12.3 flexure

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12.3.1

FLEXURAL DESIGN

Service Limit State» Transformed Section» Compute Stresses» Compare with Stress Limits

Strength Limit StateMoment - Curvature

12.3.2General Assumptions for Flexural Design

“Plane sections before bending remain plane after bending”

Equilibrium of external forces and internal stresses

Compatibility of strains

12.3.3General Assumptions for Flexural Design of Prestressed Concrete Members

Service Load Design:• Concrete is uncracked• Stress in prestressing steel is linearly proportional to strain• Iterate to determine strand pattern

- satisfy stress limits for concrete and prestressing steel

Check Strength at Critical Sections:• Concrete

- inelastic in compressive regions- tensile strength neglected

• Prestressing steel- inelastic

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12.3.4Determine Strand Pattern

Add strands until stress limits at midspan are satisfied

• Fill rows from bottom• Minimum strand spacing

- LRFD Article 5.10.3.3.1 • Minimum Cover• Minimum Cover

- LRFD Article 5.12.3

Then check stresses at ends

12.3.5Typical Strand Pattern

12.3.65.10.3.3.1 Minimum Strand Spacing

Strand Size (in.) Spacing (in.)0 6000

Minimum clear distance between starnds at ends of pretensioned girders:

• 1.33 x maximum aggregate size• 3db

0.60000.5625 Special

0.56252.00

0.50000.4375

0.50 Special1.75

0.3750 1.50

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12.3.7Minimum Concrete CoverLRFD 5.12.3

12.3.8Minimum Concrete Cover

LRFD 5.12.3

Modification factors for W/C ratio

For W/C ≤ 0.40 . . . . . . . . . . . 0.8

For W/C ≥ 0.50 . . . . . . . . . . . 1.2

Minimum cover to main bars, including epoxy-coated bars = 1.0 IN.

Minimum cover to ties and stirrups may be 0.5 IN. less than the values specified in Table for main bars, but shall not be less than 1.0 IN.

12.3.9Design for Flexure at Service Limit State

Compute Section Properties• Determine effective width of deck• Transform deck to girder concrete• Transform strand (optional)

Compute Stresses• At release• At Service Limit State

- Permanent loads only- Permanent and transient loads

Compare Stresses to Stress Limits• Concrete• Prestressing Steel

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12.3.10Transform Composite Deck Concrete to Girder Concrete

Effective deck width - (LRFD 4.6.2.6.1)

Transformed effective deck width

Use same modular ratio for short- and long-term effects

12.3.11Transform Prestressing Steel to Girder Concrete

LRFD 5.9.1.4

Section properties may be based on either the gross or transformed section

Prestressing steel may be transformed using the same procedure used for mild reinforcement

12.3.12Assumptions for Service and Fatigue Limit States

LRFD 5.7.1

The following should apply to modular ratios between steel and concrete:

• the modular ratio, n, is rounded to the nearest integer number,

• the modular ratio is not less than 6.0, and

• an effective modular ratio of 2n is applicable to permanent loads and prestress.

- intended to apply to compression reinforcement - see Std Specs Article 8.15.3.5

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12.3.13Compute Stresses at Release

Non-Composite Section (Bare Girder)

Loads• Girder dead load• Initial prestress

Top of girder

Bottom of girder

b

gdl

b

iiRb

t

gdl

t

iiRt

SM

SeP

APf

SM

SeP

APf

−+=

+−=

12.3.14Compute Stresses at Release

12.3.15Compute Stresses at Service Limit State After Losses with Permanent Loads Only

Composite Section (Girder + Deck)

Loads on Non-Composite Section• Girder, deck dead loads• Other dead loads applied before placing deck

( di h )(e.g., diaphragms)• Final prestress (after losses)

Loads on Composite Section• Barrier and future wearing surface• Other dead loads (utilities, etc.)• Vehicular live load

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12.3.16Compute Stresses at Service Limit State After Losses with Permanent Loads Only

Top of deck

tcd

cdlPtd S

Mf =

Top of girder

Bottom of girder

bcg

cdl

b

ncdlgdl

b

eePbg

tcg

cdl

t

ncdlgdl

t

eePtg

SM

SMM

SeP

APf

SM

SMM

SeP

APf

−+

−+=

++

+−=

12.3.17

Compute Stresses at Service Limit State After Losses with Permanent and Transient Loads

Top of deck

tcd

ILLcdlLPtd S

M Mf ++

+=

Top of girder

Bottom of girder

bcg

ILLcdl

b

ncdlgdl

b

eeLPbg

tcg

ILLcdl

t

ncdlgdl

t

eeLPtg

SM M

SM M

S

eP APf

SM M

SM M

S

eP APf

++

++

+−

+−+=

++

++−=

12.3.18

Compute Stresses at Service Limit State After Losses with Permanent and Transient Loads

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12.3.19Stress Limits for Prestressing Tendons

LRFD 5.9.3

For Pretensioned Construction:

• Low relaxation strand ( fpy = 0.90 fpu ):0 75f Immediately prior to transfer0.75fpu Immediately prior to transfer 0.80fpy At Service Limit State, after

losses

• Stress Relieved strand ( fpy = 0.85 fpu ):0.70fpu Immediately prior to transfer 0.80fpy At Service Limit State, after

losses

12.3.20Stress Limits for Concrete

LRFD 5.9.4.1.1 and 5.9.4.1.2

For Temporary Stresses Before Losses (Fully Prestressed Components):

• Compression: Pretensioned components

• Tension (non-segmental bridges):Precompressed tensile zone without bonded

reinforcement≤ 0.200 KSI – Other than precompressed tensile

zone, and without bonded reinforcementIn areas with bonded reinforcement sufficient to

resist concrete tensile force (fs = 0.50fy)

cif0948.0 ′

A/N

cif 60.0 ′

cif24.0 ′

12.3.21

LRFD 5.9.4.2.1

For Stresses At Service Limit State After Losses (Fully Prestressed Components):

• Compression (non-segmental bridges):

Stress Limits for Concrete

Compression (non-segmental bridges):

c

c

c

f 40.0

f 60.0f 45.0

′

′′ Permanent loads

Permanent and transient loads, and during shipping and handlingLive load and 0.5 the sum of effective prestress and permanent loads

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12.3.22Stress Limits for Concrete

LRFD 5.9.4.2.2


• Tension in precompressed tensile zone (other th t l b id )than segmental bridges):

Components with bonded prestressing tendons other than piles

Components subjected to severe corrosive conditions

Components with unbonded prestressing tendons

tension no

f0948.0

f190.0

c

c

′

′

12.3.23

LRFD 5.9.4.2.2


• Tension in other areas (segmental only):

Stress Limits for Concrete

Note other tensile stress limits for segmentally constructed bridges.

cf190.0 ′ If bonded reinforcement is provided which is sufficient to carry the tensile force in the concrete at a stress of 0.5fsy

12.3.24Control of Stresses at Ends of Pretensioned Members

The following methods can be used individually or in combination with other methods

1. Draping, Harping or Deflecting• Reduce eccentricity at ends• Raise center group of strands until stressRaise center group of strands until stress

limits are satisfied

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2. Debonding, Blanketing or Shielding• Reduce prestress force at ends by preventing

bond of selected strands with concrete• Increase number of debonded strands until

stress limits are satisfied

12.3.26Special Provisions for Debonded Strands

Std Specs 9.27.3 requires:• Development length for debonded strands is

doubled

LRFD 5.11.4.3 further requires:• Number of strands debonded ≤ 25% of totalNumber of strands debonded ≤ 25% of total

strands• Number of strands debonded in any row ≤ 40%

of total strands in that row • Exterior strands in each row must be fully

bonded • All limit states must be satisfied


3. Adding Mild Reinforcement• If tensile stress > , but not more

than , add mild reinforcement to resist 120% of the tensile force

cif0948.0 ′

cif22.0 ′

( )( )s

toptopcis f

bx2f2.1A =

where fs = 0.5 fsy = 30 KSI

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4. Adding Top Strands• Reduce moment at ends by adding

strands at the top of the girder• Can debond top strands in center

portion of the girderportion of the girder- Must provide access hole for cutting

strand

12.3.29

5. Increasing Compressive Strength of Concrete at Release,

• Increase until stress limits are satisfied

Control of Stresses at Ends of Pretensioned Members

cif ′

cif ′

• Use reasonable value for that can be achieved economically by local producers

• Maintain reasonable balance between cci fand f ′′

cif ′

12.3.30Fatigue Limit State Stress Range Requirements

LRFD 5.5.3.3

Prestressing Tendons

• 18.0 KSI for radii of curvature in excess of 30.0 FT• 10 0 KSI for radii of curvature not exceeding 12 0 FT• 10.0 KSI for radii of curvature not exceeding 12.0 FT• Linear interpolation may be used between the limits

Fatigue loading is a design truck (no lane load) with constant axle spacing of 30.0 FT.

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12.3.31Basic Assumptions for Design at Strength Limit State

LRFD 5.7.2.1

Reinforcement• Mild reinforcement yields• Prestressing steel is near or beyond yield

Concrete• Maximum usable concrete strain is 0.003 IN/IN

for unconfined concrete• Stress-strain distribution results in predictions

of strength in substantial agreement with test results

12.3.32Stress in Prestressing Steel at Nominal Flexural Resistance

LRFD 5.7.3.1

Components with bonded tendons with flexure about one axis and where fpe is not less than 0.5 fpu :

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

ppups d

ck ff 1

where:fpe = effective stress in the prestressing steel after losses fpu = specified tensile strength of prestressing steel

k =

= 0.28 for low relaxation strandfpy = yield strength of prestressing steel

⎟⎟⎠

⎞⎜⎜⎝

⎛−

pu

py

ff

04.12

12.3.33Stress in Prestressing Steel at Nominal Flexural Resistance

LRFD 5.7.3.1

c = distance from the extreme compression fiber to the neutral axis

dp = distance from the extreme compression fiber to centroid of prestressing steel

For rectangular section behavior:

where:Aps = area of prestressing steel

b = width of compression face

p

pups1c

ysyspups

df

Akbf85.0

fAfAfAc

+′

′′−+=

β

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12.3.34Concrete Stress Distribution at Nominal Flexural Resistance

LRFD 5.7.2.2

Equivalent Rectangular Stress Block • Used in lieu of “more exact” stress distributions • Stress is 0.85 fc’• Stress block extends a distance a= β1c from the

extreme compression fiber• β1varies with concrete strength fc’

12.3.35

Equivalent Rectangular Concrete Stress Distribution at Nominal Flexural Resistance

LRFD 5.7.3.2

psps

c

fATabf.C

ca

=

′==

8501β

12.3.36

LRFD 5.7.3.2

Mr = φMn

Rectangular Sections:

⎞⎛⎞⎛⎞⎛ aaa

Nominal Flexural Resistance

where:

Mr = factored resistanceMn = nominal resistancea = β1c

= depth of equivalent stress block

⎟⎠⎞

⎜⎝⎛ −′′′−⎟

⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −=

222a d fA a d fA a d fAM syssysppspsn

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12.3.37Nominal Flexural Resistance

LRFD 5.7.3.2.2

Flanged Sections:

Mn = Equation for Rectangular +

Note:

Flanged section applies when c > hf , while this applied when a > hf in Std Specs

)2h

2a(h)bb(f85.0 f

f1w'c −− β

12.3.38Nominal Flexural Resistance for Flanged Concrete Sections

(MacGregor 1988)

12.3.39

Nominal Flexural Resistance for Rectangular Concrete Sections with Compression Reinforcement

(MacGregor 1988)

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12.3.40

Unified Design Provisions – Key Concept

Strength reduction factor, φ,depends on

maximum net tensile strain, εt ,t

at nominal resistance, Mn

12.3.41

5.2 - Definitions

Net Tensile Strain - The tensile strain at nominal resistance exclusive of strains due to effective prestress, creep, shrinkage, and temperature.

12.3.42

5.2 - Definitions

Extreme Tension Steel — The reinforcement (prestressed or nonprestressed) that is farthest from the extreme compression fiber.

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12.3.43

5.2 - Definitions

dt

0.003

εt = Net tensile strain dt = Depth to extreme tension steel

εtColumnStrainBeam

12.3.44

5.2 - Definitions

εt Extreme tension steel strain at nominal resistance, due to applied loads

0 003

εt

c a = β1c C

T

Pn

Mn

0.003

12.3.45

5.2 - Definitions

Compression-Controlled Strain Limit —The net tensile strain (εt ) at balanced strain conditions. See Article 5.7.2.1.

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12.3.46

5.7.2.1 – Balanced Strain Condition

0.003

fy /Es (or 0.002)

12.3.47

5.2 - Definitions

Compression-Controlled Section — A cross section in which the net tensile strain (εt ) in the extreme tension steel at nominal resistance is less than or equal to the compression-controlled strain limit.

[Usually 0.002]

12.3.48

5.2 - Definitions

Tension-Controlled Section — A cross section in which the net tensile strain (εt ) in the extreme tension steel at nominal resistance is greater than ornominal resistance is greater than or equal to 0.005.

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12.3.49

5.5.4.2 Resistance Factors φ

P/S1.00

R.C.0.90

⎟⎠⎞

⎜⎝⎛ −+= 1

cd25.0583.0 tφ

Transition Tension -Controlled

Compression-Controlled

φ

εt = 0.002 εt = 0.005

0.75

Net Tensile Strain

⎟⎠⎞

⎜⎝⎛ −+= 1

cd15.065.0 tφ

12.3.50

10.3.3-4 – STRAIN CONDITIONS

Compression-Controlled

Tension-ControlledTransition

c ≤ 0.375 dt0.375 dt < c < 0.6 dtc ≥ 0.6 dt

12.3.51Example – R.C. Beam

a = β1c C

Tεt

c

0.00312”

3#8dt = 13.5”16”

Given: f’c = 4 ksi; fy = 60 ksiAssume steel yieldsT = Asfy = 3(0.79)60 = 142.2 kipsa = T/(0.85 f’cb) = 3.49 in. c = a/β1 = 4.1 in.Mn = T [dt-(a/2)] = 1672 in.-k = 139.3 ft-kc/dt = 4.1/13.5 = 0.304 < 0.375 orεt = 0.003 [(dt-c)/c] = 0.0069 in./in. Tension-controlledMr = φMn = 0.90 (139.3) = 125.4 ft-k

t

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12.3.52Minimum Flexural Reinforcement

LRFD 5.7.3.3.2

“Unless otherwise specified, at any section of a flexural component, the amount of prestressed and non-prestressed reinforcement shall be adequate to develop ...”:

Mr = φ Mn ≥ 1.2 Mcr

≥

where:• Mcr is based on the modulus of rupture.

uM34

12.3.53Minimum Flexural Reinforcement

LRFD 5.7.3.3.2

where:fr = modulus of rupturef = stress in concrete due to effective prestress

rcnc

cdncperccr fS1

SS M- )f (fS M ≥⎟⎟

⎠

⎞⎜⎜⎝

⎛−+=

fpe = stress in concrete due to effective prestress only (after all losses) at surface where tension is caused by applied loadsSc = composite section modulus for extreme tension fiberSb = non-composite section modulus for extreme tension fiber

Md/nc = non-composite dead load moment at the section

12.3.54

Interior Diaphragms

Intended Purposes• Improve distribution of loads between girders• Stabilize girders during construction• Temporary bracing can be used

Di d tDisadvantages• Costly to install• Research has shown that they are not

necessary for load distribution• May be detrimental in cases of side impact

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12.3.55

Lateral Stability

Lateral stability of long girders should be considered• See references by Mast• Lifting

- supported by cables- location of lifting loops has greatest effect g p g

on stability• Transportation

- bottom supported, but rotation possible- characteristics of trailer and route strongly

affect stability• Sections with large flanges are more stable

12.3.56

Moment-Curvature Analysis(Lin and Burns 1981)

12.3.57Example of Moment-Curvature

Response of Prestressed Girder(Lin and Burns 1981)

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12.3.58Calculated and Observed Moment-Curvature Responses

(Collins and Mitchell 1991)

12.3.59Moment-Curvature Analysis of a Concrete Member

(Collins and Mitchell 1991)

12.3.60Moment-Curvature Response for Different Quantities of Mild Reinforcement

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12.3.61Effect of Compression Reinforcement on Moment Capacity

(MacGregor 2005)

12.3.62Increase in Moment Capacity with Compression Reinforcement

(MacGregor 2005)

12.3.63Effect of Compression Reinforcement on Long-Term Deflections

(MacGregor 2005)

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12.3.64

Effect of Compression Reinforcement onMoment-Curvature Response of

Under-Reinforced Sections(MacGregor 2005)

12.3.65Effect of Compression Reinforcement on Moment-Curvature Response of

Over-Reinforced Sections(MacGregor 2005)

12.3 flexure

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