COMPREHENSIVE RETROFIT EXAMPLE 1
MULTI-SPAN CIP REINFORCED CONCRETE BOX GIRDER BRIDGE
1. Problem Statement
Evaluate a six-span cast-in-place reinforced concrete highway overcrossing
located in southern California for seismic retrofitting. The structure is supported
on monolithic single column bents and has a single expansion joint hinge located
in span 3. The bridge was constructed in the early 1960’s and has several
obvious seismic deficiencies including substandard transverse column
reinforcement and a minimal support length at the interior expansion joint hinge.
Use the D2 method of evaluation.
Once an evaluation has been completed and seismic deficiencies identified and
quantified, develop a retrofit strategy that will result in the minimal performance
criteria being met. Design the retrofit measures necessary to implement the
selected strategy.
2. Description of As-Built Bridge
The bridge, which is located in a seismically active region of southern California,
is on a curved horizontal alignment of 600 ft radius and has variable span
lengths. It passes over a freeway and parallel surface streets. The site class is
Type D.
The cross-section of the superstructure is of constant width and consists of five
girder stems with an overhang on one side. A raised curb with emergency
sidewalk is provided on the other side. The depth of the superstructure varies
from 7’-0” to 3’-6” with the transition occurring in span 3.
Abutments are seat type supported on 45-ton piles with approach retaining walls
provided to contain approach roadway fills. They are oriented normal to the
superstructure. The superstructure is supported on elastomeric bearings with
concrete shear keys provided to restrain transverse movement. The bearing
seat is 2’-6” in width.
The internal expansion joint hinge located in span 3 consists of an 8 inch bearing
seat with embedded steel angles for bearing. Transverse concrete shear keys
are provided, but no longitudinal cable restrainers are in place.
Internal bents are single columns of circular cross section supported on pile
footings. Columns at Bents 2 and 3 are 6’-0” in diameter while the remaining
columns are 5’-0”. The main reinforcing steel is lap spliced just above the
footings, and the column transverse steel, consisting of #5 spirals with a 5” pitch,
is lap spliced periodically.
Pile footings vary in size depending on the size of the column, and lack upper
layers of reinforcing steel to resist negative bending moments. Piles have a
design capacity of 45 tons and are effectively “pinned” at the base of the footing.
The reinforcing steel from the piles extends into the footing and can resist the
seismic uplift capacity of the piles, which is assumed to be 50% of their ultimate
compressive capacity (C
u
= 2 x C
design
).
A field inspection of the bridge revealed no deterioration or modification of the
structure. Because of the age of the concrete it is assumed to have an in-situ
strength of 5500 psi. Reinforcing steel is Grade 60.
The as-built plans for the bridge are shown in Figures E1-1 through E1-3.
3. Enhanced Procedure for Method D2 Seismic Evaluation
The procedure described in the manual for Method D2 is enhanced to include
components other than the columns.
Step 1 – Strength and Deformation Capacities
a. Hinge Force and Displacement Capacity
The expansion joint hinge force and displacement capacities are calculated
based on the details of the as-built structure. The transverse force capacity is
based on shear friction in the shear keys. The total number of #5 bars crossing
the shear plane is 16 and the bars are assumed to be Grade 60 with an expected
strength of 66 ksi. The concrete crack surface over each of the two shear keys is
16 inches by 36 inches. Therefore, the shear capacity is given by
[
]
(
)
(
)
[
]
(
)
kips
502
0
60
31
.
0
16
4
.
1
2
36
16
150
.
85
.
P
f
A
cA
V
V
u
n
cv
vf
y
c
=
+
⋅
⋅
+
⋅
⋅
=
+
μ
+
φ
=
φ
=
The displacement capacity can be calculated as the seat width minus the
expansion joint gap plus 100 mm (4 inches).
inches
0
.
3
4
1
8
c
2
g
N
s
ej
h
c
=
−
−
=
−
−
=
δ
Table E1-1 - Hinge Force and Displacement Capacities
Transverse Force
Kips
(KN)
Longitudinal
Displacement - inches
(mm)
Hinge
1
502
(2232)
3.0
(102)
b. Column and Foundation Shear Capacities
In the case of column shear capacities, both the initial and final shear strength is
considered. An example of the required calculation for Bent 2 follows.
[
]
KN
1291
kips
290
240
)
7
.
68
(
875
.
)
1157
(
2
2
tan
P
2
V
KN
2514
kips
565
)
435
.
1
(
5
)
63
.
4
72
(
)
60
)(
31
(.
2
cot
s
D
f
A
2
V
435
.
1
tan
1
cot
697
.
0
2355
.
0
01
.
0
00368
.
0
64
.
0
tan
00368
.
0
)
63
.
4
72
(
5
)
31
(.
4
D
s
A
4
01
.
)
36
(
27
.
1
32
A
A
64
.
0
A
2
A
8
.
0
6
.
1
A
A
6
.
1
tan
p
yh
h
s
25
.
0
25
.
0
ss
v
2
c
sl
t
25
.
0
t
v
25
.
0
g
t
g
v
25
.
0
g
t
e
v
⇒
=
⎟
⎠
⎞
⎜
⎝
⎛
=
α
Λ
=
⇒
=
−
π
=
θ
′′
π
=
=
θ
=
θ
=
=
⎥⎦
⎤
⎢⎣
⎡
=
θ
∴
=
−
=
′′
=
ρ
=
π
⋅
=
=
ρ
⎥
⎦
⎤
⎢
⎣
⎡
ρ
ρ
=
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
ρ
ρ
=
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
ρ
Λ
ρ
=
θ
(
)
(
)
(
)
(
)
KN
782
3
kips
850
145
290
565
85
.
0
V
V
V
85
.
0
V
KN
532
6
kips
1468
872
290
565
85
.
0
V
V
V
85
.
0
V
:
by
given
is
)
V
and
(V
capacity
shear
ultimate
final
and
initial
the
Therefore,
KN
45
6
kips
145
A
f
60
.
0
V
KN
3881
kips
872
1000
)
36
(
8
.
0
5500
61
.
3
A
f
61
.
3
V
:
is
ly)
respective
V
and
(V
concrete
the
of
on
contributi
final
and
initial
The
cf
p
s
uf
ci
p
s
ui
uf
ui
e
'
ce
cf
2
e
'
ce
ci
cf
ci
⇒
=
+
+
=
+
+
=
⇒
=
+
+
=
+
+
=
⇒
=
=
⇒
=
⋅
π
⋅
⋅
=
=
The following table includes the shear capacities for all of the columns.
Table E1-2 - Column Shear Strength Capacity
Column
Column
Length
Feet (m)
Dead Load
Axial Load
Kips (KN)
Initial Shear
Strength
Kips (KN)
Final Shear
Strength
Kips (KN)
2
20.0
(6.10)
1157
(5149)
1468
(6532)
850
(3782)
3
24.5
(7.47)
1069
(4757)
1407
(6263)
789
(3512)
4
17.0
(5.18)
448
(1994)
1006
(4475)
576
(2565)
5
19.4
(5.91)
457
(2034)
996
(4431)
567
(2521)
6
21.7
(6.62)
545
(2425)
1001
(4455)
572
(2545)
Similarly, the foundation shear capacity is calculated based on the capacity of the
piles in shear plus the capacity provided by passive pressure on the face of the
pile cap. The ultimate lateral capacity of a single pile is assumed to be 40 kips
(178 KN) based on physical testing of similar piles. The shear capacities for Bent
2 and 3 are calculated as follows:
( )
(
)
KN
1068
kips
240
15
4
3
2
4
2
dW
3
h
2
W
P
)
cap
(
H
KN
4450
kips
1000
40
25
40
N
)
piles
(
H
p
c
p
c
⇒
=
⋅
+
⋅
=
′
⋅
=
=
⇒
=
=
⋅
=
The total shear capacities of all pier foundations are summarized below
Table E1-3 - Pier Foundation Capacities
Shear Capacity
Kips (KN)
Pier Foundation
Longit. Trans.
2 and 3
1240
(5518)
1240
(5518)
4, 5 and 6
832
(3702)
832
(3702)
c. Abutment Force and Displacement Capacities
Abutment force capacities are governed either by the capacity of shear keys or
the capacity of the piles and wingwalls. The shear key capacity is calculated in a
manner similar to those for the hinge and is summarized below.
i. Shear
Keys:
[
]
(
)
(
)
[
]
(
)
kips
630
60
31
.
0
16
4
.
1
2
36
30
150
.
85
.
0
P
f
A
cA
V
V
u
n
cv
vf
y
c
=
⋅
⋅
+
⋅
⋅
=
+
μ
+
φ
=
φ
=
ii.
Piles and Wingwalls
In the case of the piles the calculation for Abutment 1 is performed as follows:
kips
520
40
13
)
40
(
N
V
p
=
p
=
⋅
=
The wingwall capacity is equal to the capacity of one wingwall in shear. In this
case the wingwall is 14 feet high and 12 inches thick (d=9”). Therefore:
kips
190
074
.
0
2
9
12
14
85
.
0
f
2
Hd
V
w
c
'
=
⋅
⋅
⋅
⋅
⋅
=
⋅
φ
=
kips
710
190
520
V
V
V
abut
=
p
+
w
=
+
=
The displacement capacity in the longitudinal direction depends on the geometry
of the seat and the nominal amount of expansion (g
e
= 1”). In this case:
inches
25
4
1
30
c
2
g
N
s
e
c
abut
=
−
−
=
−
−
=
δ
Similarly for Abutment 7:
inches
25
kips
576
V
abut
abut
=
δ
=
Table E1-4 - Abutment Force and Displacement Capacities
Transverse Force
Kips (KN)
Abut
Shear
Keys
Piles and
Wingwalls
Longitudinal
Displacement -
inches (mm)
1
630
(2804)
710
(3160)
25.0
(635)
7
630
(2804)
576
(2563)
25.0
(635)
Step 2 – Nonlinear Static Pushover Analysis
In a displacement-based approach, the first step in a nonlinear static pushover
analysis is to assess the deformation capacity of various ductile elements such
as columns. One method is to perform moment-curvature computer analyses
based on allowable strains. For this problem, simplified methods presented in
the retrofit guidelines for determining allowable plastic rotations of the columns
are used. These depend on the limit state being investigated.
In this bridge, the unconfined splices that are typical of bridge construction prior
to 1971 must be considered. The transverse spiral reinforcing in the column is
lap spliced (a substandard detail) and will be subject to failure as soon as the
outer concrete cover spalls. Therefore, a compression failure in unconfined
concrete should be investigated. Shear failure is another possibility that could
limit ductile response. The following calculations are performed for Bent 2.
( )
( ) ( )
(
)
( )
(
)
(
)(
)
( )
(
)
(
)(
)
( )
72
15
.
96
in
0.2216
0.2216D
c
error
and
trial
by
2216
.
0
32
.
1
D
d
2
1
D
c
2
1
f
f
5
.
0
A
f
P
1
D
c
bending)
al
longitudin
(for
in
99
.
20
25
.
1
00207
.
4400
12
10
08
.
0
L
bending)
transverse
(for
in
59
.
30
25
.
1
00207
.
4400
12
20
08
.
0
L
d
4400
L
08
.
0
L
rad/in
00006348
.
0
44
.
1
69
.
2
2
2
72
29000
60
2
D
E
f
2
725
.
0
'
c
y
t
g
'
c
e
p
p
b
y
p
s
y
y
=
=
=
⇒
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎣
⎡
α
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
′
−
−
ρ
+
β
=
=
+
=
=
+
=
ε
+
=
=
−
−
−
=
′
=
φ
(
)
(
)
bending)
nal
(longitudi
rad
00524
.
0
99
.
20
0002498
.
L
bending)
e
(transvers
rad
00764
.
0
59
.
30
0002498
.
L
rad/in
0002498
.
00006348
.
96
.
15
005
.
c
p
p
p
p
p
p
y
cu
p
=
=
φ
=
θ
=
=
φ
=
θ
=
−
=
φ
−
ε
=
φ
The splice section is evaluated as follows:
in
42
L
in
41
25
.
1
4225
66000
032
.
0
d
f
f
032
.
0
l
lap
b
'
ce
ye
s
=
=
=
=
Therefore, this is a "long” splice and, by inspection, unconfined compression will
control.
Although the final shear capacity is sufficient to resist shear demands in the
transverse direction, shear failure could occur in the longitudinal direction due to
shear capacity degradation resulting from flexural yielding. The amount of plastic
rotation that is allowed will be limited because of this. This is calculated as
follows.
(
)( )
( )
(
)(
)
(
) ( )
( )(
)( )( )
ft
kip
091
,
11
12
72
144
27
.
28
5
.
5
316
.
0
180
.
0
316
.
0
0517
.
0
1
1153
.
0
D
A
f
A
f
P
A
f
P
A
f
P
A
f
P
1
D
A
f
M
M
Therefore,
1153
.
0
2
6
.
0
1
316
.
0
72
18
.
65
5
.
5
66
5
.
1
01
.
0
32
.
0
2
1
A
f
P
D
D
f
f
K
D
A
f
M
316
.
0
27
.
28
71
.
24
85
.
0
85
.
0
5
.
0
A
A
5
.
0
A
f
P
180
.
0
5
.
5
66
5
.
1
01
.
f
f
A
f
P
0517
.
0
144
27
.
28
5
.
5
1157
A
f
P
ksi
5
.
5
f
Where
A
f
P
A
f
P
A
f
P
A
f
P
1
D
A
f
M
D
A
f
M
2
g
'
c
2
g
'
c
bcc
g
'
c
to
g
'
c
bcc
g
'
c
e
g
'
c
bo
po
o
g
'
c
bcc
'
c
su
t
shape
g
'
c
bo
g
cc
g
'
c
bcc
'
c
su
t
g
'
c
to
g
'
c
e
'
c
2
g
'
c
bcc
g
'
c
to
g
'
c
bcc
g
'
c
e
g
'
c
bo
g
'
c
po
=
÷
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛
−
−
−
−
=
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎣
⎡
⎟
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎜
⎝
⎛
−
−
−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
=
⎟
⎠
⎞
⎜
⎝
⎛ −
+
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
κ
+
′
ρ
=
=
⎟
⎠
⎞
⎜
⎝
⎛
=
αβ
=
−
=
⎟
⎠
⎞
⎜
⎝
⎛
−
=
ρ
−
=
=
=
=
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎣
⎡
⎟
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎜
⎝
⎛
−
−
−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
Therefore, the limitations on flexural yielding based on shear strength for Bent 2
degradation is:
(
)
(
20
.
99
)
0
.
00656
radians
(Does
not
control)
0003124
.
0
L
rad/in
0003124
.
0
00006348
.
0
2
852
1470
1109
1470
5
2
V
V
V
V
5
kips
1109
10
11091
L
M
V
p
p
p
y
f
i
m
i
p
p
m
=
=
φ
=
θ
=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+
⎟
⎠
⎞
⎜
⎝
⎛
−
−
=
φ
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
−
=
φ
=
=
=
The deformation capacity of all as-built columns is summarized below.
Column Deformation Capacity (Longitudinal)
Column
Yield
Curvature
radians/in
(radians/m)
Ultimate
Curvature
radians/in
(radians/m)
Plastic
Moment
Kip ft
(KN m)
Plastic
Hinge
Length
inches (m)
Plastic
Rotation
radians
2
.0000698
(.00275)
.000291
(.01147)
11095
(15053)
22.1
(.561)
.00643
3
.0000698
(.00275)
.000300
(.01182)
10963
(14874)
24.3
(.617)
.00729
4
.0000856
(.00337)
.000339
(.01336)
7121
(9661)
20.7
(.526)
.00786
5
.0000856
(.00337)
.000378
(.01489)
7132
(9676)
21.8
(.554)
.00825
6
.0000856
(.00337)
.000361
(.01422)
7238
(9820)
22.9
(.582)
.00828
Column Deformation Capacity (Transverse)
Column
Yield
Curvature
radians/in
(radians/m)
Ultimate
Curvature
radians/in
(radians/m)
Plastic
Moment
Kip ft
(KN m)
Plastic
Hinge
Length
inches (m)
Plastic
Rotation
radians
2
.0000698
(.00275)
.000291
(.01147)
11095
(15053)
31.7
(.805)
.00923
3
.0000698
(.00275)
.000300
(.01182)
10963
(14874)
36.0
(.914)
.01083
4
.0000856
(.00337)
.000380
(.01497)
7121
(9661)
28.8
(.732)
.01096
5
.0000856
(.00337)
.000378
(.01489)
7132
(9676)
31.1
(.790)
.01178
6
.0000856
(.00337)
.000361
(.01422)
7238
(9820)
33.4
(.848)
.01204
Once the deformation capacity of the potential plastic hinge has been
determined, a longitudinal displacement capacity evaluation of the entire bridge
is determined through a longitudinal “push-over” analysis. In this type of analysis
the columns are modeled as non-linear elements. The frame, which is modeled
in 2-dimensions, is incrementally displaced in the longitudinal direction until the
maximum allowable plastic rotation is achieved in the plastic hinge zones. The
displacement at which this occurs is identified as the displacement capacity, Δ
ci
.
The transverse displacement capacity is determined by a transverse “push-over”
analysis of each bent. Both the longitudinal and transverse “push-over” models
include non-linear foundation springs for both rotational and translational
movement.
1. Longitudinal “Push-over” Analysis
a. Computer Models
The computer model used for the longitudinal “push-over” analysis is shown in
Figure E1-4. This model was analyzed using the DRAIN2DX computer program
that was originally developed at the University of California at Berkeley. The
non-linear elements used to model the potential plastic hinges in the reinforced
concrete column are based on a tri-linear interaction curve (i.e. Shape Code 3).
This tri-linear curve is selected to match the actual interaction curve in the vicinity
of the axial load. The actual interaction curve is calculated using the computer
program YIELD, one of several that can be used for this purpose. The tri-linear
curves used in this analysis are shown in Figures E1-5a and E1-5b.
101 102201202 203301302 303 304 306 307 401 402 403 501 502 503601602 210 220 310 320 410 420 510 520 610 630 230 330 305 430 530 620
Rigid Links (Typ)
Nonlinear Foundation Element (Typ)
Nonlinear Beam Element (Typ) Slaved Nodes at Hinge
Interaction Diagram - 5 ft Column
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Nominal Moment - kip ft
Nominal Moment - kip ft
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A
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Interaction Diagram Interaction Diagram
Tri-linear curve fit for DRAIN2DX
Tri-linear curve fit for DRAIN2DX
