Application of Fracture Emechanics to Concrete Arne Hillerborg

8/10/2019 Application of Fracture Emechanics to Concrete Arne Hillerborg

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DIVISION OF BUILDING MATERIALS

LUND INSTITUTE

OF

TECHNOLOGY

APPLICATION O FRACTURE

MECHANICS TO CONCRETE

Summary of a series of lectures 988

Arne Hillerborg

REPORT

TVBM 3030

LUND SWEDEN 988


2/30

GODEN:

LUTYDG/ TYBM-3030)/1-28/ 1988)

PPLIC TION OF

FR CTURE

MECH NICS

TO

CONCRETE

Summary of a series

of

lectures

988

rne Hillerborg

ISSN 0348-7911

REPORT TYBM-3030

LUND SWEDEN 988


3/30

PPLIC TION OF FR CTURE MECH NICS

TO

CONCRETE

with

emphasis

on

the

work

performed at

Lund Ins t of

Tech

Arne Hil le rborg

Div. o f

Bui ld ing Mater ia l s Lund In s t .

o f

Tech .

Lund Sweden.

Abst rac t

Dif fe rent approaches to the

appl i ca t ion

o f f r ac t u re

mechanics

to

concre te

a re

d i scussed with an emphasis

on

the models based

on

s t r a i n

sof ten ing

and

s t r a i n

l oca l i za t ion

p a r t i c u l a r l y

t he

f i c t i -

t i ous

crack

model. Examples o f r e s u l t s o f t he t heo re t i ca l analyses

a re

demonst ra ted and some

prac t i ca l

conclus ions

a re drawn. Future

resea rch i s a l so

commented.

In t roduc t ion

Convent ional f r ac t u re

mechanics

i s

mainly

based on t he theory o f

e l a s t i c i t y

and

t

i s used

for

s tudy ing

the s t a b i l i t y and

propaga t ion

o f

ex i s t i ng

c racks . The

modern form o f

a pp l i c a t i on

o f f r ac t u re

mechanics to concre te which

was f i r s t

developed in Lund

d i f f e r s

from

t he convent ional

f r ac t u re

mechanics

in both these respec t s .

In t he ba s i c form o f convent ional f r ac t u re

mechanics

t i s

assumed

t h a t t he s t re s se s

and

s t r a in s t end

towards

i n f i n i ty a t crack t i p

Fig .

1 . This

i s

of cours e no t

r e a l i s t i c . In

s p i t e o f

t h i s the

t heo re t i ca l r e s u l t s

based

on t h i s

assumption

in many

cases

l ead to

r e a l i s t i c conclus ions . In o the r cases t h i s i s not t he case .

This

has

s ince

long ~ e n

recognized

and in convent ional

f r ac t u re mechanics

many methods

have been

advised

to overcome

t h i s

problem.

Some o f

these methods a re s i mi l a r to t he methods

now

app l i ed to concre te

even though

the r e

a re

fundamental di f fe rences .

The most impor tan t p r a c t i c a l

d i f fe rence

between convent ional

f r ac t u re

mechanics and

the

modern

appl i ca t ion

to

concre te

i s

t h a t

convent ional

f r ac t u re

mechanics has

never

been app l i ed t o s t ruc tu re s


4/30


5/30

3

O

K

/ y

V Ttx

x

Fig .

1 s t r e s s

d i s t r ibu t ion accord ing to

t he

theory o f e l a s t i c i t y .

where

a

i s the crack length

for a

c rack a t an edge, o r h a l f t he

c rack

l ength fo r

an

i n t e r i o r c rack) , G

i s

t he s t r e s s

which

would

have ac ted if t he re had been no crack , and

Y

i s

a

dimensionless fac -

t o r , which

depends

on the

type

o f s t ruc tu re ,

t he loading condi t ions ,

and to some ex ten t to the

crack

l ength . The value of Y i s o f e n

approximate ly

2.

According to t h i s formal

s t r e s s

d i s t r i b u t i o n t he s t r e s s with in a

dis tance

Xl exceeds the t e n s i l e

s t r eng t h f

t

The s t r e s s d i s t r i b u t i o n

cannot be

v a l i d

with in t h i s pa r t .

The

l a rge r the value

i s o f X l

the l e s s accura te a re the conclus ions drawn

by

means o f LEFM

As t he s t r e s s approaches

i n f i n i t

y, t he analys

i s

of crack

s t a b i l i t y

and crack propagat ion cannot be based on a comparison wi th t he

s t r eng t h o f t he

mater ia l .

Ins tead it i s necessary to

in t roduce a new

c r i t e r i on ,

which says t h a t the

crack

wi l l

s t a r t propagat ing

when the

s t r e s s i n t ens i t y f ac t o r K reaches a c r i t i c a l

value ,

th cr i t i c l

s tress in tens i ty

K

c

which i s

assumed

to be

a

mate r ia l p rope r ty .

t

can

be not iced t ha t convent ional f rac ture mechanics can only

t r e a t

problems concerned

with an exis t ing c rack , as Kbecomes

zero

when

t he

c rack length

a

i s

zero.

For uncracked

mate r ia l

t he

ordinary

theory

of s t r eng th of ma te r ia l s has to be used, with

a

comparison


6/30


7/30


8/30

6

thus contract . At the

post-peak

stage

an

increase in the to ta l

deformation

corresponds to a decrease in

deformation

for most

par ts

of the specimen, but an increase in the deformation within the f rac

tu re zone.

The

term s t ra in local iza t ion i s

of en used to

character i

ze th i s

behaviour.

The

increase in deformation i s local ized to the

f racture zone, whereas

no

fur ther

increase

in

s t ra in takes

par t

out

side th i s zone.

Another term,

which i s of

en used

to

describe

the

stress-deformation

a t the post-peak

s tage

i s

s t ra in

sOftening, which

means

tha t the

s t re ss

decreases as the

average s t ra in or ra ther the deformation)

increases.

In

Fig.

2 the stress-deformation curves from the o u ~ gauges are

shown on

the assumption t ha t the

fracture zone i s s i tua ted

within

gauge-length C. The sum of the deformations in gauges B, C and D i s

equal

to

the

deformation

in

gauge

A.

From

the

f igure

t

i s evident t ha t

the curves

from the

different

gauges

are

different

with

the

exception

of

gauges

B

and

D

which

are equal . Thus the s t ress-deformat ion

re la t ion

cannot be expressed

by

a

s ingle

curve,

as

th i s

re la t ion depends

on the gauge-length

and

on the posi t ion

of the

gauge

with

respect to the f racture zone.

I t

i s t h u s not possib le to find a general s t re ss - s t ra in curve for a

mater ia l ,

including

the

descending

branch.

This fac t i s of en ne

glected .

Many examples

can be

found

in the

l i t t e r a tu re where equa

t ions for such s t ress -s t ra in curves for concrete have been proposed.

A general

descrip t ion

of the

stress-deformation

proper t ies

can be

given by means of

two

curves according to Fig. 3, one s t ress -s t ra in

G-E) curve for s t ra ins smaller than the s t ress a t the peak

point ,

and one

stress-deformation

G-w) curve

for

the

addit ional deforma

t ion w within the f racture zone, caused

by

the damage within t h i s

zone.

The

general

equat ion for the

deformation 81 on a

gauge

length

l i s then given by

~ l

=

El

w


9/30

7

where i s taken from the ~ E c u r v e and w from the rr-w-curve. The

l a t t e r va lue

i s on ly used where t he re

i s

a

f rac ture

zone with in the

gauge

l e ng th i n quest ion . In

o t he r

cases

it equals zero . In t he

post -peak region

the

va lue

o f

i s

taken

from

the

unIoading branch.

One

e s s e n t i a I

proper t

y of t he

~ w c u r v e

i s t he a rea below t he curve ,

as

t h i s a rea i s a

measure o f

t he energy

which i s

absorbed per u n i t

a rea

o f

t he f r ac t u re zone

dur ing

a

t e s t

t o f a i lu re .

This

va lue i s

us ua l ly ca l l ed t h e

fracture

energy more

cor rec t f r ac t u re

energy per

u n i t a rea , and i s denoted by G

F

For

t he

behaviour

o f

a

specimen

it

i s

of

importance

how

much o f

the

deformat ion

t h a t i s due to

t he

rr-E-curve,

and how much t h a t i s due

to the G-w-curve. In

o t he r

words

it

i s important how t he s e curves

a re r e l a t e d

to each o t he r . One

way

o f defin ing

t h i s r e l a t i o n

i s

by

t ak ing t he r a t i o

between

a deformation equal to GF/f

t

from t he G-w

curve

and a d e f o r m a t i o ~ f t /E from t he rr-E-curve. This y i e l ds a value

which

i s

ca l l ed t he ch r c ter is t ic length

ch

of t he

mate r ia l

GF/f

t

cor responds

to

t he maximum

deformat ion w i

t he

shape o f

t he

curve

had been

a

r e c t a ng le , whereas

f t / E i s the maximum s t r a i n

i

t he rr-E-curve were a s t r a i gh t l i ne , see

Fig.

4.

The c h a ra c t e r i s t i c

l eng th Ich i s a mate r ia l

prope r t y , wh i ch cannot

be d i r e c t l y measu

red ,

bu t

which

i s ca l cu l a t ed

from

t he measured

va lues o f

E, G

F

and

f t

Fig . 4.


10/30

The f r a c t u r e

zone.

The

development

of t he f r a c tu r e zone s t a r t s by t he formation

o f

microcracks ,

which

make

t h i s zone

weaker. At

t h i s

f i r s t

s t age

the

zone

may

comprise a ce r t a in

l eng th

in the

s t r e s s

d i r ec t ion , and the

a dd i t i ona l de forma t ion corresponds to t he t he sum of the addi t iona l

deformat ions

wi th in

these

microcracks .

As t he

deformat ion i nc reases , it ge t s

more

and

more l oca l i zed ,

and

in r e a l i t y p r a c t i c a l l y a l l the addi t iona l deformat ion happens

within

a

narrow

zone, which

may

bes t

be descr ibed as

an

i r r e g u l a r

crack ,

which

changes

d i rec t i on ,

b i fu r c a t e s

e tc ,

depending

on

t he

inhomoge

n i t i e s o f t he ma t e r i a l . Thus

t he width

o f t he f r ac t u re zone ( the

s i z e

in

t he s t r e s s di rec t ion)

can

be assumed to be p ra c t i c a l l y zero.

One

way

o f express ing t h i s i s simply

to

assume

as

a formal model

t h a t

t he

f r a c t u r e zone i s

a

crack wi th t he

width w

and with t he

abi -

litY

to t r a n s f e r s t r e s s e s accord ing to t he IT-w-curve.

A crack with

t he a b i l i t y t o t r a n s f e r s t r e s s e s i s o f course no r e a l crack , but

j u s t a

hypo the t i c a l model.

t

can t he r e fo r e

be sa i d

t h a t t he crack

i s

f i c t i t i ous .

This model has

been

c a l l e d the

f i c t i t i o u s

crack

model .

The

model was

f i r s t

publ ished by

Hil le rborg

e t

a l

(1976) ,

and

it has l a t e r

been fu r t he r

developed and

app l i ed in

many pUbl ica

t i ons ,

e g

by Petersson (1981)

and Gustafsson

(1985) .

However

from t he po i n t o f

view o f p r a c t i c a l a pp l i c a t i on it

does

not

mat te r whether t he a dd i t i ona l deformat ion

in

t he f r ac tu re zone i s

assumed

to

t a ke p a r t

in a

f i c t i t i ous crack o r if it i s

assumed

to

be d i s t r i b u t e d

on

a

ce r t a in

l ength ,

as

long

as

t h i s

l e ng th i s smal l

in

comparison

with o the r dimensions o f t he s t ruc tu re . Thus t he r e i s

hard ly any p ra c t i c a l di f fe rence between t he

f i c t i t i o u s

c rack

model ,

and

t he crack

band model

proposed by

Bazant and

Oh

(1983) ,

where t h e y assume a d i s t r ibu t ion on a l eng th equal to t imes the

maximum aggrega t

e

s i ze . These

models

a re

o f

e n implemented i n to

a

f i n i t e e lement

scheme

in

d i f f e ren t

ways, bu t

t h i s

i s

ano ther ques

t i o n ,

which

w i l l

be

commented upon l a t e r .


11/30

9

The

concentrat ion

of

the

f racture zone to a

th in

band

or

a single

crack

i s

typica l for t ensi le f racture of concrete . t i s not accom

panied by any s ignif icant l a te ra l deformations

or s t resses ,

and thus

it

corresponds

to

a

very

simple

one-dimensional

s t ress

and deforma

t ion s ta te . Therefore the

a-w-curve can

be

assumed to be

a

material

proper ty, which

i s

ra ther insens i t ive to the shape of the st ructure

and

the

general s t re ss s t a t e , as long as a l l other s t re s ses e g a

possible

perpendicular

compressive st ress are low

compared

to the

st rength.

For metals

the s t re ss s t a t e

i s

much more complicated,

as

the

yie l

ding

of

metals gives

r i se

to

a

complicated

three-dimensional

s t re s s

s ta te and

l a t e ra l deformations. Fracture mechanics of

metals

cannot

be t rea ted with the same mode l t ha t

i s

used for concrete.

For concrete

in compression the f racture zone also develops in

another

way than in tension, as crushing i s accompanied by l a te ra l

deformations.

The compression f racture

i s

much more complicated

than

t ens i le f racture. Some kind

of descending rr-w-curve exis t s , but

probably

it cannot

be

regarded to be a weIl defined mater ia l pro

perty. The maximum

s t re ss as

weIl as

the

shape

of

the curve can

be

expected

to depend on e g confinement through s t i r rups or s t ra in

gradients .

The appl icat ion

of

the

fracture mechanics

ideas to com-

press ive f rac ture

i s

s t i l l an

unexplored

but in te res t ing domain. I t

wil l

only

be very shor t ly commented upon a t the end of th i s

paper.

Material propert ies .

The s t ress -s t ra in

curve

in tension

for

ordinary concrete

deviates

ra the r l i t t l e

from

a s t ra igh t l ine .

Thus

for most prac t ica l applica

t ions th i s re la t ion

can be

assumed to

be a s t ra igh t l ine . All appl i

cat ions

so far

seem

to

have

been

based

on th i s assumption.

The st ress-deformation

curve

corresponding to

the descending

branch

can

be measured by

means of

modern t e s t

equipment. The

shape

of t h i s

curve

i s

now

re la t ive ly

weIl

known

I t

has

the

general

shape

shown

in

Fig.

5. One important parameter

of

the

curve

i s the

enclosed


12/30

Fig . 5 .

Ordinary

shape

o f t he

~ w c u r v e

for

concre te .

1

o 5

1

15

w[pml

a rea , which equals

the f r ac t u re

energy

G

F

Another

impor tant

proper

t y may be t he i n i t i a l s lope o f t he

curve .

This proper ty and ts

p ra c t i c a l s ign i f i c a nc e

has no t

h i t h e r t o been s tud ied in

de ta i l .

The f r ac t u re energy G

F

can su i t ab ly be

determined

by means o f a

s imple bending

t e s t

on a

notched beam according to

a RILEM

recommen

da t ion ,

Fig . 6.

The load-deformat ion curve i s recorded in t he

t e s t .

The

t o t a l

energy,

t h a t i s absorbed dur ing t h e t e s t , i s equal to

the

sum of t he

a reas A l

A

2

and

A

3

. Al i s measured

in

t he

diagram,

A

2

i s

ca l cu l a t ed

as F

1

S

0

where F l i s a cen t r a l force g iv ing t he same

bending

moment

as the weight of t he beam

and t he loading equipment.

A

3

i s assumed to

be

equal to A

2

. This t o t a l energy i s div ided by the

a rea

b(h -a ) , t ha t

has been

f rac tured ,

in order to

ge t

t he value of

G

F

Fig .

6.

Tes t for the

determinat ion o f

the

f r ac tu re energy G

F

according to RILEM

Recommendation 1985).


13/30


14/30

12

3

M

2

f f t ~ t . .

f

Fracture zone depth f

Fig . 7. s t r e s s development in an

unnotched

beame

t he beame At t he same t ime t he s t r e s s

across

t h i s zone decreases as

t he

addi t iona l deformation inc reases . At the upper

end

o f t he

f r ac -

t u r e zone t he s t r e s s i s equal to f

t

By means o f a s u i t a b l e numerica l ana lys i s

the

development o f

t he

s t r e s s d i s t r i b u t i o n

i n t he beam can be

fol lowed

as t he f r ac tu re

zone

grows. The cor responding

development

o f t he bending moment can be

ca l cu l a t ed

as

it i s shown in Fig . 7 as weIl

as

t he def l ec t i on .

Exact ly t he same procedure

can

be fol lowed a l so if t he beam conta ins

a

crack from

t he beginning F ig . 8 which

i s the

case t r e a t e d

by

convent ionaI f r ac t u re

mechanics.

In

t h a t

case t he f r ac tu re

zone

s t a r t

to grow a l ready as soon as a

load

i s appl ied due to

t he

s t r e s s

concent ra t ion

a t t he

crack t i p .

The growth i s

however

slow in

the

beginning .

t

can be

shown t ha t the depth

o f

t he f r ac tu re zone

a t

low

l oads

in t h i s

case i s approximately propor t iona l to

t he

square

o f

t he

moment.


15/30

13

Fig. 8. s t r e ss development in a notched beame

In

more complicated cases t i s possib le tha t the f racture zone

s t a r t s

ins ide a

s t ruc ture

and grows in

two

di rec t ions . This may for

instance

happen

with some shear cracks in reinforced

beams or

with

sp l i t t ing

cracks

under

concentrated forces

l ike

end

anchors

for

pres t ress ing tendons.

s ize parameters. Bri t t leness number.

The charac te r i s t i c length lch i s a material

property.

I f a characte-

r i s t i c

s ize of a s t ructure

is divided by l ch th i s

gives a

dimen-

s ionless ra t io , which re la tes a property of

the

s t ruc ture

to a pro-

per ty of the mater ia l .

Fora

beam the depth d i s of en chosen as the

charac te r i s t i c

s ize , and

the ra t io

then i s d / l

ch

This

ra t io i s

sometimes

cal led

r i t t leness number as t gives an indicat ion of

the br i t t l eness

of

the st ructure. The higher the br i t t l eness number

the more b r i t t l e the s t ructure .

As

wil l be

demonstrated

la ter ,

the re la t ive

s t rength

of a

s t ruc ture ,

expressed as a r a t i o between a formal

s t re ss

a t maximum load and f t ,

i s

a

funct ion of

d /l

ch

In

t h i s

way many

resul t s

of

the

theore t ica l

analyses can be given in

dimensionless general

diagrams.


16/30

14

Appl ica t ion by

means of

f i n i t e element ana lys i s .

The app l i ca t i on by

means

o f t he f i n i t e element

method

FEM)

i s r a t

h e r s t ra ight forward i

the f r ac t u re zone

fo l lows

the

direc . t ion o f

t he

element

boundaries.

The

f r ac t u re zone

then can

be

model led

e i t

he r

a s a

sepa ra t ion between elements t he f i c t i t i o u s crack

model

o r

as a change in s t r e s s - s t r a in

prope r t i e s

of a row o f elements the

c rack

band model .

In t he f i c t i t i o u s

crack model

it may

fo r

ins tance

be su i t ab le t o

ca l cu la t e t he

fo rces

in

the

node

po in t s between the e lements . When

such

a

fo rce

reaches

a

value

corresponding

to t he

t ens i l e

s t rength ,

a

s e pa ra t i on between t he elements i s assumed, Fig . 9. Forces a re

in t roduced

between the separa ted

node

po in t s . The va lues

of these

fo rces depend

on the

separa t ion

dis tances

w according to

t he

a-w-curve

for the m a te r ia l ,

see

Fig .

3.

In t h i s way t he development

o f

t he

f r ac tu re

zone and the

corresponding fo rce s and deformat ions

can be fo l lowed.

In

t he

c rack

band

model

t he

formal

s t r e s s - s t r a in -cu rve

fo r

an

e l e

ment ,

where

t he f r ac t u re

zone passes , i s

s imply

determined according

to

t he

genera l formula

Fig . 9. The format ion

and growth o f

a

f r a c t u r e zone, modelIed

by

means o f

a success ive

s e pa ra t i on

o f node

po in t s .


17/30

15

where

El

i s

the

formal

average s t r a i n

in

t he

element

E

and

w

a re

in

accordance with

the

p rope r t i e s

o f

t he

mat e r i a l and

l i s t he s i ze of

the

element in

the d i rec t i on

o f

t he

t e n s i l e s t re s s .

In t he types of analyses descr ibed above t h e

f rac t u re

zones and the

c racks

propagate

along

d i sc re t e

c racks o r

bands.

Therefore t h i s

type

of approach i s

ca l l ed

the dis re te crack approach

This

approach i s

more

d i f f i c u l t

to apply

when t he f r ac tu re zone does no t fo l low along

t he d i r ec t ion

o f

t he element boundaries bu t c rosses the elements a t

skew angles . ce r t a in p o s s i b i l it i e s e x i s t

for

t he a pp l i c a ti on o f t he

d i s c r e t e

c rack approach

to

such cases but

the s e a r e

compl ica ted

and

w i l l

no t

be

discussed

here .

When t he d i r ec t i on

o f

t he f r ac tu re

zone

does

not

fol low along the

element boundar ies t

i s e a s i e r to apply t he smeared crack approach

In t h i s approach a formal s t r e s s - s t r a i n r e l a t i o n

i s

assumed for the

mat e r i a l j u s t l i k e in the c rack

band

model

descr ibed

above. Even

though t h i s

approach

may be

formal ly e a s i e r

to

apply t invo lves

some

problems

and

r i s k s

o f mi s i n t e rp re t a t i ons .

When t he formal s t r e s s - s t r a i n

r e l a t i o n s h a l l be c a l c u l a t e d

t he

value

o f t he

l eng th

l i s

not

weIl def ined when t he d i r ec t i on o f t he t e n s i

l e s t r e s s forms a skew angle wi th

t he element

d i rec t i ons . This l eads

to an uncer ta in ty in the de te rmina t ion o f t h i s

r e l a t i o n which

gives

r i s e to

an

uncer ta in ty in t he r e s u l t s .

st ll worse however i s t ha t t he p rope r t i e s

o f

many types

o f

f i n i t e

e lements a re

not

su i t ab l e

for

de s c r ib ing t he growth o f a f r ac t u re

zone in a r e a l i s t i c way. When t he s t r a i n in an element r eaches the

descending

branch

in the s t r e s s - s t r a i n

curve

t h i s cor responds to a

nega t ive

modulus o f e l a s t i c i t y fo r fu r the r deformat ions . Such

an

e lement may then be coupled

more o r

l e s s in p a r a l l e l l with

an

ad j o i

n ing

element with a pos i t i ve

modulus

o f e l a s t i c i t y .

The

r e s u l t may

be very discont inuous s t r e s s and s t r a i n

d i s t r i b u t i o n s which

a re un-

r e a l i s t i c . In

order

to

avoid

l a r g e mi

s takes

t

i s

necessa ry to be

very ca re fu l

when

the

smeared c rack

approach

i s

used. n u n c r i t i c a l

app l i ca t i on

may

l ead

t o qu i t e un re l i ab l e

r e s u l t s . Af

t e r t h i s

warning


18/30

16

has been

given,

the smeared approach

wi l l

no t be discussed any

fu r

t h e r i n

t h i s

paper .

Appl ica t ion

to t he bending o f beams.

All t he

re su l t s which

wi l l

be shown beolw a re

based

on t he a pp l i c a

t i o n

o f the f i c t i t i ous

crack model.

The assumed r r -E-re la t ion i s a l

ways

a s t r a ig h t

l i n e whereas two d i f f e r e n t r e l a t i ons have been used

fo r t he G-w-rela t ion according to

Fig .

10.

The

s ing l e s t r a ig h t l i n e

denoted SL , i s t he s imples t pos s ib l e assumption

fo r

the numerical

ana lyses

whereas

t he

b i l i n e a r r e l a t i o n

i s meant

to

be

a

good

app

roximation for the

r ea l

shape. t i s

denoted

C fo r concre te .

E

w

w

Fig .

10. Simpl i f ied assumptions r egard ing

mate r ia l proper t i es .

A s imple example i s t he ben t un re in fo rced beam, without o r with a

notch

(crack)

on

the t ens i l e s i de . Figs . 11-14 show

r e s u l t s

o f such

ana lyses . The s t r e ng th

o f t he

beam i s

expressed

as

a

formal

bending

s t r e s s

a t

f a i l u r e

fo r t he

unnotched

beam

and fo r t he notched beam

f

ne t

6Mjb(d-a)2

where

M

i s

t he

maximum

moment

and

a

i s

t he

notch

depth.


19/30

17

The va lue of the r a t i o f f / f

t

can

be taken

from

diagrams o f

t he

type

shown in Fig . 7, and

t he

r a t i o f ne t / f

t

from

Fig.

8.

Fig .

11 shows

t he v a r i a t i o n

of

t he

f l exura l s t rength

modulus

o f

rupture , MOR

with d / I ch From

t h i s

diagram t can

e g

be

seen

t h a t

t he r a t i o

between f l exu ra l

s t r eng t h

and t e n s i l e

s t rength

for a

100

mm deep beam with Ich = 400 mm can be

expected to

be about

1 . 6 ,

which i s in a

r easonab le

agreement with exper ience .

wi th t he

model

t

i s

a l so

pos s ib l e to

study the in f luence of

sh r in -

kage

s t r e s s e s .

Fig . 12 shows

an

example of t h i s , where

t he

shr inkage

s t r a in s

have been

assumed

to

have

values of

an

order

which

can

be

expected in a normal

i n t e r i o r

s t ruc tu re .

Fig . 11. Rat io between

f l exu ra l s t r eng t h and

t en s i l e s t r eng t h .

Fig .

12. In f luence o f

shr inkage s t r e s s e s on

t he r a t i o

between

f l exu ra l

s t r eng t h

and

t en s i l e

s t rength .

3

2

0.01

trift

2.5

2.0

1.5

1.0

0.5

O

0 01

plastic

elastic brittle

0 1

)M f f ~ M u / b d 2 / 6 )

1 l 1

Initial

stress at

mid-span:

d Ich

10

0 1

10


20/30

3

f

net

f

t

2.5

2

1.S

\\

lEFM

Q :0 6\ \

d \\0.2

I

J-w.

el

I

\ \

\ \

\ \

\

\

\\

0.1

L - r r r r r - - r - - r - r - . , . . . . . , . T T - ~ ~ Y - - - -

O.OS

0.1

02 0.5

2

Fig .

13

Rat io

between

net

bending

s t r eng t h

o f a notched

beam and

t he

t e n s i l e s t r eng t h .

18

1 .01 . - - - -

l

4.0

a o

100-w: CJI

o L - L - L - L - L - ~ - - - . l - . l - _

0.5 1.0

old

Fig . 14

In f Iuence of t he

notch

depth

on t he

net

bending

s t r eng t h .

Fig 13 shows t he va r i a t ion of the

s t rength

o f a notched

beam

with

d/ Ich . In t h i s case t he

s t rength

i s very sens i t i ve to the depth for

deep beams

As

a

mat te r

o f f ac t

the s t rength for

deep beams

approa-

ches va lues

which

a re predic ted by l i n e a r e l a s t i c f r ac tu re mecha-

n i es

which means t ha t they a re inve rse ly propor t iona l to t he square

roo t o f

t he beam depth .

Fig 14 shows how t he depth of the

notch in f luences

the ne t bending

s t r eng t h

o f a notched beame For a low value of

d / Ich t he notch has

p ra c t i c a l l y no in f luence on the ne t bending s t r eng t h f

n e t

which i s

near ly

t he same as

t he

bending s t r e ng th f f o f an unnotched beame

These

beams a re sa i d to

be notch i n sens i t i ve . For h igh va lues

o f

d/ Ich t he va lue of f

ne t

decreases as soon as

t he re i s a

notch which

means

t h a t

t he s e

beams

a re notch sens i t i ve . t can be noted

t ha t

the

not h

s n s i t i v i t y

i s not a mate r ia l p rope r ty

bu t

a prope r ty

t h a t

depends on

t he

beam

depth d

as

weIl

as

on t he m a te r ia l p rope r ty Ich


21/30

1 9

In a l l t he f igures it can be seen t h a t t he

s t rength

depends

on the

s i z e d of t he beame Thus the re i s a s i ze e f fec t , which i s

explained

by

means o f

t he

f r ac t u re

mechanics approach. The s i z e

e f fec t

i s

g r e a t e r

fo r notched beams

than

fo r unnotched, p a r t i c u l a r l y for deep

notched beams.

For unnotched beams t h i s s i ze e f f e c t

increases when

shr inkage o r tempera ture

s t r e s se s

a re

ac t ing .

In

Figs . 11 and 13 it

can be

seen

t h a t t he

s t r eng t h approaches the

va lue pre d ic t e d by t he theory o f p l a s t i c i t y

for small

beams, and

t h a t it approaches t he value

pred ic t ed by t he t heory

o f e l a s t i c i t y

fo r l a rge beams. The ana lys i s covers a l l cases

between

these

ext remes

fo r

unnotched

as

weIl

as

notched

beams.

It t hus

has

a

genera l

a p p l i c a b i l i t y . A smal l value o f

the

b r i t t l e n e s s number

d / Ich gives a more p l as t i c - t ough behaviour ,

whereas

a

h igh

value

gives

a more e l a s t i c - b r i t t l e behaviour.

S e n s i t i v i t y

analys i s .

The

formal

s t rength

accord ing

to

the

above

r e s u l t s

depends

on

the

va lue

o f

d / I ch ' where Ich in i t s

t u rn

depends on E, G

F

snd

f t accor

ding t o t he

r e l a t i o n Ich =

G

F

f

t

2

The

diagrams

a r e given in

loga

r i tmic sca l e s (except Fig .

12) .

For a

small

change

in

d/ Ich

t he

r e l a t i on

can

approximately

be w r i t t e n

l n f f

=

A (1-2B) lnf

t

- BInd

BlnE

BlnG

F

where A

and

B

a re cons tan ts ,

with B

showing t he

negat ive s lope in

t he diagram a t t he s tud i e d

p a r t

of the curve .

A

d i f f e r e n t i a t i o n

gives

df

f

df t dd d

= (1-2B) f

t -

Ber BIf B

GF


22/30

20

This

express ion

shows the r e l a t i ve

change

in

the

formal s t r eng th f f

fo r

a smal l

r e l a t i ve change in

one of the

parameters . This can be

ca l l ed

the

s ens i t i v i ty

and it i s determinded from

the

s lope

-B in

t he diagrams.

t must

be

no t

ed t h a t

t he

diagrams

are

given

with

the

sca le on the

v e r t i c a l

axes t imes

as

l a rge

as

t he sca le on t he

ho r i zon t a l

ax is .

Therefore t he

s lopes ,

which

a re

measured in the

f igures ,

must

be

div ided by

4. I f fo r

example

t he

measured

s lope i s

-0 .6 , t he value o f

B

i s 0.15 , which means t h a t

t he

s e n s i t i v i t y

with

regard

to

G

p

i s 0.15

and

with

regard

to

f t 0 .7 .

In

t h i s

case an

increase

in

f t

with

10

percen t

increases

f f with

7 percen t ,

whereas

an increase in

G

p

with 10 percen t increases f f with 1.5 pe rcen t .

In

t he

same

case

an

i nc rease

in t he beam

depth

d

wi th

10

percen t

decreases f f with

1.5

percen t .

Appl i ca t ion to unre inforced conc re t e pipes .

An i n t e r e s t i ng

prac t i c a l

appl ica t ion has

been made

to t he s t r eng th

o f

unre inforced conc re t e pipes . Por t hes e t he re are e s s e n t i a l l y two

d i f f e r e n t

types

o f

f a i l u r e ,

t h a t

a re o f

i n t e r e s t ,

see

Pig.

15.

One

i s t he bending

f a i l u r e

or beam

f a i l u re ) ,

where t he pipe i s suppor

t ed

and

loaded

l i k e

a

beame The o the r

i s the crushing f a i l u r e or

r ing f a i l u re ) , where the pipe i s

loaded

and suppor ted

a long

its

l eng th . This

type o f load r e s u l t s in

bending f a i l u r e s in sec t ions

a long t he

pipe , a t

t he t op and t he

bottom,

and

a t

t he two s ides .

The

s t r u c t u r e

i s under

t hese condi t i ons

s t a t i c a l l y inde terminate , which

means

t h a t a

moment r ed i s t r i bu t ion

can t ake p lace

before t he

maximum

load

i s

reached. The amount

o f t h i s

moment

r e d i s t r i b u t i o n depends on

t he toughness

o f

t he

s t ruc tu r e ,

and t he re fo re increases with a

decrease i n s i z e .

~

~ c r

l ~

- - - - ~ - - - -

Pig.

15.

Bending

f a i l u r e and c rush ing f a i l u r e o f

a

pipe .


23/30

Fig. 16. Yariat ion

of

formal bending

strength of

a

pipe.

21

i

f

t

3 5

3

2 5

1 5

0 1 0 2

0 5

C ushing failure

5 1

dj/lch

In Fig.

16

the

var ia t ion

in

the

formal bending

s t rength

with

the

s ize of the pipe i s shownfor

these two

loading s i tua t ions

The

for-

mal s t rength f f

i s

the maximum s t re ss a t maximum load, calculated

according

to the theory of

e la s t i c i ty

From the f igure t i s evident t ha t the formal s t rength i s much

higher for the crushing fa i lure

than for

the beam

fa i lu re

The

s ize

dependence i s also much

higher for the

crushing fa i lure

The reason

for the

difference in s t rength is t ha t the sect ion depth for the

act ing moment

i s

much lower for the

crushing fa i lure

the

wall

thickness) than for the beam fa i lure the diameter

of the

pipe) . The

reason

for

the

higher

s ize

dependence

for

the

crushing

fa i lu re

i s

tha t

th i s s t ruc ture

i s

s ta t i ca l ly indeterminate.

The values according to Fig. 16 are in a good agreement with t e s t

resul t s They have

found

a

prac t ica l

applicat ion for redesign of

cer ta in

pipes.


24/30

22

Appl ica t ion to shea r

fa i lu re

o f

beams.

The

shea r

s t r eng t h

o f

re inforced

beams

without

shea r re inforcement

has

a l so

been analysed

by

means of the f i c t i t i ous crack model. This

i s a

very

compl ica ted

case as

t

depends not only on t he concre te

prope r t i e s in t e ns ion bu t a l so on concre te prope r t i e s

in

compres

s ion and shea r on t he s t ee l prope r t i e s on t he

bond

behaviour

b e t -

ween concre te

and s t e e l and

on

many

o the r

f ac t o r s .

The f r ac tu re

zone and t he

r e su l t i ng

cracks a re curved and they

can

appear in

many d i f f e r e n t pos i t i ons .

Due

to t h e

complexi ty

of the

she

a r f r ac t u re

t he a na lys i s

which

has

been performed so f a r has had to be performed on t he

bas i s

o f many

approximat ions and

s i mpl i f i ca t i ons . Thus only one crack

a t

a t ime

has been s tud i e d bu t t h i s crack has been var ied in order to f i nd

the

most dangerous

s i tua t ion .

The shape

and pos i t i on

of t he c rack

has been

assumed

in advance

for

each ca l cu l a t i on bu t a f t e rwards t

has

been checked t h a t

t h e crack

i s

near ly perpendicular

to

t he

p r i n -

c i pa l t ens i l e s t r e s s . Dowel a c t ion has not been

taken

i n t o account

nor

aggregat

e

i n t e r l ock .

The

prope r t i e s

o f

the

r e in fo rcement

bond

prope r t i e s

and f a i l u r e

in

t he

concre te

compression

zone have

been

taken i n to account .

Fig .

17.

Theore t ica l r a t i o

between

formal shea r s t r eng t h

and t e n s i l e

s t r eng t h fo r

a

beam with l ong i tud ina l

r e in fo rcement .

0 6

0 5

0 4

0 3

0 2

0 15

0 1

1.5

1

0.5

~ . 1

L/d-6

S- l

Lld-9

0 2

la

5 0 d Ilch


25/30


26/30

/V

u

.

d

=O 2m

1.4

1.2

1.0

0.8

0.6

0.4

0.2

ACI

CES

_____

-

c

al Experimental results. Survey by Kennedy/Taylor /

bl Experimental results by Leonhardt/Leonhardtl

ej Present coleulations.l

e

h O.25m. r=1.0 , l/d

=3

0.2 0.4 0.6 0.8 1.0

1.2

24

Seam depth,

d m)

Fig .

18 In f luence o f

beam

depth .

1.6

~ C E B

1.4

ACI

1.2

1.0

f ~ = 3 psi

0.8

- - - Survey

of

150 tests lHedmon et 011

o Colcul'otions l id = d/l

h=O 6 ond 2:4

o

O

0.5 lO

1.5

2.0

2.5

2

1

Survey of 479 tests/Hedman

et

ll

El Coleulatlons

l

=1.0 , d/Ich =0.6 and 2.4

fe

3000psi

o

O

3

6

9 lid

Fig .

19 In f luence o f span

to

depth

r a t i o .

Figs . 18

-

20 Comparisons

between t heo re t i ca l values

t e s t va

l ues

and

code va lues

regard ing the i n f luence o f

d i f f e r e n t

f ac t o r s on

t he

shear

s t r eng t h o f r e in fo r

ced beams

Percentage reinforcement,

9 %)

Fig .

20

In f luence o f

re inforcement r a t i o


27/30

25

where

Vu

i s

the formal shea r s t r eng t h (shear

force d iv ided

by the

c ross

sec t ion area

and k a

cons tan t .

This express ion can be r e a r -

ranged by

i n se r t i ng

t he d e f i n i t i o n Ich

= G

F

f

t

:

From

t h i s

express ion it can

be

seen

t h a t t he

shea r s t r e ng th depends

as

much

on G

F

as

o n f

t

It

i s

genera l ly

accepted t h a t

t he

t e n s i l e

s t r eng t h f t i s approximately propor t iona l

to

t he square root of the

compress ive s t rength .

Thus f

t

can

be assumed

to be propor t iona l to

t he compress ive s t rength . The

conc lus ion

from

t h i s

i s t h a t t he shear

s t r eng t h

o f

a

beam

depends

as

much

on

t he

f r ac t u re

energy

o f

t he

concre te in the

beam as

on its compress ive s t rength .

When a l abora tory

t e s t i s performed

on a

concre te s t ruc ture ,

t he

compress ive s t rength

i s

t r ad i t i ona l l y

always measured

and

repor t ed .

From

t he

above

it fol lows

t h a t

t he

f rac t u re energy should

a l so

be

measured

and reported where

shea r

t e s t s a r e performed,

as

t h i s

pro-

per t y i s as

important

as t he compress ive

s t rength .

The same may hold

a l so

for

many

othe r types o f

s t ru c tu r a l

t e s t s .

AIso in code formulas for shea r

s t r eng t h t he f r ac t u re

energy ought

to

be t aken

in to

account in

some way

o r o the r . How t h i s should be

done

i s t oo ea r l y to spec i fy , bu t

one

p o s s ib i l i t y could

be

to give

some t ype o f

cor rec t ion

f ac t o r , depending on

t he

type of concre te ,

fo r

example r educ t ion

f ac t o r s fo r

l i g h t weight concre te and

fo r high

s t r eng th concre te .

Direc t ion o f fu tu re re sea rch .

It has been demonstrated

above

t h a t t he a pp l i c a t i on

o f

f r ac tu re me-

chanics to concre te s t ruc tu re s can give impor tant c on t r ibu t ions to

t h e

unders tanding

of the

behaviour

o f

s t ruc tu re s in cases where

our

knowledge

e a r l i e r has been mainly based on empir ica l s t ud i e s , l i k e

t he r a t i o

between f l exura l and t e n s i l e s t r eng t h , the i n f luence o f

shr inkage

on

the f l exura l

s t r eng t h

and

t he

shea r

s t r e ng th

o f

beams.


28/30

26

still we a re however

only in

t he beginning

o f

a

development.

I f

we

fo r

example

look on t he a pp l i c a t i on to shear

f rac ture , the

r e s u l t s

which were demonstrated a re based on analyses where

many

r a t h e r

rough

approximat ions

have been made.

These

were pa r t ly

due

to

a

l ack

o f knowledge, e g

r egard ing

t he aggregate i n t e r loc k and the dowel

a c t ion , and p a r t

y

on t he complexi ty o f

the

problem,

which

made it

too d i f f i c u l t to t ake a l l f ac t o r i n t o account with t he ex i s t i ng

f i n i t e element program.

Thus one impor tan t

type

o f

resea rch

i s to f ind more adequate mate r i -

a l prope r t i e s

to be

i n se r t ed

i n t o t he f i n i t e element analyses . One

example

o f

such

a

resea rch

work i s mentioned below.

n

o the r

impor tan t

type

o f

resea rch i s to develop f i n i t e

element

programs, which

a re

b e t t e r adopted

to

handle t h i s

type

o f f r ac t u re

mechanics, with l oca l i za t ion o f f r ac t u re zones and s t r a i n so f t en ing .

t

i s

a l s o impor tan t

to

apply

t he

mode

l in a

sys temat ic

way to r ea l

s t ruc tu re s i n order to achieve

a

b e t t e r

unders tanding

o f d i f f e r e n t

types

o f

behaviour , e g

as

a

background

for

b e t t e r

des ign

r u l e s

and

codes . The r e s u l t s

o f

such sys temat i c analyses can pre f e r r a b ly be

given in dimension less genera l diagrams o f

the types shown above.

Presen t re sea rch

in

Lund sp r ing 1988 .

l a rge

t e s t

program

i s going on r egard ing

the

behaviour o f

a f r a c -

t u r e

zone in mixed

mode,

i e

with

shea r deformat ions and

s t r e s s e s

in

a f r ac t u re zone a f e r it has s t a r t e d in t ens ion . Some r e s u l t s were

presen ted in 1987. More sys temat i c r e s u l t s w i l l be presented a t

a

conference

in

vienna in J u ly 1988, and t he f ina l complete

r epor t

i s

expected

to

appear in 1989. One example o f a

t e s t

r e s u l t i s presen-

t ed

in

Fig . 21.

f i r s t a t tempt has a l s o been made to apply t he model with l o c a l i z a -

t i o n

and s t r a i n

hardening

to t he f r ac t u re i n t he compression zone o f


29/30

27

a re inforced beame

These

f i r s t

resul t s indicate

tha t

the s t ress -

s t ra in re la t ion

to be

used

for the

prac t ica l

design should

prefer-

rably have an ul t imate s t ra in

equal

to

kl /x

instead of

the

normally

assumed

3.5

permil le

where

k l i s a mater ia l property and x i s the

depth of

the

compression

zone.

I f

th i s conclusion i s correct t

wil l

have

a s ignif icant influence in many prac t ica l s i tua t ions .

Further research i s needed before the re su l t

i s suff ic ient ly

conf i r -

med.

In

a

t h i rd

projec t a

number of

t e s t s are being

performed

on some

simple unreinforced s t ructures in order to

check

the general appl i -

cab i l i ty

of the

fracture

mechanics

aproach.

A

wide

range

of

di f fe -

rent mater ia ls are

tes ted

par t icular ly with respect to d i f fe ren t

toughness.

Very

b r i t t l e mater ia ls

are tes ted

l ike

pure

cement pas

te

as weIl

as wery

tough

mater ia ls l i ke

f ibre

re inforced concrete

and

some mater ia ls in the intermediate range.

E

~ 1 0 0

Z

o

; .BOO

J

~ 6 0 0

Z

~ o o

.200

6.00

~

~ O

er

3.00

2.00

1.00

.000 .200

~ O O

.600 .BOO i OO i.2O

i ~ O

SHEAR

DEFORMATION. mm

r

A

I

.000

.200

.0400

.600 .BOO

1 00

1.20 1..040

Sf EAR DEFORMATION.

mm

3.00

2.00

vi

i

~ 1.00

J

z .000

-1.00

-2.00

\

3.00

.000 .200

~ O O

.600

.BOO

1.00

NORMAL DEFORMATlON.mm

Fig.

21. Example of resul t s

of t e s t s with shear deforma

t ions and

t ens i l e

deforma

t ions act ing simultaneously.

The

deformation

path

in th i s

case has been arranged to

follow a predetermined para

bola.

Maximum aggregate s ize

8

mm.


30/30

28

References

American

Concrete I n s t i t u t e

1983) Bui lding code requi rements

for

r e in fo rc e d concre te ,

ACI

318-83.

Bazant , Z.P.

and

Oh. B.H.

1983) Crack band theory

fo r f r ac t u re of

concre te .

RILEM Mater i a l s and

St ruc tures ,

Vol

16,

No 93, 155-177.

CEB/FIP Model Code

for

Concrete St ruc tu r e s

1978).

CEB Bul l e t i n

124/125-E.

H i l l e rborg ,

A.,

Modeer,

M

and

Petersson ,

P.E.

1976)

Analys is

of

c rack

format ion and crack growth

in concre te

by

means

o f

f r ac t u re

mechanics

and f i n i t e elements . Cem.

and Concre.

Res. ,

6,

773-782.

Iguro ,

M.,

Shioya,

T. , Noj i r i ,

Y. and

Akiyama,

H.

1984)

Experimen

t a l s tud ie s on shea r s t rength o f l a rge re inforced concre te beams

under uniformly d i s t r ibu ted load

in

Japanese) , Proceedings , Japan

Soc ie ty

o f c i v i l Engineers Tokyo), No. 348/V-1,

Aug. 1984,

175-184.

Also Concrete Library In t e rna t i ona l , JSCE, No. 5,

Aug.

1985,

137-154 in Engl ish) .

Gustafsson, P. J . 1985) Fracture mechanics

s t ud i e s o f non-yie ld ing

mat e r i a l s l i k e

concre te .

Report TVBM-1007,

Div.

o f

Bui ld ing M ate r i

a l s , Lund I n s t .

o f

Technology, Sweden.

Pete rsson ,

P.E. 1981)

Crack growth

and development o f

f r ac tu re

zones i n p l a in concre te and s imi la r mater i a l s . Report TVBM-1006,

Div. o f Bui ld ing Mater i a l s , Lund In s t . o f Technology, Sweden.

RILEM 1985) Determinat ion o f the f r ac t u re energy o f mortar and

concre te by means

o f

t h r e e -po in t bend

t e s t s on

notched

beams,

RILEM

Mater ia l s and St ruc tures , Vol 18,

No

106,

185-290.

Documents

Application of Fracture Emechanics to Concrete Arne Hillerborg