Prestressed Concrete Bridge Design
Basic Principles
Emphasizing AASHTO LRFD Procedures
Praveen Chompreda, Ph. D.
MAHIDOL UNIVERSITY  2009  EGCE 406 Bridge Design
Part I: Introduction
Reinforced vs. Prestressed Concrete
Principle of Prestressing
Hl P
H
istorica
l P
erspective
Applications
Classification and Types
Advantages
Design Codes
Stages of Loading
Stages of Loading
ReinforcedConcrete
Reinforced
Concrete
Recall Reinforced Concrete knowledge:
C b k
C
oncrete
is
strong
in
compression
b
ut
wea
k
in
tension
Steel is strong in tension (as well as compression)
葉ｮｮ拾
葉ｮｮ拾
ﵰ說︠ｬﰠ︠﹤
＠ﱬﹳ說
＠ﱬﹳ說
Tensile strength of concrete is neglected (i.e. zero)
RC beam alwayscrack under service load
ReinforcedConcrete
Reinforced
Concrete
Cracking moment
of an RC beam is generally
Cracking moment
of an RC beam is generally
much lower than the service moment
PrincipleofPrestressing
Principle
of
Prestressing
Prestressing is a method in which compression force is
applied to the reinforced concrete section
applied to the reinforced concrete section
.
The effect of prestressing is to reduce the tensile stress
i h i h i h h il i bl
i
n
t
h
e
sect
i
on
to
t
h
e
po
i
nt
t
h
at
t
h
e
tens
il
e
stress
i
s
b
e
l
ow
the cracking stress. Thus, the concrete does not crack!
It is then possible to treat concrete as an elastic material
The concrete can be visualized to have
2
force systems
The concrete can be visualized to have
2
force systems
‰
Internal Prestressing Forces
ｲﴠ
ｲﴠ
These 2 force systems must counteract each other
PrincipleofPrestressing
Principle
of
Prestressing
Stress in concrete section when the prestressing force is
applied at the cg of the section (simplest case)
applied at the c
.
g
.
of the section (simplest case)
PrincipleofPrestressing
Principle
of
Prestressing
Stress in concrete section when the prestressing force is
applied eccentrically with respect to the cg of the
applied eccentrically with respect to the c
.
g
.
of the
section (typical case)
Smaller Compression
+
+=
c.g.
e0
F/A
MDLy/I
MLLy/I
Small Compression
Fe0y/I
Prestressing
Force
Stress
from DL
Stress
from LL
Stress
Resultant
Cross
Section
HistoricalPerspective
Historical
Perspective
The concept of prestressing was invented
centuries ago when metal bands were
centuries ago when metal bands were
wound around wooden pieces (staves) to
form a barrel
form a barrel
.
ﵥﰠ﹤
ﵥﰠ﹤
︠ﹳ塞
北ﵰ說︠
北ﵰ說︠
︠
ﱯ＠
ﱯ＠
葉辰
HistoricalPerspective
Historical
Perspective
The concept of prestressed concreteis also not new. In
1886 a patent was granted for tightening steel tie rods in
1886
,
a patent was granted for tightening steel tie rods in
concrete blocks. This is analogous to modern day
segmental constructions
segmental constructions
.
Early attempts were not very successful due to low
strength of steel at that time Since we cannot prestress
strength of steel at that time
.
Since we cannot prestress
at high stress level, the prestress losses due to creep and
shrinkage of concrete quickly reduce the effectiveness of
shrinkage of concrete quickly reduce the effectiveness of
prestressing.
HistoricalPerspective
Historical
Perspective
Eugene Freyssinet (
1879
1962
) was the first to
Eugene Freyssinet (
1879

1962
) was the first to
propose that we should use very high strength
steel which permit high elongation of steel
steel which permit high elongation of steel
.
The high steel elongation would not be
entirely offset by the shortening of concrete
entirely offset by the shortening of concrete
(prestress loss) due to creep and shrinkage.
First prestressed concrete
brid
g
e in 1941 in France
g
First prestressed concrete
bridge in US: Walnut Lane
Bid i Pli Bil
B
r
id
ge
i
n
P
ennsy
l
van
i
a.
B
u
il
t
in 1949. 47 meter span.
ApplicationsofPrestressedConcrete
Applications
of
Prestressed
Concrete
Bridges
Slb i bildi
Sl
a
b
s
i
n
b
u
ildi
ngs
Water Tank
Concrete Pile
葉ﱬ
葉ﱬ
Offshore Platform
Nuclear Power Plant
Re
p
air and Rehabilitations
p
ClassificationandTypes
Classification
and
Types
Pretensioning v.s. Posttensioning
External v.s. Internal
ﱩ﹥沈ﱡ
ﱩ﹥
沈ﱡ
EndAnchored v.s. Non EndAnchored
Bonded v.s. Unbonded Tendon
Pt Ct
I
Pl Cit
P
recas
t
v.s.
C
as
t

I
n
Pl
ace
v.s.
C
ompos
it
e
Partial v.s. Full Prestressin
g
g
ClassificationandTypes
Classification
and
Types
Pretensionin
g
vs. Posttensionin
g
gg
In Pretension, the tendons are tensioned against some
abutments
before
the concrete is place After the
abutments
before
the concrete is place
.
After the
concrete hardened, the tension force is released. The
tendon tries to shrink back to the initial length but the
tendon tries to shrink back to the initial length but the
concrete resists it through the bond between them, thus,
compression force is induced in concrete Pretension is
compression force is induced in concrete
.
Pretension is
usually done with precast members.
ClassificationandTypes
Classification
and
Types
Pretensioned Prestressed Concrete
Pretensioned Prestressed Concrete
Casting Factory
Concrete
Mixe
r
ClassificationandTypes
Classification
and
Types
In Posttension, the tendons are tensioned afterthe
concrete has hardened Commonly metal or plastic
concrete has hardened
.
Commonly
,
metal or plastic
ducts are placed inside the concrete before casting.
After the concrete hardened and had enough strength
After the concrete hardened and had enough strength
,
the tendon was placed inside the duct, stressed, and
anchored against concrete Grout may be injected into
anchored against concrete
.
Grout may be injected into
the duct later. This can be done either as precast or
t
i
l
cas
t

i
np
l
ace.
ClassificationandTypes
Classification
and
Types
Precast Segmental
Girder to be
Posttensioned In
Posttensioned In
Place
ClassificationandTypes
Classification
and
Types
El Il P
E
xterna
l
vs.
I
nterna
l P
restressing
Prestressin
g
ma
y
be done inside or outside
gy
Linear vs. Circular Prestressing
﹥︠不
﹥︠不
葉鸞ｲ
ﬠｲ塞＠沈ﱡ
ﬠｲ塞＠沈ﱡ
Bonded vs. Unbonded Tendon
The tendon may be bonded to concrete (pretensioning
or posttensioning with grouting) or unbonded
(ii ih i) Bdi hl
(
posttens
i
on
i
ng
w
i
t
h
out
grout
i
ng
)
.
B
on
di
ng
h
e
l
ps
prevent corrosion of tendon. Unbonding allows
djtt f ti f t lt ti
rea
dj
us
t
men
t
o
f
pres
t
ress
i
ng
f
orce
a
t l
a
t
er
ti
mes.
ClassificationandTypes
Classification
and
Types
EndAnchored vs. NonEndAnchored tendons
I P d f h
I
n
P
retensioning,
ten
d
ons
trans
f
er
t
h
e
prestress
through the bond actions along the tendon; therefore,
it is nonendanchored
In Posttensionin
g
, tendons are anchored at their ends
g
using mechanical devices to transfer the prestress to
concrete
;
therefore
,
it is endanchored.
(
Groutin
g
or
;,
(g
not is irrelevant)
ClassificationandTypes
Classification
and
Types
Partial vs. Full Prestressing
P d b d b h
P
restressing
ten
d
on
may
b
e
use
d
in
com
b
ination
wit
h
regular reinforcing steel. Thus, it is something between
full prestressed concrete (PC) and reinforced
concrete (RC). The goal is to allow some tension and
cracking under full service load while ensuring
sufficient ultimate strength.
We sometimes use partially prestressed concrete
(PPC) to control camber and deflection, increase
(PPC) to control camber and deflection, increase
ductility, and save costs.
RCvsPPCvsPC
RC
vs
.
PPC
vs
.
PC
AdvantagesofPCoverRC
Advantages
of
PC
over
RC
Take full advantages of high strength concrete
Take full advantages of high strength concrete
and high strength steel
Nd l il
N
ee
d l
ess
mater
i
a
l
s
Smaller and lighter structure
No cracks
Use the entire section to resist the load
Better corrosion resistance
､ｲﭳ﹤ﱥﱡﹴ
､ｲﭳ﹤ﱥﱡﹴ
Very effective for deflection control
Better shear resistance
DesignCodesforPC
Design
Codes
for
PC
ACI318 Building Code (Chapter 18)
A
ASHTO LRFD (Chapter 5)
Other institutions
PCI –Precast/Prestressed Concrete Institute
鸞
ﹳ說﹩ﹳ
鸞
ﹳ說﹩ﹳ
StagesofLoading
Stages
of
Loading
Unlike RC where we primarily consider the
ltit ldi t t id ltil
u
lti
ma
t
e
l
oa
di
ng
s
t
age,
we
mus
t
cons
id
er
mu
lti
p
l
e
stages of construction in Prestressed Concrete
The stresses in the concrete section must remain
below the maximum limit at all times!!!
below the maximum limit at all times!!!
StagesofLoading
Stages
of
Loading
Typical stages of loading considered are Initial
d Si St
an
d S
erv
i
ce
St
ages
Initial
(
Immediatel
y
after Transfer of Prestress
)
(y)
Full prestress force
N M
( t h M
ddi
N
o
M
LL
(
may
or
may
no
t h
ave
M
DL
d
epen
di
ng
on
construction type)
Service
ﱯ
ﱯ
MDL+MLL
StagesofLoading
Stages
of
Loading
For precast construction, we have to investigate
some intermediate states during transportation
some intermediate states during transportation
and erection
Part II: Materials and
Hardwares for Prestressin
g
g
Concrete
Prestressing Steel
Prestressing Hardwares
Prestressing Hardwares
Concrete
Concrete
Mechanical properties of
concrete that are relevant
concrete that are relevant
to the prestressed
concrete design:
concrete design:
Compressive Strength
Mdl f Elii
M
o
d
u
l
us
o
f El
ast
i
c
i
ty
Modulus of Rupture
Concrete:CompressiveStrength
Concrete:
Compressive
Strength
AASHTO LRFD
For prestressed concrete, the
compressive strength should
compressive strength should
be from 2870 MPa at 28 days
For reinforced concrete
,
the
,
compressive strength should
be from 1670 MPa at 28 days
Concrete with f’
c
> 70 MPa
can be used when supported
by test data
Concrete:ModulusofElasticity
Concrete:
Modulus
of
Elasticity
AASHTO (5.4.2.4)
γ
ㄮ1
⡦
)
〮㔠
䵐M
γ
c
⡦
c
)
䵐M
γc
1.5
in kg/m3
葉
葉
For normal weight concrete,
we can use
Ec
=4800(f’c)0.5 MPa
Concrete:ModulusofRupture
Concrete:
Modulus
of
Rupture
Indicates the tensilecapacity of
concrete under bendin
g
g
Tested simplysupported
concrete beam under 4
p
oint
p
bending configuration
fr
= My/I = PL/bd2
AASHTO (5.4.2.6)
f
r
= 0.63 (f’c)0.5 MPa
Concrete:SummaryofProperties
Concrete
:
Summary
of
Properties
PrestressingTendons
Prestressing
Tendons
Prestressing tendon may be in the form of
td i d b thdd d
s
t
ran
d
s,
w
i
res,
roun
d b
ar,
or
th
rea
d
e
d
ro
d
s
Materials
High Strength Steel
Fib
Rifd Cit (l b fib)
Fib
e
r

R
e
i
n
f
orce
d C
ompos
it
e
(
g
l
ass
or
car
b
on
fib
ers
)
Tendons
Tendons
Common shapes
of prestressing
of prestressing
tendons
Most Popular
(
7wire Strand
)
(
)
PrestressingSteel
Prestressing
Steel
PrestressingStrands
Prestressing
Strands
Prestressing strandshave two grades
Gd
250
(
f
250
k
1725
MP)
G
ra
d
e
250
(
f
pu
=
250
k
si
or
1725
MP
a
)
Grade 270
(
f
pu
= 270 ksi or 1860 MPa
)
(
pu
)
Types of strands
Sd Rlid Sd
S
tresse
d R
e
li
eve
d S
tran
d
Low Relaxation Strand (lower prestress loss due to
relaxation of strand)
PrestressingStrands
Prestressing
Strands
PrestressingStrands
Prestressing
Strands
PrestressingStrands
Prestressing
Strands
Modulus of Elasticity
ﵐ﹤
ﵐ﹤
207000 MPa for Bar
Th dl f lii
Th
e
mo
d
u
l
us
o
f
e
l
ast
i
c
i
ty
of strand is lower than
that of steel bar because
strand is made from
twisting of small wires
together.
Hardwares&PrestressingEquipments
Hardwares
&
Prestressing
Equipments
Pretensioned Members
Hld
D D
H
o
ld

D
own
D
evices
Posttensioned Members
Anchorages
Sti Ah
St
ress
i
ng
A
nc
h
orage
DeadEnd Anchorage
Ducts
Posttensionin
g
Procedures
g
PretensionedBeams
Pretensioned
Beams
PretensioningHardwares
Pretensioning
Hardwares
HoldDown Devices for
Pretensioned Beams
Pretensioned Beams
PosttensionedBeams
Posttensioned
Beams
Posttension Hardwares
Sti Ah
St
ress
i
ng
A
nc
h
orage
DeadEnd Anchorage
Duct/ Grout Tube
PosttensioningHardwares

Anchorages
Posttensioning
Hardwares
Anchorages
PosttensioningHardwares

Anchorages
Posttensioning
Hardwares
Anchorages
PosttensioningHardwares

Anchorages
Posttensioning
Hardwares
Anchorages
PosttensioningHardwares

Ducts
Posttensioning
Hardwares
Ducts
PosttensioningProcedures
Posttensioning
Procedures
PosttensioningProcedures
Posttensioning
Procedures
Grouting is optional (depends on
the s
y
stem used
)
y)
Part III: Prestress Losses
Sources of Prestress Losses
Lump Sum Estimation of Prestress Loss
PrestressLosses
Prestress
Losses
Prestress force at any time is less than that during jacking
Prestress force at any time is less than that during jacking
Sources of Prestress Loss
Elastic Shortening :
Because concrete
Because concrete
shortens when the
prestressing force is
prestressing force is
applied to it. The
tendon attached to it
tendon attached to it
also shorten, causing
stress loss
PrestressLosses
Prestress
Losses
Sources of Prestress Loss (cont.)
Friction :Friction in the duct of
p
osttensionin
g
s
y
stem causes
pgy
stress at the far end to be less than that at the jacking
end. Thus, the average stress is less than the jacking stress
Anchorage Set :The wedge in the
h i lihl lk
anc
h
orage
may
set
i
n
s
li
g
h
t
l
y
to
l
oc
k
the tendon, causing a loss of stress
PrestressLosses
Prestress
Losses
Sources of Prestress Loss
(cont.)
(cont.)
Shrinkage : Concrete
shrinks over time due to
shrinks over time due to
the loss of water, leading
to stress loss on attached
to stress loss on attached
tendons
蘒
﹣
蘒
﹣
ｲﹳﵥ﹤
ﵰ略
ﵰ略
ﱥ葉ﱯ︠
ﹳ
ﹳ
PrestressLosses
Prestress
Losses
Sources of
Prestress Loss
Prestress Loss
(cont.)
Stl Rlti
St
ee
l R
e
l
axa
ti
on
:
Steel loss its stress
with time due to
with time due to
constant
elongation the
elongation
,
the
larger the stress,
the larger the loss
the larger the loss
.
TimeLineofPrestressLoss
Time
Line
of
Prestress
Loss
SH
Posttensionin
g
FR
AS
SH
CR
RE
g
J
acking
f
Initial
f
Effective
f
ES
RE
f
pj
f
pi
f
pe
SH
Pretensioning
Jki
ES
SH
CR
RE
(AS
RE)
Pretensioning
J
ac
ki
ng
(against
abutment)
Initial
f
Effective
f
ES
RE
Release
(
cuttin
g
RE)
abutment)
fpj
f
pi
f
pe
(g
strands)
Instantaneous LossesTimeDependent Losses
PrestressLoss
–
ByTypes
Prestress
Loss
By
Types
Pretensioned
Posttensioned
Pretensioned
Posttensioned
InstantaneousElastic ShorteningFriction
A S
A
nchorage
S
et
Elastic Shortening
Time
Dependent
Shrinkage (Concrete)
Creep (Concrete)
Shrinkage (Concrete)
Creep (Concrete)
Relaxation (Steel)Relaxation (Steel)
PrestressLoss

Pretensioned
Prestress
Loss
Pretensioned
PrestressLoss

Posttensioned
Prestress
Loss
Posttensioned
LumpSumPrestressLoss
Lump
Sum
Prestress
Loss
Pretress losses can be very complicate to
tit i it dd ft
es
ti
ma
t
e
s
i
nce
it d
epen
d
s
on
so
many
f
ac
t
ors
In t
yp
ical constructions
,
a lum
p
sum estimation of
yp,p
prestress loss is enough. This may be expressed
in terms of:
in terms of:
Total stress loss (in unit of stress)
Percentage of initial prestress
LumpSumPrestressLoss
Lump
Sum
Prestress
Loss
A. E. Naaman
(with slight modifications)
–
not including FR, AS
A. E. Naaman
(with slight modifications)
not including FR, AS
Start with 240 MPa for Pretensioned Normal Weight
Concrete with Low Relaxation Strand
Add 35 MPa for StressRelieved Strand or for Lightweight
Concrete
Dd
35
MP f P
D
e
d
uct
35
MP
a
f
or
P
osttension
Pt L (fi
f) (MP)
Types of
Prestress
Types of Concrete
P
res
t
ress
L
oss
(f
p
i

f
pe
) (MP
a
)
StressRelieved
Strand
Low Relaxation
Strand
Strand
Strand
PretensionedNormal Weight Concrete
Lihtiht Ct
275
310
240
275
Li
g
ht
we
i
g
ht C
oncre
t
e
310
275
PosttensionedNormal Weight Concrete240205
Lightweight Concrete275240
LumpSumPrestressLoss
Lump
Sum
Prestress
Loss
ACIASCE Committee (Zia et al. 1979)
拾ﵡﵵﴠﱯ說ﵡ鸞ﵥ
拾ﵡﵵﴠﱯ說ﵡ鸞ﵥ
ﵡﴠﱯ
ﵐ
殺
ｦ﹣
ﵐ
ﵒ便
﹤
ﱯﱡｮ
﹤
﹤
﹤
ｮｲﵡﰠ不﹣
ﱩ﹣
ﱩ﹣
LumpSumPrestressLoss
Lump
Sum
Prestress
Loss
T.Y. Lin & N. H. Burns
S f L
P f L (%)
S
ource
o
f L
oss
P
ercentage
o
f L
oss
(%)
PretensionedPosttensioned
Elastic Shortening (ES)41
Creep of Concrete (CR)65
Shrinkage of Concrete (SR)76
Steel Relaxation (R2)
8
8
Steel Relaxation (R2)
8
8
Total2520
Note: Pretension has larger loss because prestressing is usually
done when concrete is about 1

2 days old whereas Posttensioning
done when concrete is about 1

2 days old whereas Posttensioning
is done at much later time when concrete is stronger.
LumpSumPrestressLoss
Lump
Sum
Prestress
Loss
AASHTO LRFD (for CR SR R2) (5953)
AASHTO LRFD (for CR
,
SR
,
R2) (5
.
9
.
5
.
3)
LumpSumPrestressLoss
Lump
Sum
Prestress
Loss
AASHTO LRFD (Cont.)
率ﰠ＠鸞拾ﱡ
率ﰠ＠鸞拾ﱡ
pspy
Af
PPR
=
PPR = 10 for Prestressed Concrete
p
spysy
PPR
AfAf
=
+
PPR = 1
.
0 for Prestressed Concrete
PPR = 0.0 for Reinforced Concrete
ｲ﹩ｳ
Δ
f
⤠楳慬捵污瑥搠慳)
ｲ﹩ｳ
Δ
f
灅p
⤠楳慬捵污瑥搠慳)
2
00
psps
iG
i
EE
FeMe
F
ff
⎡⎤
Δ
⎢⎥
00
,
i
psps
iG
i
pEScgpFG
c
cici
ff
EEAII
+
Δ
==+−
⎢⎥
⎣⎦
Stress of concrete at the c.g. of tendon due to prestressing force and dead load
Part IV: Allowable Stress
Desi
g
n
g
Stress Inequality Equation
Allowable Stress in Concrete
Allowable Stress in Prestressing Steel
Feasible Domain Method
Envelope and Tendon Profile
Basics:
SignConvention
Basics:
Sign
Convention
In this class, the following convention is used:
Tensile Stress in concrete is ne
g
ative
(

)
g(
)
Compressive Stress in concrete is positive (+)
Positive Moment:
In some books
,
the si
g
n convention for stress ma
y
be
,gy
opposite so you need to reverse the signs in some
formula!!!!!!!!!
Basics:
SectionProperties
Basics:
Section
Properties
c.
g
.
o
f
Prestressin
g
Tendo
n
Concrete Cross

I
K
g
fg
Area: Aps
Concrete Cross

Sectiona Area: A
c
K
t
Kb
yt
(
ab
s
)
e
(

)
Zt
Center of Gravity of
Concrete Section
h
Concrete Section
(c.g.c)
(abs)
y
b
(abs)
c.g. of Prestressing Tendon
Area: Aps
Basics:
SectionProperties
Basics:
Section
Properties
Moment of Inertia, I
Moment of Inertia, I
2
IydA=
∫
Rectangular section about c.g. Ixx
= 1/12*bh3
A
Ix’x’
= Ixx
+ Ad2
y
t
and y
b
are distance from the c.g. of section to
y
t
and y
b
are distance from the c.g. of section to
top and bottom fibers, respectively
說﹡ﰠﵯﱵ蘒
說﹡ﰠﵯﱵ
Zt
= I/yt
Zb
= I/yb
Basics:
SectionProperties
ﭥ說ﴠ
Basics:
Section
Properties
Kern of the section
,
k
,
is the distance from c
.
g
.
where compression force will not cause any
i i h i
tens
i
on
i
n
t
h
e
sect
i
on
Cid Tp Fib
Cid Btt Fib
C
ons
id
er
T
o
p Fib
er
(Get Bottom Kern, k
b)
C
ons
id
er
B
o
tt
om
Fib
er
(Get Top Kern, k
t)
0
0
t
Fey
F
AI
=−
0
0
b
Fey
F
AI
=+
c
AI
I
c
AI
I
0b
ct
I
ek
Ay
==
0t
cb
I
ek
Ay
=−=
Note: Top kern has negative value
Basics:
GeneralDesignProcedures
Basics:
General
Design
Procedures
Select Girder type, materials to be used, and
b f ti td
num
b
er
o
f
pres
t
ress
i
ng
s
t
ran
d
s
Check allowable stresses at various sta
g
es
g
Check ultimate moment strength
Check cracking load
ﬠ
ﬠ
Check deflection
StressinConcreteatVariousStages
Stress
in
Concrete
at
Various
Stages
StressInequalityEquations
Stress
Inequality
Equations
We can write four equations based on the stress at the
We can write four equations based on the stress at the
top and bottom of section at initial and service stages
No.CaseStress Inequality Equation
IInitialTop
⎛⎞
=−+=−+≥
⎜⎟
⎝⎠
minmin
1
ioo
ii
tti
cttcbt
Fee
FMFM
σσ
AZZAkZ
IIInitialBottom
minmin
1
ioo
ii
bci
cbbctb
Fee
FMFM
σσ
A
ZZAkZ
⎛⎞
=+−=−−≤
⎜⎟
⎝⎠
IIIServiceTop
cbbctb
⎝⎠
⎛⎞
=−+=−+≤
⎜⎟
⎝⎠
maxmax
1
oo
i
tcs
ttbt
FeMeM
F
F
σσ
AZZAkZ
!
IVService
Bottom
⎝⎠
c
tt
c
bt
AZZAkZ
⎛⎞
=+−=−−≥
⎜⎟
⎝⎠
maxmax
1
oo
bts
FeMeM
FF
σσ
AZZAkZ
!
Bottom
⎜⎟
⎝⎠
bts
cbbctb
AZZAkZ
AllowableStressinConcrete
Allowable
Stress
in
Concrete
AASHTO LRFD (5.9.4) provides allowable stress in
concrete as functions of com
p
ressive stren
g
th at that
pg
time
ﹳ磻ｬﱯﱩﵩ
ﹳ磻ｬﱯﱩﵩ
省ﵥ鸞
ｳ
省ﵥ鸞
ｳ
Compression
Tension
Tension
„
Service (After All Losses)
Ci
C
ompress
i
on
Tension
AllowableStressinConcrete
Allowable
Stress
in
Concrete
Immediately after Prestress Transfer (Before Losses)
Immediately after Prestress Transfer (Before Losses)
Using compressive strength at transfer, f’ci
Allbl i 060 f’
All
owa
bl
e
compress
i
ve
stress
=
0
.
60 f’
ci
Allowable tensile stress
AllowableStressinConcrete
Allowable
Stress
in
Concrete
At service (After All Losses)
At service (After All Losses)
Compressive Stress
AllowableStressinConcrete
Allowable
Stress
in
Concrete
At service (After All Losses)
At service (After All Losses)
Tensile Stress
Allowable
StressinConcrete

Summary
Allowable
Stress
in
Concrete
Summary
Sta
g
e
W
hereLoadLimitNote
g
InitialTension
at Top
Fi+MGirder
0.58√f’ciWith bonded reinf…
0.25√f’ci Without bonded
> 1.38 MPareinf.
Compression
at Bottom
Fi+MGirder
0.60 f’ci
ServiceCompression
at Top
F+MSustained
0.45f’c*
0.5(F+MSustained
)+MLL+IM
0.40f’c*
F+MSustained+MLL+IM
0.60Øwf’c*
Tension F+M
Sustained+0.8MLL+IM
0.50√f’cNormal/ Moderate
at Bottom(Service III Limit State)
exposure
0.25√f’cCorrosive exposure
0
Ubdd d
0
U
n
b
on
d
e
d
ten
d
on
* Need to check all of these conditions (cannot select only one)
AllowableStressinPrestressingSteel
Allowable
Stress
in
Prestressing
Steel
ACI and AASHTO code specify the allowable
t i th ti tl t jki d ft
s
t
ress
i
n
th
e
pres
t
ress
i
ng
s
t
ee
l
a
t j
ac
ki
ng
an
d
a
ft
er
transfer
AllowableStressinPrestressingSteel
Allowable
Stress
in
Prestressing
Steel
AASHTO
LRFD
LRFD
(5.9.3)
AllowableStressinPrestressingSteel
Allowable
Stress
in
Prestressing
Steel
ACI

318
(
2002
)
AllowableStressinPrestressingSteel
Allowable
Stress
in
Prestressing
Steel
AllowableStressDesign
Allowable
Stress
Design
There are many factors affecting the stress in a
prestressed girder
prestressed girder
Prestressing Force (Fi
or F)
L f d (0)
L
ocation
o
f
prestress
ten
d
on
(
e
0)
Section Property (A, Zt
or Zb, kt
or kb)
External moment, which depends on
The Section used
(
dead load
)
()
Girder Spacing (larger spacing larger moment)
Slab Thickness (larger spacing thicker slab)
Stages of construction
AllowableStressDesign
Allowable
Stress
Design
For bridges, we generally has a preferred section type
for a given range of span length and we can select a
for a given range of span length and we can select a
girder spacing to be within a reasonable range
Sections
Sections
AASHTO Type
I

VI Sections
I

VI Sections
ftm
5015
7523
100
30
100
30
15046
Sections
Sections
AASHTO Type IVI Sections (continued)
BridgeGirderSections
Bridge
Girder
Sections
BridgeGirderSections
Bridge
Girder
Sections
AllowableStressDesign
Allowable
Stress
Design
For a given section, we need to find the
biti f ti f (F
F hih
com
bi
na
ti
on
o
f
pres
t
ress
i
ng
f
orce
(F
i
or
F
,
w
hi
c
h
depends on the number of strands), and the
location of strands (in terms of e
0) to satisfy
these equations
these equations
Possible methods:
Keep trying some number of strands and locations
(
Trial & Error
)
()
We use “Feasible Domain” Method
FeasibleDomain

Equations
Feasible
Domain
Equations
We can rewrite the stress inequality equations and add one more
We can rewrite the stress inequality equations and add one more
equation to them
No
Case
Stress Inequality Equation
No
.
Case
Stress Inequality Equation
IInitialTop
(
)
⎛⎞
≤+−
⎜⎟
0min
1
btit
e
kM
σ
Z
IIInitialBottom
(
)
⎜⎟
⎝⎠
0min
btit
i
e
σ
F
(
)
⎛⎞
≤++
⎜⎟
1
ekM
σZ
IIIServiceTo
p
(
)
≤++
⎜⎟
⎝⎠
0mintcib
i
ekM
σZ
F
(
)
⎛⎞
≥
⎜⎟
1
kM
Z
p
IV
Service
(
)
⎛⎞
≥
+−
⎜⎟
⎝⎠
0maxbcst
e
kM
σ
Z
F
(
)
⎛⎞
1
!
IV
Service

Bottom
V
Pil Lii
(
)
⎛⎞
≥++
⎜⎟
⎝⎠
0max
1
ttsb
ekMσZ
F
(
)
V
P
ract
i
ca
l Li
m
i
t
(
)
00,min
7.5
bcb
mp
eeydycm≤=−=−
FeasibleDomain
–
GraphicalInterpretation
Feasible
Domain
Graphical
Interpretation
FeasibleDomain
Feasible
Domain
Feasible domain tells you the possible location and
prestressing force at a given sectionto satisfy the stress
inequality equation
We usually use feasible domain to determine location
d i f h iil i (
an
d
prestress
i
ng
f
orce
at
t
h
e
most
cr
i
t
i
ca
l
sect
i
on
(
e.g.
midspan of simplysupported beams)
葉
critical section
葉
critical section
,
we need to find the location for the tendon at other
points
to satisfy stress inequalities
points
to satisfy stress inequalities
We use the prestressing envelopeto determine the
location of tendon alon
g
the len
g
th of the beam
(
tendon
gg(
profile)
Envelope

Equations
Envelope
Equations
We use the same equation as the feasible domain, except that we’ve
already known the F or Fi and want to find e
0
at different points along
already known the F or Fi and want to find e
0
at different points along
the beam
No
Case
Stress Inequality Equation
No
.
Case
Stress Inequality Equation
IInitialTop
(
)
⎛⎞
≤+−
⎜⎟
0min
1
btit
e
kM
σ
Z
IIInitialBottom
(
)
⎜⎟
⎝⎠
0min
btit
i
e
σ
F
(
)
⎛⎞
≤++
⎜⎟
1
ekM
σZ
IIIServiceTo
p
(
)
≤++
⎜⎟
⎝⎠
0mintcib
i
ekM
σZ
F
(
)
⎛⎞
≥
⎜⎟
1
kM
Z
p
IV
Service
(
)
⎛⎞
≥
+−
⎜⎟
⎝⎠
0maxbcst
e
kM
σ
Z
F
(
)
⎛⎞
1
!
IV
Service

Bottom
V
Pil Lii
(
)
⎛⎞
≥++
⎜⎟
⎝⎠
0max
1
ttsb
ekMσZ
F
(
)
V
P
ract
i
ca
l Li
m
i
t
(
)
00,min
7.5
bcb
mp
eeydycm≤=−=−
Envelope

Equations
Envelope
Equations
We then have 5 main equations
鸞雷ｷｵ﹤
葉省
鸞雷ｷｵ﹤
葉省
Ｉ
雷鸞稜磻
III
an
d IV
prov
id
e
th
e
upper
b
oun
d
o
f
e0
(
use
max
i
mum
of the two)
III F+M
III
a
uses
F+M
Sustained
IIIb uses 0.5(F+M
Sustained)+MLL+IM
雷北
כֿ
雷北
﹥
כֿ
ﱌשּ
IV uses F+MSustained+0.8MLL+IM
V is a practical limit of the e
(it is also the absolute
V is a practical limit of the e
0
(it is also the absolute
lower bound)
Envelope&TendonProfile
Envelope
&
Tendon
Profile
Envelope&TendonProfile
Envelope
&
Tendon
Profile
Envelope&TendonProfile
Envelope
&
Tendon
Profile
Note
Th d fl f d b
Th
e
ten
d
on
pro
f
i
l
e
o
f
pretensione
d
mem
b
ers
are
either straight or consisting of straight segments
The tendon profile of posttensioned member may be
one strai
g
ht tendon or smooth curved, but no shar
p
gp
corners
Envelope&TendonProfile
Envelope
&
Tendon
Profile
There is an alternative to draping the strands in
tid b
pre
t
ens
i
one
d
mem
b
er
W
e
p
ut
p
lastic sleeves around some strands at
pp
supports to prevent the bond transfer so the
prestress force will be less at that section
prestress force will be less at that section
Part II: Ultimate Stren
g
th
g
Desi
g
n
g
Concrete and Prestressing Steel Stresses
Cracking Moment
Cracking Moment
Failure Types
Ali f M
Rtl Sti
A
na
l
ys
i
s
f
or
M
n
–
R
ec
t
angu
l
ar
S
ec
ti
on
TSection
Ali f M
T
Si
A
na
l
ys
i
s
f
or
M
n
–
T

S
ect
i
on
Load
–
Deflection
–
ConcreteStress
Load
Deflection
Concrete
Stress
Load

Deflection
Load
Deflection
1 & 2: Theoretical camber (upward deflection) of
prestressed beam
3: Self weight + Prestressing force
4: Zero deflection
p
oint
(
Balanced
p
oint
)
with uniform
p(p)
stress across section
5: Decompression point where tension is zero at the
b fb
b
ottom
f
i
b
e
r
6: Cracking point where cracking moment is reached
7: End of elastic range (the service load will not be larger
than this)
8
Yildi f i l
8
:
Yi
e
ldi
ng
o
f
prestress
i
ng
stee
l
9: Ultimate strength (usually by crushing of concrete)
PrestressingSteelStress
Prestressing
Steel
Stress
PrestressingSteelStress
Prestressing
Steel
Stress
The
p
restressin
g
steel stress increases as the load
pg
increases
Crackin
g
of beam causes a
j
um
p
in stress as additional
gjp
tension force is transferred from concrete (now
cracked) to prestressing steel
At ultimate of prestressed concrete beam, the stress in
steel is somewhere between
y
ield stren
g
th
f
py
and
yg
f
py
ultimate strength
f
pu
Stress is lower for unbonded tendon because stress is
distributed throughout the length of the beaminstead of
just one sectionas in the case of bonded tendon
At ultimate, the effect of prestressing is lost and the
section behaves
j
ust like an RC beam
j
CrackingMoment
Cracking
Moment
Concrete cracks when bottom fiber reaches the tensile
capacity (modulus of rupture)
capacity (modulus of rupture)
f
r
= 0.63 (f’c)0.5 MPa (5.4.2.6)
CrackingMoment
Cracking
Moment
The moment at this stage is called “cracking moment”
which depends on the geometry of the section and
which depends on the geometry of the section and
prestressing force
1
ocrocr
br
FeMeM
FF
σf
AZZAkZ
⎛⎞
=+−=−−=
⎜⎟
⎝⎠
Solve the above equation to get
M
cbbctb
AZZAkZ
⎝⎠
Solve the above equation to get
M
cr
()
crotrb
MFekfZ=−−
()
crotrb
Note: Need to input f
r
and kt
as negative values !!!
FailureTypes
Failure
Types
This is similar to RC
Fracture of steel after concrete cracking. This is a sudden
failure and occurred because the beam has too little
reinforcement
Crushing of concrete after some yielding of steel. This is
called tension

controlled.
called tension
controlled.
Crushing of concrete before yielding of steel. This is a
brittle failure due to too much reinforcement It is called
brittle failure due to too much reinforcement
.
It is called
overreinforced or compressioncontrolled.
FailureTypes
Failure
Types
AnalysisforUltimateMomentCapacity
Analysis
for
Ultimate
Moment
Capacity
Analysis assumptions
Pl l f bd (l
Pl
ane
section
remains
p
l
ane
a
f
ter
b
en
d
ing
(l
inear
strain
distribution)
Perfect bond between steel and concrete (strain
com
p
atibilit
y)
py)
Concrete fails when the strain is equal to 0.003
Tensile strenth f cncrete is nelected at ltimate
Tensile stren
g
th
o
f c
o
ncrete is ne
g
lected at
u
ltimate
Use rectangular stress block to approximate concrete
stress distribution
AnalysisforUltimateMomentCapacity
Analysis
for
Ultimate
Moment
Capacity
Recall from RC Design that the followings must
b tif
t ll ti
tt ht h
b
e
sa
ti
s
f
y
a
t
a
ll ti
mes
no
ma
tt
er
w
h
a
t h
appens:
EQUILIBRIUM
STRAIN COMPATIBILITY
ﱳ＠ﱤ︠﹣
ﱳ＠ﱤ︠﹣
AnalysisforUltimateMomentCapacity
Analysis
for
Ultimate
Moment
Capacity
For equilibrium, there are commonly 4 forces
C
C
ompression
in
concrete
Com
p
ression in Non
p
restressed reinforcement
pp
Tension in Nonprestressed reinforcement
ﹳ說︠葉﹦ｲﵥﹴ
ﹳ說︠葉﹦ｲﵥﹴ
For concrete compression, we still use the ACI’s
rectangular stress block
rectangular stress block
RectangularStressBlock
Rectangular
Stress
Block
RectangularStressBlock
Rectangular
Stress
Block
0.85
'28 MPa
'28
1
c
f
f
≤
⎧
⎪
⎛⎞
⎪
⎛⎞
1
'28
0.850.0528'56 MPa
7
1
1
c
c
f
βf
⎪
−
⎛⎞
⎪
=−≤≤
⎨
⎜⎟
⎝
⎛⎞
⎜⎟
⎠
⎪
⎝⎠
'56 MPa
0.65
c
f
⎝
⎠
⎪
≥
⎩
⎝⎠
⎪
β
1
is equal to
0
85
for
f
’
<
28
MPa
β
1
is equal to
0
.
85
for
f
c
<
28
MPa
It decreases 0.05 for ever
y
7 MPa increases in
f
’
c
y
f
c
Until it reaches 0.65 at f’
c
> 56 MPa
AnalysisforUltimateMomentCapacity
Analysis
for
Ultimate
Moment
Capacity
For tension and compression in nonprestressed
ift d th thi i RC
re
i
n
f
orcemen
t
,
we
d
o
th
e
same
thi
ng
as
i
n
RC
design:
Assume that the steel yield first; i.e.
T
s
=
A
s
f
y
or C
s
=
A
s
’
f
y
’
T
s
A
s
f
y
or C
s
A
s
f
y
Check the strain in reinforcement to see if they
actually yield or not if not calculate the stress based
actually yield or not
,
if not
,
calculate the stress based
on the strain at that level & revise the analysis
to find new value of neutral axis depth c
to find new value of neutral axis depth
,
c
Ts
= Asfs
= AsEsεs
= A
sEs∙ 0.003(cd)/c
AnalysisforUltimateMomentCapacity
Analysis
for
Ultimate
Moment
Capacity
For tension in
prestressing steel we
prestressing steel
,
we
observe that we
cannot
assume the
cannot
assume the
behavior of
prestressing steel
prestressing steel
(which is high strength
tl) t b lti
s
t
ee
l) t
o
b
e
e
l
as
ti
c
perfectly plastic as in
h f l
th
e
case
o
f
stee
l
reinforcement in RC
AnalysisforUltimateMomentCapacity
Analysis
for
Ultimate
Moment
Capacity
At ultimate of prestressed concrete beam, the stress in
steel is clearly not the yield strength but somewhere
steel is clearly not the yield strength but somewhere
between yield strength fpy
and ultimate strength fpu
W lld i
f
W
e
ca
ll
e
d i
t
f
ps
The true value of stress is difficult to calculate (generally
requires nonlinear moment

curvature analysis) so we
g
enerall
y
estimate it usin
g
semiem
p
irical formula
gyg
p
ACI Bonded Tendon or Unbonded Tendon
AASHTO Bonded Tendon or Unbonded Tendon
UltimateStressinSteel:
f
Ultimate
Stress
in
Steel:
f
p
s
AASHTO LRFD Specifications
﹤
﹤ｮ鸞
鸞
f
p
>
0
5
f
p
﹤
﹤ｮ鸞
鸞
f
p
e
>
0
.
5
f
p
u
f
c
⎛⎞⎛⎞
1;21.04
p
y
pspu
pp
u
f
c
ffkk
df
⎛⎞⎛⎞
=−=−
⎜⎟⎜⎟
⎜⎟⎜⎟
⎝⎠⎝⎠
Note: for
p
reliminar
y
desi
g
n, we ma
y
conservativel
y
pp
⎝⎠⎝⎠
灹gyy
慳獵浥a
f
ps=
f
py
(5.7.3.3.1)
For Unbonded tendon, see 5.7.3.1.2
UltimateStressinSteel:
f
Ultimate
Stress
in
Steel:
f
p
s
AnalysisforUltimateMomentCapacity
Analysis
for
Ultimate
Moment
Capacity
Notes on Strain Compatibility
The strain in top of concrete at ultimate is 0.003
We can use similar triangleto find the strains in concrete
or reinforcin
g
steelat an
y
levels from the to
p
strain
g
yp
We need to add the tensile strain due to prestressing
(occurred before casting of concrete in pretensioned or
(occurred before casting of concrete in pretensioned or
before grouting in posttensioned) to the strain in
concrete at that level to get the true strain of the
concrete at that level to get the true strain of the
prestressing steel
Maximum&MinimumReinforcement
Maximum
&
Minimum
Reinforcement
Mi Rift (
5
7
3
3
1
)
M
ax
i
mum
R
e
i
n
f
orcemen
t (
5
.
7
.
3
.
3
.
1
)
The maximum of nonprestressed and prestressed
reinforcement shall be such that c/de ≤ 0.42
c/de
= ratio between neutral axis depth (c) and the
centroid depth of the tensile force (d
e)
Minimum Reinforcement (5.7.3.3.2)
Th ii f td d td
Th
e
m
i
n
i
mum
o
f
nonpres
t
resse
d
an
d
pres
t
resse
d
reinforcement shall be such that
ØM
1
2
M
(M
ki )
ØM
n
>
1
.
2
M
cr
(M
cr
=
crac
ki
ng
moment
)
,
o
r
ØMn
> 1.33Mu
(Mu
from Strength Load Combinations)
ResistanceFactor
φ
Resistance
Factor
φ
Rit Ft Ø
Section Type
R
es
i
s
t
ance
F
ac
t
or
Ø
RC and PPC
PPC with
PC
RC and PPC
w/ PPR< 0.5
PPC with
0.5< PPR< 1
PC
(PPR= 1.0)
Under

Reinforced Section
090
090
100
Under

Reinforced Section
c/de ≤ 0.42
0
.
90
0
.
90
1
.
00
O
Rifd Sti
Nt
070
070
O
ve
r

R
e
i
n
f
orce
d S
ec
ti
on
c/de > 0.42
N
o
t
Permitted
0
.
70
0
.
70
Note: if c/d
e
> 0.42 the member is now considered a
e
compression member and different resistance factor applies (see
5.5.4.2)
AASHTO des nt ermit the se f er
reinfrced RC
AASHTO d
o
es n
o
t
p
ermit the
u
se
o
f
ov
er

reinf
o
rced RC
(defined as sections with PPC < 0.5) sections
RectangularvsT

Section
Rectangular
vs
.
T
Section
Most prestressed concrete
beams are either ISha
p
ed or T
p
shaped (rarely rectangular) so
they have larger compression
flange
flange
If the neutral axis is in the
flange we called it
rectangular
flange
,
we called it
rectangular
section behavior. But if the
neutral axis is below the flan
g
e
g
of the section, we call it T
section behavior
This
has
nothing
to
do
with
the
overall shape of the section !!!
RectangularvsT

Section
Rectangular
vs
.
T
Section
If it i T
Sti bhi th t l f idth
If it i
s
a
T

S
ec
ti
on
b
e
h
av
i
or,
th
ere
are
now
t
wo
va
l
ue
o
f
w
idth
s,
namely b(for the top flange), and bw
(web width)
W
e need to consider nonuniform width of rectangular stress
block
RectangularvsT

Section
Rectangular
vs
.
T
Section
We generally assume that the section is rectangular first and
We generally assume that the section is rectangular first and
check if the neutral axis depth (c) is above or below the
flange thickness
h
f
flange thickness
,
h
f
Note:ACI method checks a=ß1cwith hf, which may give
lihtl difft lt h
<
h
bt
>
h
s
li
g
htl
y
diff
eren
t
resu
lt
w
h
en
a
<
h
f
b
u
t
c
>
h
f
T

SectionAnalysis
T
Section
Analysis
We divide the compression side into 2 parts
Oh f fl (dh
b
b
)
O
ver
h
anging
portion
o
f fl
ange
(
wi
d
t
h
=
b

b
w
)
Web
p
art
(
width = b
w
)
p(
w
)
T

SectionAnalysis
T
Section
Analysis
From equilibrium
11
0.85'0.85'()''
cwcwfpspssysy
fb
β
捦bbβhAfAfAf+−=+−
For preliminary analysis, or first iteration, we may assume fps
= fpy
and solve for c
and solve for c
1
''0.85'()
085'
p
sysysycwf
AfAfAffbb
β
h
c
晢
β
+−−−
=
1
0
.
85'
cw
fb
β
T

SectionAnalysis
T
Section
Analysis
For a more detailed approach, we recall the equilibrium
11
0.85'0.85'()''
cwcwfpspssysy
fb
β
捦bbβhAfAfAf+−=+−
⎛⎞
Substitute
1
pspu
c
ffk
d
⎛⎞
=−
⎜⎟
⎜⎟
⎝⎠
, Rearrange and solve for c
''085'()
AfAfAffbb
βh
p
d
⎜⎟
⎝⎠
+−−−
=
+
1
1
''0
.
85'()
085
'
/
p
spusysycwf
AfAfAffbb
βh
c
fb
βkAfd
+
1
0
.
85/
cwpspup
fb
βkAfd
T

SectionAnalysis
T
Section
Analysis
Moment Capacity (about a/2)
'''
222
npspspsyssys
aaa
MAfdAfdAfd
⎛⎞⎛⎞⎛⎞
=−+−−−
⎜⎟⎜⎟⎜⎟
⎝⎠⎝⎠⎝⎠
1
222
085
'
()
f
f
h
fbb
βha
⎝⎠⎝⎠⎝⎠
⎛⎞
+
−−
⎜⎟
1
0
.
85()
2
cw
f
fbb
βha
+
⎜⎟
⎝⎠
T

SectionAnalysisFlowchart
T
Section
Analysis
Flowchart
T

SectionAnalysisFlowchart
T
Section
Analysis
Flowchart
T

Section
T
Section
In actual structures, the section is
p
erfect T or I sha
p
es 
pp
t
here are some tapering flanges and fillets. Therefore, we
need to idealized the true section to simplify the analysis.
Little accuracy may be lost.
We need this for ultimate analysisonly. We should use
t
he true section property for the allowable stress analysis/
design
Part III: Com
p
osite Beam
p
Typical Composite Section
Com
p
osite Section Pro
p
erties
pp
Actual, Effective, and Transformed Widths
Allowable Stress Design
Stress Inequality Equation, Feasible Domain, and Envelope
Cracking Moment
Uli M Ci
Ul
t
i
mate
M
oment
C
apac
i
ty
Composite
Composite
Composite generally means the use of two
difft til i ttl lt
diff
eren
t
ma
t
er
i
a
l
s
i
n
a
s
t
ruc
t
ura
l
e
l
emen
t
s
Exam
p
le: Reinforced Concrete
p
Concrete –carry compression
Stl Rift
ti
St
ee
l R
e
i
n
f
orcemen
t
–carry
t
ens
i
on
Exam
p
le: Carbon Fiber Com
p
osite
pp
Carbon Fiber –carry tension
E Resin Matri
hld the fibers in lace
E
poxy
Resin Matri
x
–
h
o
ld the fibers in
p
lace
CompositeBeam
Composite
Beam
In the context of bridge design, the word
it b th f t difft
compos
it
e
b
eam
means
th
e
use
o
f t
wo
diff
eren
t
materials between the beam and the slab
Steel Beam + Concrete Slab
ﰠﹳ說
ﰠﹳ說
Concrete in slab carries compression
﹣ﴠ不
ｮ
﹣ﴠ不
ｮ
ﬠ﹣ﱡｲﵡﰭ﹣
P
restresse
d C
oncrete
b
eam
carr
i
es
tens
i
on
Concrete in slab carries compression
TypicalCompositeSections
Typical
Composite
Sections
TypicalCompositeSections
Typical
Composite
Sections
Slab may be cast:
El
l
E
ntire
l
y
castinp
l
ace
with removable
formwor
k
Using precast panel
as a formwork the
as a formwork
,
the
pour the concrete
topping
topping
WhyComposite?
Why
Composite?
There are some benefits of using precast
lt
e
l
emen
t
s
Save Time
Better Quality Control
There are some benefits of putting the composite
slab
ｶ磻ﹴ葉ﱥﵥﹴ
ｶ磻ﹴ葉ﱥﵥﹴ
Quality control is not that important in slabs
ParticularDesignAspects
Particular
Design
Aspects
There are 3 more things we need to consider specially
for composite section (on top of stuffs we need to
for composite section (on top of stuffs we need to
consider for noncomposite sections)
Tfi f Si
T
rans
f
ormat
i
on
o
f S
ect
i
on
Actual width vs. Effective width vs. Transformed width
Composite Section Properties
Loadin
g
Sta
g
es
gg
Allowable Stress Design
S
h
o
red
vs. U
n
s
h
o
red
Be
a
m
s
So vs. Uso as
Horizontal Shear Transfer
CompositeSectionProperties
Composite
Section
Properties
There are 3 value of widths we will use:
Al dh f h (b) Th
A
ctua
l
wi
d
t
h
o
f
t
h
e
composite
section
(b)
:
Th
is
is
equal to the girder spacing
Effective width of the composite section (be)
ｲﵥｦｭｮ
ｲﵥｦｭｮ
CompositeSectionProperties
Composite
Section
Properties
Effective Width
The stress distribution across the width are not uniform –the
farther it is from the center, the lesser the stress.
To simplify the analysis, we assume an effective width where the
stress are constant throughout
We also assume the effective width to be constant along the span.
CompositeSectionProperties
Composite
Section
Properties
Effective Width
s
(AASHTO LRFD
4.6.2.6.1)
be
be
ts
bf
bw
boverhang
⎧
b
Exterior
Girder
Interior
Girder
⎧
=
⎨
⎩
'max
/2
w
w
f
b
b
b
Exterior BeamInterior Beam
'/26
bt
+
⎧
'
12
bt
+
⎧
Ⱪn≥
,
✯26
浩n
2
ws
e
eextoverhang
bt
b
bb
+
⎧
⎪
=+
⎨
⎪
12
min
/4
ws
e
bt
bs
L
+
⎧
⎪
=
⎨
⎪
2
/8L
⎪
⎩
/4
L
°
¯
CompositeSectionProperties
Composite
Section
Properties
Transformed Width
Transformed Width
Typically the concrete used for slab has lower strength
h d f i
t
h
an
concrete
use
d f
or
precast
sect
i
on
Lower strength Lower modulus of elasticity
Thus, we need to use the concept of transformed
section to transform the slab material to the precast
section to transform the slab material to the precast
material
,,
'
'
cCIPCcCIPC
trecee
Ef
bbnbb
Ef
==≅
,,
'
trecee
cPPCcPPC
Ef
Modular Ratio, usually
< 1.0
CompositeSectionProperties
Composite
Section
Properties
Transformed Width
CompositeSectionProperties
Composite
Section
Properties
Summary of steps for Width calculations
Actual Width
Effective Width
Transformed Width
Actual Width
b
Effective Width
be
Transformed Width
btr
Equals to girder
s
p
acin
g
Accounts for
nonuniform stress
Accounts for
dissimilar material
pg
distributionproperties
CompositeSectionProperties
Composite
Section
Properties
After we get the transformed section, we can
th llt th ti ti
th
en
ca
l
cu
l
a
t
e
o
th
er
sec
ti
on
proper
ti
es
Acc
= Ac
+ tsbt
r
ytc, ytb
Ztc, Zbc
dpc
CompositeSectionProperties
Composite
Section
Properties
Precast Cross

Composite Cross
Sectiona Area
:
A
cc
btr
Precast Cross
Sectiona Area: Ac
Sectiona Area
:
A
cc
ytc
(
b
)
y
’t
c
yt
(abs)
c.g.
Com
p
osit
e
d
pc
(
a
bs
)
y
(abs)
c.g.
Precast
h
(abs)
p
dp
pc
y
b
c
yb
(abs)
y
(abs)
Aps
Aps
Precast vs. Composite
DesignofCompositeSection
Design
of
Composite
Section
Most of the theories learned previously for the
it ti till hld bt ith
noncompos
it
e
sec
ti
on
s
till h
o
ld b
u
t
w
ith
some
modifications
We will discuss two design limit states
Allbl St Di
All
owa
bl
e
St
ress
D
es
i
gn
Ultimate Strength Design
AllowableStressDesign

Composite
Allowable
Stress
Design
Composite
OUTLINE
Shored vs. Unshored
﹥若鸞ｮ
﹥若鸞ｮ
Feasible Domain & Envelope
AllowableStressDesign

Composite
Allowable
Stress
Design
Composite
In allowable stress design, we need to consider two loading stages
as
p
revious; however, the initial moment
(
immediatel
y
after
p
(y
transfer) is resisted by the precastsection whereas the service
moment(after the bridge is finished) is resisted by the composite
section (precast section and slab acting together as one member)
We need to consider two cases of composite construction
hd
met
h
o
d
s:
Shored –beam is supported by temporary falsework when the slab is
cast The falsework is removed when the slab hardens
cast
.
The falsework is removed when the slab hardens
.
Unshored –beam is not supported when the slab is cast.
ShoredvsUnshored
Shored
vs
.
Unshored
ShoredvsUnshored
Shored
vs
.
Unshored
Moments resisted by the precastand composite sections
are different in the two cases
Fully Shored
沈不
沈不
Composite: Slab Weight, Superimposed Loads (such as asphalt
surface
)
, and Live Load
)
Unshored
沈不﹤
沈不﹤
Composite: Superimposed Loads (such as asphalt surface), and
Live Load
ShoredvsUnshored
Shored
vs
.
Unshored
FULLY SHORED
Top of precast,
not top of
i
Consider, as example, the top of precast beam
compos
i
te
()()
()()
'
oGirderSlabSDLLIMtc
tcs
cttgc
FeMMMMy
F
σσ
AZZI
+
++
=−++≤
ShoredvsUnshored
Shored
vs
.
Unshored
UNSHORED
Consider, as example, the top of precast beam
()()
'
FeMMMMy
F
++
⠩()
oGirderSlabSDLLIMtc
tcs
cttgc
FeMMMMy
F
σσ
AZZI
+
++
=−++≤
ShoredvsUnshored
Shored
vs
.
Unshored
From both case we can rewrite the stress e
q
uation as:
q
≤
()
()
oC
P
FeM
M
F
=−++
≤
()
()
'
oC
P
tcs
ctttc
σσ
AZZZ
Mp
= Moment resisted by the precast section (use Z
t, Zb)
Fully Shored: M
p =M
girder
Unshored: Mp =M
girder +M
slab
M
M id b h i i ( Z’
Z
)
M
c
=
M
oment
res
i
ste
d b
y
t
h
e
compos
i
te
sect
i
on
(
use
Z’
tc,
Z
bc
)
Fully Shored: M
c =M
slab
+ MSD +M
LL+IM
ｲ蘒
ｲ蘒
ﱌשּ
︠省塞ｮｲｴﴠｦ
︠省塞ｮｲｴﴠｦ
ﵰｳ若
StressInequalityEquations
To
p
of
p
recast,
Stress
Inequality
Equations
pp
not top of
composite
CaseStress Inequality Equation
I
Initial

Top
⎛⎞
Fee
FMFM
I
Initial

Top
II
Iiil
B
⎛⎞
=−+=−+≥
⎜⎟
⎝⎠
minmin
1
ioo
ii
tti
cttcbt
Fee
FMFM
σσ
AZZAkZ
F
FMFM
⎛⎞
II
I
n
i
t
i
a
l

B
ottom
minmin
1
ioo
ii
bci
cbbctb
F
ee
FMFM
σσ
AZZAkZ
⎛⎞
=+−=−−≤
⎜⎟
⎝⎠
MM
§∙
IIIServiceTop
1
'
pp
ococ
tcs
ctttccbttc
MM
FeMeM
FF
σσ
AZZZAkZZ
⎛⎞
=−++=−++≤
⎜⎟
⎝⎠
!
IVServiceBottom
1
pp
ococ
bts
cbbbcctbbc
MM
FeMeM
FF
σσ
AZZZAkZZ
⎛⎞
=+−−=−−−≥
⎜⎟
⎝⎠
VIServiceTop Slab
cbbbcctbbc
⎝⎠
,
,,
cCIPC
cc
tslabccsSlab
E
MM
σnσ
ZZE
==≤
,tctccPPC
ZZE
Stress at the top of the slab must also be less than the allowable compressive stress
FeasibleDomain&Envelope
Top of precast
Feasible
Domain
&
Envelope
We can rewrite the stress equations and add practical limit equation
No.CaseStress Inequality Equation
IInitialTo
p
(
)
⎛⎞
1
p
II
Initial
Bottom
(
)
⎛⎞
≤+−
⎜⎟
⎝⎠
0min
1
btit
i
ekMσZ
F
⎛⎞
1
II
Initial

Bottom
III
S
T
()
⎛⎞
≤++
⎜⎟
⎝⎠
0min
1
tcib
i
ekMσZ
F
1
Z
⎛⎞
⎛⎞
III
S
ervice
T
op
0
1
'
t
bpccst
tc
Z
ekMMσZ
FZ
⎛⎞
⎛⎞
≥++−
⎜⎟
⎜⎟
⎝⎠
⎝⎠
⎛⎞
!
IVService
Bottom
0
1
b
tpctsb
bc
Z
ekMMσZ
FZ
⎛⎞
⎛⎞
≥+++
⎜⎟
⎜⎟
⎝⎠
⎝⎠
V
Practical Limit
(
)
00,minbc
mp
eeyd≤=−
E
MM
VIServiceTop
Slab
,
,,
,
cCIPC
cc
tslabccsSlab
tctccPPC
E
MM
σnσ
ZZE
==≤
CrackingMoment

Composite
Cracking
Moment
Composite
We consider 2 cases
1. Cracking moment is less than Mp
Cracking occurs in the precast section
Cracking occurs in the precast section
The equation is the same as noncomposite section
1
ocrocr
br
FeMeM
FF
σf
AZZAkZ
⎛⎞
=+−=−−=
⎜⎟
⎝⎠
()
MFkfZ
cbbctb
AZZAkZ
⎜⎟
⎝⎠
()
crotrb
MF
e
kfZ
=−−
CrackingMoment

Composite
Cracking
Moment
Composite
II. Cracking moment is greater Mp
Ck h
C
rac
k
ing
occurs
in
t
h
e
composite
section
We find ∆
M
cr
(
moment in addition to
M
p
)
cr
(
p
)
1
pp
ocrocr
MM
FeMeM
FF
σσ
⎛⎞
ΔΔ
=+
−−
=
−−−
≥
⎜⎟
1
bts
cbbbcctbbc
σσ
AZ婚Ak婚
=+=≥
⎜⎟
⎝⎠
Z
()
bc
crotprbc
b
Z
MFekMfZ
Z
⎡⎤
Δ=−−−
⎣⎦
crcrp
MMM=Δ+
UltimateStrengthDesign

Composite
Ultimate
Strength
Design
Composite
Ultimate strength of composite section follows similar
procedure to the T

section Some analysis tips are:
procedure to the T

section
.
Some analysis tips are:
When the neutral axis is in the slab, we can use a composite T
section with flange width equals to
Effective Width
and using
f’
section with flange width equals to
Effective Width
and using
f
c
of the slab
﹥ﰠ拾葉說ﵡ鸞
﹥ﰠ拾葉ｮ
ﵡ鸞
ｲﵥ說f’c
of the precast section This is an
a
pp
roximate value but the errors to the ultimate moment
pp
capacity is small.
ShearTransfer
Shear
Transfer
To get the
com
p
osite
p
behavior, it is
very important
that the slab
and girder must
not slip past
not slip past
each other
ShearTransfer
Mechanisms
Shear
Transfer
Mechanisms
The key parameter that
determines whether these two
parts will slip past each other
or not is the shear strength at
t
he interface of slab and girde
r
This interfacial shear strength
comes from:
Friction (F =
μ
N)
Cohesion
ShearTransfer
–
Cohesion&Friction
Shear
Transfer
Cohesion
&
Friction
Cohesion is the chemical bonding of the two materials. It depends
on the cohesion factor
(
c
)
and the contact area. The
g
reater the
()g
area, the larger the cohesion force.
Friction is due to the roughness of the surface. It depends on the
friction factor or coefficient of friction (μ)and the normal force
(N). To increase friction, we either make the surface rougher
(increase μ)or increase the normal force.
N
N
V
hu
ΦVhn
=ΦμN
hu
ShearTransfer

Formula
Shear
Transfer
Formula
AASHTO LRFD (5.8.4)
The nominal shear resistance at the interface between two
concretes cast at different times is taken as:
Fri
ct
i
o
n
F
actor
Area of Concrete
Friction Factor
Compressive force normal
Area of shear reinforcement crossing the
shear plane
⎧
䍯桥獩潮
呲ansfering⁓hea
r
䍯浰re獳sve=rceo牭慬=
瑯桥慲⁰污湥
≤
⎧
=++
⎨
≤
⎩
0.2'
()
55
ccv
nhcvvfyc
fA
VcAμAfP
A
≤
⎩
5
.
5
cv
A
COHESIONFRICTION
ShearTransfer
–
Shear
Transfer
Cohesion & Friction
AASHTO LRFD (
5
8
4
2
)
AASHTO LRFD (
5
.
8
.
4
.
2
)
ShearTransfer
–
Cohesion&Friction
Shear
Transfer
Cohesion
&
Friction
The normal force in the friction
formula comes from two parts
Yielding of shear reinforcement
If k h
Avf
If
crac
k
ing
occurs
at
t
h
e
interface, there will be tension
in the steel reinforcement
in the steel reinforcement
crossing the interface. This
tension force in steel is balanced
b h i f i
N=Avf fy
b
y
t
h
e
compress
i
ve
f
orce
i
n
concrete at that interface; thus,
creating normal
“
clamping
”
creating normal clamping
force.
Permanent compressive force at
the interface
Dead Weight of the slab and
earin srface (ashalt)
wearin
g
s
u
rface (as
p
halt)
Cannot rely on Live Loads
MinimumShearReinforcement
Minimum
Shear
Reinforcement
For V
n
/A
cv
> 0.7 MPa, the crosssectional area of shear reinforcement
n
cv
crossing the interface per unit length of beam must not be less than
Width of the interface (generally
≥
0.35
v
vf
b
A
f
Width of the interface (generally
equals to the width of top flange
of girder)
If less, then we cannot use any Avffy
in the nominal shear strength
y
f
The spacing of shear reinforcement must be
≤
600 mm
Possible reinforcements are:
Sl b
S
ing
l
e
b
a
r
Stirrups (multiple legs)
Wldd i fbi
W
e
ld
e
d
w
i
re
f
a
b
r
i
c
Reinforcement must be anchored properly (bends, hooks, etc…)
UltimateShearForceatInterface
Ultimate
Shear
Force
at
Interface
There are two methods for calculating shear force per unit length at
t
he interface
(
the values ma
y
be different
)
(y)
Using Classical Elastic Strength of Materials
Factored shear force
acting on the
(
)
=
Δ
u
uh
gc
VQ
V
I
Moment of Area above the shear plane
Factored shear force
acting on the
composite sectiononly (SDL +LL+IM)
gc
Moment of Area above the shear plane
about the centroid of composite section
Moment of Inertia of the
composite section
Using Approximate Formula (C5.8.4.11)
V
TotalFactored vertical shea
r
at the section
=
u
uh
e
V
V
d
Distance from centroid of tension
steel to midde
p
th of the deck
p
UltimateShearForceatInterface
Ultimate
Shear
Force
at
Interface
The critical section for shear at the interface is generally the
section where vertical shear is the
g
reatest
g
First critical section:h/2from the face of support
May calculate at some additional sections away from the support
(which has lower shear) to reduce the shear reinforcement accordingly
Critical Section For Shear
h
Critical Section For Shear
h/2
h/2
Resistance Factor (Φ) for shear in normal weight concrete : 0.90
SomeDesignTips
Some
Design
Tips
For T and Box Sections which cover the full girder spacing
with thin concrete topping (usually about
50
mm), we may
with thin concrete topping (usually about
50
mm), we may
not need any shear reinforcement (need only surface
rou
g
henin
g)
–
need to chec
k
gg)
For ISections, we generally require some shear
reinforcement at the interface
reinforcement at the interface
We generally design the web
shear reinforcement first (not taught),
and extend that shear reinforcement through the interface. Then
and extend that shear reinforcement through the interface. Then
we check if that area is enough for horizontal shear transfer at the
interface.
If not, we need additional reinforcement
If enough, then we do nothing
FinalNotesonCompositeBehavior
Final
Notes
on
Composite
Behavior
Cit ti i d t l f td t
C
ompos
it
e
sec
ti
on
i
s
use
d
no
t
on
l
y
f
or
pres
t
resse
d
concre
t
e
sections, but also for steelsections.
﹥拾拾ｭｮ﹤ﹴ
﹥拾拾ｭｮ﹤ﹴ
ﱡﰠ說﹡ﰠﱩﰠ說︬ﱬﱯﰠ
ﱩ
說︠
FinalNotesonCompositeBehavior
Final
Notes
on
Composite
Behavior
The analysis concept is
similar to that of
b
prestressed concrete.
There are also:
Effective width and
transformed section
Shored and Unshored
Construction
Sh Tf t Itf
Sh
ear
T
rans
f
er
a
t I
n
t
er
f
ace
FinalNotesonCompositeBehavior
Final
Notes
on
Composite
Behavior
There are various ways to transfer shear at steelconcrete interface
Studs
Channels
Spirals
Studs
Channels
Spirals
FinalNotesonCompositeBehavior
Final
Notes
on
Composite
Behavior
Shear Studis one
of the most
h
common
s
h
ear
connectors –it is
welded to the top
welded to the top
flange of steel
girder
FinalNotesonCompositeBehavior
Final
Notes
on
Composite
Behavior
Steel Girder with Shear Stud
Part IV: Things I did not teach
but you should be aware of !!!
Shear Strength –MCFT
Unbonded and External Prestressing
Unbonded and External Prestressing
Anchorage Reinforcement
Camber and Deflection Prediction
Detailed Calculation of Prestress Losses
Shear
Shear
Traditionally, the shear design in AASHTO Standard
Specification is similar to that of ACI which is empirical

Specification is similar to that of ACI
,
which is empirical

based
Th il f f i d h iil
Th
e
ax
i
a
l f
orce
f
rom
prestress
i
ng
re
d
uces
t
h
e
pr
i
nc
i
pa
l
tensile stress and helps close the cracks; thus, increase
h
s
h
ear
resistance.
Shear

MCFT
Shear
MCFT
The shear resisting mechanism in concrete is very
complex and we do not clearly understand how to
complex and we do not clearly understand how to
predict it
鸞ｲﱥ
鸞ｲ
ﱥ
､復便ﵰ說︠
Th
e
actua
l
t
h
eory
is
very
comp
l
icate
d b
ut
somew
h
at
simplified procedure is used in the code
This theory is for both PC and RC
Shear
Shear
The nominal shear resistance is the sum of shear strength
of concrete steel (stirrups) and shear force due to
of concrete
,
steel (stirrups)
,
and shear force due to
prestressing (vertical component)
=++≤+0.25'
ncspcvvp
VVVVfbdV
cot
v
y
v
Afdθ
V
=
y
s
V
s
=0.083'
ccvv
V
β
晢d
MinimumTransverseReinforcement
Minimum
Transverse
Reinforcement
We need some transverse reinforcement when the
ultimate shear force is greater than ½ of shear strength
ultimate shear force is greater than ½ of shear strength
from concrete and prestressing force
>+0.5()
ucp
VφVV
If we need it the minimum amount shall be
If we need it
,
the minimum amount shall be
≥
0083
'
v
bs
Af
≥
0
.
083
vc
y
Af
f
MinimumTransverseReinforcement
Minimum
Transverse
Reinforcement
Maximum Spacing
v
<
0
125
f
’
c
For
v
u
<
0
.
125
fc
s
max
= 0.8 dv
≤
600 mm
For vu > 0.125f’c
smax
= 0.4dv ≤
300 mm
Must subtract the area of duct to the width 5.8.2.7
UnbondedorExternalPrestressing
Unbonded
or
External
Prestressing
Strain compatibility does
not a
pp
l
y
for unbonded
ppy
tendon (the strain in
steel does not equal to
t
he strain in concrete
near it)
The strain in tendon is
averaged along the
length of the beam
length of the beam
UnbondedorExternalPrestressing
Unbonded
or
External
Prestressing
View inside a boxsection
AnchorageReinforcement
Anchorage
Reinforcement
Posttensioning
anchorages
anchorages
creates very high
compressive
compressive
stress behind the
bearing plate
bearing plate
AnchorageReinforcement
Anchorage
Reinforcement
This causes large
p
rinci
p
al tensile
pp
stress in the
transverse direction,
leading to concrete
cracking
We need to
determine the
magnitude of this
stress and design
some reinforcement
for it
AnchorageReinforcement
Anchorage
Reinforcement
Methods:
Tdl
T
ra
d
itiona
l
(Approximate)
StrutandTie
Method
(
new
(
for ACI and
A
ASHTO
)
)
Finite Element
Analysis
Analysis
(complicated)
StrutandTie Method
AnchorageReinforcement
Anchorage
Reinforcement
StrutandTie Method
CamberandDeflection
Camber
and
Deflection
AASHTO does not require the deflection criteria be met
略ﵢｮ磻
略ﵢｮ磻
省ｮ拾ｴｮ
ｵ
ｵ
The structure may deflect and vibrate too much that it
cause fatigue failure (due to repetitive stress cycles)
cause fatigue failure (due to repetitive stress cycles)
,
especially in steel connections.
說ﹳ拾ﵦｲ︠磻
說ﹳ拾ﵦｲ︠磻
DetailedCalculationofPrestressLoss
Detailed
Calculation
of
Prestress
Loss
In many cases it is adequate to use the
“
Lump Sum
”
loss
葉鸞
拾ﱵﵰｳ
鸞
I
n
some
cases,
we
need to know
tl th t i
exac
tl
y
th
e
s
t
ress
i
n
the strands so we
can determine the
can determine the
camber and
deflection
deflection
Cantilever
Construction
Construction
Repair/
Rehabilitation
References
References
AASHTO (2000). AASHTO LRFD Bridge Design Specifications –SI
Units, Second Edition, 2000 Interim Revisions, American
Association of State Highway and Transportation Officials,
Washington D.C.
http://www.transportation.org
Naaman, A. E. (2005), Prestressed Concrete Analysis and Design:
Fundamentals, 2nd
Edition, Technopress3000, Ann Arbor, MI, USA
h//h
3000
h
ttp:
//
www.tec
h
nopress
3000
.com
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