Post - Tensioned Concrete Design For ACI 318-08

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26 Νοε 2013 (πριν από 3 χρόνια και 6 μήνες)

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Post
-

Tensioned Concrete

Design


For ACI 318
-
08















C
on
t
e
n
t
s







P
o
s
t
-
T
e
n
s
i
on
i
n
g
C
on
c
r
e
t
e

D
es
i
g
n
C
od
es




Chapter

1

Design

for

ACI

318
-
08


1.1

Notatio
n
s

1
-
1

1.2

D
e
sign
L
oad

Comb
i
natio
n
s

1.2.1

Initial Service

Load Comb
i
nation

1.2.2

Ser
vice
L
oad

Comb
i
nation

1.2.3

Lon
g
-
T
e
rm Service
L
oad

Comb
i
nation

1.2.4

Strength Des
i
gn Lo
a
d Co
m
bination

1
-
5

1
-
5

1
-
5

1
-
6

1
-
6

1.3

Limit on Mat
e
rial Strength

1
-
7

1.4

Strength Re
d
uction F
a
ct
or
s

1
-
7

1.5

D
e
sign Ass
u
mptions f
o
r Pres
t
r
e
s
sed Concrete

1
-
8

1.6

Serv
ic
e
ability Requ
i
re
m
en
t
s of Flexural

Memb
e
rs

1.6.1

Serviceability

Ch
e
ck at Initial

Service

Load



1
-
10



1
-
10


1.6.2

Serviceability

Checks

at Service Load

1.6.3

Serviceability Checks at Long
-
T
erm

Service
L
oad

1.6.4

Serviceability

Che
c
ks of Prestressing

Ste
el

1
-
10



1
-
11



1
-
11

1.7

Beam D
e
si
g
n

1.7.1

D
e
sign Flex
u
ral Re
i
nforc
e
ment

1.7.2

D
e
sign Be
a
m

Shear Reinf
o
rc
e
ment

1.7.3

D
e
sign Be
a
m

Tors
i
on Rei
n
forc
e
ment

1
-
12

1
-
12

1
-
23

1
-
26

1.8

Slab D
e
sign

1.8.1

D
e
sign for F
l
exure

1
-
31

1
-
31


1.8.2

Ch
ec
k for P
u
nch
i
ng She
a
r

1.8.3

D
e
sign Pu
nc
hing She
a
r Reinforce
m
ent

1
-
33

1
-
37























Post
-

T
ensioned Concrete
D
es
i
g
n

f
o
r

AC
I

318
-
08







Herein

describes

in

detail

the

v
a
rious

aspects

of

the

post
-
tensioned concrete

design

procedure
with

the

Ameri
can

code

ACI

31
8
-
0
8

[ACI

2008].

Various

notati
o
ns used

in

t
h
is

chapter

are listed

in

Tab
l
e

1
-
1.

F
o
r

referencing

to

t
h
e

pertinent

sections of

the

ACI

code

in this

chapter,

a

prefix

“ACI”

followed

by

t
he

section

n
u
mber

is

used.




1
.
1

N
o
t
a
t
i
on
s


The

following

table

identi
f
ies

the

various

notations

u
s
ed

in

this

chapter.
















N
o
t
a
t
i
on
s

1
-

1


A

2

2

2

A'

c

2

2



Post
-
Te
n
sioned

Concrete

Design



Table 1
-
1 List of S
y
mbols

Used in the

ACI 3
1
8
-
0
8 Code


cp

Area

enclosed

by

t
h
e

outs
i
de

perimeter

of

the

section,

in


A
g

Gross

area

o
f

concrete,

in


A

2

l

Total

area

of

longitudinal

r
einforcement

to

resist

torsion,

in


A
o

Area

enclo
se
d

by

the

shear

flow

path,

sq
-
in


A
oh

Area

enclo
se
d

by

the

centerline

of

the

outermost

clo
s
ed
transverse

to
r
sional

reinforcement,

sq
-
in

A
ps

Area

of

prestre
ssing

steel

in

flexural

te
n
sion

zone,

in


A
s

Area

of

tens
i
on

reinforcement,

in


s

Area

of

com
p
ression

reinforcement,

in


A

2

s(r
e
qu
i
re
d
)

Area

of

steel

required

for

t
ension

reinfo
r
cement,

in


A

/s

t

Area

of

clos
e
d

shear

reinforce
m
ent

per

unit

length

of

m
ember

for

torsion,

sq
-
in/in


A

2

v

Area

of

shear

reinforceme
n
t,

in


A

/s

2

v

Area

of

shear

reinforceme
n
t

per

unit

length

of

membe
r
,

in

/in


a

Depth

of

compression

block,

in


a

b

Depth

of

compression

block

at

balanced

condition,

in


a
max

Maximum

allowed

dep
th

o
f

compressi
o
n

block,

in


b

Width

of

member,

in


b

f

Effective

wi
d
th

of

flange

(T
-
beam

s
ect
i
on),

in


b
w

Width

of

web

(T
-
beam

section),

in


b
0

Perimeter

of

the

punching

critical

secti
o
n,

in


b
1

Width

of

t
h
e

punchi
n
g

crit
i
cal

section

in

the

direction

of
b
ending,

in

b
2

Width

of

t
h
e

punchi
n
g

crit
i
cal

section

perpendicular

to

the
direction

of

b
ending,

i
n

c

Depth

to

neu
t
ral

axis,

in


b

Depth

to

neu
t
ral

axis

at

balanced

conditions,

in




1
-

2

N
o
t
a
t
i
on
s


f'

n



Chapter 1

-

D
e
s
i
g
n
f
o
r

AC
I

3
18
-
08



Table 1
-
1 List

of S
y
mbols

Used in the

ACI 3
1
8
-
0
8 Code

d

Distance

fr
o
m

compr
e
ssion

face

to

tension

reinforcement,

in


d'

Concrete

cover

to

center

of

reinforcing,

i
n


d

e

Effective

depth

from

compression

face

t
o

centroid

of

tension

reinforcement,

in


d
s

Thickness

of

slab

(T
-
beam

section),

in


d
p

Distance

fr
o
m

extr
e
me

c
o
mpression

fiber

to

centroid

of
prestressing

s
teel,

in

E
c

Modulus

of

elasticity

of

concrete,

psi


E
s

Modulus

of

elasticity

of

reinforcement,

assumed

as

29,
0
00,
0
00

p
si

(ACI

8.5.2)


f'

c

Specified

compressive

strength

of

c
o
ncrete,

psi


ci

Specified

compressive

st
r
ength

of

concrete

at

time

of

initial
prestress,

psi


f
pe

Compressive

stress

in

concrete

due

to

effective

prestress
forces

only

(after

allowance

of

all

prestress

losses),

psi

f
ps

Stress

in

prestressing

steel

at

nominal

flexural

strengt
h
,

psi


f
pu

Specified

tensile

strength

of

prestressi
n
g

steel,

psi


f
py

Specified

yield

strength

of

prestressing

steel,

psi


f
t

Extreme

fiber

stress

in

tension

in

the

precompres
s
ed

tensile
zone

using

g
r
oss

section

propertie
s,

psi

f
y

Specified

yield

strength

of

flexural

reinforcement,

psi


f
ys

Specified

yield

strength

of

shear

reinfor
c
ement,

psi


h

Overall

depth

of

a

section,

in


h

f

Height

of

the

flange,

in



M

0

Design

moment

resistance

of

a

section

with

tendons

only,

N
-

m
m








N
o
t
a
t
i
on
s

1
-

3


n



















Post
-
Te
n
sioned

Concrete

Design



Table 1
-
1 List of S
y
mbols

Used in the

ACI 3
1
8
-
0
8 Code


M

bal

Design

moment

resistance

of

a

section

with

tend
o
ns

and

the

necessary

mi
l
d

reinforcement

to

reach

the

balanced

c
o
ndition,
N
-
mm

M

u

Factored

m
o
ment

at

section,

lb
-
in


N

c

Tension

force

in

concrete

due

to

unfacto
r
ed

dead

load

plus

live

load,

lb


P
u

Factored

axi
a
l

load

at

sect
i
on,

lb


s

Spacing

of

t
h
e

shear

reinf
o
rcement

along

the

length

o
f

the
beam,

in

T
u

Factored

torsional

moment

a
t

section,

lb
-
in


V
c

Shear

force

r
esisted

by

concrete,

lb


V
max

Maximum

permitted

total

factored

shear

force

at

a

sec
t
ion,

lb


V
u

Factored

she
a
r

force

at

a

s
e
ction,

lb


V
s

Shear

force

r
esisted

by

st
e
el,

lb





Factor

for

ob
t
aining

dep
t
h

of

compression

b
l
ock

in

c
oncrete

1


c




c


c,
max




ps s
s,
min


Ratio

of

the

maxim
u
m

to

the

minimum

dimensions

of

the
punchi
n
g

crit
i
cal

section


Strain

in

con
c
rete


Maximum

usable

compression

strain

allowed

in

extrem
e

concrete

fiber

(0.003

i
n
/in)


Strain

in

prestressing

steel


Strain

in

rein
f
o
rcing

steel


Minimum

tensile

strain

al
lo
wed

in

steel

reinforcement

at
nominal

strength

for

tensi
o
n

con
t
rolled

b
ehavior

(0.
00
5

in/in)





Strength

red
u
ction

factor


f

Fraction

o
f

u
nbalanced

moment

transferred

by

flexu
r
e


v

Fraction

of

unbalanced

moment

transfe
r
red

by

eccentricity

of
shear





1
-

4

N
o
t
a
t
i
on
s




Chapter 1

-

D
e
s
i
g
n
f
o
r

AC
I

3
18
-
08



Table 1
-
1 List of S
y
mbols

Used in the

ACI 3
1
8
-
0
8 Code




Shear

strength

reduct
ion

f
a
ctor

for

light
-
weight

concrete





Angle

of

compression

diagonals,

degrees



1
.
2

D
es
i
g
n
Lo
a
d
C
o
m
b
i
n
a
t
i
on
s


The

design

load

combinations

are

the

v
a
rious

combinations

of

the

load

ca
se
s
for

which

the

structure

needs

to

be

designed. For

A
CI

318
-
08,

if

a

st
ructure

is
subjected to

dead

(D),

li
v
e

(L),

patte
r
n

live

(PL),

snow

(S),

w
i
nd

(W),

and
earthquake

(E)

loads,

a
n
d

consideri
n
g that

wind

and

earthquake

forces

are
reversible,

t
h
e load

comb
in
ations

in

t
h
e following

sections

may

need

to

be

con
-

sidered

(ACI

9.
2.1).


For

post
-
tensioned concrete

design,

the

user

can

specify

the

prestressing

load
(PT)

by

prov
i
ding
t
h
e

ten
d
on

p
r
ofile.

The

default

load

combinat
i
ons

for

post
-
tensioning a
r
e

defined

in the

following

sections.



1
.
2
.
1

I
n
i
t
i
a
l

S
e
r
v
i
c
e

Lo
a
d
C
o
m
b
i
n
a
t
i
on


Th
e

following

load

comb
in
ation

is

used

for

checking

the

requir
e
ments

at

trans
-

fer

of

prest
r
ess

fo
r
ces,

i
n

accordance with

ACI

318
-
08

clause

18.4.1.

The
prestressing

forces

are

considered

without

any

long
-
t
erm

loses

for

the

initial
service

load

co
mbination

c
h
eck.


1.0D

+

1.0PT

(ACI

18
.
4
.1)



1
.
2
.
2

S
e
r
v
i
c
e

L
o
a
d
C
o
m
b
i
n
a
t
i
on


The

following

load

comb
i
n
ations

are

used

for

checking

t
h
e

requirements

of
prestress

for

serviceabili
t
y

in

accorda
n
ce

with

ACI

318
-
08

clauses

18.3.3,

18.4
.
2(b),

a
n
d

18
.
9.3
.
2. It

is

assumed that

all

long
-
term

losses

ha
v
e

already

oc
-

curred

at

the

service

s
t
age.


1.0D

+

1.0PT

1.0D

+

1.0L

+

1.0PT

(A
CI

18.4.2
(
b))






D
es
i
g
n

Lo
a
d

C
o
m
b
i
n
a
t
i
on
s

1
-

5




Post
-
Te
n
sioned

Concrete

Design



1
.
2
.
3

Long
-
T
e
r
m
S
e
r
v
i
c
e

Lo
a
d
C
o
m
b
i
n
a
t
i
on


The

following

load

comb
i
n
ations

are

used

for

checking

t
h
e

requirements

of
prestress

in

accordance with

ACI

318
-
0
8

clause

18.4
.2(a).

The

permanent

load
for

this

load

combination is

taken

as

50

percent

of

the

live

load.

It

is

assumed
that

all

long
-
t
erm

losses

have

already

occurred

at

the

service

stage.


1.0D

+

1.0PT

1.0D

+

0.5L

+

1.0PT


(ACI

18.4.2
(
b))



1
.
2
.
4

S
t
r
e
ng
t
h
D
es
i
g
n
Lo
a
d
C
o
m
b
i
n
a
t
i
on


The

following

load

comb
i
n
ations

are

used

for

checking

t
h
e

requirements

of
prestress

for

strength

in

accordance

with

ACI

318
-
08,

Chapters

9

and

18.


The

strength

design

combinations

requ
i
red

for

shear

design

of

beams

and
punchi
n
g

shear

require

the

f
u
ll

PT

forces

(primary

and

secondary).

Flexural

de
-

sign

requires

only

t
h
e

h
y
perstatic

(se
c
ondary) forces.

The

hype
r
static

(secon
-

dary)

forces

are

automatically determi
n
ed


by

subtracti
n
g

out

the

pri
mary

PT

m
o
ments

when

the

flexural

design

is

carried

out.


1.4D

+

1.0P
T
*


(ACI

9.2.1)


1.2D

+

1.6L

+

1.0P
T
*


(ACI

9.2.1)


1.2D

+

1.6
(
0
.
75

PL)

+

1
.0
P
T
*


(ACI

9.2.1,

13.7.6.3)


0.9D



1.6W

+1.0P
T
*


1.2D

+

1.0L



1.6W

+

1.
0
P
T
*


0.9D



1.0E

+

1.0P
T
*

1.2D

+

1.0L



1.0E

+

1.0
P
T
*


1.2D

+

1.6L

+

0.5S

+

1.
0P
T
*

1.2D

+

1.0L

+

1.6S

+

1.
0P
T
*


1.2D

+

1.6S



0.8W

+

1.
0
P
T
*

1.2D

+

1.0L

+

0.5S



1.
6
W

+

1.0P
T
*

(ACI

9.2.1)

(ACI

9.2.1)
(ACI

9.2.1)
(AC
I

9.2.1)


1.2D

+

1.0L

+

0.2S



1.0E

+

1.0P
T
*

(ACI

9.2.1)


*


Repl
a
ce

P
T

by

H

f
o
r

fl
ex
ural

d
esign

o
n
l
y






1
-

6

D
es
i
g
n

Lo
a
d

C
o
m
b
i
n
a
t
i
on
s


c

y

yt

t



Chapter 1

-

D
e
s
i
g
n
f
o
r

AC
I

3
18
-
08



The

IBC

20
0
6

basic

load

combinations

(Section

1605.2.1)

are

the

same.

Th
e
s
e
also

are

the

d
efault

design

load

combi
n
ations

whenever the

ACI

318
-

08

code

is

used.

The

user

should

use

o
t
her

approp
r
iate

load

c
o
mbinations

if
roof

live

load

is

treated

se
p
arately,

or

if

other

types

of

loads

are

present.



1
.
3

L
i
m
i
t
s

o
n
M
a
t
e
r
i
a
l

S
t
r
e
ng
t
h


The

concrete

compressive

strength,

f'

,

should

n
o
t

b
e

less

than

2500

psi

(ACI

5.1.1).

The

upper

limit

of the

reinforcement

yield

strength,

f

,

i
s

take
n

a
s
8
0
ksi

(ACI

9.4) a
n
d the
u
pper

l
i
mit

of

the reinforcement

shear

strength,

f

,

is

taken

a
s

60

k
si

(ACI

1
1.5.2).


This procedure

enforces

the

upper

material

stre
n
g
th

limits

for

flexure

and

shear

design of

beams and

slabs

or

for

torsion

design

of

beams.

T
h
e

input
material

strengths are

taken

as

the

upper

li
m
its

if

they

are

defined

in

the

material

prop
erties as being

greater

than the

limits.

The

user is

responsible

for

ensuring

that

the

mini
mum

strength

is

satisfied.



1
.
4

S
t
r
e
ng
t
h
R
e
du
c
t
i
o
n
F
ac
t
o
r
s


The strength

reduction

factors,


,

are a
p
plied

on

the
s
p
ecified

strength

to

obtain
the

design

strength

pr
o
v
id
e
d

by

a

mem
b
er.

The



f
a
c
t
ors

for

flexure,

shear,

and
torsion

are

as

follows:



t

=

0.90

for

f
lexure

(tension

controlled)

(ACI

9.3.2.1)



c

=

0.65

for

flexure

(compression

control
led)

(ACI

9.3.2.2
(
b))



=

0.75

for

s
h
ear

and

torsion.

(ACI

9.3.2.3)
The

value of



varies fr
o
m

compressio
n
-
controll
e
d

to

tension
-
cont
r
o
lled

based

on

the

maximum

tensi
l
e

strain

in

the

reinforcement

at

the

extr
e
me

edge,



(ACI

9.3.2.2
)
.


Sections

are

consid
ered

compression
-
con
t
rolled

when

the

tensile

strain

in

the
extreme

tension

reinforcement

is

eq
u
a
l

to

or

less

than

the

compression
-

controlled

strain

limit

at

th
e

time

the

concrete

in

c
o
mpression

r
e
aches

i
t
s

a
s
-

sumed

st
r
ain

limit

of



,

which

i
s

0.003.

The

compression
-
controlled

strain

c.
max




L
i
m
i
t
s

o
n

M
a
t
e
r
i
a
l

S
t
r
e
ng
t
h

1
-

7


y

t

1

c



Post
-
Te
n
sioned

Concrete

Design



limit

is

the

te
nsile

strain

in

the

reinforcement

at

the

balanced

st
r
a
i
n

condition,
which

is

taken

as

the

yield

strain

of

the

r
einforcement,

(
f

/E
)

(ACI

10.3.3).


Sections

are

tension
-
cont
r
o
lled

when

t
h
e

tensile

strain

in

the

ex
t
reme

tensi
o
n
reinforcement

is

equal

to

or

greater

than

0.0
0
5,

just

as

the

concrete

in

compres
-

sion

reaches

i
ts

assumed

s
t
rain

limit

of

0.003

(ACI

10.3.4)
.


Sections

with



between

the

two

limits

are

considered

to

be

in

a

transition

re
-

gion

between

compression
-
controlled

and

tensio
n
-
controlled

s
ections

(ACI

10.3
.
4).


When

the

section

is

tension
-
controlle
d
,


t

is

used.

When

the

s
e
ction

is

com
-

pres
sion
-
controlled,


c

is

u
sed.

When

the

section

is

in

the

transition

region,



is
linearly

interpolated

between

the

two

values

(ACI

9.3.2).




1
.
5

D
es
i
g
n
A
ss
u
m
p
t
i
on
s

f
o
r

P
r
es
t
r
esse
d
C
on
c
r
e
t
e


Strength

design

of

prestressed members for

flexure

and

axial

l
o
ads

s
hall

be
based

on

assumptions

giv
e
n

in

ACI

10
.2
.




The

strain

in

the

reinforcement

and

c
o
n
crete

shall

be

assumed

directly

pro
-

portional

t
o

t
h
e

distance

from

the

neutral

axis

(ACI

10.2.2).




The

maxim
u
m

usable

str
a
in

at

the

e
x
t
r
eme

concrete

compression

fi
ber

shall
be

assumed

equal

to

0
.0
0
3

(ACI

10.2.3).




The

tensile

strength

of

the

concrete

shall

be

neglected

in

axial

and

flexural
calculations

(ACI

10.2.5).




The

relationship

between

the

concrete

compressive

stress

distribution

and

the
concrete

strain

s
hall

be

assumed

to

be

r
e
ctangular

by

an

equivalent

rectangu
-

lar

concrete

stress

distribution

(ACI

10
.2
.7).




The

concrete

stress

of

0.8
5
f'

shall

be

assumed

uniformly

distribu
t
ed

over

an

equivalent
-
compression

z
o
ne

bounded

by

edges

of

the

cross
-
secti
on

and

a
straight line

located

paral
l
el to

the

neutral

axis

at

a

distance

a

=



c

from

the

fiber

of

maximum

compressive

strain

(ACI

10.2.7
.
1
)
.





1
-

8

D
e
s
i
g
n
A
s
s
u
m
p
t
i
on
s

f
o

r

P
r
e
s
t
r
e
ss
e
d

C
on
c
r
e
t
e


t



Chapter 1

-

D
e
s
i
g
n
f
o
r

AC
I

3
18
-
08





The

distance

f
rom

the

fiber

of

max
i
mum

strain

to

the

neutral

axis,

c

shall

be
measured

in

a

direction

pe
r
p
endicular

to

the

neutral

axis

(ACI

10.2
.
7.2).


Elastic

theory

shall

be

used

with

the

fol
l
owing

two

assumptions:




The

strains

shall

vary

linearly

with

depth

thro
u
g
h

t
h
e

entire

load

range

(ACI

18.3
.
2.1).



At

cracked

se
ctions,

the

concrete

resists

no

tension

(
A
CI

18.3
.
2.1).

Prestre
s
sed

c
oncrete

members

are

inv
e
stigated

at

the

following

three

stag
e
s

(ACI

18.3.2):




At

transfer

of prestress

fo
r
ce




At

service

loading




At

nominal

strength


The

prestre
ss
ed

flexural

members

are

c
l
assified

as

C
l
ass

U

(uncr
a
cked),

Class
T

(transition),

and

Class

C

(cracked)

based

on

f

,

the

computed

extreme

fib
e
r
stress

in

tension

in

t
h
e

precompressed

tensile zone

at

service

loa
ds

(ACI

18.3
.
3).


The

precom
p
ressed

tensile

zone

is

that

portion

of

a

p
restressed

member

where
flexural

tension,

calculated using

gro
s
s

section

properties, would

occur

under
unfactored

dead

and

live

loads

if

th
e

prestre
s
s

force

w
as

not

present.

Prestressed concrete

is

u
s
ually

design
e
d

so

that

t
h
e

prestress

force

introduces
compression

into

this

zone,

thus

effectively

reducing

the

magnitu
d
e

of

the

tensile

stress.


For

Class

U

and

Class

T

flexural

members, str
e
ss
e
s

at

service

l
o
ad

are

determined u
sing

uncracked

section propert
i
es,

while

for

Class

C

f
l
exural

members, stres
s
es

at

s
ervice

load

are

ca
l
culat
e
d

based

on

the

cracked

section

(ACI

18.3
.
4).


A

prestressed

two
-
way

slab

system

is

designed

as

Class

U

only

wi
t
h

f
t



6

ported
.

f

'
c

(ACI

R18.3.3);

otherwise,

an

over
-
stressed

(O/S)

c
o
ndition

is

re
-






D
es
i
g
n
A
ss
u
m
p
t
i
on
s

f
o
r

P
r
es
t
r
esse
d
C
o
n
c
r
e
t
e

1
-

9


ci

f

f

ci

ci



Post
-
Te
n
sioned

Concrete

Design



The

following

table

prov
i
des

a

s
u
mmary

of

the

conditions

cons
i
d
ered

for

the
various

se
ction

classes.






Assumed behavior

Prestressed



Nonpres
t
ressed

Class U

Class T

C
l
ass C


Uncracked

Transition

between
uncracked

and

cra
c
ked


Cracked


Cracked

Section

prope
r
ties

for

stress
calculation

at

service

loads

Gross sec
t
ion

18.
3
.
4

Gross sec
t
ion

18.
3
.
4

Cracked sec
t
ion

18.
3
.
4


No

requirem
e
nt

Allowable

s
t
ress

a
t transf
e
r

18.
4
.
1

18.
4
.
1

18.
4
.
1

No

requirem
e
nt

Allowable

compre
s
s
ive

stress

based
on

uncracked

sect
i
on

pro
p
ert
i
es


18.
4
.
2


18.
4
.
2


No

requirem
e
nt


No

requirem
e
nt

Tensile

stress

a
t

s
e
rvi
ce

loads

18.
3
.
3




7
.5

f

c


7.
5

f

c





f
t



12

f

c



No

requirem
e
nt


No

requirem
e
nt


1
.
6

S
e
r
v
i
cea
b
ili
t
y

R
e
qu
i
r
e
m
e
n
t
s

o
f

F
l
e
x
u
r
a
l

M
e
m
b
e
r
s



1
.
6
.
1

S
e
r
v
i
cea
b
ili
t
y

C
h
e
c
k

a
t

I
n
i
t
i
a
l

S
e
r
v
i
c
e

Lo
a
d


The

stres
s
es

in

the

concrete

immediate
l
y

after

prestress

force

tra
n
sfer

(
before
time

dependent

prestress

losses)

are

checked

against

the

following

l
i
mits:




Extreme

fiber

stress

in

compression:

0.60

f

'

(ACI

18.4.1(
a
))





Extreme

fiber

stress

in

tension:

3

'

(ACI

18.4.1
(
b))




Extreme

fiber

stress

in

tension

at

end
s

of

simply

supp
o
rted

members:

6

'



(ACI

18.4.1(
c
))




1
.
6
.
2

S
e
r
v
i
cea
b
ili
t
y

C
h
e
c
k
s

a
t

S
e
r
v
i
c
e

Lo
a
d


The

stres
s
es

i
n

the

concrete

for

Class

U

and

Class

T

prestres
s
ed

flexural

m
e
m
-

bers

at

servi
c
e

loads,

and

after

all

pres
t
ress

loss
e
s

occur,

are

ch
e
cke
d

against

the

following

limits:






1
-

1
0

S
e
r
v
i
cea
b
ili
t
y

R
e
qu
i
r
e
m
e
n
t
s

o
f

F
l
ex
u
r
a
l

M
e
m
b
e
r
s


c

c



Chapter 1

-

D
e
s
i
g
n
f
o
r

AC
I

3
18
-
08





Extreme

fiber

stress

in

compression

due

to

prestress

plus

total

load:

0.60

f

'



(ACI

18.4.2
(
b))




Extreme

fiber

stress

in

tension

i
n

the

precompre
s
s
e
d

tensile

zone

at

service
loads:




Class

U

b
e
ams

and

one
-
way

slabs:

f
t



7.5

f

'
c

(ACI

18.3.3
)




Class

U

t
wo
-
way

slabs:

f
t



6

f

'
c

(ACI

18.3.3)




Class

T

be
am
s:

7.5

f

'
c



f
t



12

f

'
c

(ACI

18.3.3)




Class

C

b
e
ams:

f
t



12

f

'
c

(ACI

18.3.3)


For

Class

C

prestres
s
ed flexural

mem
b
ers,

checks

at

service

loads

are

not

re
-

quired

by

t
h
e

code.

However,

for

Class

C

prestressed

flexural

members

not
subject

to

fatigue

or

to

a
g
gressive exposure,

the

spacing

of

b
o
n
d
ed

reinforce
-

ment

nea
r
est

the

extreme

tension

face

shall

not

exceed

that

given

by

ACI

10.6
.
4

(ACI

18.4.4).

It

is

ass
umed

that

the

user

h
a
s

c
h
ecked

the

re
q
uirements

of
ACI

10.6.4

a
nd

ACI

18.4
.
4.1

to

18.
4
.4

ind
e
pendent
l
y
.



1
.
6
.
3

S
e
r
v
i
cea
b
ili
t
y

C
h
e
c
k
s

a
t

Long
-
T
e
r
m
S
e
r
v
i
c
e

Lo
a
d


The

stres
s
es

i
n

the

concrete

for

Class

U

and

Class

T

prestres
s
ed

flexural

m
e
m
-

bers

at

long
-
t
erm

service

loads, and

after

all

prestress

losses occur, are

checked
against

the

same

limi
t
s

as

f
or

the

normal

service

load,

except

for

the

following:




Extreme

fiber

stress

in

compression

due

t
o

prestress

plus

total

load:


0.45

f

'

(ACI

18.4.2(
a
))



1
.
6
.
4

S
e
r
v
i
cea
b
ili
t
y

C
h
e
c
k
s

o
f

Pr
es
t
r
ess
i
n
g
S
t
ee
l


Perform

checks

o
n

the

tensile

stresses

in

the

prestressi
n
g steel

(
ACI

1
8
.5.1).

The

permissible

t
ensile

st
r
ess

checks,

in

all

types

of






S
e
r
v
i
cea
b
ili
t
y

R
e
qu
i
r
e
m
e
n
t
s

o
f

F
l
ex
u
r
a
l

M
e
m
b
e
r
s

1
-

1
1


py

pu

pu

py

pu



Post
-
Te
n
sioned

Concrete

Design



prestressing

s
teel,

in

terms

of

the

specified

minimum

tensile

str
e
ss

f
pu
,

and

the
minimum

yi
e
ld

stress,

f
y
,

are

summari
ze
d

as

follows:




Due

to

tend
o
n

jacking

for
c
e:

min(0.94
f

,

0.80
f

)

(ACI

18.5.1(
a
))




Immediately

after

force

tr
an
sfer:

min(0.82
f




At

anchors

and

couplers

after

force

transfer:

0.70
f

,

0.74
f

)

(ACI

18.5.1
(
b))




(ACI

18.5.1(
c
))



1
.
7

B
ea
m
D
es
i
gn


In

the

desig
n

of

prestressed concrete

beams,

c
alculates

and

reports

the required

areas

of

reinforcement

for

f
l
exure,

shear,

and

torsion

based

on

the beam
moments,

shear for
c
es,

torsion,

l
o
ad combinat
i
on

factors,

a
n
d

other

crite
ria

described

in

the

s
u
bsections

that

fo
llow.

The

reinforcement

r
equirements are

ca
l
culated

at

each

stati
o
n

along

t
h
e

length

of

the

b
e
am.


Beams

are

designed

for

major

direction

flexure,

shear,

and

torsion

only.

Effects
resulting

from

any

axial

forces

and

minor

direction

bending

t
h
at

may

exist

i
n
the

beams

must

be

investigated

indepen
d
ently

by

t
h
e

u
ser.


The

beam

design

p
r
ocedure

invo
l
ves

the

following

st
e
p
s:




Design

flexural

reinforcement




Design

shear

reinforcement




Design

torsion

reinforcement



1
.
7
.
1

D
es
i
g
n

F
l
e
x
u
r
a
l

R
e
i
n
f
o
r
ce
m
e
n
t


The

b
eam

top

and

bottom

flexural

rei
n
forcement

is

designed

at

each

station
along

the

beam.

In

designing

the

flexural

reinforcement for

the

major

moment
of

a

particular

beam

for

a particular

station,

the

f
o
ll
o
wing

steps

are

involved:




Dete
r
mine

f
a
ctored

mom
e
nt
s




Dete
r
m
ine

required

flexu
r
al

reinforcement





1
-

1
2

B
e
a
m
D
e
s
i
gn




Chapter 1

-

D
e
s
i
g
n
f
o
r

AC
I

3
18
-
08



1
.
7
.
1
.
1

D
e
t
e
r
m
i
n
e

F
ac
t
o
r
e
d

M
o
m
e
n
t
s


In

the

design

of

flexural

r
einforcement of

prestr
e
ss
e
d

concrete

b
e
ams,

the

f
actored

mome
n
ts

for

e
ach

l
o
ad

combinat
i
on at

a

part
ic
ular

be
a
m

s
ta
tion

are

ob
tained

by

factoring

the

corresponding

moments

for

different

load

cases,

with the

corresponding

l
o
ad

factors.


The

beam

is

then

designed

for

the

maximum positive

and

maximum negati
v
e
factored

m
o
ments

obtained

from

all

o
f

the

load

combinations. Positive

beam
moments

can

be

used

to

calcula
t
e

bott
o
m

reinforcement.

In

such

cases

the
beam

may

be

designed

as

a

rectangular or

a

flan
g
ed

beam.

Negative

beam
moments

can

be

used

to

calculate

top

reinforcement. In

s
u
ch

cas
e
s

the

beam
may

be

designed

as

a

rectangular

or

i
n
v
erted

flanged

beam.



1
.
7
.
1
.
2

D
e
t
e
r
m
i
n
e

R
e
qu
i
r
e
d

F
l
ex
u
r
a
l

R
e
i
n
f
o
r
ce
m
e
n
t


In

the

flexural

reinforcement design

process,

both

the tension

and

compression

reinforcement

shall be calculated
.

Compression

rei
nforcement

is

added when

the

applied

design

moment

excee
d
s

the

max
i
mum m
o
ment

c
a
pacity

of

a singly

reinfo
r
ced section.

The

user

has

the

opt
i
on

of

avoiding

the

compression
reinforcement

by

increasing

the

effective

depth,

the

width,

or

t
h
e

strength

of
the

co
ncrete.


The

design

p
rocedure

is

based

on

the

simplified

rectangular

stress

block,

as
shown

in

Figure

1
-
1

(ACI

10.2).

Fur
t
hermore,

it

i
s

assumed that

t
h
e

net

t
ensile
strain

in

the

reinforcement

shall

not

be

less

than

0
.005

(tension

controlled)
(ACI

10.3.4)
.

When

the

applied

mom
e
nt

exceeds the

moment c
a
pacity at

this
design

condi
t
ion,

the

area

of

compression

reinforcement

is

calculated

on

the
assumption

that

the

addit
i
onal

moment

will

be

carr
i
ed

by

comp
r
ession

reinforcement

and

additional

tension

reinfo
r
ceme
nt.


The

design

p
r
ocedure

used

,

for

both

rectan
g
u
lar

and

flanged

sections

(L
-

and

T
-
beams),

is

summari
z
ed in

the

subsections

that

follow.

It

is

assumed
that

the

des
i
gn

ultimate

axial

force

does

not

exceed



(0.
1
f
'

A

)

(ACI

10.3.5);

c

g

hence

all

be
a
ms

are

desi
g
n
ed

for

maj
o
r

direction

flexure,

shear,

and

torsion

only.









B
e
a
m
D
e
s
i
g
n

1
-

1
3






A

BEAM
SE
C
T
I
O
N

STR
A
IN
DI
A
G
R
A
M

STR
E
SS
DI
A
G
R
AM





a



c

c



d

d

A

b

A

n

m
a
x

s
m
i
n

c





max

s

=



Post
-
Te
n
sioned

Concrete

Design



1
.
7
.
1
.
2
.
1

D
es
i
g
n

o
f

R
ec
t
a
ngu
l
a
r

B
ea
m
s

The process

first

d
etermines

if

th
e

moment

capacity

provided

by

the

post
-
tensioni
n
g

tendons

alo
n
e

is

enough.

In

calculating

the

capacity,

it

is

ass
u
med

that

A

In

that

case,

the

moment

c
apacity



M

0

is

determined

as

f
o
llows:

=

0.





0.003

b

0
.
8
5

f


c




A

s


d


c

C
s

a




1
c



d

p

d
s

A
ps





A
s








ps


T


s


s






T
cps

T
cs


BEAM
SE
C
T
I
O
N

STR
A
IN
DI
A
G
R
A
M

STR
E
SS
DI
A
G
R
AM


Fi
gure

1
-
1

Re
c
t
an
g
u
l
a
r

B
e
a
m

D
e
s
i
gn



The

maxim
u
m

depth

of

t
h
e

compression

zone,

c

,

is

calculated

based

on

the

limitation

that the tension

reinforce
m
ent

strain

shall

not be

less than


,

which

is

equal

to

0
.
005

for

tensi
o
n
-
controlled

behavior

(ACI

10.3
.
4):











max





c

max



d

(ACI

10.2.2)





c

max





s

min




where,



c
max



s
min

=

0
.
003

(ACI

10.2.3)


=

0.0
0
5

(ACI

10.3.4)


Therefore,

the

limit

c

≤ c

is

set

for

tension
-
controll
e
d

sections.




1
-

1
4

B
e
a
m
D
e
s
i
gn


=


c

1



ps

ps

u

n

s

c

m
a
x

a

ps

ps

s



Chapter 1

-

D
e
s
i
g
n
f
o
r

AC
I

3
18
-
08



The

max
imum

allowable

depth

of

the

rectangular

compression

block,

a

given

b
y
:


m
ax

,

is



max


1

max

(ACI

10.2.7
.1
)


where



is

calculated

as:




f

'
c



4000



1
=

0.85



0
.
0
5






1000



,

0.65




1





0
.
85

(ACI

10.2.7
.3
)


The process

det
ermines

the

depth

of

the

neutral

axis,

c
,

by

imposing force

equilib
rium,

i.e.,

C

=

T
.

After

t
h
e

depth

of

t
h
e

neutral

axis

has

been

determined, the stress

in

the

p
o
st
-
tensioning

steel,

f

,

is

computed

based on

strain compatibility

for

bo
n
ded

t
endons.

For

un
b
o
nded

t
e
ndons,

t
h
e

code

equatio
n
s

are

used

to

compute

the

stress,

f

in

the

post
-
tension
i
ng

steel.


Based

on

the

calculated

f

,

the

depth

of

the

neutral

axis

is

recalc
u
lated,

and

f

is

further

updated.

After

this

iteration

p
r
ocess

has

c
o
nverged,

the

d
epth

of

the
rectangular

compression

block

is

determined

as

follows:


a



1
c




If

c



c
m
a
x

(ACI

10
.
3.4),

t
he

moment

capacity

of

t
he

section,

p
r
ovided

by

post
-
tension
i
ng

steel

only,

is

computed

as:





M

0





A

f



d



a



n

p
s

p
s



p

2







If

c



c
m
a
x






(ACI

10.3.4),

a

failure

condition

is

d
e
clared.


If

M





M

0

,


ca
l
culates

the

moment

capac
i
ty

and

the

A

required

at

the balanced

condition.

The balanced

con
d
ition

is

taken

as

the marginal

tension
controlled

c
a
se.

In that c
a
s
e,

it

is

a
s
sumed

that

the

d
e
pth

of the

neutral

axis,

c
is

equal

to

c

.

The

stress

in

the

post
-
tensioning

steel,

f

is

then

calc
u
lated

and

the

max

ps

area

of

required

tension

r
einforcement,

A

,

is

determined

by

imposing

f
o
rce

equilibrium,

i
.e.,

C

=

T
.


C



0.85

f
'
a

b





B
e
a
m
D
e
s
i
g
n

1
-

1
5


f

s

u

n

u

u

n

u

ps

u

n

n

n

s

y

s



Post
-
Te
n
sioned

Concrete

Design



bal

b
al

bal

T



A
ps

f

ps



A
s

f
s


0.85

f


a

b



A

f

b
a
l

A
bal



c

m
a
x

ps

ps

s

b
al

s


After

the

ar
e
a

of

tension

reinforcement

h
a
s

been

dete
r
m
ined,

the

c
a
pacity

of

t
he
section

with

post
-
tension
i
ng

steel

and

t
e
nsion

reinforcement

is

computed

as:




M

bal





A

f

b
a
l



d



a
ma
x







A
b
al

f

bal



d



a
m
a
x



n

ps

ps



p

2



s

s




s

2












In

that

case,

it

is

assumed

that

the

bonded

tension

r
einforcement

will

yield,
which is

true

for

most

cases. In

the

case

that

it

does

not

yield, the

stress in

the
reinforcement,

f

,

is

determined

from

the

elastic
-
p
e
r
fectly

plastic

stress
-
st
r
ain

relationship.

The

f

value

of

the

reinforceme
n
t

is

then

replac
ed

w
ith

f

in

the

preceding four

equations.

This

case

do
e
s

not

involve

any

iter
a
tion

in

determining

the

dep
t
h

of

the

neutral

axis,

c
.



1
.
7
.
1
.
2
.
1
.
1

C
a
s
e

1
:

P
o
s
t
-
t
e
n
s
i
on
i
n
g
s
t
e
e
l

i
s

a
d
e
q
u
a
t
e


When

M





M

0

,

the amount

of

pos
t
-
tensioning

s
t
eel

is

adequate

to

resist

the

design

moment

M

.

Minimum

reinforcement

is

provided

to

satisfy

ductility

re
-

quirements

(ACI

18.9.3
.
2

and

18
.
9.3
.
3),

i.e.,

M





M

0

.



1
.
7
.
1
.
2
.
1
.
2

C
a
s
e

2
:

P
o
s
t
-
t
e
n
s
i
on
i
n
g
s
t
e
e
l

p
l
u
s

t
e
n
s
i
o
n

r
e
i
n
f
o
r
ce
m
e
n
t

In

this

case,

the

amount

of

post
-
tension
i
ng

steel,

A

,

alone

is

not

s
u
fficient

to
resist

M

,

and

therefore

the

required

area

of

tension

reinforcement

is

computed

to

supplement

the

post
-
tensioning

stee
l
.

The

combination

of

po
s
t
-
tensioning

steel

and

tension

reinforcement

should

r
esult

in

a
<

a

.


When



M

0



M






M

b
al

,

max



determines the

required

area

of

tension

re
-

n

u

n

inforcement,

A

,

iteratively

to

satisfy the

design moment

M


and

reports this

re
-

s

quired

area

of

tension

reinforcement.

Since

M

u

i
s

bou
n
de
d

by



M

0


at

t
h
e


lo
wer

end

and
and

b
a
l



M

b
al


at

the

up
p
er

end,

and

b
a
l



M

0

is

associa
t
ed

with

A
s



0



M

n

is

assoc
i
ated

with

A
s



A
s

,

the

required

a
r
ea

will

f
all

within

the

range

of

0

to

A

ba
l
.




1
-

1
6

B
e
a
m
D
e
s
i
gn


u

u

u

n

f

s

s

u
s

u

n

s

s

s

s

u

u

ps

s

s

s

s

s



Chapter 1

-

D
e
s
i
g
n
f
o
r

AC
I

3
18
-
08



The

tension

reinforcement

is

to

be

placed

at

the

bottom

if

M

is

p
o
sitive,

or

at

the

top

if
M

i
s

negative.



1
.
7
.
1
.
2
.
1
.
3

C
a
s
e

3
:

P
o
s
t
-
t
e
n
s
i
on
i
n
g
s
t
e
e
l

a
n
d
t
e
n
s
i
o
n

r
e
i
n
f
o
r
ce
m
e
n
t

a
r
e

no
t
a
d
e
qu
a
t
e


When

M





M

b
al

,


com
p
ression

reinforcement

is

required

(ACI

10.3.5).

In

this

case


assum
e
s

th
at

the

depth

of

the

neutral

axis,

c
,

is

equal

to

c

.


The

values

o
f

f


and

f


reach

their

respective

balanced

condition

values,

max

b
a
l

ps


and

f

ba
l

.


The

area

of

compression

reinfor
c
ement,

A
'

,


is

then

determined

as

follows:


The

moment

required

to

be

resisted

by

compression reinforcement

and

tension
reinforcement

is:


M



M





M

b
a
l


The

required

compression

reinforcement

is

given

b
y
:


A

M

us

'
s





f

'




0.85

f

'
c



d

e



d

'



,

where



f

'



E





c
max



d

'






(ACI

10.2.2,

10.2
.
3,

1
0
.2
.4
)

s

s

c

max








f

y

c
max




The

tension

reinforcement

for

balanc
i
ng

the

compression

rein
fo
rcement

i
s
given

b
y
:



A
c
o
m



M

us

f

y



d
s



d

'




Therefore,

the

total

tension

reinforcement,

A

=

A
ba
l

+

A
c
o
m

,

and

the

tot
a
l

compression

reinforcement

is

A'

.

A

is

to

be

placed

at

the

bottom

and

A'

is

to

be

placed

at

the

top

if

M

is

positive,

and

vice

versa

if

M

is

negative.









B
e
a
m
D
e
s
i
g
n

1
-

1
7


u

n

m
a
x

s
m
i
n

c





max

,

is

=


c

1



s

=

a



Post
-
Te
n
sioned

Concrete

Design



1
.
7
.
1
.
2
.
2

D
es
i
g
n
o
f

F
l
a
ng
e
d
B
ea
m
s


1
.
7
.
1
.
2
.
2
.
1

F
l
a
ng
e
d

B
ea
m
U
nd
e
r

N
e
g
a
t
i
v
e

M
o
m
e
n
t

In

designi
n
g

for

a

factored

negative

moment,

M



(i.e.,

designi
n
g

t
o
p

reinforce
-

ment),

the

calculation

of

the

reinforcement

area

is

exactly

the

same

as

above,
i.e.,

no

flang
e
d

beam

data

is

used.



1
.
7
.
1
.
2
.
2
.
2

F
l
a
ng
e
d

B
ea
m
U
nd
e
r

P
o
s
i
t
i
v
e

M
o
m
e
n
t

The process

first

d
eterm
ines

if

th
e

moment

capacity

provided

by

the

post
-
tensioni
n
g

tendons

alo
n
e

is

enough.

In

calculating

the

capacity,

it

is

ass
u
med

that

A

In

that

case,

the

moment

c
apacity



M

0

is

determined

as

f
o
llows:

=

0.


The

maxim
u
m

depth

of

t
h
e

compression

zone
,

c

,

is

calculated

based

on

the

limitation

that the tension

reinforce
m
ent

strain

shall

not be

less than


,

which

is

equal

to

0
.
005

for

tensi
o
n
-
controlled

behavior

(ACI

10.3
.
4):










max





c

max


d





c

max





s

min



(ACI

10.2.2)


w
here,



c
max



s
min

=

0
.
003

(ACI

10.2.3)


=

0.0
0
5

(ACI

10.3.4)


Therefore,

the

limit

c

≤ c

is

set

for

tension
-
controll
e
d

section:


The

maximum

allowable

depth

of

the

rectangular

compression

block,

a

given

b
y
:


m
ax



max


1

max

(ACI

10.2
.7
.1
)


where



is

calculated

as:




f

'
c



4000



1
=0.85



0
.
0
5






1000



,

0.65




1





0
.
85

(ACI

10.2.7
.3
)








1
-

1
8

B
e
a
m
D
e
s
i
gn




A

B
E
AM
SECT
I
O
N

STR
A
IN
DI
A
G
R
AM

STR
E
SS
DI
A
G
R
AM

d


d

h

c

C

T

C






C


ps

max

max

u

n

s

ps

ps

ps



Chapter 1

-

D
e
s
i
g
n
f
o
r

AC
I

3
18
-
08




b
f


h
f




0
.
0
0
3

0
.
8
5

f


c

0.8
5

f


c




A

s


d

p







A
s








A
ps





bw

d


f


s

C
s

c

d
s






ps



s


T
s


C

f



C
w




T
w


T
f


B
E
AM
SECT
I
O
N

STR
A
IN
DI
A
G
R
AM


Fi
gure

1
-
2

T
-
B
eam

Des
ig
n

STR
E
SS
DI
A
G
R
AM


The process

determines

the

depth

of

the

neutral

axis,

c
,

by

imposing

force

equilib
r
ium,

i.e.,

C

=

T
.

After

t
h
e

depth

of

t
h
e

neutral

axis

has

been

determined,

the

stress

in

the

post
-
tension
i
ng

steel,

f

is

computed

b
a
sed

on

strain

compatibility

for

bo
n
ded

t
endons.

For

unb
o
nded

t
e
ndons,

t
h
e

code

equatio
n
s

are

used

to

compute

the

stres
s,

f

in

t
h
e

post
-
tensioning

steel.

Based

on

the

calculated

f

,

the

depth

of

t
h
e

neutral

axis is

recalculated,

and

f

is

further

updated.

After

this

iteration

process

has

converged,

the

dep
t
h

of the

rectangular

compression

block
is

determined

as

follow
s:


a



1
c




If

c



c

(ACI

10.3.4),

the

moment

capacity

of

the

section,

provided

b
y

post
-
tension
i
ng

steel

only,

is

computed

as:





M

0





A

f



d



a



n

ps

p
s



p

2










If

c > c

(
A
CI

10.3
.
4),

a

failure

condition

is

declared.


If

M





M

0

,


ca
l
culates

the

moment

capac
i
ty

and

the

A

required

at

the

balanced

condition.

The

balanced

c
ondition

is

t
a
ken

as

the

marginal

ten
-

sion
-
controll
e
d

case.

In

that

case,

it

is

assumed

that

the

depth

of

the

neutral




B
e
a
m
D
e
s
i
g
n

1
-

1
9


s

f

.

c

c

c

A

c

f

f

s

u

n

u

u

n

u

ps

ps

y

s



Post
-
Te
n
sioned

Concrete

Design



axis

c

is

equal

to

c

.

T
h
e

stress

in

the

post
-
tension
i
ng

steel,

f

,

is

then

calcu
-

max

ps

lated

and

the

area

of

required

tension

reinforcement,

A

,

is

dete
rm
ined

by

im
-

posing

f
o
rce

equilibrium,

i
.e.,

C
=

T
.




If

a



h

,

the

subsequent

calculations

for
A

are

exactly

the

same

as

previously

f

s

defined

for

the

rectangular

beam

design.

However,

in

that

case

the

width

of

the

beam

is

t
a
ken

as

b

.

Compression

re
in
forcement

is

required

if

a

>

a




If

a

>

h

,

the

calc
ulation

for

A

is

given

b
y:


max

f

s


C



0
.
85

f

'

A
comp


where

A
c
o
m

is

the

area

of

concre
t
e

in

compres
s
ion,

i.e.,



com
c




b
f

h
f



b
w


a
m
a
x



h
f




bal

b
al

bal

T



A
ps

f

ps



A
s

f
s



A
bal



0.85

f

'
c

A
com



A

bal
ps

s

b
al

s


In

that

case,

it

is

assumed

that

the

bonded

tension

r
einforcement

will

yield,
which is

true

for

most

cases. In

the

case

that

it

does

not

yield, the

stress in

the
reinforcement,

f

,

is

determined

from

the

elastic
-
p
e
r
fectly

plastic

stress
-
st
r
ain

rel
ationship.

The

f

value

of

the

reinforceme
n
t

is

then

replaced

w
ith

f

in

the

preceding four

equations.

This

case

do
e
s

not

involve

any

iter
a
tion

in

determining

the

dep
t
h

of

the

neutral

axis,

c
.


C
as
e

1
:

P
o
s
t
-
t
e
n
s
i
on
i
n
g
s
t
ee
l

i
s

a
d
e
qu
a
t
e


When

M





M

0


the

a
mount

of

po
s
t
-
tensioning

s
teel

is

adequate

to

resist

the

design

moment

M

.

Minimum

reinforcement

is

provided

to

satisfy

ductility

re
-

quirements

(ACI

18.9.3
.
2

and

18
.
9.3
.
3),

i.e.,

M





M

0

.



C
as
e

2
:

P
o
s
t
-
t
e
n
s
i
on
i
n
g
s
t
ee
l

p
l
u
s

t
e
n
s
i
o
n
r
e
i
n
f
o
r
ce
m
e
n
t

In

this

case,

the

amount

of

post
-
tension
i
ng

steel,

A

,

alone

is

not

s
u
fficient

to
resist

M

,

and

therefore

the

required

area

of

tension

reinforcement

is

computed





1
-

2
0

B
e
a
m
D
e
s
i
gn


u

n

s

u

u

u

n

f

s

s

u
s

u

n

ps

s



Chapter 1

-

D
e
s
i
g
n
f
o
r

AC
I

3
18
-
08



to

supplemen
t

the

post
-
tensioning

stee
l
.

The

combination

of

po
st
-
tensioning

steel

and

tension

reinforcement

should

r
esult

in

a

<

a

.


When



M

0



M






M

b
a
l

,

max



determines the

required

area

of

tension

re
-

n

u

n

inforcement,

A

,

iteratively

to

satisfy the

design moment

M


and

reports this

re
-

s

quired

area

of

tension

reinforcement.

Since

M

u

i
s

bou
n
de
d

by



M

0


at

t
h
e


lowe
r

end

and



M

b
a
l


at

the

up
p
er

end,

and



M

0

is

assoc
i
ated

with

A

=

0

n

and

b
a
l

n

s

b
al



M

n

is

assoc
ia
ted

with

A
s



A
s

,

the

required

area

will

fall

w
i
thin

the

range

of

0

to

A

.


The

tension

reinforcement

is

to

be

placed

at

the

bottom

if

M

is

p
o
sitive,

or

at

the

top

if
M

i
s

negative.


C
as
e

3
:

P
o
s
t
-
t
e
n
s
i
on
i
n
g
s
t
ee
l

a
n
d
t
e
n
s

i
o
n
r
e
i
n
f
o
r
ce
m
e
n
t

a
r
e

no
t

a
d
e
qu
a
t
e


When

M





M

b
a
l

,


com
p
ression

reinforcement

is

required

(ACI

10.3.5).

In

that

ca
s
e,


assumes

th
at

the

depth

of

the

neutral

axis,

c
,

is

equal

to

c

.


The

value

of

f


and

f


reach

their

res
p
ective

balanced

condition

values,

max

b
a
l

ps


and

f

ba
l

.


The

area

of

compression

reinfor
c
ement,

A
'

,


is

then

determined

as

follows:


The

moment

required

to

be

resisted

by

compression reinforcement

and

tension
reinforcement

is:


M



M





M

b
a
l


The

required

compression

reinforcement

is

given

b
y
:



A

'



M

us


,

where

s



f

'



0.85

f

'





d



d

'






f

'



E



s

c

s




c
max



d

'








(ACI

10.2.2,

10.2
.
3,

and

1
0
.2.4)

s

s

c

max








f

y

c
max




The

tension


reinforcement

for

balanc
i
ng

the

compression

rein
fo
rcement

i
s
given

b
y
:






B
e
a
m
D
e
s
i
g
n

1
-

2
1


s

s

s

u

u

ct

s



Post
-
Te
n
sioned

Concrete

Design




A
c
o
m



M

us

f

y



d
s



d

'




Therefore,

the

total

tension

reinforcement,

A

=

A
bal

+

A
c
o
m

,

and

the

to
t
a
l

compression

reinforcement

is

A'

.

A

is

to

be

placed

at

the

bottom

and

A'

is

to

s

s

s

be

placed

at

the

top

if

M

is

positive,

and

vice

versa

if

M

is

negative.



1
.
7
.
1
.
2
.
3

D
u
c
t
ili
t
y

R
e
q
u
i
r
e
m
e
n
t
s


also

checks

the

following

cond
i
tion

by

cons
i
d
er
ing

the

p
o
st
-
tensioning steel

and

tension

reinforcement

to

avoid

abrupt

failu
r
e.




M

n



1.
2
M

c
r

(ACI

18.8.2)


The

preceding

condit
i
on

is

permitted

to

be

waived

for

the

following:
(a)

Two
-
way,

u
n
bonded

post
-
t
ensioned

slabs

(b)

Flexural

m
e
mbers

with

sh
ear

and

flexural

strength

at

least

twi
c
e

that

re
-

quired

by

ACI

9.2.


These

except
i
ons

currently

are

N
OT

handled

.



1
.
7
.
1
.
2
.
4

M
i
n
i
m
u
m
a
n
d
M
ax
i
m
u
m
R
e
i
n
f
o
r
ce
m
e
n
t

The

minim
u
m

flexural

tension

reinfo
r
cement

required

in

a

beam

section

is
given

by

the

f
ollowing

lim
i
t:


A



0.0
0
4

A

(ACI

18.9.2)

s

ct


where,

A

is

the

area

of

the

cross
-
s
e
ctio
n

between

the

flexural

tension

face

and

the

center

of

gravity

of

t
h
e

gross

section.


An

upper

limit

of

0.
0
4

times

the

gross

web

area

on

both

the

tens
i
on

reinforce
-

ment

and

the

compression

reinforcement

is

imposed

upon

request

as

follows:











1
-

2
2

B
e
a
m
D
e
s
i
gn


u

c

c



Chapter 1

-

D
e
s
i
g
n
f
o
r

AC
I

3
18
-
08




0
.
4
bd

A
s






0
.
4
b
w

d


0
.0
4
bd

A

s






0
.
0
4
b
w

d

R
ec
t
a
n
g
u
l
a
r

b
e
am
Fl
a
n
ge
d b
eam
R
e
ct
a
ng
u
l
a
r

b
ea
m

Fl
a
ng
e
d

b
ea
m



1
.
7
.
2

D
es
i
g
n
B
e
a
m
S
h
ea
r

R
e
i
n
f
o
r
ce
m
e
n
t


The shear re
i
nforcement is

designed for

each

load

combination

at

each

station
along the le
n
gth of the b
e
a
m
. In desi
g
n
ing the shear reinforcement for a par
-

ticular bea
m
,

for a particular loading co
m
bination, at a particular station due to
the beam

major shear, the following ste
p
s are involved:




Dete
r
m
ine the factored sh
e
a
r force,
V
.




Dete
r
m
ine the sh
ear force,

V

that can be resis
t
ed by the concrete.




Dete
r
m
ine the shear reinf
o
rcement required to carry the balance.


The

following

three

sections

describe

in

detail

the

a
lgorithms

associated

with
these steps.



1
.
7
.
2
.
1

D
e
t
e
r
m
i
n
e

F
ac
t
o
r
e
d

S
h
e
a
r

Fo
r
c
e


In

the

design

of

the

be
a
m

shear

r
einforcement,

the

s
hear

forc
e
s
f
or

each

load
co
m
bination

at

a

particular

beam station

are

obtained

by

factori
n
g

the

correspondi
n
g

sh
e
ar

forces

for

different

lo
a
d

cases,

with

the

corresp
o
nding load
combination factors.



1
.
7
.
2
.
2

D
e
t
e
r
m
i
n
e

C
on
c
r
e
t
e

S
h
ea
r

C
a
p
ac
i
t
y


The shear force carr
i
ed by the concrete,
V

, is calcula
t
e
d as:


V

= min(
V

,

V

)

(ACI 11.3.3)

c

ci

cw


where,



V
ci






0.6





f

'
c




b
w

d

p






V
d






V
i

M

cre

M

max






1.
7





f

'
c




b
w

d




(ACI 11.3.3
.1
)




B
e
a
m
D
e
s
i
g
n

1
-

2
3




y

V

M

M

V

max

w

p

f

d

f

pe

V

d

p

V

ci

i

V

cw



Post
-
Te
n
sioned

Concrete

Design




V
cw




3.
5



f

'
c




0.3

f

pc


b

d




V

p


(ACI 11.3.3
.2
)


d

p



0.8
0
h

(ACI 11.3.
3.
1)




I

M







6



f

'



f



f




(ACI 11.3.3
.1
)

cre





t

c

pe

d






where,


=

stress

due

to

unfactored

dead

load,

at

the

extreme

fi
b
er

of

the

section where tensile stress
i
s caused

by externally applied loads,
psi


=

com
press

st
r
e
ss

in

conc
re
te

due

to

ef
f
ective

prest
re
ss

forc
e
s

only
(after allow
a
nce for all prestre
s
s l
o
sse
s
) at the extreme fiber of the
section

where

tensile

stress

is

ca
u
sed

by

externally

applied

loads,

psi


=

shear force at

the section due to unfactored

dead load,

lbs


=

vertical com
p
onent of effective prestress

force at the s
e
ction, lbs


=

no
m
inal

shear

strength

p
r
ovid
e
d

b
y

t
h
e

concrete

when

diagonal
cracking results from combined shear and moment



cre

=

moment

cau
s
ing

flexural

cracking

at

the

s
e
ction

be
c
a
use

of

exte
r
nally applied

loads



max

=

maxim
u
m

f
a
ctored

mom
e
nt

at

s
ection

because

of

externally

ap
-

plied loads


=

factored

sh
ea
r

force

at

the

s
e
ction

be
c
a
use

of

exte
r
nally

applied
loads occurring simultaneously with
M


=

no
m
inal

sh
ear

strength

p
r
ovided

b
y

t
h
e

concrete

when

diagonal
cracking results from high

principal tensile stress in the web


For

light
-
we
i
ght

concrete,

the


reduction factor


f

'
c

term

is

multiplied

by

the

shear

strength






1
-

2
4

B
e
a
m
D
e
s
i
gn


w

s

f

w







Chapter 1

-

D
e
s
i
g
n
f
o
r

AC
I

3
18
-
08



1
.
7
.
2
.
3

D
e
t
e
r
m
i
n
e

R
e
qu
i
r
e
d

S
h
ea
r

R
e
i
n
f
o
r
ce
m
e
n
t


The shear force is limi
t
ed to a maximum of:


V
max




V
c




8


f

'
c


b

d


(ACI 11.4
.7
.9
)


Given

V
,

V

,

and

V

,

the

required

shear

reinforcem
e
nt

is

calcula
te
d

as

follows

u

c

max

where,


,

the

strength red
u
ction factor, is 0.75 (ACI
9
.3.2
.
3).




If

V
u



0.
5

V
c


A
v

= 0

(ACI 11.4.6
.1
)

s




If

0.
5

V
c



V
u




V
max


A


V






V



v



u


c




(ACI 11.4.7
.1
, 11
.
4.7
.
2)

s



f

yt

d


A



0.7
5



f

'

5
0
b





v




ma
x


c

b

,



w



(ACI 11.4.6
.3
)





yt

f

yt






If
V

>

V

, a failure condition is declared (ACI 11.4
.
7.9).

u

max


For

members

with

an

effective

prestre
s
s

force

not

l
e
ss

than

40

percent

of

the
tensile

strength

of

t
he

fle
x
ural

reinforcement,

the

required

shear

reinforceme
n
t

is co
m
puted as follows (ACI 11.5
.
6.3,

1
1.5.
6
.4):








f

50




ma
x


0.75

'
c

b

,

b



A





f

w

f

w





v



mi
n




y

y



s



A
ps

f

pu

d



80

f

yt

d

b
w




If

V

exceeds

the

maximum

permitted

value

of


V

,

the

concrete

section

u

should

be increased in size (ACI 11.5.7
.
9).

max







B
e
a
m
D
e
s
i
g
n

1
-

2
5


v

u



Post
-
Te
n
sioned

Concrete

Design



Note

that

if

to
rsion

design

is

considered

and

torsion

reinforcem
e
nt

is

needed,
the

equation

given

in

ACI

11.5
.
6.3

does not

need

to

be

satisfied

i
n
dependently.
See the next section
Design of Beam To
r
sion Reinforcement

for details.


If

the

beam

depth

h

is

less

than

the

min
imum

of

10

in,

2.
5
h

,

and

0.5
b

,

the

f

w

m
ini
m
um

s
h
ear

reinforce
m
ent

given

by

ACI

11.5.6.3

is

not

enforced

(ACI

11.5
.
6.1(c)).


The

maxim
u
m

of

all

of

t
he

calcula
t
ed

A

/
s

values,

obtained

fr
o
m each

load
combination,

is

reported

along

with

the controlling

s
h
ear

force

and

associated
load combination.


The

beam

s
h
ear

reinfor
ce
m
ent

requir
e
me
nts

considered

are based

purely

on

shear

strength

cons
i
de
r
ations.

Any
m
ini
m
um stirrup

require
m
ents

to

satisfy

spacing

and

volu
m
etric

considerations

m
ust

b
e investigated

in
dep
endently

by
the user.



1
.
7
.
3

D
es
i
g
n
B
e
a
m
T
o
r
s
i
o
n

R
e
i
n
f
o
r
ce
m
e
n
t


The torsion reinforcement is designed f
o
r each design load combination at each
station

along the

length

o
f

the

bea
m
.

The

followi
n
g

steps

are

i
n
volved

in
designing
t
he shear reinforce
m
ent for a
particular stati
o
n due to

the
b
eam

torsion:




Dete
r
m
ine the factored torsion,

T
.




Dete
r
m
ine s
p
ecial
s
ection properties.




Dete
r
m
ine cr
i
tical torsion capacity.




Determine the torsion reinforcement required.



1
.
7
.
3
.
1

D
e
t
e
r
m
i
n
e

F
ac
t
o
r
e
d

To
r
s
i
on


I
n

the

design of

beam

tors
i
on

reinforcement,

the

torsions

for

each

load

combination at a particular beam station are obtained by f
a
ctoring the
correspondi
n
g torsions

for
d
ifferent

load cases

with

the

correspond
i
ng

load
co
m
bination

fac
tors (ACI 11.6.2).


In
a statical
l
y indeterminate structure

where redistribution of the torsion in a
m
e
m
ber

can

occur

due

to

redistribution

of

internal

forces

upon

cracking,

the




1
-

2
6

B
e
a
m
D
e
s
i
gn


u

u

A

h

cr

A

oh

cp

A

oh

A

o

p

cp

p

h

A

cp

oh

A

o

p

cp



Chapter 1

-

D
e
s
i
g
n
f
o
r

AC
I

3
18
-
08



design

T

i
s

permitted

to

b
e

reduced

in

accordance

w
i
th

the

code

(ACI 11.6.2
.2
).

However it is not done
aut
om
atically

t o
redistribute

the
internal

forces

and

reduce

T
.




1
.
7
.
3
.
2

D
e
t
e
r
m
i
n
e

S
p
ec
i
a
l

S
ec
t
i
o
n

P
r
op
e
r
t
i
es


For

torsion

d
esign,

special

section

properties,

s
u
ch

as

A

,

A



,

A
,

p



,

and

p



are

cp

oh

o

cp

h

calculated. These properties are descri
b
ed in the fol
lo
wing (ACI 2.1).


=

Area enclosed by

outside

peri
m
eter of

concrete cross
-
section


=

Area enclo
se
d by centerline of the outermost closed
t
ransverse
torsional reinforcement


=

Gross ar
e
a e
n
closed by shear flow path


=

Outside peri
m
eter of concrete cross
-
section


=

Perime
t
e
r of centerline of outermost closed transver
s
e
torsional reinforcement


In

calculating

the

section

properties

invol
ving

reinforcement,

such

as

A

,

A
,

oh

o

and

p

,

it

is

a
ssumed

that

t
he

distance

bet
w
e
e
n

the

centerline

of

the

outermost

closed stirrup and the oute
rm
ost concrete surface is 1.75 inches. T
h
is is
equivalent

to

1.5

inches

clear

c
o
ver

and

a

#4

stirrup.

For

torsion

design

of

flanged beam

sectio
n
s,

it

is

assumed

that

plac
i
ng

torsion

reinforcement

i
n

the

flange

area

is

inefficient.

With

this

assu
m
ption,

the

flange

is

ignored

for

torsion

rein
forcement

ca
l
culation.

However,

the

fl
a
nge

is

considered

during

T

c
alculation.

With

this

assu
m
ption,

t
he

special

properties

for

a

rectangular

beam

section

are
given as:


=

bh

(ACI 11.6.1,

2.1)


=

(
b


2
c
)(
h



2
c
)

(ACI 11.6.3
.1
, 2.1,

R11.
6
.
3
.6(b))


=

0.85 A

(ACI 11.6.3
.6
, 2.1)


=

2b +

2h

(ACI 11.6.1,

2.1)







B
e
a
m
D
e
s
i
g
n

1
-

2
7


A

oh

f

cr

p

u

cr

u

cr

p

h

A

oh

w

o

p

cp

p

h

w

cp

cp

pc

c



Post
-
Te
n
sioned

Concrete

Design



=

2
(
b


2
c
)
+
2
(
h


2
c
)

(ACI 11.6.3
.1
, 2.1)


where,

the

section

di
m
ensions

b
,

h
,

a
nd

c

are

shown

in

F
i
gure

1
-
3.

Si
m
ilarly,
the special s
e
ction properties for a flanged beam

section are given

as
:


A

=

b

h +
(
b



b

)
h

(ACI 11.6.1,

2.1)

cp

w

f

w

f


=

(
b



2c
)(
h



2c
)

(ACI 11.6.3
.1
, 2.1,

R11.
6
.
3
.6(b))


=

0.85 A

(ACI 11.6.3
.6
, 2.1)


=

2b

+ 2h

(ACI11.6.1,

2
.1)


=

2
(
h


2
c
) +
2
(
b



2c
)

(ACI 11.6.3
.1
, 2.1)


where

the

sect
ion

di
m
ensions

b

,

b

,

h
,

h

,

and

c

for

a

flanged

beam

are

shown

f

w

f

in

Figure

1
-
3.

Note

that

the

flange

wid
t
h

on

either

side

of

the

beam

web

is

li
m

ited to the smaller of
4h

or

(h


h

)
(ACI 13.2
.
4).

f

f



1
.
7
.
3
.
3

D
e
t
e
r
m
i
n
e

C
r
i
t
i
ca
l

To
r
s
i
o
n
C
a
p
ac
i
t
y


Th
e

critical

t
orsion

capacity,

T

,

for

which

the

torsion

in

the

section

can

be

ig
-

nored is calculated as:




2




T
cr







f

'
c

A





cp







cp

1



f

pc

4

f

'
c


(ACI 11.6.1
(
b))






where

A

and

p

are

the

area

and

perimeter

of

the

concr
ete

cros
s
-
s
ection

as

de
-

scribed

in

detail

in

the

previous

section;

f

is

the

co
n
crete

compr
e
ssive

st
r
ess

at

the

centroid

of

the

section;



is

the

strength

reducti
o
n

factor

for

t
orsion,

which

is

equal

to

0.75

by

default

(ACI

9.3.2.
3
);

and

f


pressive

stre
n
gth.



1
.
7
.
3
.
4

D
e
t
e
r
m
i
n
e

To
r
s
i
o
n

R
e
i
n
f
o
r
ce
m
e
n
t

is

the

specif
i
ed

c
oncrete

c
o
m
-


If

the

factored

torsion

T


is

less

than

the

threshold

limit,

T

,

t
o
r
sion

can

be

ly

ignored

(ACI

11
.
6
.
1).

In

that

case,

no

tors
i
on

reinforcement

is

r
equired.

Howeve
r,

if

T

exceeds

t
h
e

threshold

l
i
mit,

T

,

it

is

assumed

that

the

torsional

resistance

is

provided

by

closed

stirrups,

longit
u
di
-

nal bars, and

co
m
pression

d
i
agonal (ACI R11.6
.
3.6).



1
-

2
8

B
e
a
m
D
e
s
i
gn


b



2
c

h

d
s

h



2
c

h

b

h



2
c

b
w

b
f

b



2
c

h

h
f

h



2
c

h

b

h



2
c

b
w

b
f

b



2
c

h

d
s

h



2
c

h

b

h



2
c

b
w

b
f

b



2
c

h

d
s

h



2
c

h

b

h



2
c

b
w

b
f

>
T

t

t

b

l

u

cr





Chapter 1

-

D
e
s
i
g
n
f
o
r

AC
I

3
18
-
08



If
T

t
he required closed stirrup area per unit spacing,
A

/s
, is

calcula
t
e
d as:


A
t



T
u

tan



s



2

A
o

f

yt


(ACI 11.6.3
.6
)


and the requi
r
ed longi
t
udi
n
al reinforcement is calculated as:



A
l



T
u

p
h



2

A
o

f

y

tan




(ACI 11.6.3
.7
, 11
.
6.3
.
6)



c

b



2
c

c


c

h
f

b
f



c





h



2
c

h

h

h



2
c






c



b



b
w



2
c
b
w



c



Clos
e
d

Stirr
u
p

i
n

Re
ctang
u
l
ar

Beam

Clos
e
d

Stirr
u
p

i
n

T
-
Be
am

S
e
c
t
io
n


Figure

1
-
3

Closed

stirrup

and

section

dimensions

f
o
r

torsion

design




where, the m
i
nimum value of

A

/s
is taken as:


A
t

25

w


(ACI 11.6.5
.3
)

s

f

yt


and the
m
ini
m
um

value of

A

is taken as follows:









B
e
a
m
D
e
s
i
g
n

1
-

2
9


u

u

u

cr

l

t

l

f

w

u

u

w

v



Post
-
Te
n
sioned

Concrete

Design



A



5



f

c


A
cp





A
t



p



f

yt




(ACI 11.6.5
.3
)





f

y



s







h







y




In

the

preced
i
ng

expressions,



is

taken

as

45

degrees

for

prestressed

m
e
m
bers
with an effe
c
tive prestress

force less t
h
an 40 percent of the tensile strength of
the longi
t
ud
i
nal reinforce
m
e
nt; otherwise



is taken

as 37.5 degr
e
e
s.


An

upper

li
m
it

of

the

combination

of

V

and

T

that

can

be

c
a
rri
e
d

by

the

sec
-

tion is also c
h
ecked using
t
he equation:


2



V
u



2



T
u

p
h





V
c
















b

d

1.7

A
2










8

b

d

f

c




(ACI 11.6.3
.1
)




w





oh







w




For

rectangular

sections,

b

is

r
e
placed

with

b
.

If

the

combination

of

V

and

T

exceeds

this

l
imit,

a

failure

me
s
sage

is

d
ecla
r
ed.

In

that

ca
s
e,

the

c
oncrete

se
c
-

tion should be increa
s
ed in size.


When

to
rsional

reinforcement

is

required

(
T

>

T

),

the

area

of

transverse

closed

stirrups

and

the

area

of

regular

shear

stirrups

must

satisfy

t
he

following
limit.




A

A






f
c


5
0
b







v



2

t





ma
x

0.7
5


b
w

,



(ACI 11.6.5
.2
)



s

s






f

yt

f

y





If

this

equat
i
on

is

not

satisfied

with

the

originally

calculated

A

/
s

and

A

/
s
,

A

/
s

v

t

v

is

increased

t
o

satisfy

this

condition.

In

that

case,

A

the ACI Section
1
1.5
.
6.3
in
dependently.

/
s

does

not

n
e
ed

to

satisfy


The

max
i
mum

o
f

all

of

t
he

ca
l
culated

A

and

A

/
s

values

obtained

from

each

load combination is report
e
d along with
t
he controlling combination.


The

beam torsion

reinfor
c
e
m
ent

require
m
ents

considered

are based

purely

on

strength

considerat
i
ons.

Any

m
in
im
um

stirrup

re
quire
me
nts and

longitud
i
nal

reinforcement

requirements

to

satisfy

spacing

considerations

m
ust be investigated inde
p
endently by

the user.








1
-

3
0

B
e
a
m
D
e
s
i
gn




Chapter 1

-

D
e
s
i
g
n
f
o
r

AC
I

3
18
-
08




1
.
8

S
l
a
b
D
es
i
gn


Si
m
ilar to

conventi
o
nal design, th
e
slab

design procedure involv
e
s defining

sets

of

strips

in

two

m
utually

perpendicular

directions.

The

locations of

the

strips
are

usually

governed

by

t
he

locations

of

the

slab

supports.

T
he moments

for

a

particular

s
trip

are

r
ecovered

from

the

analysis

a
n
d

a

flexural design

is

completed

using

the

ultima
t
e

strength

d
e
sign

method

(ACI

318
-
08) for

prestressed

reinforced

concrete

as

d
e
scribed

in

the

following sections.

To learn
more about
t
he design strips, refer to the section entitled " Design
Fea
t
u
res" in the
Key Features and Terminolo
g
y

ma
nual.



1
.
8
.
1

D
es
i
g
n

f
o
r

F
l
e
x
u
r
e


The process

desig
n
s

the

slab

on a

strip
-
by
-
s
t
rip

basis.

T
h
e

m
o
me
nts

used

for

the
design of the slab elements are the nodal reactive m
o
ments, which

are obtained
by multip
l
ying th
e slab e
l
ement stiffness matr
i
ces by the element nodal dis
-

placement

v
e
ctors.

Those

moments

wi
l
l

always

b
e

i
n

static

equi
l
ibrium

with
the applied loads, irrespect
i
ve of the refinement of the finite eleme
n
t mesh.


The design of the slab reinforcement
f
or a

particular strip is completed at spe
-

cific locations along the length of
t
he strip. Those l
o
cations correspond to
t
he
element boundaries. Controlling reinfo
r
cement is

computed on either side
o
f
those ele
m
e
n
t boundaries. The slab flexu
r
al design procedure

for each load
co
m
bination
i
nvolves the
f
ollowing:




Dete
r
m
ine f
a
ctored mom
e
nts for each slab strip.




Dete
r
m
ine the capacity of post
-
tensioned sections.




Design flexural reinforcement for the strip.


These

thr
e
e

s
teps

a
r
e

de
sc
ribed

in

the

s
ubsection

t
hat

follow

and

are

r
e
peated
for every load combination. The maxim
u
m

reinforcement

calculated

for

the

top
and

bottom

of

the

slab

wi
t
hin

each

d
e
s
i
gn

strip,

along

with

the

corresponding

controlling load combination, is obtained and report
e
d.








S
l
a
b
D
e
s
i
g
n

1
-

3
1


A

y

y

A

f



Post
-
Te
n
sioned

Concrete

Design



1
.
8
.
1
.
1

D
e
t
e
r
m
i
n
e

F
ac
t
o
r
e
d
M
o
m
e
n
t
s

f
o
r

t
h
e

S
t
r
i
p


For each el
e
m
ent within the design strip, for each load combination, the pro
cess

ca
l
culates the nodal reactive m
o
ments. The
nodal moments are then

added
to get
t
he strip
m
oments.



1
.
8
.
1
.
2

D
e
t
e
r
m
i
n
e

C
a
p
ac
i
t
y

o
f

P
o
s
t
-
T
e
n
s
i
on
e
d

S
ec
t
i
on
s


Calculation

of

the

post
-
te
n
sioned

secti
o
n capacity

i
s

identical

to

t
hat

described
earlier for rectangular beam

sections.



1
.
8
.
1
.
3

D
es
i
g
n
F
l
ex
u
r
a
l

R
e
i
n
f
o
r
ce
m
e
n
t

f
o
r

t
h
e

S
t
r
i
p


The reinforce
m
ent c
o
m
putation for each slab design strip, given the bending
m
o
m
ent,

is

identical

to

t
he

design

of rectangular

beam

sections

described
earlier (or to

the flanged beam if the s
l
ab is ribbed). In so
m
e cases, at a given

design

section

in

a

desi
g
n

strip,

there

m
ay

be

two

or

m
ore

slab

pr
o
perties
across

the

width of

the design

strip. In that

case,

the

design

the tributary width
a
ssociated wi
t
h

each of the slab properties sepa
r
a
te
l
y using

its

tributary

bending
moment.

The

re
inforcem
e
nt

obtained

for

each

of

the tributary

wid
t
hs

is

summed

to

obtain

the

total

reinforcement

for

the

full

width of the design strip
at the considered design section. This
m
ethod is used when drop

panels

are

included. Where

openings

occur,

t
h
e

slab

wid
th

is

adjusted

ac
cordingly.



1
.
8
.
1
.
3
.
1
M
i
n
i
m
u
m

a
n
d

M
ax
i
m
u
m

S
l
a
b

R
e
i
n
f
o
r
ce
m
e
n
t

The minimum flexural t
e
nsion reinforcement required for each direction of a
slab is given

by the fol
l
owing li
m
its (ACI 7.12
.
2):



s,
m
i
n

= 0
.
00
2
0
bh

for
f

= 40

ksi or
5
0 ksi

(
ACI 7.12.2
.1
(a))



s,
m
i
n

= 0
.
00
1
8
bh

for
f

= 60

ksi

(ACI 7.12.2
.1
(b))


0.0018



60000

A

=

bh

for
f



> 60 ksi

(ACI 7.12.2
.1
(c))

s,
m
i
n

y

y








1
-

3
2

S
l
a
b
D
e
s
i
gn


t

f

c

cf

c

y

y



Chapter 1

-

D
e
s
i
g
n
f
o
r

AC
I

3
18
-
08



Reinforcement

is

not

required

in

po
sitive

moment

areas

where

f

,

the

extr
e
m
e
fiber

stress

in

tension

in

the

precompres
s
ed

tensile

zone

at

service

loads

(after

all prestre
s
s losses occurs)

does not exceed

2

'

(ACI 18.
9
.3.1).


In

positive

moment

areas

where

the

computed

tensile

stress

in

the

concrete

at


service

loads

exceeds

2


computed as:

f

'

,

the

m
ini
m
um

area

of

bon
d
ed

reinforce
m
ent

is



A
s

,
min




N

c

0.5

f

y


, where
f




60 ksi

(ACI 18.9.3
.2
)


In

negative

m
o
m
ent

areas

at

colu
m
n

supports,

the

m
ini
m
um

area

of

bonded
reinforcement in the top of slab in each direction is computed as:


A
s

,
min



0.0075

A
cf

(ACI 18.3.9
.3
)


where

A

is

the

larger

gross

cross
-
sectional

area

of

t
he

slab
-
be
a
m

strip

in

the

two ortho
go
nal equivalent fra
me
s inter
s
ecting a column in a two
-
way slab
system.


When

spacing

of

tendons exceed

54

inches,

additional

bonded

s
hrinkage

and
te
m
perature

reinforcement

(as

c
o
m
puted

above,

ACI

7.12
.
2.1)

is required

be
-

tween

the

te
n
dons

at

slab

edges,

extending

from the

slab

edge

for

a

dista
n
ce
equal to th
e
te
ndon spaci
n
g (ACI 7.12
.3
.3)


In addition, an upper li
m
it on both the tension reinforcement and

co
m
pression
reinforcement

has

been

imposed

to

be

0
.
04

times

the

gross

cross
-
s
ectional

area.

Note that the requirements

when
f

> 60

ksi currently

are not h
and
l
ed.



1
.
8
.
2

C
h
e
c
k

f
o
r

P
unch
i
n
g

S
h
ea
r


The

algorithm

for

check
i
ng

punching

shear

is

deta
i
led

in

the

section

entitled

“Slab

Punching

Shear

Check”

in

the

Key

Features

and

Terminology

ma
nual.
Only the code specific it
e
ms a
r
e d
e
scribed in the fo
ll
owing sec
tio
n
s.









S
l
a
b
D
e
s
i
g
n

1
-

3
3


M

f

u



Post
-
Te
n
sioned

Concrete

Design



1
.
8
.
2
.
1

C
r
i
t
i
ca
l

S
ec
t
i
o
n

f
o
r

P
un
c
h
i
n
g

S
h
ea
r


The

punch
i
ng

shear

is

checked

on

a

crit
i
cal

section

at

a

distance

of

d
/
2

from

the
face

of

the

support

(ACI

11.11
.
1.2).

For rectangular

colu
m
ns

and

concentrated
loads,

the

critical

area

is

taken

as

a

r
e
ctangular

area

with

the

si
d
es

parallel

to
the sides of the colu
m
ns or the point l
o
ads (ACI 11.11.1
.
3). Fi
g
ure 1
-
4 shows
the auto pu
n
ching peri
m
e
ters conside
r
ed for the various colu
m
n shape
s. The

colu
m
n location (i.e., int
e
rior, edge, corner) and the punchi
n
g peri
m
eter

m
a
y be
overwri
t
ten using the

Punching

C
h
eck Overwrites.


d

2












Interior

Column

d

2

d

2







E
dg
e

C
olu
mn


E
d
ge

C
ol
u
mn



d

2

d

2

d

2








C
ircular

C
o
l
umn

T
-
Sh
a
pe

C
o
lumn

L
-
Sha
p
e

Colu
m
n


Figure

1
-
4

Punching

Shear

Perimeters



1
.
8
.
2
.
2

T
r
a
n
s
f
e
r

o
f

U
nb
a
l
a
n
ce
d

M
o
m
e
n
t


The

fraction

of

unbalanced

m
o
me
nt

tra
n
sferred

by

flexure

is

taken

to

be



and

the

fraction

of

unbalanced

m
oment

transferred

by

eccentrici
ty

of

shear

is
taken to be



M

.

v

u


1



f





(ACI 13.5.3
.2
)

1




2

3


b
1

b
2





1
-

3
4

S
l
a
b
D
e
s
i
gn


v





x





y



v

f

vx

vy

y

vx

vy

1

2

x



Chapter 1

-

D
e
s
i
g
n
f
o
r

AC
I

3
18
-
08



= 1 −



(ACI 13.5.3
.1
)


For

flat

plates,



is

determined

from

the

following

equations

taken

f
rom

ACI

421
.
2R
-
07

[
ACI

2007]

S
eismic

Desi
g
n

of

Punch
in
g

Shear

Rei
n
forcement

in

Flat Plates
.


For interior
c
olumns,







1



1




(ACI 421.2
C
-
11)

vx






v
y



1



1





2

3



1

1





2

3


l
y

l
x




l
x

l
y





(ACI 421.2
C
-
12)


For edge col
um
ns,


= same as

for interior c
o
lu
m
ns

(ACI 421.2
C
-
13)





vy




1



1

1




2

3



l

x



l

y



0.2


(ACI 421.2
C
-
14)


= 0 when
l

/
l



0.2


For corner columns,


= 0.4

(ACI 421.2
C
-
15)


= same as

for edge col
um
ns

(ACI 421.2
C
-
16)


where

b

is

the

width

of

the

critical

section

me
asured

in

the

direction

of

t
h
e

span

and

b

is

the

width

of

the

critical

section

measured

in

the

direction

per
-

pendicular

to

the

span.

T
he

values

l

and

l

are

the

projections

of

the

shea
r
-

critical
s
ecti
o
n on
to its principal axes,

x

and
y
, respe
c
tively.



1
.
8
.
2
.
3

D
e
t
e
r
m
i
n
e

C
on
c
r
e
t
e

C
a
p
ac
i
t
y


The

concrete

punchi
n
g

sh
e
a
r

stress

capacity

of

a

two
-
way

prestressed

section

is
taken as:






S
l
a
b
D
e
s
i
g
n

1
-

3
5


pc

pc

p

s



s



30

c

p

p

c

p

p

b

pc

pc



Post
-
Te
n
sioned

Concrete

Design



v









f

'
c




0.3

f

pc




v


(ACI 11.11.
2.
2)








d










mi
n


3.
5
,





s




1.
5




(ACI 11.11.
2.
2)

p













o






where,



is

t
he

factor

us
e
d

to

compute

v

in

prestres
s
ed

slab;

b

is

the

perime
t
er

p

of

the

critical

section;

f

c

is

the

average

value

of

f

o

in

the

two

dir
e
ctions;

v

is

the

vertical

component

of

all

effective

prestress

stresses

crossi
n
g

the

critical
section; and


is a s
c
ale
f
actor based on the location of the critical section.



40

fo
r

i
n
t
e
r
ior

c
olumn
s,









fo
r

e
dge

c
o
lumns,
a
nd


20

fo
r

c
o
rne
r

co
l
um
n
s
.



(ACI 11.11.
2.
1)


The

concrete

capacity

v

computed

from

ACI

11.12.2
.
2

is

permitted

only

when

the following conditions are satisf
i
ed:




The

column

is

farther

than

four

times

the

slab

thickness

a
way

from

a
ny

dis
-

continu
o
us slab edges.




The value of

f

'

c

is taken no g
r
eater than 70 psi.




In each direc
t
ion, the

value of

f

is with
i
n

the range:


125



f



500 p
s
i


In

thin

slabs,

the

slope

of

t
he

tendon

p
r
ofile

is

hard

t
o

control

and

special

c
are
should

be

exercised

in

computing

v
.

I
n

case

of

uncertainty

between

the

design

and as
-
built profile, a reduced or zero value for

v

sho
u
ld be used.


If the preceding three conditions are not

satisfied, the concrete punching shear
stress

cap
a
c
ity of
a two
-
way prestressed

section is ta
k
e
n as the mi
n
imum of the
following
t
h
r
ee limits:












1
-

3
6

S
l
a
b
D
e
s
i
gn


c

s

0



Chapter 1

-

D
e
s
i
g
n
f
o
r

AC
I

3
18
-
08








4








2









f

'
c






c












s

d





c



mi
n





2






b

f

'
c

(ACI 1
1.11.
2.
1)






c








4







f

'
c


where,


is
t
he ratio of the

maximum to

the minimum dimensions of the criti
-

cal s
e
ction,
b

is the perimeter of the critical section, and


is a sca
l
e factor

based on the
l
ocation of the

critical sectio
n (ACI 11.
1
2.2.1).


A limit is imposed on
t
he value of

f

'

c

as:


f

'

c



100

(ACI 11.1.2)



1
.
8
.
2
.
4

D
e
t
e
r
m
i
n
e

C
a
p
ac
i
t
y

R
a
t
i
o


Given the p
u
nching shear force and the fractions of
m
o
m
ents transferred by ec
-

centricity of

shear

about the two axes,
the shear

stress is

co
m
puted ass
u
m
ing
linear

variation

along

the perimeter

of

the

critical

section.

The

ratio

of

the
ma
xi
m
u
m shear stress a
n
d the concrete punchi
n
g

shear stress capacity is re
-

ported as the punchi
n
g

shear capacity ratio .



1
.
8
.
3

D
es
i
g
n

P
u
nch
i
n
g

S
h
ea
r

R
e
i
n
f
o
r
ce
m
e
n
t


The

use

of

s
hear

studs

as

shear

reinfo
r
cement

in

sl
a
bs

is

permit
t
ed,

provided
that the ef
f
e
c
tive depth of the slab is

greater

than or

equal to 6 in
c
hes, and not
less than 16 ti
me
s the shear reinforcement bar diameter (ACI 11.11.
3). If the
slab thickness does not
m
eet these

re
q
uire
m
ents, the punching
s
hear reinforce
-

m
ent is not
d
esigned and
t
he slab thick
n
ess should be

increased by
t
he user.


The

algorithm

for

designing

the

required

punching shear

reinforce
m
ent

is

used
when the p
unching shear capacity ratio exceeds unity. The

Critical Section for
Punching

S
h
ear

and

Tra
n
sfer

of

Unba
l
anced

M
oment

as

described

in

t
he

ear
-





S
l
a
b
D
e
s
i
g
n

1
-

3
7


V

o

V

o

o

= 6

'

= 8

'



Post
-
Te
n
sioned

Concrete

Design



lier

sections

re
m
ain

unchanged.

The

d
e
s
i
gn

of

p
u
nc
h
ing

shear

reinforcement

is
carried out as described in the subsections that follo
w
.



1
.
8
.
3
.
1

D
e
t
e
r
m
i
n
e

C
on
c
r
e
t
e

S
h
ea
r

C
a
p
ac
i
t
y


The

concrete

punch
i
ng

s
h
ear

stress

ca
p
acity

of

a

two
-
way

prestressed

section
with punch
in
g

shear reinforcement is as previousl
y

deter
m
ined, but

li
m
ited to:


v
c





2



v
c





3


f

'
c


f

'
c

for shear lin
k
s

(ACI 11.11.
3.
1)


for shear studs

(ACI 11.11.
5.
1)



1
.
8
.
3
.
2

D
e
t
e
r
m
i
n
e

R
e
qu
i
r
e
d

S
h
ea
r

R
e
i
n
f
o
r
ce
m
e
n
t


The shear force is limi
t
ed to a maximum of:



max

f

c

b

d

fo
r

shear links

(ACI 11.11.
3.
2)



max

f

c

b

d

fo
r shear studs

(ACI 11.11.
5.
1)


Given
V
,
V

, and
V

, the required shear

reinforcem
e
nt is ca
l
culat
e
d as follows,

u

c

max

where,


,

the

strength red
u
ction factor, is 0.75 (ACI
9
.3.2
.
3).


A


V
u




V
c



v



s



f

y
s

d


A

f
'


(ACI 11.4.7
.1
, 11
.
4.7
.
2)




v




2

c

b

for shear studs

s

f

y




If
V

>

V

, a failure condition is

declared.

(ACI 11.11.
3.
2)

u




If

V

max


exceeds

the

maximum

permitted

value

of


V



,

the

concrete

section

u

sh
ould

be increased in size.

max



1
.
8
.
3
.
3

D
e
t
e
r
m
i
n
e

R
e
i
n
f
o
r
ce
m
e
n
t

A
rr
a
ng
e
m
e
n
t


Punching

sh
e
a
r

reinforcement

in

the

vicinity

of

rectangular

colu
m
ns

should

be
arranged

on

peripheral

li
n
es,

i
.
e.,

lines

runni
n
g

para
l
lel

to

and

at

constant

dis
-



1
-

3
8

S
l
a
b
D
e
s
i
gn



Inter
i
or


E
d
g
e


Co
r
ner


peripheral

o
f









y

x

x




Chapter 1

-

D
e
s
i
g
n
f
o
r

AC
I

3
18
-
08



tances

from the

sides

of

t
h
e

colu
m
n.

Figure

1
-
5

shows

a

typical

arrange
m
ent

o
f
shear reinforcement in the vicinity of

a rectangular interior, edge, and corner
column.


The distance

between the

column face

and

the first line of shear reinforcement
shall

not

exceed

d/2

(ACI

R11.3.3, 11.11.5.2).

The

spacing

bet
w
een

adja
c
ent
shear reinfor
c
ement in the first line of shear reinforcement shall
n
ot exceed 2
d
me
a
sured in a direction parallel to the column face (
A
CI 11.
11.3.3).


Punching

sh
e
a
r

reinforce
m
ent

is

m
ost

eff
e
ctive

n
e
ar

column

c
orners

where
there

are

concentrations

of

shear

str
e
s
s
.

Therefore,

the

minimum

number

of
lines

of

shear

reinforcement

is

4,

6,

and

8,

for

cor
n
er,

edge,

and

interior

col
-

u
m
ns respectivel
y.


T
y
p
i
cal

Studrail
(only

first

and

la
s
t
stu
d
s

sh
o
w
n
)

O
u
term
o
st
peripheral

l
i
ne
of

stu
d
s

Outermost
peripheral

l
i
ne
o
f

s
tu
d
s



F
ree
edge

d

2

d

2

I
y

g
y

g
x

s
0


s
0


s
0

I
y

x

I

x


Critical
section
c
e
nt
r
o
id

g
x


d

2




Fr
e
e

e
d
g
e


y

x

y

F
ree

edge


I
x

I
x

I
x


Critical

section
centroid


Inter
i
or

Column


E
d
g
e

Column


Co
r
ner

Col
u
mn


F
i
gure

1
-
5

T
y
p
i
cal

a
rr
a
ng
e
ment

o
f

sh
e
a
r

s
t
uds

a
n
d

cr
i
ti
cal

sec
ti
o
ns

ou
t
s
i
d
e
shea
r
-
rei
n
for
c
ed
z
o
ne



1
.
8
.
3
.
4

D
e
t
e
r
m
i
n
e

R
e
i
n
f
o
r
ce
m
e
n
t

D
i
a
m
e
t
e
r
,

H
e
i
gh
t
,

a
n
d

S
p
a
c
i
ng


The

punchi
n
g

shear

reinforcement

is

m
o
st

effective

when

the

anchorage

is
close

to

the

top

and

bott
o
m

surfaces

of

the

slab.

T
h
e

cover

of

anchors

shou
l
d
not

be

less

t
h
an

the

m
inimum cover

specified

in

ACI

7.7 plus

half of

the

di
-

ame
t
er of the flexural

reinforcement.


Punching

shear

reinforcement

in

the

fo
r
m

of

shear

studs

is

generally

available
in 3/8
-
,

1/2
-
,

5/8
-
, and

3/4
-
inch dia
m
eters.





S
l
a
b
D
e
s
i
g
n

1
-

3
9



o

o

s

o



When

specifying

shear

studs,

the

distance,

s
,

bet
w
e
e
n

the

column

face

and

t
h
e
first

pe
ripheral

line

of

shear

studs

should

not

be

s
m
aller

than

0.3
5
d
.

The

limits

of
s

and the spacing,
s
, between the per
i
pheral lines are specified
a
s:




0.5
d

(ACI 11.11.
5.
2)




0
.
75
d
s







0
.
5
0
d


g



2
d

(ACI
11.11.
5.
3)
f
o
r

f
o
r




u



6





u



6




f

'
c

f

'
c



(ACI
11.11.
5.
2)


o

s

o





The limits of
s

and the spacing,
s
, between the links are specified
a
s:




0.5
d

(ACI 11.11.
3
)


s



0
.
5
0
d

(ACI 11.11.
3
)








































1
-

4
0

S
l
a
b
D
e
s
i
gn