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
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