Nonlinear Redundancy Analysis
of Steel Tub Girder Bridge
Analysis Report
Bala Sivakumar PE
Feng Miao Ph.D
HNTB Corporation
Graziano Fiorillo
Dr. Michel Ghosn Ph.D.
The City College of New York / CUNY
OCTOBER 2013
NON-LINEAR PUSH-OVER ANALYSES OF STEEL TUB GIRDER BRIDGE
SEPTEMBER 2013
NON-LINEAR ANALYSIS OF STEEL TUB GIRDER BRIDGE
This document describes the model and covers the details of the redundancy analysis for a three-span
two-girder-steel tub bridge example using ABAQUS. ABAQUS is very powerful commercial software
with the capability to solve complex plasticity problems. The finite element analysis for this report is used
to investigate the intact bridge system’s response and to show how the bridge’s behavior will change with
different damage scenarios where one of the tub beams fractures near the mid-span of the second span.
For all the analysis cases except the negative moment in this report, the bridge is loaded by two-side-byside HS-20 trucks near midpoint of the middle span. For the case with negative moment, only one truck is
placed in each span with two adjacent spans loaded. The details of the setup and the numerical results are
given below.
Figure 1- Original Model of Sample Bridge
1. Bridge Description
A 3-D finite element model is used to analyze the behavior of superstructure of the steel tub-girder bridge.
As given in Figure 1, two steel tub girder/deck system has the top slab width of 42’ 10 3/4 ”. The threespan continuous bridge is 540 ft long and the span lengths are all equal 180 ft. These three spans form a
part of a much larger multi-span bridge and they are extracted for the purpose of demonstrating the
redundancy of continuous span steel tub bridges. The effective slab thickness is considered as 9”. The
depth of the tub-girder is 6’ 9”. In this study, concerned with the effect of vertical load, the substructure is
not modeled. In the deck, the unconfined concrete strength is assumed to be 4000psi and the reinforcing
steel is taken as Grade 60.
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NON-LINEAR PUSH-OVER ANALYSES OF STEEL TUB GIRDER BRIDGE
SEPTEMBER 2013
Figure 2- Typical Deck Section
The steel Tub Section consists of two web plates with the thickness of ¾” each sloped with the grade 4:1.
The bottom flange is 2” thick plate with the total width 5’7 ½”. To simplify the model, the top plates of
the top flange of the tub are not modeled in this report. Note that this generic section is considered at the
both negative and positive flexure zones. The top flange is considered as continuously braced under the
deck connected through the studs and the bottom flange is detailed as non-compact in the design using the
additional diaphragms and stiffeners. Therefore, the global nonlinear analysis of the girder does not
include the local failure modes of the flanges.
The tub girders are braced externally at quarter locations along the span. The angle sections L6.6.1
are explicitly modeled using shell elements. These external bracings are expected to fail when the
superstructure is subjected to an eccentric loading such as when the vehicles are pushed towards one side
of the structure. In addition to provide the transverse stiffening at the external brace locations, the internal
K-brace system is also modeled using angle sections L6.6.1.
The concrete deck strength is assumed to be 4.0 ksi and the steel box yielding strength is 50 ksi. The
stress strain curves for the concrete and steel are plotted in the Figure 3.
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NON-LINEAR PUSH-OVER ANALYSES OF STEEL TUB GIRDER BRIDGE
SEPTEMBER 2013
Concrete Nonlinear Material
5
Stress(ksi)
4
3
2
1
0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
-3
Strain(10 in/in)
(a) Stress-strain curve for concrete
Steel Nonlinear Material
70
Stress(ksi)
60
50
40
30
20
10
0
0
0.03
0.06
0.09
0.12
0.15
0.18
Strain(in/in)
(b) Stress-strain curve for steel
Figure 3- Nonlinear Material Data used in the Shell Elements
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4.0
NON-LINEAR PUSH-OVER ANALYSES OF STEEL TUB GIRDER BRIDGE
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2. Intact Pushdown Analysis under Vertical Loads in the span 2: Model 1
Figure 4- Model 1 with vertical loads at the Mid-span
The Model 1 is the base analysis that represents the case with no damage in tub sections. The vehicle load
is represented by 2-inch2 area loads at the mid-span to represent the condition where both HS-20 vehicles
are leaned towards one girder. The vertical pushdown analysis results are presented in terms of the total
vehicle axle load and the recorded displacement in the exterior girder at the midspan. The pushdown
analysis is started after the dead load is applied and the final dead load stresses are the starting point for
the pushdown analysis. Therefore, at the end of the initial staging analysis, the pushdown curve starts at a
vertical displacement equal to 0.028 ft at Step 0. The relationship between the maximum displacement
and the live load is plotted in Figure 5 and Figure 6 shows the stress distribution contour when the bridge
fails. Steel near the support first yields and with the increment of live load, the concrete in the middle
span starts to yield and then some part of concrete crushes when the bridge reaches its maximum live
loading at 2706 kips.
According to NCHRP 406 report (Ghosn and Moses, 1998), the redundancy can be evaluated by the ratio
of LFu/LF1 for the ultimate limit state. Here, LFu is expressed the number of two side-by-side HS20 trucks
required to cause the system failure. LF1 gives the number of two side-by-side HS20 trucks leading to the
first member failure. The maximum load effect LFu is equivalent to 18.8 times the effect of two side-byside HS20 trucks. In this bridge, LF1=(R-D)/LL=(33500 kip-ft - 2084.3 kip-ft)/1895 kip-ft=16.6 for the
first member that fails in positive bending. The first failure actually takes place in negative bending with
LF1=(R-D)/LL=(12638 kip-ft – 4842.8 kip-ft)/806 kip-ft=9.7. Here R is the plastic moment capacity of
the mid-span steel box section obtained using the program XTRACT; D is the dead load effect; LL is the
live load effect of two side-by-side HS20 trucks.
With the concrete crushing, the bridge begins to be unloaded. And at the same time, the live load is much
more distributed to the supports and then some of them ruptures when the bridge is unloaded to 2511
kips. Once some steel on the supports ruptures, the live load is redistributed much more to the middle
span and bridge failure mechanism forms, which can be seen in Figure 6. With the rupture of steel in the
middle span, the bridge collapses with displacement 1.72 ft. When the bridge collapses, the cross frames
right under the center loading plastically buckles.
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NON-LINEAR PUSH-OVER ANALYSES OF STEEL TUB GIRDER BRIDGE
SEPTEMBER 2013
Original Model 1
3000
Live Load(kips)
2500
2000
1500
1000
500
0
0
0.5
1
1.5
2
Displacement(ft)
Figure 5- Vertical Pushover Analysis of the Original Model with No Deficiency.
(a) Concrete crushing and Steel yielding
(b) Cross Frame Plastic Buckling
Figure 6- Deformed Shape and stress distribution contour
3. Vertical Load Analysis without cross frames: Model 2
In order to study how the cross frames affect the bridge behavior, Model 2 is created where all the cross
frames are removed from the system. The relationship between the maximum displacement and the live
load is plotted in Figure 7 and Figure 8 shows the stress distribution contour when the bridge fails.
Steel on the support and the concrete in the middle span yields first. With the increment of live load, the
steel in the middle span starts to yield and then some part of concrete crushes when the bridge reaches its
maximum live loading at 1726 kips. The maximum load effect LFu is equivalent to 12 times the effect of
two side-by-side HS20 trucks. With the concrete crushing, the bridge begins to be unloaded and the load
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NON-LINEAR PUSH-OVER ANALYSES OF STEEL TUB GIRDER BRIDGE
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is more redistributed to the supports. Finally, with steel in the middle span ruptures, the failure
mechanism forms, which leads to the collapse of the bridge at the displacement at 1.53ft. Comparing with
the bridge model 1 with cross frames, the capacity reduces 36%. We can conclude that with the proper
design of cross frames, the bridge capacity can be significantly improved.
Model 2 without Cross Frames
Live Load(kips)
2000
1600
1200
800
400
0
0
0.5
1
1.5
2
Displacement(ft)
Figure 7- Vertical Pushover Analysis of Model 2 without cross frames
Figure 8- Deformed Shape and stress distribution contour
4. Vertical Load Analysis with Damage Scenario: Model 3
The finite element analysis for this model is to investigate the damage scenario where one of the tub
beams fractures and the continuity of the element is compromised at the mid-span. A 0.5-ft. wide fracture
is induced in the bottom flange near the mid-span of the exterior girder. This fracture cut all the way from
the bottom flange throughout the two webs to the bottom of the deck where the deck is still continuous at
the mid-span. This damaged structure is identified as Model 3. The main purpose of this model is to study
the capacity of the system to carry some load if some section fractures.
The relationship between the maximum displacement and the live load is plotted in Figure 9 and Figure
10 shows the stress distribution contour when the bridge fails. With the increment of live load, some part
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NON-LINEAR PUSH-OVER ANALYSES OF STEEL TUB GIRDER BRIDGE
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of concrete crushes when the bridge reaches its live loading at 1940 kips. The maximum load effect for
this damage scenario LFd is equivalent to 13.5 times the effect of two side-by-side HS20 trucks. With the
concrete crushing, the load is much more redistributed to the supports and the bridge can continue to carry
more live loads. Finally, with steel on the supports ruptures and more concrete crushing takes place in the
middle span, the failure mechanism forms, which leads to the collapse of the bridge at the displacement at
1.18 ft and maximum load is 1941 kips. When the bridge collapses, the cross frames right under the
center loading linearly buckles. It can be seen that the fractured bridge’s capacity will be reduced 28%
comparing with the intact bridge.
Model 3
Live Load(kips)
2500
2000
1500
1000
500
0
0
0.3
0.6
0.9
1.2
1.5
Displacement(ft)
Figure 9- Vertical Pushover Analysis of Model 3 with fracture
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NON-LINEAR PUSH-OVER ANALYSES OF STEEL TUB GIRDER BRIDGE
SEPTEMBER 2013
(a) Concrete Crushing and Steel rupturing
(b) Cross Frame Buckling
Figure 10- Deformed Shape and stress distribution contour
5. Vertical Load Analysis with Damage Scenario: Model 4
The finite element analysis for this model is to investigate how the fractured bridge behavior will be
affected if we totally remove all the cross frames. This model is identified as Model 4, which is based on
damaged Model 3. The relationship between the maximum displacement and the live load is plotted in
Figure 11 and Figure 12 shows the stress distribution contour when the bridge fails.
With the increment of live load, some part of concrete crush when the bridge reaches its maximum
loading at 1242 kips. The maximum load effect LFd is equivalent to 8.6 times the effect of two side-byside HS20 trucks. With the concrete crushing, the bridge begins to be unloaded. Finally, with steel
ruptures and more concrete crushing in the middle span, the failure mechanism forms, which leads to the
collapse of the bridge at the displacement at 1.42 ft. Comparing with the bridge model 3 with cross
frames, the capacity reduces 36%. We can conclude that with the proper design of cross frames, the
fractured bridge capacity can be significantly improved.
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NON-LINEAR PUSH-OVER ANALYSES OF STEEL TUB GIRDER BRIDGE
SEPTEMBER 2013
Model 4
Live Load(kips)
1500
1200
900
600
300
0
0
0.3
0.6
0.9
1.2
1.5
Displacement(ft)
Figure 11- Vertical Pushover Analysis of Model 4 with fracture
Figure 12- Deformed Shape and stress distribution contour
6. Vertical Load Analysis with Damage Scenario: Model 5 and Model 6
The finite element analysis for this model is to investigate how the capacity of fractured bridge will be
affected if one or three cross bracings are removed. This model is identified as Model 5 (one bracing
removed) and Model 6 (three bracings removed), which is based on Model 3. The relationship between
the maximum displacement and the live load is plotted in Figure 13 and Figure 14 shows the stress
distribution contour when the bridge fails.
With the increment of live load, some part of concrete in the middle span and the steel on the supports
begin to yield. Finally, with steel ruptures on the supports and more concrete crushing in the middle span,
the failure mechanism forms, which leads to the collapse of the bridge. The maximum load for removing
one bracing is 1879 kips and 1685 kips if remove three bracings. The maximum load effects LFd are
equivalent to 13 and 11.7 times the effect of two side-by-side HS20 trucks for model 5 and model 6,
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NON-LINEAR PUSH-OVER ANALYSES OF STEEL TUB GIRDER BRIDGE
SEPTEMBER 2013
respectively. When the bridge collapses, all the other cross frames are still in the linear stage. Comparing
with the bridge model 3 with all cross frames, the capacity reduces 3% and 13% if we remove one bracing
and three bracings, respectively. We can conclude that the capacity of fractured bridge will change very
little if we only remove one bracing. However, the change will become larger if we remove three
bracings. Therefore, it is important to properly design cross bracings so that a fractured bridge can benefit
from the lateral support of cross bracings.
Model 5_Remove Cross Frames
Live Load(kips)
2000
1500
Remove one
CrossFrame
Remove three
CrossFrames
1000
500
0
0
0.3
0.6
0.9
1.2
1.5
Displacement(ft)
Figure 13- Vertical Pushover Analysis of Model 5 and Model 6
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NON-LINEAR PUSH-OVER ANALYSES OF STEEL TUB GIRDER BRIDGE
SEPTEMBER 2013
(a) One Bracing Removed
(b) Three Bracings Removed
Figure 14- Deformed Shape and stress distribution contour
7. Vertical Load Analysis with Damage Scenario: Model 7
The finite element analysis for this model is to investigate the second damage scenario.
The second damage scenario has 15ft. wide fracture cut through the whole depth of one web and the half
bottom near the mid-span of the exterior girder. However, the deck over the fracture is still continuous
near the mid-span. This damaged structure is identified as Model 7. The main purpose of this model is to
study the capacity of the system to carry some load with different types and width of fractures. The
relationship between the maximum displacement and the live load is plotted in Figure 15 and Figure 16
shows the stress distribution contour when the bridge fails.
With the increment of live load, some part of concrete in the middle span and the steel on the supports
begin to yield. Finally, with steel ruptures on the supports and more concrete crushing in the middle span,
the failure mechanism forms, which leads to the collapse of the bridge. The maximum load is 2530 kips.
The maximum load effect LFd is equivalent to 17.6 times the effect of two side-by-side HS20 trucks.
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NON-LINEAR PUSH-OVER ANALYSES OF STEEL TUB GIRDER BRIDGE
SEPTEMBER 2013
When the bridge collapses at the maximum displacement 1.70ft, the cross frames close to the damage
zone plastically buckles. Comparing with Model 3 where two webs cut with 0.5.ft fracture width, Model
7 can carry 30% more load even with wide fracture but with only one web cut. And the capacity of
Model 7 is only 6.5% less than the intact bridge.
Figure 15- Vertical Pushover Analysis of Damaged Model 7
(a) Concrete Crushing and Steel Rupturing
(b) Cross Frame Buckling
Figure 16- Deformed Shape and stress distribution contour
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NON-LINEAR PUSH-OVER ANALYSES OF STEEL TUB GIRDER BRIDGE
SEPTEMBER 2013
8. Vertical Load Analysis with Damage Scenario: Model 8
The finite element analysis for this model is to investigate the third damage scenario.
An 80-ft wide fracture is induced in the whole depth of one web in the third damage scenario. However,
the deck over the fracture is still continuous. This damaged structure is identified as Model 8. The main
purpose of this model is to study the capacity of the system to carry some load with very wide damage
due to possible collision. The relationship between the maximum displacement and the live load is plotted
in Figure 17 and Figure 18 shows the stress distribution contour when the bridge fails.
With the increment of live load, some part of concrete in the middle span and the steel on the supports
begin to yield. Finally, with steel ruptures on the supports and more concrete crushing in the middle span,
the failure mechanism forms, which leads to the collapse of the bridge. The maximum load is 2103.9 kips.
The maximum load effect LFd is equivalent to 14.6 times the effect of two side-by-side HS20 trucks.
When the bridge collapses at the maximum displacement of 1.17ft, the cross frames closing to the damage
zone plastically buckle. Comparing with Model 7, it can be seen that the wider fracture reduces the
capacity by 20%. And the capacity of Model 7 is 28.6% less than the intact bridge.
Figure 17- Vertical Pushover Analysis of Damaged Model 8
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NON-LINEAR PUSH-OVER ANALYSES OF STEEL TUB GIRDER BRIDGE
SEPTEMBER 2013
Figure 18- Deformed Shape and stress distribution contour
9. Vertical Load Analysis for negative moment over the supports: Model 9
The AASHTO LRFD manual assumes that all steel box girders are non-compact in negative bending. The
finite element analysis for this model is used to verify whether this assumption is correct or not and if the
negative section has any ductility. For this analysis, the bridge is loaded by two trucks in one lane with
only one HS20 truck applied in one span and the other truck applied in the adjacent span. The bridge
reaches its maximum loading at 2149.5 kips when some steel ruptures over the support and some concrete
crushes near the loading areas. The relationship between the maximum displacement and the live load is
plotted in Figure 19. It can be seen from Figure 19 that negative section is compact and it has some
ductility. The 2149.5 kip capacity represents the ability of the bridge to carry 14.9 times the effect of the
HS-20 trucks.
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NON-LINEAR PUSH-OVER ANALYSES OF STEEL TUB GIRDER BRIDGE
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Negative Moment
2500
Live Load(kips)
2000
1500
1000
500
0
0
0.5
1
Displacement(ft)
1.5
Figure 19- Vertical Pushover Analysis of Damaged Model 9
10. Redundancy Analysis and Comparisons
The finite element analysis shown in this report indicated that the capacity of the continuous steel tub
bridge depends on the damage scenarios and cross frames. Table 1 summarizes the redundancy ratios for
the different models. According to NCHRP 406, a redundancy ratio for the originally intact bridge
subjected to overloading should produce a redundancy ratio LFu/LF1 greater than 1.3 to be considered
sufficiently redundant. Damaged bridges should give LFd/LF1 ratio of 0.50 or higher.
The redundancy ratios compare the maximum capacity of the system to that of the first member to fail. If
two trucks are loaded in the middle span, the first member fails in positive bending when the weight of
these trucks is incremented by a factor at LF1=16.6. However, when one truck is loaded in each of two
adjacent spans, the first member fails in negative bending at a load factor LF1=6.94. This LF1
corresponds to the first yielding of the section at the support which is used by AASHTO as the failure
criterion for the strength limit of steel box girder sections in negative bending. The redundancy ratios are
assembled in Table 1. The results show that this bridge provides high levels of redundancy for the
ultimate limit state due to overloading of the originally intact bridge. That is if the bridge is evaluated for
the strength limit state using traditional methods where the analysis is performed using a linear elastic
model and the ultimate member capacity is evaluated using the code approach where yielding is the limit
for the negative bending section, the approach will vastly under predict the load carrying capacity of the
system. Even if the bridge were to be badly damaged due to fatigue fracture or other extreme events, the
bridge is found to be highly redundant able to withstand a considerable amount of load before collapse. It
is noted that all the damaged bridge model were evaluated by placing the load in the middle span only. A
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NON-LINEAR PUSH-OVER ANALYSES OF STEEL TUB GIRDER BRIDGE
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more complete analysis should also include possible loading in two spans. However, because of the
ductility of the bridge members even in negative bending regions, the bridge is expected to still be able to
carry a significant load for all loading conditions.
Table 1 summary table for the redundancy ratio of the steel box-girder bridge
Analysis Case Model
LFu/LF1
Model 1
2.71
Model 2
---
1.74
Model 3
---
1.95
Model 4
---
1.24
Model 5
---
1.89
Model 6
---
1.68
Model 7
---
2.54
Model 8
---
2.10
Model 9
2.15
---
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LFd/LF1
NON-LINEAR PUSH-OVER ANALYSES OF STEEL TUB GIRDER BRIDGE
SEPTEMBER 2013
CONCLUSIONS
The system redundancy of a steel box-girder bridge was evaluated through a set of non-linear static
analyses using ABAQUS. The analysis results show that the bridge system’s capacity is very high and it
definitely relies on the cross frames and the damage scenarios. It can be concluded as follows:
(1) The proper design of cross frames can greatly affect the bridge system’s capacity. In this report,
the removal of the bracings would reduce the capacity by up to 36%.
(2) For the damaged bridge model 3, the half-foot fracture cuts all the way from the bottom flange
through two webs until the bottom of the deck. The capacity is 28% lower than that of the intact
model 1.
(3) For damage model 7, the fracture is 15 ft wide and the bridge’s capacity is only 6.5% less
comparing with the intact bridge model because the fracture is only cut through one web;
however, if the fracture is even wider to 80 ft, the bridge’s capacity is greatly reduced to be 77%
of the intact bridge.
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