Size effect in concrete under compression

Arghya Deb

Associate Professor

Department of Civil Engineering

IIT Kharagpur

Concrete

• Composite: cement, sand, coarse aggregate.

• Cement: silicates and aluminate of lime(Ca). In presence of water, acquires adhesive & cohesive properties.

• Water necessary for hydration of cement (~25% by wt.)

• Additional water necessary for workability. Total water content ~ 40%-60% by wt. of cement.

• This water is not chemically bound – can evaporate depending on relative humidity.

• Porosity of concrete due to evaporating free water as well as air voids (~2% of total volume)

Concrete: main damage mechanisms

• In concrete, “weaker planes” occur at the interface of the cement mortar and the aggregate: as a result of bleeding, shrinkage etc.

• The micro cracks that appear at the interface tend to propagate along the aggregate surfaces.

• These micro cracks can combine to form macro cracks.

• In addition there can be “mortar cracks” which run through the matrix material, as well as “aggregate cracks” which tend split apart the aggregates.

Concrete: anisotropy

Casting Direction

Accumulation of weak planes or voids under large aggregate

particles

Concrete: compression behaviour

• The propagation of internal micro-cracks and micro-voids

is reflected in the macroscopic stress-strain behavior of

concrete.

• For instance, under uniaxial compression, growth of micro

cracks aligned to the direction of loading leads to stress

softening.

c

c

Direction of external loading

tensile crack

Concrete: tensile behavior

• Tensile strength of concrete ~ one-tenth compressive

strength

• Under uniaxial tension, propagation of micro cracks

along a plane normal to the loading direction leads to

strain softening behavior.

• The presence of steel reinforcement reduces crack

widths: stiffening effect on the post peak behavior.

t

t

Stress –strain response of Concrete: effect of

confinement

• Under confinement compressive strength of concrete

increases.

• Confining pressure acts to prevent crack propagation

and leads to more ductile response.

c

c

Uniaxial Compression

Biaxial Compression

Triaxial Compression

Size Effect in Concrete

300 mm

150 mm

150 mm

150 mm

Compressive strength of a cube is about 20% higher

than a cylinder made of the same concrete mix.

The LEFM size effect: • Size effect: variation of nominal strength with characteristic

size of a structural member.

• Effect of specimen size on nominal strength in metals

loaded in tension is well known.

• Within elastic range, nominal strength , where d is

the characteristic size of the specimen.

• This is true for self similar specimens.

dn

1

Self Similar Specimens:

Griffith’s energy approach: • The fracture work is given as where

is the newly cracked area

• Energy for fracture work is supplied by the strain energy

: volume of neighbourhood contributing to crack growth

• Using self-similarity and energy balance:

22doGaoGAGW ffff

VE

W n 1

2

1 2

3232 1

2

11

2

1do

Eao

EW nn

dWW nf

1

V

AGW ff fW A

LEFM not generally applicable to concrete:

• Cracks in concrete are very different in geometry from

the idealized sharp cracks of linear elastic fracture

mechanics.

• Crack front is highly irregular, blunted by a zone of

distributed micro cracking that precedes it, known as the

fracture process zone (FPZ).

• In large structures, size of the fracture process zone is

negligible compared to the dimensions of the specimen:

LEFM size effect law can be used to determine the

influence of size on tensile strength of large specimens.

The Real Crack vs. The Fracture Process Zone

Experimental Evidence for size effect in concrete

• Concrete specimens do exhibit size effect in uniaxial

tension and flexural loading.

• If a LEFM based size effect is not applicable, the only

explanation is by means of a statistical size effect.

Z.P. Bazant and J Planas (1997)”Fracture and size effect in concrete and other quasibrittle materials” CRC Press LLC.

The case for a statistical size effect: • Concrete is a heterogeneous material with randomly

distributed voids and flaws.

• Weibull distribution best approximates the tensile

strength distribution in concrete specimens.

• Tensile failure occurs at places where tensile stresses

exceeds tensile strength.

Material structures at nano/micro/meso scale for cement and concrete (Van Mier 2007)

Material strength distribution:

• Weibull distribution function:

• δ: scale parameter, β: shape parameter.

Weibull function for different β at δ=4 (1.25<x<10.25) (Van Mier et al. 2002)

x

ex

xf

1

)(

Stress distribution: • More extensive the region of high tensile stresses,

greater the chance of initiation of tensile fracture.

• If the stress gradient reduces, the region of high tensile

stress increases: this increases chances of tensile

damage & leads to lower structural strength.

• Stress gradients are known to be inversely proportional

to structural size.

• As structure increases in size, stress gradient becomes

smaller: size of region with high tensile stresses increase

The influence of stress gradient:

• Increase in size of highly stressed region results in a

decrease of structural strength with size.

Statistical Size effect: • Concisely: is the number of flaws in specimen

the characteristic specimen size

the gradient in the tensile stress σ

the size of region with

dDN

dDD

ddN

nnn

11,

1

1

1

N

d

D )(9.0 saycr

Asymptotic behaviour at very large sizes: • The reduction of strength with size is not indefinite.

• Beyond a certain size the size vs. structural strength

curve tends to asymptote horizontally.

• Once a structure is sufficiently large, the size of the

region with high tensile stress is sufficiently big.

• It is sure to contain the critical mass of voids and flaws

necessary to precipitate failure.

• For sizes larger than this size, there is no further

reduction in strength.

Statistical vs Energetic Size effect: • There is little doubt that a statistical size effect exists in

concrete. Does it explain the entire size effect observed

in concrete specimens?

• For reasons mentioned earlier – for a long time it was

thought that a fracture mechanics based energetic size

effect could not describe the size effect in concrete.

• Bazant’s work in this area, in particular Bazant’s paper,

Size effect in blunt fracture: Concrete, rock, metal”,

(ASCE, Journ. of Engg. Mech.,110(4),518-535,1984)

however led to a major reconsideration.

Bazant’s Size effect law: • The major innovation was the idea of a ‘crack band’.

• Crack propagation depends on the geometry and stress

state in the FPZ, approximated by a crack band, rather

than the length of the open crack behind the FPZ.

• FPZ is modeled as a band of fixed width, , n is

an integer and is the max. aggregate size.

ac ndw ad

t

Bazant’s Size effect law: • When the material in the crack band can transmit no

further stresses, the crack band becomes part of the

macro crack which further propagates by .

• Work of fracture per unit advance of crack band (fracture energy)

• The strain energy required for fracture has two sources:

strain energy in stress relief zone surrounding the crack

band and strain energy in fracture process zone (FPZ).

c

t

t

ccf

EE

EwG

21

2

a

Bazant’s Size effect law: • Strain energy released is given by

• Enforcing energy balance i.e. , it can be shown that

(*)

2

22

2

2

2

),()(2

1

)(2

1

bdd

aw

d

af

bd

P

E

zonerycontributoofvolumebd

P

EW

c

c

c

bGa

Wf

(**)

a

tn

d

d

B

0

1

constants are and 0B

Asymptotes to LEFM: • If becomes vanishingly small, the crack band

geometry would approach the ideal crack of LEFM.

• Function in Eqn.(*) becomes a function of only and

Eqn(**) becomes

• If size of structure is very large, the term ,

LEFM size dependence is recovered.

cw

fd

a

d

Cn

aa d

d

d

d

00

1

Asymptotes to limit criterion of failure: • On the other hand, if the size of the structure is very

small, then contributes little to the denominator of

Eqn.(**).

• Experimental results have generally agreed with

Bazant’s size effect relation.

ad

d

tn B

Bazant’s Size effect law:

Size effect in compression: • Concrete under compression also exhibits size effect.

• Gonnerman in 1925 was the first to observe size effect in

nominal strength of cylinders and cubes.

• In the absence of definitive knowledge about the size

effect, codes introduce factors of safety that account for

the variation of compressive strength with size.

• Usually a factor of 1.2( BS 1881: Part120; BSI 1983) is

used to convert cylinder strength to cube strength

Effect of loading environment: • Failure mode in concrete specimens under compression

strongly depends on the loading environment.

• Stiff i.e. nearly rigid plattens provide end restraints to

lateral motion through shear stresses

• These shear stresses act to confine the concrete in a

conical zone beneath the loading platten.

Effect of loading environment;

• If a flexible loading platten is used, such that the lateral

strains in the platten are more than in concrete – shear

stresses act outward from the center of the specimen

• Results in tensile stresses in the mortar. If tensile

stresses exceed the local tensile strength: growth of

nearly vertical splitting cracks.

Effect of loading environment;

• These very different loading-platten dependent failure

modes complicate an already murky scenario. The

questions that can be posed therefore are:

• Is there an energetic size effect for compressive loading

with rigid plattens?

• Is there an energetic size effect for compressive loading

with flexible plattens?

Effect of loading environment;

Bazant’s Size effect in Compression:

• Existence of energetic size effect in compression depends

on the configuration of the crack ( Bazant and Xiang

(1997).

• Splitting cracks oriented along loading direction do not

affect axial stress distribution: results in unchanged global

energy release and hence causes no size effect.

• Inclined cracks will alter the axial stress field and will

change global energy release rate: result in size effect.

Bazant’s Size effect in Compression:

Inclined cracks:

• Bazant and Xiang (1997) considered an inclined crack

band, formed by localization of large number of splitting

cracks of length ‘h’ whose growth had been arrested.

• The microslabs between the cracks were considered to

be short columns: global failure occurs when these

columns buckle.

• Because of buckling of the micro-slabs, the stress, and

consequently the strain energy, in the material on both

sides of the crack band is reduced.

Size effect in Compression:

Bazant’s assumption

Bazant’s size effect law in compression: • In the crack band itself, there is accretion in strain

energy due to bending of the micro-slabs, but no further

increase of strain energy due to axial compression.

• The crack band grows due to formation of further

splitting cracks.

• By equating rate of loss of strain energy to the energy

required for crack band growth, the following size

dependence relation was obtained:

5

2dn

Experimental evidence:

• Experimental evidence for a compressive size effect that

follows this law is sketchy.

• Bazant and Kwon(1993)’s experiments on reinforced

concrete prisms seem to support this relation.

• Other experiments, however, e.g. those by Van Mier

(1997) do not show any effect of size on the peak stress.

• Numerical simulations by Dros and Bazant (1989),

Bazant and Ozbolt (1992), Cusatis and Bazant (2006)

have all failed to capture any significant size effect in

concrete specimens under axial compression.

Lack of experimental evidence

No size effect for splitting crack growth:

• In a homogeneous material, splitting crack growth under

unconfined compression is impossible.

• Splitting crack growth in concrete due to tensile cracks

that develop at aggregate interfaces.

• These cracks release the lateral stresses at the interface

but do not affect the primary vertical stresses.

• Cracks are random, their sizes small, they do not affect

the vertical stresses: Bazant & Planas suggest they do

not affect global energy release & cause no size effect.

Crack growth mechanism

Vonk’s experiments: • Experimental study by Vonk(1992) however showed

existence of size effect for splitting crack growth with

flexible plattens

Vertical Splitting Cracks: no size effect at all?

• Bazant’s reasoning for suggesting that splitting crack

growth causes no size effect at all, is not clear.

• Energy reduction due to splitting crack may be local and

random, but it does causes release in stored elastic

strain energy and loss in lateral stress carrying capacity

in the specimen

• Since splitting crack initiation and growth strongly

depends on the existence of inhomogenieites, whose

distribution is random, it seems quite probable that a

statistical size effect would exist.

Numerical simulations:

• Numerical simulations to understand the size effect in

compression were performed using the ABAQUS finite

element code.

• The concrete was modelled using the concrete damage

plasticity constitutive relations (Lubliner et al. (1989)).

• This is a non-associative plasticity model, based on a

modified version of the Drucker Prager yield criterion and

the hyperbolic Drucker Prager flow criterion.

Numerical modeling of concrete:

• It accounts for the different constitutive response of

concrete in tension and compression (fracture dominated

vs. plasticity dominated)

• Accounts also for the pressure sensitivity of the

constitutive response.

• Damaged states in tension and compression are

characterized by different equivalent plastic strains in

tension and compression.

Modified Drucker Prager yield criterion:

• The yield criterion is written as

Effective stress tensor

Hydrostatic component

Deviatoric component

Mises stress in terms of the effective stress

• α is a material constant depending on the ratio of the

yield stresses in equi-biaxial and uniaxial compression

)())(3(1

1),( II

pc

pppqF εεεσ

)1( d σσ

)(3

1 σtracep σIS p

S:S2

3q

Modified Drucker Prager yield criterion:

• is a material constant depending on the ratio of

evaluated at yield in triaxial tension and triaxial

compression

• depends on the ratio of the cohesive stress in

compression and tension.

q

p

Effect of Confinement

• External confinement of concrete increases strength

and ductility significantly

• In elastic zone, the behavior of confined and unconfined

columns is very similar. On plastification small stress

increment causes large radial expansion

• Confining pressure minimizes the expansion and growth

of tensile cracks in concrete & results in higher failure

loads

• Does confining pressure affect the size-dependent

response of concrete specimens?

Confinement by FRP wrap

concrete

epoxy

fabric

Model geometry, loading and

boundary conditions

Axial load

Dia=0.15m

h=0.3m

Fiber reinforced polymers (FRP)

• Fiber reinforced polymer (FRP) composites are typically

made of fibers such as glass, aramid, and carbon

embedded in a polyester or vinyl ester resin matrix.

• The strength of FRP depends on the elastic properties of

the fiber and matrix, their relative volumes, and length

and orientation of fibers within matrix.

• GFRP: Made of glass which is not as strong as carbon

fiber. It is much cheaper and significantly less brittle.

• CFRP: Made of carbon fibers about 5-10μm in diameter.

It has high modulus, high tensile strength.

• AFRP: Made of aramid fiber.

•FRP composites are corrosion resistant, lightweight, and

have high strength.

•FRPs are commonly used in aerospace, automotive,

marine, and construction industries.

•Applications include the construction of FRP bridge deck

systems, concrete decks with reinforcing FRP rebar, and

the strengthening and repair of existing structures.

Property ranges for different type of FRPs

Type of FRP Ultimate

tensile stress

(Mpa)

Elastic

modulus

(Gpa)

Fiber

content

(% by wt)

Thickness

(mm)

GFRP laminate 400-1800 20-55 50-80 0.4-2.0

CFRP Laminate 1200-2250 120-250 65-75 0.1-1.9

AFRP laminate 1000-1800 40-125 60-70 0.025-0.29

Preparation of FRP wrapped specimens

Wrapped specimens after testing

Effect of confinement: a LEFM based explanation

• If wrap stiffness above a certain threshold value, the

confining pressure activated will prevent localized failure

and result in strength gain.

• Similar results from LEFM: Sato and Hashida (2006)

studied the fracture toughness of rocks deep inside the

earth, and therefore subjected to large confining

pressures.

• They report that the apparent fracture toughness for rocks

increases substantially due to increased confining

pressures.

Effect of confinement: a LEFM based explanation

• The following relationship was proposed for the fracture

toughness and the confining pressure if the cohesive

stress ( ) acting in the fracture process zone was

uniform:

is the fracture toughness of the unconfined rock

is the fracture toughness after confinement

is the magnitude of the confining pressure

cc

p p

K

K

0

2

1

c

cK

pK

0p

Summary

• Size effect in tension: closely resembles Bazant’s law.

• Size effect in compression: combination of energetic &

statistical effects

• Bazant’s model in compression: overpredicts size

effect for inclined crack band, ignores statistical size

effect for splitting crack growth.

• Numerical studies on unconfined and confined

concrete confirm these conclusions

Reference • R. Dandapat, A. Deb and S. K. Bhattacharyya (2012 Lo alized failure in FRP

wrapped cylindrical olu s , ACI Structural Journal, 109 (4), 445-456

• Z. P. Baža t and J. Planas (1998 F a tu e and Size effect in concrete and other

quasibrittle ate ials , CRC Press LLC.

• J.G.M. Van Mier, (1997 F a tu e Processes of concrete: assessment of material

parameters for fracture odels CRC Press Inc.

• J.G.M. Van Mier, (2007 Multi-scale interaction potentials (F-r) for describing

fracture of brittle disordered materials like cement and o ete , International

Journal of Fracture. 143(1), 41-78.

• J.G.M. Va Mie et al. Fracture mechanisms in particle composites: statistical

aspects in lattice type analysis , Me ha i s of ate ials. , -724.

• ). P. Baža t “ize effe t i lu t f a tu e: Co ete, o k, etal. Jou al of Engineering Mechanics, ASCE,110(4), 518-535.

• BS-1881 Part-120: Testing concrete method for determination of the compressive

strength of concrete cores, British Standards Institute, 1983.

• M.D. Kotsovos Effe t of testi g te h i ues o the post ulti ate ehaviou of o ete i o p essio . Mate ials a d “t u tu es, RILEM, , -12.

• Z. P. Baža t a d Y. Xia g “ize effe t i o p essio f a tu e: “plitti g a k a d p opagatio , Jou al of E gi ee i g Me ha i s, A“CE, .

• R. Vo k “ofte i g of o ete loaded i o p essio , PhD thesis, Ei dhove University of Technology, Eindhoven.

Reference • Z.P. Baža t and Y.W. Kwon(1994 Failu e of sle de a d sto ky ei fo ed o ete

olu s: tests of size effe t , Mate ials a d “t u tu es,27,79-90.

• P. Droz and Z.P. Baža t(1989 No lo al a alysis of sta le states a d sta le paths of p opagatio of da age shea a d . I C a ki g a d Da age, “t ai Lo alizatio a d “ize effect, J. Mazars and Z.P. Bazant, eds, Elsevier Applied Science, London.183-207.

• Z.P. Baža t and J.Ozbolt(1992 Co p essio failu e of uasi-brittle material: Nonlocal

i opla e odel Jou al of E gi ee i g Me ha i s, A“CE,118(3).2484-2504.

• J. Lubliner et al.(1989 A plasti da age odel fo o ete I te atio al Jou al of solids and structures,25(3) 299-326.

• Y. Xiao and H. Wu(2000 Co p essive ehavio of o ete o fi ed y a o fi e o posite ja kets , Jou al of Mate ials i Civil E gi ee i g, A“CE,12(2),139-146.

• J.G.M. Van Mier, (1984 “t ai Softening of Concrete Under Multiaxial Loading

Co ditio s , PhD Thesis, Eindhoven University of Technology, Netherlands.

• H.G. Gonnerman (1925 Effe t of size and shape of test specimen on compressive

strength of concrete. Proc., ASTM, West Conshohocken, PA, 237-250.

• Cusatis, G. and Z.P. Baža t(2006 “ize effect on compression fracture of concrete with or

without V-notches: a numerical meso-mechanical study , Computational Moedelling of

Concrete Structures-Meschke,de Borst, Mang & Bicanic (eds), Taylor & Francis Group,

London.

• K. Sato and T. Hashida (2006 Cohesive crack analysis of toughness increase due to

confining p essu e Pure and Applied Geophysics,163,1059-1072.