40609 prestressed

Prestressed Concrete Bridge DesignBasic Principles Emphasizing AASHTO LRFD Procedures

Praveen Chompreda, Ph. D.

MAHIDOL UNIVERSITY | 2009 | EGCE 406 Bridge Design

Part I: Introduction

Reinforced vs. Prestressed ConcretePrinciple of PrestressingH l PHistorical PerspectiveApplicationsClassification and TypesAdvantagesDesign CodesStages of LoadingStages of Loading

Reinforced ConcreteReinforced Concrete

Recall Reinforced Concrete knowledge:C b k Concrete is strong in compression but weak in tensionSteel is strong in tension (as well as compression)Reinforced concrete uses concrete to resist Reinforced concrete uses concrete to resist compression and to hold the steel bars in place, and uses steel to resist all of the tensionuses steel to resist all of the tensionTensile strength of concrete is neglected (i.e. zero)RC beam always crack under service load

Reinforced ConcreteReinforced Concrete

Cracking moment of an RC beam is generally Cracking moment of an RC beam is generally much lower than the service moment

Principle of PrestressingPrinciple of Prestressing

Prestressing is a method in which compression force is applied to the reinforced concrete sectionapplied to the reinforced concrete section.The effect of prestressing is to reduce the tensile stress i h i h i h h il i b l in the section to the point that the tensile stress is below the cracking stress. Thus, the concrete does not crack!It is then possible to treat concrete as an elastic materialThe concrete can be visualized to have 2 force systemsThe concrete can be visualized to have 2 force systems

Internal Prestressing ForcesExternal Forces (from DL LL etc )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)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 applied eccentrically with respect to the c.g. of the section (typical case)

Smaller Compression

+ + =c.g.

e0

F/A MDLy/I MLLy/I Small CompressionFe0y/I

PrestressingForce

Stressfrom DL

Stressfrom LL

StressResultant

Cross-Section

Historical PerspectiveHistorical Perspective

The concept of prestressing was invented centuries ago when metal bands were centuries ago when metal bands were wound around wooden pieces (staves) to form a barrel form a barrel.

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

Historical PerspectiveHistorical Perspective

The concept of prestressed concrete is also not new. In 1886 a patent was granted for tightening steel tie rods in 1886, a patent was granted for tightening steel tie rods in concrete blocks. This is analogous to modern day segmental constructionssegmental constructions.Early attempts were not very successful due to low strength of steel at that time Since we cannot prestress 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 shrinkage of concrete quickly reduce the effectiveness of prestressing.

Historical PerspectiveHistorical PerspectiveEugene Freyssinet (1879 1962) was the first to Eugene Freyssinet (1879-1962) was the first to propose that we should use very high strength steel which permit high elongation of steel steel which permit high elongation of steel. The high steel elongation would not be entirely offset by the shortening of concrete entirely offset by the shortening of concrete (prestress loss) due to creep and shrinkage.

First prestressed concrete bridge in 1941 in FrancegFirst prestressed concrete bridge in US: Walnut Lane B id i P l i B il Bridge in Pennsylvania. Built in 1949. 47 meter span.

Applications of Prestressed ConcreteApplications of Prestressed Concrete

BridgesSl b i b ildiSlabs in buildingsWater TankConcrete PileThin Shell StructuresThin Shell StructuresOffshore PlatformNuclear Power PlantRepair and Rehabilitationsp

Classification and TypesClassification and Types

Pretensioning v.s. PosttensioningExternal v.s. InternalLinear v s CircularLinear v.s. CircularEnd-Anchored v.s. Non End-AnchoredBonded v.s. Unbonded TendonP t C t I Pl C itPrecast v.s. Cast-In-Place v.s. CompositePartial v.s. Full Prestressingg

Classification and TypesClassification and TypesPretensioning vs. Posttensioningg g

In Pretension, the tendons are tensioned against some abutments before the concrete is place After the abutments before the concrete is place. After the concrete hardened, the tension force is released. The tendon tries to shrink back to the initial length but the tendon tries to shrink back to the initial length but the concrete resists it through the bond between them, thus, compression force is induced in concrete Pretension is compression 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 concrete has hardened. Commonly, metal or plastic ducts are placed inside the concrete before casting. After the concrete hardened and had enough strength 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 anchored against concrete. Grout may be injected into the duct later. This can be done either as precast or

t i lcast-in-place.


Precast Segmental Girder to be Posttensioned In Posttensioned In Place


E l I l PExternal vs. Internal PrestressingPrestressing may be done inside or outsideg y

Linear vs. Circular PrestressingPrestressing can be done in a straight structure such as Prestressing can be done in a straight structure such as beams (linear prestressing) or around a circular structures such as tank or silo (circular prestressing)structures, such as tank or silo (circular prestressing)

Bonded vs. Unbonded TendonThe tendon may be bonded to concrete (pretensioning or posttensioning with grouting) or unbonded ( i i i h i ) B di h l (posttensioning without grouting). Bonding helps prevent corrosion of tendon. Unbonding allows

dj t t f t i f t l t tireadjustment of prestressing force at later times.


End-Anchored vs. Non-End-Anchored tendonsI P d f h In Pretensioning, tendons transfer the prestress through the bond actions along the tendon; therefore, it is non-end-anchoredIn Posttensioning, tendons are anchored at their ends gusing mechanical devices to transfer the prestress to concrete; therefore, it is end-anchored. (Grouting or ; , ( gnot is irrelevant)


Partial vs. Full PrestressingP d b d b h Prestressing tendon may be used in combination with regular reinforcing steel. Thus, it is something between full prestressed concrete (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 camber and deflection, increase (PPC) to control camber and deflection, increase ductility, and save costs.

RC vs PPC vs PCRC vs. PPC vs. PC

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

N d l i lNeed less materialsSmaller and lighter structureNo cracksUse the entire section to resist the loadBetter corrosion resistanceGood for water tanks and nuclear plantGood for water tanks and nuclear plant

Very effective for deflection controlBetter shear resistance

Design Codes for PCDesign Codes for PC

ACI-318 Building Code (Chapter 18)AASHTO LRFD (Chapter 5)

Other institutionsPCI – Precast/Prestressed Concrete InstitutePTI – Post-Tensioning InstitutePTI Post Tensioning Institute

Stages of LoadingStages of Loading

Unlike RC where we primarily consider the lti t l di t t id lti l ultimate loading stage, we must consider multiple

stages of construction in Prestressed ConcreteThe stresses in the concrete section must remain below the maximum limit at all times!!!below the maximum limit at all times!!!


Typical stages of loading considered are Initial d S i Stand Service Stages

Initial (Immediately after Transfer of Prestress)( y )Full prestress force N M ( t h M d di No MLL (may or may not have MDL depending on construction type)

ServicePrestress loss has occurredPrestress loss has occurredMDL+MLL


For precast construction, we have to investigate some intermediate states during transportation some intermediate states during transportation and erection

Part II: Materials and Hardwares for Prestressingg

ConcretePrestressing SteelPrestressing HardwaresPrestressing Hardwares

ConcreteConcrete

Mechanical properties of concrete that are relevant concrete that are relevant to the prestressed concrete design:concrete design:

Compressive StrengthM d l f El i iModulus of ElasticityModulus of Rupture

Concrete: Compressive StrengthConcrete: Compressive Strength

AASHTO LRFD

For prestressed concrete, the compressive strength should compressive strength should be from 28-70 MPa at 28 daysFor reinforced concrete, the ,compressive strength should be from 16-70 MPa at 28 daysConcrete with f’c > 70 MPa can be used when supported by test data

Concrete: Modulus of ElasticityConcrete: Modulus of Elasticity

AASHTO (5.4.2.4)E = 0 043γ 1.5(f’ )0.5 MPaEc 0.043γc (f c) MPa

γc1.5 in kg/m3

f’c in MPaf c in MPa

For normal weight concrete, we can useEc =4800(f’c)0.5 MPa

Concrete: Modulus of RuptureConcrete: Modulus of Rupture

Indicates the tensile capacity of concrete under bendinggTested simply-supported concrete beam under 4-point pbending configurationfr = My/I = PL/bd2

AASHTO (5.4.2.6)fr = 0.63 (f’c)0.5 MPa

Concrete : Summary of PropertiesConcrete : Summary of Properties Prestressing TendonsPrestressing Tendons

Prestressing tendon may be in the form of t d i d b th d d dstrands, wires, round bar, or threaded rods

MaterialsHigh Strength SteelFib R i f d C it ( l b fib )Fiber-Reinforced Composite (glass or carbon fibers)

TendonsTendons

Common shapes of prestressing of prestressing tendons

Most Popular (7-wire Strand)( )

Prestressing SteelPrestressing Steel

Prestressing StrandsPrestressing Strands

Prestressing strands have two gradesG d 250 (f 250 k 1725 MP )Grade 250 (fpu = 250 ksi or 1725 MPa)Grade 270 (fpu = 270 ksi or 1860 MPa)( pu )

Types of strandsS d R li d S dStressed Relieved StrandLow Relaxation Strand (lower prestress loss due to relaxation of strand)

Prestressing StrandsPrestressing Strands

Prestressing StrandsPrestressing Strands Prestressing StrandsPrestressing Strands

Modulus of Elasticity197000 MPa for Strand197000 MPa for Strand207000 MPa for Bar

Th d l f l i i The modulus of elasticity of strand is lower than that of steel bar because strand is made from twisting of small wires together.

Hardwares & Prestressing EquipmentsHardwares & Prestressing Equipments

Pretensioned MembersH ld D DHold-Down Devices

Posttensioned MembersAnchorages

St i A hStressing AnchorageDead-End Anchorage

DuctsPosttensioning Proceduresg

Pretensioned BeamsPretensioned Beams

Pretensioning HardwaresPretensioning Hardwares

Hold-Down Devices for Pretensioned BeamsPretensioned Beams

Posttensioned BeamsPosttensioned Beams

Posttension HardwaresSt i A hStressing AnchorageDead-End AnchorageDuct/ Grout Tube

Posttensioning Hardwares - AnchoragesPosttensioning Hardwares Anchorages Posttensioning Hardwares - AnchoragesPosttensioning Hardwares Anchorages

Posttensioning Hardwares - AnchoragesPosttensioning Hardwares Anchorages Posttensioning Hardwares - DuctsPosttensioning Hardwares Ducts

Posttensioning ProceduresPosttensioning Procedures Posttensioning ProceduresPosttensioning Procedures

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

Part III: Prestress Losses

Sources of Prestress LossesLump Sum Estimation of Prestress Loss

Prestress LossesPrestress LossesPrestress force at any time is less than that during jackingPrestress force at any time is less than that during jackingSources of Prestress Loss

Elastic Shortening :Because concrete Because concrete shortens when the prestressing force is prestressing force is applied to it. The tendon attached to it tendon attached to it also shorten, causing stress loss

Prestress LossesPrestress Losses

Sources of Prestress Loss (cont.)Friction : Friction in the duct of posttensioning system causes p g ystress at the far end to be less than that at the jacking end. Thus, the average stress is less than the jacking stress

Anchorage Set : The wedge in the h i li h l l k anchorage may set in slightly to lock

the tendon, causing a loss of stress


Sources of Prestress Loss (cont.)(cont.)

Shrinkage : Concrete shrinks over time due to shrinks over time due to the loss of water, leading to stress loss on attached to stress loss on attached tendonsCreep : ConcreteCreep : Concreteshortens over time under compressive stress compressive stress, leading to stress loss on attached tendonsattached 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 AS

SHCRRE

g

Jacking

f

Initial

f

Effective

f

ES RE

fpj fpi fpe

SHPretensioning

J ki ES

SHCRRE

(ASRE)

Pretensioning

Jacking (against

abutment)

Initial

f

Effective

f

ES RERelease

(cutting

RE)

abutment)

fpjfpi fpe

( gstrands)

Instantaneous Losses Time-Dependent Losses

Prestress Loss – By TypesPrestress Loss By Types

Pretensioned PosttensionedPretensioned Posttensioned

Instantaneous Elastic Shortening FrictionA SAnchorage SetElastic Shortening

Time-Dependent

Shrinkage (Concrete)Creep (Concrete)

Shrinkage (Concrete)Creep (Concrete)

Relaxation (Steel) 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 ti t i it d d f testimate since it depends on so many factors

In typical constructions, a lump sum estimation of yp , pprestress loss is enough. This may be expressed in terms of:in terms of:

Total stress loss (in unit of stress)Percentage of initial prestress

Lump Sum Prestress LossLump Sum Prestress LossA. E. Naaman (with slight modifications) – not including FR, ASA. E. Naaman (with slight modifications) not including FR, AS

Start with 240 MPa for Pretensioned Normal Weight Concrete with Low Relaxation StrandAdd 35 MPa for Stress-Relieved Strand or for Lightweight Concrete D d 35 MP f PDeduct 35 MPa for Posttension

P t L (f i f ) (MP )Types of Prestress Types of Concrete

Prestress Loss (fpi-fpe) (MPa)

Stress-Relieved Strand

Low Relaxation StrandStrand Strand

Pretensioned Normal Weight ConcreteLi ht i ht C t

275310

240275Lightweight Concrete 310 275

Posttensioned Normal Weight Concrete 240 205Lightweight Concrete 275 240

Lump Sum Prestress LossLump Sum Prestress Loss

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

T f Maximum Prestress Loss

(fpi fpe) (MPa)Types of Prestress Types of Concrete

(fpi-fpe) (MPa)

Stress-Relieved Strand

Low Relaxation StrandStrand Strand

Pretensioned Normal Weight ConcreteLightweight Concrete

345380

276311Lightweight Concrete 380 311


T.Y. Lin & N. H. BurnsS f L P f L (%)Source 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 8Steel Relaxation (R2) 8 8

Total 25 20

Note: Pretension has larger loss because prestressing is usually done when concrete is about 1-2 days old whereas Posttensioning done when concrete is about 1-2 days old whereas Posttensioning is done at much later time when concrete is stronger.

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


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

ps pyA fPPR =

PPR = 1 0 for Prestressed Concreteps py s y

PPRA f A f

=+

PPR = 1.0 for Prestressed ConcretePPR = 0.0 for Reinforced Concrete

Elastic Shortening Loss (Δf ) is calculated as:Elastic Shortening Loss (ΔfpES) is calculated as:

20 0ps ps i Gi

E E Fe M eFf f⎡ ⎤

Δ ⎢ ⎥0 0

, i

ps ps i GipES cgp F G

cci ci

f fE E A I I+Δ = = + −⎢ ⎥

⎣ ⎦

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 ConventionBasics: Sign Convention

In this class, the following convention is used:Tensile Stress in concrete is negative (-)g ( )Compressive Stress in concrete is positive (+)Positive Moment:

Positive Shear:Positive Shear:

In some books, the sign convention for stress may be , g yopposite 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 PropertiesMoment of Inertia, IMoment of Inertia, I

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

A

Ix’x’ = Ixx + Ad2

yt and yb are distance from the c.g. of section to yt and yb are distance from the c.g. of section to top and bottom fibers, respectivelySectional modulus Z (or S)Sectional modulus, Z (or S)

Zt = I/yt

Zb = I/yb

Basics: Section PropertiesKern of the section k is the distance from c g

Basics: Section PropertiesKern of the section, k, is the distance from c.g. where compression force will not cause any

i i h itension in the section

C id T p Fib C id B tt FibConsider Top Fiber(Get Bottom Kern, kb)

Consider Bottom Fiber(Get Top Kern, kt)

00 tFe yFA I

= − 00 bFe yFA I

= +cA II

cA II

0 bc t

Ie kA y

= = 0 tc b

Ie kA y

= − =

Note: Top kern has negative value

Basics: General Design ProceduresBasics: General Design Procedures

Select Girder type, materials to be used, and b f t i t dnumber of prestressing strands

Check allowable stresses at various stagesgCheck ultimate moment strengthCheck cracking loadCheck shearCheck shearCheck deflection

Stress in Concrete at Various StagesStress in Concrete at Various Stages

Stress Inequality EquationsStress Inequality EquationsWe 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 EquationI Initial-Top ⎛ ⎞

= − + = − + ≥⎜ ⎟⎝ ⎠

min min1i o oi it ti

c 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


⎛ ⎞= + − = − − ≤⎜ ⎟

⎝ ⎠

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

!

Bottom ⎜ ⎟⎝ ⎠

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 p gtimeConsider the following limit states:Consider the following limit states:

Immediately after Prestress Transfer (Before Losses)Immediately after Prestress Transfer (Before Losses)CompressionTensionTension

Service (After All Losses)C iCompressionTension

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

Using compressive strength at transfer, f’ci

All bl i 0 60 f’Allowable compressive stress = 0.60 f’ci

Allowable tensile stress

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

Allowable Stress in ConcreteAllowable Stress in ConcreteAt 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 > -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)

Allowable Stress in Prestressing SteelAllowable Stress in Prestressing Steel

ACI and AASHTO code specify the allowable t i th t i t l t j ki d ft stress in the prestressing steel at jacking and after

transfer


AASHTO LRFD LRFD (5.9.3)

Allowable Stress in Prestressing SteelAllowable Stress in Prestressing SteelACI-318 (2002)ACI 318 (2002)


Allowable Stress DesignAllowable Stress Design

There are many factors affecting the stress in a prestressed girderprestressed girder

Prestressing Force (Fi or F)L f d ( 0)Location of prestress tendon (e0)Section Property (A, Zt or Zb, kt or kb)External moment, which depends on

The Section used (dead load)( )Girder Spacing (larger spacing larger moment)Slab Thickness (larger spacing thicker slab)

Stages of construction


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

SectionsSections

AASHTO Type I-VI SectionsI-VI Sections

ft m50 1575 23100 30100 30150 46

SectionsSections

AASHTO Type I-VI Sections (continued)

Bridge Girder SectionsBridge Girder Sections Bridge Girder SectionsBridge Girder Sections


For a given section, we need to find the bi ti f t i f (F F hi h combination 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 equationsthese equationsPossible methods:

Keep trying some number of strands and locations (Trial & Error)( )We use “Feasible Domain” Method

Feasible Domain - EquationsFeasible Domain EquationsWe 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 EquationI Initial-Top ( )⎛ ⎞

≤ + −⎜ ⎟0 min1

b ti te k M σ Z

II Initial-Bottom

( )⎜ ⎟⎝ ⎠

0 minb ti ti

e σF

( )⎛ ⎞≤ + +⎜ ⎟

1e k M σ Z

III Service-Top

( )≤ + +⎜ ⎟⎝ ⎠

0 mint ci bi

e k M σ ZF

( )⎛ ⎞≥ ⎜ ⎟1k M Zp

IV Service

( )⎛ ⎞≥ + −⎜ ⎟⎝ ⎠

0 maxb cs te k M σ ZF

( )⎛ ⎞1

!

IV Service-Bottom

V P i l Li i

( )⎛ ⎞≥ + +⎜ ⎟⎝ ⎠

0 max1

t ts be k M σ ZF

( )V Practical Limit ( )0 0 ,min 7.5 b c bmpe e y d y cm≤ = − = −

Feasible Domain – Graphical InterpretationFeasible Domain Graphical Interpretation Feasible DomainFeasible Domain

Feasible domain tells you the possible location and prestressing force at a given section to satisfy the stress inequality equationWe usually use feasible domain to determine location

d i f h i i l i ( 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 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 inequalitiespoints to satisfy stress inequalitiesWe use the prestressing envelope to determine the location of tendon along the length of the beam (tendon g g (profile)

Envelope - EquationsEnvelope EquationsWe use the same equation as the feasible domain, except that we’ve already known the F or Fi and want to find e0 at different points along already known the F or Fi and want to find e0 at different points along the beam

No Case Stress Inequality EquationNo. Case Stress Inequality EquationI Initial-Top ( )⎛ ⎞

≤ + −⎜ ⎟0 min1

b ti te k M σ Z

II Initial-Bottom

( )⎜ ⎟⎝ ⎠

0 minb ti ti

e σF

( )⎛ ⎞≤ + +⎜ ⎟

1e k M σ Z

III Service-Top

( )≤ + +⎜ ⎟⎝ ⎠

0 mint ci bi

e k M σ ZF

( )⎛ ⎞≥ ⎜ ⎟1k M Zp

IV Service

( )⎛ ⎞≥ + −⎜ ⎟⎝ ⎠

0 maxb cs te k M σ ZF

( )⎛ ⎞1

!

IV Service-Bottom

V P i l Li i

( )⎛ ⎞≥ + +⎜ ⎟⎝ ⎠

0 max1

t ts be k M σ ZF

( )V Practical Limit ( )0 0 ,min 7.5 b c bmpe e y d y cm≤ = − = −

Envelope - EquationsEnvelope Equations

We then have 5 main equationsI & II provide the lower bound of e (use minimum of the I & II provide the lower bound of e0 (use minimum of the two)III d IV id th b d f ( i III and IV provide the upper bound of e0 (use maximum of the two)

III F+MIIIa uses F+MSustained

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

IIIc uses F+M +MIIIc uses F+MSustained+MLL+IM

IV uses F+MSustained+0.8MLL+IM

V is a practical limit of the e (it is also the absolute V is a practical limit of the e0 (it is also the absolute lower bound)

Envelope & Tendon ProfileEnvelope & Tendon Profile Envelope & Tendon ProfileEnvelope & Tendon Profile

Envelope & Tendon ProfileEnvelope & Tendon Profile

NoteTh d f l f d b The tendon profile of pretensioned members are either straight or consisting of straight segmentsThe tendon profile of posttensioned member may be one straight tendon or smooth curved, but no sharp g pcorners

Envelope & Tendon ProfileEnvelope & Tendon Profile

There is an alternative to draping the strands in t i d bpretensioned member

We put plastic sleeves around some strands at p psupports to prevent the bond transfer so the prestress force will be less at that sectionprestress force will be less at that section

Part II: Ultimate Strength gDesigng

Concrete and Prestressing Steel StressesCracking MomentCracking MomentFailure TypesA l i f M R t l S tiAnalysis for Mn – Rectangular SectionT-SectionA l i f M T S iAnalysis for Mn – T- Section

Load – Deflection – Concrete StressLoad Deflection Concrete Stress

Load - DeflectionLoad Deflection

1 & 2: Theoretical camber (upward deflection) of prestressed beam3: Self weight + Prestressing force4: Zero deflection point (Balanced point) with uniform p ( p )stress across section5: Decompression point where tension is zero at the b f bbottom fiber6: Cracking point where cracking moment is reached7: End of elastic range (the service load will not be larger than this)8 Yi ldi f i l8: Yielding of prestressing steel9: Ultimate strength (usually by crushing of concrete)

Prestressing Steel StressPrestressing Steel Stress

Prestressing Steel StressPrestressing Steel StressThe prestressing steel stress increases as the load p gincreasesCracking of beam causes a jump in stress as additional g j ptension force is transferred from concrete (now cracked) to prestressing steelAt ultimate of prestressed concrete beam, the stress in steel is somewhere between yield strength fpy and y g fpyultimate strength fpu

Stress is lower for unbonded tendon because stress is distributed throughout the length of the beam instead of just one section as in the case of bonded tendonAt ultimate, the effect of prestressing is lost and the section behaves just like an RC beamj

Cracking MomentCracking Moment

Concrete cracks when bottom fiber reaches the tensile capacity (modulus of rupture)capacity (modulus of rupture)

fr = -0.63 (f’c)0.5 MPa (5.4.2.6)

Cracking MomentCracking Moment

The moment at this stage is called “cracking moment” which depends on the geometry of the section and which depends on the geometry of the section and prestressing force

1o cr o crb r

Fe M e MF Fσ fA Z Z A k Z

⎛ ⎞= + − = − − =⎜ ⎟

⎝ ⎠

Solve the above equation to get M

c b b c t bA Z Z A k Z⎝ ⎠

Solve the above equation to get Mcr

( )cr o t r bM F e k f Z= − −( )cr o t r b

Note: Need to input fr and kt as negative values !!!

Failure TypesFailure Types

This is similar to RC

Fracture of steel after concrete cracking. This is a sudden failure and occurred because the beam has too little reinforcementCrushing of concrete after some yielding of steel. This is called tension-controlled.called tension controlled.Crushing of concrete before yielding of steel. This is a brittle failure due to too much reinforcement It is called brittle failure due to too much reinforcement. It is called overreinforced or compression-controlled.

Failure TypesFailure Types Analysis for Ultimate Moment CapacityAnalysis for Ultimate Moment Capacity

Analysis assumptionsPl l f b d (l Plane section remains plane after bending (linear strain distribution)Perfect bond between steel and concrete (strain compatibility)p y)Concrete fails when the strain is equal to 0.003Tensile stren th f c ncrete is ne lected at ltimateTensile strength of concrete is neglected at ultimateUse rectangular stress block to approximate concrete stress distribution

Analysis for Ultimate Moment CapacityAnalysis for Ultimate Moment Capacity

Recall from RC Design that the followings must b ti f t ll ti tt h t hbe satisfy at all times no matter what happens:

EQUILIBRIUM

STRAIN COMPATIBILITY

They also hold in Prestressed Concrete!They also hold in Prestressed Concrete!


For equilibrium, there are commonly 4 forcesC Compression in concreteCompression in Nonprestressed reinforcementp pTension in Nonprestressed reinforcementTension in Prestressed reinforcementTension in Prestressed reinforcement

For concrete compression, we still use the ACI’s rectangular stress blockrectangular stress block

Rectangular Stress BlockRectangular Stress Block Rectangular Stress BlockRectangular Stress Block

0.85 ' 28 MPa' 28 1

cff

≤⎧⎪

⎛ ⎞⎪ ⎛ ⎞1

' 280.85 0.05 28 ' 56 MPa7

11

cc

fβ f⎪ −⎛ ⎞⎪= − ≤ ≤⎨ ⎜ ⎟

⎝⎛ ⎞⎜ ⎟⎠⎪ ⎝ ⎠

' 56 MPa0.65 cf⎝ ⎠⎪

≥⎩

⎝ ⎠⎪

β1 is equal to 0 85 for f ’ < 28 MPaβ1 is equal to 0.85 for f c < 28 MPa

It decreases 0.05 for every 7 MPa increases in f ’cy f c

Until it reaches 0.65 at f ’c > 56 MPa


For tension and compression in nonprestressed i f t d th thi i RC reinforcement, we do the same thing as in RC

design:Assume that the steel yield first; i.e. Ts = Asfy or Cs = As’fy’Ts Asfy or Cs As fyCheck the strain in reinforcement to see if they actually yield or not if not calculate the stress based actually yield or not, if not, calculate the stress based on the strain at that level & revise the analysisto find new value of neutral axis depth c to find new value of neutral axis depth, c Ts = Asfs = AsEsεs = AsEs· 0.003(c-d)/c


For tension in prestressing steel we prestressing steel, we observe that we cannot assume the cannot assume the behavior of prestressing steel prestressing steel (which is high strength t l) t b l tisteel) to be elastic-

perfectly plastic as in h f l the case of steel

reinforcement in RC


At ultimate of prestressed concrete beam, the stress in steel is clearly not the yield strength but somewhere steel is clearly not the yield strength but somewhere between yield strength fpy and ultimate strength fpu

W ll d i fWe called it fps

The true value of stress is difficult to calculate (generally requires nonlinear moment-curvature analysis) so we generally estimate it using semi-empirical formulag y g p

ACI Bonded Tendon or Unbonded TendonAASHTO Bonded Tendon or Unbonded Tendon

Ultimate Stress in Steel: fUltimate Stress in Steel: fps

AASHTO LRFD Specifications For Bonded tendon only (5 7 3 1 1) and for fp > 0 5fpFor Bonded tendon only (5.7.3.1.1) and for fpe > 0.5fpu

fc⎛ ⎞ ⎛ ⎞1 ; 2 1.04 py

ps pup pu

fcf f k kd f

⎛ ⎞ ⎛ ⎞= − = −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

Note: for preliminary design, we may conservatively

p p⎝ ⎠ ⎝ ⎠

p y g y yassume fps=fpy (5.7.3.3.1)

For Unbonded tendon, see 5.7.3.1.2

Ultimate Stress in Steel: fUltimate Stress in Steel: fps Analysis for Ultimate Moment CapacityAnalysis for Ultimate Moment Capacity

Notes on Strain Compatibility

The strain in top of concrete at ultimate is 0.003We can use similar triangle to find the strains in concreteor reinforcing steel at any levels from the top straing y pWe need to add the tensile strain due to prestressing(occurred before casting of concrete in pretensioned or (occurred before casting of concrete in pretensioned or before grouting in posttensioned) to the strain in concrete at that level to get the true strain of the concrete at that level to get the true strain of the prestressing steel

Maximum & Minimum ReinforcementMaximum & Minimum Reinforcement

M i R i f t (5 7 3 3 1)Maximum Reinforcement (5.7.3.3.1)The maximum of nonprestressed and prestressed reinforcement shall be such that c/de ≤ 0.42c/de = ratio between neutral axis depth (c) and the centroid depth of the tensile force (de)

Minimum Reinforcement (5.7.3.3.2)Th i i f t d d t d The minimum of nonprestressed and prestressed reinforcement shall be such thatØM 1 2M (M ki ) ØMn > 1.2Mcr (Mcr = cracking moment), orØMn > 1.33Mu (Mu from Strength Load Combinations)

Resistance Factor φResistance Factor φR i t F t Ø

Section Type

Resistance Factor Ø

RC and PPC PPC with PCRC and PPC w/ PPR < 0.5

PPC with 0.5< PPR < 1

PC(PPR = 1.0)

Under-Reinforced Section 0 90 0 90 1 00Under-Reinforced Sectionc/de ≤ 0.42

0.90 0.90 1.00

O R i f d S ti N t 0 70 0 70Over-Reinforced Sectionc/de > 0.42

Not Permitted

0.70 0.70

Note: if c/de > 0.42 the member is now considered a e compression member and different resistance factor applies (see 5.5.4.2)AASHTO d es n t ermit the se f er reinf rced RC AASHTO does not permit the use of over-reinforced RC (defined as sections with PPC < 0.5) sections

Rectangular vs T-SectionRectangular vs. T Section

Most prestressed concrete beams are either I-Shaped or T-pshaped (rarely rectangular) so they have larger compression flange flange If the neutral axis is in the flange we called it rectangular flange, we called it rectangular section behavior. But if the neutral axis is below the flange gof the section, we call it T-section behavior

This has nothing to do with the overall shape of the section !!!


If it i T S ti b h i th t l f idth If it is a T-Section behavior, there are now two value of widths, namely b (for the top flange), and bw (web width)We need to consider nonuniform width of rectangular stress block


We generally assume that the section is rectangular first and We generally assume that the section is rectangular first and check if the neutral axis depth (c) is above or below the flange thickness hfflange thickness, hf

Note:ACI method checks a=ß1c with hf, which may give li htl diff t lt h < h b t > hslightly different result when a < hf but c > hf

T-Section AnalysisT Section Analysis

We divide the compression side into 2 partsO h f fl ( d h b b )Overhanging portion of flange (width = b-bw)Web part (width = bw)p ( w)


From equilibrium

1 10.85 ' 0.85 ' ( ) ' 'c w c w f ps ps s y s yf b β c f b b β h A f A f A f+ − = + −

For preliminary analysis, or first iteration, we may assume fps = fpyand solve for cand solve for c

1' ' 0.85 ' ( )0 85 '

ps y s y s y c w fA f A f A f f b b β hc

f b β+ − − −

=10.85 'c wf b β


For a more detailed approach, we recall the equilibrium

1 10.85 ' 0.85 ' ( ) ' 'c w c w f ps ps s y s yf b β c f b b β h A f A f A f+ − = + −

⎛ ⎞Substitute 1ps pu

cf f kd

⎛ ⎞= −⎜ ⎟⎜ ⎟

⎝ ⎠, Rearrange and solve for c

' ' 0 85 ' ( )A f A f A f f b b β h

pd⎜ ⎟⎝ ⎠

+ − − −=

+1

1

' ' 0.85 ' ( )0 85 ' /

ps pu s y s y c w fA f A f A f f b b β hc

f b β kA f d+10.85 /c w ps pu pf b β kA f d


Moment Capacity (about a/2)

' ' '2 2 2n ps ps p s y s s y sa a aM A f d A f d A f d⎛ ⎞ ⎛ ⎞ ⎛ ⎞= − + − − −⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠

1

2 2 2

0 85 ' ( ) ff

hf b b β h a

⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎛ ⎞+ − −⎜ ⎟10.85 ( )

2c w ff b b β h a+ ⎜ ⎟⎝ ⎠

T-Section Analysis FlowchartT Section Analysis Flowchart

T-Section Analysis FlowchartT Section Analysis Flowchart T-SectionT SectionIn actual structures, the section is perfect T or I shapes -p pthere are some tapering flanges and fillets. Therefore, we need to idealized the true section to simplify the analysis. Little accuracy may be lost.

We need this for ultimate analysis only. We should use the true section property for the allowable stress analysis/ design

Part III: Composite Beamp

Typical Composite SectionComposite Section Propertiesp pActual, Effective, and Transformed WidthsAllowable Stress DesignStress Inequality Equation, Feasible Domain, and EnvelopeCracking MomentUl i M C iUltimate Moment Capacity

CompositeComposite

Composite generally means the use of two diff t t i l i t t l l tdifferent materials in a structural elementsExample: Reinforced Concretep

Concrete – carry compressionSt l R i f t t iSteel Reinforcement – carry tension

Example: Carbon Fiber Compositep pCarbon Fiber – carry tensionE Resin Matri h ld the fibers in lace Epoxy Resin Matrix – hold the fibers in place

Composite BeamComposite Beam

In the context of bridge design, the word it b th f t diff t composite beam means the use of two different

materials between the beam and the slabSteel Beam + Concrete Slab

Steel beam carries tensionSteel beam carries tensionConcrete in slab carries compression

Prestressed Concrete Beam (high strength concrete) Prestressed Concrete Beam (high-strength concrete) + Concrete Slab (normal-strength concrete)

P d C b i iPrestressed Concrete beam carries tensionConcrete in slab carries compression

Typical Composite SectionsTypical Composite Sections

Typical Composite SectionsTypical Composite Sections

Slab may be cast:E l l Entirely cast-in-place with removable formwork

Using precast panel as a formwork the as a formwork, the pour the concrete toppingtopping

Why Composite?Why Composite?

There are some benefits of using precast l telementsSave TimeBetter Quality ControlCheaperCheaper

There are some benefits of putting the composite slab

Provide continuity between elementsProvide continuity between elementsQuality control is not that important in slabs

Particular Design AspectsParticular Design Aspects

There are 3 more things we need to consider specially for composite section (on top of stuffs we need to for composite section (on top of stuffs we need to consider for noncomposite sections)T f i f S iTransformation of Section

Actual width vs. Effective width vs. Transformed widthComposite Section Properties

Loading Stagesg gAllowable Stress DesignShored vs. Unshored BeamsS o vs. U s o a s

Horizontal Shear Transfer

Composite Section PropertiesComposite Section Properties

There are 3 value of widths we will use:A l d h f h (b) Th Actual width of the composite section (b): This is equal to the girder spacingEffective width of the composite section (be)Transformed width of the composite section (b )Transformed width of the composite section (btr)

Composite Section PropertiesComposite Section PropertiesEffective Width

The stress distribution across the width are not uniform – the farther it is from the center, the lesser the stress.To simplify the analysis, we assume an effective width where the stress are constant throughout We also assume the effective width to be constant along the span.

Composite Section PropertiesComposite Section PropertiesEffective Width s(AASHTO LRFD - 4.6.2.6.1)

bebets bf

bwboverhang

⎧ bExterior Girder

Interior Girder

⎧= ⎨

⎩' max

/ 2w

wf

bb

b

Exterior Beam Interior Beam

' / 2 6b t+⎧ ' 12b t+⎧,int

,

' / 2 6min

2

w se

e ext overhang

b tb

b b+⎧

⎪= + ⎨⎪

12min

/ 4

w s

e

b tb s

L

+⎧⎪= ⎨⎪2

/ 8L⎪⎩ / 4L⎪

⎩

Composite Section PropertiesComposite Section PropertiesTransformed WidthTransformed Width

Typically the concrete used for slab has lower strength h d f ithan concrete used for precast section

Lower strength Lower modulus of elasticityThus, we need to use the concept of transformed section to transform the slab material to the precast section to transform the slab material to the precast material

, ,''

c CIPC c CIPCtr e c e e

E fb b n b b

E f= = ≅

, ,'tr e c e ec PPC c PPCE f

Modular Ratio, usually < 1.0


Transformed Width


Summary of steps for Width calculations

Actual Width Effective Width Transformed WidthActual Widthb

Effective Widthbe

Transformed Widthbtr

Equals to girder spacing

Accounts for nonuniform stress

Accounts for dissimilar material p g

distribution properties


After we get the transformed section, we can th l l t th ti tithen calculate other section properties

Acc = Ac + tsbtr

ytc, ytb

IIgc

Ztc, Zbc

dpc


Precast Cross-Composite Cross-Sectiona Area: Acc

btrPrecast CrossSectiona Area: Ac

Sectiona Area: Acc

ytc

( b )y’tc

yt

(abs) c.g. Composite dpc

(abs)y

(abs)

c.g. Precast

h (abs)

pdp

pc

ybc

yb

(abs)

y(abs)

Aps Aps

Precast vs. Composite

Design of Composite SectionDesign of Composite Section

Most of the theories learned previously for the it ti till h ld b t ith noncomposite section still hold but with some

modificationsWe will discuss two design limit states

All bl St D iAllowable Stress DesignUltimate Strength Design

Allowable Stress Design - CompositeAllowable Stress Design Composite

OUTLINEShored vs. UnshoredStress Inequality EquationStress Inequality EquationFeasible Domain & Envelope

Allowable Stress Design - CompositeAllowable Stress Design Composite

In allowable stress design, we need to consider two loading stages as previous; however, the initial moment (immediately after p ( ytransfer) is resisted by the precast section whereas the service moment (after the bridge is finished) is resisted by the compositesection (precast section and slab acting together as one member)We need to consider two cases of composite construction

h dmethods:Shored – beam is supported by temporary falsework when the slab is cast The falsework is removed when the slab hardenscast. The falsework is removed when the slab hardens.Unshored – beam is not supported when the slab is cast.

Shored vs UnshoredShored vs. Unshored Shored vs UnshoredShored vs. Unshored

Moments resisted by the precast and composite sections are different in the two casesFully Shored

Precast: Girder WeightPrecast: Girder WeightComposite: Slab Weight, Superimposed Loads (such as asphalt surface), and Live Load)

UnshoredPrecast: Girder Weight and Slab WeightPrecast: Girder Weight and Slab WeightComposite: Superimposed Loads (such as asphalt surface), and Live Load

Shored vs UnshoredShored vs. UnshoredFULLY SHORED

Top of precast, not top of

iConsider, as example, the top of precast beam composite

( ) ( )( ) ( ) 'o Girder Slab SD LL IM tct cs

c t t gc

Fe M M M M yFσ σA Z Z I

++ += − + + ≤

Shored vs UnshoredShored vs. UnshoredUNSHOREDConsider, as example, the top of precast beam

( ) ( ) 'Fe M M M M yF + +( ) ( )o Girder Slab SD LL IM tct cs

c t t gc

Fe M M M M yFσ σA Z Z I

++ += − + + ≤

Shored vs UnshoredShored vs. UnshoredFrom both case we can rewrite the stress equation as:q

≤( )( )o CPFe MMF

= − + + ≤( )( )

'o CP

t csc t t tc

σ σA Z Z Z

Mp = Moment resisted by the precast section (use Zt, Zb)Fully Shored: Mp = Mgirder

Unshored: Mp = Mgirder + Mslab

M M i d b h i i ( Z’ Z )Mc = Moment resisted by the composite section (use Z’tc, Zbc)Fully Shored: Mc = Mslab + MSD + MLL+IM

Unshored: M = M + MUnshored: Mc = MSD + MLL+IM

We can also write similar equation for stress at the bottom of We can also write similar equation for stress at the bottom of composite beam

Stress Inequality Equations Top of precast, Stress Inequality Equations p pnot top of composite

Case Stress Inequality EquationI Initial-Top ⎛ ⎞Fe eF M F MI Initial-Top

II I i i l B

⎛ ⎞= − + = − + ≥⎜ ⎟

⎝ ⎠min min1i o oi i

t tic t t c b t


FF M F M⎛ ⎞II Initial-Bottom min min1i o oi ib ci

c b b c t b


⎛ ⎞= + − = − − ≤⎜ ⎟

⎝ ⎠M M⎛ ⎞III Service-Top 1

'p po c o c

t csc t t tc c b t tc

M MFe M e MF Fσ σA Z Z Z A k Z Z

⎛ ⎞= − + + = − + + ≤⎜ ⎟

⎝ ⎠!IV Service-Bottom

1p po c o cb ts

c b b bc c t b bc

M MFe M e MF Fσ σA Z Z Z A k Z Z

⎛ ⎞= + − − = − − − ≥⎜ ⎟

⎝ ⎠VI Service-Top Slab

c b b bc c t b bc⎝ ⎠

,, ,

c CIPCc ct slab c cs Slab

EM Mσ n σZ Z E

= = ≤,tc tc c PPCZ Z E

Stress at the top of the slab must also be less than the allowable compressive stress

Feasible Domain & Envelope Top of precastFeasible Domain & EnvelopeWe can rewrite the stress equations and add practical limit equation

No. Case Stress Inequality EquationI Initial-Top

( )⎛ ⎞1p

II Initial Bottom

( )⎛ ⎞≤ + −⎜ ⎟

⎝ ⎠0 min

1b ti t

i

e k M σ ZF

⎛ ⎞1II Initial-Bottom

III S T

( )⎛ ⎞≤ + +⎜ ⎟

⎝ ⎠0 min

1t ci b

i

e k M σ ZF1 Z⎛ ⎞⎛ ⎞III Service-Top

01

't

b p c cs ttc

Ze k M M σ ZF Z

⎛ ⎞⎛ ⎞≥ + + −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

⎛ ⎞

!

IV Service-Bottom 0

1 bt p c ts b

bc

Ze k M M σ ZF Z

⎛ ⎞⎛ ⎞≥ + + +⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

V Practical Limit ( )0 0 ,minb cmpe e y d≤ = −

EM MVI Service-Top Slab

,, ,

,

c CIPCc ct slab c cs Slab

tc tc c PPC

EM Mσ n σZ Z E

= = ≤

Cracking Moment - CompositeCracking Moment Composite

We consider 2 cases1. Cracking moment is less than Mp

Cracking occurs in the precast sectionCracking occurs in the precast sectionThe equation is the same as noncomposite section

1o cr o crb r

Fe M e MF Fσ fA Z Z A k Z

⎛ ⎞= + − = − − =⎜ ⎟

⎝ ⎠

( )M F k f Z

c b b c t bA Z Z A k Z⎜ ⎟⎝ ⎠

( )cr o t r bM F e k f Z= − −

Cracking Moment - CompositeCracking Moment Composite

II. Cracking moment is greater Mp

C k h Cracking occurs in the composite sectionWe find ∆Mcr (moment in addition to Mp)cr ( p)

1p po cr o crM MFe M e MF Fσ σ⎛ ⎞Δ Δ

= + − − = − − − ≥⎜ ⎟1b tsc b b bc c t b bc

σ σA Z Z Z A k Z Z

= + = ≥⎜ ⎟⎝ ⎠

Z ( )bccr o t p r bc

b

ZM F e k M f ZZ

⎡ ⎤Δ = − − −⎣ ⎦

cr cr pM M M= Δ +

Ultimate Strength Design - CompositeUltimate Strength Design Composite

Ultimate strength of composite section follows similar procedure to the T-section Some analysis tips are:procedure to the T-section. Some analysis tips are:

When the neutral axis is in the slab, we can use a composite T-section with flange width equals to Effective Width and using f’section with flange width equals to Effective Width and using f cof the slabWhen the neutral axis is in the precast section we may use a When the neutral axis is in the precast section, we may use a Transformed Section and f’c of the precast section - This is an approximate value but the errors to the ultimate moment ppcapacity is small.

Shear TransferShear Transfer

To get the composite pbehavior, it is very important that the slab and girder must not slip past not slip past each other

Shear Transfer MechanismsShear Transfer Mechanisms

The key parameter that determines whether these two parts will slip past each other or not is the shear strength at the interface of slab and girder

This interfacial shear strength comes from:

Friction (F = μN)Cohesion

Shear Transfer – Cohesion & FrictionShear Transfer Cohesion & Friction

Cohesion is the chemical bonding of the two materials. It depends on the cohesion factor (c) and the contact area. The greater the ( ) garea, the larger the cohesion force.

Friction is due to the roughness of the surface. It depends on the friction factor or coefficient of friction (μ) and the normal force (N). To increase friction, we either make the surface rougher (increase μ) or increase the normal force.

NN

Vhu

ΦVhn =ΦμN

hu

Shear Transfer - FormulaShear Transfer FormulaAASHTO LRFD (5.8.4)The nominal shear resistance at the interface between two concretes cast at different times is taken as:

Friction Factor

Area of Concrete

Friction Factor

Compressive force normal

Area of shear reinforcement crossing the shear plane

⎧Cohesion

Transfering Shear Compressive force normal to shear plane

≤⎧= + + ⎨ ≤⎩

0.2 '( )

5 5c cv

nh cv vf y c

f AV cA μ A f P

A≤⎩ 5.5 cvACOHESION FRICTION

Shear Transfer –Shear Transfer Cohesion & Friction

AASHTO LRFD (5 8 4 2)AASHTO LRFD (5.8.4.2)

Shear Transfer – Cohesion & FrictionShear Transfer Cohesion & FrictionThe normal force in the friction formula comes from two parts

Yielding of shear reinforcementIf k h

Avf

If cracking occurs at the interface, there will be tension in the steel reinforcement in the steel reinforcement crossing the interface. This tension force in steel is balanced b h i f i

N=Avf fy

by the compressive force in concrete at that interface; thus, creating normal “clamping” creating normal clamping force.

Permanent compressive force at the interface

Dead Weight of the slab and earin s rface (as halt)wearing surface (asphalt)

Cannot rely on Live Loads

Minimum Shear ReinforcementMinimum Shear ReinforcementFor Vn/Acv > 0.7 MPa, the cross-sectional area of shear reinforcement n cvcrossing the interface per unit length of beam must not be less than

Width of the interface (generally

≥0.35 v

vfbA

f

Width of the interface (generally equals to the width of top flange of girder)

If less, then we cannot use any Avffy in the nominal shear strength

yf

The spacing of shear reinforcement must be ≤ 600 mm

Possible reinforcements are:S l bSingle bar

Stirrups (multiple legs)W ld d i f b iWelded wire fabric

Reinforcement must be anchored properly (bends, hooks, etc…)

Ultimate Shear Force at InterfaceUltimate Shear Force at Interface

There are two methods for calculating shear force per unit length at the interface (the values may be different)( y )

Using Classical Elastic Strength of Materials

Factored shear force acting on the ( )

=Δ u

uhgc

V QV

I Moment of Area above the shear plane

Factored shear force acting on the composite section only (SDL +LL+IM)

gc Moment of Area above the shear plane about the centroid of composite sectionMoment of Inertia of the

composite section

Using Approximate Formula (C5.8.4.1-1)

V Total Factored vertical shear at the section= u

uhe

VVd Distance from centroid of tension

steel to mid-depth of the deckp

Ultimate Shear Force at InterfaceUltimate Shear Force at Interface

The critical section for shear at the interface is generally the section where vertical shear is the greatestg

First critical section: h/2 from the face of support

May calculate at some additional sections away from the support (which has lower shear) to reduce the shear reinforcement accordingly

Critical Section For Shear

h

Critical Section For Shear

h/2 h/2

Resistance Factor (Φ) for shear in normal weight concrete : 0.90

Some Design TipsSome Design Tips

For T and Box Sections which cover the full girder spacing with thin concrete topping (usually about 50 mm), we may with thin concrete topping (usually about 50 mm), we may not need any shear reinforcement (need only surface roughening) – need to checkg g)For I-Sections, we generally require some shear reinforcement at the interfacereinforcement at the interfaceWe generally design the web shear reinforcement first (not taught), and extend that shear reinforcement through the interface. Then and extend that shear reinforcement through the interface. Then we check if that area is enough for horizontal shear transfer at the interface.

If not, we need additional reinforcementIf enough, then we do nothing

Final Notes on Composite BehaviorFinal Notes on Composite BehaviorC it ti i d t l f t d t Composite section is used not only for prestressed concrete sections, but also for steel sections.

Benefits is that the slab helps resists compression and helps prevent Benefits is that the slab helps resists compression and helps prevent lateral torsional buckling of the steel section, as well as local buckling at the compression flange.g p g

Final Notes on Composite BehaviorFinal Notes on Composite Behavior

The analysis concept is similar to that of

b

prestressed concrete. There are also:

Effective width and transformed sectionShored and Unshored ConstructionSh T f t I t fShear Transfer at Interface


There are various ways to transfer shear at steel-concrete interface

Studs ChannelsSpirals Studs ChannelsSpirals


Shear Stud is one of the most

h common shear connectors – it is welded to the top welded to the top flange of steel girder


Steel Girder with Shear Stud

Part IV: Things I did not teach but you should be aware of !!!

Shear Strength – MCFTUnbonded and External PrestressingUnbonded and External PrestressingAnchorage ReinforcementCamber and Deflection PredictionDetailed Calculation of Prestress Losses

ShearShear

Traditionally, the shear design in AASHTO Standard Specification is similar to that of ACI which is empirical-Specification is similar to that of ACI, which is empirical-basedTh i l f f i d h i i l The axial force from prestressing reduces the principal tensile stress and helps close the cracks; thus, increase h shear resistance.

Shear - MCFTShear MCFT

The shear resisting mechanism in concrete is very complex and we do not clearly understand how to complex and we do not clearly understand how to predict itAASHTO LRFD (5 8 3) uses new theory called AASHTO LRFD (5.8.3) uses new theory, called “modified compression field theory (MCFT)”Th l h l d b h The actual theory is very complicated but somewhat simplified procedure is used in the codeThis theory is for both PC and RC

ShearShear

The nominal shear resistance is the sum of shear strength of concrete steel (stirrups) and shear force due to of concrete, steel (stirrups), and shear force due to prestressing (vertical component)

= + + ≤ +0.25 'n c s p c v v pV V V V f b d V

cotv y vA f d θV = ysV s

= 0.083 'c c v vV β f b d

Minimum Transverse ReinforcementMinimum Transverse Reinforcement

We need some transverse reinforcement when the ultimate shear force is greater than ½ of shear strength ultimate shear force is greater than ½ of shear strength from concrete and prestressing force

> +0.5( )u c pV φ V V

If we need it the minimum amount shall beIf we need it, the minimum amount shall be

≥ 0 083 ' vb sA f≥ 0.083v cy

A ff

Minimum Transverse ReinforcementMinimum Transverse Reinforcement

Maximum SpacingFor v <0 125f’cFor vu <0.125f c

smax = 0.8 dv ≤ 600 mm

For vu > 0.125f’c

smax = 0.4dv ≤ 300 mm

Must subtract the area of duct to the width 5.8.2.7

Unbonded or External PrestressingUnbonded or External Prestressing

Strain compatibility does not apply for unbonded pp ytendon (the strain in steel does not equal to the strain in concrete near it)The strain in tendon is averaged along the length of the beamlength of the beam

Unbonded or External PrestressingUnbonded or External Prestressing

View inside a box-section

Anchorage ReinforcementAnchorage Reinforcement

Post-tensioning anchorages anchorages creates very high compressive compressive stress behind the bearing platebearing plate


This causes large principal tensile p pstress in the transverse direction, leading to concrete crackingWe need to determine the magnitude of this stress and design some reinforcement for it


Methods:T d l Traditional (Approximate)Strut-and-Tie Method (new (for ACI and AASHTO))Finite Element Analysis Analysis (complicated)

Strut-and-Tie Method


Strut-and-Tie Method

Camber and DeflectionCamber and Deflection

AASHTO does not require the deflection criteria be metBut excessive camber and deflection causes uneven rides But excessive camber and deflection causes uneven rides and the impression that the structure is not strong enoughenoughThe structure may deflect and vibrate too much that it cause fatigue failure (due to repetitive stress cycles) cause fatigue failure (due to repetitive stress cycles), especially in steel connections.Vibrations may cause discomfort to drivers on bridgeVibrations may cause discomfort to drivers on bridge

Detailed Calculation of Prestress LossDetailed Calculation of Prestress LossIn many cases it is adequate to use the “Lump Sum” loss In many cases, it is adequate to use the Lump Sum loss

I In some cases, we need to know

tl th t i exactly the stress in the strands so we can determine the can determine the camber and deflectiondeflection

Cantilever ConstructionConstructionRepair/ Rehabilitation

ReferencesReferences

AASHTO (2000). AASHTO LRFD Bridge Design Specifications – SI Units, Second Edition, 2000 Interim Revisions, American Association of State Highway and Transportation Officials, Washington D.C. http://www.transportation.org

Naaman, A. E. (2005), Prestressed Concrete Analysis and Design: Fundamentals, 2nd Edition, Technopress 3000, Ann Arbor, MI, USAh // h 3000http://www.technopress3000.com

40609 prestressed

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