkijkkijkij ,,,*** , (14)

where the tensors uijk*

and pijk*

can be found elsewhere [19].

In order to check the LEFM criterion for crack propagation, we have to calculate the SIF or the energyrelease rate, if necessary. In the next section the numerical procedures adopted will be explained.

4. Crack propagation formula and its incorporation in the boundary element code

In the theoretical paper [5], Nielsen proposes a simplified fracture mechanics model for crack growth

of quasibrittle materials, based on an energy balance equation. The newly developed formula was evaluated

by Olsen [21], for studying the crack propagation in three point bending concrete beams. The numerical

procedure requires determination in advance of the change of the elastic strain energy with crack length a

(i.e. dW/da function), which can be done by a simple FEM calculation. We begin with short presentation of

the main points of the theory and the assumptions made.

Consider a plane body with an internal crack of length a, subjected to load, applied at the boundary. At

the critical moment, when the crack is on the verge of extension, the following energy balance equation isvalid, as stipulated by Griffith, [21]:

dW/da+GFb=0, (15)

where Wthe strain energy stored in the body of width b, GFis the fracture energy and for the sake of

simplicity only the case of displacement controlled systems will be examined in this paper.

As stated above, the first term of equation (15) is called driving force, whereas the fracture energy

GF can be viewed as a crack resisting, material parameter. Using Irwin's idea for introducing a small, but


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w0 2

leff

w0 2

a'a

a

x

t

roposed stress distribution

of cohesive forces

elastic

x

the tip of the effective crack

the real stress distribution

Figure 8 The stress field for concrete around crack tip and ope-

ning displacements with and without cohesive stresses

finite nonlinear zone in the crack tip [1], and making some approximate energy consideration, Nielsen

suggests the following equations, [5]:

2t

2I

pf

K1a

= , (16)

2t

2Ipeff

f

K40a40l

.. == . (17)

where ap' is the length of the fracture

process zone of the model, leffis the

effective crack length term, KIis the

stress intensity factor andftis tensile

strength the concrete, see figure 8.

The length of the fracture zone ap'

is obtained by the well known Irwin's

equilibrium conditions, see [5] and

[21], under the following two

simplifying assumtions: (1) the type of(w) relation for the softening curve

is not taken into account; (2) in

considering an elastic-brittle material,

the real distribution of the cohesive

stresses is presented as rectangular

along a part ap' of the actual fracture

zone ap. Therefore, as seen from

equation (16), we get an Irwin type approximation forap'. As for is the effective crack length term leff , an

assumption is made to take into account the work of cohesive stresses by using a simplified crack opening

displacements formula, according to LEFM (see reference [5], equation (3.8)).

Now, having the effective crack length term leff, obtained by this extremely simplified analysis,

Nielsen suggests to consider the energy available on the basis of an effective crack length aeff=a+ leff,instead of the real crack length a. Bearing in mind that for the possible crack growth the increment leffis a

positive function of both crack length and displacement u, starting from equation (15) and after some

alterations, we get the following final equation:

+

=

a

l1

a

WbG2

u

l

a

W

du

da

eff

F

eff

. (18)

Equation (18) is theEnergy Balance Crack Propagation formula for a displacement controlled system.In fact, that is a first order differential equation with respect to the crack length function a, where the

derivative W/a is taken at a+ leff, while leff/a may be taken at a. It is impossible to solve the differentialequation (18) analytically, so as in reference [21], we explore the fourth order Runge Kutta method. It is

very important to note, that the crack propagation formula takes into account the plastic behaviour near the

crack tip by putting (17) in it, although it is based on the calculation of elastic energy W(u,a) and its

derivative W/a. The latter term is used for the calculation of the stress intensity factor in (17) by exploringthe well known LEFM relation, see also equation (2). The accuracy of the adopted numerical procedure for

the solution of (18) depends on the number of "loading" steps, which is actually the number of divisions of


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11

the imposed displacement u from its initial value u0 to its final one umax. After the current solution step is

performed, having the increment of the real crack length a, we calculate the effective crack length aeff as:

aeff=a+leff+a, (19)

where a includes the initial crack length plus sum of increments a, as calculated in the previous steps.

Within every step, using the current value of displacement u , the calculated value ofWand updated

aeff, we calculate the force responseFfrom the relation W=Fu/2. It is presumed in this paper, that only one

boundary link is moving and the crack path is known in advance (symmetric fracture mode I). In such a way

we are able to obtain the force-displacement curve for the whole (u0- umax) interval.

The described numerical procedure is implemented into a boundary element program called

BEPLANE. To be more readable, the relevant flow chart given in Figure 9, is supplied with someequations.

START: u =uo; a =aoLinear Solution(LS)K =..

effeff laa +=

u = u; a =aeffdetermination ofF = F(u,aeff

Runge Kutta method

+

+

=

a

l1

a

WbG

u

l

a

W

du

da

eff

F

eff

aaanew +=

maxuu >

STOPYes

BEPLANE LS KI=..u = u + du; a = anew

32 LS for 1 loading step

= 1,4 calculation of k

( )4321 kk2k2k

6

1a +++=

2t

2I

eff

K40l

,=

o

Figure 9 The flow chart

for the

implemented

numerical

procedure


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12

To perform a full, fourth order Runge Kutta procedure within one displacement step, 32 linear elastic

solutions are needed for different values of arguments a and u. Full details of methods and procedures for

calculation of derivatives within BEPLANE can be found in reference [15].

As a first illustration of the potential of the theoretical model and the boundary element program, we

do a numerical simulation on a three point bending concrete beam. The input data is taken from reference

[21] and is given in the Table 1. A vertical displacement at the middle of the beam (0 2 mm) is imposed toTable 1

E = 42210 N/mm2; = 0,2; GF= 0,0957 N/mm; ft= 6,86 N/mm2; L=8D; D = 100 mm; ao = 50 mm

Point u [mm] F[N] a + a [mm] leff= 0,4 pa [mm] pa [mm] aeff=a + a + pa [mm]

A 0.08 1066.85 50.146 1.746 4.3646 54.510

B 0.15 1572.95 52.280 6.0816 15.203 67.482

C 0.19 1349.97 57.285 9.3906 23.476 80.761

simulate the behaviour of the displacement controlled system. The main purpose of this numerical test is toshow and estimate the continuous change of few important geometrical parameters for the three "state"

pointsA, B and C. That is why only the middle part of the beam is shown in Figure 10 (a) and the

parameters plotted in a real scale are as follows: areal=acur+a, (fourth column); leff= 0,4a'p(fifth column);

a'p(sixth column) and aeff=a + a +a'p , (last column). The force-displacement relationship is presented in

Figure 10 (b) and the state pointsA, B and Care shown in order to make some conclusions. PointA

indicates the end of elastic behaviour and the beginning of the pre-critical process zone development. The

length of the real crack is 50.146 mm and the effective crack length - 54.51 mm. The length of the fracture

zone is just 4.3646mm which takes about 9 % of the ligament. The critical state, corresponding toFmax, is

C

B

A

0

200

400

600

800

1000

1200

1400

1600

1800

0 0.05 0.1 0.15 0.2

Displacement [mm]

Force [N]

Figure 10 (a) Real scale graphical presentation for areal, leffand a'pfor state points A, B and C of the

three point bend speciment; (b) Force-displacement diagram illustrating points A, B and C

(b)

A B C

leffpa

a + da

leff

a + da

Dleff

a + da

pa

pa


13/20


14/20

14

fracture mechanics theoretical model has the potential to simulate the behaviour of concrete beam within

the framework of the limitations, already mentioned.

Analysis of a concrete beam with complex geometry and openings

The objective of the next numerical test is to demonstrate the versatility of the BE approach. The

program is able to solve plane objects having arbitrary contour geometry including openings. By using themulti-domain techniques approach, different material properties can be considered. A symmetrical concrete

beam with four openings is drawn in Figure 13. Geometrical and material data are also enclosed.

Due to symmetry half of the beam is considered. In addition, the beam is divided into two sub-domains. The

boundary element mesh, which includes the contours of the openings, the boundary conditions symbols and

the deformed shape (for a certain solution step) are shown in Figure 14, in the manner they are displayed in

the postprocessor of BEPLANE. Again a vertical displacement of the middle point is performed.

Beam and Fracture data:

E = 30000 N/mm2

; = 0,2; ft= 3,3 N/mm2

; GF= 0,1 N/mm; b=50 mm

400400 mm

75

125

25

25

50

25

25

50 150 50 75 200 100 100

75

Figure 13 A three-point beam loaded symmetrically having an initial crack of length a0 =0,25D

A - B100.3

0

250

500

750

1000

1250

1500

1750

0 0.15 0.3 0.45 0.6 0.75

Force

BEM - Crack Growth formula FEM - cohesion solution Experiment

Figure 12 Comparison of results for load-deflection curves for beam A-B100.3

Displacement


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15

Other graphics, taken from the BEPLANE postprocessor are given in Figure 15. We call it "monitor

process"window, and it is specially designed for the program user to follow the computational process on-

line. It consists of four windows, as follows:

Figure 14 The BE mesh and the deformed shape of the beam as displayed in BEPLANE

Figure 15 The BEPLANE on-line "monitor process" window


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16

(1) BEPLANE logo and some input data;(2) On-line appearing plot of the force-displacement curve. From this curve the consumed fracture

energy is calculated as follows, [21]:

Gcalc=A /b/a , (20)

where A is the external energy supplied, which can be calculated from the area under force-displacement

curve, b is the thickness of the body and a is the length of the fully developed part of the crack;(3) G-verification window where on-line calculated values of fracture parameterGcalcF are plotted

against the input parameterGF. During the computational process the two current values are displayed on

the right side of the monitor;

(4) A graphic of the real crack growth a against the corresponding displacement value u. As seen

from the particular graphic, in the displacement range between (0 1 mm) no substantial crack growth is

observed. As the load approaches its critical (peak) value, the gradient of the crack growth function has

gradually increased. Additional useful data appears on the right side of the monitor to help the user, such as

the current stress intensity factorKI and others.

The force-displacement relation graphic for the beam into consideration is shown in Figure 16.

Another solution for the same beam is made using ANSYS cohesive model and the relevant curve is also

plotted in the figure with thinner line. It can be seen (assuming the cohesive solution is reliable enough) that

the crack propagation formula applied in conjunction with boundary elements is able to predict the peakload (the difference is within 4 %) and the ascending branch of the curve. As in the case of the previous

example, the theory is only able to predict a part of the descending branch of the load deflection curve.

It should be mentioned however, that the particular beam geometry was especially chosen as a purely

"academic" test. In reality, the onset of crack propagation in such a concrete beam is likely to develop at the

corner points of the openings in line with the development of the major crack. The present approach

considers the development of one crack only.

E = 30000; G f= 0.1; ft = 3.3; = 0.2; b = 50 mm

1014.987

1061.27

0

200

400

600

800

1000

1200

0 0.05 0.1 0.15 0.2 0.25 0.3

Displacement

Force

BEPLANE

ANSYS

Figure 16 Force-displacement curves from present method and ANSYS cohesive approach


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17

Size effect prediction numerical test

Since the presented theory determines the load carrying capacity (Fmax) of the concrete structures upon

the conditions and limitations described, it should be able to predict the size effect, i.e. it should be able to

give the load carrying capacity as a function of absolute values of a chosen geometrical parameter (that

could beD dimension in our case), characterizing the size of the structure. As in reference [21], it is

convenient to use a common valueB, as a measure of brittleness of the structure:

EG

DfB

F

2t= , (21)

where the parameters in the right side of equation (21) are already defined.

Table 2

The size effect is rigorously defined through comparison of geometrically similar structures of different

D sizes, provided all other dimensions keep the proportions. The same effect might be achieved by using

theB parameter (or any component ofB) instead ofD. We consider again the concrete beam from Figure 13

and keeping the standard values of the parametersD,ftand modulusE, we vary Gf from 10000 to 0.01.

10000 3.30 30000 25 100 3.63E-06 1197.260318 1.547973 plastic

100 3.30 30000 25 100 0.000363 1196.946628 1.547567 plastic

10 3.30 30000 25 100 0.00363 1194.613960 1.544551 plastic

1 3.30 30000 25 100 0.0363 1171.497187 1.514663 quasibrittle0.5 3.30 30000 25 100 0.0726 1148.687751 1.485172 quasibrittle

0.2 3.30 30000 25 100 0.1815 1089.848953 1.409098 quasibrittle

0.1 3.30 30000 25 100 0.363 1014.986815 1.312306 quasibrittle

0.05 3.30 30000 25 100 0.726 913.432510 1.181004 quasibrittle

0.02 3.30 30000 25 100 1.815 735.105509 0.950439 brittle

0.01 3.30 30000 25 100 3.63 599.736424 0.775417 brittle

GF ft E ao Fmax /ftD B Type ofbehaviour

-0.12-0.08

-0.04

0

0.04

0.08

0.12

0.16

0.2

-5.6 -4.8 -4 -3.2 -2.4 -1.6 -0.8 0

Log (B)

Log(

/ft)

LEFM

plastic

quasibrittle

Figure 17 Structural size effectrepresented by failure load versus brittelness number B


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18

A set of ten numerical solutions is performed with the present theory for different values ofGf and the load

carrying capacityFmax is obtained, see Table 2. The results are also depicted in Figure 17, where brittleness

numberB versus (/ft) in a logarithmic scale is shown. Note that following [21],Fmaxhas been given a

dimensionless form as a Navier stress along the depth (D-a0).

Before making comments on the results from Table 2 and Figure 17, it is instructive to plot and

examine the force-displacement graphics for the same concrete beam. That is done in the Figure 18 fromwhere the different modes of failure are clearly indicated.

Let us first consider the two solutions corresponding to smallest values of fracture energy, GF=0.01and

GF=0.02, see the last two rows of the Table 2, the respective points from Figure 17 and relevant curves

from Figure 18. The failure modes are typical for a perfectly brittle structure and it can be easily proven that

the values ofFmaxcan be simply related to the SIF, calculated by using principles of the LEFM, [21]. Note

also, that the two points from the Figure 17 are placed on the inclined straight line of slope -1/2, which is a

well-known fact from the size effect theory of LEFM, [1]. Therefore, concrete structures having parameters

within the same range, can be classified as brittle, i.e. they experience a brittle type of behaviour.

Absolutely the same situation will apply for concrete structures with bigD dimension, but having the same

B parameter. The simplified methods of LEFM can be applied in such a case and the size effect law is valid.

The presented theory is able to easily cover this type of behaviour.On the other hand it could be noted from Table 2 that for the first three solutions theFmax values are

almost equal and the respective failure modes are typical for plastic type of behaviour. That corresponds tobig values of fracture energy (GF=10,100, 10000 ) or smaller absolute values of characteristic dimensionD.

No size effect is observed in this case, so any nonlinear classical failure theory, which uses some type of

strength limit or failure surface in terms of strain or stress, may be applied. Therefore, the nominal strength

for these type of structures is independent from the structural size and we classify them asplastic structures.

E = 30000; ft = 3.3; = 0.2; b =50

0

200

400

600

800

1000

1200

1400

0 0.05 0.1 0.15 0.2 0.25 0.3

Displacement

Force

GF = 10000

GF = 100

GF = 10

GF = 1

GF = 0.5

GF = 0.2

GF = 0.1

GF = 0.05

GF = 0.02

GF = 0.01

Figure 18 Force-displacement curves for the beam of figure 13 varying the fracture energy GF

brittle

plastic

quasibrittle


19/20

19

It is clear from the above discussion that for concrete structures with small absolute dimensions (plastic

type of behaviour) plasticity methods can be applied, whereas for concrete structures with big dimensions

(brittle type of behaviour) the LEFM methods are more appropriate. Therefore, between those two extremes

when a concrete structures of a medium size is to be analyzed, a transitional behaviour should be expected.

That is illustrated as a solid curve in Figure 17. We classify this type of behaviour as quasibrittle. The

nonlinear methods of fracture mechanics are applied in this case and they give best results. For values ofGF

between 0.05 1, the present solutions clearly show this tendency. There is no size effect observed but thenominal strength varies with the change of parameterB through changes ofGForD .

The conclusion, made from the results of this numerical test, is that the present theoretical model is

able to perform analysis of concrete structures for the whole range of fracture mechanics parameters and

structural sizes.

6. Concluding remarks

In this paper the simplified fracture mechanics theoretical model of Nielsen is implemented in the

boundary element program BEPLANE. The numerical procedure is integrated in the software program in

such a way that the final solution is obtained in one run and there is no need for calculations in advance.

The model has shown to be promising for mode I fracture analysis of unreinforced concrete. For a variety of

example problems reasonable agreement with the experimental data has been found, not only for the peakload but also in the pre-peak nonlinear zone. The numerical approach allows considering plane concrete

structures with arbitrary contour geometry and openings. It was demonstrated that by means of the present

approach any type of structural behaviour might be considered starting from ideally brittle through

quasibrittle and plastic. A general deficiency of the approach is that only a part of the post-peak softening

branch is covered by the solution. Also at present, the method does not include treatment of more then one

discrete crack, which can be a topic of future work.

REFERENCES

[1]Baant, Z., J. Planas, Fracture and Size Effect in Concrete and Other QuasibrittleMaterials,

CRC Press, LLC, (1998).

[2] Karihaloo B. L., Fracture mechanics & structural concrete, Longman Scientific & Technical, (1995).[3]Shah, S., Swartz, S., and Ouyang., C., Fracture Mechanics of Concrete: Applications of Fracture

Mechanics to concrete, Rock, and Other Quasibrittle Materials , John Wiley & Sons, Inc, (1995).

[4] Gospodinov, Gospodin, "Review of the Computational Models for Cracking of Quasibrittle

Materials",(in Bulgarian), Annuaire de lUniversite dArchitecture, de Genie Civil et de Geodesie,

Sofia, Jubilee Session, (2002).

[5]Nielsen, M. P.: An Energy Balance Crack Growth Formula, Bygningsstatiske Meddelelserm

Edited by Danish Society for Structural Science and Engineering, Volume 61, No 3-4, pp. 1-125, (1990).

[6]Ngo, D., and Scordelis, A. C., Finite Element Analysis of Reinforced Concrete Beams,

Journal of the American Concrete Institute, Vol. 64, No 3, pp.152-163, March (1967).

[7]Rashid, Y. R., Analysis if Prestressed Concrete Pressure Vessels, Nuclear Engineering

and Design, Vol. 5, No 4, pp. 334-344, (1968).

[8]Baant, Z. P., and Cedolin, L., Blunt Crack Band Propagation in Finite Element Analysis,

Journal of the Eng. Mechanics Division. ASSCE, Vol. 105, No EM2, pp. 297-315, (1979).

[9] M. Jirsek and B. Patzk, Models for quasibrittle failure: Theoretical and computational aspects,

Second European Conference on Computational Mechanics,(ECCM), Cracow, Poland, June (2001).

[10] Rots, J. G. and Blaauwendraad, J., Crack modeling for concrete: Discrete or smeared? Fixed,

multi-directional or rotating? HERON, vol. 34, No 1, Delft UT, (1989).

[11] Rots, J. G. et al., Smeared crack approach and fracture localization in concrete, HERON,

vol. 30, No 1, Delft UT, (1985).


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[12] Ingraffea, A. R., and Saouma, V., Numerical modeling of discrete crack propagation in reinforced

and plain concrete, in Fracture Mechanics of Concrete, EAFM, eds. G. Sih and A. diTommaso, (1985).

[13] de Borst, R., and Muhlhaus, H., Continuum models for discontinuous media, in: Fracture Processes

in Concrete, Rock and Ceramics, van Mier, J. and al. eds., RILEM, E&FN Spon, 1991.

[14] Akesson,Magnus, Implementation and Application of Fracture Mechanics Models for Concrete

Structures, PhD thesis, Chalmers UT, Goteborg, Sweden, (1996).

[15]Kerelezova, Irina, Numerical Modeling of Quasibrittle Materials by Means of Fracture MechanicsApproach, PhD thesis, University of Arch., Civil Eng. & Geodesy, Depart. of Civil Eng., Sofia, (2002).

[16] Jenq, Y., and Shah, S., A Two Parameter Fracture Model for Concrete, J. Eng. Mech.,

Vol. 111, No 4, pp 1227-41, (1985).

[17] Hilleborg, A., Modeer, M. and Petersson, P. E. Analysis of crack formation and crack growth by

means of fracture mechanics and finite elements. Magazine of Concrete Research, Vol. 6, (1976).

[18] Baant, Z. P., and Oh B., Crack band theory for fracture of concrete,

RILEM Mat. Struct., 16(93); 155-177, (1983).

[19]Aliabadi, M. and D. Rooke, Numerical Fracture Mechanics. Kluwer Academic Publishers, (1991).

[20] Gospodinov, G., and P. Drakaliev, On the application of the boundary element method in the plane

problems of the theory of elasticity, Journal of theoretical and applied mechanics, Sofia, vol. 2, (1990).

[21] Olsen, D. H., Concrete fracture and cracks growth- a fracture mechanics approach, PhD thesis,

Department of Structural Engineering, DTU, Series R, No 42, (1998).

Acknowledgment: The authors would like to express their gratitude to Professor M. P. Nielsen and

Dr. T. C. Hansen for their support and useful discussions during the course

of this work.

boundary elements modeling of plain concrete notched beams by means of crack propagation...

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