nonlinear redundancy analysis of steel tub girder...

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

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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-by-

side 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 three-

span 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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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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(a) Stress-strain curve for concrete

(b) Stress-strain curve for steel

Figure 3- Nonlinear Material Data used in the Shell Elements

Concrete Nonlinear Material

0

1

2

3

4

5

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Strain(10-3in/in)

Stress(ksi)

Steel Nonlinear Material

0

10

20

30

40

50

60

70

0 0.03 0.06 0.09 0.12 0.15 0.18

Strain(in/in)

Stress(ksi)


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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-by-

side 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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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

Original Model 1

0

500

1000

1500

2000

2500

3000

0 0.5 1 1.5 2

Displacement(ft)

Liv

e L

oad(k

ips)


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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.

Figure 7- Vertical Pushover Analysis of Model 2 without cross frames


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

Model 2 without Cross Frames

0

400

800

1200

1600

2000

0 0.5 1 1.5 2

Displacement(ft)

Liv

e L

oa

d(k

ips)


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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.

Figure 9- Vertical Pushover Analysis of Model 3 with fracture

Model 3

0

500

1000

1500

2000

2500

0 0.3 0.6 0.9 1.2 1.5

Displacement(ft)

Liv

e L

oad(k

ips)


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(a) Concrete Crushing and Steel rupturing

(b) Cross Frame Buckling



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-by-

side 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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Figure 11- Vertical Pushover Analysis of Model 4 with fracture


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,

Model 4

0

300

600

900

1200

1500

0 0.3 0.6 0.9 1.2 1.5

Displacement(ft)

Liv

e L

oad(k

ips)


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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.

Figure 13- Vertical Pushover Analysis of Model 5 and Model 6

Model 5_Remove Cross Frames

0

500

1000

1500

2000

0 0.3 0.6 0.9 1.2 1.5

Displacement(ft)

Liv

e L

oa

d(k

ips)

Remove oneCrossFrame

Remove threeCrossFrames


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(a) One Bracing Removed

(b) Three Bracings Removed



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.



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.


B.2-12

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



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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.



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.



B.2-14


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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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

0

500

1000

1500

2000

2500

0 0.5 1 1.5

Live Load(kips)

Displacement(ft)

Negative Moment


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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 LFd/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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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.

nonlinear redundancy analysis of steel tub girder...

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