ABC Introducción Al Diseño y Construcción de Elementos Pretensados Para Puentes

8/11/2019 ABC Introduccin Al Diseo y Construccin de Elementos Pretensados Para Puentes

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PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard

Pre-Stressed ConcreteStructures

Basic Concepts

ABC - Introduccin al Diseo y

Construccin de Elementos

Pretensados para Puentes

Centro de Transferencia de Tecnologa en Transportacin

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2

Basic Principles of Reinforced Concrete Design

Deflection issue

2


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Deflection Response of RC Beams

force

Deflection at Midspan 3


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4

First Cracks Appear, End of Elastic beam behavior. The principles learned during the

Mechanical of Material Courses are not longer useful.

Response of member

Top face feels compression

Bottom face feels tensionConcrete weak in tension

Starts cracking soon

P0

First Cracks

Concrete Beam Behavior Initial Stage


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5

New Cracks Appear. The first cracks open more and more each time that the loadincrease . There is not sign of failure of the concrete located in the compression side of

the beam ( Top Side of the beam for this load condition)

P1

Existing Cracks OpenNew Cracks Appear

Concrete Beam Behavior Intermediate Stage


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6

Pu

Comb teeths

Steel Reinforcement

Concrete Beam Behavior Ultimate Stage

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7

At ultimate load ( Pu) The resisting moment can't be computed using the principles

of mechanics of materials due that the section has suffer a considerable inelasticdeformation. For this case the resisting moment shall be computed using equilibrium

principles. Or in other words the resisting moment can easily computed at any

section i as the value of C or T multiplied by z

C

Tz

i

Concrete Beam Behavior Ultimate Stage


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8

Stresses Induced by the Acting Load


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Example

Given the beam section illustrated below, and assuming that the beam is simple supported andhas a span of 50 ft, find the maximum live load that can be applied to this beam when the beam

is reinforced with the balanced reinforcement and when it is reinforced with the limit steel

reinforcement that correspond to the tension-Controlled Sections .

28

12

As

fy = 60 ksi

f!c= 5 ksi

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h

bw

As

0.85fc

a = 1ca/2

C

T

dd

a

2

c

0.003

#s > #y

Concrete Design Basic Concepts

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11

Mn= (C or T) (d ) = Asfy(d )a

2

a

2

Mn= Asfy(d )

0.5As fy

0.85f!cb

Concrete Design Basic Concepts

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12

Balanced Strain Condition

h

bw

Asb

0.85fc

#s = #y= fy/Es

ab= 1cbCb

T

dt dtab2

cb

0.003

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cb= 0.592dt

As= 0.5031fcbdt/ fy

$b= 0.5031fc/ fy

Mnb= $b b dt fy( dt )

0.5 $b dt fy

0.85 fc

cb

dt=

0.0030.003 + #y

#s = #y= fy/Es

#s = #y= .00207 ;fy = 60 ksi

Balanced Strain Condition

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14

Tension-Controlled Sections

h

bw

As

0.85fc

#t = 0.005

at= 1ctCt

T

dt dtat2

ct

0.003

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15

ct= 0.375dt

As= 0.3191fcbdt/ fy

$t= 0.3191fc/ fy

Mnt= $t b dt fy (dt )

0.5 $t dt fy

0.85fc

Tension-Controlled Sections

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$b= 0.5031fc/ fy

$

t= 0.3191fc/ fy

$b= 0.503*0.8*5 ksi / 60 ksi = 0.0335

$b= 0.319*0.8*5 ksi / 60 ksi = 0.0213

Steel for Balanced and Tension-Controlled Sections

16

PRE STRESSED CONCRETE SEMINAR


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Mnb= $b b dt fy( dt )0.5 $b dt fy

0.85 fc

Mnt= $t b dt fy (dt )

0.5 $t dt fy

0.85fc

Mnb= 0.33512*25.5*60 ( 25.5 )

0.5*0.335*25.5*60

0.85*5

Mnt= 0.21312*25.5*60 ( 25.5 )

0.5*0.213*25.5*60

0.85*5

Mnb=11975 k-in = 998 k-ft ; Mu = %Mnb = 0.65*998 = 648 k-ft

Mnt

= 8472 k-in = 706 k-ft ; Mu

= %Mnt

= 0.9*706 = 635.5 k-ft

Balanced and Tension-Controlled Moments

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hmin= 50/16 = 3.125 ft = 37.5 > 28 ; L / hmin= 50/ 3.125 = 16

WLL-bal= 0.61 k/ft

wDL=b*h*&=(12*28)/144*.15 k/ft3 = 0.336 k/ft

Mu ={(1.2*(wDL + wSup-Imp)+1.6wLL) * L2}/8

wLL= (8*Mu/L2-1.2*(wDL + wSup-Imp)/1.6

WLL-tens= 0.59 k/ft

ACI - Table 9.5(a) Minimum thickness, h

MemberSimply

supportedOne end

continuousBoth endscontinuous

Cantilever

Beams or ribbedone-way slabs

/16 /18.5 /21 /8

WSup-Imp=0.560 k/ft

WLL-bal+ WSup-Imp= 1.17 k/ft WLL-tens+ WSup-Imp = 1.15 k/ft

Live loads

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WLL-bal-38= 0.52 k/ft

wDL=b*h*&=(12*38)/144*.15 k/ft3 = 0.475 k/ft

Mu ={(1.2*(wDL + wSup-Imp)+1.6wLL) * L2}/8

wLL= (8*Mu/L2-1.2*(wDL + wSup-Imp)/1.6

WLL-tens-38= 0.49 k/ft

WSup-Imp=0.560 k/ft

Vs. WLL-bal-28= 0.61 k/ft

Vs. WLL-tens-28= 0.59 k/ft

It is curious, I can obtaining the

capacity needed with a 12x 28

beam but probably if I use this one,

the deformation will be greater than

the code limits. If I increase the beam

size the opposite situation occurs. I

am wondering if something differentcould be done???

Result Discussions

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28

12

As

SAFE SUPERIMPOSED SERVICE LOAD

Wsup-Imp + WLL-bal-38=0.56 k/ft + 0.52 k/ft =1.08 k/ft

38

12

As

Reinforced ConcretePre-stressed

Concrete

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21

Dr. Wendy

46'

36'

5.75'

1.5'5' 1.5'2'

5.75'5.75' 5.75' 5.75' 5.75' 5.75' 2.25'3.50'

Clearly here we have a problem. If I want tomaximize the capacity of the structure to

sustain very heavy Live Loads Like in bridges,I need to reduce the effect of the dead load.

But If I reduce the size of the elements thedeflections will increase and I will obtain

deflections values that will be greater than thevalues prescribed by the code. Let me see if

pre-stressing the structure I solve the problem.

Result Discussions

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22

Pre-Stressed Effects



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Pre-Stressed Load Application Procedure

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Post-tensioned Load Application Procedure

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25

PRE-STRESSED LOADAPPLICATION PROCEDURE

Equivalent Pre-Stressed Effects

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Equivalent Pre-Stressed Effects

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27

Pre-Stressing, Post-Tensioning Advantage

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Pre-Stressing Construction Advantage

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Pre-Stressing Construction Advantage

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Pre-Stressing Element Connections

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Pre-Stressed Slabs

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PRE-STRESSED CONCRETE SEMINAR


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.Pre-Stressed Slabs

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Pre-Stressed Slabs

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Pre-Stressed Slabs

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Pre-Stressed Slabs

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Pre-Stressed Slabs

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PRE STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard

Pre-Stressed Slabs

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Pre-Stressed Slabs

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Pre-Stressed Slabs

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Pre-Stressed Slabs

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Pre-Stressed Slabs

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Pre-Stressed Beams

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Pre-Stressed Beams

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Pre-Stressed Beams

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Pre-Stressed Beams

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By: Dr. Daniel A. Wendichansky Bard

Pre-Stressed Beams

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Pre-Stressed Beams

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Pre-Stressed Beams

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Pre-Stressed Columns

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Post-Tensioned Beams

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Post-Tensioned Anchorages

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Post-Tensioned Bars

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Pre-stressed Building Examples

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Pre-stressed Building Examples

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Materials for Reinforced and

Pre-stressed Concrete

ABC - Introduccin al Diseo yConstruccin de Elementos


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Presentation- References

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Compressive Strength of Concrete

Compressive strength

68

6

12

ASTM C 31

ASTM C 39



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69

Compressive strength Curve

Strain

Stress

f"c

Ec= wc1.533#f"c


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Compressive strength Curves

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Tensile strength Varies between 8% and 15% of the compressive

strength

71

Tension Strength of Concrete

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Modulus of rupture (flexural test)

72

b

h

fr=6M

bh2


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Modulus of rupture (flexural test)

ASTM C 78 Standard Test Method for Flexural

Strength of Concrete (Using Simple Beam withThird-Point Loading)

ASTM C 293 Standard Test Method for Flexural

Strength of Concrete (Using Simple Beam With

Center-Point Loading)

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Split cylinder test

!

d

P


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Split cylinder test

75

!

d

fct=2P

$!d

P


75

PRE-STRESSED CONCRETE SEMINARB D D i l A W di h k B d


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Split cylinder test

ASTM C 496 Standard Test Method for Splitting

Tensile Strength of Cylindrical Concrete Specimens

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Relationship between compressive and

tensile strengths

Tensile strength increases with an increasein compressive strength

Ratio of tensile strength to compressive

strength decreases as the compression

strength increases

Tensile strength %#f"c

77

Tension and Compression Strength Relationship

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Relationship between compressive and

tensile strengths

Mean fct= 6.4#f"c For deflections (Eq. 9-10):

fr= 7.5#f"c

For strength (ACI 11.4.3.1):

fr= 6#f"c

78

Tension and Compression Strength Relationship

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PRE-STRESSED CONCRETE SEMINARB Dr Daniel A Wendichansk Bard


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Shrinkage

Creep

Thermal expansion

79

Shrinkage, Creep and Thermal Expansion

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PRE-STRESSED CONCRETE SEMINARBy: Dr Daniel A Wendichansky Bard


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Shrinkage

Shrinkage

Shortening of concrete during hardening and

drying under constant temperature Moisture diffuses out of the concrete

Exterior shrinks more than the interior

Tensile stresses in the outer layer and

compressive stresses in the interior

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Shrinkage

ShrinkageStrain

Time

Shrinkage

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Shrinkage

When not adequately controlled, can cause:

Unsightly or harmful cracks Large and harmful stresses

Partial loss of initial prestress

Reinforcement restrains the development of

shrinkage

82

Shrinkage

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Creep

Strain

Time

Load applied

Elastic strain

Creep strain

Load removed

Elastic recovery

Creep recovery

Permanent

Deformation

Creep

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Creep strains can lead to

Increase in deflections with time

Redistribution of stresses

Decrease in pre-stressing forces

Creep

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Thermal expansion

Coefficient of thermal expansion or

contractionAffected by:

Composition of the concrete

Moisture content of the concrete

Age of the concrete

Thermal Expansion

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Thermal Expansion

Thermal expansion

Coefficient of thermal expansion or contraction

Normal weight concrete

Siliceous aggregate: 5 to 7 x 10-6strain/F

Limestone/calcareous aggregate: 3.5 to 7 x 10-6strain/F

Lightweight concrete

3.6 to 6.2 x 10

-6

strain/F

A value of 5.5 x 10-6strain/F is satisfactory

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Reinforcing Steels

Deformed Bar Reinforcement

Welded Wire Reinforcement

Pre-stressing Steel

87 87



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ASTM A 615, Specification for Deformed andPlain Carbon-Steel Bars for ConcreteReinforcement

ASTM A 706, Specification for Low-Alloy SteelDeformed and Plain Bars for ConcreteReinforcement

ASTM A 996, Specification for Rail-Steel and

Axle-Steel Deformed Bars for ConcreteReinforcement

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y a e e d c a s y a d


90

fs= Ess

fs= fy

90



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y y

Steel GradeMinimum yield

stress, fy(ksi)

Ultimate strength

(ksi)

40 40 70

50 50 80

60 60 90

75 75 100


Bar Designation Area (in.2) Weight (plf) Diameter (in.)

No. 3 0.11 0.376 0.375

No. 4 0.20 0.668 0.500

No. 5 0.31 1.043 0.625

No. 6 0.44 1.502 0.750

No. 7 0.60 2.044 0.875

No. 8 0.79 2.670 1.000

No. 9 1.00 3.400 1.128

No. 10 1.27 4.303 1.270

No. 11 1.56 5.313 1.410

No. 14 2.25 7.650 1.693

No. 18 4.00 13.600 2.257

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y y

Pre-stressing Steel

Strands

Seven-wire

Three- and Four-wire Wire

Bars Plain

Deformed

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y y

Pre-stressing Steel

ASTM A 416, Standard Specification for Steel

Strand, Uncoated Seven-Wire for Prestressed

Concrete

ASTM A 421, Standard Specification for

Uncoated Stress-Relieved Steel Wire for

Prestressed Concrete

ASTM A 722, Standard Specification forUncoated High-Strength Steel Bars for

Prestressing Concrete

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y y

Pre-stressed 7-wire strand

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Prestressing Steel

7-Wire Strands, fpu= 270 ksi

Nominal Diameter(in.)

Area (in.2) Weight (plf)

3/8 0.085 0.29

7/16 0.115 0.40

1/2 0.153 0.52

9/16 0.192 0.65

3/5 0.217 0.74

9696



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Prestressing Steel

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Design Aid 11.2.5in PCI Design

Handbook, 6th Ed.

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Prestressing Steel

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Prestressing Steel

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100

Losses in Pre-stressedConcrete

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The fraction of the jacking force (Pj) that is eventually transferred to the

concrete after releasing the temporary anchor or withdrawal of the

hydraulic jack is the Pi.

This force also keeps on decreasing with time due to time-dependent

response of constituent materials; steel and concrete and reduced to a

final value known as effective pre-stressing force, !Pi.

Pre-stressed Losses

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Pre-stressed Losses

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Topics Addressed

Pre-stressed Losses

Elastic Shortening

Creep of Concrete Shrinkage of Concrete

Steel Relaxation

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Essentially, the reduction in the pre-stressing force can be grouped intotwo categories:

Immediate elastic loss during the fabrication or construction process,

including elastic shortening of the concrete, anchorage losses, and

frictional losses.

Time-dependent losses such as creep and shrinkage and those due to

temperature effects and steel relaxation, all of which are determinable at

the service-load limit state of stress in the pre-stressed concrete element.

An exact determination of the magnitude of these losses, particularly the

time-dependent ones, is not feasible, since they depend on a multiplicity

of interrelated factors. Empirical methods of estimating

Pre-stressed Losses

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losses differ with the different codes of practice or recommendations,

such as those of the Pre-stressed Concrete Institute, the ACIASCE joint

committee approach, the AASHTO lump-sum approach, the Comite

Eurointernationale du Beton (CEB), and the FIP (FederationInternationale de la Precontrainte). The degree of rigor of these

methods depends on the approach chosen and the accepted practice of

record

Pre-stressed Losses

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Aps= total area of pre-stressing steel

e = eccentricity of center of gravity of pre-stressing steel with respect to center of

gravity of concrete at the cross-section considered

Ec= modulus of elasticity of concrete at 28 days

Eci= modulus of elasticity of concrete at time prestress is applied

Es= modulus of elasticity of prestressing steel. Usually 28,500,000 psi

fcds= stress in concrete at center of gravity of prestressing steel due to all

superimposed permanent dead loads that are applied to the member after it has

been pre-stressedfcir= net compressive stress in concrete at center of gravity of prestressing steel

immediately after the pre-stress has been applied to the concrete.

fcpa = average compressive stress in the concrete along the member length at the center

of gravity of the pre-stressing steel immediately after the prestress has been

applied to the concrete

NOTATION

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fg = stress in concrete at center of gravity of prestressing steel due to weight of

structure at time prestress is applied

fpi= stress in prestressing steel due to Ppi = Ppi/Aps

fpu= specifi ed tensile strength of prestressing steel, psi

Ic= moment of inertia of gross concrete section at the cross-section considered

Md = bending moment due to dead weight of member being prestressed and to any

other permanent loads in place at time of prestressing

Mds= bending moment due to all superimposed permanent dead loads that are applied

to the member after it has been prestressed

Ppi = prestressing force in tendons at critical location on span after reduction for

losses due to friction and seating loss at anchorages but before reduction for ES,

CR, SH, and RE

NOTATION (Cont.)

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For members with bonded tendons:

ES = KesEs fcir/Eci

where Kes= 1.0 for pre-tensioned members

Kes= 0.5 for post-tensioned members where tendons are tensioned in

sequential order to the same tension.With other post-tensioning procedures, the value for Kes may vary from

0 to 0.5.

fcir= Kcir fcpi fg

Where: Kcir= 1.0 for post-tensioned members

Kcir= 0.9 for pre-tensioned members


ES = KesEs fcpa/Eci

Elastic Shortening of Concrete (ES)

(1)

(2)

(1a) 108



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Creep Loss (CR)

In general, this loss is a function of the stress in the concrete at the

section being analyzed. In posttensioned, nonbonded members, the loss

can be considered essentially uniform along the whole span. Hence, an

average value of the concrete stress between the anchorage points can be

used for calculating the creep in posttensioned members.

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CR = KcrEs* (fcir- fcds) / Ec

Where: Kcr= 2.0 for pre-tensioned members

Kcr= 1.6 for post-tensioned members

For members made of sand-lightweight concrete the foregoing values of

Kcr should be reduced by 20 percent.

For members with unbonded tendons:

CR = Kcr

Es

/ Ec

* fcpa

Creep of Concrete (CR)

In general, this loss is a function of the stress in the concrete at the

section being analyzed. In post-tensioned, non-bonded members, the loss

can be considered essentially uniform along the whole span. Hence, an

average value of the concrete stress between the anchorage points can be

used for calculating the creep in post-tensioned members.

(3)

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As with concrete creep, the magnitude of the shrinkage of concrete is

affected by several factors. They include mixture proportions, type of

aggregate, type of cement, curing time, time between the end of

external curing and the application of pre-stressing, size of the member,

and the environmental conditions. Size and shape of the member also

affect shrinkage. Approximately 80% of shrinkage takes place in the first

year of life of the structure. The average value of ultimate shrinkage

strain in both moist-cured and steam-cured concrete is given in the ACI

209 R-92 Report.For post-tensioned members, the loss in pre-stressing due to elastic

shortening and shrinkage is somewhat less since most of the elastic

shortening and some shrinkage has already taken place before post-

tensioning.

Shrinkage of Concrete (SH)

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SH = 8.2 10-6 KshEs(1 - 0.06 V/S) (100 - RH)

where Ksh= 1.0 for pretensioned members

Kshis taken from Table for post-tensioned members.

Shrinkage of Concrete (SH)

(4)

TABLE 1

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RE = [Kre - J (SH + CR + ES)] C

where the values of Kre, J, and C are taken from Tables 2 and 3.

Relaxation of Tendons (RE)

(5)

TABLE 2

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TABLE 3

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Annual Average Relative Humidity

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AASHTO Lump-Sum Losses

Approximated Methods for Losses Computations

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119


P t d L


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120

Pre-stressed Losses

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For the simply supported double-tee shown below, estimate loss of

prestress using the procedures as outlined earlier under Computation ofLosses. Assume the unit is manufactured in Green Bay, WI.

EXAMPLE

121


EXAMPLE


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EXAMPLE

122


EXAMPLE


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EXAMPLE

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124


EXAMPLE


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EXAMPLE

125


EXAMPLE


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EXAMPLE

126


EXAMPLE


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EXAMPLE

Table 2

Table 2

Table 3

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Pre-stressed concrete DesignService Design

130



130



Introduction to the Service Design Concepts


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131

Introduction to the Service Design Concepts

Serviceability Limit State Design (SLS)is design of prestressed concretemember at the service load stage. The SLS design is related with the

working stress design method because the allowable stress limit

controls the design process. SLS design approach, control the selection

of thegeometrical dimensionand thelayout of the prestressing steelregardless if it is pre-tensioned or post-tensioned member. After the SLS

design is satisfied then the other factor such as shear design, torsion

design, ultimate strength design, control of deflection and

cracking are satisfied. Different with reinforced concrete design, in

prestressed concrete design several load stage must be checked such

as whenload transfer stage,service load stageandultimate load stage.

131


Basic Assumptions in the Service Design Method


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There are three basic concepts may be applied to design the prestressed

concrete member using, as follows:

1.The material is assumed as anelastic composite Material.

2.Thecompressive stress is carried by the concrete material and the

tensile stress is carried by the prestressing steel.

3.Pre-stressed concrete section is assumedun-cracked due to service

load

132

Basic Assumptions in the Service Design Method

132


Loading Stages


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133

Pre-stressed concrete design must consider all the loading stages

started from the initial pre-stressing force until the limit state of

failure. For each loading stage theactual stress must be checkedand

compare with the allowable stress defined by the code. At the

ultimate stage the flexural capacity must be compared with the

ultimate bending moment or shear. For serviceability limit statedesign thetwo loading stages may governs the design, as follows :

1.

Initial Loading , a loading stage where the prestressing force

is transferred to the concrete without any external loading

except the self weight of the structural member.

2.

Final Service Loading, a loading stage where the prestressing

force already reduced by several lossesand full service load is

applied.

Loading Stages

133


Loading Stages


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Complete loading stages forServiceability limit state design of

prestressed concrete member are as follows :

Initial Loading

a.Initial Prestressing Force.

b. Self Weight (DL)

Final Service Loading

a.Full Dead Load (SDL) + Effective Prestressing Force.

b.Full Service Load (DL+SDL+LL) + Effective Prestressing Force.

The loading stages for of prestressed concrete member at Ultimate limit

state designare as follows :

Full Ultimate Load (!DL + "SDL + #LL)Where: ! " #are combination Factors

134

Loading Stages

134


Initial Loading Stage


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THE TRANSFER STAGE

The intial pre-stresing forcePi is acting in the concrete element, afterthe Transfer stage occurs. The process of transferring the force acting

in the hydraulic jack to the concrete element is called The transfer

stage The external load acting at this stage is normally the self

weight (DL) of the structural member. After the force acting in the

hydraulic jack is transferred, the concrete element isunder one of

several critical conditions for the reasons described as it follow

follows :

1. Thepre-stressing force is maximumbecause the losses has not yet

occurred.

2. The acting external loading isminimum.

3. Theconcrete strength is minimumas the concrete is still young.135

Initial Loading Stage

135


Service Loading Stage


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Service load stage is a loading stage where thepre-stress long term loss

already taken. The pre-stress long term loss already taken is the creep

loss, shrinkage loss and steel relaxation loss. The external loading at this

stage is thefull service load these are self weight (DL), superimposed

deadload (SDL) and live load (LL).This stage is criticalbecause of as follows :

1. Thepre stressing force is minimumbecause all the losses already

taken.2. The applied loading is maximumbecause all the service loads already

applied.

3.The pre stressing force at this loading stage is designated asPe.

136

Service Loading Stage

136


Stresses and Moments Sign Convention


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137

Happy Beam is (+)Sad Beam is (-)

Mxz=Bending Moment

MxzMxz+ x

- x

Stresses and Moments Sign Convention

x

yt(+) y

b(-) y

I

yM ttop

)()()(

++

=+!

I

yM tbot

)()()(

!+

=!"

0(-) e

0(+) e

137


Stresses Sign Convention


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Stresses in Beams

Combined Bending and Axial Loading

Axial Load in Beams

x

+)x-top

Mxz

Mxz

P

x

+

)

xa

NN

+)x-top

+) xa

NN Px

top = +)x-top +

+)xa

) x-bottom

bottom = x-bottom + + xa

Stresses Sign Convention

138


Whats Look Different ????


139/206

What s Look Different ????

Mxz=Bending Moment

MxzMxz

+ x

- x

There is nothing new, however to beconsistent with many authors, now

Compression stresses are positive and

tension stresses are negative

139


Stress Diagram


140/206

140

The following figure shows the stress diagram for each loading stage, as

follows :

Stress Diagram

Pre-Stress Self -Weight Losses Sup- Imp + Live Load

Initial Service

140


Loading- Stresses


141/206

141

ci

bDLb0ii

ti

DL0ii

II

P

II

P

yMy)e(P

A

yMy)e(P

A

i

i tt

!!

!!

="+=

=+"=

!"#

#"$

!ci = Initial concrete compressive stress (transfer stage)

!ti = Initial concrete tensile stress ( (transfer stage)!cs = Service concrete compressive stress (service load stage)

!ts = Service concrete tensile stress (service load stage)

MDL = Moment due to the beam self weight

MSDL= Moment due to superimposed dead load

MLL = Moment due to acting live load

Loading StressesInitial Loading Stress Expression

Final Service Loading - Stress Expression

ts

cs

tttt

IIII

PIIII

P

bLLbSDLbDLb0

LLSDLDL0

yMyMyMy)e(P

A

s

yMyMyMy)e(P

A

s

e

e

e

e

!!

!!

="""+=

=+++"=

!"#

#"$

141


Kern Center Concepts


142/206

!ciall = concrete compressive stress (transfer stage)

!

tiall = concrete tensile stress (transfer stage)

!csall = concrete compressive stress (service load stage)

!tsall = concrete tensile stress (service load stage)

ALLOWABLE SERVICE STRESSES DENOMINATION

IyNe

AN )(0 !

+=

The Max eccentricity limit so that no tensile stress will develop any

where in the member could be found by setting != 0 ;or:

or: ;

0))(

1(1

2 =!+

r

ye

A

by

bS

)(t

kt

e !== xt(-) k

b(+) k

t(+) y

b(-) yty

bS

)(bk

be; +==

"iiLLSDLDLDLFFPFP;MMM

maxM;M

minM

eei !"=++== ==

Kern Center Concepts

142


Central Kern


143/206

143

Central kern is a region which is an axialcompressive force of any magnitude will not

produce any tension at this section.

The central kern is depends to the cross

section. The central kern is independent to

the applied compressive force and allowable

stress.

Central Kern

143


Basic Concepts


144/206

Basic Concepts


ts

cs

tttt

IIII

P

IIII

P

bLLbSDLbDLb0

LLSDLDL0

yMyMyMy)e(P

A

s

yMyMyMy)e(P

A

s

e

e

e

e

!!

!!

="""+=

=+++"=

!"#

#"$

ci

bDLb0ii

ti

DL0ii

II

P

II

P

yMy)e(P

A

yMy)e(P

A

i

i tt

!!

!!

="+=

=+"=

"#$

$#%

Initial Loading Stress Expression

GeometryRelated

Code

Related

GeometryLimits

144


Pre-stressed Equations to Remember


145/206

145

AllowDL0

ci

bDLb0ii

AllowDL0

ti

DL0ii

citt

titb

tt

S

P

II

P

S

P

II

P

M)]

k)(

e(1[

A

yMy)e(P

A

M)]

k

e(1[

A

yMy)e(P

A

ii

ii

!!!

!!!

"#

#

+$=#+=

%+#$=+#=

=

=

!"#

#"$

Pre stressed Equations to Remember


Be Careful !!!!!

145


Sign Convention Summary


146/206

The sign convention for pre stressing force is as follows :

1. The pre-stressing force is always inplus (+) sign.

The sign convention for concrete stress is as follows :1. Plus sign (+)is used to forcompressive concrete stress.

2.

Minus sign (-)is used fortensile concrete stress.

The sign convention for bending moment is as follows :1.

Plus sign (+) is used to for positive bending moment which is compression in the top fiber andtension in the bottom fiber.

2. Minus sign (-) is used to for negative bending moment which is tension in the top fiber andcompression in the bottom fiber.

The sign convention for pre-stressing eccentricity is as follows :

1.

The eccentricity ispositive (+)if goesdown ward / below the neutral axis.

2. The eccentricity isnegative (-) ifgoesup ward / above the neutral axis.

The sign convention for bending moment due to pre-stressing eccentricity is as follows :1. The bending moment ispositive (+)if the eccentricity goesup ward / above the neutral axis.

2.

The bending moment is negative (-) if the eccentricity goes down ward / below the neutralaxis.

Sign Convention Summary

146


Loading Stages - Stresses


147/206

147

AllowLLSDLDL0

bLLbSDLbDLb0

AllowLLSDLDL0

LLSDLDL0

ts

t

ts

ts

b

cs

cstttt

bbb

e

e

ttt

e

e

SSS

PIIII

eP

csSSS

PIIII

eP

MMM)]

k)(

e(1[

A

s

yMyMyMy)e(P

A

s

MMM)]

k

e(1[

A

s

yMyMyMy)e(P

A

s

!!!

!!

!!!

!!

"###

#

+$

=###+=

%+++#$=

=+++#=

==

=

!"#

!"#

#"$

#"$

Using the the previous equations will be re-writted as:

AllowDL

ci

bDLbii

AllowDL

ti

DLii

ci

t

ti

b

tt

t

ii

t

ii

S

P

II

P

SP

IIP

M

k

e

A

yMyeP

A

Mke

AyMyeP

A

!!!

!!!

"#

#

+$=#+=

%+#$=+#=

=

=

)])(

(1[)(

)](1[)(

00bot

00top

oad g Stages St essesInitial Loading Stress Expression


147


Loading Stages - Stresses


148/206

148

Allow0

Allow

0

ts

t

i

ts

b

ics

bSmax

M)]

k)(

e(1[

A

Fs

tSmax

M

)]k

e

(1[A

Fscs

!!!

!!!

"##

+$

$%

&+#$

$%

=

==

=

"#$

$#%

Allow0

ci

Allow0

ti

cit

i

ti

min

b

i

t

i

t

i

S

F

S

F

minM

)]k)(

e

(1[A

M)]

k

e(1[

A

!!!

!!!

"##

+$=

%+#$

=

==

!"#

#"$



g g

148


Loading Stages- Exentricity


149/206

149

)F1(

)F

1(

bM()ke

tM()ke

S

S

Allow

0

Allow

0

cimin

timin

i

t

i

b

!"

!"

#

#

+!

$!

+

+

Initial Loading Excentricity

Final Service Loading - Excentricity Expression

)

F

1(

)F1(

bM()ke

tM()ke

S

S

Allow

0

Allow0

tsmax

csmax

i

t

i

b

!

!

"

"

+!

#

$!

#

+

+

%

%

g g y

149


Loading Stages- Forces


150/206

150

Initial Loading Force Expression

Final Service Loading - Force Expression

)/)F

)/)F

ti

bi

ke(b

M(

ke(tM(

0

Allow

0

Allow

S

S

cimin

timin!

!

"

"

#

#

+

!

=

=

)/)F

)/)F

ti

bi

ke(b

M(

ke(tM(

0

Allow

0

Allow

S

S

tsmax

csmax

!

!

"

"

#

#

+$

!$

=

=

g g

150


Loading Stages - Forces


151/206

151

Initial Loading ForceExpression

Final Service Loading - Force Expression

)/)F1

)/)F

1

bM(ke(

tM(ke(

S

S

Allow

0

Allow

0

cimin

timin

t

i

b

i

!

!

"

"

+

#

#

#

=

=

)/)F

1

)/)F1

bM(ke(

tM(ke(

S

S

Allow

0

Allow0

tsmax

csmax

t

i

b

i

!

!

"

"

+

#

$#

$

$

=

=

g g

151


Service Condition Allowable Stresses


152/206

152

As explained before that pre-stressed concrete design in the

serviceability limit state is controlled by the allowable stress determined

by the code. The allowable stress is both for allowable stress inconcrete

and allowable stress in pre-stressing steel. There are at least four stress

limitations, as follows :

1. Tensile stress and compressive stress at transfer stage.

2. Tensile stress and compressive stress at service load stage.

152




153/206

153

Allowable concrete stresses for transfer stage and

service load stage

fci= allowable concrete compressive stress (transfer stage)fti= allowable concrete tensile stress (transfer stage)

fc= allowable concrete compressive stress (service load stage)ft = allowable concrete tensile stress (service load stage)

fci= initial concrete compressive strengthfc= 28 days concrete compressive strength 153




154/206

154

Allowable steel stresses for transfer stage and

service load stage

154


Design of Flexural Members


155/206

155

The design process is more difficult than analysis process because the

unknown variables are more than when the section is analyzed. In thedesign process When need to find the minimum section to satisfy the

allowable stress. After the section is determined then the actual stress

also must be checked. Two governs condition are the maximum pres-

tressing force with minimum external load and minimum pre-stressingforce with maximum external load.

ECCENTRIC PRESTRESSING

The figure below shows the stress history of the eccentric pre-stressing

system. Two stress conditions are used to determine the minimum

section modulus. For eccentric pre-stressing the minimum section is

controlled by maximum eccentricity at mid span.155


Stress History


156/206

y

156


Stress History


157/206

157

The followings are the explanation of the stress history, asfollows :

The actual stress due to pre-stressing and external load is shown

with dashed line. Due to initial pre-stressing the stress will be

tension in the top fiber (must be less than allowable tensile stress

fti ) and compression in the bottom fiber (must be less than

allowable compressive stress fci ).Due to effective pre-stressing

the stress will be compression in the top fiber (must be less than

allowable compressive stress fc ) and tension in the bottom fiber

(must be less than allowable tensile stress ft ).The top fiber iscontrolled by the initial allowable tensile stress fti and allowable

compressive stress fc. The bottom fiber is controlled by the

initial allowable compressive stress fci and allowable tensile

stress ft

y

157


Stress History


158/206

158 158


Stress History


159/206

159

As previously explained the top fiber is controlled by the initial

allowable tensile stress fti and allowable compressive stress fc. So the

section modulus for the top fiber St will be derived based on the two

allowable stresses above.


Allow0

ti ti

min

b

i

t

M)]

k

e(1[

A S

Pi!!! "+#$==!"#

Allow0 cs

tSmax

M)]

k

e(1[

A

Fs

b

i

cs!!! "+#$

$%

= =

"#$


159


Stress History


160/206

Assuming that all pre-stressed losses have occurred leads to a

loading for which:

1

tSmin

M)]

k

e(1[

A

F

b

i 0 !=+"##$

Which can also be written as :

1

tSmin

M

tSmin

M

tSmin

M)]

k

e(1[

A

F][

b

i 0 !="#+"$$#

# "

but: Allow0

ti

min

b

][t

M)]

k

e(1[

A S

Pi!=+"#

Therefore:

1)1(

t

M

Smin

ti

Allow !! ="#+"160


Stress History


161/206

161

Now if we add a moment of amplitude "M to the section the

corresponding additional stress on the top fiber will be :

t

M

Stop

!="!

And the resulting stress due to the initial stress at the top plus the

additional stress due to the moment "M must be less than or equal tothe allowable compressive stress

Allow

cs1 !!! "#+!"#

Using previous equations it results that :

AllowAllowCS

t

S

M)1(

tS

Mmin

ti !! "

#+$%+$

161


Preliminary Sizing


162/206

162

The previous equation can be expressed as:

Allowti

Allowcs

M)1(min

M

tS

!"#"

$+!#=

By similarly examining the state of stress on the bottom fibber, itcan be shown that:

Allowts

Allowci

M)1(min

M

bS

!"#!

$+#"=

Note that:

MMM !+=minmax 162



163/206

163 163



164/206

164 164



165/206

165 165




166/206

166

AllowDL0

ci

bDLb0ii

AllowDL0

ti

DL0ii

ci

tt

titb

tt

S

P

II

P

S

P

II

P

M)]

k)(

e(1[

A

yMy)e(P

A

M)]

k

e(1[

A

yMy)e(P

A

ii

ii

!!!

!!!

"#

#

+$=#+=

%+#$=+#=

=

=

!"#

#"$


Be Careful !!!!!

166




167/206

167

AllowLLSDLDL0

bLLbSDLbDLb0

AllowLLSDLDL0

LLSDLDL0

tst

ts

ts

bcs

cs

tttt

bbb

e

e

ttt

e

e

SSS

P

IIII

P

csSSS

P

IIII

P

MMM)]

k)(

e(1[

A

s

yMyMyMy)e(P

A

s

MMM)]

k

e(1[

As

yMyMyMy)e(P

A

s

e

e

!!!

!!

!!!

!!

"###

#

+$

=###+=

%+++#$=

=+++#=

==

=

!"#

!"#

#"$

#"$


Be Careful !!!!!

167


Pre-Stressed Design The Procedure


168/206

168

The pre-stressed concrete design start by suggesting cross sections

that could be solution to the problem. Having these sections the next

step will be to determine the required pre-stressing force and its

eccentricity at the critical section. After the pre-stressing force is

determined this pre-stressing force is used along the span and we must

compute the eccentricity at other location so there is no allowable

stress is exceeded. The common method is by computing the

eccentricity envelope which is the maximum location that produces

concrete stress less than allowable value. The spreadsheet thataccompany this presentation will be used to clarify the whole

concepts.

168


Pre-Stressed Concrete Design - Example


169/206

This example was taken from the Pre-stresses concrete Design Book

by Antoine Naaman. In the example the author has used Z for thesection modulus insteadof the typical mechanic of materials Books

that normally use S.

169




170/206

170




171/206

171




172/206

172




173/206

173




174/206

174




175/206

175




176/206

176

Ultimate Flexural DesignCapacity



176


Load Combinations


177/206

U = 1.4 (D + F)

U = 1.2 (D + F + T) + 1.6 (L + H) + 0.5 (Lror S or R)

U = 1.2D + 1.6 (Lror S or R) + (1.0L or 0.8W)

U = 1.2D + 1.6W + 1.0L + 0.5(Lror S or R)

U = 1.2D + 1.0E + 1.0L + 0.2S

U= 0.9D + 1.6W + 1.6H

U= 0.9D + 1.0E + 1.6H

177


Strength Design


178/206

Strength design is based using the rectangular stress block

The stress in the pre-stressing steel at nominal strength, fps, can bedetermined by strain compatibility or by an approximate empiricalequation

For elements with compression reinforcement, the nominal strengthcan be calculated by assuming that the compression reinforcementyields. Then verified.

The designer will normally choose a section and reinforcement and

then determine if it meets the basic design strength requirement:

!Mn" M

u

178


Stress Block Theory The Whitney stress block is a simplified stress distribution


179/206

The Whitney stress block is a simplified stress distribution

that shares the same centroid and total force as the realstress distribution

=

179


Equivalent Stress Block b1Definition


180/206

"1= 0.85

when fc < = 4,000 psi

"1= .85 0.05 {fc (ksi) 4(ksi)}

"1= .65 when fc > 8,000 psi

180


Strength Design Flowchart


181/206

1

2

3 4

5

181


Find depth of compression block


182/206

1

182


Depth of Compression Block


183/206

Where:Asis the area of tension steel

Asis the area of compression steel

fyis the mild steel yield strength

a =A

s ! f

y "A '

s! f '

y

.85 ! f 'c!b

Assumes

compression

steel yields

183


Flanged Sections


184/206

Checked to verify that the compression block is truly rectangular2

184


Compression Block Area


185/206

If compression block is rectangular, the flangedsection can be designed as a rectangular beam

Acomp

= a !b

= =

185


Compression Block Area


186/206

If the compression block is not rectangular (a> hf),

=

to find a

Af =(b !b

w) "h

f

Aw

=Acomp

!Af

a =A

w

bw

186


Determine Neutral Axis


187/206

From statics and strain compatibility

c=

a / !1

187


Check Compression Steel

3


188/206

Verify that compression steel has reached yield usingstrain compatibility

3 'c d> !

3

188


Compression Comments


189/206

By strain compatibility, compression steel yields if:

If compression steel has not yielded, calculation for amust be revised by substituting actual stress for yieldstress

Non prestressed members should always be tensioncontrolled, therefore c / dt< 0.375

Add compression reinforcement to create tesnion

controlled secions

c > 3 ! d'

189


Moment Capacity

2 equations


190/206

2 equations

rectangular stress block in the flange section

rectangular stress block in flange and stem section4

190


Strength Design Flowchart


191/206

1

2

3 4

5

191


Stress in Strand

f - stress in the strand after losses


192/206

fse stress in the strand after losses

fpu- is the ultimate strength of the strandfps- stress in the strand at nominal strength

This portion of the flowchart is dedicated to determining the stress in the prestress reinforcement

1

#p= (Aps)/( bdp) Pre-stressed Reinforcement

192


Stress in Strand


193/206

Typically the jacking force is 65% or greater

The short term losses at midspan are about 10% or less

The long term losses at midspan are about 20% or less

fse ! 0.5 " fpu

193


Stress in Strand

fpu - is the ultimate strength of the strand


194/206

fpu is the ultimate strength of the strand

fps- stress in the strand at nominal strength

This portion of the flowchart is dedicated to determining the stress in the prestress reinforcement

5

$= (As*fy)/( bdfc) Non Pre-stressed Reinforcement

$= (As*fy)/( bdfc) Non Pre-stressed Reinforcement

#p= (Aps)/( bdp) Pre-stressed Reinforcement

194


Stress in Strand

Prestressed Bonded reinforcement


195/206

Prestressed Bonded reinforcement

%p= factor for type of prestressing strand, see ACI 18.0

= .55 for fpy/fpunot less than .80

= .45 for fpy/fpunot less than .85

= .28 for fpy/fpunot less than .90 (Low Relaxation Strand)

#p= prestressing reinforcement ratio

fps =f

pu! 1"

#p

$1

%p!

fpu

f 'c

+d

dp

& " & '( )'

(

))

*

+

,,

-

.//

0

122

195


Compression Block Height

Assumes compression


196/206

Where

Aps- area of prestressing steel

fps - prestressing steel strength

a =A

ps ! f

ps +A

s ! f

y "A '

s! f '

y

.85!

f 'c!

b

Assumes compression

steel yields

Prestress component

196


Flexural Strength Reduction Factor


197/206

See figure next Page

197


Flexural Strength Reduction Factor


198/206

Based on primary reinforcement strain Strain is an indication of failure

mechanism

Three Regions

c/dt = > 0.6&= 0.70 with spiral ties

&=0.65 with stirrups

c/dt


199/206

0.002 < '< 0.005 at extremesteel tension fiber, or

0.375 < c/dt < 0.6

&= 0.57 + 67(') or&

= 0.48 + 83(') with spiralties

&= 0.37 +0.20/(c/dt) or

&

= 0.23 +0.25/(c/dt)&

with stirrups

199


Strand Slip Regions


200/206

ACI Section 9.3.2.7

where the strand em

ABC Introducción Al Diseño y Construcción de Elementos Pretensados Para Puentes

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Transcript of ABC Introducción Al Diseño y Construcción de Elementos Pretensados Para Puentes