Part I: Introduction

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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ﰠ︠﹤
@ﱬﹳ說
@ﱬﹳ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
.


ﵥﰠ﹤

ﵥﰠ﹤
︠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

End-Anchored v.s. Non End-Anchored

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
n-p
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

End-Anchored vs. Non-End-Anchored 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 non-end-anchored

In Posttensionin
g
, tendons are anchored at their ends
g
using mechanical devices to transfer the prestress to
concrete
;
therefore
,
it is end-anchored.
(
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﹤ﱥﱡﹴ

、イﭳ﹤ﱥﱡﹴ

Very effective for deflection control

Better shear resistance
DesignCodesforPC
Design

Codes

for

PC

ACI-318 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 28-70 MPa at 28 days

For reinforced concrete
,
the
,
compressive strength should
be from 16-70 MPa at 28 days

Concrete with f’
c
> 70 MPa
can be used when supported
by test data
Concrete: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 simply-supported
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 
(
7-wire 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

Dead-End Anchorage

Ducts

Posttensionin
g
Procedures
g
PretensionedBeams
Pretensioned

Beams
PretensioningHardwares
Pretensioning

Hardwares

Hold-Down Devices for
Pretensioned Beams
Pretensioned Beams
PosttensionedBeams
Posttensioned

Beams

Posttension Hardwares
Sti Ah

St
ress
i
ng
A
nc
h
orage

Dead-End 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ﹳﵥ﹤
ﵰ略
ﵰ略


ﱥ葉ﱯ︠
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 LossesTime-Dependent 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 Stress-Relieved 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
)
Stress-Relieved
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

ACI-ASCE Committee (Zia et al. 1979)

拾ﵡﵵﴠﱯ說ﵡ鸞ﵥ

拾ﵡﵵﴠﱯ說ﵡ鸞ﵥ

ﵡﴠﱯ


ﵐ

殺




ヲ﹣



ﵐ
ﵒ便
﹤
ﱯﱡ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﹡ﰠﵯﱵ蘒

說﹡ﰠﵯﱵ



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
IInitial-Top
⎛⎞
=−+=−+≥
⎜⎟
⎝⎠
minmin
1
ioo
ii
tti
cttcbt
Fee
FMFM
σσ
AZZAkZ
IIInitial-Bottom
minmin
1
ioo
ii
bci
cbbctb
Fee
FMFM
σσ
A
ZZAkZ
⎛⎞
=+−=−−≤
⎜⎟
⎝⎠
IIIService-Top
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ﱯ﹧ﱩﵩ

ﹳ磻ャﱯ﹧ﱩﵩ

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 I-VI 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
IInitial-Top
(
)
⎛⎞
≤+−
⎜⎟
0min
1
btit
e
kM
σ
Z
IIInitial-Bottom
(
)
⎜⎟
⎝⎠
0min
btit
i
e
σ
F
(
)
⎛⎞
≤++
⎜⎟
1
ekM
σZ
IIIService-To
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 simply-supported 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
IInitial-Top
(
)
⎛⎞
≤+−
⎜⎟
0min
1
btit
e
kM
σ
Z
IIInitial-Bottom
(
)
⎜⎟
⎝⎠
0min
btit
i
e
σ
F
(
)
⎛⎞
≤++
⎜⎟
1
ekM
σZ
IIIService-To
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


雷北
﹥

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
T-Section
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 compression-controlled.
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@ﱤ︠﹣

ﱳ@ﱤ︠﹣
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(c-d)/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
semi-em
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 I-Sha
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

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

ﰠﹳ說

ﰠﹳ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

cast-in-p
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
§∙
IIIService-Top
1
'
pp
ococ
tcs
ctttccbttc
MM
FeMeM
FF
σσ
AZZZAkZZ
⎛⎞
=−++=−++≤
⎜⎟
⎝⎠
!
IVService-Bottom
1
pp
ococ
bts
cbbbcctbbc
MM
FeMeM
FF
σσ
AZZZAkZZ
⎛⎞
=+−−=−−−≥
⎜⎟
⎝⎠
VIService-Top 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
IInitial-To
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
VIService-Top
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 cross-sectional 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 (C5.8.4.1-1)
V
TotalFactored vertical shea
r
at the section
=
u
uh
e
V
V
d
Distance from centroid of tension
steel to mid-de
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 I-Sections, 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
ﱡﰠ說﹡ﰠﱩ﹧ﰠ說︬ﱬﱯﰠ
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 steel-concrete 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 box-section
AnchorageReinforcement
Anchorage

Reinforcement

Post-tensioning
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)

Strut-and-Tie
Method
(
new
(
for ACI and
A
ASHTO
)
)

Finite Element
Analysis
Analysis
(complicated)
Strut-and-Tie Method
AnchorageReinforcement
Anchorage

Reinforcement
Strut-and-Tie 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