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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
Principle of PrestressingPrinciple of Prestressing
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)
Principle of PrestressingPrinciple of Prestressing
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.
Classification and TypesClassification and Types
Pretensioned Prestressed ConcretePretensioned Prestressed ConcreteCasting Factory
ConcreteMixer
Classification and TypesClassification and Types
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.
Classification and TypesClassification and Types
Precast Segmental Girder to be Posttensioned In Posttensioned In Place
Classification and TypesClassification and Types
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.
Classification and TypesClassification and Types
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)
Classification and TypesClassification and Types
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!!!
Stages of LoadingStages of Loading
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
Stages of LoadingStages of Loading
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
Prestress LossesPrestress Losses
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
Prestress LossesPrestress Losses
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
Lump Sum Prestress LossLump Sum Prestress Loss
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)
Lump Sum Prestress LossLump Sum Prestress Loss
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
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
!
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
Allowable Stress in Prestressing SteelAllowable Stress in Prestressing Steel
AASHTO LRFD LRFD (5.9.3)
Allowable Stress in Prestressing SteelAllowable Stress in Prestressing SteelACI-318 (2002)ACI 318 (2002)
Allowable Stress in Prestressing SteelAllowable Stress in Prestressing Steel
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
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 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
Allowable Stress DesignAllowable Stress Design
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!
Analysis for Ultimate Moment CapacityAnalysis for Ultimate Moment Capacity
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
Analysis for Ultimate Moment CapacityAnalysis for Ultimate Moment Capacity
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
Analysis for Ultimate Moment CapacityAnalysis for Ultimate Moment Capacity
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
Analysis for Ultimate Moment CapacityAnalysis for Ultimate Moment Capacity
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 !!!
Rectangular vs T-SectionRectangular vs. T 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
Rectangular vs T-SectionRectangular vs. T Section
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)
T-Section AnalysisT Section Analysis
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
T-Section AnalysisT Section Analysis
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
T-Section AnalysisT Section Analysis
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
Composite Section PropertiesComposite Section Properties
Transformed Width
Composite Section PropertiesComposite Section Properties
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
Composite Section PropertiesComposite Section 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
Composite Section PropertiesComposite Section Properties
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
Fe eF M F Mσ σA Z Z A k Z
FF M F M⎛ ⎞II Initial-Bottom min min1i o oi ib ci
c b b c t b
Fe eF M F Mσ σA Z Z A k Z
⎛ ⎞= + − = − − ≤⎜ ⎟
⎝ ⎠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
Final Notes on Composite BehaviorFinal Notes on Composite Behavior
There are various ways to transfer shear at steel-concrete interface
Studs ChannelsSpirals Studs ChannelsSpirals
Final Notes on Composite BehaviorFinal Notes on Composite Behavior
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
Final Notes on Composite BehaviorFinal Notes on Composite Behavior
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
Anchorage ReinforcementAnchorage Reinforcement
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
Anchorage ReinforcementAnchorage Reinforcement
Methods:T d l Traditional (Approximate)Strut-and-Tie Method (new (for ACI and AASHTO))Finite Element Analysis Analysis (complicated)
Strut-and-Tie Method
Anchorage ReinforcementAnchorage Reinforcement
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
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