Size Effect in Concrete Structures

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Chapter 2

Size Effect in Concrete Structures

ROLF

ELIGEHAUSEN JOSKO

OZBOLT

Institut für Werkstoffe im Bauwesen Universität Stuttgart

Pjaffenwaldring 4

7000

Stuttgart 80 Germany

BSTR CT

The size effect for notched-tension specimens three-point bend

specimens pull-out headed anchor specimens and beams loaded in

torsion are calculated using a 2D and 3D finite element program. The

program is based on the nonlocal microplane model. The calculated

fai/ure loads are compared with previously obtained experimental

results. Test results and calculated data are compared with the recently

proposed size effect

law.

Results

of tests

and analysis exhibit signijicant

size effect that should be taken into account in design practice. It is

demonstrated that the nonlocal microplane model used in a 2D and 3D

finite element code can correctly predict fai/ure loads for similar

specimens of different sizes.

1. INTRO UenON

The size effect in concrete structures is a weil known phenomenon.

For example the bending strength decreases with i n r e ~ i n g specimen

height. Another example

is

the shear strength

of

concrete beams

without shear reinforcement. Kani

[1] was

one of the first to

demonstrate that the shear strength

of

identical concrete beams

decreases with increasing beam depth and that the shear design

provisions used at that time were unsafe for larger

beams This

size

effect can weil be explained by fracture mechanics because the

17



8

ROLF EUGEHAUSEN O ~ K O OZBOLT

fracture in a concrete structure is driven by the stored elastic energy

that is released globally from the entire structure. However,

before

faHure, microcracking in the concrete causes deviations of the

siz

effect from the geometrical size effect known from linear

elastic

fracture mechanics LEFM), because for normal geometrical sizes the

fracture process zone is relatively large with regard to the geometry

of

the structure and therefore the size effect can only be

correctly

calculated using nonlinear fracture mechanics NLFM).

In numerical analysis it is very difficult

to

model damage

and

fracture processes in materials such as concrete correctly. t present,

three different material models for strain-softening damage exists:

1) Continuum models used together with fracture mechanics.

2) Random particle model, in whicb tbe microstructure is imagined

to

consist

of

randomly arranged rigid aggregate pieces

with

elastic-softening interactions between them.

3) Micro-finite element models, in which the matrix as weil as

the

aggregate pieces in concrete are subdivided into many

finite

elements, whose inelastic behaviour and cracking as weil

as

interface bond failures are taken into account.

The last two material models automatically take into account

tbe

structural size effect but they are still extremely demanding of

computer time and cannot be used in structural analysis. Therefore,

the continuum material models must be formulated in such a way that

they are capable of describing fracture of the structure in a correct

way.

In

the present study a number of finite element analyses are

performed to investigate the structural size effect. A continuum

material model, called the nonloeal mieroplane model based on tbe

smeared crack approach is used. Numerical studies of the structural

size effect are presented and discussed for plain concrete specimens

loaded in centric tension, three-point bending, axisymmetric pull-out

of headed anchors and beam torsion. Tbe calculated results are

compared with available test data as weil as with the

size effect

law

recently proposed by BaZant [2].

2. SUMMARY OF NONLOCAL MICROPLANE MODEL FOR

CONCRETE

The microplane models were initiated by Taylor [3], who suggested

the principle for

the

modelling

of

plasticity

of

polycrystalline metals.



SIZE EFFECf IN CONCRETE STRUcruRES

19

In that approach, developed in detail

by

Batdorf Budianski

[4]

and

others, the plastic slips were calculated independently on various

crystallographic planes based on the resolved shear stress component,

and were then superimposed to obtain the plastic microstrain. Later

this approach was extended, under the name multilaminate model, to

the modelling of non-softening plastic response of soils or rocks [5].

Recently [6]-[9], this approach was extended to include strain

softening of concrete, and

was

renamed more generally as the

microplane model in recognition of the fact that the approach is not

limited to plastic slip but can equally

weIl

describe cracking and

strain-softening damage. To prevent instability due to strain softening,

the microplanes must be constrained kinematically rather than stati

caIly

in which case the use of the principle of virtual work must

replace the direct superposition of the plastic strains

as

used in the slip

theory.

In the present study the microplane model originally developed by

Baiant Prat [9],

is

slightly modified and implemented into a 20 and

3D

finite element code. The basic hypotheses used in the model are as

folIows:

Hypothesis

I The

strains on any microplane represent the resolved

components of the macrostrain tensor ij kinematic constrains).

Hypothesis II Each microplane resists not only normal strain EN

but also shear strain

ET.

Shear strain

is

split into two mutually

perpendicular in-plane components and as a consequence the shear

stress vector

is

not parallel with the shear strain vector. This

is

the

main modification in contrast to the originally proposed microplane

model [9], where stress and strain vectors are parallel.

Hypothesis

The normal microplane strain

is

split into volu

metrie and deviatoric components, i.e.

EN =

Ev

+

Eo.

Hypothesis W The stress-strain relation for each microplane

is

path independent as long as there

is

no unloading on this microplane

for that component. Ouring each unloading and reloading, which

is

defined separately in each microplane, the curves of the stress and

strain differences from the state at the start of the unloading are also

path independent. Thus, all the macroscopic path dependence is

produced

by

various combinations of loading and unloading on all the

microplanes.

Hypothesis V -

The volumetrie, deviatorie and shear responses on

eaeh microplane are mutually independent.



20

ROLF EUGEHAUSEN O ~ K O

O ~ O T

These five hypotheses were shown to allow an excellent representation

of nonlinear test data for concrete in 1D, 2D and 3D stress-strain

states [9 .

basie requirement for a continuum model for a brittle heteroge

neous material such as concrete

is

that it must correctly display

the

consequences of heterogeneity of the microstructure. A continuum

constitutive model lumps the average response of a certain charac-

teristic volume of the material (Fig. 1). In essence, one

may

distinguish two types of interactions among the particles or damage

sites in the microstructure, which must be somehow manifested in

the

continuum model: (1) Interaction at a distance

mong v rious

sites

(e.g. between A and B, Fig. 1); and (2) interaction mong v rious

orient tions (see angle er in Fig. 1).

The interactions at a distance control the localization

of

damage.

They are ignored in the classical, local continuum models

but

are reftected in nonlocal models [10]. The nonlocal aspect

is

a

requisite for a realistic description

of

the size effect, as well as for

the modelling of fracture propagation in the form öf a crack

band.

According to the nonlocal concept, the stress at a point depends not

only on the strain at the same point but also on the strain field

in

a

certain neighbourhood of the point [11]-[14]. For strain-softening

behaviour, this concept

was

introduced

by

Baiant

et

al

[15]. In

the

current study, an effective form of the nonlocal concept, in which all

variables that are associated with strain softening are nonlocal and

all

other variables are local, is used. The originally proposed nonlocal

concept [10], is here modified

by

introducing additional weighting

functions that control averaging into the directions

of

the

main

principle stresses . . n important advantage of this formulation, called

h

Fig. Interaction among the various

orientations and interaction at a

distance.



SIZE EFFECf

IN

CONCRETE STRUcnJRES

21

nonlocal damage or nonlocal continuum with local strain, is that the

differential equations of equilibrium as well as the boundary conditions

are of the same form as in the local continuum theory, and that there

exist no zero-energy periodic modes of instability.

The key parameter in the nonlocal concept used is the characteristic

length l over which the strains are averaged, because it has a

significant inftuence on the results of the analysis. Baiant Pijaudier

Cabot

[16]

assumed that this length

is

a material parameter which can

be correlated with G

F

and approximately taken as 3d

a

da = maximum

aggregate size) in the case of uniaxial macroscopic stress-strain state.

However, presently in general 3D stress-strain situations, the concrete

fracture property cannot be measured and correlated with the charac

teristic length. As a consequence,

l

is

difficult

to

interpret as a

material parameter depending on the concrete mix only, but may be

inftuenced by other parameters as weil. Further studies are needed to

clarify whether this length depends only on the concrete composition

or also on other parameters such as strain conditions in the failure

zone. Therefore in the current study, in general, the characteristic

length was determined such that together with the assumed tension

strain-softening relationship, the failure load of a certain type of

specimen with given size was correctly matched. Then, in the analysis

of the specimens of different sizes, this characteristic length was taken

as constant. Only in the analysis of the pull-out specimen was the

characteristic length taken arbitrarily as

l

= 12 mm.

In a preceding paper [17], the nonlocal microplane model as weil as

an effective numerical iterative algorithm for the loading steps, that is

used in the finite element code, is described in detail.

3

NUMERICAL STUDIES

To demonstrate that the nonlocal microplane model, implemented in

the 20 and 3D finite element code, can correctly predict the failure

load of plain concrete specimens of different sizes, numerical studies of

four cases are presented Fig. 2):

1) notched tension specimen;

2)

three-point bend specimen;

3)

pull-out specimen with headed anchors;

4) torsion of short beams.



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SIZE EFFECT IN CONCRETE STRUCfURES

23

In all examples, specimens of

three

different sizes, in the size ratio

1 :2: 4, with geometrically similar shapes are used. Exception is made

in example 3) where a size ratio of

1:

3: 9 is studied. Cases 1) and 2)

are analysed using four-node plane stress isoparametric finite elements

with two by two integration rule. The analysis of the pull-out specimen

is made using four-node axisymmetric finite elements with two by two

integration rule. Finally, in the torsion problem, eight-node finite

elements are employed with two by two by two smallest specimen)

and three by three by three integration rule, respectively. All

specimens in the analysis are loaded by prescribing displacement

increments in each loading step.

Microplane model parameters in cases 1), 2) and 4) are deter

mined so that they represent

the

concrete properties used in

the

experiments.

In

all the examples the shape of the tension stress-strain

curve is determined so that the fracture energy G

F

for the unit

area

of

a specimen

of

length 360 mm is approximately 0·1

NImm

Example 2) has

been

analysed by Bafant Ozbolt [17],

but

was

re-analysed using slightly different material parameters. Example 3)

has been analysed by Eligenausen Ozbolt [18], and results are

shown from this work.

Example

l ) -The notched tension specimen shown in Fig. 2 a) is

considered. This type

of

specimen was tested by Baiant Pfeiffer

[ 9]

using concrete with maximum aggregate size da = 12·7 mm. The depth

of the smallest specimen was d = 38·1 mm, the depth of the notch was

always

1/6

of the

depth

of the specimen and the thickness of the

specimen was constant for all sizes,

b

=

9

mm.

In

the analysis only

one half of the specimen is modelIed. The finite element meshes are

shown in Fig. 3 in the deformed state .

The

characteristic length

is

taken as

I

c

=

3d

a= 38·1 mm.

The

microplane model parameters are

taken so that the calculated tension strength is t = 2·70 MPa.

In

the

experiment the estimated average tension strength, calculated on the

basis of the measured uni axial compressive strength, was

t

=

2·69 MPa.

The

characteristic length is chosen such that together with

the microplane material parameters, the average failure load of the

specimen with

d

= 76·2 mm is matched.

In Fig. 4 the nominal stresses

at

failure related

to

the total

area

a

= Fulbd F

u

= peak load), obtained in the numerical analysis and

in the experiments average values) are compared with the size effect



24

ROLF EUGEHAUSEN

81

JOSKO OZBOLT

....-

-

\

....-

r --..

-

4d 3

1

-

....-

.

-

\

4d 3

1

-...-

1 -

-

-

/

1/

1 -

1

-

1/

....-

l

4d 3

1

1/

1 \

1/

r\.

L. -l.2

-

I }

~

}

•

v=-

~

.

-

J..

~

h

.....

fJ

h

fJ

t

h

J

p

l

1

l

1

11

't

Fig.

3

Deformed finite element meshes and fracture process zone shaded

areas) at peak load for the notched tension specimen.



SIZE

EfFEcr IN

CONCRETE SfRUCIlJRES

0.10 ..----------------------

NOTCHED

TENSION

SPECIMEN

d= 38

. 1,

76.2 and 152.4 mm; b= 19 mm

0.05 -

1 = 2 70 MPa; JN=Fu/bd

0.00

+ - - - - ~ : - - - : - : - - _ _ : _ : _ _ _ : _ - ' - - - - - - - - - l

strength criteria

~ O . 0 5

- - - ....b_

LEFM

: :;

e -0.10

o

t l ,

s -0 .15-

o test

data (average ) ,

t: calculated da

ta

' , ,

, 0

-0

.

20

-

---s ize effect law

B= 1.124

d

o

104 67

t:

-0 .

25

+-- - . - - - - - - . - - - . - - ,- - - , - - - - , .. . - , - - - r -- - , . . . - - -

-0.49

-0

.39

-0.29

-0.19 -0 .09 0.01 0 . 11 0.21

log

(d /d

o

25

Fig. 4 Comparison between calculated and measured failure loads with size

effect law for the notched tension specimen.

law is

proposed by Baiant

[ ]:

C N =

Bft l

+

ß -1f2

ß=dld

o

1)

The optimum values for the parameters

Band

d

o

are obtained

by

linear regression

of

the numerical results (Fig. 5). In Fig. 6 the

nominal stresses

at

failure (numerical and experimental results) are

plotted as a function

of

the specimen depth in normal scale.

I t can be seen from Figs

4-6

that the numerical and experimental

results indicate a size effect: the nominal stresses at peak load decrease

with increasing specimen depth, that me ans the absolute failure loads

increase approximately by a factor

of

1·5 when doubling the depth,

much

less than the increase in failure area.

In Fig. 3 the shaded areas indicate zones where the tensile stress at

peak load exceeds approximately 75 of the uniaxial tensile strength.

According to the assumed stress-strain relationship, the concrete

starts to exhibit nonlinear behaviour for stresses C

>

O·

75ft. Therefore

the shaded area can be assumed as the size

of

the fracture process

zone at peak load. Note that the scale for the specimens of different

sizes

is

inversely proportional to the specimen depth.

I t

is

evident from



26

ROLF EUGEHAUSEN t JOSKO OZBOLT

2 . 0 0 ~ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

NOTCHED TENSION

SPECIMEN

d=

38.1

76.2

and

152.4 mm;

b= 19 m m

1.75

1,=

2.70

MPa;

JN=Fu/bd

~

~ . 5 0

~

0

: 1 25

Y = A X C

A =0.007558 ; C = 0.79 t1

1.00

- I - - - , - . , - - .. - - - , . - T - - . , - -T - - , - - . . - - - , - . . .. . - - , - - - - i

30

50 70 90 0

130

SO

X d

mm)

Fag

5 Linear regression analysis of the calculated peak loads for the notched

tension specimen

1.3 - r - - - - - - - - - - - - - - - - - - - - - - - .

NOTCHED TENSION

SPECIMEN

1.2

d= 38.1 76.2

and 152.4

mm; b= 19 m m

1,= 2.70

MPa;

JN=Fu/bd

I .

I

1.0

0 9

~

~ 0 8

b

0.7

o

test

data

average)

0.6 6 calculated data

0.5

size ellect

law

B= 1.124 d

o

=

104.67

0 . 4 - - - . - - r - - . - - - , - - . , - - . , - - - - , ~ - , _ - _ 1

20

0 40.0

60.0

80.0

100.0 120

0

140.0 160.0 180.0

200.0

d mm)

Fig 6 Comparison between calculated and measured failure loads with

size

effect law for the notched tension specimen shown in normal scale



SIZE EFFECT

IN

CONCRETE STRUCfURES

27

Fig.

3 that, relative to the specimen size, the fracture process zone

decreases with increasing specimen depth. This is a consequence

of

the

fact

that the volume

of

the nonlocal continuum over which the strains

are averaged

is

constant and therefore this volume is, relative to the

specimen size,

sm

aller if the size of the specimen is larger.

In Fig. 7 axial strain profiles across the symmetry line

of

the

specimen at the start of the analysis and at peak load are plotted for all

sizes. This figure clearly indicates that the strain distribution over the

cross-section is more uniform if the size

of

the specimen is

sm

aller

Therefore with decreasing depth, the stresses in the critical section are

more uniformly distributed and the average stress increases.

Summarizing, the size effect can be explained by two effects: 1) The

size of the fracture process zone relative to the specimen size decreases

with increasing specimen depth; 2) because of 1) the strain and

stress distribution becomes less uniform with increasing member

depth, resuIting in a decrease of the nominal stress at peak load.

Example

2)-The three-point bend specimen shown in Fig. 2 b)

was

tested by ahnt Pfeiffer [19], using concrete with maximum

0 60

-y---------------------.....

NOT HED

TENSION SPECIMEN

-

STRAIN

PROFILE

d=

38 1

.

76 2 and

152.4

mm;

b= 9 mm

4>

,

,

,

,

,

,

Fig.

7 Strain redistribution in the critical cross-section of the notched tension

specimen.



28

ROLF ELIGEHAUSEN

&.

O ~ K O OZBOLT

1 25

d

1

F 2

'

E

t

v

-

1

P

v

~ p

\.

I I

Jo i

t 25d

f

F 2

I'\..

\

P

L

\

I

e

/

1 25d

Fig. 8 Deformed finite element meshes and fracture process zone shaded

areas) at

peak

load for the three-point bend specimen.




29

aggregate size da = 12·7 mm. The geometry

of

the specimens basically

was the same as in the case of Example (1), except that the deptb of

the smallest specimen size was d = 76·2 mm and the thickness

of

all

specimens

was

b =

38

mm. In the analysis, again only one half of the

specimen is modelIed. The finite element meshes are shown in Fig. 8

in

the deformed state. The characteristic length is taken as

lc =

3d

a

,

the

microplane model parameters are chosen so that the tension strength

is

t = 2·

74

MPa. In tbe experiments the average estimated tension

strength was t = 2·90 MPa. The material parameters were taken such

that the average failure load of the specimen with d = 152·4 mm is

matched.

In Figs 9-11 the nominal bending stresses at peak load according to

tbe theory of elasticity, related to tbe total depth d aN = 15F

u

/ 4bd),

obtained numerically and experimentally are compared witb each

other and with Baiant's size effect law. Again, calculated results and

experimentally measured data exhibit a very strong size effect, weil

known for bending specimens [20J [21J. According to Fig. 11 the

ben ding strength for a specimen with

d

=

76· 2 mm is

aN

=

1· t

This

relatively small bending strength is due to the notch, because the

0.20 ~ - - - - - - - - - - - - - - - - - - - . . . . .

0 10

3 POINT

BEND

SPECIMEN

d=

76.2,

152.4

and 304.8

mm;

b= 38.0

ft= 2.74 MPa; JN= 15F

u

4bd

0.00 ~ ~ l

-0.10

'"":l

~ 0 . 2 0

\..

~ 0 . 3 0

.;;;;..

~ 0 . 4 0

-0.50

-0.60

o

: .

strength criteria

'1 --

,

test data average) 0

calculated da ta

tI . ,

,

,

size effect law ' ,

B= 2.666

do= 33

.644

-0 .

70 __ _ ~ ~ _ ~ _ . - - - - - - r - _ ~ - . - _ . . . . . - - - - _ - - . - - . J

Fig.9

-0.10

0.11

0.31

0.51

0.71

0.90

l f f

log

d/do)

Comparison between calculated and measured failure loads with sire

effect law for the three-point bend specimen.



3

ROLF EUGEHAUSEN ; O ~ K O OZBOLT

~ o ~ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ,

1.8

t 5

~ 1 . 3

: :

~ . 0

.....::

:...

0.8

0.5

3-POINT BEND SPECIMEN

d= 76.2,

152.4 and

304.8

mm

b 38

mm

f t=

2.74

MPa;

(JN=

15F

v

/4bd

Y = A X C

A

= 0.004182

: C =

0.1407

0 . 3 4 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

60 120 180

240 300

360

X

=

d

mm)

Fig. 1 Linear regression analysis of

the

calculated peak loads for tbe

three point bend specimen.

2.40

. . . - - ------------------------------ .

2 .00

1.60

_ 1.20

~

b'

0.80

0.40

3-POINT

BEND SPECIMEN

d= 76.2. 152.4 and 304.8

mm

b=

38

f t= 2.74 MPa; JN=

15F

N

/4bd

o test data (average)

A calculated data

size effect

law

B= 2.666 d

o

33.644

o

A

0.00 - - - - . . . . . - - - - - - . - - - - . . . . , . - - - - - , - - - ~ - - - _ . _ - - - -

50.0

100

.0 50.0 200 .0

250

.0 300. 0 350.0 400.0

d

mm)

Fig. 11 Comparison between calculated and measured failure loads with

size

effect law for

the

three point

bend

specimen. shown

in

normal scale.



SIZE EFFECf IN CONCRETE

STRUcruRES

31

strength related to tbe net area is

C1

N

= : 2· 5ft wbicb agrees witb tbe

value expected for unnotcbed beams.

The sbaded areas in Fig. 8 indicate tbe size of tbe fracture process

zones. As in tbe previous example the relative size

of

the fracture

process zone decreases with increasing member depth. When in

addition the strain and stress distribution over the critical cross-

section

is

analysed one comes to tbe same explanation for the size

effect

as

in tbe case of tbe tension specimen. However the size effect

is

mucb more pronounced tban in notcbed tension specimens because

the size of tbe fracture process zone relative to tbe member depth

is

smaller.

In Fig.

12

tbe nominal bending strengtbs related to the value for

d

=

100 mm

are plotted as a function of tbe member depth. The

numerical values compared are calculated for tbe net member deptb

d.

=

5/6d,

witb predictions according to different proposals valid for

unnotcbed specimens. The test results

by

Heilmann [20] and Malcov

&

Karavaev [21] agree rather weIl. The bending strength decreases from

C1N =2ft for d = 100

mm

to C1N =

1·

ft for d = 1000 mm. According to

tbe CEB Model Code [22] tbe bending strength

is

only C1N = l 5ft for

d =100 mm

but approaches

C1N = ·0ft

for larger specimens. The

3.00.-----------------------------------------

2.50 ,

\

2 .

00

.

o

Heilmann 1969)

Malcov

Karavaev 1968)

CEB MC90 1990)

Size eltec law notched specifnen)

Numencal

results net

area)

b:

1.50

,

-

1.00 +----------------=-=------------------1

0.50 - - . - ~ - , - _ _ r _ _ _ , - - r _ . _ _ - r - _ , _ - r ~ _ , _ - - r _ _ _ r - - - , r - - r _ . . - _ _ r _ i

50 150 250

350 450

550 650 750 850 950

d mm)

Fig.

12

Relative bending strength as a function

of

the member depth.



32

ROLF ELIGEHAUSEN

; O ~ K O

OZBOLT

numerical results for notched specimens agree roughly with the other

predietions; however, when extrapolating them

by

the size effect

law

to larger specimens the nominal bending strength

is

mueh lower than

the centrie tension strength. This is in contradietion to the experimen-

tal results for unnotehed specimens. This is probably due to the

fact

that the size effeet law was adjusted to fit the results

of

notched

specimens. Therefore unnotehed specimens

of

different sizes should be

analysed and the resulting size effeet law should

be

compared with test

results.

According to the CEB Model Code 1990 (MC90) [22], the ultimate

bending moment

of

large specimens

d 5=

1 m) inereases in proportion

to

d

•

In contrast to this, the size effect law and linear elastie fracture

meehanics prediet an inerease

of

M

in proportion to d

1

•

S

• This means

that the failure moment calculated according

to

MC90 might

be

unconservative for large specimens.

xample 3)-The

conerete cone failure load

of

headed anehors

embedded in a large conerete block is studied.

The

geometry of

the

specimen

is

shown in Fig. 2(e).

t is

correlated with the embedment

depth d. The smallest embedment depth

is

d = 50 mm. The distance

between support and anehor is 3d so that an unrestrieted formation of

the failure cone

is

possible. The axisymmetrie finite element mesh,

shown in Fig.

3 (deformed shape), is constant in all analysed cases,

Le. the elements are scaled in proportion to

d.

Contaet between

anehor and conerete in the direction

of

loading exists under the head

of the anehor only. To account for the restraining effeet

of the

embedded anehor, the displaeements

of

the eonerete surfaee along the

anehor in the vicinity

of

the head are fixed in the direetion perpen-

dicular to the load direetion. Except at supports, all other nodes at the

conerete surface are supposed to be free. Mieroplane model para-

meters are taken so that the calculated tension strength

is

approxi-

mately

t =

3 MPa and the uniaxial compression strength

is f =

40 MPa.

The eharacteristie length

of

the nonlocal continuum is taken as

L

= 2

mm. Pulling

out

of

the anehor

is

performed by preseribing

displacements at the bottom

of

the head.

According to Eligehausen Ofbolt [18], the conerete eone failure

load ean be calculated with Bafant's size effeet law

ß

l

o

(2)



~ f

t

~


33

~ ~

Fig. 13 Finite element mesh for the

headed stud specimen, shown in de-

formed shape at peak load.

where

u

represents load at failure including size effect,

N

a failure

load without size effect, and dis embedment depth.

Band

d

o

are again

obtained using linear regression analysis

of

the numerical results (Fig.

14 .

N

the ultimate load with no size effect,

is

caIculated using the

formula

(3)

where c represents the concrete compression strength, a

is

a factor to

calibrate caIculated failure loads with measured values and to ensure

the dimensional correctness of eqn (3). Equation (3)

is

proposed by

ACI 349, Appendix B (1978) [23], for the prediction

of

the concrete

cone failure load.

In Fig.

5

the results

of

the analysis are plotted and compared with

the size effect law (eqn

2».

Tbe coefficient

a in

eqn (3)

is

fixed such



34

ROLF EUGEHAUSEN

t

O ~ K O OZBOLT

7.0....--------------------

6.0

5.0

4

•

O

~

t:

ta. 3.0

-..:.

11

:...

2.0

PULL-OUT

AXISYMMETRIC)

d=

50,

150

and

: ßO

mm; a=

F

N

=

2.9 sqrtU.J

-'1 /z

F

u

=

F

N

B t+d7d

o

J

y= AX C

4=

0.Ot2;

C=

0.52

t.O

0

0.0

- I - - - . - - - . - - - - . - - - r - - - - - - r - - . - - - - r - ~ - _ j

o tOO

200 300 400

500

X = d mmJ

Fig. 4 Linear regression analysis of the calculated peak loads for the headed

stud specimen.

PULL-OUT

(AXISYMMETRICJ

O.tO d= 50, 150 and 450 mm; a= 3d;

Je=

3.0

MPa; J.=

40.0

MPa.

-

-O.tO

-0.50

-

strength criteria

-

-

-

-

size

eJJect

law

F

u

=

F

N

B{t d/doT,jJ

= t 387 ; cl, = 4,3.33

F

N

= 2.90 sqrt J.J er

o calculated

0 . 7 0 ~ . . ~

0.00

0.25

0.50 0.75 t .00

t .25

t.50

log

{d/doJ

Fag 15 Comparison between calculated peak loads

and

size effect law for

the

headed stud specimen.



SIZE EFFECf IN CONCRETE STRUCI1JRES

35

that the numericaJly obtained failure load for anehors with an

embedment depth

d

=50 mrn is predicted correctly.

s

can be seen

from Fig 15, the concrete cone fai/ure loads exhibit a strong size

effeet, because the numerical resuJts are dose to the LEFM solution.

In Fig. 16 the results of the analysis are compared with different

failure load equations. The relative failure loads are shown as a

funetion of the embedment depth. The failure load for an embedment

depth d = 150 mm is taken as the reference value. Plotted are the

relative failure loads according

to

the size effeet law (eqn 2», a

formula that neglects

the

size effeet (eqn

3»,

and a formula derived

on the basis of linear elastie fraeture meehanics (eqn 4» [24]:

(4)

In eqn (4),

al

is a constant and is Young s modulus. The fracture

loads predieted by

eqn

(4) agree

rather

weIl with test results [24].

Assuming no size effect, the failure loads should inerease in proportion

to

d

2

, that means by a faetor of nine, when tripling the embedment

depth. The results of the analysis show that the inerease of the failure

load s mueh less (approximately by a factor of 5·7 . Therefore the size

effeet should be taken into account in the design of anehorages,

otherwise the failure loads are underestimated for smaIl embedment

depths (Fig. 16(a»

and

are

overestimated for large embedment depths

(Fig. 16 b». The agreement between the size effeet formula and the

formula based on linear elastie fraeture meehanics

is

good in the entire

embedment range. This could be expeeted on the basis of Fig. 15. The

size

effect has also been observed in tests by Bode Hanenkamp

[25]

and

by

Eligehausen et

al

[26]. According to these authors, the failure

load inereases in proportion to d

1

•

S

•

The relative shapes

of

the fracture cone for three different embed-

ment depths, estimated from the numerical analysis

at

peak load, are

plotted in Fig. 17.

In

Fig. 18 the distribution

of

the tensile stresses

perpendieular

to

the failure cone surface are shown as a funetion of

the ratio lh/lhmax where

lh

represents the distance from the anehor and

lhmu is the failure cone radius taken from Fig. 17. These distributions

are estimated from the results of the numerical analysis.

From Figs 17 and 18 the size effeet can be explained as folIows. With

inereasing embedment

depth

the ratio of the diameter of the failure

cone to embedment depth decreases, i.e. the effeetive relative cone

surface

area

deereases as weil. Furthermore, the average stress over

the failure surface also decreases with increasing embedment depth



1

1 2 -r-----------------------

1 0

o

no

size

eJJect Jormula

l inear Jracture mechanics

-

LEFM

size eJJect Jormula

-

NLFM

numerical results

/

/

'

/ '

/,

/,

/ , ,

/,,'

/ , ,

/ , '

/ , '

/ , '

/ , '

/

,

/

,

/

,

O . O - - - - - - - - - - . - - - - - - - - - - - r - - - - - - - - - - . - - - - - - - - - ~

3

9

8

17

l

·6

}

, ' -

;:

0

~

~

..;.

~ 4

C§

'3

tS

t:: 2

~

1

60

90 120

EM EDMENT DEPTH mm)

a)

no

size

eJJect

Jormula

l inear Jracture mechanics

-

LEFM

size eJJect

Jormula -

NLFM

numerical

resulls

150

,

,

,

,

/

/

,-

,-

,

, ,

, '

, ' /

, ' /

, , , , ,

, l

V

. . . r ~

° 3 4 0 ~ ~ - - 9 r O - - . - - - - . - - . - - - . - - . - - . - - . - - . - - 3 - 9 r O - - . - ~ 4 5 0

150

21 27

33

EM EDMENT DEPTH mm)

b)

Fig 6

Prediction of the failure loads for the headed stud

spe imen

according to different proposals.



SIZE EFFECI' IN CONCRETE STRUCI1JRES

. 4 , - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ~

:

;::

1 2

I

0 1 ,,--_

ti

~ 0 8

I >

-l

c:

:: 0.6

Q

~

0 4

:.:

::

~ 0 .2

Q

4 0

.0

d=

50

mm

d=

ISO

mm

d=

450

mm

peak

load

d= 50 mm

d=

ISO

mm

d= 450 mm

2 4

0 0 + - - a f 6 . . . - - . - - ~ - - : _ r : _ - 4 - _ _ : : _ c _ - - - - r - -

0 0

0 4

0 8 1 2

RELATIVE

HORIZONTAL CRACK

LENCTH ViA ')

37

F &g. 8 Tensile stress distribution along tbe cone surface at peak load in

axisymmetric pull-out.



38

ROLF EUGEHAUSEN ; O ~ K O OZBOLT

because tbe stress distribution is more triangular as in tbe c se

of a

large embedment deptb and more parabolic in tbe case of

smaller

embedments.

Example

4)-The

sbort beams loaded in torsion Fig.

2 d»

were

tested by BaZant et al [27] using concrete witb maximum aggregate

size da

=

4·8 mm. Tbe deptb

of

tbe smallest specimen

was

38·1

mm.

Tbe finite element mesbes are plotted in Fig.

19.

Tbe

same

mesb

is

used for tbe small and middle-sized specimens 72

finite

elements), wbile for tbe largest specimen tbe number of

finite

elements is increased 176 finite elements). For tbe beam

with

d =

38·1 mm, eight integration points are used in eacb finite

element

wbile in tbe middle-sized and tbe largest specimen

27 integration

points are used. To avoid localization due to concentrated

loads

imposed at tbe beginning and at tbe end of tbe specimen, tbe

first nd

last cross-sections of tbe finite element mesb are supposed to

behave

linear elastically. Tbe cbaracteristic lengtb is taken

as

lc = 15 mm

the

microplane model parameters are cbosen so tbat tbe calculated tension

strengtb is

=

2·60 MPa and tbe uniaxial compression strength is

d= 381 mm

d=

76.2

mm

d= 152 4

mm

YJg

19 Finite element meshes used in the analysis

of

the torsion

specimen.



SIZE

EfFECf IN

CONCRETE STRUCI1JRES

39

k =

43

MPa. Tbe average estimated tension strength in the experi

ments was t = 2·70 MPa. Material model parameters are obtained on

the basis of fitting the average experimental failure load for the

smallest specimen.

In Fig. 20 the nominal torsion stresses at peak load calculated on the

basis of linear elastic theory, O = M

t

/(0·208d

3

) with

Mt

= peak torsion

moment, are compared with the average experimental values and the

size

effect

law

The optimum values for the parameters

Band d

o

are

found

by

linear regression analysis of tbe numerical resuIts (Fig. 21).

Figure 22 represents a similar comparison in nonlogarithmic scale. As

in tbe previous examples, experimentally and numerically obtained

failure loads exhibit significant size effect.

To explain in detail the reason for tbe size effect in tbis complicated

stress-strain state, further studies are required.

In tbe present numerical analysis and tbe tests, tbe concrete

composition

was

constant. However, note tbat in practice tbe maxi

mum

aggregate size is not constant, and tbat for larger structures

coarser aggregates are often used. In this case the size effect sbould be

less

pronounced than that found in tbis study. This can be seen from

0 . 5 ~ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ,

0. 0

TORSION SPECIMEN

d=

38. t, 76.2 and 152.4 n l-

Je 2.60

MPa; a

N

Mt/O.208d:

0.05

- 0 0 0 ~ - - - - - - - - - - ~ - - - - - - - - - - - s ~ t ~ r - e n - - g ~ t h - - c - r t ~ · t ~ e - r t 7 · a - - ~

-0.05

~ - 0 . 0

-0. 5

:. ; -e-- _

2.-0.20

_

t;.

g> 0.25

....

-c J

t;.

test data

(average) -

........

o

calculated data

........

0

.

30

-0.40

.... t;.

----s ize

eJJect la.w

B= 3.474, d

o

31.89 19 ,

-0.35

-0.45

-_- . -_- . -_-- . -_- - . -_- ._-- , -_- - , -_- - , -_- - - ,_- .J

-0.20 -0. 0 -0.00 0.10 0.20 0.30 0.40 0.50 0.60

0.70

log d/rio)

Fig.

20 Comparison between calculated and measured failure loads with size

effect law for the beam loaded in torsion.



40

ROLF EUGEHAUSEN

.

JOSKO OZBOLT

0.75

~ - - - - - - - - - - - - - - - - - - - - -

0.50

0.25

>

TORSION

SPE IMEN

d= 38. I, 76.2 and 152.4

m' }

1,=

2.60

MPa; JN= Mc 0.208d

Y = A X C

A = 0.002598 ; C = 0.08285

0.00

- I - - - - - - r - - - r - - - - . . - - - - - . - - .. - - - - - - - - - r - - - - - - . - - . - - I

30 50

70

90 f fO f30 f50

X = d

(mm)

Fig. 2 Linear regression analysis of the calculated peak loads for the beam

loaded in torsion.

3.60 -r----------------------

3.20

2.80

2.40

TORSION

SPE IMEN

d= 38.1, 76.2 and 152.4 m l7l

1,= 2.60

MPa; JN= Mc 0.208ä

~ 2 0 0

\

b

f.60

A

f.20

A test data

(average)

o calculated data

0.80

size

ellect law

B= 3.474, da= 31.89

0.40

- - - , - - - , - - - - , - - - - - , - - . . , - - - - , - - - - - - - , . - - - - , - - -1

20.0

40.0 60.0

80.0 fOO.O f20.0 f40.0 f60.0 f80.0 200 0

d

(mm)

F tg. 22 Comparison between calculated and measured failure loads with size

effect law for the beam loaded in torsion shown in normal scale.



SIZE

EFFEcr

IN CONCRETE

STRUcruRES

.75

.50

.25

o

Kennedy (1967)

Kani

(1969)

Leonhardt

(1961)

Bazant cl Kazemi (1990)

Taylor - fully scaled

[ 972)

Chana - fully scaled 1981)

. .

...

1.00

t . . . = . . . . . ~ ; ; ; ; : _ I

~ 0 7 5

0.50

o

0.25 - j ---- ;--- -- . --- ---r-- -- . -- --- j

100

200 300

400 500

600 700

800

900

1000

mm)

41

Yag 23

Relative shear strength of beams without shear reinforcement

as

a

function

of

the member depth.

Fig.

23

which shows the relative shear stresses at peak load (shear

failure) of beams without shear reinforcement as a function of the

member depth. The shear strength for slabs with = 250 mm is taken

as

a reference value. In Fig.

23

test results of Leonhardt Walter

[28] Kani [1] and Kennedy [29] and the size effect law, as proposed by

Baiant Kazemi [30], are plotted. In these investigations the

concrete mix was constant. As can be seen, the relative shear strength

decreases significantly with increasing member depth. Taylor

[31]

tested fully scaled specimens that scaled all parameters, incIuding the

aggregate size. The shear strength did not decrease significantly with

increasing specimen size. H6Wever, Chana [32] who also tested fully

scaled specimens found that influence of the member depth on the

shear strength was almost the same

as

in the investigations with

constant concrete mix.

4. CONCLUSIONS

The results of the present numerical study on the behaviour of plain

concrete structures under different loading conditions demonstrate



42

ROLF ELiGEHAUSEN O ~ K O OZBOLT

tbat tbe peak loads exhibit a significant size effect. Therefore, the

increase of tbe failure load is mucb less tban tbe increase of tbe failure

surface area. Tbis is

in

accordance witb experimental evidence.

Similar

results can be expected in otber cases wbere tbe concrete tension

strengtb plays a dominant role, sucb as a bond between defonned

reinforcing bars and concrete, frame corners, puncbing, etc.

Tbe analysis demonstrates tbat tbe microplane material model based

on tbe nonlocal strain concept is capable of correctly predicting the

bebaviour of concrete structures in respect of fracture processes, peak

load and size effect. Since tbe microplane model is a fully 3D material

model it can be effectively used in 2D and 3D finite element codes.

Tbe fact tbat in tbe numerical analysis tbe size effect is calculated

correctly is due to tbe nonlocal strain concept.

Bcdant s size effect law

or

a suitably simplified formula can predict

size effect ratber weil in a small range of dimensions. But to cbeck this

law

in a broader range, tests

of

very large structures are required.

Furtber studies are needed to clarify tbe inftuence of tbe concrete

mix on tbe size effect. Furtbermore, design provisions sbould

e

evaluated, wbicb take tbe practical conditions into account, and

which

sbould be incorporated in codes.

Tbe size effect

in

concrete structures is significant and sbould

e

taken into account in tbe design codes.

REFEREN ES

[IJ Kani, G. N., How safe are our large concrete beams? CI

Journol

Proceedings, 64 (1967) 128-41.

[2J

Bahnt Z.

P. Size effect in blunt fracture: Concrete, rock, metaI. J

Eng.

Mechanics (ASCE), 110(4) (1984) 518-35.

[3J

Taylor, G. 1., Plastic strain in metals. J

Inst. Metals 62 (1983) 307-24.

[4J

Batdorf,

S.

B. Budianski,

B.

A Mathematical Theory

of

Plasticity

Based on the Concept of Slip. NACA TNI871, April,

1949.

[5J Zienkiewicz, O. C. Pande, G. N. Time-dependent multi-laminate

model of rocks-a numerical study

of

deformation and failure of rock

masses.

Int.

J

Num. Anal. Meth. in Geomechanics

1 (1977) 219-47.

[6J Bafant, Z. P. Gambarova, P. G. Crack shear in concrete: crack band

microplane model. J Struc. Eng. (ASCE), 110(10) (1984) 2015-35.

[7J Bafant, Z. P., Microplane model for strain-controlled inelastic be·

haviour. In

Mechanics 0/ Engineering Materials

ed. C. S. Desai R. H.

Gallager. John Wiley Sons, Chichester and New York,

1984,

Chap.

4

pp.

45-59.



SIZE EFFECf IN CONCREfE

STRUcruRES

43

[8] Bafant, Z. P. & Oh B.-H., Microplane model for progressive fracture of

concrete and rock. J. Eng. Mechanics (ASCE), 111(4) (1985) 559-82.

[9] Bafant, Z. P. Prat, P. C. Microplane model for brittle-plastic

material--Parts l nd

11

J. Eng. Mechanics (ASCE), 114(10) (1988)

1672-1702.

[ O}

Baf.ant,

Z.

P. & Pijaudier-Cabot, G. Nonlocal continuum damage,

localization instability and convergence. J. Applied Mechanics (ASME),

55 (1988) 287-93.

[11] Kröner, E. Interrelations between various branches of continuum

mechanics. In

Mech.

of

Generalized Continua,

ed. E. Kröner. Springer,

W Berlin, 1968, pp. 33a-40.

[12] Eringen, A. C. Edelen, D. G.

D.

On nonlocal elasticity. In .

J. Eng.

Sd. 1 (1972) 233-48.

[13]

Krumhansl, J. A. Some considerations of the relations between solid

state physics and generalised continuum mechanics. In Mech.

of

General-

ized Continua,

ed. E. Kröner. Springer, W. Berlin, 1968, pp. 298-331.

[14} Levin, V. M. The relation between mathematical expectation of stress

and strain tensor in elastic micro-heterogeneous media. Prikl. Mat.

Mehk., 3S (1971) 694-701 (in Russian).

[15]

Baf.ant, Z.

P.

Belytschko, T. B. Chang, T. P., Continuum model for

strain softening. J. Eng. Mechanics (ASCE), 110(12) (1984) 1666-92.

[16] Baf.ant, Z. P. Pijaudier-Cabot,

G.

Measurement of characteristic

length of nonlocal continuum. J. Eng. Mechanics (ASCE), 115(4) (1989)

755-67.

[17} Bafant,

Z.

P. O bolt,

J.,

Nonlocal microplane model for fracture,

damage and size effect in structures. Report 89-1O/498n Center for

Concrete and Geomaterials, Northwestem University, Evanston, 1989,

33 pp. Also

J. Eng. Mechanics

(ASCE), (in press).

[18] Eligehausen,

R.

O bolt,

J.

Size effect in anchorage behaviour. Paper

presented at Proceedings of the Eighth European Conference

on

Fracture-Fracture Behaviour and Design of Materials and Structures,

Torino, Italy, 1-5 October 1990

[19] Baf.ant, Z. P. & Pfeiffer,

p

A. Determination of fracture energy from

size effect and brittleness number. ACI

Materials Journal,

84 (1987)

463-80.

[20]

Heilmann,

H.

G. Beziehungen zwischen

Zug-und

Druckfestigkeit des

Betons.

Beton,

2 (1969) 68-72 (in German).

[21]

Malkov, K. Karavaev, A. Abhängigkeit der Festigkeit des Betons auf

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sabemssungen.

Wissenschaftliche Zeitschrift der Technischen Universität

Dresden, 17(6) (1968) 1545-7.

[22] Comite Euro-International du Beton, CEB-FlP Model Code 1990, First

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Bulletin d Information Nos

195

and

196, CEB, Lausanne, March,

1990

[23] ACI 349-76: Code Requirements for Nuclear Safety Related Concrete

Structures. ACI

Journal,

7S (1978) 329-47.

[24] Eligehausen,

R.

Sawade, G. A fracture mechanics based description

of the pull-out behavior

of

headed studs embedded in concrete.

Fracture



44

ROLF ELIGEHAUSEN . O ~ K O OtoO T

Mechanics 0/ Concrete

Structures-RILEM

Report, ed. L. Elfgren.

Chapman and Hall, London, 1989, pp. 281-99.

[25] Bode, H. Hanenkamp, W., Zur Tragfahigkeit von Kopbolzen bei

Zugbeanspruchung. Bauingenier 60 (1985) 361-7 (in German).

[26] Eligehausen, R. Fuchs, W. Mayer, B. Tagverhalten von

Dübelbefestigungen bei Zugbeanspruchung. Betonwerk

+

Fertigteil-

Technik 12 (1987) 826-32, and 1 (1988) 29-35, (in Gennan and

English).

[27] Bafant,

Z. P.

Sener S. Prat,

P.

Size effect test of torsional

failure

1

plain and reinforced concrete beams. Materials and Structures 21 1980)

425-30.

[28] Leonhardt, F. Walter, R. Beiträge zur Behandlung der

Schub-

probleme im Stahlbetonbau, Beton

und

Stahlbeton (Berlin),

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