2D Fibre Reinforced Concrete (FRC) elements:

from material to structural design

Paolo Martinelli 📧 paolo.martinelli@polimi.it

OUTLINE

• INTRODUCTION

• FRC CLASSIFICATION

• NOVEL INDIRECT TENSILE TESTS

• THE REDISTRIBUTION FACTOR

• REFERENCE TETS

• RESULTS AND DISCUSSION

• AN EXAMPLE OF FRC FLAT SLAB REALIZED IN ITALY

• CONCLUSIONS

FRC MATERIAL

(1) Fibre-Reinforced Concrete (FRC) is a composite material characterized by a cement matrix and discrete fibres (discontinuous). The matrix is made of either concrete or mortar. Fibres can be made of steel, polymers, carbon, glass or natural materials.

STRUCTURAL AND OTHER USES OF FRC IN MODEL CODE 2010

(2) Fibre materials with a Young’s Modulus which is significantly affected by time and/or thermo-hygrometrical phenomenon are not covered in Model Code for the structural use. (3) Mixtures of different types and/or sizes of fibres can also be used (called hybrid fibre reinforced concrete). (4) FRC for structural applications means the use of design constitutive laws to consider the post-cracking residual strength provided by fibre reinforcement. Other cases, like early age crack-control or fire resistance, are considered non structural use of FRC. (5) For structural use, a minimum mechanical performance of FRC must be guaranteed. (6) Fibres can be used to improve behaviour in SLS since they can reduce crack spacing and crack width, thereby improving durability.

STRUCTURAL APPLICATIONS

• industrial pavements

• harbour and airport pavements

• shotcrete linings

• tunnel segments

• precast sewer pipes

• retaining structures

• foundation slabs

• slabs on piles

• flat slabs (limited)

BEHAVIOUR IN COMPRESSION

TENSILE AND FLEXURAL BEHAVIOUR

0.00 0.10 0.20displacement w (mm)

0.0

1.0

2.0

3.0

4.0

5.0

ave

rag

e t

en

sio

n

t (M

Pa

)

zoom w = 0.20 mm

TRA0 med

TRA4 med

TRA8 med

0.00 2.50 5.00deflection f (mm)

0.0

4.0

8.0

12.0

loa

d P

(K

N)

zoom w = 5.00 mm

FLE0 med

FLE4 med

FLE8 med

P [kN]

N

[mm]

w[mm]

w We use residual tensile strength after cracking!

Vf [%]

0.0 0.4

0.8

0.8

0.4

0.0

0 5 1 0

6

3

0

P [kN]

P

30

60

0.8

0.0

[mm]

Vf [%]

Vf [%]

FRC Classification

A unique standard for both the behaviors

P

P P

PPcr crP

crack formation

crack

crack formation

localization

Depending on the fibre content the stable crack propagation progressively grows …

2

sp

j

,2

3

hb

lFf jR

EN 14651

hsp = 125 mm

b = 150 mm

fR,1 fR,3

U.L.S.

Reference test

fL S.L.S.

PARAMETER IDENTIFICATION IN UNIAXIAL TENSION

Classification according to Model Code 2010

Number / letter

Ex: 3c

a: 0.5< fR3,k/fR1,k < 0.7 b: 0.7 < fR3,k/fR1,k < 0.9 c: 0.9 < fR3,k/fR1,k < 1.1 d: 1.1 < fR3,k/fR1,k < 1.3 e: fR3,k/fR1,k >1.3

STANDARD TEST FOR CLASSIFICATION

CLASSIFICATION ACCORDING TO MODEL CODE 2010

(5) Fibre reinforcement can substitute (also partially) conventional reinforcement at ultimate limit state if the following relationships are fulfilled: fR1k/fLk > 0.4; fR3k/fR1k > 0.5

CMOD (mm)

N

fLK

0.5 2.5 0

fR1k fR3k

MINIMUM REQUIREMENT ACCORDING TO PERFORMANCE BASED DESIGN

mix-constituent Dosage (kg/m3) fresh state performance

Cement type 42.5 472 Slump flow diameter (mm) 680 (t = t0)

650 (t = t0 + 1h)

Fly ash 45 T50 (sec) 2.8

Water 216 (w/b = 0.42)

superplasticizer 6 (lt/m3)

V-funnel time TV (sec) 4.7 (t = t0)

8.0 (t = t0 + 1h)

Sand 0-4 mm 850 Cube compressive strength (N/mm2)

Gravel 4-8 mm 886 7 days 56.2

Steel fibers 65/35 50 28 days 71.0

13 beams cast: 9 from the center, 4 from the side!

EFFECT OF CASTING

0 1 2 3 4 5

CMOD (mm)

0

4

8

12

N (

N/m

m2)

center-cast specimens

side-cast specimens

0.5 mm

fR1

fR1k = 5 N/mm2

fR3

fR3k = 4.1 N/mm2

a

b

c

d

e

fR3k = 0.5 fR1k

fR3k = 0.7 fR1k

fR3k = 0.9 fR1k

fR3k = 1.1 fR1k

fR3k = 1.3 fR1k

2.5 mm

FRC

class 5b

EFFECT OF CASTING

fL = 6.3 fL = 5.5

fR1 = 8.3 fR1 = 6.0

fR3 = 7.2 fR3 = 4.9

nf = 280 nf = 209

EFFECT OF CASTING

fFtu = fR3/3

0)2.05.0( 13

3

RRFts

u

FtsFtu fffCMOD

wff

145.0 RFts ff

CONSTITUTIVE MODELS FOR UNIAXIAL TENSION

NON-STANDARDIZED TEST DOUBLE EDGE WEDGE SPLITTING TEST (DEWS) – POLITECNICO DI MILANO

INDIRECT TENSILE TEST

Compression of a cylindrical specimen in FRC,

notched on the side surface and loaded through

two steel cylinders. Crack opening measured via

two clip gauges.

𝐹𝑆𝑃 = 𝑃 ∙cos 𝜗 − 𝜇 sin 𝜗

sin 𝜗 + 𝜇 cos 𝜗

𝜎𝑁 =𝐹𝑆𝑃

𝑡 ℎ𝑠𝑝

Pure tensile stress compared to the Brazilian test

STEEL CYLINDER

CLIP GAUGE

NON-STANDARDIZED TEST DEWS TEST - STAGES

STAGE 1 Linear elastic behavior

STAGE 2 Cracking of the specimen

STAGE 3 Pull-out and activation of

the residual strength

NON-STANDARDIZED TEST BARCELONA TEST – UNIVERSIDAD POLITECNICA DE CATALUNYA

INDIRECT TENSILE TEST

Compression of a cylindrical specimen in FRC through

two steel punches in contact with the upper and

bottom faces of the specimen. A strain gauge to

measure the chain crack width.

NON-STANDARDIZED TEST BARCELONA TEST - STAGES

STAGE 1 Linear elastic behavior

STAGE 2 Cracking of the specimen

and cone formation

STAGE 3 Sinking of the cone and the mobilization of the

residual strength

The cylindrical specimens tested with DEWS and BCN tests were directly cored from the shallow beams

following different directions in order to have a complete representation of the beam.

Cylindrical cores cut in

specimens with dimensions of

100Ø100mm (BCN) and

100Ø50mm (DEWS)

BCN 56 tested specimens

DEWS 112 tested specimens

SHALLOW BEAM

Cylindrical core

SPECIMEN

48 cylindrical cores extracted

from 4 shallow beams

SHALLOW BEAM

DEWS AND BCN SPECIMENS

fcftl [MPa] fR0.5 [MPa] fR1.5 [MPa] fR2.5 [MPa]

Average tensile stress DEWS

2.92 0.82 0.62 0.49

Average tensile stress BCN

2.86 1.17 0.57 0.44

Percentage variation

2 % 29 % 8 % 10 %

• Peak points coincident

• Very similar trends in the pull-out

region

• At 0.5 mm, percentage difference of

~30%

BCN 56 tested specimens

DEWS 112 tested specimens

DEWS AND BCN COMPARISON RESPONSES

BCN TEST DEWS TEST

fcftl [MPa] fR0.5 [MPa] fR1.5 [MPa] fR2.5 [MPa]

COV – BCN 7 % 26 % 35 % 34 %

COV - DEWS 19 % 69 % 72 % 76 %

BCN DEWS

• COVBCN<< COVDEWS

DEWS & BCN MECHANICAL COMPARISON MEAND AND COV

DIFFERENT FRACTURED AREAS

ABCN = 42x3= 126 cm2

ADEWS = 5x6= 30 cm2

𝐴𝐵𝐶𝑁

𝐴𝐷𝐸𝑊𝑆≅ 4

DEWS & BCN MECHANICAL COMPARISON FRACTURED AREA

• Coring of the shallow beams in z direction

• Cored cylinders were subdivided into ‘’top’’ and

‘’bottom’’ specimens

• In total 64 and 32 specimens were obtained

from DEWS and BCN, respectively

In the phase of the fresh concrete, tendency of the fibers to move, by gravitational effect, to the lower regions of the formwork. Influenced by vibrating concrete and addition of superplasticizers.

DEWS & BCN MECHANICAL COMPARISON FIBER SEGREGATION

Inhomogeneities in the fiber concentrations

DEWS & BCN MECHANICAL COMPARISON FIBER SEGREGATION

BCN tests DEWS tests

fR0.5 [MPa] fR1.5 [MPa] fR2.5 [MPa]

BOTTOM DEWS 1.01 0.81 0.67

TOP DEWS 0.73 0.63 0.59

Percentage variation

28 % 22 % 12 %

fR0.5 [MPa] fR1.5 [MPa] fR2.5 [MPa]

BOTTOM BCN 1.23 0.62 0.44

TOP BCN 0.85 0.43 0.33

Percentage variation

30 % 30 % 25 %

k mf f ks

0

1

2

3

4

5

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

CTODm [mm]

No

min

al S

tre

ss

sN [

MP

a]

Load vs. Central deflection

0

50

100

150

200

250

300

350

0 1 2 3 4 5 6

Central deflection [mm]

Lo

ad

[k

N]

Slab P22 50/0,75 Vf=0,38 %

Slab P21 50/0,75 Vf=0,38 %

Slab P20 50/0,75 Vf=0,38 %

P = P (fm)

P = P (fk)

THE KEY ROLE OF THE SCATTERING

FdRdRd fPKP

4.1

,

,

max,

max,

kFtu

mFtu

m

kMC

f

f

P

PK

Rd

The factor KRd, depends:

Model Code definition:

Resistant load amplified for structures able to significantly redistribute the applied

load:

2 5

V KRd = KRd (V/V0 , Pmax/Pcr)

V0

DEFINITION OF REDISTRIBUTION FACTOR (KRd)

kFtu

mFtu

u

mrand

u

krand

Rdf

f

P

PK

,

,

,

,

3

u

k

u

krand

RdP

PK

hom,

,

4

ULS definition

ALTERNATIVE MEASURES FOR KRd FACTOR (DUCTILE STRUCTURES)

Effects of structural redundancy on the topological heterogeneities of FRC:

48.1,1,1

0.1112

1.0-1,5.0

10

1010

kV

V

V

V

V

V

u

uu

CNR DT 204/2006

fk = fm - k s

=

halV csu ,2.11

hallV cscs

u 21

1

,4.1

halV cs

u 2,3.11

Redistribution resources

1. Nonlinear finite element approach (NLFE)

2. Yield line method (YL)

3. Empirical approach

Approaches 1 and 2 take into account the expected heterogeneity of the mechanical characteristics

The structure is subdivided into homogeneous sub-regions where several material properties are randomly assigned.

APPROACHES TO COMPUTE KRd FACTOR

• Only the parameters that identify the post-peak tensile constitutive law

(fR1, and fR3 or feq1 and feq2) are assumed as stochastic variables, whereas

the other quantities are assumed to be deterministic.

• A normal distribution of the stochastic variables is considered; mean and

standard deviation are calculated using bending test results on standard

notched specimens.

• The stochastic variables are used to build a set of linear elastic-

bilinear(linear) softening models, randomly assigned to the slab sub-

regions.

NLFE AND YL APPROACHES

1 XX FxxFxXP

1

, 1

2

2*1

n

XXS

n

XX

n

i i

X

n

i i

X

XmXU

Fractile definition x:

Normal distribution

• Characteristic value can be determined by

11

UXXU

X

X FmxFmx

• Mean mX and variance 2X are not known. Only their estimates can be

obtained by the samples.

[Approach proposed by Zupan et al. (2007)]

STATISTICAL APPROACH IN STRUCTURAL DESIGN

1ˆ, xXP

*

,ˆ

XSXX

• The approach adopted controls the probability of the characteristic value

estimate (confidence interval).

• Once prescribed the confidence interval , the characteristic value

estimate can be determined from:

• The estimate of the characteristic value is:

,X̂

• The characteristic value is distributed by non-central t distribution only in

the case of normally distributed X

n 3 4 5 7 10 20 50 100

-3.125 -2.681 -2.463 -2.250 -2.104 -1.932 -1.811 -1.758

STATISTICAL APPROACH IN STRUCTURAL DESIGN

Models construction by considering heterogeneous materials: discrete vs continuous

approach

Example of CDF discretization with a Heaviside step function: steps introduced at

percentiles = 0.025, 0.15, 0.5, 0.85, 0.975 5 material combinations

NLFE AND YL APPROACHES

Example of slab subdivision for the random heterogeneous material assignation

NLFE APPROACH

• NLFE: automatically includes both fractured volume involved in the failure process and the stability of the system

• YL: The ultimate moment per unit width mu is a function of the stochastic variables, as consequence the ultimate load Pu results to be a random quantity.

• YL is able to take into account only the fractured volume involved in the failure process and not the stability of the system for the evaluation of KRd.

• YL: can be used only if the bending response of the cross-section is ductile.

NLFE AND YL APPROACHES

7.15.01 f

ct

f

G AK

EMPIRICAL APPROACH

REFERENCE TESTS – GEOMETRY (i)

REFERENCE TESTS – GEOMETRY (ii)

Slabs on ground

[Tests carried out by Sorelli et al.

(2006)]

Elevated slab

[Tests carried out by

Parmentier et al. (2014)]

REFERENCE TESTS – MATERIALS

Mix design Component Dosage (kg/m3)

Slab on ground Elevated slab

Cement (type 1) 345 (42.5R) 200 (32.5N)

Cement (type 2) - 200 (52.5R)

Aggregate 1 621 (0-4 mm) 850 (0-4 mm)

Aggregate 2 450 (4-15 mm) 400 (4-8 mm)

Aggregate 3 450 (8-15 mm) 440 (6-14 mm)

Water 190 215

Hooked end fibres 30 70

Plasticizer 3.8 1.6

Slab on ground Elevated slab

df (mm) 0.6 1.0

lf (mm) 30 60

Aspect ratio (–) 50 60

Tensile strength (MPa) 1100 1450

Type Hooked–end Hooked–end

Steel fiber properties

Slab on ground* Elevated slab

(Casting C2)

Specimen # 7 11

fL,m [MPa] 4.06 (15.6%) 4.54 (11.0%)

fR1,m [MPa] 3.07 (19.8%) 6.39 (24.9%)

fR3,m [MPa] 2.73 (20.5%) 6.37 (20.3%)

fR1,m/fR1,k 1.70 1.89

fR3,m/fR3,k 1.74 1.62

fFts,m/fFts,k 1.70 1.89

fFtu,m/fFtu,k 1.78 1.48

kn 2.08 1.89

Class “1.5c” “3e”

SLAB FRACTURE PROPERTIES

Slabs on ground Elevated slab

EXPERIMENTAL CRACK PATTERNS

RESULTS AND DISCUSSION (i)

• NLFE: provide a P- (or P-COD) curves Pu,rand is then extracted

• YL: provide directly (and only) Pu,rand

Slab on ground numerical response for homogeneous and heterogeneous random material

RESULTS AND DISCUSSION (ii)

a) slab on ground (SoG) case with yield line method (YL);

b) slab on ground case with nonlinear finite element (NLFE);

c) elevated slab case with YL method

Material probability density function and ultimate load histogram

RESULTS AND DISCUSSION (iii)

Redistribution factors KRd

Slab on ground Elevated slab

KRd3 YL 1.66 (1.4) 1.47 (1.4)

NLFE 1.69 (1.4) -

KRd4 YL 1.58 (1.4) 1.57 (1.4)

NLFE 1.48 (1.4) -

KfG 1.45 1.7

kFtu

mFtu

u

mrand

u

krand

Rdf

f

P

PK

,

,

,

,

3

u

k

u

krand

RdP

PK

hom,

,

4

EXPERIMENTAL CAMPAIGN ON KRD FACTOR

Phase 1 (Rep. 2)- 20 slabs

Phase 2 (Rep. 12)

12φ12 in each direction

PRELIMINARY RESULTS

fct-fl = 5.24 (6%) fR1k = 4.25

(20%)

fR2k = 4.73 (22%) fR3k = 3.45

(17%) fR4k = 2.31 (20%)

Slab at ground level of about 240 m2 realized in SFRC covering the underground story.

AN EXAMPLE OF FRC FLAT SLAB BUILT IN ITALY

SFRC particularly convenient because allows significant optimizations of the reinforcement in terms

of casting simplification, performance increase (like depth reduction), quality of the casting and

durability due to crack opening control.

In conjunction with steel fibers, traditional steel high bond

bars can be introduced in the critical regions to enhance the robustness of the structures and to activate

suitable ductile failure mechanisms at the onset of collapse.

AN EXAMPLE OF FRC FLAT SLAB BUILT IN ITALY

Fibers

Due to their random distribution, they act in

every direction inside the plate and this means

that they are able to limit the occurrence of

cracks or to limit the opening regardless of the

area in which the crack propagates and by its

orientation. Furthermore, even the fibers non-

orthogonal to the crack plane are able to limit

the occurrence of the crack.

Conventional steel rebar

The reinforcing bars are valuable in

regions where the stress concentrations

are elevated, localized and well oriented.

However, they are arranged according to

precise directions (they may not limit the

occurrence of cracks parallel to their

direction) and are arranged with a

discrete and not continuous spacing.

DESIGN PHILOSOPHY

SFRC mix design

Component Dosage (kg/m3)

Preliminary mix Final mix

Cement type CEM II

/A–LL 42.5R 340 370

Filler - 150

Gravel 565 409

Washed sand 1088 993

Sifted sand 184 244

Water 170 185

Super-plasticizer 2.72 5.6

Straight steel fiber 30 35

MATERIAL CHARACTERISTICS

• Check of the fiber content at the fresh state from 7 different truck mixers: 33 kg/m3 - COV = 20.7% (design

value 35 kg/m3)

• = 2358 kg/m3

• Rcm = 55.6 MPa on 7 cubes

Three-point bending test results

Final cast

MATERIAL CHARACTERISTICS

fL fR1 fR2 fR3 fR4

[MPa] [MPa] [MPa] [MPa] [MPa]

mean (mx) 5.518 4.784 4.629 4.063 3.649

st. dev. (sx) 0.300 1.018 1.013 0.777 0.668

fk (normal) 4.865 2.568 2.426 2.373 2.194

my 1.707 1.547 1.512 1.385 1.279

sy 0.056 0.208 0.223 0.205 0.196

fk (log–normal) 4.876 2.987 2.790 2.557 2.348

Final cast

The material was classified as “2.5c”

Three-point bending test results:

nominal strengths

Reinforcement disposal in the

elevated slab (10@30-35 cm)

Preliminary casting of the lift-core

compartment foundation base

EXECUTION MODALITIES

Casting of the elevated slab (4

different casting points)

Curing conditions of the casting

carried out with water flooding of

about 2 cm for 1 week

EXECUTION MODALITIES

Intrados (left) and extrados (below)

views of the SFRC elevated slab

EXECUTION MODALITIES

• Barcelona test confirms its reliability for production control and DEWS test

appears very effective to evaluate the effects of fiber orientation.

• The indirect tensile tests (BCN and DEWS tests) show similar results

(differences <10%) when the cracks are stabilized.

• Operational aspects: the BCN test has a rapid execution time (about 30 min.

for test). DEWS test is characterized by a slower procedure (about 1 hour

per test). RILEM test is similar to the BCN test for time execution but adopts

unwieldy specimens.

CONCLUSIONS (i)

• The load bearing capacity of a redundant structure is significantly higher

than that computable by means of the characteristic parameters obtained

with a notched bending test on a small specimen as that suggested by

Standard CEN. This experimental evidence can be justified by the

heterogeneity neglected in the computational approach.

• The approaches investigated confirmed the physical intuition that a FRC

redundant structure has a bearing capacity associated to the average

strength rather than to the characteristic one.

• Different proposals according to Model Code 2010 suggest that it is

necessary to take into account both the volume of fracture involved in the

collapse and the redistribution ability of the structure.

CONCLUSIONS (ii)

• Some variations could be expected with a 3D heterogeneous analysis.

• Although the simplified proposal of considering only the volume involved

in the failure process is an attracting solution, the dependency on the

redistribution ability appears also a not negligible parameter.

• A suitable calibration taking into account several boundary conditions and

reinforcement detailing of a redundant structure could be very helpful to

verify the reliability of the proposed approaches.

• The simple cases analysed highlight that the limit selected as upper bound

in the Model Code should be a choice in favour of safety.

CONCLUSIONS (iii)

• A 240 m2 SFRC elevated slab was designed and built in a two–stories family

house in Erba (first example in Italy). The experience has demonstrated

that the use of SFRC can simplify and keep faster the executive

procedures.

• Cost reduction of about 15%.

• Not negligible segregation effect was detected by coring the shallow

beams, and looking the results of two simplified tests based on indirect

tension (Barcelona test and DEWS test).

CONCLUSIONS (iv)

THANK YOU FOR YOUR ATTENTION!