Page 1

Fracture Mechanics of Concrete Structures,

Proceedings FRAMCOS-2, edited by Folker H. Wittmann,

AEDIFICATIO Publishers, D-79104 Freiburg (1995)

EXPERIMENTAL OBSERVATIONS OF CONCRETE

HAVIOR UNDER UNIAXIAL COMPRESSION

Y-H. Lee, K. Willam, and

Kang,

CEAE-Department, University of Colorado, Boulder, Colorado, USA

Abstract

This paper presents novel experimental observations of

post-peak response phenomena when cylindrical concrete

are loaded under deformation control in uniaxial compression.

focus are degradation measurements of stiffness and strength ~~.~~~~.,.....

unloading and reloading cycles considering the effects of

conditions on cylindrical specimens of different height.

tence of a unique focal point (pole) of stiffness degradation ex-

plored for the description of elastic concrete degradation.

dominantly axial splitting failure is interpreted on the meso-level

order to overcome the shortcomings of macroscopic failure '-A.'V..._,.__, .... _..

tions.

1 Introduction

Tensile fracture in cementitious materials such as mortar and con-

397

crete is normally studied in the form of mode I fracture experiments

and in the form of direct and indirect tension tests. On the other

hand well-posed mixed mode fracture studies are far scarcer, quite

opposite to the standard compression test which is widely used for

the characterization of mechanical concrete properties in spite of its

index character. fact, little effort has been expended to correlate

the failure processes in tension and compression in order to extract

more fundamental properties from post-peak measurements of stiff-

ness and strength in uniaxial compression. The lack of quantitative

experimental data has hampered the development of coupled elastic

plastic damage formulations for concrete if one considers the date of

the classic load-unload-reload compression tests by Sinha, Gerstle

and Tulin (1964) at the University of Colorado, Boulder.

It is widely recognized (Van Mier 1984 and 1986, Vonk 1992,

and Rokugo and Koyanagi 1992) that the degradation of strength

and stiffness of concrete in uniaxial compression is accompanied by

highly localized

in the form of axial splitting. In contrast to

the conical mode shear failure, compressive splitting must resort

to subtle explanations in terms of (a) fracture mechanics arguments

considering initial microdefects (Nemat-Nasser and Hori, 1993), (b)

mesomechanical arguments which account for the heterogeneity of

concrete, and ( c) boundary effects which induce tension along the

line of the Brazilian test configuration. Slate and Hover (1984)

showed pervasive internal crack growth up to peak by studying the

interior of concrete specimens which were loaded up to a certain

level and which were subsequently unloaded. From their experimen-

tal observations it is believed that energy dissipation in the pre-peak

regime is a global continuum-dominated process which may be at-

tributed to microcracking throughout the entire specimen. On the

other hand, energy dissipation in the post-peak regime is a localized

surface dominated fracture process after coalescence of macrocracks

in the peak regime. short, concrete failure in direct compression

is a very complex process which entails tensile debonding due to

mismatch in the aggregate-cement paste composite, as well as shear

faulting in the form of crack bridging. Thereby the fundamental

question is whether compressive failure is really another manifesta-

tion of tensile cracking, or an independent failure process of mixed

398

mode shear. The fracture argument is here addressed via the charac-

teristic length argument which is observed from tests on specimens

with different height.

Another issue in this paper is the identification of a unique

point of stiffness degradation. This locus is determined using ex-

perimental observations between stiffness and strength, and between

stiffness and permanent deformation. Through regression analysis

of experimental data a secant relation is developed which paves

way to combine elastic degradation and plastic softening.

On a final note the axial splitting failure mode observed the

experiment is discussed in terms of the fracture mechanics of

tial microdefects and microstructural argument which account for

the heterogenity of concrete.

this conjunction, shortcomings

macroscopic failure descriptions are contrasted with failure modes

observed in experiments.

2 Experimental setup

2.1 Testing equipment

All experiments were carried out with a general purpose MTS com-

pression and tension apparatus comprised of a standard 110 kips

( 489 kN) loading frame and function generator units. The averaged

axial deformation measurements of two transducers which were at-

tached between upper and lower loading platens were used as feed-

back signal for servo-control (Fig. l(a)). The transducers were 200

hrdc LVDT with ±0.2 inch nominal linear range and 15 V /inch

sensitivity. Lateral deformations were measured by four transduc-

ers attached at midheight of the specimen at 90° intervals around

the circumference (Fig. 1 (b)). Four 100 hrdc LVDT with ±0 .1

nominal linear range and 54 V /inch sensitivity were used for this

purpose. Data for all experiments were monitored and stored by

a data acquisition system developed at the University of Colorado

Structural Research Laboratory.

2.2 Test specimens

The tests were performed on cylindrical specimens of d == 3.0 in

diameter and h ==5.4, 3.6, 1.8 in height. Normal strength and

high strength concrete specimens were cast using two different

399

axial LVDT

(a)

upper loading platen

aluminum circular plate

base aluminum plate

lower loading platen

lateral L VDT

tt'c.

lateral L VDT

steel plate

NC.

steel plate

HC.

base aluminum plate I d = 3 in

I

(b)

Figure 1: Test Set-up in Direct Compression Test (a): Axial

transducers (b): Lateral transducers

proportions. The mix proportions resorted to a water-cement ra-

tio of W / C == 0. 65 and W / C == 0 .4 for normal strength and high

strength concrete, a mix of C:S:G == 1:2.63:2.14 and 1:1.28:1.31 for

normal and high strength concrete, and a maximum aggregate size

of d == 3/8 in in both cases. ASTM C150 Type I Portland Cement

was used. The cylinders were cast in steel molds and consolidated

by hand tamping. The molds were removed approximately 24 hours

after casting. The cylinders were then cured in a fog room until test-

ing. The specimens, having an initial length of H == 6 in, were cut to

their nominal lengths using a concrete saw. The ends of each spec-

imen were milled with a diamond grinding wheel to the prescribed

height with a tolerance of l::l.H == ±0.02 in.

2.3 Boundary conditions

boundary condition in the uniaxial compression test depends

primarily on the type of loading platen and the specific interface

between specimen and loading platen. In the experiments fixed

loading platens were used to stabilize the post-peak behavior. A

very elaborate triaxial compression test set-up was developed by van

Mier ( 1986) and by Vonk ( 1992) using steel brushes to minimize the

interface friction. In the current compression test a special provision

400

was adopted to minimize the interface friction which was developed

by Slowik et al. (1993) for direct tension tests.

their experi-

ments the normal strength test cylinder was attached to two high

strength concrete specimens of the same diameter. Since the elastic

properties of the two concretes do not differ significantly, the lateral

restraint at the end surfaces of the test specimen is minimized. Sim-

ilarly to that tension set-up, the short test specimens (h == 1.8 in)

were placed between two high strength concrete cylinders of 1.8

height lubricating instead of gluing the platen interfaces (Fig. 1 (a)).

the taller specimens (h == 5.4 and 3.6 in) the platen-specimen

interfaces were prepared by lubrication with "grease". The peak

strengths of the three specimen geometries are summarized in Ta-

ble 1 which indicates negligible difference of axial strength values.

Without the special provision in Fig. 1, the h == 1.8 in specimen

yielded f~ == 5. 75 ksi, which is 30% above the other values. In order

to extract the deformation on the actual concrete specimen from

the total deformation, a high strength specimen of

in height

was tested up to 4.5 ksi under cyclic loading prior the real test.

The loading and unloading curves were approximated by 5th and 5th

order polynomials through regression analysis.

calibrations

were used to extract deformations of the normal strength specimen

from axial LVDT readings depending on the loading condition.

Table 1: Peak strength

local fracture energies

Stiffnesses (ksi) corresponding

Specimen

Peak

to stress levels at 4 ksi

height

strength

(pre-peak) and peak strength (ksi)

(inch)

(ksi)

4.0 (pre-peak)

peak strength

unloading reloading unloading reloading

5.4

4.45

5200

3850

5200

2940

3.6

4.45

5130

3700

5130

2940

1.8

4.65

5080

3800

4900

2720

3 Experimental observations

3 .1 Fracture energy effect

Local fracture

energy

(kip· in/in2

)

unloading reloadir

69.66

78.98

67.53

71.77

83.11

84.30

The post-peak behavior of concrete under uniaxial compression

a surface- dominated fracture process due to excessive lateral de-

formation which results in axial splitting. If the lateral restraint

at the platen interface is minimized, this phenomenon is similar

401

to Mode I type tensile cracking in uniaxial tension except for the

axial compression.

view of the constant fracture energy release

argument in uniaxial tension, the question arises whether frac-

is preserved in compression irrespective of specimen

This question studied with experiments on three spec-

heights, h == 5.4, 3.6, and 1.8 in (1.8, 1.2, and 0.6 aspect

with the same cross-section ( d==3 in). During the test a con-

deformation rate of l.67x 10-5 inch/sec was applied in all the

tests irrespectively of specimen height. Table 1 shows that the peak

is independent of the specimen height due to the special

in Fig. 1 for

h == 1.8 in specimen. Fig. 2 illustrates the

.............. L .... ...., ... stress versus axial and lateral strain response of the

~~-·~~~~~~ hei hts.

indicates close a reement of stiffness u to

4.0

3.0

2.0

.o

5 _4 inch specimen

-

3 _6 in.ch specimen

h --1_8 inch specimen

I !

.n n· . '!

!""

1,! ~

I

I ,,

I

1,1..

. "r""""v,,

)./

""~ ..

ii

{~ .... f..¥

{/

/

25.0

2: Nominal Stress vs. Axial

Lateral Strains

strength when interface friction is minimized, while the post-

peak behavior diverges for the three specimen heights. The h ==

1.8 specimen failed

axial splitting within concentric rings of

cylindrical specimens intersected orthogonally by radial cracks.

the

splitting mode of

short test specimen

the high

concrete caps at both ends. Fig. 3

nominal stress versus axial

lateral deformation

contrast to Fig. 2 the agreement of the post-peak

supports the argument of fracture energy irrespectively of

specimen height. The local fracture energy values are listed in

1 which were extracted from the area in the post-peak regime

402

depending on the loading

(see Fig. 4).

The lateral deformations

an another key to

constant fracture energy

rate in the post-peak

as a

function of specimen height.

5 illustrates lateral versus ax-

ial deformations and lateral versus axial strains, where

strain values were

dividing averaged values

-

5 _4 i::nch specimen

- - -

3 _6 i:n.c:h specimen

-

- -

1_8 inch specimen

LO

Figure 3: Nominal

eral deformation measurements. Close scrutiny points

ratio between lateral

axial deformations which is ~~~~L~~~

than the elastic Poisson's

and which may be called a

tious Poisson's ratio".

that the axial '-"-'--'.l.V' ............ u"J-•V'.L.LU

proportional to the lateral

irrespective

height. This leads an

observation

pa ti on measured along

is transferred

lateral crack expansion at

fracture energy

is interesting to note

between

lateral

deformations is about uz/ Ua = for

specimen heights

in accordance of factor 8 to 12 between the uniaxial ron.in---.-r"""'" 001

and the tensile strength. Based on this observation

stresses induce lateral

cracking under traction-free

boundary condition which may explained by microstructural ar-

guments of cement paste-aggregate composites. For more .............. , .... i-._.. .... ..,

we consider the response the 5.4 in height specimen

403

continuum (or glohal)

/

fracture energy

local fracture energy

Axial deformation (in)

Figure 4: Continuum (Global) and Local Fracture Energies

ratio between lateral and axial deformation remains constant

starting at Ua = -15 x 10-3 in axial compaction, which approxi-

mately corresponds to the inflection point in the post-peak regime

400.0

350.0

300.0

--

250.0

-~ 200.0

~

~ 150.0

100.0

50.0

0.0

--\:

~~ =--= ::: :::: ::::::::

\ \'\ -

-

1.8 inch specimen

'

,\,

'$-i.\...

·,

i

\

'I

5o.o

40_.o

30.0

_ 20.0

;o.o

Ax1al deformauon (*-10 10)

0.0

~~~~---~~,~~

120.0

i

/

.;-El

I

i

/

I

,·

/)

/

"

A

/

--1·

/

/

,;

/

f

I

/1

,,. ;

/

;

i

;

5.0

10.0

15.0

20.0

Axial strain (*-1 o')

100.0

80.0

60.0

40.0

20.0

0.0

25.0

Figure 5: Lateral vs. Axial Deformation and Lateral vs. Axial

Strain Response

when a a = -3.3 ksi. If we recall that localized cracking is formed

peak strength, energy dissipation this range is the combined

of axial splitting and plastic shear dissipation. This is why

the slope between lateral and axial deformations shows the extreme

transition of v = 0.18 in the pre-peak and Vdef = 12 in the post-peak

regime. When passing through a a = -3.3 ksi, the fracture process

results in purely lateral expansion of vertical cracking very similar

the tensile test. If we assume that the plastic energy dissipation

due to axial splitting is much larger than that due to shear fault-

a constant characteristic length corresponding to the specimen

404

height averages the latter part of the softening regime.

3.2 Stiffness degradation

3.2.1 Pre-peak regime

Unloading and reloading stiffnesses were measured at the nominal

stress level of aa == 2.0, 3.0, 3.5, and 4.0 ksi for the three specimen

heights. Up to 85% of peak strength all stiffness showed the same

properties shown in Table 1. Table 1 indicates that the reloading

stiffness at peak level is reduced to 76% which is the average of three

different specimens, while unloading stiffnesses show little changes.

According to Slate and Hover ( 1984) the mortar cracks are bridged

between bond cracks at 70% to 90% of peak strength. Subsequently

cracks coalesce as the stress is further increased, elastic damage in

terms of unloading stiffness is negligible, while elastic damage

terms of reloading stiffness is significant due to progressive microc-

rack formation.

3.2.2 Post-peak regime

degradation properties in the post-peak regime are described

terms of stiffness-strength, stiffness-fracture energy, and stiffness-

plastic deformation.

(a) Stiffness versus Strength: The stiffness degradation during un-

loading and reloading is shown in Fig. 6( a) for the three specimen

heights plotting normalized stiffness (Ed/ Eo) versus normalized ax-

ial strength ( ac/ JD respectively, where Eo refers to the stiffnesses

at peak. The linear degradation relationship between normalized

stiffness and the corresponding strength values is striking.

(b) Stiffness versus Fracture Energy: The normalized fracture en-

ergy ( Gj / Gj max) values are shown in Fig. 6(b) which depicts lin-

earity between Gj and the normalized stiffness. Thereby the local

fracture energy was evaluated in terms of the areas under the axial

stress-deformation curves between unloading (reloading) stiffness at

peak strength and that at unloading. Here G

1

c

denotes the total

max

fracture energy of the entire softening regime.

( c) Stiffness versus Plastic Deformation: Finally, the stiffness degra-

dation is plotted versus normalized permanent deformation Fig.

6( c). The permanent deformation at the unloading point was nor-

405

malized by the total permanent deformation at the last loading cy-

cle, while all permanent deformations were zeroed at peak strength.

This data reduction led to a hyperbolic relationship between stiff-

ness and plastic deformation which is single-valued for all three

specimen heights.

1.0

<>

0.8 -

g*

~

<>.lfl

~ *

~ 0.6 -

t5

*

.£;!

~

~

0

0.4 -

@

o+

:.a

<!;"" *

~

-.#~@ *

0.2

0.0

0.0

0.8

~

EB-

~ 0.6

s .£;!

~ 0.4

§

z

0.2

0.2

0.4

0.6

0.8

Normalized axial stress (a_,,./f.)

(a)

D

-t> ~ D

*

+ <>

*

-

LO

0.0 .__~_,__~~~__._~-~~

0.0

0.2

0.4

0.6

0.8

1.0

Normalized permanent deformation (u/'""'!uP'0

ta1)

~

~

s .£;!

~

:.a

~

LO

<>

0.8

~

-g

0.6

*6*

0

<>-El

6 D

0

0.4

<:±>Q:"l.D

*

+<>

* ~

0.2

0.0

0.0

0.2

0.4

0.6

0.8

LO

Normalized local fracture energy (G,c/G,cm~)

(b)

o 5.4 inch specimen (unloading)

o 3.6 inch specimen (unloading)

<> 1.8 inch specimen (unloading)

6 5.4 inch specimen (reloading)

+ 3.6 inch specimen (reloading)

* 1.8 inch specimen (reloading)

Figure 6: Compressive Concrete Response: (a) Stiffness versus

strength, (b) Stiffness versus Fracture Energy, and

( c) Stiffness versus Permanent Deformation

Focal point

The issue, whether there exists a unique focal (pole) point of stiff-

ness degradation is essential for a coherent description of elastic

degradation in concrete. A simple analytical procedure is proposed

based on experimental observation to determine the location of the

focal point, see Fig. 7. the plot Epl denotes the permanent strain

at peak strength, Ep2 the plastic strain of the peak strength to the

current stress a, Ee the elastic recovery strain, E the total strain,

E( E) the degrading secant modulus of elasticity, Emax the modulus

of elasticity at peak strength, and a max the peak strength. If we as-

sume existence of a focal point in the stress-strain plane, the secant

406

stress-strain relation may be expressed as follows:

C5 -

e5o = E(c)(E - Eo)

(1)

where E = Ee + Epl + Ep2, and Epl =constant. In order to find

unknown initial strain and stress values, Eo and e5o, in Eqn. (1) one

more equation is required. From the initial condition of C5-E diagram

marked as line (a) in Fig. 7, the following expression is derived

(2)

where Ep2 = 0. The values of Ee and Ep2 are determined from

experimental observations in Fig. 6. The linear relation of stiffness

and strength in Fig. 6(a) may be cast into

E

C5

-= a -

(3)

Emax

C5max

where a defines the slope between E~ax and u:ax. The relation

Figure 7: Illustration of a Focal Point

tween stiffness and plastic deformation may be approximated by the

hyperbolic relation in Fig. 6( c):

E

c

(4)

E

-~+'

max

f max

c

p

where c is a parameter which is calibrated from experiments. From

the geometry of triangle (2) in Fig. 7 the secant stress-strain rela-

tion yields Ee = !fr. Combining Eqn. (3) and the hyperbolic formu-

lation in Eqn. ( 4) and the geometry of Fig. (7), Ee and Ep2 may

407

U,.LU.U,lJ'l_;'Ul as follows:

1 amax

Ee==---,

aEmax

Ep2 ==

_

max(Emax

l)

-CEp

- - -

E

(5)

is interesting to note that a == 1 based on

(3) and (5) which

agrees well with the plot Fig. 6( a) due to the linear relationship

between stiffness and strength degradation.

determination of

focal point reduces the solution of

two

(1)

(2) with respect to Eo and ao. Since amax

EmaxEe and from the geometry in Fig. 7,

(

Emax

)

Emax

)

( )

EQ ~ - 1 == Epl

- 1 - Ep2

6

It is noted in

( 6) that Ep2 has to be a function of ( E]lx

1)

cancel on both sides, otherwise a unique focal point can

not be found. Because of

Ep2 was characterized by hyperbolic

expression in Eqn. ( 4), which leads to the initial strain

-

0

EQ -

Ep

max

(7)

where

leads

0

== Epi,

constant. Substituting Eqn. (5) and (8) into (2)

the initial stress

E

max

ao == -C maxEp

(8)

Substituting Eqns. (7) and (8) into (1 ), the generalized secant for-

mulation for a is

(9)

where Ed(E) == E(E). Furthermore, the permanent strain Ep and

the degradation modulus elasticity Ed( E) can be expressed using

Eqns. (3) and (9) as follows:

(10)

For verification

proposed procedure to determine a unique focal

point Eqns. (7), (8), (9), and (10) are used to predict the behavior

of the h == 5.4 specimen. From regression analysis of three spec-

imen heights the mean value of c is known to be 0.261 and 0.287

for reloading and unloading stiffnesses, respectively, and c == 0.286

for reloading in the case of the h == 5.4 in specimen. The per-

manent strain Ep

0

== -0.66 x io-3

, the maximum permanent strain

408

7_0

o_o

o_o

_5_0

-14_0

Figure 8:

LO

Focal

oint

Stress-strain curve

Degraded stiffness

-1 o_o

-6-0

-2_0

Axial strain (* 103

)

(a)

2.0

-LO

-2_0

-4_0

-5_0

Proposed formula

Degraded stiffness

Experiment

-10_0

-8_0

-6-0

-4_0

-2_0

Axial strain (* 103

)

(b)

o_o

Comparison of Focal Point with Experimental Result:

(a) Focal Point of 5.4 Inch Specimen and (b) Compar-

ison of Focal Point with 5.4 inch Specimen

0 5 .4 inch specimen

CJ 3.6 inch specimen

* 1.8 inch specimen

- -

Proposed formula

inch specimen.

inch specimen

* l .8 inch specimen

- -

Proposed formula

o_s

Normalized axial stress <==u./f.,)

(a)

o_s

=

o_o

o_o

0_2

0_4

o_6

o_s

LO

Normalized permanent deformation (EPra.:a.i/eP104a1)

(b)

Figure 9: Experimental Result Predicted with the Proposed

Formula for Focal Point: (a) Prediction of Fig. 6

(a) and (b) Prediction of Fig. 6( c)

409

4

of elasticity measured at the

the

a max == -4.45

results of the

stress-strain ~.L~,,.....~ ,~~.L~ and

IJUJ.L . .LUV'.L.L with

experimental

.....,....., ..... .L ...... r-, is a common failure mode of many brit-

compressive loadings, microde-

u.. ............ ....,.LJI.'-./...,, inclusions, and

fail

tensile

large confining pressure, brittle materials

may undergo plastic flow accompa-

the plastic deformation should be the

they

not even

small

of con-

demonstrate that

caused by nucleation at

grow in

compress10n.

order

splitting

compression, the

crack

initially

and Bombolakis (1963),

later quantified analytically and confirmed experimentally by

Nemat-Nasser and Hori (1982), is adopted. The model assumes a

410

a =O

22

(a-1)

(a-2)

-r,.., vs. eo

y=45', Tj=0.45, r=IO.s

Incipient Cracking Angle. 9 0

Figure 10: ( a-1) Preexisting

and straight ~~~·~i~~

P

1

Q , ( a-2) A representative tension crack

splitting forces (b )Tr{) VS.

/ ==

preexisting flaw with

r -1- 0

cracks under biaxial compression. However since

culations are very laborious curved cracks,

were

with equivalent straight cracks; Hori and Nemat-Nasser

shown in Fig. lO(a).

A 2-d elasticity boundary-value problem associated

shown in Fig. lO(a-1), was formulated in terms

gral equations and solved

see N emat-N asser

(1982). The boundary conditions on the flaw PP

1

are Uy+==

== Txy - == -Tc+ rya y, where Tc denotes the cohesive stress, 'I]

tional coefficient in a range of 0.3-0.6, Uy the displacement

y-direction, a y the normal stress, and Txy the shear stress on

Under uniaxial compression, if

amount of

applied, then the crack grows

crack extension is attained .

......... ,'-' ....... u...., ... micromechanism of

brittle specimen.

In Fig. lO(a-2), the force F

flaw on cracks PQ and P

1 Q1

•

crack

a

representing cracks at the flaw tips, which are

caused by a frictional motion of the preexisting

to farfiek compressive stresses a. The force Fis

411

Decision of Incipient Cracking Angle 0 0

based on Initial Crack Angle 0

Initial Kink Crack Angle, e

Figure 11:

driving

*

1 . 2

T = - -a- Sln r'\/ -

T

2

I

C

2cr* sine

I<1 = ---;===

1

2cr* sin ()

1

TrB = J2IT[{ \/7r(l + l*) +

+

l

2

angle,

with the kink crack

lO(b)

1=45°.

seen

the initial kink

412

Crack Propagation 0 0 from the Kink Crack

based on inclination of flaw

(6=60)

- --- -

r-30

----- r-45

----~ r-60

incipient cracking angle,

2cT* and the

Q1

are

(11)

-2cr* cos e 1 -Jri , 2(()

)

- '2 7r crsm

-1

()0

3

]

cos 2 + 4 cos 2) = 0

com press10n,

pressive loading, see

based on graphs

values. Assuming

crack angle () == 60°,

Fig. lO(b) or 11( a).

()

0

(

2

) is attained as

processes, the

ing. In this ~~~~,~~~~~~

is attained in

interpretation.

5

trol on

specimens.

fracture energy

issue of

Acknowledgements

The first two authors wish to thank

experiments.

AFOSR under

Boulder.

413

References

Brace, W.F. and Bombolakis, E.G. (1963) A note on brittle crack

growth in compression..J. Geophys. Res., 68, 3709-3731.

Hori, H. and Nemat-Nasser, S. (1986) Brittle failure in compression:

splitting, faulting, and brittle-ductile transition. Phil. Trans.

Roy. Soc. Lond., 319, 1549, 337-374.

Nemat-Nasser, S. and Hori, M. (1993) Micromechanics: Overall

Properties of Heterogeneous Materials.North-Holland.

Rokugo, K. and Koyanagi, W. (1992) Role of compressive fracture

energy of concrete on the failure behavior of reinforced concrete

beams, in Applications of Fracture Mechanics to Reinfor-

ced Concrete (ed. A. Carpinteri), Elsevier Applied Science,

London, 437-464.

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