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Concrete science and technology

Bridge between materials science, concrete

technology, execution and design of concretestructures

Lecture 10.09.2010

Mix design

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September 17, 2010 2

Cement

When happiness was a common thing

Sand Aggregate

Water

B25

B35B45

Concrete

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It has becomecomplicated

Gravel

Sand

Cement

Water

Slag

Fly ash

Silica fume

FillersSteel FibersPolymer fibersAccelerator

Retarder

Stabilizer

Air entrainer

Superplasticizer

Water

entrainer

Recycled aggregates

B15

B35

B45

B55

B65

B75

B85

B95

B105

B120

B200

B800

Light weight Concrete

Selfcompacting concrete

Fibre reinforced concrete

Property defined concrete

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Mix design procedureDutch standard procedure (ordinary concrete mixtures)

Phases in mix design process1. Choice of materials

2. Determination of composition of cement paste

3. Determination of aggregate (sand and gravel)

4. Check of volumes and mass (uitleveringsberekening)5. Check of chloride content

fccmfcck

Pre-phase:

1. Designer defines required strength (C/B value) and specific mass

2. Location and environment determine environmental class

3. Adopted execution method determines consistency

C-value

B-value

B-value

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Mix design procedureDutch standard procedure

Phases in mix design process

1. Choice of materials Aspects:

1. Availability2. Properties of raw materials (chemical, physical, mechanical)3. Price

2. Determination of composition of cement paste

1. Strength of the concrete

2. Norm strength of the cement paste

3. Water/binder ratio

4. Air content

5. Water and cement content

3. Determination of percentages sand and gravel (Workability!)1. Fuller method

2. Method of Rengers-Anthonisse (Currently used Dutch mixtures)

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Mix design procedureDutch standard procedure Design criteria

Strength

Workability (consistency classes 1, 2, 3,4)

Particle size distribution aggregate

Max. particle size aggregate

Percentage of fine material (< 250 m)

Amount of water

Durability

Specific mass (type of aggregate) Aesthetics (Color)

Chloride content (check)

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Strength

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Transfer of forces inside skeleton

matrix

External loading

Strength:

1. Matrix2. Aggregate

3. Interfacial zone

Strength

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Norm strength of cement vs. Concrete strength

External loading

Strength

Concrete strength

Coarse aggregate

Sand

Mortar

Cement paste

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Relation cylinder/cube compressive strength

C-value

B-value

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Strength classes ENV 206 and VBT

ENV 2006 VBT 1996 (Dutch)

Strength class Char. Cylindercompr. Strength

[MPa]

Char. Cubecompr. Strength

[MPa]

Strength class Char. Cubecompr. Strength

[MPa]

-

C12/15

-

12

-

15

B5

B15

5

15

C16/20

C20/25

16

20

20

25

-

B25

-

25C25/30

C30/37

25

30

30

37

-

B35

-

35

C35/45

C40/50

35

40

45

50

B45

-

45

-

C45/55

C50/60

45

50

55

60

B55

-

55

-

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Concrete strength and norm strength N

c

w/c

bN.a(N)fccm +=

N = Norm strength of cement

Norm strength: mortar prisms 160x40x40 mm

s/c = 3:1, w/c = 0.5

a = 0.8 (0.75 - 0.85)

b = 25 (20 - 25)

c = 45 (40 50)

fccm

Strength

B-value

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Strength

class

Compressive strength (Norm strength)

N/mm2Begin

setting

min.

Shape

preser-

vation

mm

Initial strength

After 28 days

After 2

days

After 7

days

32.5

32.5 R

--

> 10

> 16

--

> 32.5

> 32.5

< 52.5

< 52.5

> 60 < 10

42.5

42.5 R

> 10

> 20

--

--

> 42.5

> 42.5

< 62.5

< 62.5

52.5

52.5 R

> 20

> 30

--

--

> 52.5

> 52.5

--

--

> 45

Boundary values: strength, begin setting and shape preservation

Table 2.6

(Norm strength: mortar prisms 160x40x40 mm, s/c = 3:1, w/c = 0.5)

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Type of

cement

code Norm strength N of cement [N/mm2]

1 day 2 days 3 days 28 days

Portland

cement

CEM I 32.5 R

CEM I 42.5 R

CEM I 52.5 R

10

19

29

17

30

39

25

35

44

48

58

63

Portland-

flyash

cementCEM II/B V32.5 R 13 22 25 49

Blast

furnace

slagcement

CEM III/A 32.5

CEM III/A 42.5

CEM III/B 32.5 LH

CEM III/BA 42.5

7

8

5

8

14

17

10

17

19

22

14

25

46

59

48

58

Guide values for mean norm strength N of currently used cements

Table 10.8

Strength

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cw/c

bN.a(N)fccm +=

Relationship betweenw/c and compressivestrength of concrete fordifferent strength

classes (= Normstrengths) of cement

Compre

ssivestren

gth

[MPa]

Strength

Water/cement ratio

Norm strength cementCompressivestrength

(N/mm2)

Strength

class

Compressive strength (Norm strength)N/mm2

Beginsettin

g

min.

Shapepreser

-vation

mm

Initial strength

Af ter 28 daysAfter 2

days

After 7

days

32.5

32.5 R

--

> 10

> 16

--

> 32.5

> 32.5

< 52.5

< 52.5

> 60 < 10

42.5

42.5 R

> 10

> 20

--

--

> 42.5

> 42.5

< 62.5

< 62.5

52.5

52.5 R

> 20

> 30

--

--

> 52.5

> 52.5

--

--

> 45

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Strength depends on maximum aggregate size

Biggerparticles result

in lowerstrength

(Generally because of

more intensivemicrocracking)

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Mixture parameters vs. Mixture properties

Parameter Strength Workability Durability

Water/cement ratio X

Type of cement X

Amount of cement

Aggregate (Fineness modulus)

Amount of fines (< 250 m)

Max. particle diameter aggregate XSlump (experimental)

Air content

Water demand

Curing regime

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Workability

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Workability

Classification:

No slump concrete (aardvochtig beton)

Semi-Plastic (half plastisch)

Plastic (plastisch)

Flowable (vloeibaar beton)

Influencing factors: Particle grading of aggregate (sieve line)

Maximum particle diameter

Percentage fine material (< 250 m)

Amount of water

(Super)plasticizers

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Consistency / Workability

Consistency

Consistency of the concrete determines:

Plasticity

Cohesion

Internal friction

Stability

Consistency tests:1. Compacting factor

2. Vebe test

3. Slump test

4. Flow table test

5. Viscometer

y stress

Rateof flow

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Walz vessel

Consistency

Consistency of sticky mixtures

Betoniek, 13/08

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Slump test (consistency 2, 3)

Consistency Cone of Abrams

De Rooij

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Consistency classes (Dutch)

Class Consistency Compacting

Factor

V = (400/400-s)

Slump test

mm

Flow table

test

mm

1 No slump > 1.26 < 40 ---

2

Half plastic

(semi -) 1.25 1.11 50 - 90 150 - 350

3 Plastic 1.1 1.05 100 - 150 360 - 480

4 Flowable --- > 160 490 - 600

Less preferable method

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Particle size of materials

Material size [m]

Gravel 4000 - 63000 Sand: Coarse 500 - 4000

Fine 63 - 200

Fly ash 1 - 200

Cement 0.5 100 Silica fume 0.1 0.15

Particle grading

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September 17, 2010 25Sand: fine, middle and coarse

fine

coarse

Particle size [mm]

% passing

middle

Particle grading

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Boundaries for workable mixtures

Mixtures 0 - 16 mm

Sieve opening [mm]

Cum.sieveres

iduein%(v/v)

A

B

C

Particle grading

Fine

coarse

0.125 0.250 0,500 1 2 C4 C8 C16

0

10

20

30

40

50

60

70

80

90

100

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Boundaries for workable mixtures

Fine

coarse

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Particle grading aggregate

Fineness modulus (sand and gravel):

100

residusievecumulativesumF =s

Sieves according

NEN 2560

Sieve residue (sand)

per sieve Cumulative

C4

2 mm

1 mm100 m

150 m

125 mrest

5.7

24.5

21.326.8

17.5

3.3

0.9

5.7

30.2

51.578.3

95.8

99.1

--

total 100.0 Fs

Particle grading

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Amount of sand and gravel in mixtureSand percentage Ps according to Rengers-Anthonisse:

Ps = 10 Fs + 28 + 0.05 z 0.08 C

Fs = Fineness modulus of the sand

z = Slump value [mm]

C = Cement content [kg/m3]

Gravel percentage: Pg = 100 - Ps

Note: Rengers-Anthonisse is applicable for currently used Dutch mixtures

Particle grading

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Maximum particle size Dmax

Maximum particle Dmax: largest sieve

diameter of the coarsest particle group

Criteria originate from:

1. Mixing, transport, pouring

2. Reinforcement layout

3. Cover on reinforcement

4. Free space between prestressing ducts

Particle grading

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Amount of fine material: < 250 m

Fine material (< 250 m) consists of: Cement Fillers and fine sand Air bubbles (Air entraining agents)

Minimumamount of fine material is related tomaximumparticle diameter

Maximum particle Dmax[mm]

Minimum amount of fine material (< 250 m)per m3 concrete

[l]

8

1631.5

140

125115

Table 10.5

Particle grading

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Measured air content of fresh

paste

% (v/v)

Effective air content per m3 concrete, to be

considered as fine material

[l]

2

3

4

56

--

10

20

3040

Table 10.6 Effective air content, to be considered as fine material

Air content in concrete

Air content affects workability, strengthand durability

1% air results in a strength reduction of about 5%

Air entraining agent

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Largest sieve [mm] 8 16 31.5 63

Grade area A-B A-C A-B A-C A-B A-C A-B A-C

Consistency 1

(slump < 40 mm)

Compaction factor > 1.26175 195 160 180 150 170 140 155

Consistency 2(slump 50 90 mm) 192 213 180 200 165 185 155 170

Consistency 3

Slump 100 150 mm 205 225 195 218 180 200 168 190

Guide values for water demand W of concrete [kg/m3 concrete]

Table 10.11

For consistency 4no guide values are given.

Higher consistency only by using (super)plasticizers

Dont add water to achieve consistency 4 Water demand

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Water/cement ratio X X

Type of cement X

Amount of cement (X)

Aggregate (Fineness modulus sand Fs) X

Amount of fines (< 250 m) X

Max. particle diameter aggregate X XSlump (experimental) X

Air content X

Water demand X

Curing regime

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DurabilityEnvironmental classes

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Durability of concrete

Town of Bellefontaine, State of Ohio, USA, 1891

50 year concrete pavement

75 year concrete pavementGeorge Bartholomew

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Oresund Bridge (2000)

15 mm

Concrete

crack

Photo: Ingenieur 2007

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Durability & Environmental classes

Rebar corrosion

Carbonation-induced corrosion

Chloride-induced corrosion

ASR Alkali-Silica Reaction

Sulphate attack (sea water)

Leaching processes (acid attack)

Main points of concern:

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H2O, CO2

CaCO3

pH = 7 - 9

pH = 13

pH

Consumption

of Ca(OH)2

Rebar corrosion!

Rebarw

Carbonationinduced rebar corrosion

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Design lifetime of large infrastructural works:

100 - 150 years

Eastern Scheldt storm surge barrier

Chloride exposure

Freeze-thaw

Drying & wetting

(Micro)cracking

Risk of corrosion ofreinforcing steel

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Environmental classes (NEN)

Class Description of environment (Current Dutch code)

1

2

3

4

5

Dry

Humid

Humid in combination with de-icing salts

Sea water

Aggressive;

a: weakb: moderate

c: strong

d: very strong

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Type of aggressive

substance

No weak moderate strong Very

strong

pH>6.5 6.5 5.5 5.5 4.5 4.5 4.0 < 4.0

CO2 (dissolves lime)

[mg CO2/l] < 15 15-30 30-60 60-100 >100

Ammonium

[mg NH4+/l] < 15 15-30 30-60 60-100 >100

Magnesium

[mg NH4+/l]

< 100 100-300 300-1500 1500-3000 >3000

Sulphates

[mg SO42-/l]

< 200 200-600 600-3000 3000-6000 >6000

Aggresivity of solutions in water

Table 10.2

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Environmental classes

Class Description of environment (European code)

XO

XC

XD

XS

XF

XA

NO risk of corrosion or attack

Carbonation initiated corrosion

Chloride induced corrosion, not from sea water (De-icing)

Chloride induced corrosion - Sea water

Freeze-thaw attack, with and without de-icing salts

ChemicalAttack


1

2

34

5

Dry

Humid

Humid in combination with de-icing saltsSea water

Aggressive; a: weak, b: moderate, c: strong, d: very strong

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Environmental classes NEN-EN 206-1

Class Description of environment

XO

X0

NO risk of corrosion or attack

Concrete without reinforcement, except classes XF and XA concrete with

reinforcement: very dry environment

XC

XC1

XC2

XC3

XC4

Carbonation initiated corrosion

Dry of constant wet

Wet, rarely dry

Intermediate moisture condition

Alternating wet and dry

XDXD1

XD2

XD3

Chloride induced corrosion, not from sea water (De-icing)Dry of constant wet

Wet, rarely dry

Alternating wet and dry

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Environmental classesEnvironmental classes Houses and buildingsHouses and buildings

Facade

Cellar wallBalcony

Cellar wall Cellar deck

Cellar f loor

Floors, walls (inside)

Fresh water

E i t l l NEN EN 206 1

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Environmental classes NEN-EN 206-1

Class Description of environment

XS

XS1

XS2

XS3

Chloride induced corrosion - Sea water

Salt containing air

Constant under water

Tidal zone, splash zone

XF

XF1

XF2XF3

XF4

Freeze-thaw attack, with and without de-icing salts

Non-saturated water, without de-icing salt

Non-saturated water, with de-icing saltSaturated water, without de-icing salt

Saturated water, with de-icing salt or sea water

XA

XA1

XA2

XA3

ChemicalAttack

Weak aggressive environment

Moderate aggressive environment

Severe aggressive environment

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Environmental classes Houses and buildings

Coastal area

Facade

Cellar wall

Floors, walls (inside)

Cellar floor slab

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Environmental classes Industrial buildings (XA)

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Environmental classes

Industrial buildings and infrastructural works (XA, XF)

D bilit it i

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Durability criteria

Consequences for mix design

2. Water/cement (binder) ratio

General: Denser cement paste gives higher durability

1. Type of cement

3. Cement content

4. Curing (Give it a week)

Prevent early evaporation

Better curing gives higher degree of hydration

Keep temperature low


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1

2

3

4

5

Dry

Humid

Humid in combination with de-icing salts

Sea water

Aggressive;

a: weak

b: moderate

c: strong

d: very strong

Criteria for concrete mixtures for different environmental classes

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Microlab Faculty of Civil Engineering and Geosciences

Environmental class

1 2 3 4 5a 5b 5c,d

+aea +aea

W/c (w/b)

Plain

Reinforced

Prestressed

--

0.65

0.60

0.70

0.55

0.55

0.55

0.55

0.55

0.45

0.45

0.45

0.55

0.55

0.55

0.45

0.45

0.45

0.55

0.55

0.55

0.50

0.50

0.50

0.45

0.45

0.45

Min. cement/ binder

[kg/m3]

Plain

Reinf. & prestress

Grading A-B

Grading A-C

Discontineous

150

260

260

260

200

280

280

280

280

280

300

300

280

280

300

300

280

280

300

300

300

300

300

300

300

300

300

300

Table 10.3 aea = air entraining agent


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Environmental class

1 2 3 4 5a 5b 5c,d

+aea +aea

Min. air content

[%] for aggr. Diam.

D = 63 mm

D = 31.5 mm

D = 16 mmD = 8 mm

--

--

----

--

--

----

3.0

3.5

4.05.0

--

--

----

3.0

3.5

4.05.0

--

--

----

--

--

----

--

--

----

--

--

----

Type of cement Sulphate

resistant

BFSC

Sulphate resistant

cement

Table 10.3 (cont.)

aea = air entraining agent (luchtbelvormer)

Mixture parameters vs Mixture properties

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Water/cement ratio X X X

Type of cement X X

Amount of cement (X) X

Aggregate (Fineness modulus) X X

Amount of fines (< 250 m) X

Max. particle diameter aggregate X X X

Slump (experimental) X

Air content X X

Water demand X

Curing regime X

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Mix design

Mixture for specified environment

Example

Mix design - Example

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Project: Infrastructure construction Mass concrete

Environmental class 2 (Dutch code)

Materials: Sand and gravel given (see table)


Grading sand and gravel (in stock)

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Grading sand and gravel (in stock)

Sieve according NEN2560

Cumulative sieve residue [%]

sand gravel

C 31.5

C16

C8

C4

2 mm

1 mm

500 m

250 m

125 m

--

--

--

3

9

28

60

94

100

--

28

70

94

100

100

100

100

100Fineness modulus F 2.94 6.92


6% sand < 250 m

Mix design

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Mix design

The unknowns are:

Concrete strength

Type of cement

w/c

Amount of water

Amount of cement

Amount of air

Amount of sand

Amount of gravel

Air?

Water?

Cement?

Sand?

Gravel? workability

durability

Mix design - Considerations

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Mix design Considerations

Mass concrete: Low heat cementCEM III/B (Slag: 66-80%)

Required strength: B25 (= C20/25)

fcck= 25 MPafccm = 32 MPa

Pouring with container: Consistency class 2


Class Consistency Compacting

Factor

V = (400/400-s)

Slump test

z

mm

Flow table

test

mm

1 No slump > 1.26 < 40 ---

2Half plastic

(semi -)1.25 1.11 50 - 90 150 - 350

3 Plastic 1.1 1.05 100 - 150 360 - 480

4 Flowable --- > 160 490 - 600

Slump z = 50 90 mm

z = 70 mm

Mix design

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Mix design

The unknowns are:

Concrete strength B25

Type of cement CEMIII-B

w/c .

Amount of water .

Amount of cement .

Amount of air .

Amount of sand .

Amount of gravel .

Air?

Water?

Cement?

Sand?

Gravel?

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Strength


Guide values for mean norm strength N of currently used cements

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Type of

cement

code Norm strength N of cement [N/mm2]

1 day 2 days 3 days 28 days

Portland

cement

CEM I 32.5 R

CEM I 42.5 R

CEM I 52.5 R

10

19

29

17

30

39

25

35

44

48

58

63

Portland-

flyashcement CEM II/B V32.5 R 13 22 25 49

Blast

furnace

slag

cement

CEM III/A 32.5

CEM III/A 42.5

CEM III/B 32.5 LHCEM III/BA 42.5

7

8

58

14

17

1017

19

22

1425

46

59

4858

Table 10.8


Concrete strength and Norm strength N

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c

w/c

bN.a(N)fccm +=

N = Norm strength of cement = 48 MPa (for CEM III/B)

a = 0.8; b = 25; c = 45

Required w/c = 0.65

(Note: Norm strength refers to mortars with w/c=0.5)

fccm


Check whether w/c is OK in view of durability requirements!! (Table 10.3)

a = 0.8 (0.75 - 0.85)

b = 25 (20 - 25)

c = 45 (40 50)

Mix design

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Mix design

The unknowns are:



w/c 0.65

Amount of water .

Amount of cement . Amount of air .

Amount of sand .

Amount of gravel .

Air?

Water?

Cement?

Sand?

Gravel?

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Strength


Strength gives first indication about w/c (w/c = 0.65)

Is this w/c also OK for durability?


Table 10 3

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Environmental class

1 2 3 4 5a 5b 5c,d

+aea +aea

W/c (w/b)Plain

Reinforced

Prestressed

--

0.65

0.60

0.70

0.55

0.55

0.55

0.55

0.55

0.45

0.45

0.45

0.55

0.55

0.55

0.45

0.45

0.45

0.55

0.55

0.55

0.50

0.50

0.50

0.45

0.45

0.45

Min. cement/binder [kg/m3]

(Plain concrete)

Reinf. & prestress

Grading A-B Grading A-C

Discontineous

150

260

260

260

200

280

280

280

280

280

300

300

280

280

300

300

280

280

300

300

300

300

300

300

300

300

300

300

Table 10.3

aea = air entraining agent


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g g

c

w/c

bN.a(N)fccm +=

N = Norm strength of cement = 48 MPa (for CEM III/B)

a = 0.8; b = 25; c = 45

(Note: Norm strength refers to mortars with w/c=0.5)

Required w/c = 0.65

For Env.Class 2 (table 10.3) w/c = 0.55

For sake of safety: w/c = 0.53

fccm


Mix design

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Mix design

The unknowns are:



w/c 0.65 0.53

Amount of water .

Amount of cement . Amount of air .

Amount of sand .

Amount of gravel .

Air?

Water?

Cement?

Sand?

Gravel?

Air content and water demand

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Air content: No requirements

Assume: Va< 2%, e.g. 1%

Then no effect on strength!

Water demand: Depends on aggregate grading!

(Table 10.11)


Guide values for water demand W of concrete [kg/m3 concrete]

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Largest sieve [mm] 8 16 31.5 63

Grade area A-B A-C A-B A-C A-B A-C A-B A-C

Consistency 1(slump < 40 mm)

Compaction factor > 1.26175 195 160 180 150 170 140 155

Consistency 2

(slump 50 90 mm) 192 213 180 200 165 185 155 170

Consistency 3

Slump 100 150 mm205 225 195 218 180 200 168 190

Table 10.11

For consistency 4no guide values are given.

Higher consistency only by using (super)plasticizers

Dont add water to achieve consistency 4

Air content and water demand

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Air content: No requirements

Assume: Va< 2%, e.g. 1%

Then no effect on strength!

Water demand (Table 10.11)Consistency class 2

Dmax < 32 mm

Grading lines A-B

Vw = 165 l/m3


Cement content and paste volume

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Cement content C:

W/C = 0.53

Vw = 165 l/m3 = 165 kg/m3

Note: Environmental Class 2 (Table 10.3): C> 280 kg/m3: OK

C = 311 kg/m3


Table 10.3

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Environmental class

1 2 3 4 5a 5b 5c,d

+aea +aea

W/c (w/b)

Plain

Reinforced

Prestressed

--

0.65

0.60

0.70

0.55

0.55

0.55

0.55

0.55

0.45

0.45

0.45

0.55

0.55

0.55

0.45

0.45

0.45

0.55

0.55

0.55

0.50

0.50

0.50

0.45

0.45

0.45

Min. cement/ binder

[kg/m3]

Plain

Reinf. & prestress

Grading A-B

Grading A-C

Discontineous

150

260

260

260

200

280

280

280

280

280

300

300

280

280

300

300

280

280

300

300

300

300

300

300

300

300

300

300

aea = air entraining agent

Mix design

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The unknowns are:



w/c 0.65 0.53

Amount of water 165 l.

Amount of cement 311 kg/m3

Amount of air 10 l (=1%)

Amount of sand .

Amount of gravel .

Air?

Water?

Cement?

Sand?

Gravel?

Cement content and paste volume

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Cement content C:

W/C = 0.53

Vw = 165 l/m3 = 165 kg/m3

Note: Environmental Class 2 (Table 10.3): C> 280 kg/m3: OK

C = 311 kg/m3

Cement paste volume:

Slag cement: C = 311 kg/m3 (311/2950) = 0.102 m3

Water: Vw= 165 kg/m3 = 0.165 m3

Air: Va = 1% = 0.010 m3

Cement paste volume Vcp = 0.280 m3

Sand + coarse aggregate V = 0.720 m3

Amount of sand and gravel

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Sand percentage Ps according to Rengers-Anthonisse:

Ps = 10 Fs + 28 + 0.05 z 0.08 C

Fs = 2.94 (Fineness modulus of sand)

z = 70 mm (Slump value between 50 and 90 mm)

C = 311 kg/m3 (Cement content)

Ps = 10 2.94 + 28 + 0.05 70 0.08 311

Ps = 36%

Gravel percentage: Pg = 100 36 = 64%


Check amount of fine material < 250 m

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Fine material (< 250 m) consists of Cement (0.102 l)

Fillers and fine sand (see sieve analysis: 6% of sand)

Air bubbles (Air entraining agents) (--)

Maximum particle Dmax

[mm]

Minimum amount of f ine material ( 115 l OK


Sieve according NEN2560

Cumulative sieve residue [%]

sand gravel

C 31.5

C16

C8

C4

2 mm

1 mm

500 m250 m

125 m

--

--

--

3

9

28

6094

100

--

28

70

94

100

100

100100

100

Fineness modulus F 2.94 6.92

Mix design

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The unknowns are:


Type of cement CEMIII-B w/c 0.65 0.53

Amount of water 165 l.

Amount of cement 311 kg/m3

Amount of air 10 l (=1%)

Amount of sand 260 l.

Amount of gravel 460 l.

Air?

Water?

Cement?

Sand?

Gravel?

Mix design Summary

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RequirementsStrength B25

Environmental class 2

Consistency class 2

Mixture composition

W/c 0.53

Cement 311 kg/m3

Water 165 kg/m3

Air 10 l

Sand 36% of 720 l = 260 l

Gravel 64% of 720 l = 460 l


In case of new types of mixtures: Always test mixtures!!

b35 tanim.pdf

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