example 9.2 – part iv pci bridge design manual bulb “t” (bt-72) three spans, composite deck...
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EXAMPLE 9.2 – Part IVPCI Bridge Design Manual
EXAMPLE 9.2 – Part IVPCI Bridge Design Manual
BULB “T” (BT-72)
THREE SPANS, COMPOSITE DECK
LRFD SPECIFICATIONS
Materials copyrighted by Precast/Prestressed Concrete Institute, 2011. All rights reserved. Unauthorized duplication of the material or presentation
prohibited.
UNFACTORED SHEARS AND MOMENTS LIVE LOAD ENVELOPE INCLUDES IM FACTOR
UNFACTORED SHEARS AND MOMENTS LIVE LOAD ENVELOPE INCLUDES IM FACTOR
Midspan Values, symmetrical
STRENGTH LIMIT STATEPositive Moment Zones
STRENGTH LIMIT STATEPositive Moment Zones
Since the dead loads and live loads produce stresses of the same sign, use maximum load factors.
Find Mu at midspan.
Mu = 1.25 DC + 1.5 DW + 1.75 (LL+IM)
= 1.25(1391 + 2127 + 73) + 1.5 (128) + 1.75 (2115)
= 8382 k-ft.
This is the applied FACTORED load at midspan of a center span interior beam.
Consider the section at strength limit state. Assume the stress block is entirely within the deck
slab (a< 7.5”). If this is true, treat as a rectangular section.
STRENGTH LIMIT STATE
Positive Moment Zones
Because the stress block is assumed to be in the flange (slab), the properties of the SLAB
concrete are used. Therefore, there is no need to “transform” the slab concrete to beam
concrete. The actual effective width of 144” is used, not the transformed width.
STRENGTH LIMIT STATE
Positive Moment Zones
If the stress block had fallen into the precast beam:
The section would be treated as a “T” beam.
The section would be assumed to be made of “beam” concrete, so the beam concrete properties
would be used. For this reason, the transformed slab and haunch widths would also be used.
STRENGTH LIMIT STATE
Positive Moment Zones
Equilibrium:
Compression = Tension
0.85 fc’ b a = Aps fps
STRENGTH LIMIT STATE
Positive Moment Zones
ps pup
c ps ps
c ps pup
ps pu
puc ps
p
cf f k
d
f ba A f
a c
cf b c A f k
d
A fc
ff b kA
d
1
1
1
1
0.85 '
0.85 ' 1
0.85 '
The value of fps can be found from:
Then:
(Eq’n 5.7.3.1.1-1)
STRENGTH LIMIT STATE
Positive Moment Zones
ps pu
puc ps
p
A fc
ff b kA
d
10.85 '
c = depth of neutral axis
b = width of compression block (flange in this case)
Aps = area of TENSILE prestressing steel
dp = depth to centroid of tensile prestressing steel
k = a constant for the prestressing steel
k = 0.28 for low relaxation steel
Reminder: When a < ts , the stress block is in the slab. Use the effective width of the slab NOT the
transformed width and use 1 of the slab concrete.
STRENGTH LIMIT STATE
Positive Moment Zones
If there is mild (nonprestressed) tensile steel, As, and mild compression steel, As’, and
both yield (strength of fy ), the equation for c becomes:
ps pu s y s y
puc w ps
p
A f A f A fc
ff b kA
d
1
' '
0.85 '
c s y s y ps pup
cf b c A f A f A f k
d
1.85 ' ' ' 1
Rectangular section assumed. (Eq’n 5.7.3.1.1-4)
STRENGTH LIMIT STATE
Positive Moment Zones
Reminder of previously calculated values:
Aps = 44 strand(0.153 in2) = 6.72 in2
fpu = 270 ksi
dp = beam depth+haunch+slab-ybs
= 72” + 0.5” + 7.5” – 5.82” = 74.18”
ybs = distance from bottom of beam to centroid of prestressing steel.
b = 144”
fc’ = 4.0 ksi
1 = 0.85 (for 4 ksi concrete)
k = 0.28 for low relaxation strand
STRENGTH LIMIT STATE
Positive Moment Zones
1
2
2
1
0.85 '
6.732 270
2700.85(4 )(0.85)(144") 0.28(6.732 )
74.18"4.30"
0.85(4.30") 3.65" 7.5
ps pu
puc ps
p
A fc
ff b kA
d
in ksi
ksiksi in
c
a c in
OK – Stress Block in Flange – Rect. Section
STRENGTH LIMIT STATE
Positive Moment Zones
Find the approximate stress in the prestressing steel:
1
4.30(270 ) 1 0.28
74.18
265.6
ps pup
ps
ps
cf f k
d
f ksi
f ksi
This equation gives the approximate stress in the prestressing steel. A more accurate value can be
found from strain compatibility (see the PCI Design Handbook)
STRENGTH LIMIT STATE
Positive Moment Zones
From Equilibrium:
2
* ( / 2)
* ( / 2)
2
3.65"(6.732 )(265.6 ) 74.18"
2
129372 10780
n ps ps p
n
n
Moment Compression d a
Tension d a
aM A f d
M in ksi
M k in k ft
STRENGTH LIMIT STATE
Positive Moment Zones
22)('85.0
2'''
22
1f
fwcsys
sysppspsn
hahbbf
adfA
adfA
adfAM
The complete moment equation, assuming prestressed and nonprestressed tensile steel, compression steel and a flanged
section (T beam) is given by Eq’n 5.7.3.2.2-1 :
ds is the depth from the compression fiber to the tensile mild steel and ds’ is the depth from the compression fiber to
the compression mild steel, bw is web width and hf is flange thickness.
In this design example: As = As’= 0 and b = bw , so the equation simplifies to Mn = Apsfps(dp – a/2)
STRENGTH LIMIT STATE
Positive Moment Zones
1.0
8382
1.0(10780 )
u r n
u
n
M M M
for precast
M k ft
k ft M
OK for Strength Limit State in (+) Moment Zones IF the section is tension
controlled.
STRENGTH LIMIT STATE
Positive Moment Zones
STRENGTH LIMIT STATENegative Moment Zones
STRENGTH LIMIT STATENegative Moment Zones
Since the dead loads and live loads produce stresses of the same sign, use maximum load factors.
Also, only the loads carried as by the continuous span cause negative moment. Thus, the only DC
is the barrier:
Mu = 1.25 DC + 1.5 DW + 1.75 (LL+IM)
= 1.25(-197) + 1.5 (-345) + 1.75 (-2328)
= -4837 k-ft.
This is the applied FACTORED load over the pier of a center span interior beam.
In the negative moment area, the bottom of the bottom flange is the compression area (use BT72
concrete properties). The tensile steel is placed in the deck.
STRENGTH LIMIT STATE
Negative Moment Zones
Consider the section at strength limit state. Assume the stress block is entirely within the bottom
flange (a< 6”). If this is true, treat as a rectangular section.
STRENGTH LIMIT STATE
Negative Moment Zones
The general equation for equilibrium is:
ps pu s y s y
puc ps
p
A f A f A fc
ff b kA
d
1
' '
0.85 '
c s y s y ps pup
cf b c A f A f A f k
d
1.85 ' ' ' 1
STRENGTH LIMIT STATE
Negative Moment Zones
a = As fy / 0.85 fc’ b
This is the general equation with Ap = As’ = 0 and
a= 1c.
STRENGTH LIMIT STATE
Negative Moment Zones
n ps ps p s y s
fs y s c w f
a aM A f d A f d
ha aA f d f b b h
1
2 2
' ' ' 0.85 '( )2 2 2
Once again, the complete moment equation, assuming prestressed and nonprestressed tensile
steel, compression steel and a flanged section (T beam) is given by Eq’n 5.7.3.2.2-1:
In the negative moment case, Ap = As‘= 0 and b = bw , so the equation simplifies to:
Mn = Asfs(ds – a/2)
STRENGTH LIMIT STATE
Negative Moment Zones
Mn = As fy (ds – a/2)
In this equation, As and a are unknown.
STRENGTH LIMIT STATE
Negative Moment Zones
Although As and “a” are unknown, it is possible to estimate “a”. Since a is usually an
order of magnitude smaller than d, even a gross error in “a” leads to a small error in the
moment arm,
ds -a/2, and a reasonably accurate estimate of As.
STRENGTH LIMIT STATE
Negative Moment Zones
,min
4837
48375374
0.9
u
un imum
M k ft
M k ftM k ft
= 0.9 for reinforced concrete in flexure IF tension controlled.
The bottom flange is 6” high. We want to keep the assumption of a rectangular section and it
is better to overestimate “a” as this will underestimate the moment arm.
Assume a = 6”
STRENGTH LIMIT STATE
Negative Moment Zones
n s y s
s
s
aM A f d
k ft A ksi
A in
2
2
6"5374 (12) (60 ) 76.25"
2
14.67
If the steel centroid is in the center of the slab:
ds = 72” + 0.5” + 7.5”/2 = 76.25”
STRENGTH LIMIT STATE
Negative Moment Zones
The bars should be placed within the lesser of:
The effective flange width = 144”
1/10 of the average length of adjacent spans=
[(119+120)/2](12)/10 = 143”
Note that if the 1/10 span controls, additional, nonstructural bar is required in the areas outside of 1/10 span width
but within the effective width. The amount of bar required is 0.4% of the area outside of the width determined by
1/10 span.
In this case, the two are practically equal. It would be impractical to supply additional steel in the extra 1/2 inch on
side with the required amount of steel, which would be 0.03 in2
.
STRENGTH LIMIT STATE
Negative Moment Zones
The bars are be placed within the effective width of 144”. The bridge design handbook suggests:
#5 @ 12” Top mat = 9 bars X 0.31 in2
= 2.79 in2
#4 @ 12” Bottom mat = 9 bars X 0.22 in2
= 1.99 in2
#7 Split between top and bottom mat
= 18 bars x 0.6 in2
= 10.8 in2
Total = 15.6 in2
#7 placed between each #4 on bottom mat and between each #5 on top mat. Spacing is 8 in c/c.
Now ds = 75.6 in.
STRENGTH LIMIT STATE
Negative Moment Zones
As =15.6 in2
:
2
0.85 '
15.6 60
0.85 7 26"
6.04 6
s y
c
A fa
f b
in ksi
ksi
in in
This is close enough to 6 inch that the section can be assumed a rectangle.
STRENGTH LIMIT STATE
Negative Moment Zones
2
2
6"0.9(15.6 )(60 ) 75.6"
2
61158 5096
4837
n s y
n
n
n u
aM A f d
M in ksi
M in k ft k
M k ft M
OK, IF the section is tension controlled.
Check Maximum Moment:
STRENGTH LIMIT STATE
Negative Moment Zones
Tension controlled, compression controlled and transition sections are defined by the strain
in the extreme tension steel at Mn. Strain in the extreme tensile steel is defined in the
diagram.
This definition applies to prestressed and non-prestressed steel.
STRENGTH LIMIT STATE
TENSION CONTROLLED SECTIONTENSION CONTROLLED SECTION
• A section is TENSION CONTROLLED if the extreme steel strain > 0.005.
• This applies to prestressed and non-prestressed steel.
• If tension controlled– = 1 for prestressed– = 0.9 for non-prestressed– is interpolated for partially prestressed.
COMPRESSION CONTROLLED SECTION
COMPRESSION CONTROLLED SECTION
• A section is COMPRESSION CONTROLLED if the extreme steel strain < balanced.
• For non-prestressed steel, the limit is fy/Es
• For prestressed steel, the limit is 0.002.• If compression controlled, = 0.75 for
members with ties or spirals.
TRANSITION SECTIONTRANSITION SECTION
• A transition section is has an extreme tensile steel strain between tension and compression controlled.
• For transition sections, is interpolated based in extreme tensile steel strain.
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
1.05
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007
Extreme Steel Strain
Ph
i F
act
or
CompressionControlled Transition
Tension Controlled
Prestressed:Strain = 0.004Phi = 0.92
Prestressed
Reinforced
Definition of for prestressed steel and GR 60 non-prestressed steel.
SECTION FACTORS
Check for Tension Control:
The tension control limit can be found as:
So a section is tension controlled if:
0.003 0.005 0.003
0.375
0.375
t
t
t
d c
c
d
c
d
SECTION FACTORS
In the prestressed girder (positive moment zone), it was found that:
c = 4.30 inches dt = 72+7.5-2 = 77.5 in
c/ dt = 4.30/77.5 = 0.055 < 0.375
Tension controlled
= 1.0
SECTION FACTORS
In the reinforced (negative moment) section
c = a/1 = 6.04/.7 = 8.60”
from the calculation of Mn
Assume 2.5 inches clear cover to the #5 top mat steel:
dt = 72 + 7.5 – 2.5 – (5/8)/2=76.7 in
c/dt = 8.60 / 76.7 = 0.112 < 0.375
Tension Controlled
SECTION FACTORS
Article 5.7.3.3.2 requires:
Mn =Mr > lesser of 1.2 Mcr or 1.33Mu
Mcr = Cracking Moment
1.33Mu = 1.33 (8381 k-ft) = 11150 k-ft
MINIMUM STEEL POSITIVE MOMENT SECTION
MINIMUM STEEL POSITIVE MOMENT SECTION
bccr r pb bc d / nc
b
r c
pe pe cpb
b b
b
bc
d / nc
SM f f S M
S
f . f ' Modulus of rupture
P P ef Stress at precast tensile fiber
A S
S Section Modulus to tensile fiber noncomp.
S Section Modulus to tensile fiber comp.
M noncomp. DL mome
1
0 37
nt
Equation 5.7.3.3.2-1
MINIMUM STEEL POSITIVE MOMENT SECTION
bccr r pb bc d nc
b
r
b
pb
d nc g s
bc
cr
SM f f S M
S
f ksi ksi
S in
f ksi
M M M k ft k in
S in
M
/
3
/
3
1
0.37 7 0.980
14915
1072 1072(30.78)3.610
767 149151391 2127 3518 42216
20545
205450.980 3.610 20545 42216 1
14915
783
cr
k in k ft
M k ft
66 6530
1.2 7837
MINIMUM STEEL POSITIVE MOMENT SECTION
Since 1.2Mcr = 7837 k-ft < 1.33Mu = 11147 k-ft
1.2Mcr controls.
Mn = 10649 k-ft > 1.2Mcr = 7837 k-ft OK
MINIMUM STEEL POSITIVE MOMENT SECTION
MINIMUM STEEL NEGATIVE MOMENT SECTION
MINIMUM STEEL NEGATIVE MOMENT SECTION
bccr r pb bc d nc
b
r
d nc
bc
pb
cr
cr u
n r cr
SM f f S M
S
f ksi ksi
M
S in
f
M ksi in k in k in
M k ft M k ft k ft
M M k ft M OK
/
/
3
3
1
0.37 4 0.740
0
57307
0
0.740 57307 42407 3534
1.2 4241 1.33 1.33 4837 6433
5115 1.2
DEVELOPMENT LENGTH OF PRESTRESSING STEEL
DEVELOPMENT LENGTH OF PRESTRESSING STEEL
d ps pe b
d
d
f f d
ksi ksi in
in ft
2
3
21.6 265.6 159.2 0.5
3
127 10.6 .
The term = 1.6. (5.11.4.2)
The prestressing steel must be embedded 10.6 ft from the point of maximum stress. The point of
maximum stress is at midspan and the embedment is 59.5 ft. OK
Eq’n 5.11.4.2-1
CONTROL OF CRACKING BY DISTRIBUTED REINFORCING IN THE SLAB
CONTROL OF CRACKING BY DISTRIBUTED REINFORCING IN THE SLAB
7002e
cs s
s df
According to Article 5.7.3.4 the spacing of the mild steel reinforcement in the layer closest
to the tension face shall satisfy equation 5.7.3.4-1.
e = Exposure factor = 1.00 for Class 1 exposure 0.75 for Class 2 exposure condition
fs = Tensile stress in steel reinforcement at the service limit state, ksi
dc= Thickness of concrete cover measured from extreme tension fiber to center of the flexural reinforcement located closest therto
βs=
h= Overall height of the section, in
10.7( )
c
c
d
h d
CONTROL OF CRACKING BY DISTRIBUTED REINFORCING IN THE SLAB
To find fs, the cracked moment of inertia is needed:
2 2
290005.7
5072
5.7 15.6 89.2
s
c
s
E ksin
E ksi
nA in in
CONTROL OF CRACKING BY DISTRIBUTED REINFORCING IN THE SLAB
2
2
22 3
120 6 2 10 4.5 6 89.2
2
1 4.520 6 3 2 10 4.5 6
2 3
6 89.2 75.62
3 254.2 7441 0
23
i ii
Ax A x
in in in in in x in x
inin in in in in in
xin x in in
in x in x in
x in
Use bottom as reference point:
CONTROL OF CRACKING BY DISTRIBUTED REINFORCING IN THE SLAB
3 2
3 2
3 22
4
120 6 20 6 23 3
121 1
2 10 4.5 10 4.5 23 7.536 2
16 23 89.2 75.6 23
3
330350
cr
cr
I in in in in in in
in in in in in in
in in in in in
I in
CONTROL OF CRACKING BY DISTRIBUTED REINFORCING IN THE SLAB
Mserv = -197k-ft - 345 k-ft - 2328 k-ft
= -2870 k-ft = - 34400 k-in
If 2.5 in cover and Class 1 ( = 1)
dc = 2.94 in = 2.5 + (7/16)
ss
k in in inf ksi
in4
34400 75.6 235.7 31.3
330350
2.941 1.054
0.7 80 2.94
700(1.0)2 2.94 15.3
1.054 31.3
s
in
in in
s in inksi
CONTROL OF CRACKING BY DISTRIBUTED REINFORCING IN THE SLAB
So maximum spacing =
15. 3 inches for Class 1 ( = 1)
10.0 inches for Class 2 ( = 0.75)
Actual spacing is 8 inches c/c.
OK
CONTROL OF CRACKING BY DISTRIBUTED REINFORCING IN THE SLAB