Prestressed Concrete Design - Mahidolmucc.mahidol.ac.th/~egpcp/Handout406/40610 Prestressed1.pdf ·...

Prestressed Concrete Bridge DesignBasic Principles Emphasizing AASHTO LRFD Procedures

Praveen Chompreda, Ph. D.

EGCE 406 Bridge Design

MAHIDOL UNIVERSITY

2010

1© 2010 | Praveen Chompreda

Part I: Introduction

Reinforced vs. Prestressed ConcretePrinciple of PrestressingH l PHistorical PerspectiveApplicationsClassifications and TypesRC vs. PC vs. PPCDesign CodesStages of LoadingStages of Loading


Reinforced ConcreteReinforced Concrete

Recall that in Reinforced Concrete… Concrete is strong in compression but weak in tension Concrete is strong in compression but weak in tension Steel is strong in tension (as well as compression) Reinforced concrete uses concrete to resist compression and to hold Reinforced concrete uses concrete to resist compression and to hold

the steel bars in place, and uses steel to resist all of the tension Tensile strength of concrete is neglected (i.e. assumed zero) An RC beam always crack under the service load


Reinforced ConcreteReinforced ConcreteCracking moment of an RC beam is much lower than the service moment

4© 2010 | Praveen Chompreda Source: MacGregor and Wight (2005).

Principle of PrestressingPrinciple of Prestressing

Prestressing is a method in which compression force is applied to the reinforced concrete section.

The effect of prestressing is to reduce the tensile stress in the section to the point that the tensile stress is below the cracking stress. Thus, the

t d t k!concrete does not crack! It is then possible to treat prestressed concrete as an elastic material The concrete can be visualized to have 2 force systems The concrete can be visualized to have 2 force systems

Internal Prestressing Forces External Forces (from DL, LL, etc…)( , , )

These 2 force systems must counteract each other



Stress in concrete section when the prestressing force is applied at the c.g. of the section (simplest case)



Stress in concrete section when the prestressing force is applied eccentrically with respect to the c.g. of the section (typical case)

Smaller Compression

+ + =c.g. + + =

S ll C i

e0

F/A MDLy/I MLLy/I Small Compression

PrestressingF

Stressf DL

Stressf LL

StressR l

Cross-S i

Fe0y/I

Force from DL from LL ResultantSection


Historical PerspectiveHistorical Perspective

The concept of prestressing was invented centuries ago when metal bands were wound around wooden

( ) f b l pieces (staves) to form a barrel.

The metal bands were

Source: Wikipedia (2006)

tighten under tensile stress, which creates compression between the staves –between the staves allowing them to resist internal liquid pressure

8© 2010 | Praveen Chompreda Source: Naaman (2004)

Historical PerspectiveHistorical Perspective

Source: Naaman (2004)

The concept of prestressed concrete is also not new. In 1886, a patent was granted for tightening steel tie rods in concrete blocks. This is analogous to modern day segmental constructions.

E l tt t t f l d t th l t th f t l t Early attempts were not very successful due to the low strength of steel at that time. Since we cannot prestress at high stress level, the prestress losses due to creep and shrinkage of concrete quickly reduce the effectiveness of prestressing.


Historical PerspectiveHistorical Perspective Eugene Freyssinet (1879 1962) was the first to propose Eugene Freyssinet (1879-1962) was the first to propose

that we should use very high strength steel which permit high elongation of steel. The high steel elongation would not be entirely offset by the shortening of concrete (prestress loss) due to creep and shrinkageand shrinkage.

First prestressed concrete bridge Source: Wikipedia (2006)

First prestressed concrete bridge in 1941 in France

First prestressed concrete bridge in US: Walnut Lane Bridge in in US: Walnut Lane Bridge in Pennsylvania. Built in 1949. 47 meter span.


Applications of Prestressed ConcreteApplications of Prestressed Concrete

Bridges Slabs in buildingsg Water Tank Concrete Pile Thin Shell Structures Offshore Platform Nuclear Power Plant Repair and Rehabilitations




Classifications and TypesClassifications and Types

Pretensioning v.s. Posttensioning External v s Internal External v.s. Internal Linear v.s. Circular

End-Anchored v.s. Non End-Anchored Bonded v.s. Unbonded Tendon Precast v.s. Cast-In-Place v.s. Composite Partial v.s. Full PrestressingPartial v.s. Full Prestressing


Classifications and TypesClassifications and Types Pretensioning vs Posttensioning Pretensioning vs. Posttensioning

In Pretension, the tendons are tensioned against some abutments beforethe concrete is place. After the concrete hardened, the tension force is p ,released. The tendon tries to shrink back to the initial length but the concrete resists it through the bond between them, thus, compression f i i d d i P i i ll d i h force is induced in concrete. Pretension is usually done with precast members.



Pretensioned Prestressed ConcretePretensioned Prestressed ConcreteCasting Factory

ConcreteMixer



In Posttension, the tendons are tensioned after the concrete has hardened. Commonly, metal or plastic ducts are placed inside the concrete before casting. After the concrete hardened and had enough strength, the tendon was placed inside the duct, stressed, and anchored against concrete Grout may be injected into the duct later This can be against concrete. Grout may be injected into the duct later. This can be done either as precast or cast-in-place.



Precast Segmental Girder to be Posttensioned In Posttensioned In Place




E l I l P i External vs. Internal Prestressing Prestressing may be done inside or outside

Li Ci l P i Linear vs. Circular Prestressing Prestressing can be done in a straight structure such as beams (linear

prestressing) or around a circular structures, such as tank or silo prestressing) or around a circular structures, such as tank or silo (circular prestressing)

Bonded vs. Unbonded Tendon The tendon may be bonded to concrete (pretensioning or posttensioning

with grouting) or unbonded (posttensioning without grouting). Bonding helps prevent corrosion of tendon Unbonding allows readjustment of helps prevent corrosion of tendon. Unbonding allows readjustment of prestressing force at later times.



End-Anchored vs. Non-End-Anchored tendons In Pretensioning, tendons transfer the prestress through the bond In Pretensioning, tendons transfer the prestress through the bond

actions along the tendon; therefore, it is non-end-anchored In Posttensioning, tendons are anchored at their ends using mechanical

devices to transfer the prestress to concrete; therefore, it is end-anchored. (Grouting or not is irrelevant)



Partial vs. Full Prestressing Prestressing tendon may be used in combination with regular Prestressing tendon may be used in combination with regular

reinforcing steel. Thus, it is something between full prestressedconcrete (PC) and reinforced concrete (RC). The goal is to allow some tension and cracking under full service load while ensuring sufficient ultimate strength.

We sometimes use partially prestressed concrete (PPC) to control We sometimes use partially prestressed concrete (PPC) to control camber and deflection, increase ductility, and save costs.


RC vs. PPC vs. PCRC vs. PPC vs. PC



© 2010 | Praveen Chompreda 21Source: Naaman (2004)



Advantages of PC over RCAdvantages of PC over RC Take full advantages of high strength concrete and high Take full advantages of high strength concrete and high

strength steel Need less materials Smaller and lighter structure No cracks Use the entire section to resist the load Better corrosion resistance

G d f r ater tanks and n clear lant Good for water tanks and nuclear plant

Very effective for deflection controlB tt h i t Better shear resistance


Design Codes for PCDesign Codes for PC

ACI-318 Building Code (Chapter 18)( p )

AASHTO LRFD (Chapter 5)(Chapter 5)

Other related institutions PCI – Precast/Prestressed Concrete Institute PTI – Post-Tensioning Institute


Design PrinciplesDesign Principles

In RC, we primarily design the member for either service limit states (Working stress design method), or ultimate limit states (Working stress design method), or ultimate limit states (Ultimate strength design).

In PC both service limit states and ultimate limit states must In PC, both service limit states and ultimate limit states must be checked. In service limit states section must have stresses below the allowable In service limit states, section must have stresses below the allowable

stress limits In ultimate limit states, the moment and shear capacity must be greater p y g

than the ultimate (factored) loads.

© 2010 | Praveen Chompreda 25

Stages of LoadingStages of Loading

Unlike RC where we primarily consider the capacity of the structure at one stage (i.e. during service), we must consider multiple stages of construction in Prestressed Concrete

The stresses in the concrete section must remain below the maximum li it t ll ti !!!limit at all times!!!




Stages of LoadingStages of Loading

Typical stages of loading considered Typical stages of loading considered

Initial (Immediately Transportation/ Service( yafter Prestress Transfer)• Full prestress force

pErection• Partial loss of

prestress force

• Prestress loss has fully occurredDL SDL LL• Full prestress force

• May or may not include DL

prestress force• DL• Different support

• DL+SDL +LL

(depending on construction type)

ppconditions during erection from serviceservice


Part II: Materials and Hardwares for Prestressingg

ConcretePrestressing SteelPrestressing SteelPrestressing Hardwares


ConcreteConcrete

Mechanical properties of concrete that are relevant concrete that are relevant to the prestressed concrete design includes:g Compressive Strength (f’c ) Modulus of Elasticity (Ec)y ( c) Modulus of Rupture (fr)



Concrete: Compressive StrengthConcrete: Compressive Strength

AASHTO LRFD

For prestressed concrete, f’c at 28 days should be 28-70 MPay

For reinforced concrete, f’c at 28 days should be 16-70 MPay

Concrete with f’c > 70 MPa can be used only when supported by test data


Concrete: Modulus of ElasticityConcrete: Modulus of Elasticity

Modulus of elasticity can be obtained directly from test or

ti t d f i estimated from compressive strength (AASHTO secion 5.4.2.4)

E = 0 043γ 1 5(f’ )0 5 MPa Ec = 0.043γc1.5(f c)0.5 MPa

γc in kg/m3

f’ in MPa f c in MPa

For normal weight concrete, we can use a simplified equationcan use a simplified equationEc =4800(f’c )0.5 MPa


Concrete: Modulus of RuptureConcrete: Modulus of Rupture

Indicates the tensile capacity of concrete under bendingg

Tested simply-supported concrete beam under 4-point bending p gconfiguration

fr = My/I = PL/bd2

Modulus of rupture can also be estimated from compressive strength (AASHTO section 5.4.2.6) fr = 0.63 (f’c)0.5 MPa


Concrete : Summary of PropertiesConcrete : Summary of PropertiesSource: Naaman (2004)

33

Prestressing TendonsPrestressing Tendons

Prestressing tendon may be in the form of strands, wires, round bar, or threaded rodsround bar, or threaded rods

Materials High Strength Steel High Strength Steel Fiber-Reinforced Polymer (FRP) Composites (glass or carbon fibers)


TendonsTendons

Common shapes of prestressing of prestressing tendons

Most Popular (7-wire Strand)( )

35© 2010 | Praveen ChompredaSource: Naaman (2004)

Prestressing SteelPrestressing Steel



( )

Prestressing StrandsPrestressing Strands

Prestressing strands have two grades Grade 250 (f = 250 ksi or 1725 MPa) Grade 250 (fpu 250 ksi or 1725 MPa) Grade 270 (fpu = 270 ksi or 1860 MPa)

Types of strands Types of strands Stressed Relieved Strand Low Relaxation Strand (lower prestress loss due to relaxation of Low Relaxation Strand (lower prestress loss due to relaxation of

strand)



Source: AASHTO (2000)


Prestressing StrandsPrestressing StrandsSource: Naaman (2004)



Modulus of Elasticity 197000 MPa for Strands 197000 MPa for Strands 207000 MPa for Bars

The modulus of The modulus of elasticity of strand is lower than that of lower than that of steel bar because strand is made from strand is made from twisting of small wires together. g



Hardwares & Prestressing EquipmentsHardwares & Prestressing Equipments

Pretensioned Members Hold-Down Devices Hold Down Devices

Posttensioned Members Anchorages Anchorages

Stressing Anchorage Dead-End Anchorageg

Ducts Posttensioning Proceduresg



Pretensioned BeamsPretensioned Beams


Pretensioning HardwaresPretensioning Hardwares

Hold-Down Devices for Pretensioned BeamsPretensioned Beams

Source: Naaman (2004) Source: Naaman (2004)


( )

Posttensioned BeamsPosttensioned Beams

Source: VSL (2006)

Posttension HardwaresSt i A h

Source: VSL (2006)

Stressing Anchorage Dead-End Anchorage Duct/ Grout Tube


Posttensioning Hardwares - AnchoragesPosttensioning Hardwares Anchorages

45© 2010 | Praveen Chompreda Source: VSL (2006)





Posttensioning Hardwares - DuctsPosttensioning Hardwares Ducts

Source: VSL (2006)


Posttensioning ProceduresPosttensioning Procedures


Posttensioning ProceduresPosttensioning Procedures

Grouting is optional (depends on the system used)( p y )


Part III: Prestress Losses

Sources of Prestress LossesL S E i i f P LLump Sum Estimation of Prestress Loss


Prestress LossesPrestress Losses

Prestress force at any time is less than that during jacking Sources of Prestress Loss

Anchorage Set (AS)

Creep of Concrete

(CR)

Friction (FR)

( ) (CR)

Shrinkage of Concrete (FR) of Concrete

(SH)

Prestress L

Elastic Shortening

Prestress Relaxation

LossShortening

(ES)Relaxation

(RE)


Prestress LossesPrestress Losses Sources of Prestress Loss Sources of Prestress Loss

Elastic Shortening : Caused by concrete shortening y gwhen the prestressing force is applied. The tendon tt h d t it l h t attached to it also shorten,

causing a stress loss



Sources of Prestress Loss (cont.) Friction : Friction in the duct of posttensioning system causes stress at

h f d b l h h h j ki d Th h the far end to be less than that at the jacking end. Thus, the average stress over the entire tendon is less than the jacking stress

Source: VSL (2006)

Anchorage Set : The wedge in the

( )

g ganchorage may set in slightly to lock the tendon, causing a loss of stress



Sources of Prestress Loss (cont.) Shrinkage : Concrete shrinks g

over time due to the loss of water, leading to stress loss

tt h d t don attached tendons Creep : Concrete shortens

over time under compressive over time under compressive stress, leading to stress loss on attached tendons



Sources of Prestress Loss


Prestress Loss (cont.)

St l R l ti Steel Relaxation : Steel loss its stress with time due to with time due to constant elongation the elongation, the larger the stress, the larger the lossthe larger the loss.


Time Line of Prestress LossTime Line of Prestress Loss

SHPosttensioning

FR ASES

SHCRRE

Jacking

f j

Initial

f i

Effective

f

ES

fpj fpi fpe

SHPretensioning

Jacking ES

SHCRRE

(ASRE)

Pretensioning

Jacking (against

abutment)

Initial

f

Effective

f

ESRelease (cutting

)

RE)

fpjfpi fpestrands)

Instantaneous Losses Time-Dependent Losses57

Prestress Loss – By TypesPrestress Loss By Types

Pretensioned PosttensionedPretensioned PosttensionedInstantaneous Elastic Shortening Friction

Anchorage SetAnchorage SetElastic Shortening

Time-Dependent Shrinkage (Concrete) Shrinkage (Concrete)Time Dependent Shrinkage (Concrete)Creep (Concrete)Relaxation (Steel)

Shrinkage (Concrete)Creep (Concrete)Relaxation (Steel)( ) ( )


Prestress Loss - PretensionedPrestress Loss Pretensioned


Prestress Loss - PosttensionedPrestress Loss Posttensioned


Lump Sum Prestress LossLump Sum Prestress Loss

Pretress losses can be very complicate to estimate since it depends on so many factorsdepends on so many factors

In typical constructions, a lump sum estimation of prestress loss may be accurate enough This may be expressed in terms of:may be accurate enough. This may be expressed in terms of: Total stress loss (in unit of stress) Percentage of initial prestress Percentage of initial prestress

Some common methods Naaman Naaman ACI-ASCE T Y Lin T.Y. Lin


Lump Sum Prestress LossLump Sum Prestress Loss A E Naaman Method – not including FR AS A. E. Naaman Method – not including FR, AS

Start with 240 MPa for Pretensioned Normal Weight Concrete with Low Relaxation StrandRelaxation Strand

Add 35 MPa for Stress-Relieved Strand or for Lightweight Concrete Deduct 35 MPa for Posttension

Types of Prestress Loss (fpi-fpe) (MPa)

Types of Prestress Types of Concrete Stress-Relieved

StrandLow Relaxation

Strand

P d N l W h C 275 240Pretensioned Normal Weight ConcreteLightweight Concrete

275310

240275

Posttensioned Normal Weight ConcreteLightweight Concrete

240275

205240



ACI-ASCE Committee Method (Zia et al. 1979) This is the Maximum Loss that you may assume This is the Maximum Loss that you may assume

Types of P

Types of Concrete

Maximum Prestress Loss (fpi-fpe) (MPa)

Prestressyp

Stress-Relieved Strand

Low Relaxation Strand

P t i d N l W i ht C t 345 276Pretensioned Normal Weight ConcreteLightweight Concrete

345380

276311



T.Y. Lin & N. H. Burns Method

S f L P f L (%)Sources of Loss Percentage of Loss (%)

Pretensioned Posttensioned

Elastic Shortening (ES) 4 1

Creep of Concrete (CR) 6 5

Shrinkage of Concrete (SR) 7 6

Steel Relaxation (R2) 8 8

Total 25 20Source: Lin and Burns (1981)

Note: Pretension has larger losses because prestressing is usually done when concrete is about 1-2 days old; while posttensioning is done at much later time

when concrete is stronger.64© 2010 | Praveen Chompreda

Lump Sum Prestress LossLump Sum Prestress Loss AASHTO LRFD (for CR SR R2) (5 9 5 3) AASHTO LRFD (for CR, SR, R2) (5.9.5.3)

65© 2010 | Praveen Chompreda Source: AASHTO (2000)


AASHTO LRFD (Cont.) Partial Prestressing Ratio (PPR) is calculated as: Partial Prestressing Ratio (PPR) is calculated as:

ps pyA fPPR

A f A f

PPR = 1.0 for Prestressed Concrete

ps py s yA f A f

PPR = 0.0 for Reinforced Concrete

Elastic Shortening Loss (∆fpES) is calculated as:

20 0

, i

ps ps i GipES cgp F G

E E Fe M eFf fE E A I I

cci ciE E A I I Stress of concrete at the c.g. of tendon due to prestressing force and dead load


Part IV: Allowable Stress Designg

Stress Inequality EquationAllowable Stress in ConcreteAllowable Stress in Prestressing SteelFeasible Domain MethodEnvelope and Tendon Profile


Basics

Sign ConventionConcrete Section PropertiesConcrete Section PropertiesOverview of Design Procedures


Basics: Sign ConventionBasics: Sign Convention

In this class, the following convention is used: Tensile Stress in concrete is negative (-)

C i S i i i i ( ) Compressive Stress in concrete is positive (+) Positive Moment:

P i i Sh Positive Shear:

I b k h i i f b i In some books, the sign convention for stress may be opposite so you need to reverse the signs in some formula!!!!!!!!!


Basics: Section PropertiesBasics: Section Propertiesc.g. of Prestressing TendonConcrete Cross-

IK

g f gArea: Aps

Concrete Cross-Sectiona Area: Ac

Kt

Kbyt

(abs) e (-)

Zt

Zb

( )

kt (-)

( )

Center of Gravity of Concrete Sectionh Zb

yb

kb (+)e (+)

Concrete Section(c.g.c)(abs)

yb

(abs)

c.g. of Prestressing TendonArea: Aps


Basics: Section PropertiesBasics: Section Properties Moment of Inertia, I,

2I y dA Rectangular section about c.g. Ixx = 1/12×bh3

I I + Ad2

A

Ix’x’ = Ixx + Ad2

yt and yb are distance from the c.g. of section to top and bottom fibers respectivelybottom fibers, respectively

Sectional modulus, Z (or S) Z = I/y Zt = I/yt

Zb = I/yb


Basics: Section PropertiesBasics: Section Properties Moment of Inertia for typical sections



( )

Basics: Section PropertiesBasics: Section Properties




Basics: Section Properties Kern of the section, k, is the distance from c.g. where compression force

Basics: Section Propertiesg p

will not cause any tension in the section

Consider Top Fiber(Get Bottom Kern kb)

Consider Bottom Fiber(Get Top Kern k )

00 tFe yFA I

00 bFe yF

(Get Bottom Kern, kb) (Get Top Kern, kt)

cA IIe k

0cA I

I k

0 bc t

e kA y

0 t

c b

e kA y

N T k h i lNote: Top kern has negative valueSource: Nawy (2000)




Basics: DepthsBasics: Depths

Definitions of depths used



General Design ProceduresGeneral Design Procedures

Check Check shear

Check Ultimate moment strength

cracking load

S G

Check allowable stresses at various stages

Select Girder type and number/ location of strands






Allowable Stress

Stress in concrete at various stagesS i li iStress inequality equationAllowable stressesSections


Stress in Concrete at Various StagesStress in Concrete at Various Stages

82© 2010 | Praveen Chompreda Source: Nawy (2000)

Stress in Concrete at Various StagesStress in Concrete at Various Stages


83

( )

Stress Inequality EquationsStress Inequality Equations We can write four equations based on the stress at the We can write four equations based on the stress at the

top and bottom of section at initial and service stages

No. Case Stress Inequality Equation

I Initial-Top

min min1i o oi i

t tic t t c b t

Fe eF M F Mσ σA Z Z A k Z

II Initial-Bottommin min1i o oi i

b cic b b c t b

Fe eF M F Mσ σA Z Z A k Z

III Service-Top

c b b c t b

max max1o oit cs

t t b t

Fe M e MFFσ σA Z Z A k Z!

IV Service-Bottom

c t t c b tA Z Z A k Z

max max1o o

b tsFe M e MF Fσ σ

A Z Z A k Z

!

b tsc b b c t bA Z Z A k Z


Allowable Stress in ConcreteAllowable Stress in Concrete

AASHTO LRFD (5.9.4) provides allowable stress in concrete as functions of compressive strength at that timep g

Consider the following limit states:g Immediately after Prestress Transfer (Before Losses)

Compressionp Tension

Service (After All Losses)( ) Compression Tension



Allowable compressive stress in concrete is used to control creep, which causes prestress loss over timecreep, which causes prestress loss over time

Allowable tensile stress in concrete is used to prevent Allowable tensile stress in concrete is used to prevent cracking, which reduces the usable section (remember that once the concrete cracks it can no longer support tensile once the concrete cracks, it can no longer support tensile stress, even at levels smaller than tensile strength)


Allowable Stress in ConcreteAllowable Stress in Concrete Immediately after Prestress Transfer (Before Losses) Immediately after Prestress Transfer (Before Losses)

Using compressive strength at transfer, f’ci

Allowable compressive stress = 0 60 f’ Allowable compressive stress = 0.60 f ci

Allowable tensile stress

87© 2010 | Praveen Chompreda Source: AASHTO (2000)

Allowable Stress in ConcreteAllowable Stress in Concrete At service (After All Losses) At service (After All Losses) Compressive Stress




At service (After All Losses) At service (After All Losses) Tensile Stress



Allowable Stress in Concrete - SummaryAllowable Stress in Concrete SummaryStage Where Load Limit Noteg

Initial Tension at Top

Fi+MGirder -0.58√f ’ci With bonded reinf…

-0.25√f ’ci Without bonded f ci> -1.38 MPa reinf.

Compression at Bottom

Fi+MGirder 0.60 f ’ci

Service Compression at Top

F+MSustained 0.45f ’c *

0.5(F+MSustained)+MLL+IM 0.40f ’c *

F+MSustained+MLL+IM 0.60Øwf ’c *

Tension F+MSustained+0.8MLL+IM -0.50√f ’c Normal/ Moderate at Bottom (Service III Limit State) exposure

-0.25√f ’c Corrosive exposure

0 U b d d d0 Unbonded tendon* Need to check all of these conditions (cannot select only one) 90

Allowable Stress in Prestressing SteelAllowable Stress in Prestressing Steel

Both ACI and AASHTO code specify the allowable stress in the prestressing steel at jacking and after transferthe prestressing steel at jacking and after transfer Prevents accidental rupture during jacking Control long-term relaxation Control long term relaxation



AASHTO LRFD LRFD (5.9.3)

92Source: AASHTO (2000)

Allowable Stress in Prestressing SteelAllowable Stress in Prestressing Steel ACI-318 (2008)ACI 318 (2008)





Allowable Stress DesignAllowable Stress Design

There are many factors affecting the stress in a prestressed girderp g Prestressing Force (Fi or F) Location of prestress tendon (e0) Section Property (A, Zt or Zb, kt or kb) External moment, which depends on

The Section used (dead load)

How to Start the Design? The Section used (dead load)

Girder Spacing (larger spacing larger moment) Slab Thickness (larger spacing thicker slab)

the Design?

Stages of construction Composite/ Noncomposite behavior


Allowable Stress DesignAllowable Stress Design

For bridges, we generally has a preferred section type for a given range of span length and we can select a girder spacing given range of span length and we can select a girder spacing to be within a reasonable range


SectionsSections

AASHTO Type I-VI Sections

ft m

50 15

75 23

100 30100 30

150 46


SectionsSections

AASHTO Type I-VI Sections (continued)



Bridge Girder SectionsBridge Girder Sections


Bridge Girder SectionsBridge Girder SectionsSource: Nawy (2000)

100

Feasible Domain & Envelopep


Feasible DomainFeasible Domain

For a given section, we need to find the combination of prestressing force (Fi or F, which depends on the number of prestressing force (Fi or F, which depends on the number of strands), and the location of strands (in terms of e0) to satisfy these equationsq

Possible methods: Trying to select some number of strands and locations (Trial & Error) Trying to select some number of strands and locations (Trial & Error) Using “Feasible Domain” Method

Graphical Method Graphical Method


Feasible Domain - EquationsFeasible Domain Equations We can rewrite the stress inequality equations and add one more We can rewrite the stress inequality equations and add one more

equation to them

No Case Stress Inequality EquationNo. Case Stress Inequality Equation

I Initial-Top

0 min

1b ti te k M σ Z

F

II Initial-Bottom

iF

0 min

1t ci be k M σ Z

F

III Service-Top

iF

01

b te k M σ Z !

IV Service-Bottom

0 maxb cs te k M σ ZF

0 max1

t ts be k M σ ZF

!

V Practical Limit

0 maxt ts bF

0 0 ,min 7.5 b c bmpe e y d y cm ,mp


Feasible Domain – Graphical InterpretationFeasible Domain Graphical Interpretation



( )

EnvelopeEnvelope

Feasible domain tells you the possible location and prestressing force at a given section to satisfy the stress inequality equation

We usually use feasible domain to determine the location and prestressing force at the most critical section (e.g. midspan of simply-supported beams)

After we get the prestressing force at the critical section, we need to find the location for the tendon at other points to satisfy stress inequalities

We use the prestressing envelope to determine the location of tendon along We use the prestressing envelope to determine the location of tendon along the length of the beam (tendon profile)


Envelope - EquationsEnvelope Equations We use the same equations as those in the feasible domain, except that we’ve

already known the F or F and want to find e at different points along the beamalready known the F or Fi and want to find e0 at different points along the beam

No. Case Stress Inequality Equation

I Initial-Top

0 min

1b ti te k M σ Z

F

II Initial-Bottom iF

0 min

1t ci be k M σ Z

FIII Service-Top

0 t c biF

0 max1

b cs te k M σ Z!

IV Service-Bottom

0 maxb cs te σF

0 max1

t ts be k M σ Z

!

V Practical Limit

0 maxt ts be k M σ ZF

0 0 min 7.5 b c be e y d y cm 0 0 ,minb c bmpy y


Envelope - EquationsEnvelope Equations

We then have 5 main equations

I & II provide the lower bound of e0 (use minimum of the two)

III and IV provide the upper bound of e (use maximum of the two) III and IV provide the upper bound of e0 (use maximum of the two)

IIIa uses F+MSustained

IIIb uses 0.5(F+MSustained)+MLL+IM

IIIc uses F+MSustained+MLL+IMSustained LL IM

IV uses F+MSustained+0.8MLL+IM

V l l f h ( l h b l l b d) V is a practical limit of the e0 (it is also the absolute lower bound)


Envelope & Tendon ProfileEnvelope & Tendon Profile



( )


109Source: Naaman (2004)


Notes The tendon profile of pretensioned members are either straight or The tendon profile of pretensioned members are either straight or

consisting of straight segments The tendon profile of posttensioned member may be one straight

tendon or smooth curve, but no sharp corners



Alternative to draping the strands at ends, we can put plastic sleeves around some around some strands at supports to prevent the bond transfer so the prestress force will be less at that will be less at that section


Prestressed Concrete Design - Mahidolmucc.mahidol.ac.th/~egpcp/Handout406/40610 Prestressed1.pdf ·...

Documents

Transcript of Prestressed Concrete Design - Mahidolmucc.mahidol.ac.th/~egpcp/Handout406/40610 Prestressed1.pdf ·...