40609 Prestressed

Prestressed Concrete Bridge Design Basic Principles

Part I: Introduction

Emphasizing AASHTO LRFD Procedures

Praveen Chompreda, Ph. D.

Reinforced vs. Prestressed Concrete Principle of Prestressing H Historical lP Perspective Applications Classification and Types Advantages Design Codes Stages of Loading

MAHIDOL UNIVERSITY | 2009 | EGCE 406 Bridge Design

Reinforced Concrete

Reinforced Concrete

Recall Reinforced Concrete knowledge:

Concrete is strong in compression but C b weakk in tension Steel is strong in tension (as well as compression) Reinforced concrete uses concrete to resist compression and to hold the steel bars in place, and uses steel to resist all of the tension Tensile strength of concrete is neglected (i.e. zero) RC beam always crack under service load Cracking moment of an RC beam is generally much lower than the service moment

Principle of Prestressing

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


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

These 2 force systems must counteract each other


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

Historical Perspective

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

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

Smaller Compression

c.g.

+

e0

=

F/A CrossSection

+ Fe0y/I

Prestressing Force

MDLy/I

MLLy/I

Stress from DL

Stress from LL

Small Compression Stress Resultant

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



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

Applications of Prestressed Concrete

Bridges Sl b in Slabs i buildings b ildi Water Tank Concrete Pile Thin Shell Structures Offshore Platform Nuclear Power Plant Repair p and Rehabilitations

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

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

Classification and Types

Pretensioning v.s. Posttensioning External v.s. Internal Linear v.s. v s Circular End-Anchored v.s. Non End-Anchored Bonded v.s. Unbonded Tendon P Precast t v.s. Cast-In-Place C t I Pl v.s. Composite C it Partial v.s. Full Prestressingg



Pretensioningg vs. Posttensioningg

In Pretension, the tendons are tensioned against some abutments before the concrete is place place. After the concrete hardened, the tension force is released. 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. concrete Pretension is usually done with precast members. Pretensioned Prestressed Concrete Casting Factory


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

Concrete Mixer

Classification and Types Precast Segmental Girder to be Posttensioned In Place


E External l vs. IInternall Prestressing P

Linear vs. Circular Prestressing

Prestressingg mayy be done inside or outside 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,

Bonded vs. Unbonded Tendon

The tendon may be bonded to concrete (pretensioning or posttensioning with grouting) or unbonded ( (posttensioning i i without ih grouting). i ) B Bonding di hhelps l prevent corrosion of tendon. Unbonding allows readjustment dj t t off prestressing t i force f att later l t times. ti


Partial vs. Full Prestressing

Prestressing tendon P d may be b used d in combination b withh 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 ductility, and save costs.


End-Anchored vs. Non-End-Anchored tendons

IIn P Pretensioning, tendons d transfer f the h prestress through the bond actions along the tendon; therefore, it is non-end-anchored In Posttensioning, g tendons are anchored at their ends using mechanical devices to transfer the prestress to concrete;; therefore,, it is end-anchored. ((Groutingg or not is irrelevant)

RC vs vs. PPC vs. vs PC

Advantages of PC over RC

Take full advantages of high strength concrete and high strength steel

Need N d less l materials i l Smaller and lighter structure No cracks Use the entire section to resist the load Better corrosion resistance Good for water tanks and nuclear plant

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

Other institutions

PCI – Precast/Prestressed Concrete Institute PTI – Post Post-Tensioning Tensioning Institute

Very effective for deflection control Better shear resistance

Stages of Loading

Design Codes for PC

Unlike RC where we primarily consider the ultimate lti t lloading di stage, t we mustt consider id multiple lti l stages of construction in Prestressed Concrete The stresses in the concrete section must remain below the maximum limit at all times!!!

Stages of Loading

Typical stages of loading considered are Initial and d Service S i Stages St Initial ((Immediatelyy after Transfer of Prestress))

Full prestress force N MLL (may No ( or may nott have h MDL depending d di on construction type)

Service

Prestress loss has occurred MDL+MLL

Stages of Loading

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

Part II: Materials and Hardwares for Prestressingg Concrete Prestressing Steel Prestressing Hardwares

Concrete

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

Compressive Strength M d l off Elasticity Modulus El i i Modulus of Rupture

Concrete: Compressive Strength

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

Concrete: Modulus of Elasticity

AASHTO (5.4.2.4) Ec = 0.043γ 0 043γc1.5(f (f’c)0.5 MPa

Concrete: Modulus of Rupture

γc1.5 in kg/m3 ff’c in MPa

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

Indicates the tensile capacity of concrete under bendingg Tested simply-supported p concrete beam under 4-point bending configuration fr = My/I = PL/bd2 AASHTO (5.4.2.6)

Concrete : Summary of Properties

fr = 0.63 (f’c)0.5 MPa

Prestressing Tendons

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

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

Tendons

Prestressing Steel

Common shapes of prestressing tendons

Most Popular Æ ((7-wire Strand))

Prestressing Strands

Prestressing strands have two grades

Grade G d 250 (fpu = 250 ksi k or 1725 MPa) MP ) Grade 270 ((fpu = 270 ksi or 1860 MPa))

Types of strands

SStressed d Relieved R li d SStrand d Low Relaxation Strand (lower prestress loss due to relaxation of strand)




Modulus of Elasticity

Hardwares & Prestressing Equipments

Pretensioned Members

H ld D Hold-Down Devices D

Posttensioned Members

Anchorages

Stressing St i A Anchorage h Dead-End Anchorage

Ducts Posttensioningg Procedures

Pretensioned Beams

197000 MPa for Strand 207000 MPa for Bar

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

Pretensioning Hardwares

Posttensioned Beams

Hold-Down Devices for Pretensioned Beams

Posttension Hardwares

Posttensioning Hardwares - Anchorages

Stressing St i A Anchorage h Dead-End Anchorage Duct/ Grout Tube



Posttensioning Hardwares - Ducts

Posttensioning Procedures

Posttensioning Procedures

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

Prestress Losses

Part III: Prestress Losses

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

Sources of Prestress Losses Lump Sum Estimation of Prestress Loss

Prestress Losses

Sources of Prestress Loss (cont.)

Friction : Friction in the duct of p posttensioningg system y causes stress 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 anchorage h may set in i slightly li h l to llockk the tendon, causing a loss of stress

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

Prestress Losses

Sources of Prestress Loss (cont.) Shrinkage : Concrete shrinks over time due to the loss of water, leading to stress loss on attached tendons Creep : Concrete shortens over time under compressive stress, stress leading to stress loss on attached tendons

Prestress Losses

Time Line of Prestress Loss Posttensioningg

Sources of Prestress Loss (cont.)

FR

Jacking

AS ES

Initial

fpj

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

Pretensioning (AS RE) J ki Jacking (against abutment)

fpj

fpi

Release

ES

( (cutting g strands)

Instantaneous Losses

Prestress Loss – By Types Pretensioned Instantaneous

TimeDependent

SH CR RE

fpe

Initial

SH CR RE

fpi

Elastic Shortening

Friction A Anchorage Set S Elastic Shortening

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

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

Effective fpe

Time-Dependent Losses

Prestress Loss - Pretensioned Posttensioned

Effective

Prestress Loss - Posttensioned

Lump Sum Prestress Loss

Pretress losses can be very complicate to estimate ti t since i it d depends d on so many factors f t In typical yp constructions,, a lump p sum estimation of prestress loss is enough. This may be expressed in terms of:


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


A. E. Naaman (with slight modifications) – not including FR, AS

Start with 240 MPa for Pretensioned Normal Weight Concrete with Low Relaxation Strand Add 35 MPa for Stress-Relieved Strand or for Lightweight Concrete D d Deduct 35 MPa MP ffor Posttension P

Types of Prestress Pretensioned

P t Prestress Loss L (fpi-fpe) (f i f ) (MP (MPa)) Types of Concrete

Stress-Relieved Low Relaxation Strand Strand

Normal Weight Concrete Li ht i ht C Lightweight Concrete t

275 310

240 275

Posttensioned Normal Weight Concrete Lightweight Concrete

240 275

205 240

ACI-ASCE Committee (Zia et al. 1979)

This is the Maximum Loss that you may assumed

T Types off Prestress Pretensioned

Types of Concrete Normal Weight Concrete Lightweight Concrete

Maximum Prestress Loss (fpi fpe) (MPa) (fpi-fpe) Stress-Relieved Low Relaxation Strand Strand 345 380

276 311



T.Y. Lin & N. H. Burns S Source off Loss L

AASHTO LRFD (for CR CR, SR SR, R2) (5.9.5.3) (5 9 5 3)

P Percentage off Loss L (%) Pretensioned

Posttensioned

Elastic Shortening (ES)

4

1

Creep of Concrete (CR)

6

5

Shrinkage of Concrete (SR)

7

6

Steel Relaxation (R2)

8

8

25

20

Total

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


AASHTO LRFD (Cont.)

Partial Prestressing Ratio (PPR) is calculated as:

PPR =

Aps fpy Aps fpy + As fy

PPR = 1.0 1 0 for Prestressed Concrete PPR = 0.0 for Reinforced Concrete

Elastic Shortening Loss (ΔfpES) is calculated as:

ΔfpES =

E ps Eci

Part IV: Allowable Stress Design g

fcgp,Fi +G

E ps ⎡ Fi Fi e02 MG e0 ⎤ = + − ⎢ ⎥ Eci ⎣ A c I I ⎦

Stress of concrete at the c.g. of tendon due to prestressing force and dead load

Stress Inequality Equation Allowable Stress in Concrete Allowable Stress in Prestressing Steel Feasible Domain Method Envelope and Tendon Profile

Basics: Sign Convention

Basics: Section Properties

In this class, the following convention is used:

c.g. g off Prestressingg Tendon Area: Aps

Concrete CrossSectiona Area: Ac

Tensile Stress in concrete is negative g (-) () Compressive Stress in concrete is positive (+) Positive Moment:

yt ((abs))

e ((-)) kt (-)

Positive Shear:

h (abs)

kb (+)

e (+)

Center of Gravity of Concrete Section (c.g.c)

I Kt Kb Zt Zb

yb (abs)

In some books,, the sign g convention for stress mayy be opposite so you need to reverse the signs in some formula!!!!!!!!!


Moment of Inertia, I

I = ∫ y dA 2

A

Rectangular section about c.g. Ixx = 1/12*bh3 Ix’x’ = Ixx + Ad2

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

Zt = I/yt Zb = I/yb

c.g. of Prestressing Tendon Area: Aps


Kern of the section, section k, k is the distance from cc.g. g where compression force will not cause any tension i iin the h section i Consider C id T Topp Fib Fiber (Get Bottom Kern, kb)

0=

F Fe0 y t − Ac I

e0 =

I = kb Ac y t

Consider C id Bottom B tt Fiber Fib (Get Top Kern, kt)

0=

F Fe0 y b + Ac I

e0 = −

I = kt Ac y b

Note:Top kern has negative value

Basics: General Design Procedures

Stress in Concrete at Various Stages

Select Girder type, materials to be used, and number b off prestressing t i strands t d Check allowable stresses at various stages g Check ultimate moment strength Check cracking load Check shear Check deflection

Stress Inequality Equations

Allowable Stress in Concrete

We can write four equations based on the stress at the top and bottom of section at initial and service stages

No.

Case

Stress Inequality Equation

I

Initial-Top

F Fe M F ⎛ e ⎞ M σ t = i − i o + min = i ⎜ 1 − o ⎟ + min ≥ σ ti Ac Zt Zt Ac ⎝ kb ⎠ Zt

II

Initial-Bottom

F Fe M F ⎛ e σ b = i + i o − min = i ⎜ 1 − o Ac Zb Zb Ac ⎝ kt

III

Service-Top

IV

! ServiceBottom

F Feo Mmax Fi ⎛ eo ⎞ Mmax σt = − + = ≤ σ cs ⎜1− ⎟ + Ac Zt Zt Ac ⎝ k b ⎠ Zt

⎞ Mmin ≤ σ ci ⎟− ⎠ Zb

F Feo Mmax F ⎛ eo σb = + − = ⎜1− Ac Zb Zb Ac ⎝ kt

⎞ Mmax ≥ σ ts ⎟− Zb ⎠

AASHTO LRFD (5.9.4) provides allowable stress in p strength g at that concrete as functions of compressive time Consider the following limit states:

Immediately after Prestress Transfer (Before Losses)

Compression Tension

Service (After All Losses)

Compression C i Tension


Immediately after Prestress Transfer (Before Losses)

Using compressive strength at transfer, f’ci

At service (After All Losses) Compressive Stress

Allowable All bl compressive i stress = 00.60 60 f’ci Allowable tensile stress



At service (After All Losses) Tensile Stress

Allowable Stress in Concrete - Summary Stage g Initial

Where Tension at Top

Load Fi+MGirder

Compression Fi+MGirder at Bottom Service Compression F+MSustained at Top 0.5(F+MSustained)+MLL+IM Tension at Bottom

Limit

Note

-0.58√f’ci

With bonded reinf…

-0.25√f’ci > -1.38 MPa

Without bonded reinf.

0.60 f’ci 0.45f’c

*

0.40f’c

*

F+MSustained+MLL+IM

0.60Øwf’c

*

F+MSustained+0.8MLL+IM (Service III Limit State)

-0.50√f’c

Normal/ Moderate exposure

-0.25√f’c

Corrosive exposure

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

Allowable Stress in Prestressing Steel

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


ACI-318 ACI 318 (2002)


AASHTO LRFD (5.9.3)


Allowable Stress Design

There are many factors affecting the stress in a prestressed girder

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

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

Stages of construction

Sections

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


AASHTO Type I-VI Sections

ft 50 75 100 150

m 15 23 30 46

Sections

AASHTO Type I-VI Sections (continued)

Bridge Girder Sections

Bridge Girder Sections


Feasible Domain - Equations

For a given section, we need to find the combination bi ti off prestressing t i fforce (Fi or F, F which hi h depends on the number of strands), and the location of strands (in terms of e0) to satisfy these equations Possible methods:

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

We can rewrite the stress inequality equations and add one more equation to them

No No.

Case


I

Initial-Top

⎛1⎞ e0 ≤ k b + ⎜ ⎟ Mmin − σ ti Zt ⎝ Fi ⎠

)

II

Initial-Bottom

⎛1⎞ e0 ≤ kt + ⎜ ⎟ Mmin + σ ci Zb ⎝ Fi ⎠

)

III

Service-Topp

IV

ServiceService Bottom

V

P Practical i l Li Limit i

(

(

(

)

(

)

⎛ 1⎞ e0 ≥ k b + ⎜ ⎟ Mmax − σ cs Zt ⎝F ⎠

⎛ 1⎞ e0 ≥ kt + ⎜ ⎟ Mmax + σ ts Zb ⎝F ⎠

e0 ≤ ( e0 )mp = y b − dc ,min = y b − 7.5 cm

!

Feasible Domain – Graphical Interpretation

Feasible Domain

Envelope - Equations

I II III

Envelope - Equations

We 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 the beam

No No.

Case Initial-Top Initial-Bottom Service-Top p

IV

ServiceService Bottom

V

P Practical i l Li Limit i

Feasible domain tells you the possible location and prestressing force at a given section to satisfy the stress inequality equation We usually use feasible domain to determine location and d prestressing i force f at the h most critical i i l section i (e.g. ( midspan of simply-supported beams) After we get the prestressing force at the critical section, section we need to find the location for the tendon at other points to satisfy stress inequalities We use the prestressing envelope to determine the g of the beam (tendon ( location of tendon alongg the length profile)

Stress Inequality Equation ⎛1⎞ e0 ≤ k b + ⎜ ⎟ Mmin − σ ti Zt ⎝ Fi ⎠

)

⎛1⎞ e0 ≤ kt + ⎜ ⎟ Mmin + σ ci Zb ⎝ Fi ⎠

)

(

(

(

⎛ 1⎞ e0 ≥ k b + ⎜ ⎟ Mmax − σ cs Zt ⎝F ⎠

(

⎛ 1⎞ e0 ≥ kt + ⎜ ⎟ Mmax + σ ts Zb ⎝F ⎠

We then have 5 main equations

I & II provide the lower bound of e0 (use minimum of the two) III and d IV provide id the th upper bound b d off e0 (use ( maximum i of the two)

)

)

e0 ≤ ( e0 )mp = y b − dc ,min = y b − 7.5 cm

!

IIIa uses F+MSustained III IIIb uses 0.5(F+MSustained)+MLL+IM IIIc uses F+MSustained+MLL+IM IV uses F+MSustained+0.8MLL+IM

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

Envelope & Tendon Profile




Note

The tendon Th d profile f l off pretensioned d members b are either straight or consisting of straight segments The tendon profile of posttensioned member may be g tendon or smooth curved, but no sharpp one straight corners

There is an alternative to draping the strands in pretensioned t i d member b We pput pplastic sleeves around some strands at supports to prevent the bond transfer so the prestress force will be less at that section

Load – Deflection – Concrete Stress

Part II: Ultimate Strength g Design g Concrete and Prestressing Steel Stresses Cracking Moment Failure Types A l i ffor Mn – Rectangular Analysis R t l Section S ti T-Section A l i ffor Mn – TAnalysis T Section S i

Load - Deflection

1 & 2: Theoretical camber (upward deflection) of prestressed beam 3: Self weight + Prestressing force 4: Zero deflection ppoint (Balanced ( ppoint)) with uniform stress across section 5: Decompression point where tension is zero at the b bottom fiber fb 6: Cracking point where cracking moment is reached 7: End of elastic range (the service load will not be larger than this) 8 Yielding 8: Yi ldi off prestressing i steell 9: Ultimate strength (usually by crushing of concrete)

Prestressing Steel Stress

Prestressing Steel Stress

Cracking Moment

The pprestressingg steel stress increases as the load increases Crackingg of beam causes a jump j p in stress as additional tension force is transferred from concrete (now cracked) to prestressing steel At ultimate of prestressed concrete beam, the stress in g fpy and steel is somewhere between yyield strength ultimate 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 tendon At ultimate, the effect of prestressing is lost and the section behaves jjust like an RC beam

Cracking Moment

F Feo Mcr F ⎛ eo + − = ⎜1− Ac Zb Zb Ac ⎝ kt

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

fr = -0.63 (f’c)0.5 MPa (5.4.2.6)

Failure Types

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

⎞ Mcr = fr ⎟− ⎠ Zb

Solve the above equation to get Mcr

Mcr = F (eo − kt ) − fr Zb Note: Need to input fr and kt as negative values !!!

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

Failure Types

Analysis for Ultimate Moment Capacity

Analysis assumptions


Recall from RC Design that the followings must b satisfy be ti f att allll times ti no matter tt what h t happens: h


For equilibrium, there are commonly 4 forces

EQUILIBRIUM

Compression in concrete C Compression p in Nonprestressed p reinforcement Tension in Nonprestressed reinforcement Tension in Prestressed reinforcement

STRAIN COMPATIBILITY

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

They also hold in Prestressed Concrete!

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

Rectangular Stress Block

Rectangular Stress Block 0.85 f 'c ≤ 28 MPa ⎧ ⎪ ⎛ f ' − 28 ⎞ ⎛ 1 ⎞ ⎪ β1 = ⎨0.85 − 0.05 ⎜ c ⎟ ⎜ 1 ⎟ 28 ≤ f 'c ≤ 56 MPa 7 ⎝ ⎠⎝ ⎠ ⎪ f 'c ≥ 56 MPa 0.65 ⎩⎪

β1 is equal to 0.85 0 85 for f ’c < 28 MPa It decreases 0.05 for everyy 7 MPa increases in f ’c Until it reaches 0.65 at f ’c > 56 MPa


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

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


For tension in prestressing steel steel, we observe that we cannot assume the behavior of prestressing steel (which is high strength steel) t l) tto bbe elasticl ti perfectly plastic as in the h case off steell reinforcement in RC


At ultimate of prestressed concrete beam, the stress in steel is clearly not the yield strength but somewhere between yield strength fpy and ultimate strength fpu W called We ll d iit fps The true value of stress is difficult to calculate (generally requires nonlinear moment-curvature analysis) so we ggenerallyy estimate it usingg semi-empirical p formula

Ultimate Stress in Steel: fps

⎛ c fps = fpu ⎜ 1 − k ⎜ dp ⎝

ACI Æ Bonded Tendon or Unbonded Tendon AASHTO Æ Bonded Tendon or Unbonded Tendon

Ultimate Stress in Steel: fps

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

⎞ ⎛ fpy ; k = 2 ⎟⎟ ⎜⎜ 1.04 − fppu ⎠ ⎝

⎞ ⎟⎟ ⎠

Note: for ppreliminaryy design, g we mayy conservativelyy assume fps=fpy (5.7.3.3.1)

For Unbonded tendon, see 5.7.3.1.2


Notes on Strain Compatibility The strain in top of concrete at ultimate is 0.003 We can use similar triangle to find the strains in concrete or reinforcingg steel at anyy levels from the topp strain We need to add the tensile strain due to prestressing (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 prestressing steel

Resistance Factor φ

Maximum & Minimum Reinforcement

M i Maximum Reinforcement R i f t (5 (5.7.3.3.1) 7 3 3 1)

The maximum of nonprestressed and prestressed reinforcement shall be such that c/de ≤ 0.42 c/de = ratio between neutral axis depth (c) and the centroid depth of the tensile force (de)

Minimum Reinforcement (5.7.3.3.2)

The minimum Th i i off nonprestressed t d and d prestressed t d reinforcement shall be such that ØMn > 1.2M 1 2Mcr (Mcr = cracking ki moment), ) or ØMn > 1.33Mu (Mu from Strength Load Combinations)

R it Resistance Factor F t Ø Section Type

RC and PPC w/ PPR < 0.5

PPC with 0.5< PPR < 1

(PPR = 1.0)

Under-Reinforced Section c/de ≤ 0.42

0 90 0.90

0 90 0.90

1 00 1.00

Over-Reinforced O R i f d SSection ti c/de > 0.42

Nott N Permitted

0 70 0.70

0 70 0.70

Rectangular vs. vs T T-Section Section

Most prestressed concrete p or Tbeams are either I-Shaped shaped (rarely rectangular) so they have larger compression flange If the neutral axis is in the flange we called it rectangular flange, section behavior. But if the g neutral axis is below the flange of the section, we call it Tsection behavior This has nothing to do with the overall shape of the section !!!

PC

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


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


T-Section T Section Analysis

We divide the compression side into 2 parts

Overhanging O h portion off flange fl (width ( d h = b-b b bw ) Web ppart (width ( = bw )

We generally assume that the section is rectangular first and check if the neutral axis depth (c) is above or below the flange thickness, thickness hf Note:ACI method checks a=ß1c with hf, which may give slightly li htl diff differentt result lt when h a < hf but b t c > hf



From equilibrium

0.85f 'c bw β1c + 0.85f 'c (b − bw )β1hf = Aps fps + As fy − As ' fy '

0.85f 'c bw β1c + 0.85f 'c (b − bw )β1hf = Aps fps + As fy − As ' fy '

For preliminary analysis, or first iteration, we may assume fps = fpy and solve for c

c=

Aps fy + As fy − As ' fy '− 0.85f 'c (b − bw )β1hf 0 85f 'c bw β1 0.85

For a more detailed approach, we recall the equilibrium

⎛ ⎜ ⎝

Substitute fps = fpu ⎜ 1 − k

c=

c dp

⎞ ⎟⎟ , Rearrange and solve for c ⎠

Aps fpu + As fy − As ' fy '− 0.85 0 85f 'c (b − bw )β1hf 0 85f 'c bw β1 + kAps fpu / d p 0.85


T-Section T Section Analysis Flowchart

Moment Capacity (about a/2) a⎞ a⎞ a⎞ ⎛ ⎛ ⎛ Mn = Aps fps ⎜ d p − ⎟ + As fy ⎜ ds − ⎟ − As ' fy ' ⎜ ds '− ⎟ 2⎠ 2⎠ 2⎠ ⎝ ⎝ ⎝ h ⎞ ⎛ +0.85 0 85f 'c (b − bw )β1hf ⎜ a − f ⎟ 2⎠ ⎝

T-Section T Section Analysis Flowchart

T Section T-Section

In actual structures, the section is pperfect T or I shapes p there 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

Composite

Part III: Composite p Beam

Composite generally means the use of two diff different t materials t i l iin a structural t t l elements l t Example: p Reinforced Concrete

Typical Composite Section Composite p Section Properties p Actual, Effective, and Transformed Widths Allowable Stress Design Stress Inequality Equation, Feasible Domain, and Envelope Cracking Moment Ul i Ultimate M Moment C Capacity i

Composite Beam

In the context of bridge design, the word composite it bbeam means the th use off ttwo diff differentt materials between the beam and the slab

Steel Beam + Concrete Slab

Steel beam carries tension Concrete in slab carries compression

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

Prestressed P d Concrete C beam b carries i tension i Concrete in slab carries compression

Concrete – carry compression St l R Steel Reinforcement i f t – carry tension t i

Example: p Carbon Fiber Composite p

Carbon Fiber – carry tension E Epoxy Resin Matrix Matri – hold h ld the fibers in place lace

Typical Composite Sections

Typical Composite Sections

Slab may be cast:

Why Composite?

EEntirely l cast-in-place l with removable formwork Using precast panel as a formwork formwork, the pour the concrete topping

There are some benefits of using precast elements l t

There are some benefits of putting the composite slab

Particular Design Aspects

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

Loadingg Stages g

Actual width vs. Effective width vs. Transformed width Composite Section Properties Allowable Stress Design SShored o vs. U Unshored s o Beams a s

Horizontal Shear Transfer

Save Time Better Quality Control Cheaper

Provide continuity between elements Quality control is not that important in slabs

Composite Section Properties

There are 3 value of widths we will use:

Actuall width A d h off the h composite section (b): (b) This Th is equal to the girder spacing Effective width of the composite section (be) Transformed width of the composite section (btr)


Effective 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.

Effective Width (AASHTO LRFD - 4.6.2.6.1)

boverhang ⎧ b b 'w = max ⎨ w ⎩bf / 2

Exterior Girder

Exterior Beam

be,ext


Typically the concrete used for slab has lower strength than h concrete used d for f precast section i Lower strength Æ Lower modulus of elasticity Thus, we need to use the concept of transformed section to transform the slab material to the precast material

btr = be nc = be

Ec ,CIPC Ec ,PPC

≅ be

Modular Ratio, usually < 1.0

f 'c ,CIPC f 'c ,PPC

⎧b 'w / 2 + 6ts ⎪ = + min ⎨ boverhang 2 ⎪ L/8 ⎩ be,int

Transformed Width

ts

bw Interior Girder

Interior Beam

⎧b 'w + 12ts ⎪ be = min ⎨ s ⎪ L/4 ⎩


Transformed Width

s be

be

bf



Summary of steps for Width calculations

Actual Width b Equals to girder p g spacing

Effective Width be Accounts for nonuniform stress distribution

Composite CrossSectiona Area: Acc

Precast Cross CrossSectiona Area: Ac

ytc ((abs) b)

yt (abs)

yy’tc (abs)

dp

Aps

dpc

Most of the theories learned previously for the noncomposite it section ti still till hold h ld but b t with ith some modifications We will discuss two design limit states

Aps

Precast vs. Composite

btr

ybc (abs)

yb (abs)

Acc = Ac + tsbtr ytc, ytb Igc Ztc, Zbc dpc

Design of Composite Section

c.g. p Composite c.g. Precast

h (abs)

Transformed Width btr Accounts for dissimilar material properties


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

Allowable All bl St Stress D Design i Ultimate Strength Design

Allowable Stress Design - Composite

OUTLINE Shored vs. Unshored Stress Inequality Equation Feasible Domain & Envelope

Allowable Stress Design - Composite

In allowable stress design, we need to consider two loading stages as pprevious; however, the initial moment ((immediatelyy after transfer) is resisted by the precast section whereas the service moment (after the bridge is finished) is resisted by the composite section (precast section and slab acting together as one member) We need to consider two cases of composite construction methods: h d

Shored vs vs. Unshored

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


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

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

Unshored

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


FULLY SHORED Consider, as example, the top of precast beam σt =


Top of precast, not top of composite i

F Feo (MGirder ) (MSlab + MSD + MLL +IM )y 'tc − + + ≤ σ cs Ac Zt Zt Igc

σt =


UNSHORED Consider, as example, the top of precast beam F Feo (MGirder + MSlab ) (MSD + MLL +IM )y 'tc − + + ≤ σ cs Ac Zt Zt Igc

Stress Inequality Equations

From both case we can rewrite the stress equation q as:

Case σt =

Mp = Moment resisted by the precast section (use Zt, Zb)

F Feo (MP ) (MC ) − + + ≤ σ cs Ac Zt Zt Z 'tc

Fully Shored: Mp = Mgirder Unshored: Mp = Mgirder + Mslab

Mc = M Moment resisted i d bby the h composite i section i ((use Z’tc, Zbc)

Fully Shored: Mc = Mslab + MSD + MLL+IM Unshored: Mc = MSD + MLL+IM

I

Initial-Top

II

I ii lB Initial-Bottom

III Service-Top ! IV Service-Bottom VI Service-Top Slab

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

Top p of precast, p not top of composite

Stress Inequality Equation σt =

Fi Fi eo Mmin Fi ⎛ eo ⎞ Mmin − + = ≥ σ ti ⎜1− ⎟ + Ac Zt Zt Ac ⎝ kb ⎠ Zt

σb =

Fi Fi eo Mmin Fi ⎛ eo ⎞ Mmin + − = ≤ σ ci ⎜1− ⎟ − Ac Zb Zb Ac ⎝ k t ⎠ Zb

σt =

F Feo M p Mc F ⎛ eo ⎞ M p Mc − + + = + ≤ σ cs ⎜1− ⎟ + Ac Zt Zt Ztc Ac ⎝ k b ⎠ Zt Z 'tc

σb =

F Feo M p Mc F ⎛ eo + − − = ⎜1 − Ac Zb Zb Zbc Ac ⎝ kt

σ t ,slab =

⎞ M p Mc − ≥ σ ts ⎟− ⎠ Zb Zbc

Mc M E nc = c c ,CIPC ≤ σ cs,Slab Ztc Ztc Ec ,PPC

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

Feasible Domain & Envelope

Top of precast

We can rewrite the stress equations and add practical limit equation

No. I

Case


Initial-Top p

II

Initial Bottom Initial-Bottom

III

S Service-Top T

⎛1⎞ e0 ≤ k b + ⎜ ⎟ Mmin − σ ti Zt ⎝ Fi ⎠ ⎛1⎞ e0 ≤ kt + ⎜ ⎟ Mmin + σ ci Zb ⎝ Fi ⎠ ⎞ Z ⎛ 1 ⎞⎛ e0 ≥ k b + ⎜ ⎟ ⎜ M p + Mc t − σ cs Zt ⎟ Z 'tc ⎝ F ⎠⎝ ⎠

(

)

(

)

IV

ServiceBottom

⎞ Z ⎛ 1 ⎞⎛ e0 ≥ kt + ⎜ ⎟ ⎜ M p + Mc b + σ ts Zb ⎟ Zbc ⎝ F ⎠⎝ ⎠

V

Practical Limit

e0 ≤ ( e0 )mp = y b − dc ,min

VI

Service-Top Slab

σ t ,slab =

!

F Feo M p ΔMcr F ⎛ eo + − − = ⎜1− Ac Zb Zb Zbc Ac ⎝ kt

ΔMcr =

⎞ M p ΔMcr − ≥ σ ts ⎟− Z Z b bc ⎠

Zbc ⎡F (eo − kt ) − M p ⎤⎦ − fr Zbc Zb ⎣

Mcr = ΔMcr + M p

σb =

F Feo Mcr F ⎛ eo ⎞ Mcr + − = = fr ⎜1− ⎟ − Ac Zb Zb Ac ⎝ k t ⎠ Zb

Mcr = F (eo − kt ) − fr Zb

Ultimate Strength Design - Composite

Cracking C k occurs in the h composite section We find ∆Mcr ((moment in addition to Mp) σb =

Cracking occurs in the precast section The equation is the same as noncomposite section

Mc M E nc = c c ,CIPC ≤ σ cs,Slab Ztc Ztc Ec ,PPC

II. Cracking moment is greater Mp

We consider 2 cases 1. Cracking moment is less than Mp

Cracking Moment - Composite

Cracking Moment - Composite

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

When the neutral axis is in the slab, we can use a composite Tsection with flange width equals to Effective Width and using ff’c of the slab When the neutral axis is in the precast section section, we may use a Transformed Section and f’c of the precast section - This is an pp value but the errors to the ultimate moment approximate capacity is small.

Shear Transfer Mechanisms

Shear Transfer

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

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:

Shear Transfer – Cohesion & Friction

Shear Transfer - Formula

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

Friction (F = μN) Cohesion

AASHTO LRFD (5.8.4) The nominal shear resistance at the interface between two concretes cast at different times is taken as: Friction Factor

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.

Area of Concrete Transfering Shear Cohesion

Area of shear reinforcement crossing the shear plane Compressive force normal to shear plane

⎧≤ 0.2f 'c Acv Vnh = cAcv + μ( Avf fy + Pc ) ⎨ 5 5 Acv ⎩ ≤ 5.5 COHESION FRICTION

N

Vhu ΦVhn =ΦμN



AASHTO LRFD (5.8.4.2) (5 8 4 2)

Minimum Shear Reinforcement

For Vn/Acv > 0.7 MPa, the cross-sectional area of shear reinforcement crossing the interface per unit length of beam must not be less than

0.35bv Avf ≥ fy

If less, then we cannot use any Avffy in the nominal shear strength The spacing of shear reinforcement must be ≤ 600 mm Possible reinforcements are: Single S l bbar

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

Stirrups (multiple legs) W ld d wire Welded i fabric f bi

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

The normal force in the friction formula comes from two parts Yielding of shear reinforcement If cracking k occurs at the h interface, there will be tension in the steel reinforcement crossing the interface. This tension force in steel is balanced b the by h compressive i force f in i concrete at that interface; thus, creating normal “clamping” clamping force. Permanent compressive force at the interface Dead Weight of the slab and wearing earin surface s rface (asphalt) (as halt) Cannot rely on Live Loads

Avf

N=Avf fy

Ultimate Shear Force at Interface

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

Using Classical Elastic Strength of Materials

Vuh =

( ΔVu ) Q Igc

Moment of Inertia of the composite section

Factored shear force acting on the composite section only (SDL +LL+IM) Moment of Area above the shear plane about the centroid of composite section

Using Approximate Formula (C5.8.4.1-1)

Vuh =

Vu de

Total Factored vertical shear at the section Distance from centroid of tension p of the deck steel to mid-depth

Ultimate Shear Force at Interface

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

Some Design Tips

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 h/2

h/2

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

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

Final Notes on Composite Behavior

Composite C it section ti iis used d nott only l ffor prestressed t d concrete t sections, but also for steel sections. Benefits is that the slab helps resists compression and helps prevent lateral torsional buckling of the steel section, as well as local bucklingg at the compression p flange. g

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


The analysis concept is similar to that of prestressed concrete. There are also:

Effective width and transformed section Shored and Unshored Construction Sh Shear Transfer T f att IInterface t f

b


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

Spirals

Studs

Final Notes on Composite Behavior Shear Stud is one of the most common shear h connectors – it is welded to the top flange of steel girder

Channels


Part IV: Things I did not teach but you should be aware of !!! Shear Strength – MCFT Unbonded and External Prestressing Anchorage Reinforcement Camber and Deflection Prediction Detailed Calculation of Prestress Losses Steel Girder with Shear Stud

Shear

Shear - MCFT

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

Shear

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

Minimum Transverse Reinforcement

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

Vn = Vc + Vs + Vp ≤ 0.25f 'c bv dv + Vp Vs =

Av fy dv cot θ s

Vc = 0.083 β f 'c bv dv

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

Vu > φ0.5(Vc + Vp )

If we need it, it the minimum amount shall be

Av ≥ 0.083 0 083 f 'c

bv s fy

Minimum Transverse Reinforcement

Maximum Spacing

Unbonded or External Prestressing

For vu <0.125f <0 125f’cc smax = 0.8 dv ≤ 600 mm

For vu > 0.125f’c smax = 0.4dv ≤ 300 mm

Strain compatibility does pp y for unbonded not apply tendon (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 beam

Must subtract the area of duct to the width 5.8.2.7

Unbonded or External Prestressing

Anchorage Reinforcement

View inside a box-section

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


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


Methods:

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


Camber and Deflection

Strut-and-Tie Method

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

Detailed Calculation of Prestress Loss

References

In many cases, cases it is adequate to use the “Lump Lump Sum Sum” loss

IIn some cases, we need to know exactly tl the th stress t in i the strands so we can determine the camber and deflection

Cantilever Construction Repair/ Rehabilitation

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, USA h // http://www.technopress3000.com h 3000

40609 Prestressed

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