Prestressed concrete : The design principleดร...

Prestressed concrete : Prestressed concrete : Prestressed concrete : Prestressed concrete : The design principleThe design principleThe design principleThe design principle

ดร. สุนิติ สุภาพDr. Suniti [email protected]@yahoo.com

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R d d b kR d d b kRecommended booksRecommended booksP t d C tPrestressed ConcreteAnalysis and Design:F d t l Fundamentals A.A. E. Naaman E. Naaman ((20052005))((20052005))

Prestressed concrete structures structures M.P. Collins & D. M.P. Collins & D. Mitchell Mitchell Mitchell Mitchell ((19971997))

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Concept of prestressed concreteConcept of prestressed concreteConcept of prestressed concreteConcept of prestressed concrete

In conventional RC structures, the flexural cracks may develop at early stages of loading.

service service

ultimate ultimate

In order to reduce such cracks from developing, a

Reinforced concrete Prestressed concrete

concentric or eccentric force is imposed in the longitudinal direction of the structural element.

Such an imposed force is called “prestressing force”.3

Reinforced vs Prestressed ConcreteReinforced vs Prestressed ConcreteReinforced vs. Prestressed ConcreteReinforced vs. Prestressed Concrete In PC, the entire concrete section is In PC, the entire concrete section is

generally active in resisting the load, while in RC only the uncracked part of y p fthe section is active.

Since PC is crackless, it provides better Since PC is crackless, it provides better protection than RC against corrosion of the steel in aggressive environmental t e stee agg ess ve e v o e ta and it is more suitable for fluid-retaining structures such as tanks and gnuclear vessels.

Prestressed concrete (fully or partially) Prestressed concrete (fully or partially) provides the means for effective deflection control, especially under long-f , p y gterm sustained loading.

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Reinforced vs Prestressed ConcreteReinforced vs Prestressed ConcreteReinforced vs. Prestressed ConcreteReinforced vs. Prestressed Concrete

PC h b tt h i t PC has better shear resistance than reinforced concrete, due to the slope of the tendons near the the slope of the tendons near the supports and the pre-compression which reduces diagonal tension which reduces diagonal tension. Thus it will require fewer stirrups.It i ft l i d th t PC It is often claimed that PC structures have a inherent safety since they undergo the most since they undergo the most severe loading during initial tensioning of the steel If they pass tensioning of the steel. If they pass this test, they are likely to perform well under future service loads.well under future service loads.

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Why we prefers PC for Why we prefers PC for y py pbridge constructions??bridge constructions??

High-strength concrete and high-tensile steel, besides being economical, make for slender sections.

Prestressed concrete bridges can be designed without any tensile stresses under service loads, thus resulting in a crack-free structure.

Post-tensioned PC finds extensive applications in long-span continuous girder bridges of variable cross-section.

Especially in Thailand, PC bridges can be constructed with minimum labor skills.

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Prestressed forcesPrestressed forcesPrestressed forcesPrestressed forces There are three types of tendon force shall be considered.

(1) Jacking force (Pj) Measured form hydraulic jack. (2) Initial force (P )After initial (short term) losses (2) Initial force (Pi)After initial (short term) losses (3) Effective force (Pe)After all losses

Pj Pi Pe

Initial losses Long term losses

Jacking Jacking forceforce

InitialInitialforceforce

EffectiveEffectiveforceforce

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Analysis stagesAnalysis stagesAnalysis stagesAnalysis stages There are two stages needed for prestressing concrete.

(1) Service stage WDM To ensure that during service life the stresses in the section To ensure that during service life the stresses in the section

are controlled in service limit stages.

(2) Ultimate stage SDM To ensure that at the ultimate stage, the PC section can be

resisted the ultimate load according to strength limit stages.

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How to analyze prestressed concrete?How to analyze prestressed concrete?How to analyze prestressed concrete?How to analyze prestressed concrete?P Mc

PC concrete can be PC concrete can be analyzed by analyzed by 2 2 stagesstages.. A

PA

IMc

f +

(1) Elastic stage : The stresses due to apply loads

Superposition method

shall be investigated by mean of conventional “strength of I

McAP

comb

material” or “combined load concept”

(2) Ultimate stage : The C( ) g

ultimate capacity of PC section can be evaluated

jd

similarly to reinforced concrete structure.

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Combined load conceptCombined load conceptCombined load conceptCombined load concept

Stresses on the section of PC can be analyzed by mean of superposition.

C.G. of girderb

w

g

C G

ct

cb

he

C.G. prestress steel

wP1 wM

PeP

Pe3

2e

3

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Combined stresses conceptCombined stresses conceptCombined stresses conceptCombined stresses conceptAxial stress Flexural stress Flexural stress

+P/A -Pect /I

Axial stress<prestress>

Flexural stress<prestress>

+Mct /I

Flexural stress<external load>

P/A – Pect/I + Mct/I

Combined stresses

-Pey /I y My /I y

+ + =

+Pecb /I -Mcb /I P/A + Pecb/I - Mcb/I

1 2 31 2 3

PP MMcPecPf

IPec

APf p

IMcfw IIA

f

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Example Example 11Example Example 11 Determine the stresses at extreme fibers of beam

section at mid-span of beam in figure 1. P = 1,625 kN and Live load = 45 kN/m.

600wLL = 45 kN/m

450 mm

450 mm

900

250 mm

10 m.

Fig.1

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Example Example 11Example Example 11

Step 1 : Calculate the sectional properties- Area : A = 600x900 = 540,000 mm2.,- Inertia : I = (1/12)bh3 = (1/12)x600x9003

= 3 65x1010 mm4= 3.65x1010 mm4. Step II : Maximum force due to external load

- Weight of beam : wg = 0.6x0.9x24 = 12.96 kN/m- Mg= wgL2/8 = 12.96x102/8 = 162 kN-mg g

= 1.62x108 N-mmM = w L2/8 = 45*102/8 = 562 2 kN m- MLL = wLLL2/8 = 45*102/8 = 562.2 kN-m

= 5.62x108 N-mm

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Example Example 11Example Example 11 Step III : combine loads : Prestress load + Mg

- Top fiber :ft= (P/A) – (Pec/I) + Mgc/I

= (1.65x106/5.4x105) – (1.65x106x250x450/3.65x1010)+ (1.62x108x450/3.65x1010)

= -0.04 Mpa (Tension!!)p ( )

- Bottom fiber :Ft = -1.48 Mpa (O.K.!!)

fb= (P/A) + (Pec/I) – Mgc/I= (1.65x106/5.4x105) + (1.65x106x250x450/3.65x1010) (1.65x10 /5.4x10 ) (1.65x10 x250x450/3.65x10 )

- (1.62x108x450/3.65x1010)= 6 15 Mpa (compression!!) 6.15 Mpa (compression!!)

Fb = +21.0 Mpa (O.K.!!) 14

Example Example 11Example Example 11 Step III : combine loads : Prestress load + Mg + MLL

- Top fiber :ft= (P/A) – (Pec/I) + (Mg+MLL)c/I

= (1.65x106/5.4x105) – (1.65x106x250x450/3.65x1010) (( 62 62) 08 4 0/3 6 010)+ ((1.62+5.62)x108x450/3.65x1010)

= 6.91Mpa (Compression!!)

- Bottom fiber :f (P/A) + (P /I) (M +M ) /I

Ft = 18.0 Mpa (O.K.!!)

fb= (P/A) + (Pec/I) – (Mg+MLL)c/I= (1.65x106/5.4x105) + (1.65x106x250x450/3.65x1010)

((1 62+5 62)x108x450/3 65x1010)- ((1.62+5.62)x108x450/3.65x1010)= -0.80 Mpa (tension!!)

Fb = -3.16 Mpa (O.K.!!)15


The stress diagram : Prestress + Mg

+3.07 -5.09 +2.0 -0.04

+ + =+ + =

+3.07 +5.09 -2.0 +6.156.15

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The stress diagram : Prestress + Mg + MLL

+3.07 -5.09 +8.93 +6.91

+ + =+ + =

+3.07 +5.09 -8.93 -0.800.80

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Prestressing methodsPrestressing methodsPrestressing methodsPrestressing methods

• Generally, there are two method for prestressing.• Depend on tensioning of prestressing strands which are induced before or after concrete hardened.

(1)Pre-tension system(2)Post-tension system

Prestressing at Prestressing at sitesite Prestressing at Prestressing at yardyard

PostPost--tension tension PrePre--tension tension

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PrePre tension systemtension systemPrePre--tension systemtension system In pre-tension, the prestressing strands will be tensioned at

yard before concrete hardened.When concrete strong enough the stands are released form When concrete strong enough, the stands are released form anchorage beds.

The prestressing force is The prestressing force is transferred to member through friction

4

through friction. The image part with relationship ID rId2 was not found in the file.

11

23

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PostPost tension systemtension systemPostPost--tension systemtension system In post-tension, the concrete element will be cast in site of

construction. However, the duct for inserting the prestressing strands are

prepared initially. When the concrete

hardened, the prestressing 4

tendon are jacked and jacking load is transferred by anchorages.

1

2

320

P stP st tensi n : tensi n : Bond Bond vsvs Unbonded tendon Unbonded tendon PostPost--tension : tension : Bond Bond vsvs Unbonded tendon Unbonded tendon

Bonded tendons represent the traditional post-tensioning technique.F l f f d h Firstly prestressing force is transferred to the concrete by end anchors, but in finish structure the tendons interact directly with the concrete by bondtendons interact directly with the concrete by bond.

Cement grout injected into the ducts provides the bondbond.

Unbonded tendons consist of single strands in Unbonded tendons consist of single strands in plastic sheaths or without grouting.

The strands are greased for reduction of friction and The strands are greased for reduction of friction and protection against corrosion.

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Unbonded considerationUnbonded considerationUnbonded considerationUnbonded consideration Advantages- Small dimension- No injection- Low friction losses- Light stressing equipment

Disadvantages- Lower ultimate strengthLower ultimate strength- Lower stiffness

Large cracks concentration

Need non-prestressing steels

- Large cracks concentration- Need corrosion protection

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PrePre tension vs Posttension vs Post tensiontensionPrePre--tension vs. Posttension vs. Post--tensiontension A post tension is suitable for longer span rather than A post-tension is suitable for longer span rather than

pretension because of transpiration problem. F i l h f i d i f For special shape of girder section, cost for new constructing a new formworks and anchorage bed foundation needed to consideringfoundation needed to considering.

Post-tensioned member may require less total prestressing f h ll i d bforce than equally strong pre-tensioned member.

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PrePre tension vs Posttension vs Post tensiontensionPrePre--tension vs. Posttension vs. Post--tensiontensionP i i ll h d b l d h h Post-tensioning allows the tendons to be placed through structural elements in smooth curves of any desired path.

Pre-tensioned tendons can be employed in other than straight paths, but not without expensive plat facilities and somewhat complicated pconstruction procedures.

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PrePre tension vs Posttension vs Post tensiontensionPrePre--tension vs. Posttension vs. Post--tensiontension Because post-tension tendons can be installed in holes

preformed in precast concrete elements or segment, they can be used to prestress a number of small precast elements to form a single large structural

bmember.

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Pretensioned vs PostPretensioned vs Post tensionedtensionedPretensioned vs. PostPretensioned vs. Post--tensionedtensioned The cost of post-tensioned tendons is significantly

greater than the cost of pretensioned tendons, because of the larger amount of labor required in placing, stressing, and grouting (where applicable)

d d ll h f l post-tensioned tendons, as well as the cost of special anchorage devices and stressing equipment.

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Effect of stresses at the end spansEffect of stresses at the end spansEffect of stresses at the end spansEffect of stresses at the end spans

The straight strand is the most common type of prestressing concrete.

However, the excessive stresses may be induced, especially, in the case of end spans of girder which the counter balance force due to applied load is rare.

Need Stress due to r Need, sufficient!!

Need, insufficient!!

Stress due to prestress

s at

fibe

r

Stress due to Mg and MLL

Stre

ss

g LLdistance

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section at end-span of beam in figure 2. P = 1,650 kN and consider only self weight of beam

600

450 mm

450 mm

900

250 mm

Fi 2

10 m.

Fig.2

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Example Example 22Example Example 22 combine loads (At end span): Prestress load + Mg

- Top fiber :pft = (P/A) – (Pec/I) + Mgc/I

= (1.65x106/5.4x105) – (1.65x106x250x450/3.65x1010) + 0( ) ( )

= -2.04 Mpa (Tension!!)

F = 1 48 M ( h k!!)

- Bottom fiber :

Ft = -1.48 Mpa (check!!)

Bottom fiber :fb = (P/A) + (Pec/I) + Mgc/I

= (1.65x106/5.4x105) + (1.65x106x250x450/3.65x1010) + 0 (1.65x10 /5.4x10 ) (1.65x10 x250x450/3.65x10 ) 0

= 8.15 Mpa (compression!!)

Fb = +21.0 Mpa (O.K.!!)

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Tendon profilesTendon profilesTendon profilesTendon profiles The high tensile stress at the ends of girder may be

reduced by decreasing the tendon eccentricity.

e Harped tendon 1

(Draped = depress)

e Harped tendon 1

e Parabolic tendon

e Debonded tendone

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Parabola tendonsParabola tendonsParabola tendonsParabola tendons

Need r Need, sufficient!!

O K

Stress due to prestress

s at

fibe

r

O.K.

Stress due to M and MLL

Stre

ss

Mg and MLLdistance

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Harped tendonsHarped tendonsHarped tendonsHarped tendons

Need Stress due to

r Need, sufficient!!

Seem O K

prestresss

at fi

ber

Seem O.K.


Stre

ss

Mg and MLLdistance

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Debonded tendonsDebonded tendonsDebonded tendonsDebonded tendons

Need Stress due to

tr Need, sufficient!!

Seem O K

prestresss

at fi

ber

Seem O.K.


Stre

ss

Mg and MLLdistance

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section at end-span of beam in figure 2. Pend = 1,155 kN (Debonded at the ends) consider to self weight onlyg y

600

450 mm

600

900

450 mm250 mm

10

Fig.2

10 m.

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Example Example 33Example Example 33 combine loads (At end span): Prestress load + Mg

- Top fiber :pft = (P/A) – (Pec/I) + Mgc/I

= (1.16x106/5.4x105) – (1.16x106x250x450/3.65x1010) + 0( ) ( )

= -1.43 Mpa (Tension!!)

F = 1 48 M (O K !!)

- Bottom fiber :

Ft = -1.48 Mpa (O.K.!!)

Bottom fiber :fb = (P/A) + (Pec/I) – Mgc/I

= (1.16x106/5.4x105) + (1.16x106x250x450/3.65x1010) + 0 (1.16x10 /5.4x10 ) (1.16x10 x250x450/3.65x10 ) 0

= 5.70 Mpa (compression!!)

Fb = +21.0 Mpa (O.K.!!)

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Pretension : Pretension : H p t dH p t dPretension : Pretension : Harp strandsHarp strands

• Need harping devices• Need of modifying of • Need of modifying of anchorage bed Strong foundationStrong foundation

• Shall be consider hold down effectdown effect

• Increase shear capacity• Reduce number of Reduce number of prestressing strands.

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Pretension : Pretension : D b d d t dD b d d t dPretension : Pretension : Debonded strandsDebonded strands

• Simple tendon profile.• No needed to modifying • No needed to modifying of anchorage bed

• Consider about number • Consider about number of debonded

• Consider shear strengthConsider shear strength• Can not reduce the number of strandsnumber of strands

• Development length design shall bedesign shall beconsidered.

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Tendon profile by FEM programTendon profile by FEM programTendon profile by FEM programTendon profile by FEM program

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Tendon profile by FEM programTendon profile by FEM programTendon profile by FEM programTendon profile by FEM program

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T d fil b FEM T d fil b FEM Tendon profile by FEM programTendon profile by FEM program

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Stress analysis by FEM programStress analysis by FEM programStress analysis by FEM programStress analysis by FEM programDead load moment (M ) Dead load stress (M )Dead load moment (Mg) Dead load stress (Mg)

Live load stress (MLL)Live load Moment (MLL)

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Stress analysis by FEM programStress analysis by FEM programStress analysis by FEM programStress analysis by FEM programPrestressing moment (Pe)Prestressing moment (Pe)

Prestressing stresses (ft & fb)

Prestressing force (P)

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St l i b FEM St l i b FEM Stress analysis by FEM programStress analysis by FEM program

Combined stress Mg + MLL + Mpre

PrestressLive load Combined43

D t il l tDetail plot

44

Tendon profile effects (prestressing only)Tendon profile effects (prestressing only)

End Mid

0.25 m.

0.25 m.

End Mid

0.25 m.

End Mid

0.25 m.0.10 m.

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Tendon profile effects (Combined)Tendon profile effects (Combined)

End Mid

0.25 m.

0.25 m.

End Mid

0.25 m.

End Mid

0.25 m.0.10 m.

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D i id ti D i id ti S ti tiS ti tiDesign consideration : Design consideration : Section propertiesSection properties

Simplified section C G of section (c vs cb) C.G of section (ct vs. cb) Moment of inertia (I) Section modulus (St and Sb)

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Simplified sectionSimplified sectionSimplified sectionSimplified section Naaman (2005) recommended to simplify of girder

geometry based on equivalent area.

bt bt

h ??A1 ht = ??

b xh = A1

A2

btxht = A1

A2

True section Equivalent section

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Transform areaTransform areaTransform areaTransform area C.G. of section

x1

AAy

y ii

1 2x1

y1

x2

A

A3y1

x3

3

y2y

AAx

x ii

y3

Gross moment of inertia

x

2)( iixix yyAII 2)( iiyiy xxAII

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Example Example 44Example Example 44 Find the section modulus of girder and composite

girder as shown below.

0.60.20

1.85

0.200.15

0.200.15

0.20 0.20

1.40 1.40

0.150.20

0.6

0.150.20

girder Girder+deck (composite) 50

Example Example 44Example Example 44 Divided section

26 7

1

43

54

51


52


8

26 7

8

6 7

1

354

53


54


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Section modulus (S)Section modulus (S)Section modulus (S)Section modulus (S) The ratio of “I/c” called “section modulus”. Therefore, the stress equation can be rewritten asq

MPePf

F fib

SSAf

For top fiber

t SM

SPe

APf

F b fib

tt SSA

For bottom fiber

MPePf bb

b SSAf

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D i id ti D i id ti M t i l ti M t i l tiDesign consideration Design consideration : Material properties: Material properties

Concrete N i l Non-prestressing steel Prestressing steel

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M h i l ti f M h i l ti f C tC tMechanical properties of Mechanical properties of ConcreteConcrete

Unit weight Unit weight Poisson’s ratio Compressive strength Compressive strength Modulus of elasticity

T il h Tensile strength Combined stresses Confined concrete

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U it i ht f C tU it i ht f C tUnit weight of ConcreteUnit weight of Concrete Normal concrete have a density of around Normal concrete have a density of around

2,300 to 2,600 kgf/m.3.

For calculating dead loads, the weight if structural concrete is often taken to be structural concrete is often taken to be 24 or 25 kN/m.3, which includes an allowance for presence of steel reinforcement.

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C t ClC t ClConcrete ClassConcrete Class The concrete is classified by compressive strength as The concrete is classified by compressive strength as

follows:

Low strength concrete : f ’c <= 20 MPa Moderate strength concrete : 20 < f ’c < 40 MPa g f c

High strength concrete : f ’c => 40 MPa

• In order to reduce the member section and size of foundation of Baiyoke tower2 High size of foundation of Baiyoke tower2, High Strength Concrete was selected.

• Concrete strength of prestressed is found Concrete strength of prestressed is found 30 – 50 Mpa

• LRFD required that the strength of LRFD required that the strength of concrete shall not be less than 28 MPa

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C i t thCompressive strengthTh C i h (f ’ ) f The Compressive strength (f ’c) of concrete is determined by test to failure 28-days-old 150 mm. by 300 300 mm. concrete cylinder at a specific rate loading.

Compressive failure test Stress-strain l ti hiCompressive failure test relationship

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Stress-strain curve of concrete

2 parabolic curve

cocco

c

co

ccf

f

2pHognestad (1951) = 0.15

cuccococu

cocc

c

f

f

1

(MPa) 0 0 4f’ li

cocu

(MPa) • 0 – 0.4f’c : linear• > 0.7f’c : losses stiffness

cf11

f’c

0 7f’

c

ccocf f

f11

( / )

0.7f’c

0.4f’c

When corresponding strain is required!! (mm./mm.)

q

co = 0.002 cu = 0.003 62

M t i l i t M t i l i t Material requirement : Material requirement : Properties of concrete

Modulus of elasticity

Ec = 4800*sqrt(f’c) AASHTO LRFD

f ’ci = compressive strength at transfer (Mpa) li d i l fi ld (f ll h d i d)= cylindrical test field (follow the designed)

= 0.80f’c may be used for designing f ’c = compressive

strength at 28 daysg y

63

Modulus of Modulus of Elasticity Elasticity (ACI)(ACI)Modulus of Modulus of Elasticity Elasticity (ACI)(ACI) For many years the modulus of elasticity of concrete y y y

was approximate adequately as 1,000f’c by ACI code.

R tl ACI318 d th d l f l ti it f Recently ACI318 proposed the modulus of elasticity for normal weight concrete as

Ec = 4,700(f’c)0.5 (MPa)

ACI363 proposed the following equation for High t th tstrength concretes :

(f’ )0 5 5 ( P ) Ec = 3,320(f’c)0.5 + 6,895 (MPa)

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R t f l diR t f l diRate of loadingRate of loading It should be noted that the shape of the

stressstress--strain curvestrain curve for various concretes of the same cylinder strength under various condition of loading, varies considerably.

Cylinder strengthCylinder strengthf’ = 21 MPaf’c = 21 MPaat 56 days

65

Repeated Compressive LoadRepeated Compressive LoadRepeated Compressive LoadRepeated Compressive Load Repeated high-intensity compressive loading produces a

pronounced hysteresis effect in the stress-strain curve.

Monotonic compressive loadMonotonic compressive load

From tests indicted that the envelope curve was almost identical to the curve obtained from monotonic load identical to the curve obtained from monotonic load application

66

Compressive Strength in Thailand Compressive Strength in Thailand Compressive Strength in Thailand Compressive Strength in Thailand Compressive strength at 28 Day

(MPa)

Cube : 150x150x150 mm. Cylinder : 150x300 mm.18 1421 1824 2128 2432 2835 30

Cylinder = (approx.) 0.87*Cube(British standard)(American standard)

38 3240 3542 3845 40

67

Poisson’s RatioPoisson’s RatioPoisson s RatioPoisson s Ratio The ratio between the transverse strain and the strain in

the direction of applied uniaxial loading, referred to as Poisson’s ratio.

For concrete, it is ,usually found to be in the range 00 15 15 in the range 00..15 15 to 00..2020.

At high compressive stresses the transverse strains increase rapidly, owing to internal cracking strains increase rapidly, owing to internal cracking parallel to the direction of loading.

68

Tensile StrengthTensile Strength The tensile strength of

Tensile StrengthTensile Strength

concrete varies from about 8 to 15% of its compressive strength, f ’c.

The tensile strength of concrete doesn’t vary concrete doesn t vary in direct proportion to its compressive strengthits compressive strength.

It does however vary It does, however, vary approximately in proportion

h f f ’to the square root of f ’c.69

Tensile StrengthTensile Strengthgg Tensile strength is quite difficult to measure with direct

axial tension l ads beca se f r blems in ri in stress axial tension loads because of problems in gripping, stress concentrations and aligning the loads.

As a result of these problems two rather indirect tests Direct tensile testDirect tensile test Modulus of ruptureModulus of rupture Splitting TestSplitting Test

As a result of these problems, two rather indirect tests have been developed to measure concrete’s tensile strength strength.

These are the modulus of rupturemodulus of rupture and the splitsplit--cylinder cylinder f pf p pp yytesttest.

70

M t i l i t M t i l i t P i f P i f Material requirement : Material requirement : Properties of concreteProperties of concrete

Modulus of rupture fr (Mpa)

Concrete matrix

DetailsDetails Modulus of rupture Modulus of rupture

D fl ti d l b 0 63 t(f’ )Deflection and clamber 0.63sqrt(f’c)Minimum reinforcement 0.97sqrt(f’c)

Sh d 0 52 (f’ )

Thermal expansion () : 10.8x10‐6/oC

Shear design 0.52sqrt(f’c)

p ( ) / Shrinkage at 28 day : 200 Micron Shrinkage for one year of drying : 500 Micron Shrinkage for one year of drying : 500 Micron

71

Tensile strength Tensile strength (ACI)(ACI)gg ( )( ) Table shows approximate formulas for tensile

strength obtained from three different approaches.

Normal weight(MPa)

Direct test 0.33(f’c)0.5

Splitting test 0.53(f’c)0.5

M d l f 0 5

Based in hundreds of

Modulus of rupture 0.50(f’c)0.5

tests, the code providesa modulus of rupture fr = 0.5(f’c)0.5

od s of pt e

ff 500 cr ff 50.0

72

Strength under Combined StressStrength under Combined Stressgg In many structural situations, concrete is subjected

simultaneously to various stresses acting in various directions.

By methods of mechanic of materials, these stresses can be y ,transformed to the principal stressesprincipal stresses, tension or compression.

Pp

M

P1P2

w

UniaxialUniaxial BiaxialBiaxial TriaxialTriaxialC

C CT

T TC

T CC C

T

CC T C CT

73

Biaxial StressBiaxial StressBiaxial StressBiaxial Stress In this case, the stresses act in one plane and the third

principal stress is zeroprincipal stress is zero.

Kupfer, H. et al (Kupfer, H. et al (19691969)) concluded that strength of concrete subjected to biaxial compressionbiaxial compression may be as much as 27% higher than uniaxial strength.

The strength of biaxial tensionbiaxial tensionis approximately equal to theuniaxial tensile strength.

However, the combination ofthe combination of, fftensile & compressive loadstensile & compressive loadsreduce both the tensile &

In the picture, fu = f’c

reduce both the tensile &compressive stresses at failure.

74

Biaxial stressBiaxial stressBiaxial stressBiaxial stress Vecchio and Collin (Vecchio and Collin (19861986)) quantified the biaxial stress

(compression affected by tension) and present the equation.

fff

cc fff

1max2 1708.0

Hsu (Hsu (19931993) ) refers to this phenomenon as “compression “compression Hsu (Hsu (19931993) ) refers to this phenomenon as compression compression softening”softening”

75

TriaxialTriaxial compressive stress behaviorcompressive stress behaviorTriaxialTriaxial compressive stress behaviorcompressive stress behavior The strength and ductility of concrete are greatly increased

under conditions of triaxial compression.

Richart, F. E. et al. (Richart, F. E. et al. (19281928)) found the following relationship for concrete cylinder loaded axially to failure while subjected y y jto confining fluid pressure.

f ’cc = f ’c + 4.1fl 1

Where f ’cc = Confined compressive strengthf ’ U fi d i hf ’c = Unconfined compressive strengthfl = Lateral confining pressure

76

T i i lT i i l i t b h ii t b h i The figure shows the axial stress strain curves obtained by

TriaxialTriaxial compressive stress behaviorcompressive stress behavior The figure shows the axial stress-strain curves obtained by

compression of concrete cylinderconcrete cylinder confined by fluid pressurefluid pressure.

77

C t fi d b i f tC t fi d b i f tConcrete confined by reinforcementConcrete confined by reinforcement In practice, concrete may be confined by transverse

reinforcement, commonly in the form of closely spaced steel spiralsspirals or hoopshoops.

At low levels of stresslow levels of stress in the concrete, the transverse reinforcement is hardly stress; hence the concrete is concrete is reinforcement is hardly stress; hence the concrete is concrete is unconfinedunconfined.

Reinforced concreteC l fi d b Column confined by various techniques

The concrete becomes confinedconfined when at stresses stresses approaching the uniaxial strengthapproaching the uniaxial strength.

78

C t fi d b i f tC t fi d b i f t IyengarIyengar et al. (et al. (19701970)) tested three sets of concrete cylinder Concrete confined by reinforcementConcrete confined by reinforcement

confined by circular spirals, each set was for a different unconfined compressive strength.

The increase in strengthand ductility with content

of confining steel is significant1 2 3 g g

79

C t fi d b i f tC t fi d b i f t Tests have demonstrated that circular spirals confine

Concrete confined by reinforcementConcrete confined by reinforcement Tests have demonstrated that circular spirals confine

concrete much more effectively than rectangular or square hoopshoops.

Tie Column Spiral Column

80

C t fi d b i f tC t fi d b i f t The reason for the difference between the confinement by

Concrete confined by reinforcementConcrete confined by reinforcement

spirals & hoops is illustrated by Fig.

Ci l i l b f h i h i i l h Circular spirals, because of their shape, are in axial hoop tension and provide a continuous confining pressurecontinuous confining pressure around th i fthe circumference.

In contrast square hoops can apply only confining reactions only confining reactions In contrast, square hoops can apply only confining reactions only confining reactions near the cornersnear the corners of the hoops tends to bend the sides outwards. outwards.

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C fi d C i t thC fi d C i t thConfined Compressive strengthConfined Compressive strength In a recent study by KapposKappos A J ( A J (19911991)) the confined In a recent study by KapposKappos, A. J. (, A. J. (19911991)) the confined

compressive strength can be obtained by multiply KK.

KK = 1 + (w)b f ’cc = KKf ’c

h = f /f ’

fc

f’cc where w = wfyw/f ’cf’c

f cc

a = 0.55, b = 0.75 forc

a = 1.00, b = 1.00 for

a = 1.25, b = 1.00 for

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Effect of Temperature ChangeEffect of Temperature ChangeEffect of Temperature ChangeEffect of Temperature Change Like most material, concrete expands with increasing expands with increasing

temperaturetemperature and contracts with decreasing temperaturecontracts with decreasing temperature.

Th ff t f h l 40Pa)

The effects of such volume changes are similar to those

40

30

/ซม.2 )

T = 70°C

T = 400°C

ngth

(MP

caused by shrinkage.

Th ff f h l Th ff f h l

20

แรงอดั,f’ c(กก./ T = 600°CT = 200°C

sive

str

en

The coefficient of thermal The coefficient of thermal expansion (expansion ()) and contraction

10หน่วย

แ

T = 800°C

Com

pres

s

varies somewhat, depending upon the type of aggregate and richness of the mix

00 0.01 0.02 0.03 0.04 0.05

ความเครยีด, c (ซม./ซม.)

C

Stain, c (mm./mm.)

upon the type of aggregate and richness of the mix.

It is generally within 10x10‐6 /Cog y / .

83

NonNon prestressing reinforcementprestressing reinforcementNonNon--prestressing reinforcementprestressing reinforcement

The nominal yield strength excess of 520 Mpa shall not be used for design purposes.

Bars with yield strengths less than 420 Mpa shall be used only with the approval of the owner.

The modulus of elasticityE of steel reinforcing Es, of steel reinforcing shall be assumed

200 000 Mpas 200,000 Mpa.

84

Idealized of stressIdealized of stress strain relationshipsstrain relationshipsIdealized of stressIdealized of stress--strain relationshipsstrain relationshipsfs (Mpa)

f

fsufsb MPa

MPaff

y

y

y

ysh 400;

300;514

fyyy

MPaf ysh 300;14.0

(mm /mm )

MPaf ysu 400;12.0

s (mm./mm.)

y sush sb

0shsy

ys

y

ss

fE

0ysu ff 5.1

susshysushsu

shsys ffff

)(

sbssusbsususb

susu fff

)(1

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Tension stiffeningTension stiffeningTension stiffeningTension stiffening When the steel is embedded in concrete, the behavior

is different than for the bare steel bars. The tensile strain, of steel between cracks, is reduced.

This phenomenon is “tension stiffening”.

This effect also tends This effect also tends to round off the sharp k if th l tiknee if the elastic-perfectly plastic behavior.

86

Prestressing reinforcementsPrestressing reinforcementsPrestressing reinforcementsPrestressing reinforcements There are two popular prestressed reinforcement has

been used. (1) Prestressing strands (2) Prestressing bars( ) g

The most commonly used type is 7 wire strand The most commonly used type is 7-wire strand. There are two different types of strand are produced.

(1) stress-relieved strand, and (2) stress-relaxation strand

87

Production of Production of ff77--wire strandwire strand

88

StressStress strain curve of PC strandsstrain curve of PC strandsStressStress--strain curve of PC strandsstrain curve of PC strands

89

How to find the yield point of strand??How to find the yield point of strand??How to find the yield point of strand??How to find the yield point of strand??

0.2% offset (AS1310)1% strain (ASTM A421)

90

Material re irement : Material re irement : Prestressing steelMaterial requirement : Material requirement : Prestressing steel Tensile and yield strengths for these steels may be taken as y g y

specified in Table.

Ep = 197,000 Mpa (strand)E = 207 000 Mpa (bar)Ep 207,000 Mpa (bar)

91

Stress limits for prestressing tendonStress limits for prestressing tendonStress limits for prestressing tendonStress limits for prestressing tendon The tendon stress at the strength and extreme event The tendon stress at the strength and extreme event

limit stages shall not exceed the tensile strength limit specified in the Tablep

92

StSt t i f t i f G dG d 270 270 t d t d StressStress--strain curve for strain curve for GradeGrade270 270 strand strand

(M ) (Mpa)

fpu = 1860 MPa

fpy = 0.90fpu= 1674 MPa

pu

fps bonded prestressed= 1674 MPa

fpe = 0.65fpy1088 M

fps Unbonded prestressed

= 1088 Mpa(approximate)

(mm /mm ) (mm./mm.)

93

StressStress strain modelingstrain modelingStressStress--strain modelingstrain modeling Low-relaxation strands (f = 1860 MPa) Low-relaxation strands (fpu = 1860 MPa)

fEf

975.00250 pu

pspspps fEf

10.010))188(1(025.0

Stress-relieved strands (fpu = 1860 MPa)

pups

pspps fEf

167.06 ))121(1(

97.003.0

Untreated strands (fpu = 1655 MPa)

p

( pu )

fEf

103.0 pu

pspspps fEf

5.02))106(1(03.0

94

ส ั ิ ั 7 ส้ ( ) 420 2534คุณสมบตของลวดอดแรง 7 เสน (คลายแรง) มอก 420-2534

เกรด เส้นผ่านศูนย์กลาง

(มม.)

พืน้ที่หน้าตดั

(มม.2)

แรงดงึที่จุดประลัย

(กก.)

แรงดงึที่จุดคราก

(กก.)

1725 9.53

12 70

51.61

92 90

9,070

16 320

8,163

14 68812.70

15.24

92.90

139.35

16,320

24,490

14,688

22,041

1860 9.53 54.84 10,043 9,387

12.70

15.24

98.71

140.00

18,730

26,580

16,857

23,922

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Design consideration : Design consideration : Elastic analysisElastic analysisDesign consideration : Design consideration : Elastic analysisElastic analysis

Stage of loads transfer Stress limits LRFD load combinations for elastic analysis LRFD load combinations for elastic analysis Composite section Estimate number of prestressing strands

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Service stage checkingService stage checkingService stage checkingService stage checking There are two service stage needed to consider.

(1) At transfer At transfer (After short term looses) : The PC girder alone shall be evaluated with deck and gconstruction weight.

(2) At service At service : (After long term looses) The residual stress in PC girder alone by former load shall be stress in PC girder alone by former load shall be considered with composite PC section, girder and deck, by superimposed dead load and live load.by superimposed dead load and live load.

97

Design criteria : Design criteria : STDSTD vsvs LRFDLRFDDesign criteria : Design criteria : STDSTD vsvs LRFDLRFD

Allowable concrete stress : TransferTransferSTD (Mpa) LRFD (Mpa)STD (Mpa) LRFD (Mpa)

Compression in girder

0 60f’ 0 60f ’0.60f ci 0.60f ciTension in girder

-Without bond reinforcementWithout bond reinforcement

Min{0.25sqrt(f’ci), 1.38} Min{0.25sqrt(f’ci), 1.38}-With bond reinforcementWith bond reinforcement

0.63sqrt(f’ci) 0.63sqrt(f’ci)

where fs = min{0.5fy, 206.84}

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Allowable concrete stress : ServiceServiceSTD (Mpa) LRFD (Mpa) – COM 1STD (Mpa) LRFD (Mpa) – COM 1

Compression in girder

(a) Condition 1 (a) Condition 1(a) Condition 10.60f’c

(a) Condition 10.45f’c

(b) Condition 2 (b) Condition 2( )0.40f’c

( )0.60wf’c

(c) Condition 30 40f’

(c) Condition 30 40f’0.40f c 0.40f c

Generally use w = 1.0 (web and flange slenderness ratio)99


Allowable concrete stress : ServiceServiceSTD (Mpa) LRFD (Mpa) – COM 3 – 80% live loadSTD (Mpa) LRFD (Mpa) – COM 3 – 80% live load

Tension in girder

With bonded reinforcement With bonded prestressing stand and reinf - With bonded reinforcement0.5*sqrt(f’c)

- With bonded prestressing stand and reinf. with Moderate corrosion

0.50sqrt(f’c)- For severe corrosive

exposure conditions0.25*sqrt(f’c)

- With bonded prestressing stand & reinf. with severe corrosion0.25sqrt(f’c)q (f c) 0.25sqrt(f c)

- With unbonded prestressing standNo tension is allowed

100

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