abc introducción al diseño y construcción de elementos pretensados para puentes
TRANSCRIPT
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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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PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
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Basic Principles of Reinforced Concrete Design
Deflection issue
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PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Deflection Response of RC Beams
force
Deflection at Midspan 3
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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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PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
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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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PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
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Pu
Comb teeths
Steel Reinforcement
Concrete Beam Behavior Ultimate Stage
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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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PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
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Stresses Induced by the Acting Load
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PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
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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PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
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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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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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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Tension-Controlled Sections
h
bw
As
0.85fc
#t = 0.005
at= 1ctCt
T
dt dtat2
ct
0.003
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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
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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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PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
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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PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
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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PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
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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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Pre-Stressed Effects
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PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Pre-Stressed Load Application Procedure
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PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Post-tensioned Load Application Procedure
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PRE-STRESSED LOADAPPLICATION PROCEDURE
Equivalent Pre-Stressed Effects
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PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
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Equivalent Pre-Stressed Effects
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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-Stressing Element Connections
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Pre-Stressing Element Connections
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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 Columns
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Post-Tensioned Beams
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Post-Tensioned Beams
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Post-Tensioned Beams
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Post-Tensioned Beams
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Post-Tensioned Beams
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Post-Tensioned Beams
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Post-Tensioned Beams
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Post-Tensioned Anchorages
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Post-Tensioned Anchorages
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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
Pretensados para Puentes
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Centro de Transferencia de Tecnologa en Transportacin
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Presentation- References
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Compressive Strength of Concrete
Compressive strength
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6
12
ASTM C 31
ASTM C 39
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Compressive strength Curve
Strain
Stress
f"c
Ec= wc1.533#f"c
Compressive Strength of Concrete
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Compressive strength Curves
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Compressive Strength of Concrete
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Tensile strength Varies between 8% and 15% of the compressive
strength
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Tension Strength of Concrete
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Modulus of rupture (flexural test)
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b
h
fr=6M
bh2
Tension Strength of Concrete
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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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Tension Strength of Concrete
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Split cylinder test
!
d
P
Tension Strength of Concrete
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Split cylinder test
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!
d
fct=2P
$!d
P
Tension Strength of Concrete
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By: Dr. Daniel A. Wendichansky Bard
Split cylinder test
ASTM C 496 Standard Test Method for Splitting
Tensile Strength of Cylindrical Concrete Specimens
76
Tension Strength of Concrete
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By: Dr. Daniel A. Wendichansky Bard
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
77
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By: Dr. Daniel A. Wendichansky Bard
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
78
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By: Dr. Daniel A. Wendichansky Bard
Shrinkage
Creep
Thermal expansion
79
Shrinkage, Creep and Thermal Expansion
79
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By: Dr. Daniel A. Wendichansky Bard
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
80 80
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By: Dr. Daniel A. Wendichansky Bard
Shrinkage
ShrinkageStrain
Time
Shrinkage
81
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By: Dr. Daniel A. Wendichansky Bard
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
82
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By: Dr. Daniel A. Wendichansky Bard
Creep
Strain
Time
Load applied
Elastic strain
Creep strain
Load removed
Elastic recovery
Creep recovery
Permanent
Deformation
Creep
83
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By: Dr. Daniel A. Wendichansky Bard
Creep strains can lead to
Increase in deflections with time
Redistribution of stresses
Decrease in pre-stressing forces
Creep
84
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By: Dr. Daniel A. Wendichansky Bard
Thermal expansion
Coefficient of thermal expansion or
contractionAffected by:
Composition of the concrete
Moisture content of the concrete
Age of the concrete
Thermal Expansion
85
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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
86 86
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By: Dr. Daniel A. Wendichansky Bard
Reinforcing Steels
Deformed Bar Reinforcement
Welded Wire Reinforcement
Pre-stressing Steel
87 87
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By: Dr. Daniel A. Wendichansky Bard
Deformed Bar Reinforcement
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
88 88
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By: Dr. Daniel A. Wendichansky Bard
89
Deformed Bar Reinforcement
89
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y a e e d c a s y a d
Deformed Bar Reinforcement
90
fs= Ess
fs= fy
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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
Deformed Bar Reinforcement
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
91
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y y
Pre-stressing Steel
Strands
Seven-wire
Three- and Four-wire Wire
Bars Plain
Deformed
92 92
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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
93 93
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y y
Pre-stressed 7-wire strand
94
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95
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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
97
Design Aid 11.2.5in PCI Design
Handbook, 6th Ed.
97
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98
Prestressing Steel
98
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99
Prestressing Steel
99
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100
Losses in Pre-stressedConcrete
100
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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
101
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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
103 103
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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
104
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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
105
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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
106
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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.)
107
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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
For members with bonded tendons:
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.
109
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For members with bonded tendons:
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)
(3a) 110
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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)
111
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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
112
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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
113
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Relaxation of Tendons (RE)
TABLE 3
114
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Relaxation of Tendons (RE)
115
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Relaxation of Tendons (RE)
116
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Annual Average Relative Humidity
117
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AASHTO Lump-Sum Losses
Approximated Methods for Losses Computations
118
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119
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P t d L
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120
Pre-stressed Losses
120
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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
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EXAMPLE
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EXAMPLE
122
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EXAMPLE
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EXAMPLE
123
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124
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EXAMPLE
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EXAMPLE
125
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EXAMPLE
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EXAMPLE
126
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EXAMPLE
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EXAMPLE
Table 2
Table 2
Table 3
127
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128
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129
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Centro de Transferencia de Tecnologa en Transportacin
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Pre-stressed concrete DesignService Design
130
ABC - Introduccin al Diseo yConstruccin de Elementos
Pretensados para Puentes
130
Centro de Transferencia de Tecnologa en Transportacin
PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
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
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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
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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
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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
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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
PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
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
PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
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
PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
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
PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Whats Look Different ????
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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
PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Stress Diagram
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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
PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Loading- Stresses
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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
PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Kern Center Concepts
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!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
PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Central Kern
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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
PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Basic Concepts
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Basic Concepts
Final Service Loading - Stress Expression
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
!!
!!
="+=
=+"=
"#$
$#%
Initial Loading Stress Expression
GeometryRelated
Code
Related
GeometryLimits
144
PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Pre-stressed Equations to Remember
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145
AllowDL0
ci
bDLb0ii
AllowDL0
ti
DL0ii
citt
titb
tt
S
P
II
P
S
P
II
P
M)]
k)(
e(1[
A
yMy)e(P
A
M)]
k
e(1[
A
yMy)e(P
A
ii
ii
!!!
!!!
"#
#
+$=#+=
%+#$=+#=
=
=
!"#
#"$
Pre stressed Equations to Remember
Initial Loading Stress Expression
Be Careful !!!!!
145
PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Sign Convention Summary
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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
PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Loading Stages - Stresses
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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
Final Service Loading - Stress Expression
147
PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Loading Stages - Stresses
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148
Allow0
Allow
0
ts
t
i
ts
b
ics
bSmax
M)]
k)(
e(1[
A
Fs
tSmax
M
)]k
e
(1[A
Fscs
!!!
!!!
"##
+$
$%
&+#$
$%
=
==
=
"#$
$#%
Allow0
ci
Allow0
ti
cit
i
ti
min
b
i
t
i
t
i
S
F
S
F
minM
)]k)(
e
(1[A
M)]
k
e(1[
A
!!!
!!!
"##
+$=
%+#$
=
==
!"#
#"$
Initial Loading Stress Expression
Final Service Loading - Stress Expression
g g
148
PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Loading Stages- Exentricity
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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
!
!
"
"
+!
#
$!
#
+
+
%
%
g g y
149
PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Loading Stages- Forces
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150
Initial Loading Force Expression
Final Service Loading - Force Expression
)/)F
)/)F
ti
bi
ke(b
M(
ke(tM(
0
Allow
0
Allow
S
S
cimin
timin!
!
"
"
#
#
+
!
=
=
)/)F
)/)F
ti
bi
ke(b
M(
ke(tM(
0
Allow
0
Allow
S
S
tsmax
csmax
!
!
"
"
#
#
+$
!$
=
=
g g
150
PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Loading Stages - Forces
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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
PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Service Condition Allowable Stresses
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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
PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Service Condition Allowable Stresses
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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
PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Service Condition Allowable Stresses
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154
Allowable steel stresses for transfer stage and
service load stage
154
PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Design of Flexural Members
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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
PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Stress History
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y
156
PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Stress History
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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
PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Stress History
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158 158
PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Stress History
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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.
Initial Loading Stress Expression
Allow0
ti ti
min
b
i
t
M)]
k
e(1[
A S
Pi!!! "+#$==!"#
Allow0 cs
tSmax
M)]
k
e(1[
A
Fs
b
i
cs!!! "+#$
$%
= =
"#$
Final Service Loading - Stress Expression
159
PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Stress History
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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
PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Stress History
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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
PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Preliminary Sizing
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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
!"#!
$+#"=
Note that:
MMM !+=minmax 162
PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
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163 163
PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
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164 164
PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
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165 165
PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Pre-stressed Equations to Remember
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166
AllowDL0
ci
bDLb0ii
AllowDL0
ti
DL0ii
ci
tt
titb
tt
S
P
II
P
S
P
II
P
M)]
k)(
e(1[
A
yMy)e(P
A
M)]
k
e(1[
A
yMy)e(P
A
ii
ii
!!!
!!!
"#
#
+$=#+=
%+#$=+#=
=
=
!"#
#"$
Initial Loading Stress Expression
Be Careful !!!!!
166
PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Pre-stressed Equations to Remember
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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
!!!
!!
!!!
!!
"###
#
+$
=###+=
%+++#$=
=+++#=
==
=
!"#
!"#
#"$
#"$
Final Service Loading - Stress Expression
Be Careful !!!!!
167
PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Pre-Stressed Design The Procedure
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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 SEMINARBy: Dr. Daniel A. Wendichansky Bard
Pre-Stressed Concrete Design - Example
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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
PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Pre-Stressed Concrete Design - Example
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170
PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Pre-Stressed Concrete Design - Example
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171
PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Pre-Stressed Concrete Design - Example
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172
PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Pre-Stressed Concrete Design - Example
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173
PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Pre-Stressed Concrete Design - Example
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174
PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Pre-Stressed Concrete Design - Example
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175
PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Centro de Transferencia de Tecnologa en Transportacin
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176
Ultimate Flexural DesignCapacity
ABC - Introduccin al Diseo yConstruccin de Elementos
Pretensados para Puentes
176
PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Load Combinations
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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
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PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Strength Design
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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
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PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Stress Block Theory The Whitney stress block is a simplified stress distribution
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The Whitney stress block is a simplified stress distribution
that shares the same centroid and total force as the realstress distribution
=
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PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Equivalent Stress Block b1Definition
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"1= 0.85
when fc < = 4,000 psi
"1= .85 0.05 {fc (ksi) 4(ksi)}
"1= .65 when fc > 8,000 psi
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Strength Design Flowchart
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1
2
3 4
5
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PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Find depth of compression block
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1
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PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Depth of Compression Block
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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
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PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Flanged Sections
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Checked to verify that the compression block is truly rectangular2
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PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Compression Block Area
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If compression block is rectangular, the flangedsection can be designed as a rectangular beam
Acomp
= a !b
= =
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PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Compression Block Area
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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
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PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Determine Neutral Axis
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From statics and strain compatibility
c=
a / !1
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PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Check Compression Steel
3
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Verify that compression steel has reached yield usingstrain compatibility
3 'c d> !
3
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PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Compression Comments
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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'
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PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Moment Capacity
2 equations
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2 equations
rectangular stress block in the flange section
rectangular stress block in flange and stem section4
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PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Strength Design Flowchart
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1
2
3 4
5
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PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Stress in Strand
f - stress in the strand after losses
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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
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PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Stress in Strand
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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
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PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Stress in Strand
fpu - is the ultimate strength of the strand
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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
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PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Stress in Strand
Prestressed Bonded reinforcement
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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
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PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Compression Block Height
Assumes compression
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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
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PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Flexural Strength Reduction Factor
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See figure next Page
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PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Flexural Strength Reduction Factor
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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
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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
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PRE-STRESSED CONCRETE SEMINARBy: Dr. Daniel A. Wendichansky Bard
Strand Slip Regions
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ACI Section 9.3.2.7
where the strand em